Making Do o dle Obsolete: Applying
auction mechanisms to meeting scheduling
A thesis presented
by
Chang Xu
to
Applied Mathematics
in partial ful llment of the honors requirements for
the degree of Bachelor of Arts Harvard College
Cambridge, Massachusetts
April 1, 2010
Abstract
Current systems for coordinating meetings between people are easily manipulated and
potentially produce suboptimal schedules. We propose and evaluate the idea of using an
auction system with a virtual currency to generate better schedules between agents who
list preferences over meeting times. By developing a computer-based schematic for
modeling commitments, we demonstrate that an auction system provides better incentives
for participants to truthfully report preferences, which leads to utility gains of 11% on
average over a poll-based system like Doodle. This thesis provides an initial proof of
validity for using an auction-based mechanism to more easily determine the optimal shared
schedule for a set of meetings. By doing this, it makes progress towards the goal of
designing a strategyproof and optimal system for scheduling meetings.
Acknowledgments
I thank David Parkes for being a supportive advisor, an encouraging guide, and an
inspiring mentor, without whom this thesis could not have been possible. I am incredibly
honored by the amount of support I have received from him. My accomplishments in this
thesis are due to his aid in all aspects, from its very inception to its nal edits.
I thank Batool Ali for helping me formulate ideas, implement the model, and structure the
write-up. I thank Ang Li for reading drafts and providing helpful feedback, Weiqi Zhang for
conversations that contributed to the initial design, and Saba Zaidi for moral support. I
thank Martin Lysy for assistance with the elegantly terse and undocumented language of
R. Lastly, I thank my family and friends.
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i Acknowledgments . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . iii
1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Related
Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Summary of Contributions . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 6 1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Generating the Problem 9 2.1 Modeling Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.1.1 Standard Preferential Attachment Model . . . . . . . . . . . . . . . 10 2.1.2 Modi ed
Preferential Attachment Model . . . . . . . . . . . . . . . . 15 2.1.3 Kronecker Graphs . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Modeling Meetings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2.1 Meetings . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2.2 Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 27 2.2.3 Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.2.4 Priorities . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2.5 Schedules . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 31
iii
CONTENTS iv
2.2.6 Heuristic for Evaluating Assignments . . . . . . . . . . . . . . . . . . 33
3 Auction Design and Agent Strategies 38 3.1 Auction Mechanism . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 38
3.1.1 Choosing an Assignment . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1.2 Adjusting
Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1.3 Termination . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 41 3.1.4 Multiple Periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42 3.1.5 Wealth and Banking Rule . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2 Poll-Based Mechanism: Doodle . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2.1 Translating
\Bids" into Poll Responses . . . . . . . . . . . . . . . . . 43 3.2.2 Choosing an Assignment . . . . . .
. . . . . . . . . . . . . . . . . . . 44 3.2.3 \Wealth" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3 Bidding Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3.1 Strategies that
Only Depend on Oneself . . . . . . . . . . . . . . . . 45 3.3.2 Strategies that Speculate on Other
Agents . . . . . . . . . . . . . . . 47 3.3.3 Hybrid Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49 3.3.4 Irrational Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4 Empirical Game Theory 51 4.1 Evaluating Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Finding Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2.1 Payo Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2.2 Evolutionary
Search for Equilibria . . . . . . . . . . . . . . . . . . . 55
5 Experimental Results 58 5.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.1.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.1.2 Experimental
Parameters . . . . . . . . . . . . . . . . . . . . . . . . 59
CONTENTS v
5.2 Validity of Auction Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.2.1 Assignment
Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.2.2 Banking Rule . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 62 5.2.3 Comparability of Auction and Doodle Mechanisms . . . . . . . . . . 63
5.2.4 Sanity Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.2.5 Comparing Di erent
Utilities . . . . . . . . . . . . . . . . . . . . . . 66 5.2.6 Random Agent Networks Robustness Check
. . . . . . . . . . . . . . 66
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.3.1 Auction: Preliminary
Results and New Strategies . . . . . . . . . . . 68 5.3.2 Interpreting Pure Equilibria . . . . . . . . .
. . . . . . . . . . . . . . 70 5.3.3 Navigating Di erent Mixed Nash Equilibria . . . . . . . . . . . . . . 72
5.3.4 Doodle: Sel sh Strategy Dominates . . . . . . . . . . . . . . . . . . 73
6 Discussion 76 6.1 Auction vs. Doodle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.2
Critique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.3 Open Questions and Future
Research . . . . . . . . . . . . . . . . . . . . . . 81
7 Conclusion 83
Bibliography 85
Chapter 1
Introduction
Gone are the days when my daily life consisted simply of school, track practice, homework,
dinner, violin, and sleep. Now, after almost four years of constantly having too much to do
and too little time to partake in everything I nd interesting, I have acquired the habit|
common around this neighborhood|of meticulously keeping to-do lists and perpetually
planning trajectories, from class to work to extracurriculars to dinner meeting to study
group, nally to sleep, maybe, and the cycle starts again. The main complication: most items
on that list require multiple people deciding on mutually agreeable times.
Scheduling is a di cult and common problem, whose complexity stems from coordinating
meeting times among people with di ering calendars. Currently, people schedule meetings on
an ad hoc basis through e-mails, Outlook Calendar, or the latest innovation, Doodle, found at
http://www.doodle.com. However, these systems consume time to create, produce schedules
that are far from optimal, and are inexible when plans change| meetings need to be cancelled
and rescheduled all over again. They lack the expressiveness to understand and
accommodate people’s preferences for how to use their time.
The problem of scheduling meetings would be much simpler if people were not rational
beings trying to manipulate the system for their own gain by choosing times most favorable to
themselves. The existing mechanisms are easily manipulated because they do not incen1
CHAPTER 1. INTRODUCTION 2
tivize participants to tell the truth about their availability. A person may be very selective
about the times that they claim to be available in order to guarantee that meetings will occur
during those times, which may be far less convenient for the other attendees.
The goal of this thesis is to propose and evaluate the idea of using an auction system
with a virtual currency to generate better schedules between agents who have preferences
over meeting times. I demonstrate that an auction system, as opposed to a poll-based
system like Doodle, provides better incentives for each participant to report accurate
preferences by allowing them to better express their choices through placing bids. All else
being equal, agents can expect utility gains of 11% on average. I further establish that the
most e ective strategies under an auction mechanism do not require any knowledge about
others, while acting sel shly often has an adverse e ect on one’s utility. Underneath the
auction mechanism, people are incentivized to be more generous with their time, whereas in
the poll-based system, sel sh participants bene t the most. This thesis provides an initial
proof of validity for using an auction-based mechanism to more easily determine the optimal
shared schedule for a set of meetings. By doing this, it makes progress towards the goal of
designing a strategyproof and optimal system for scheduling meetings.
1.1 Motivation
Traditional ways to schedule meetings involve a tradeo between the time required to reach
a decision and the degree to which individuals’ schedules can be kept private. This tradeo is
due to misaligned incentives between individual participants and is the underlying reason that
these systems are not optimal.
Ad hoc methods, such as using e-mails to request meeting times, respect privacy but are
ine cient. In such a method, the meeting organizer suggests a time and the participants
either accept, or proceed to nd a better time by trial-and-error. People do not need to
reveal their schedules, and the outcome is acceptable to everyone involved. However,
when trying to reach a consensus in groups of more than just two or three people, this
approach
CHAPTER 1. INTRODUCTION 3
fails many times before arriving at an acceptable solution, quickly becoming cumbersome.
On the other end of the spectrum, methods that depend on syncing participants’ calendar schedules, such as Microsoft Outlook, are e cient but do not o er any privacy. A
meeting organizer decides on a mutually convenient time after examining each attendee’s
schedule. While an organizer is guaranteed to nd a time where the maximum number of
participants are available, this system is undesirable in that it requires everyone to make their
schedules viewable by the public. Further, it does not accommodate the notion of priority;
availability at a certain time does not necessarily imply a willingness to attend at that time.
The concerns about privacy are mitigated in practice because each user can control the level
of granularity his schedule is revealed to the public, from everything to free-busy information
to nothing at all. However, since the organizer depends on the speci city and accuracy of her
knowledge of others’ schedules to choose a candidate meeting time, her judgment is only as
good as the information provided by the other participants.
Doodle o ers a compromise between privacy and e ciency. This 2007 Swiss software
startup provides an online tool for poll-based meeting scheduling. The organizer sets up
the poll with an initial list of suggested times, participants indicate their availabilities,
and nally the organizer chooses a time based on the information collected. It is relatively
e cient in that it is easy to determine the optimal time slot, but it requires participants to
provide availability information for the entire set of possible times. It maintains some
privacy in that participants can give false information, perhaps to understate the degrees of
freedom in their schedules. In fact, rational agents would not be truthful. For a rational
agent to guarantee the best outcome for herself, assuming that everyone else truthfully
reports their availability, she should only choose the most convenient time slots. However,
if all agents behave in this manner, the problem can be over-constrained and the meeting
may never be scheduled. Therefore the dominant strategy is a delicate balance between
sel shness and practicality, to ensure that a time is chosen while simultaneously trying to
choose an optimal time. Scheduling meetings on Doodle is a strategic game, but the
system
CHAPTER 1. INTRODUCTION 4
lacks the expressiveness to understand and synthesize people’s preferences, thus potentially
leading to suboptimal outcomes.
The popularity of tools like Doodle for the student group president and Outlook for
corporations indicates a clear demand for an e cient scheduler to facilitate collaboration while
maintaining privacy. An ideal, strategyproof version of such an application would require busy
go-getters to merely report their commitments and the priority of each. Further, it would
guarantee that these individuals would receive their optimal schedules only if they reported
their preferences truthfully. Such an accomplishment would reduce, if not eliminate, the need
to manage one’s schedule for everyone from workers at rms to busy faculty members and
students at college campuses.
1.2 Related Work
This thesis ties together various topics in economics and computer science, especially
mechanism design and game theory.
The intuitions for the ideas we explore are motivated by the area of mechanism design in
economic theory, which studies designs that maximize social welfare. An important
criterion characterizing these mechanisms is whether they can be strategically
manipulated, that is, if representing preferences which deviate from the truth may lead to a
better outcome. A mechanism is incentive compatible, or strategyproof, when such a
manipulation is not possible [5]. This property is desirable in many settings involving
practical market designs, both for markets without money, such as the labor market for
medical residents, and those with money, like the auction for radio spectrums [7, 8, 9]. This
auction mechanism for radio spectrums is designed in a such way that straightforward
bidding leads to the optimal outcome, and further, it is the dominant strategy if goods are
substitutes [9]. Over the course of a decade, this mechanism was expanded and improved
to absorb increasing areas of complexity in the problem space. The e cacy of the auction
model applied to radio spectrums, previously distributed freely, shows that such a
mechanism is a viable choice in
CHAPTER 1. INTRODUCTION 5
designing a market for a commodity such as time. The impact of the described variations on
welfare outcomes indicate positive prospects for such a market.
On the other hand, Budish and Cantillon [1] make the case that strategyproof
mechanisms should not be an inexible design requirement. They show that manipulations of
a course scheduling mechanism at the Harvard Business School has generated better
welfare outcomes than Random Serial Dictatorship, the only known strategyproof mechanism
in multi-unit assignment problems. Further, they argue that choosing to manipulate the
mechanism over truthfully reporting preferences has caused meaningful welfare loss.
In nding practical designs for markets, I have followed their advice by choosing an auction
mechanism to investigate the problem space of scheduling meetings. In particular, it is not
immediately apparent whether such a mechanism is manipulable and further leaves room for
use of a virtual currency to induce truthful reporting. Both of these requirements are not
satis ed by Doodle.
More closely related to a market for scheduling meetings is Sutherland’s example of a
futures market for computer time, a continuous auction scheme that allocated a PDP-1
computer to Harvard University users in the 1960s [10]. Its design combines insights from
similar systems at the Department of Defense and MIT’s Lincoln Laboratory. I explore
Sutherland’s mechanism as a variation. Even though the auction was carried out by users
writing their bids on transparent paper with colored felt-tipped pens, its successful
implementation is a proof of concept that the idea of people bidding on times is not too
outlandish to actually accomplish.
The implementation of these ideas is informed by generative models in computer science
literature to develop realistic and tractable models of agents, meetings, other commitments,
and their preferences. In generating agent networks, we consider approaches that are
theoretically elegant, like the preferential attachment model [4], as well as one that is based
on tting to real-world networks, by Leskovec et al. [2]. In the absence of models developed
speci cally for people who have meetings, I implement an original model derived from
these
CHAPTER 1. INTRODUCTION 6
theoretical approaches, as well as the aforementioned models, and discuss the merits of
each.
I draw upon empirical game theory in choosing methodology to evaluate the optimality of
the auction mechanism and strategies played in equilibrium. Reeves et al. [6] apply
algorithms from evolutionary game theory to analyze market-based scheduling for tasks on a
shared resource and derive symmetric mixed Nash equilibia for constrained instances. They
are constrained to using these algorithms because the strategic space for bidding agents is
far too complex for game theoretic analysis. Our experiment faces the same challenges since
each one of many agents may choose from a variety of bidding strategies. Borrowing the
approach presented by Reeves et al. to analyze smaller cases allows us to develop insights
into the larger problem.
1.3 Summary of Contributions
Current systems of scheduling meetings produce seemingly suboptimal schedules because
they do not su ciently incentivize participants to be truthful in reporting time preferences. This
thesis develops a computer-based schematic of a real-life application that captures the
complexity of people’s preferences, anticipates possible strategic manipulations, and
attempts to make the best decisions in arranging meetings.
In order to determine the most preferred arrangement of people’s commitments, I rst
develop a heuristic function which maps these commitments, weighted by priority, into their
scheduling preferences. Individuals collaborating on projects share commitments and
therefore need to agree on times. Thus, it is logical to model the interconnectedness of
schedules over the social and professional connections between people. To this end, I
implement and compare three di erent models for generating realistic friendship networks,
two based on preferential attachment and one based on Kronecker graphs.
Next, I design the auction mechanism through which people can bid on the time at which a
certain meeting is to take place. I use heuristic bidding strategies played in the auction by
CHAPTER 1. INTRODUCTION 7
simulated agents to model how individuals choose preferred times. The simulation becomes
analogous to a strategic game, where each agent’s goal is to obtain an optimal schedule for
himself. An agent may attempt to achieve the optimal outcome by bidding with various
strategies and so distorting his reported preferences. In order to measure the e cacy of the
auction mechanism, I implement an alternate, poll-based, mechanism as a baseline. This
poll-based mechanism simulates Doodle and attens preferences into binary ordinal
information.
I apply the methods of empirical game theory to analyze the performance of the two
mechanisms, determine which strategies are most e ective, and consider the impact of
di erent monetary policies. Collecting results from many simulations, I calculate expected
payo matrices of agents playing certain strategies against others and iteratively seek
symmetric mixed Nash equilibria. Strategies that depend only on information known about
oneself perform relatively well in equilibrium vis- a-vis manipulative strategies that speculate
on information about others. I compare the expected payo s in equilibrium to demonstrate
that the auction mechanism creates schedules which satisfy the preferences of more
individuals than the poll-based mechanism; further, it incentivizes truth-telling in reporting
preferences.
1.4 Outline
The remainder of this thesis is as follows: Chapter 2 lays out the process to generate
instances of the meeting scheduling problem, sequentially creating the agent network,
meetings, and projects to encapsulate the entire set of commitments agents face. Using
the prioritization of these projects, I put forward a heuristic valuation of an agent’s
preferences over his use of time. Chapter 3 describes the design of an auction mechanism
that systematically schedules these meetings, as well as bidding strategies with which
agents represent, or misrepresent, their preferences in hope of optimizing their outcome.
Figure 1.1 summarizes this model thus far.
CHAPTER 1. INTRODUCTION 8
START
§2 Generating the Problem §2.1 Modeling Agent Networks
standard preferential
modified preferential
attachment
attachment
Kronecker graphs
irrationalmixed speculate onother agents only depend on oneself
§2.2 Modeling Meetings
meetings
doodleauction
projects
SCHEDULES
priorities
Figure 1.1: High-level research methodology.
Chapter 4 presents a framework for applying empirical game theory in evaluating t
outcomes. Chapter 5 provides proof to validate the assumptions behind the model
presents the experimental results. Chapter 6 discusses the results, addresses ma
critiques, and o ers directions for future work. Chapter 7 o ers concluding thoughts
values
§3 Simulating the Auction §3.3
Bidding Strategies
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Chapter 2
Generating the Problem
In this chapter I develop a model for the meeting scheduling problem, encompassing a social
network of agents, meetings they share with each other, other commitments, as well as their
preferences for time based on priorities. I provide a brief de nition here and describe each
component in detail in the following sections.
De nition 2.0.1. A meeting scheduling problem is a model given by the 4-tuple, MSP =
fA;M;T;vg, where
A= fA;Egis the set of agents. Represented as a graph, nodes correspond to agents and
edges correspond to relationships where meetings can occur. This de nition is expanded
in Section 2.1.
M= fM;K;P; ; gis the set of commitments mapped to the agents, including the sets of
meetings, tasks, and projects, along with the parameters for the maximum number of
attendees at a meeting and the maximum priority level of a project. This de nition is
expanded in Section 2.2.
Tis the set of time slots in a given period where the meetings may be scheduled.
vis the value function that describes the preferences of agents for attending meetings at
di erent times. This de nition is expanded in Section 2.3.
9
CHAPTER 2. GENERATING THE PROBLEM 10
2.1 Modeling Agents
People who are connected in some way are more likely to have meetings with each other
and so their schedules are interdependent. In order to develop a realistic model of people’s
various commitments, we need to rst simulate a social network of agents. I capture the
relationships between the people who may wish to meet, such as friends, acquaintances,
and colleagues, in a directed graph GA= (A;E), where agents are vertices Aand \friendship"
links are edges E.
De nition 2.1.1. (Friendship) Let the friendship of agent ito jbe denoted by the edge (i;j) 2E.
Edges are asymmetric.
We explore three models in describing relationships among agents. We rst present two
symmetric models based on the preferential attachment and uses undirected graphs. Later,
we present an asymmetric model based on Kronecker graphs and uses directed graphs.
Asymmetry is useful in the case where everyone wants to meet with Bill Gates, but he may
not necessarily reciprocate the inclination. We will use fi;jgto represent undirected edges and
(i;j) to represent directed edges from agent-vertex ito j.
2.1.1 Standard Preferential Attachment Model
A social network can be modeled by adding agents one at a time and inserting links to
existing agents. This approach lends itself nicely to the preferential attachment model, which
has been shown to approximate certain power-law-like distributions such as that of web links,
income, le sizes, and mutations within a certain genera of plants [4]. We apply this model to
build the friendship network.
Most models are variations upon the following scheme, as presented by Mitzenmacher [4].
Each newcomer to an existing network makes one friend, where the choice of the friend is
more likely to be a popular agent but can also be more random. Our implementation di ers
from the standard model in the following aspects:
CHAPTER 2. GENERATING THE PROBLEM 11
Friendship links between agents are undirected: we do not distinguish between inlinks,
where the edge is directed towards the agent, and outlinks, where the edge is directed
away from the agent. Otherwise everyone would only have one friend, which makes for
an uninteresting meeting scheduling problem.
Links must be between two di erent agents and cannot be to oneself; self links have no
correlation to real scenarios.
By the same reasoning, the base case is two connected agents instead of one agent with
a link to himself.
First, we create agents 1;2, and add the edge f1;2g. Then, for each subsequent agent i,
with probability <1, we add the edge fi;jg, where jis chosen uniformly at random from the
existing agents, 1 j<i. With probability 1 , we add the edge fi;jgwhere jis chosen with
probability proportional to the number of friends jcurrently has, represented by its degree,
d(j).
Figure 2.1 illustrates a small network of 16 agents generated by this model. The initial
agents in the population are likely to become overwhelmingly popular, but the vast majority
have only a few friends.
Next, we show that despite the aforementioned di erences, our implementation are in line
with the standard model in that it leads to a social network where the number of friends is
distributed as a power law in equilibrium. We adapt the proof by Mitzenmacher [4].
De nition 2.1.2. (Power Law Distribution) A nonnegative random variable Xis said to have a
power law distribution if
aPr[X x] cx
for some constants c>0 and a>0. Here, f(x) g(x) represents that the limit of the ratio
CHAPTER 2. GENERATING THE PROBLEM 12
0
12
4 11 14 3 5
6 13 9 8 10 15
7 12
Figure 2.1: An agent network generated by the standard preferential attachment model with
the parameters jAj= 16 and = 0:3. Agents are represented as circles labeled with their
indices, and friendship links are undirected edges.
goes to 1 as xgrows large, limx!1
cxaPr[X x] = 1:
where
the
meaning is clear, be the number of agents with j friends when th
P
tagents in the system. Then, for j>1, the probability that Xj
roof. Let Xj(t), or simply Xj
increases is the sum of the probabilities that a newcomer chooses to befrie
with j1 friends by (a) choosing at random and (b) choosing proportionately
of friends:
Xj1t +
(1 )(j1)Xj12t :
Note that the second denominator, the total number of friends in the system, is 2tin our
implementation because each additional agent adds a degree of 1 for himself as well as for
his friend. Similarly, the probability that Xjdecreases is:
Xjt + (1 )jXj2t :
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al
so
wi
th
o
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e
fri
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n
d.
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h
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di
er
e
nti
al
e
q
u
ati
o
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th
at
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cri
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gr
o
wt
h
of
X
S
u
p
p
os
e
in
th
e
st
e
a
dy
st
at
e,
Xj
(t)
=
cjt
,
th
at
is,
a
g
e
nt
s
wi
th
j
fri
e
n
ds
co
ns
tit
ut
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a
fr
ac
tio
n
cj
of
th
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to
tal
a
g
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nt
s.
T
h
e
n
w
e
ca
n
su
cc
es
si
ve
ly
so
lv
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fo
r
e
ac
h
c,
b
e
gi
n
ni
n
g
wi
th
c1
(c
0=
0
tri
vi
all
y)
:j
dX1dt = 1 (1 + )X1
1
1
j
for
= 23 +
j
c
1
2t
c=
1
c
1
+
2
More generally, using the equation for dX =dt, we nd a recurrence describing c
CHAPTER 2. GENERATING THE PROBLEM 14
j1
j12t j12
Xt +
j2t
j
j
j>1,
X t+
j
(1 )(j1)X
cj
dXjdt =
= c
(2 + 2 +
j(1 )) = c
(1 )jX
j
1 + (1 )(j1)c
(2 + (j1)(1 )) c
j1
cj
cj
cj1 = 1 3 2 + j(1 )
We use this recurrence to determine exactly. For large j,
cj
= 1 3 2 + j(1 ) 1
3 1
1
cj
j
cj1
Asymptotically, for the above to hold, we have cj
cj
3
k= Pj kZ1 cj
c
3 1
1
0
2 1
k
k
jk
for some constant c0
for some constant c. We consider the behavior of the tail distribution, c, and con rm that it
converges to a power law. We have:
cj
dj c
j=k
Xcj3 1
. Since tail behavior converges to a power law, the distribution is a power
law.
Figure 2.2 illustrates a large network of 30,000 agents generated by this model. The
linear relationship in the log-log plot con rms that it is a power law distribution.
The standard preferential attachment model has the advantage in that it has been
well-studied. However, a power law distribution implies that there is a great disparity in the
popularity of agents, a few of whom are extremely popular. Another method that produces
a more equal distribution of friends may potentially provide a more realistic model of
individual popularity. We explore a modi ed model in the next section.
CHAPTER 2. GENERATING THE PROBLEM 15
2 5 10 20 50 100 200
5 20 50 200 1000 5000
Frequency
1 10 100 1000 10000
Number of Agents
Number of Friendship Links
(b) Histogram with logarithmic binning
1 10 100 1000 10000
Number of Friendship Links
(a) Log-log plot of number of friends per agent
Figure 2.2: A large (jAj= 30;000) network generated by the standard preferential attachment
model shows an asymptotically linear relationship between the frequency in the number of
friends per agent to the number of agents, which is characteristic of a power law distribution.
Given = 0:3, we expect c1jAj= (2=(3 + ))jAj= 0:606 30000 = 18;181 agents to have only
one friend, and the actual number is very similar: 18,124. In the loglog plot in Figure 2.2a,
results are noisy in the tail towards the right-hand side because the sample size for the
people with most number of links is small, so statistical uctuations are relatively large. We
applied logarithmic binning in Figure 2.2b to smooth out the irregularities and more clearly
show the linear relationship.
2.1.2 Modi ed Preferential Attachment Model
Instead of allowing each agent to only make one friend when entering the network, we allow
them to possibly make more. As before, we rst create two connected agents and add
subsequent agents in turn. For each new agent i, we consider adding the edge fi;jgfor each
existing agent jwith a probability that is positively related to the degree of j. This probability is
given by:Pr[fi;jg2E] = si1 k=0 d(k) Pd(k) :where d(j) denotes the number of friends jcurrently
has, and 0is a parameter for the degree of sociability within agent network. Normalizing by
the total number of friendships and taking the square root allow us to achieve a more realistic
distribution.
Figure 2.3 illustrates a small network of 16 agents generated by this model, and Figure 2.4
illustrates a larger network of 5,000 agents. We do not see power law behavior in
CHAPTER 2. GENERATING THE PROBLEM 16
neither the histogram nor the log-log plot, and instead we see more similarity between the
popularity of agents.
14
0
12
5
15
1
2
3
10
4
13
9
7
6
11
8
0=
0:3.Figure 2.3: An agent network generated by the modi ed preferential attachment
model with the parameters jAj= 16 and
1 5 10 50 100 500 5000
Number of Friendship Links
(b) Log-log plot of number of friends per agent
0 50 100 150
Number of Friendship Links
(a) Histogram of number of friends per
agent
1 2 5 10 20 50 100
Number of Agents
0.000 0.010 0.020
Percent of Agents
Figure 2.4: A large (jAj= 5;000) network generated by the modi ed preferential attachment
model with the sociability parameter 0= 0:3. This network does not show an asymptotically
linear or near-linear behavior, suggesting that the distribution is neither power law nor
lognormal.
CHAPTER 2. GENERATING THE PROBLEM 17
2.1.3 Kronecker Graphs
An alternate approach is to consider the properties of the network as a whole, and assign
relationships between agents in a way that satis es those properties. The rationale is that one
community of friends is likely to be similar to another community of the same scale, and a
smaller community can be imagined as a microcosm of the entire graph. Leskovec et al.
propose to model networks with this top-down approach using Kronecker graphs, named
after a non-standard matrix operation. Using data from existing networks to t the parameters
of the Kronecker graphs model, they claim to be able to simulate all observed structural
properties of certain real networks with synthetic counterparts [2]. We apply their algorithm,
KronFit, to t a data set from the online blogging community LiveJournal, and use the
resulting parameters to generate networks with similar properties but of arbitrary sizes.
De ning Kronecker Graphs
The idea is to use the Kronecker product of matrices to generate graphs. Starting with an
initiator graph K1, with N1nodes and Eedges, we can recursively produce successively larger
graphs K2;K31;::: such that the kthgraph, Kk, will have Nk= Nk 1nodes. Since the adjacency
matrix does not need to be symmetrical, Kronecker graphs are inherently directed graphs.
First, let us de ne the Kronecker product of matrices.
De nition 2.1.3. (Kronecker Product of Matricies) Given matrices A = [a0] and B, of sizes
n mand n m0i;jrespectively, the Kronecker product of matrices, AB, is of
CHAPTER 2. GENERATING THE PROBLEM 18
dimensions (n n0) (m m0) and is given by,
AB=
0 BB B B B B B@a1;1B
a1;2B
aB aa2;1B
a2;21;mB
aB .. .n;1B a. . .n;2.
. .B
a2;m. . .n;mB1 CC C C
C C CA
Then, the Kronecker product of two graphs is the Kronecker product of their adjacency
matrices.
De nition 2.1.4. (Kronecker Product of Graphs) Given graphs Gand Hwith adjacency matrices
A[G] and A[H] respectively, the Kronecker product of graphs, GH, is the graph with the
adjacency matrix A[G] A[H].
Note that if each node in GHis represented as an ordered pair X, with ia node of Gand ja
node of H, then an edge would join Xijto Xklijif and only if (X) is an edge of Gand (Xj;Xl) is an
edge of H. In other words,i;Xk
(Xij;Xkl) 2GH i (Xi;Xk) 2Gand (Xj;Xl) 2H
We generate a graph by iteratively using the Kronecker product to produce a growing
sequence of its adjacency matrices.
De nition 2.1.5. (Kronecker Power) The kth
Kronecker power of K1, K[k]
1
k),
is given by,
= Kk = K1 K1
:::K1
k1
K1
K[k] 1
(abbreviated to K
=K
k times| {z }
where A[K1
De nition 2.1.6. (Kronecker Graph) The Kronecker graph of order kis de ned by the
adjacency matrix A[K] is the Kronecker initiator adjacency matrix.
[k] 1],
CHAPTER 2. GENERATING THE PROBLEM 19
Kronecker graphs are self-similar in the way they are constructed. In producing Kfrom Kk1,
we would expand each node of Kk1by converting it into a copy of K1k, then we join these
copies together according to the adjacencies in Kk1. The underlying intuition for this process
is very natural. Leskovec et al. explains, \one can imagine it as positing that communities
within the graph grow recursively, with nodes in the community recursively getting expanded
into miniature copies of the community. Nodes in the sub-community then link among
themselves and also to nodes from other communities."
So far these Kronecker graph products are deterministic given some K, but we also want to
allow for randomness so that each part of the graph is similar to, but not a carbon copy of,
the next. The solution is to use a stochastic initiator matrix, P11, where each entry indicates
the probability that agent iis friends with agent j. It follows that each entry of Pkis similarly
stochastic. Thus, given the same set of initial parameters in the form of the stochastic initiator
matrix, we can generate many di erent graphs that share similar properties.
Now that we have a framework for growing a Kronecker graph from a set of initial
parameters, P1, we need to develop an algorithm that can best t those parameters from real
networks.
Fitting Initial Parameters
Given a real network, the KronFit algorithm [2] nds the parameters of P11that are most
likely to generate it. The intuition is to directly match the adjacency matrices of real network
Gand its synthetic counterpart K. If the adjacency matrices are similar, then similarity in the
statistical and structural properties will follow. If P(GjP) is the likelihood that a given
P1generated graph G, then the desired P11is the parameter values that maximize this
likelihood under the stochastic Kronecker graphs model. For standard notation, let P.
Formally, we are solving:
P(Gj ):
argmax
CHAPTER 2. GENERATING THE PROBLEM 20
For convenience we work with the log-likelihood l( ) and solve for the maximum likelihood
estimator^ = argmax l( ), where l( ) is de ned as:
l( ) = logP(Gj ) = log X = log
X P(Gj ; )P( j )P(Gj ; )P( )
where P(Gj ; ) is the likelihood that a given initiator matrix and permutation gave rise to
the real graph G. We can calculate this probability by modeling edges as independent
Bernoulli random variables parameterized by the parameter matrix , such that each entry
Puvof P= Pk= [k]gives the probability of edge (u;v) appearing. The likelihood is:
u; v]); (2.1)
(1 P[
w
h
e
r
e
(u;v)=2G
P(Gj ; ) =
Y(u;v)2GP[ u; v] Y
as the ith
@ @ l( ) = P @P(Gj ; )P( ) P=
P @ 0P(Gj ; 0)P( @logP(Gj ; ) = X@ 0)P(Gj ; )P( )
P(Gj )@logP(Gj ; ) @ P( jG; )
w
e
d
e
n
o
t
e
i
element of the permutation , and P[i;j] as the element at row
iand column jof matrix P.
To nd the that maximizes the likelihood, a grid search is complete but ine cient.
Instead, we take the log-likelihood of one particular , compute its gradient, then use it to
update the current estimate of and move towards a solution of higher likelihood. The
gradient is given by:
To avoid getting stuck in local minima, KronFit uses random restarts. Although these
equations provide a framework for nding the optimal stochastic initiator
matrix to t G, naive approaches to evaluating them requires a running time that is
CHAPTER 2. GENERATING THE PROBLEM 21
infeasible for large problems. KronFit applies sampling and approximation techniques to
evaluate P(Gj ) in linear time O(E).
The rst di culty arises from the node correspondence problem. If graph Ghas a set of N
nodes, each with a unique label, then there are N! di erent permutations of labels to nodes.
Since two isomorphic graphs that have di erent node labels should have the same likelihood,
then in computing the likelihood P(G) one has to consider all node correspondences P(G)
=P 2P(Gj )P( ), where the sum is over all N! permutations of Nnodes. Moreover, as equation
(2.1) takes O(N2) and needs to be evaluated N! times, just calculating one likelihood P(Gj )
requires O(N!N) running time. KronFit surmounts the rst challenge by sampling instead of
evaluating all of them, which reduces the running time to O(kN).
The second di culty is that naively calculating the log-likelihood l( ) and its gradient
@ @ 2l( ) take time quadratic in the number of nodes. We observe that real graphs are
sparse, the number of edges is not quadratic but rather almost linear in the number of nodes,
E N. So KronFit rst calculates the likelihood of a graph with zero edges and then corrects
for the edges that actually appear in G, using Taylor’s approximation to further speed up the
process. The gradient can also be e ciently calculated by exploiting the fact that two
consecutive permutations and 0di er only at two positions. Thus given the gradient from the
previous step, one only needs to account for the swap of two rows and columns of the
gradient matrix @P=@ to update to the gradients of individual parameters.
Fitting to Data
Leskovec et al. [2] use KronFit to demonstrate that stochastic Kronecker graphs is a
practical model to t a variety of real networks, including emails, blogs, collaborations, and
citations. The network data set we found to be the closest to our purpose is that of
friendship links and community membership on LiveJournal [3]. LiveJournal is a free
on-line community
CHAPTER 2. GENERATING THE PROBLEM 22
with almost 10 million members, a signi cant fraction of whom are highly active. We use the
KronFit algorithm to nd the optimal initial parameters for the LiveJournal data, P1=
[0:8967;0:5978;0:5978;0:09963]. Figure 2.5 illustrates a small network of 16 agents
generated by this model with the tted parameters.
13
3
11
7
0
6
9
15
10
4
8
2
12
Figure 2.5: An agent network generated with the Kronecker graphs model, with jAj= 2= 16.
The initial parameters, P14= [0:8967;0:5978;0:5978;0:09963], were tted to the social network
of friendship links and community membership on LiveJournal [2].
Figure 2.6 illustrates a large network of 1,048,576 agents generated using Kronecker
graphs. The log-log plot of the distribution of number of friends per agent shows a nearlinear
asymptotic relationship, though it appears to be more irregular than that of the standard
preferential attachment model.
The Kronecker graphs model o ers several advantages. First, it allows us to obtain
parameters using real network data. Second, it simulates properties observed in various
real networks. Third, it is easily scalable as large networks can be tted as well as
generated
CHAPTER 2. GENERATING THE PROBLEM 23
1 5 10 50 100 500
Number of Friendship Links
(b) Histogram with logarithmic binning
1e+00 1e+02 1e+04 1e+06
Number of Friendship Links
(a) Log-log plot of number of friends per agent
Figure 2.6: A large (jAj= 220
1
2.2 Modeling Meetings
People meet with people they know.
= 1048576) agent network generated with the Kronecker
graphs model. The initial parameters, P= [0:8967;0:5978;0:5978;0:09963], were tted to the
social network of friendship links and community membership on LiveJournal [2]. We applied
logarithmic binning in Figure 2.6b to smooth out the irregularities on the right.
in linear time. For these reasons, we use agent networks generated with the stochastic
Kronecker graphs model for the rest of the paper.
Frequency
5e+01 5e+02 5e+03 5e+04
1 100 10000
Number of Agents
In this section, we describe our model for meetings, other commitments, as well as an
agent’s prioritization of them in order to realistically simulate his preferences in scheduling.
We make the following assumptions. Though some are simplifying, they do not limit the
ability of our model to generalize to most real-world scenarios.
In the real world, each attendee in a meeting is likely to be connected to another
attendee, and people meet through a chain of acquaintances. So in our model, we
impose the condition that there must exist a walk in the friendship graph among the
attendees of a meeting.
Commitments that are not meetings a ect an agent’s availability for meetings.
CHAPTER 2. GENERATING THE PROBLEM 24
These are commitments that require only one agent, so we call them tasks.
Some commitments have deadlines.
After their deadlines, it would not make sense to complete them at all.
Commitments are frequently related.
Much of the di culty in scheduling arise from dependency relationships between items
on the to-do list, i.e. grocery shopping must precede cooking dinner. We capture these
dependencies by grouping tasks and meetings into projects, where they must be
completed in the designated sequence. We assume that agents are indi erent between
schedules as long as tasks are completed in order of dependency and before the
deadline, if one exists.
A project may consist of tasks leading up to a meeting.
We identify the relationship between tasks and meetings because we are mainly
interested in meeting scheduling. Whether tasks are considered as work leading up to a
meeting, or as work resulting from a meeting, the two cases are symmetrical. Without
loss of generality, we choose the rst interpretation. An extension that allows for
dependency relationships between projects would capture the case where there may
be work leading up to and resulting from a meeting.
Each meeting or task takes one unit of time.
This assumption, though unrealistic, is generalizable. Longer tasks can be thought of a
combination of tasks of unit length within the same project, where they are related by
dependency relationships. But to account for a meeting that spans multiple units of
time, we can again turn to an extension that allows for dependency relationships
between projects.
Projects have priorities.
CHAPTER 2. GENERATING THE PROBLEM 25
These priorities translate to the value an agent gains from completing a task or
attending a meeting, so the tasks and the meeting making up the project inherit those
priorities.
There are no externalities between projects.
We assume that projects capture all dependency relationships between commitments,
so an agent does not perceive additional (or diminished) value from completing two
projects.
An agent’s utility does not decrease if he is not able to complete a commitment.
The value of an unscheduled commitment is 0.
2.2.1 Meetings
A meeting is de ned in two parts: the agents who are requested to attend and the deadline it
should occur by, if one exists. Meeting attendees are chosen such that there exists a
sequence of connections between each pair.
De nition 2.2.1. (Meetings) Let meetings be modeled by the bipartite graph GM= (M[A;E).
Each edge (i;j) 2EMMwhere i2M, j2Adenotes that meeting ihas agent jas an attendee. If (i;j)
2E, then there must exist a walk between all j2R(i) in the friendship network GAM, where R(i)
denotes the range of i.
In the interest of realism, we include a capacity constraint that takes into account both the
number of time slots in a period as well as how busy agents are on average in the network. It
aims to prevent some agents from having an unsatis able amount of requests for meetings.
De nition 2.2.2. (Capacity Constraint) The capacity constraint limits the number of
CHAPTER 2. GENERATING THE PROBLEM 26
meetings an agent is requested to attend, given by:
d(i) max 2jTj; jMj
;
jAj
.
where d(i) denotes the degree of agent-vertex i, for i2GM
Implementation
The implementation is as follows. For each meeting,
1. Choose the number of attendees by drawing a random integer from [2; ].
2. Choose the rst attendee from agents with non-zero number of friends.
3. Choose the second attendee from the friends of the rst attendee.
4. Choose each subsequent attendee from the combined pool of friends of the attendees
without discarding duplicates. Thus, the probability that an agent is chosen is proportional to
the number of friends he has who are also attending the meeting.
5. If at any step the chosen agent is at capacity, choose another agent.
6. This process terminates in one of three ways:
(a) If enough attendees have been added, then continue onto the next meeting.
(b) If the number of attendees required is greater than the number of unique friends of all
the attendees combined, the process stops and no more agents are added. Continue
onto the next meeting.
(c) If no other agent from the list of friends is able to entertain another meeting request, the
process stops. If there is only 1 attendee so far, then start over from step 1 with the current
meeting.
0.00 0.05 0.10 0.15 0.20
Density
CHAPTER 2. GENERATING THE PROBLEM 27
Due to step 4, agents with more friends are likely to have more meetings than agents who
are not as well-connected. Then, because of the latter two termination conditions, even
though we had initially drawn the number of attendees from an uniform distribution, the
distribution of the actual number of attendees per meeting is skewed right. On average,
each meeting is likely to have slightly less than =2 attendees. Figure 2.7 shows the
distribution of meetings per agent. Given the parameters in this scenario, some agents
reach their capacity of 40 meetings, while some, mostly friendless agents, do not have
meetings. There is a local maximum at slightly less than 10, because agents have a little
below
jMjjAj = 2:5 4 =
10
2
meetings on average.
0 10 20 30 40
meetings
Figure 2.7: Number of meetings per agent with parameters jAj= 1;024 agents, jMj= 4;096
meetings, jTj= 20 time slots, and a maximum of = 5 agents per meeting.
2.2.2 Tasks
Tasks encapsulate the commitments that require only one agent. We further divide tasks
into two types: xed tasks if they have to be done at a designated time, and exible tasks if
they do not, but a exible task may have a deadline.
CHAPTER 2. GENERATING THE PROBLEM 28
= (K[A;EK
K
De nition 2.2.4. (Fixed Tasks) Fixed tasks is a subset K xed
xed !Tdenote the time that task k2K xed
xed
is assigned.
exible
= ;.
e nition 2.2.3. (Tasks) Let tasks be modeled by the bipartite graph GK
De nition 2.2.5. (Flexible Tasks) Flexible tasks is a subset K
[Kexible
= Kand K xed \Kexible
We have the identities: K
). Each edge (i;j)
D 2E, where i2K
all i2K.
K. Let the time function t(k) : K
K.
2.2.3 Projects
Projects group together meetings and tasks that depend on each other such that they must
be completed in the designated sequence, and they are either all scheduled or none at all.
At the beginning of each time period, we generate a static system of projects for each
agent. We only consider the following types of projects and ignore other combinations that
can be equivalently expressed in a way that has been captured. For example, a series of
exible tasks that must occur before a xed task is equivalent to a type 1 project which
describes the xed task, and a type 5 project which describes the series of exible tasks that
must occur before a certain deadline.
1. A xed task, e.g. a class, a conference, sleep
2. A meeting without a deadline, e.g. club meeting, doctor’s appointment
3. A meeting with a deadline, e.g. lunch at a restaurant before a gift card expires
4. A series of exible tasks, e.g. a series of chores
5. A series of exible tasks with a deadline, e.g. a problem set in parts, due on a certain
date
6. A series of exible tasks followed by a meeting, e.g. work related to a group project
CHAPTER 2. GENERATING THE PROBLEM 29
7. A series of exible tasks followed by a meeting with a deadline, e.g. work related to a
group project due on a certain date
In order to formalize the concept of projects, we must rst develop the notion of
precedence of items within a project.
De nition 2.2.6. (Precedence) Given iand k, iprecedes kif imust be completed before k,
denoted i k. There are three sets of values for i;k:
1. Between tasks: i;k2Kexible
2. From task to meeting: i2Kexible;k2M
3. Before a deadline: i2Kexible
[M;k2T
Precedence is clearly transitive: if a band b c, then a c.
De nition 2.2.7. (Projects) Let a project be modeled by a series of precedence constraints
between tasks and meetings, all assigned to the same agent, x. There are two sets of values
for xi:1 x2 xn
1. If there is only one item, then it can be a xed task, a exible task, or a meeting. n= 1:
xi2K[M
2. If there are two items or more, then they can be a series of exible tasks, perhaps ending
with a meeting. n>1: xi2Kexiblefor i2f1;:::;n1g, xn2Kexible[M
De nition 2.2.8. (Project Function) Let the project function r(i;j) : K[M !f0;1g represent whether
i;jare in the same project:
r(i;j) = 1 i (i j) _(j i);
where _represents the disjunction, OR.
CHAPTER 2. GENERATING THE PROBLEM 30
Implementation
For each agent, we iterate through all their assigned meetings and create projects until every
meeting is associated with a project. As a result, for an agent, the number of projects is
correlated with the number of meetings; people who have many meetings often have more
work that are associated with those meetings as well as work that are unassociated. In an
extreme case, an agent that does not have any meetings would also not have any projects.
However, this scenario is trivially reasonable because this agent does not need to coordinate
with the schedule of any other agent, so he is free to make his own optimal schedule.
We generate projects with the following state machine, as illustrated in Figure 2.8. From
the start state, with probability s;f, we have a type 1 project, which is a xed task with a time
randomly chosen from T. With probability s;m, we have a type 2 or 3 project, which consists of
the next unassociated meeting. With probability 1 , we have a project of type 4, 5, 6, or 7, all
starting with a exible task. The exible task is then followed by a deadline with probability t;ds;c,
a meeting with probability t;ms;m, and another exible task with probability 1 . The meeting is
followed by a deadline with probability m;dt;dt;m. The latter task is followed by a deadline with
probability t;d, and so on. All the probabilities lie between 0 and 1.
Figure 2.9 presents some of the properties of the project generator. In the presence of the
capacity constraint on meetings, agents take on a reasonable number of commitments.
2.2.4 Priorities
De nition 2.2.9. (Priority) The priority pi(j) of a meeting or a task, j, is its level of importance to
agent i, as well as the inherent value that igains from its completion.
The priority of each project is a random integer drawn from [1; ]. Each component of a
project then inherits the priority. The value for the entire project is additive.
CHAPTER 2. GENERATING THE PROBLEM 31
γ _{s,f}
Meeting γ _{m,d}
γ _{t,d}
γ _{s,m}
Deadlin
e
Fixed
Task
1- γ _{s,c}γ _{s,m}
S
T
A
R
T
γ _{t,m}
1- γ _{t,d}- γ _{t,m}
Flexible Task
Figure 2.8: Project generator represented as a nite state automaton. Arrows between states
are labeled with the respective transition probabilities. Beginning at the \START" state, the
project generator may terminate at any of the end states denoted by double circles.
2.2.5 Schedules
Given a set of commitments, the challenge is to t them optimally into a schedule. We
assume that agents prefer to complete a project than partially do many projects. So if a
meeting is scheduled, then all tasks preceding it should be scheduled if possible. If a meeting
is not scheduled, then none of the preceding tasks should be scheduled. However, if more
than one meeting is scheduled at the same time for an agent, the agent will choose to go to
the meeting with the higher priority. We rst de ne meeting assignments, the building blocks
of schedules.
De nition 2.2.10. (Assignment) The assignment function s(m;t) : M T !f0;1gdenotes whether
meeting mwill occur at time t.
De nition 2.2.11. (Schedule) A schedule Sifor agent iis a set of non-overlapping
assignments of meeting and tasks to time slots. Since the algorithm does not prevent the
possibility that two meetings requesting the same agent to be present may be scheduled at
the same time, only one of the meetings can be included in the agent’s schedule.
CHAPTER 2. GENERATING THE PROBLEM 32
0 20 40 60
tasks
(b) Number of tasks per agent
2 4 6 8 10
time
(a) Length of time per project
0 10 20 30 40 50 60
flexible
(d) Number of exible tasks per agent
0 2 4 6 8 10 12
fixed
(c) Number of xed tasks per agent
Figure 2.9: Projects generated with parameters jAj= 1;024, jMj= 4;096, and jTj= 5.
De nition 2.2.12. (Value of the Schedule) The value of a schedule Sifor agent iis the sum of
the priority levels of the scheduled meetings and tasks:
v(Sij2Si) = X\(M[K)pi(j):
Given a set of meeting assignments and their preceding tasks that also need to be part of
the schedule, an agent would try to construct the optimal schedule with the remaining free
0.00 0.02 0.04 0.06
0.00 0.02 0.04 0.06
Density
Density
0.0 0.4 0.8 1.2
0.0 0.1 0.2 0.3 0.4 0.5
Density
Density
time. Higher priority tasks take precedence over lower priority ones. Ties are broken
between tasks of equal values by rst scheduling xed tasks, because it may be possible to
accommodate the exible task at another time. Before an agent coordinates his schedule
CHAPTER 2. GENERATING THE PROBLEM 33
with others, he has an ideal schedule in mind.
De nition 2.2.13. (Ideal Schedule) The ideal schedule S ifor agent iis a schedule that
maximizes the value by accommodating the highest priority projects in the available time.
We use a greedy algorithm to determine an upper bound for the value of the ideal
schedule S i. Ignoring all precedence constraints, we rank all meetings and tasks in order of
their priorities. The value of S i, then, is at most the sum of the jTjhighest items.
2.2.6 Heuristic for Evaluating Assignments
Finally, we construct an agent’s preferences by evaluating his values for meetings if they
were to be held at various times.
De nition 2.2.14. (Value of an Assignment) The preferences of an agent are given by the
value function vi(m;t) : A M T ![0;pi(m)], which denotes the expected value that the
assignment s(m;t) = 1 would add to agent i’s schedule. The value of the assignment is
positive, and it is at most the priority level of the associated meeting:
0 vi(m;t) pi(m):
jMjThis
is hard to exactly determine. Strong dependencies exist between meetings such that it
is di cult to isolate the value each assignment contributes to the overall schedule. An exact
but naive methodology to make such an evaluation, albeit still only in expected value, is to
search through all the schedules that include s(m;t) = 1 and calculate the average of their
values, weighted by the probability of each schedule. However, since there are at most
jTjschedules, of which jTjjMj1include the desired assignment, the exponential running time is
infeasible for large problems.
We turn to heuristics. The obvious choice for an admissible heuristic is simply the priority
level of the meeting. However, the result is uninteresting because it does not give
CHAPTER 2. GENERATING THE PROBLEM 34
any useful information about the marginal value that a particular time contributes to the
assignment.
Instead, we propose a heuristic by analyzing the best case scenario: the ideal schedule
S j. If a meeting is assigned to a time slot that precludes all those of higher priority from
being completed before hand, a schedule containing this particular assignment is likely to
have less overall value than a schedule containing the assignment of a higher priority
meeting to the same time slot. This heuristic is a lower estimate of value in that it would
exceed the actual value that the assignment adds to the schedule. To formalize this heuristic
for assignment valuation, we introduce the notion of failed tasks.
De nition 2.2.15. (Number of Failed Tasks) The number of failed tasks, fti(m;t), for
assignment s(m;t) = 1 of agent iis the number of tasks that cannot be completed before time
tout of:
the number of exible tasks that precede meeting min the same project,
the number of exible tasks that are part of higher priority projects, and
the number of xed tasks of higher priorities assigned to a time before t.
We provide an example to make this concept more clear.
Example 2.2.1. Let jTj= 5; = 5. Consider Sam, who has the following projects:
project i exible task i, meeting i (deadline 3) - priority 2 project
ii xed task i (time 2) - priority 5 project iii exible task ii (deadline
3) - priority 4 project iv exible task iii, meeting ii (deadline 5) priority 4 project v xed task (time 4) - priority 1
Table 2.1: Projects for Sam
We consider each meeting and each time slot in order. For meeting i, exible task i
precedes it, and exible tasks ii and iii are of higher priority projects iii and iv, respectively,
CHAPTER 2. GENERATING THE PROBLEM 35
which gives ft(i;0) = 3;ft(i;1) = 2. After time 2, we also have to account for xed task i, which
increases the number of xed tasks, giving ft(i;2) = 2;ft(i;3) = 1.
For meeting ii, exible task iii precedes it, but there are no exible tasks of higher priority
projects, giving ft(ii;0) = 1;ft(ii;1) = 0. After time 2, we have to again account for xed task i,
except there is enough time to complete both exible task iii and xed task i, which gives
ft(ii;2) = ft(ii;3) = ft(ii;4) = 0. Table 2.2 summarizes the calculations.
Time 0 Time 1 Time 2 Time 3 Time 4 Meeting i 3
2210
Meeting ii 1 0 0 0 0
Table 2.2: Number of failed tasks, ft(m;t), for Sam
For each assignment s(m;t), if tis late enough to allow all preceding and higher priority
tasks to be completed beforehand, the value of the assignment is the maximum value
derived from attending meeting m, which is its priority level pi(m). However, if tis earlier, we
model the value of the assignment as a decreasing function at the rate of the number of
failed tasks, leveling o at a minimum of 0. The aforementioned heuristic value, v(m;t), can
be then expressed in terms of the number of failed tasks, given by the following function.
Because the context is clear, we drop the subscripts of iand let Kexibleand K0 i xedonly include
tasks assigned to agent i.
v0(m;t) = max(0;p(m) !ft(m;t)) (2.2)
ft1(m;k) + ft2 X
k2K(m;k) + X xed ft3(t;k) 1A t1 A
k2Kexible
where
ft(m;t) = max 0@
0;0 @
(m;k)
= (k m) (2.4)
ft1
(m;k)
= :r(k;m) ^(p(k) >p(m)) (2.5)
ft2
ft3(k;t) = (k t) ^(p(k) >p(m)) (2.6)
CHAPTER 2. GENERATING THE PROBLEM 36
!is a weight constant. ^represents the conjunction, AND. :represents negation. Figure 2.10
illustrates this heuristic function.
(m;t) against the timeline. t2
1,
vi
i(m;t) is a function increasing linearly in t.
0
Figure 2.10: Heuristic function v0 i
1
2,
vi(m;t) = pi
0
marks the point after which
there would be 0 failed tasks. tmarks the point before which there would have been too many
failed tasks to make such an assignment not worthwhile. For t<t(m;t) = 0. For t>t(m).
In-between, v
Example 2.2.2. Let != 1. Continuing the previous example, we subtract the number of failed
tasks from the priority level of the respective meeting to nd the values for the corresponding
assignments, v(m;t), with a minimum of zero. Table 2.3 presents the resulting values.
Time 0 Time 1 Time 2 Time 3 Time 4 Meeting i 0
0012
Meeting ii 3 4 4 4 4
Table 2.3: Heuristic values, v
(m;t), for Sam
If the values of failed preceding tasks, failed higher priority tasks, or failed time constraints
di er in importance, then heuristic function in equation (2.3) can be generalized
CHAPTER 2. GENERATING THE PROBLEM 37
by including di erent weights !1;!2;!3:
ft0(m;t) = max 0; Xk2K !1ft1(m;k) + !2ft2(m;k) + !3ft3(t;k) ! t! (2.7)
Finally, we account for xed tasks occurring at time t. Given a choice between attending a
meeting or completing a xed task at time t, an agent would choose the one with the higher
priority. So if there is a xed task at tof higher priority than v0(m;t), the agent will complete
the task instead, and the assignment s(m;t) then has a de facto value of 0. If there is a xed
tasks at time tof lower priority, the agent will never choose the xed task, so the heuristic
value function is not a ected. The nal heuristic value function is:
v(m;t) = 8 >> <0; ^> >
:v0k2K xedg(k;t)(m;j);
otherwise
where g(k;t) = (t(k) = t) ^(p(k) >p(m))
Example 2.2.3. Finishing the earlier example, we adjust the values for time slot 2, when Sam
has a xed task with a priority level of 5. For all assignments with values less than 5, we
lower them to 0 because Sam would rather complete the xed task than attend the meeting
at that time. Although Sam has another xed task, ii, at time 4, since it is of priority 1, he
would rather attend the meeting so the value does not change. Table 2.4 presents
Sam’s nal heuristic values for each assignment.
Time 0 Time 1 Time 2 Time 3 Time 4 Meeting i 0
0012
Meeting ii 3 0 4 4 4
Table 2.4: Final heuristic values, v(m;t), for Sam
In a larger scope, all the participants of meetings i and ii have similar tables with their own
values of v.
Chapter 3
Auction Design and Agent
Strategies
In this chapter, I present the designs of the auction mechanism as well as that of a
comparable poll-based mechanism. I distinguish between the online application, Doodle, and
the model it inspires, doodle, using the capitalization of \d." Next, I describe di erent
strategies agents employ to place bids based on their true preferences. There is a natural
real-world analogy: strategic bidding is like people trying to misrepresent their present
commitments and the importance of each in order to obtain the best schedule.
3.1 Auction Mechanism
The auction mechanism is a deterministic mechanism that takes as input a set of bids by
agents over the set of assignments, and produces as output a set of chosen assignments
which are to become part of schedules for each agent. The currency with which agents place
bids is virtual, and each agent has a certain amount of wealth.
De nition 3.1.1. (Wealth) The wealth widenotes the amount of virtual wealth owned by agent
i.
38
CHAPTER 3. AUCTION DESIGN AND AGENT STRATEGIES 39
We experiment with two variations of the initial allocation of wealth. In one, agents start
with an identical amount, while in another, agents start with random amounts. For simplicity,
we do not allow borrowing so agents may not overspend their wealth.
De nition 3.1.2. (Bid) The bid bi(m;t) : A M T ![0;wi] denotes the amount of virtual wealth agent
iis willing to pay if meeting mis scheduled at time t|in other words, if the assignment s(m;t) is
chosen.
The auction sequentially schedules meetings until quiescence, which is when there are
no more admissible, or strictly positive, bids. The resulting set of chosen assignments is the
meeting schedule. The auction allows for revisions of values and consequently bids after
each assignment choice so that agents may update their heuristic values to account for the
scheduled meeting. Figure 3.1 illustrates a high-level ow chart for this process, and the
following sections describe it in more detail.
3.1.1 Choosing an Assignment
The auction chooses the assignment s(m;t) with the highest overall bids:
i2A bi(m;t); (3.1)
X
arg max
s(m;t)
and assigns the associated meeting mto time t.
3.1.2 Adjusting Values
The auction process is sequential with value revision between assignments, because each
agent’s heuristic values would change as a result of the meeting that has recently been
scheduled, m.
Speci cally, if mis a high priority meeting, then the agent would have already allotted time in
his schedule so his values for other meetings do not need to change. However, if m is a
low priority meeting, then the agent needs to adjust the values of all higher priority
CHAPTER 3. AUCTION DESIGN AND AGENT STRATEGIES 40
START
are there unscheduled meetings?
yes
no are there positive
bids?
no
yes
is there a next period?
yes choose an assignment adjust
adjust wealth
values
no
END
Figure 3.1: High-level approach to setting up the auction mechanism.
meetings in order to allocate time to participate in mas well as to complete its preceding
tasks. Speci cally, the agents’ values are a ected in three ways:
1. For every assignment with meeting m, all agents should have value 0:
s(m;t) =)vi(m;t) = 0 8i2A;t2T
2. For all other meetings that the attendees of meeting mneed to attend, the assignments
at time tshould have value 0:
s(m;t) =)v(l;t) = 0 8i2A;(i;m) 2GM;l2M
CHAPTER 3. AUCTION DESIGN AND AGENT STRATEGIES 41
3. For all higher priority meetings of attendees of meeting m, we adjust their values to
account for the preceding tasks of meeting m. We update the number of failed tasks in
equation (2.3) by introducing ft4(m;k) to include this addition:
k2KXexible ft1(m;k) + ft2(m;k) + ft4k2K(m;k) + X xed
ft(m;t) = max 0@ 0;0 ft4
@
1
A
3.1.3 Termination
ft3(k;t) 1A t
(3.2)
where
(m;k) = s(m;t) ^(p(k) <p(m))
Value revision between assignments minimizes their overlap and the chance that an
agent would be requested to attend more than one meeting at the same time. Conicts
cannot be avoided because an assignment may be chosen even if an agent has a value of
0, as long as the other attendees value it positively.
The auction continues until quiescence. There are no more admissible bids when either all
meetings have been scheduled, or no agent is willing to place a positive bid on any
unscheduled meetings. A meeting remains unscheduled if no agent has the reserve
capacity in their schedule to accommodate, or wish to accommodate, this meeting and its
preceding tasks. An agent perceives the heuristic value to be 0 for an assignment
associated with an unscheduled meeting for two reasons:
The weighted number of failed tasks at time texceeds the value of this meeting,
indicating that the agent would rather work on higher priority projects. Thus a 0 bid from
an agent signals that he might not be able to fully prepare for the meeting.
The agent is unable to attend this meeting due to a previously scheduled meeting at the
same time. By implication, the auction mechanism cannot guarantee that scheduled
meetings will be fully attended.
CHAPTER 3. AUCTION DESIGN AND AGENT STRATEGIES 42
3.1.4 Multiple Periods
Each period concludes when the auction is quiescent. The auction starts again in another
period, with the same agents but a di erent set of dynamically generated projects. If a period
is analogous to a week, then repeating over many periods is similar to agents continuously
requesting meetings with each other on a weekly basis for a month, year, or even inde nitely.
3.1.5 Wealth and Banking Rule
At the end of each period, we return the agents’ bids so that wealth does not disappear in the
system, and allow agents to carry over their wealth from period to period. Wealth is
analogous to an agent’s power to inuence current or future meeting decisions. It is a relative
measure of inuence because an increase in wealth in the entire system amounts to ination,
so only uctuations in their relative levels of wealth have actual impact. An agent who places a
large bid may get his favorite time slot for the meeting he cares about this period, but when
he becomes poor next period, he will no longer be able to dominate the decision-making.
After each assignment is chosen, we deduct each agent’s bid for the assigned meeting
from his wealth. We do not return wealth to agents until the period ends, so an agent may not
immediately apply it toward bids for other meetings. Otherwise, if an agent knows that
mechanism has the potential to perpetually replenish his wealth, he would not optimize the
distribution of his wealth over bids for all assignments, but only for that of the next chosen
assignment, which is a di erent question entirely.
We propose two variations to the banking rule. The rst, redistributive, version divides each
agent’s bids for each of the chosen assignments equally among the other attendees of the
meeting. The aggregate result over all chosen assignments along with any unused wealth
then becomes the agents’ wealth for the next period. Under this banking rule, one’s wealth
is likely to change from period to period.
CHAPTER 3. AUCTION DESIGN AND AGENT STRATEGIES 43
A second, non-redistributive version returns each agent’s wealth at the end of each
period. The inspiration for this is taken from a review for a futures market for computer time
by Sutherland [10], which auctioned the use of the PDP-1 computer to Harvard University
users in the 1960s. Under this banking rule, wealth is held arti cially constant throughout the
periods, so e ectively, its only purpose is to guarantee that an agent does not bid beyond his
means.
3.2 Poll-Based Mechanism: Doodle
In order to assess the auction model, which is nuanced enough to understand a complex set
of preferences expressed in the form of bids, we also simulate a poll-based meeting
scheduling mechanism that only understands binary input in the form of poll responses.
Given that agents have the same underlying preferences, we can compare the relative utility
of agents in equilibrium as well as di erences in the e ectiveness of bidding strategies
between the two models.
Named after the popular application, the \doodle model" allows participants of a meeting
to view a list of possible times and check o \yes" or \no" for each. The meeting is
subsequently scheduled at the time that is convenient for the most number of people. Beside
the mechanism for choosing an assignment, the only other notable di erence in this model
versus the auction model is the fact that \wealth," representing the relative decision-making
powers of agents, is inherently allocated equally.
3.2.1 Translating \Bids" into Poll Responses
Agents must convert their reported preferences from cardinal information into ordinal
information by changing bids into \yes" or \no" poll responses. If each agent ihas a
threshold bid ci, above which he would be willing to attend meetings at the speci ed time,
and below which he would not, poll responses are then given by the following de nition.
CH
APT
ER
3.
AU
CTI
ON
DE
SIG
N
AN
D
AG
ENT
STR
ATE
GIE
S
44
D
e
nit
io
n
3.
2.
1.
T
h
e
p
oll
re
sp
o
ns
e
li(
m
;t)
:
A
M
T
!f
0;
1
g
d
e
n
ot
es
w
h
et
h
er
a
g
e
nt
iw
is
h
es
to
at
te
n
d
m
e
eti
n
g
m
at
ti
m
e
t:
8 >> <1; b> > :i(m;t) ci0; otherwise (3.3)
li(m;t) =
for some threshold ci.
In our implementation, the threshold ci is given by the agent’s wealth
wi
ci = wid(i) ; i2GM :
di
vi
d
e
d
by
th
e
n
u
m
b
er
of
m
e
eti
n
gs
h
e
is
re
q
u
es
te
d
to
at
te
n
d:
The
intui
tion
is to
gua
rant
ee
that
the
age
nt
only
agr
ees
to
atte
nd
mee
ting
s at
time
slot
s
with
high
er
valu
e
than
if he
had
distr
ibut
ed
his
wea
lth
equ
ally
amo
ng
his
mee
ting
s.
3.
2.
2
C
h
o
o
si
n
g
a
n
A
ss
ig
n
m
e
nt
T
h
e
d
o
o
dl
e
m
o
d
el
ch
o
os
es
th
e
as
si
g
n
m
e
nt
w
h
er
e
th
e
hi
g
h
es
t
p
er
ce
nt
a
g
e
of
at
te
n
d
e
es
is
a
bl
e
to
b
e
pr
es
e
nt
:
arg max
s(m;t)
T
hen,
just
as
in
the
auct
ion
mod
el, it
allo
ws
age
nts
P
i2A
(m;t)
d(m)
li
; m2GM
:
to
upd
ate
their
heu
risti
c
valu
es
afte
r
eac
h
choi
ce.
This
proc
ess
is
rep
eate
d
until
quie
sce
nce,
whe
n
non
e of
the
age
nts
are
willi
ng
to
atte
nd
any
mor
e
mee
ting
s.
3.
2.
3
\
W
e
al
th
"
T
h
e
d
o
o
dl
e
m
o
d
el
is
d
e
m
oc
ra
tic
in
d
es
ig
n:
ev
er
yo
n
e’
s
pr
ef
er
e
nc
es
ar
e
e
q
u
all
y
im
p
or
ta
nt
in
th
e
se
ns
e
th
at
e
ac
h
vo
te
is
co
u
nt
e
d
o
nc
e.
E
ss
e
nti
all
y,
a
g
e
nt
s
h
av
e
id
e
nti
ca
l
\w
e
alt
h,
"
a
n
d
th
e
sy
st
e
m
o
p
er
at
es
u
n
d
er
th
e
n
o
nre
di
st
ri
b
uti
ve
b
a
nk
in
g
ru
le
w
h
er
e
e
ac
h
a
g
e
nt’
s
w
e
alt
h
is
CHAPTER 3. AUCTION DESIGN AND AGENT STRATEGIES 45
returned to him at the end of the period.
3.3 Bidding Strategies
If values represent an agent’s true preferences, then bids constitute the way those
preferences are portrayed to others. We characterize systematic ways to distort true
preferences as bidding strategies, and take inspiration from how people in real life might try
to obtain schedules that are more closely aligned with their values. This leads to a de nition
of rationality.
De nition 3.3.1. A strategy is rational if an agent places higher bids for assignments with
higher values, and lower bids for assignments with lower values. Otherwise, it is irrational.
We allow saving, since wealth is carried over from period to period. At the same time, we
assume that agents do not have an incentive to save, so that if he receives his most favored
assignments, he will want to have exactly spent all of his wealth. If, however, an agent
wishes to be sure to save a part of his wealth, then we can easily generalize to this scenario
by only apportioning x% of his wealth on bids.
In this section, we present a few general types of rational and irrational bidding strategies
that agents might employ.
3.3.1 Strategies that Only Depend on Oneself
These bidding strategies only depend on the value of the assignment, independent of any
information from other agents.
Truthful
The truthful strategy involves agent ibidding an amount of wealth that is proportional
CH
APT
ER
3.
AU
CTI
ON
DE
SIG
N
AN
D
AG
ENT
STR
ATE
GIE
S
46
t
o
t
h
e
v
a
l
u
e
o
f
e
a
c
h
a
s
s
i
g
n
m
e
n
t
:
bi(
m
;t)
/vi
(
m
;t)
(3
.4
)
Threshold
T
h
e
th
re
sh
ol
d
st
ra
te
gy
in
vo
lv
es
a
g
e
nt
ibi
d
di
n
g
o
nl
y
if
th
e
va
lu
e
is
a
b
ov
e
a
m=
wi
X max bi(m;t) (3.5)
t
th
re
sh
ol
d
ci:
m=
bi(m;t) /
wi
A sensible threshold ci
wi
m=
X max bi(m;t) (3.9)
t
X 8 >> <v> > :i(m;t); pi(m) ci0; otherwise (3.6)
max
is the prio
be able to squeeze into his idea
allocate more wealth into biddin
Sel sh
The sel sh strategy involves age
value for assignments with high
for assignments with low values
unwilling to attend meetings at t
bidder bids proportionately to th
some reasonable k:
bi(m;t) /(vi(m;t))k; for k2f2;3;:::g (3.
tX
wi
bi(m;t) ;max
jTj
b (3.11)
i
s
(
m
adjustment
; step
t and
assignment
)
An agent can never overspend their wealth due to the value
in the
auction mechanism, i.e. if an agent bids all his wealth for an
the
assignment is subsequently chosen, then he can only bid 0 for the rest of his
assignments.
3.3.2 Strategies that Speculate on Other Agents
These bidding strategies modify the value of the assignment depending on certain
characteristics of other agents.
Speculate on Popularity
Agents with more friends have more opportunities to be requested to attend meetings,
so they have a greater need to choose which meetings to attend. If an agent would like
to attend a meeting with a more popular agent, then he might have to bid more in order
to ensure that the meeting takes place. For example, if one is soliciting the help of Bill
Gates, with whom many people wants to meet, then one would need to bid higher to
increase the likelihood that an assignment of this meeting will be chosen
CHAPTER 3. AUCTION DESIGN AND AGENT STRATEGIES 47
= max mX
(m;t) !
Aggressive
The aggressive strategy involves agent iexaggerating his bids across the board. He
normalizes his bids with respect to the sum of the average values over all
assignments of each meeting, as opposed to the sum of the maximum values. To
prevent any single bid from being greater than his wealth, we normalize either by the
sum of the average values or the maximum value over all assignments, whichever is
greater. This strategy is given by:
bi(m;t) /vi(m;t) (3.10)
CHAPTER 3. AUCTION DESIGN AND AGENT STRATEGIES 48
earlier by the mechanism, or at all.
agent ithat speculates on popularity would multiply the value of each assignment by
the number of friends of the most popular agent requested to be at the meeting. For agents
j2A;i6= j;(m;j) 2E
bi(m;t) /vij(m;t) maxd(j)
(3.12
)
m= X max bi(m;t) (3.13)
t
wi
M:An
Speculate on Wealth
Wealthier agents have more decision-making power, so another agent may need to bid more
in order to be heard. A lower bid may not be inuential but still needs to be paid, acting like a
sunk cost.
M:An
agent ithat speculates on wealth would multiply the value of each assignment by the
amount of wealth of the richest agent at the meeting. For agents j2A;i6= j;(m;j) 2E
bi(m;t) /vij(m;t) maxwj (3.14)
m= X max bi(m;t) (3.15)
t
wi
Speculate against Wealth
To the contrary, it might be counterproductive to overbid on meetings when wealthier agents
are present because the bidder might simply not have the wealth to compete. The wealth
contrarian strategy would instead bid more on meetings where other attendees are poorer so
that the the bidder can exert more inuence.
There are two variations. For one, an agent ithat speculates against the maximum wealth
would divide the value of each assignment by the amount of wealth of the
CHAPTER 3. AUCTION DESIGN AND AGENT STRATEGIES 49
richest agent at the meeting. For agents j2A;i6= j;(m;j) 2EM:
b (m;t) max(m;t) /
(3.16
i vijwj
)
m= X max bi(m;t) (3.17)
wi
t
M:For the second variation, an agent ithat speculates against the total wealth would divide
the value of each assignment by the combined wealth of all other agents at the meeting.
For agents j2A;i6= j;(m;j) 2E
j wj
bi(m;t) / vi
(3.18
)
wi (m;t) Pm= bi(m;t) (3.19)
Xmaxt
I implemented the second variation because it also takes into account of the number of
attendees at a meeting. The total wealth contrarian strategy would bid less when there
are more people because there is potentially more disagreement.
3.3.3 Hybrid Strategies
These bidding strategies are hybrids of the aforementioned strategies; the actual strategy
employed changes depending on the circumstances.
Hybrid Depending on Wealth
A rich agent may bid sel shly to try to dominate others’ preferences, while a poor agent
may bid conservatively by betting more on meetings with other poor agents so that his bids
would actually matter. An agent chooses di erent strategies depending on his wealth at the
time. If given a threshold, a wealthier bidder may bid sel shly, while the same bidder, now
poorer, may play the wealth contrarian strategy|or vice versa. These are just two examples
of hybrid strategies; others are possible.
CHAPTER 3. AUCTION DESIGN AND AGENT STRATEGIES 50
3.3.4 Irrational Strategies
As sanity checks, these bidding strategies serve to demonstrate that the auction mechanism
indeed function as expected by favoring the more rational strategies above.
Overly Aggressive
Similar to the aggressive strategy, except the overly aggressive bidder would always
bid by normalizing to the maximum value over all assignments. If his favorite
assignment is chosen, then he would be left with 0 wealth for all other assignments,
which would be an unfortunate predicament. The strategy is given by:
bi(m;t) /vi(m;t) (3.20)
m= X max bi(m;t) (3.21)
wi
t
While this strategy is technically still rational, since agent istill bids more for assignments
with higher values, it is de nitely an irresponsible use of wealth.
Inverse
The inverse strategy involves agent ibidding proportional to the inverse of the value of the
assignments:
biv(m;t) / 1i(m;t) (3.22)
m= X max bi(m;t) (3.23)
wi
t
This strategy is purely irrational because it would bid more for less important meetings and
less for more important ones.
Chapter 4
Empirical Game Theory
In this chapter, I present methodology for evaluating the auction and the doodle models.
First, I quantify the happiness of the agents. Then, I introduce the replicator dynamics
algorithm from evolutionary game theory to determine the bidding behavior of agents in
equilibrium.
4.1 Evaluating Utility
Schedules can be distinguished by the utility, or the degree of happiness, they provide the
agent. However, as there are many ways to quantify the concept, it is worthwhile to explore a
few and decide on the most appropriate interpretation later. We propose three de nitions, one
absolute and two relative notions of utility.
De nition 4.1.1. The absolute utility of agent iis the value of his schedule Si, which is the sum
of the priority levels of the scheduled meetings and tasks:
uabs(Si) = v(Sij2Si) = X\(M[K)pi(j):
Absolute utility penalizes the agents with fewer meetings because they would never be
able to achieve a high level of utility.
51
CH
APT
ER
4.
EM
PIRI
CAL
GA
ME
THE
OR
Y
52
D
e
nit
io
n
4.
1.
2.
T
h
e
ac
tu
al
uti
lit
y
of
a
g
e
nt
iis
th
e
ra
tio
of
th
e
va
lu
e
of
hi
s
sc
h
e
d
ul
e
Si
to the value of his ideal schedule S i:
uact(Si) = v(Si) v(S i) =
D
e
nit
io
n
4.
1.
3.
T
h
e
h
e
ur
ist
ic
uti
lit
y
of
a
g
e
nt
iis
th
e
ra
tio
of
th
e
su
m
of
th
e
va
lu
es
of
th
e
as
si
g
n
m
uabs(S)
v(S ii) :
e
nt
s
at
th
e
ti
m
e
th
ey
w
er
e
ch
os
e
n
to
th
e
va
lu
e
of
hi
s
id
e
al
sc
h
e
d
ul
e
S
i.
W
e
us
e
th
e
h
e
ur
ist
ic
va
lu
es
fr
o
m
e
q
u
ati
o
n
(3
.2
):
(
ivi) :
m
;
t
u
The relative utilitieshe are designed to )be more personalized metrics in that an agent is
u
v
happy if he is closer to( his own ideal schedule.
There are three major di erences between
(
S
S
i
the two measures. Heuristic
utility accounts for the heuristic values of the assignments,
)
v(m;t), as opposed to =
the actual values of the associated meetings, pii(m). It does not
include the values of tasks
in order to isolate the impact of the bidding strategies on
P
meeting scheduling. Lastly, if two meetings are scheduled at the same time, actual utility
would only include the value of the meeting that the agent actually attends, while heuristic
utility would include the heuristic values of both assignments.
m
2
S
i
Thus, heuristic utility can be considered as an evaluation of the e ectiveness of the
bidding process, rewarding the strategies that are able to quickly lock in the most desirable
assignments and selecting for schedules that places higher priority meetings before lower
priority ones. On the other hand, actual utility inherently assumes that agents are indi erent
between two schedules as long as the same commitments are present, and as such, it is
perhaps a more realistic evaluation of an agent’s happiness.
Collectively, the various utilities represent a holistic picture of the overall e ectiveness of
the auction mechanism in creating desirable schedules for everyone. Figure 4.1
presents the distributions of happiness over the agent population when everyone bids
truthfully.
CHAPTER 4. EMPIRICAL GAME THEORY 53
It compares the three de nitions of utility in applying to both small and large meeting
scheduling problems. We assume that truthful bidders are symmetrical, so we average their
utilities.
Although the three notions of utility result in distinct distributions over the population, they
are similar in that the average utility of agents negatively correspond with how busy they are.
It is more di cult to coordinate amongst agents who are more constrained. Furthermore, the
similarity in graphs with 8 agents and 32 agents, while maintaining the ratio of meetings to
agents, suggests that modeling a smaller number of agents can be generalized to a larger
population. We proceed with all three de nitions, then determine the best choice by
evaluating their relative merits in practice.
4.2 Finding Nash Equilibria
Next, we establish a methodology for evaluating the e ectiveness of strategies vis- a-vis
each other and exploring which strategy, or a mix of strategies, would agents employ in the
long run. We construct a symmetric game where the payo s for playing a particular strategy
depend only on the other strategies played, but not on who is playing them, by randomly
assigning strategies to agents given a strategy pro le, which is a xed proportion of each
strategy to be used in the population. Over many instances of the meeting scheduling
problem, we estimate the payo matrix over all possible strategy pro les for each set of
permissible strategies. Finally, we iteratively search for the symmetric mixed Nash
equilibria, which are the proportions of each strategy in the mix that agents are expected to
play in evolutionarily stable state. We use the replicator dynamics algorithm as described
by Reeves et al. [6] to tackle the large strategy space for these games. Although it is a
restricted approach in that only a certain set of permissible strategies are considered each
time, it is state of the art and generalizable to larger sets, though including the complete set
would not be computationally feasible.
CHAPTER 4. EMPIRICAL GAME THEORY 54
0 10 20 30 40
Absolute Utility
0 10 20 30 40
Absolute Utility
(a) Absolute utility over 200 periods with parameters jAj
= 8;jMj = 16;jTj = 8; = 5
(b) Absolute utility
32;jMj = 64;jTj = 8;
0.0 0.2 0.4 0.6 0.8 1
0.0 0.2 0.4 0.6 0.8 1.0
Actu
al
Utilit
y
(c) Actual utility over 200 periods with parameters jAj = 8;jMj = 16;jTj = 8;
with parameters jAj = 32;jMj = 64;jTj = 8; = 5
0.0 0.2 0.4 0.6 0.8 1.0
Heuristic Utility
Heuristic Utility
Density
01234567
Density
0 5 10 15 20
0.00 0.02 0.04 0.06 0.08 0.10 0.12
(e
)
Figure
4.1: The
H
if all agents are
e
ur
ist
ic
uti
lit
y
ov
er
2
0
0
p
er
io
ds
wi
th
p
ar
a
m
et
er
s
jA
j=
8;j
Mj
=
1
6;j
Tj
=
8;
=
5
Density
Density
Density
01234
Density
0 5 10 15 20 25
0.00 0.01 0.02 0.03 0.04 0.05 0.06
0.0 0.2 0.4 0.6 0.8 1.0
(f) Heuristic utility o
32;jMj = 64;jTj = 8;
distribution of absolute, actual, and heuristic
bidding truthfully.
CHAPTER 4. EMPIRICAL GAME THEORY 55
4.2.1 Payo Matrix
Payo matrices represent the expected payo s of each strategy given a set of permissible
strategies, when played in various proportions in the population.
De nition 4.2.1. (Payo ) The absolute, actual, or heuristic payo po(s) of a bidding strategy
sto an agent is the absolute, actual, or heuristic utility, respectively.
De nition 4.2.2. (Strategy Pro le) A strategy pro le for a game is a list of speci c strategies
that each player follows.
Given a set of permissible strategies Sfor agents A, the payo matrix consists of an entry
for each of thejAj+jSj1 jAj possible strategy pro le combinations. Each entry is anjAj-vector of
expected payo s associated with a particular strategy pro le. For example, if jSj= 3 and jAj= 4,
then a strategy pro le could be ftruthful, truthful, threshold, sel shg.
To estimate each entry of an expected payo matrix, we generate many instances of the
meeting scheduling problem and randomly assign strategies to agents from the strategy
pro le. Each period, we simulate the auction protocol with the given strategies, and each
agent maintains the same strategy for a number of periods in a trial. In each trial, the agents
are symmetric with respect to their strategy, so we average the payo s between those playing
the same strategies. We repeat for a number of trials, with the agents facing di erent
problems and assigned to play di erent strategies for each trial. Because the payo matrices
are empirically determined, its values are only estimates and thus require many trials to
establish their reliability.
4.2.2 Evolutionary Search for Equilibria
Given the payo matrix, we can iteratively search for expected asymptotic behavior of
agents in the form of a symmetric mixed Nash equilibrium. The process is motivated by
evolution, where successful strategies are rewarded by being played in greater proportions
over the population at the expense of poorly performing strategies over time.
CHAPTER 4. EMPIRICAL GAME THEORY 56
We choose an initial population proportion for each pure strategy in the set of permissible strategies, and update
them in successive generations with respect to their performance. The proportion pg(s) of the population playing
strategy s2Sin generation gis given by:
pg(s) /pg1(s)(EPs
W):
is the expected payo for pure strategy sagainst jAj1 players all playing mixed strategies
according to the population proportions. W is a lower bound on payo s which serves as a
I
dampening factor, for instance, the minimum value in the payo matrix.
n this equation, EPs
To calculate the expected payo EPsfrom the payo matrix, we average the payo s for
sin the pro les in which it appears, weighted by the probabilities of those pro les. If the
strategy sis not played in some pro les, then EPsneeds to be normalized by the sum of
the probabilities of the pro les in which sdoes appear. Let nsdenote the number of
players playing strategy s. Then:
(n1;:::;ns EPs = P(n1;:::;ns>0;:::;njSj)
>0;:::;n) P
) ; (4.1)
Pr(n1;:::;njSj j j) po(s) Pr(n1;:::;
1
n
where the probability of a particular pro le (n
(1)n jSj) is given by:
: (4.2)
g1
Pr(n1;:::;njSj) = N!n1!:::njSj
1
;:::;n
:::p
(jSj) j j
! pg1
We unpack Equation (4.2). Since pg1 (i) is the probability that an agent would play strategy siin generation g1, pg1(i)niis the
probability of niagents playing strategy si. Then:
pg1(1)n1 :::pg1(j j)njSj
is the probability of the agents playing the particular mix of strategies if the rst n1agents played s1, the rst
n2agents played s2, and so on. In other words, it is the probability of
CHAPTER 4. EMPIRICAL GAME THEORY 57
a pro le if order mattered. Since order does not matter, the probability of the particular pro le
(n1;:::;njSj) is the product of the probability of a pro le if order mattered and the multinomial
coe cient, which counts the number of distinct ways the given proportion of strategies can be
assigned to jAjagents.
Finally, to obtain the proportion of agents playing strategy sin generation g, we normalize
every pg(s) by dividing each by their sum,
pg(s) = pg1(s)(EPspg1s(s)(EPsW) PW) (4.3)If this population update process reaches a xed
point, then the nal population proportions are a candidate symmetric mixed strategy
equilibrium. The xed point is determined by calculating the sum of the changes in population
proportions from generation g1 to g, and stopping when it becomes 0. We verify directly that
the candidate is indeed a static Nash equilibrium by checking that the evolved strategy is a
best response to itself. Specifically, we calculate the expected payo s given the population
proportions and verify that they are the same for all the strategies.
For a given population of agents, we repeat this process for all possible initial proportions
of strategies in order to prevent perhaps getting stuck in local minima or cycles, and in
order to enumerate the possible equilibria if multiple exists. Note that this replicator
dynamics algorithm [6] is constrained by the number of strategies explored at a time in the
payo matrix. If a strategy has an initial proportion of 0, then it can never be brought into
the mix of strategies, thus it will remain at 0.
Chapter 5
Experimental Results
In this chapter, I present various metrics and sanity checks to show that the auction
mechanism functions as expected, and to establish a credible comparison with the doodle
mechanism. Then, we empirically analyze the e cacy of di erent de nitions of utility, explore
equilibrium behavior of various strategies, and compare variations in the design of the
mechanisms.
5.1 Experiments
5.1.1 Experimental Setup
I implemented the meeting scheduling problem and the auction and doodle mechanisms in
Python. Large scale experiments are conducted on the HPC clusters of the School of
Engineering and Applied Sciences Instructional Research and Computing. The cluster
consists of 28 nodes; each node has 16 GB of RAM and they are connected via high-speed
in niband interconnect. Smaller experiments are conducted on a local machine with 2.16 GHz
processor and 1 GB of RAM. For data analysis, I used R.
58
CHAPTER 5. EXPERIMENTAL RESULT S 59
5.1.2 Experimental Parameters
We optimize parameter settings in our experiments to drive more signi cant results or simply
for practicality.
Each experiment is run over a social network of 8 agents generated by the Kronecker
graphs model. A network of 8 agents is large enough such that the Kronecker graphs
model is distinguishable from the preferential attachment models, but it is still small enough
to be practical to use in running many trials. Figure 5.1 illustrates the network we use.
6
1
4
25
3
0
7
Figure 5.1: An agent network generated by the Kronecker graphs model for 8 agents used in
our experiments. The initial parameters, P1= [0:8967;0:5978;0:5978;0:09963], were tted to
real network data from the online blogging community Livejournal.
Other parameters are set to allow strategies to be most distinguished from each other. By
recognizing that the di erences in the utility of agents stem from variation in schedules that
they obtain, we look for parameter settings that would lead to schedules with the greatest
degree of di erentiation. We x the strategies of 7 agents but vary the strategy
CHAPTER 5. EXPERIMENTAL RESULT S 60
of the 8thagent, then simultaneously run di erent experiments such that every game is
identical except for the strategy of the last agent. We compare the resulting schedules and
calculate the percentage of meetings that are assigned to di erent times.
We observe that strategies matter more when the problem is more constrained,
suggesting setting a high ratio of the number of meetings to the number of agents, fewer time
slots per period, more attendees per meeting, and so on. However, these parameters have
limited degrees of freedom because the problem would become infeasible if it is too
constrained. On the other hand, we nd that raising the maximum priority level also allows
strategies to matter more. Increasing the range of the priority levels of projects from 1 to 100
as opposed to from 1 to 5 creates more disagreement among agents over the relative values
of meetings. More disagreement leads to greater variation in utility, because if a meeting is
really important to one agent but the opposite for another agent, then the rst would be
ecstatic if it is scheduled but the second would only mope.
Thus, unless otherwise noted, the experiments are run with the following parameters.
jAj= 8 agents,
jMj= 16 meetings,
Over jTj= 15 time slots per period,
With a maximum of = 5 attendees per meeting, and
A maximum priority of = 100.
The project generator Figure 2.8 has the following transition probabilities:
s;f;s;m;t;t;t;d;m;d
= 0:3
s;t;t;m
= 0:4
The number of periods and whether agents receive equal or random amounts of initial
wealth do not have discernible impact on di erentiating the strategies. So for most
experiments, we initially give agents random amounts of wealth drawn from [0;20]. In the
interest
CHAPTER 5. EXPERIMENTAL RESULT S 61
of practicality, they are run for 50-100 periods each so that we can run more trials in the
same amount of time.
For each experiment, each of the 8 agents chooses from a set of 3 strategies, giving
jAj+ jSj1jAj
= 8+318 =
108 = 45
di erent possible strategy pro les. We limit the sets to 3 strategies each to keep the number
of trials required to generate a payo matrix reasonable.
Experimental resource is limited by CPU cycles and time. Reeves et al. [6] worked on the
problem of allocating computer time to di erent tasks, and required 45 million trials over 21
strategy pro les to achieve an accurate expected payo matrix, taking weeks of CPU time.
Our problem is in a di erent domain so the problem is of a di erent scale. Even so, we do not
have the time to achieve such accuracy, but we believe that the ideas and the results
presented here are valid and robust to more trials.
5.2 Validity of Auction Design
In designing the auction model, we have made various assumptions. In this section we show
that these design decisions are reasonable and meaningful.
5.2.1 Assignment Choice
Recall that the auction and doodle mechanisms choose the assignment with the highest
overall bids and the highest fraction of bids, respectively. The implicit assumption in both
cases is that agents would place bids over enough meetings to make the assignment choice
meaningful. For instance, if agents only bid on 1 meeting and are not willing to bid on more
meetings, then assignments are essentially chosen by random selection.
This is not the case. Figure 5.2 shows that truthful agents initially do bid over a substantial
subset of their meetings. For this experiment, jTj= 15 implies that an agent
CHAPTER 5. EXPERIMENTAL RESULT S 62
could attend at most 15 meetings. Figure 5.2a shows that most agents bid on at least one
assignment over each of their meetings. Figure 5.2b shows that even the busiest agents
initially place bids on assignments ranging 34%, approximately 10, meetings, while agents
with only 1 or 2 meetings almost always would bid on both of them. For all other agents
in-between, there is a monotone decreasing relationship between the percentage of
meetings bid on and the total number of meetings. Since agents on average bid on enough
meetings initially, selecting assignments by highest overall bids is meaningful.
0 5 10 15 20 25 30
Total number of meetings for the agent
0.0 0.2 0.4 0.6 0.8 1.0
Percent of meetings bid on
(a
)
Hi
st
o
gr
a
m
of
th
e
p
er
c
e
nt
of
m
e
et
in
g
s
an agent bids on
0.0 0.2 0.4 0.6 0.8 1.0
Percent of meetings bid on
0 2 4 6 8 10 12
Density
(b
has
)
S
Figure
5.2: Given parameters jAj= 8;jMj= 32;jTj= 15 and all a
c
strategy, agents place bids on a signi cant proportion of mee
at
validating
the design of choosing an assignment with the high
te
highest
fraction of bids, in the auction and doodle mechanism
rp
lo
t
of
th
5.2.2
Banking Rule
e
p
er also assume that the redistributive banking rule is e ectiv
We
c
among
agents to allow for variation in their relative decision-m
e
traces
the wealth of agents throughout many periods. In this
nt
of
with
10 units of wealth and bid with the same strategy. The g
m
represents
the agent who does not have any friends and con
e
wealth
remains
untouched. Other agents’ wealth uctuate ove
et
Figure
5.3a shows that if all agents are truthful bidders, the ri
in
g
s
a
n
a
g
e
nt
bi
d
s
o
n
v
er
s
u
s
th
e
to
ta
l
n
u
m
b
er
of
m
e
et
in
g
s
th
at
h
e
CHAPTER 5. EXPERIMENTAL RESULT S 63
0 20 40
0 20 40 60 80 100
Periods
(a) Wealth over time for agents all bidding truthfully
0 20 40
Virtual wealth (units)
Virtual wealth (units)
rich, and the poor remains poor. Further investigation reveals that the agents who have a lot
of meetings tend to be rich, because they need to spread their wealth among more meetings,
so they often would receive more from their peers than the bids they have placed, allowing
them to get richer over time. Since wealth is nite in the system, the poor cannot keep getting
poorer, and neither can the rich get richer inde nitely. On the other hand, Figure 5.3b shows
that this does not need to be the case. If agents are aggressive bidders, their levels of wealth
uctuate more widely.
0 20 40 60 80 100
Periods
(b) Wealth over time for agents all bidding aggressively
Figure 5.3: Wealth of jAj= 8 agents all bidding with the same strategy for 100 periods with
initial wealth w= 10. Each agent is depicted in a di erent color. Wealth better resembles a
random walk in the case of all agents bidding aggressively, as in Figure 5.3b, than the case
of all agents bidding truthfully, as in Figure 5.3a.
5.2.3 Comparability of Auction and Doodle Mechanisms
Before we can analyze the results from the auction and doodle models, we need to
establish that they are comparable. Speci cally, we need to show that if di erences exist
between the utilities of agents under the two models, they do not stem from systematic
di erences in the
CHAPTER 5. EXPERIMENTAL RESULT S 64
schedules that are produced, but rather on the optimality in the choices of which meetings to
schedule and at which times. Table 5.1 shares some statistics. In every metric, the auction
and doodle mechanisms are virtually indistinguishable in their abilities to schedule meetings
at times that attendees are able to attend.
Mechanisms Statistics Auction Doodle Of requested meetings, % thatare scheduled
78.47% 78.38%
Of scheduled meetings, % agents 82.40% 82.96%
7.19 7.15
that could attend
Units of time (out of 15) agents
spent in meetings each period
Table 5.1: Average statistics from running auction and doodle mechanisms each for 100
periods with parameters jAj= 8;jMj= 32;jTj= 15; = 5. The two mechanisms seem to be
comparable.
5.2.4 Sanity Checks
Finally, we conduct sanity checks to verify that the auction mechanism is able to discriminate
between rational strategies from irrational ones.
We let 8 agents choose from a set of permissible strategies, S= ftruthful, overly
aggressive, and inverseg, a reasonable strategy, an irresponsible strategy, and an irrational
strategy. The inverse strategy is irrational because it places higher bids on less favored time
slots than more favored ones. The overly aggressive strategy is technically rational in that it
does not distort the order of preferences, but it is irresponsible because it causes the agent
to spend his wealth quickly. We repeat each strategy pro le for trials of 500 periods each.
The results in Figure 5.4 show that in all three de nitions of payo s, the truthful strategy
clearly dominates in its ability to yield better payo s. The auction mechanism is able to
distinguish strategies and rewards better ones.
It is interesting to note that the payo matrices di er in which strategy they consider to
CHAPTER 5. EXPERIMENTAL RESULT S 65
be worse. The inverse strategy is the least reasonable in terms of actual and absolute
payo s, but the overly aggressive strategy is the least reasonable in terms of heuristic
payo s. This suggests that we should compare the di erent notions of utility.
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0.35 0.40 0.45 0.50 0.55
Expected Payoffs
280 300 320 340 360 380
Expected Payoffs
Figure 5.4: Using the auction mechanism, visual representations of payo matrices
generated for 8 agents with S= ftruthful, inverse, overly aggressiveg, one reasonable
strategy and two irrational ones. The strategy pro les are arranged in lexicographic order
from left to right. The payo s for each strategy pro le are represented by a set of red, blue,
and green bars between the gray lines. Error bars indicate one standard error above and
below the mean, and they are calculated as the sample standard deviation over the square
root of the size of the sample. The truthful strategy dominates in all three matrices.
However, the inverse strategy is the worst in terms of absolute and actual payo s, while the
overly aggressive strategy is the worst in terms of heuristic payo s.
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CHAPTER 5. EXPERIMENTAL RESULT S 66
5.2.5 Comparing Di erent Utilities
Figure 5.4 also suggests that various notions of utility may not reach the same equilibria, the
decision to choose a de nition is essential for valid results. Actual utility prevails as the most
reasonable de nition for both theoretical and practical reasons. First, we believe that actual
utility is the most accurate model for the happiness agents achieve. It is a personalized
metric unlike the absolute utility, and it emphasizes the end result rather than a means to an
end unlike the heuristic utility. Actual utility is also the most capable of distinguishing
outcomes in terms of more signi cant p-values. Finally, Figure 5.4 shows that it is the most
rational of the three de nitions. Actual utility acutely punishes a purely irrational strategy, the
inverse strategy, from a merely irresponsible one, the overly aggressive strategy.
Hence from this point on, \utility" refers to \actual utility" unless speci ed.
5.2.6 Random Agent Networks Robustness Check
In our experiments, we choose to only use one agent network because given the amount of
randomization in each of our trials, randomly generated agent networks would introduce
unnecessary noise in our sample data, thus requiring even more trials for accuracy. A
complication with this decision stems from the fact that although agent networks are
generated from the same initial parameters, its stochastic nature implies that there is still a
fair amount of variation among these networks. This is particularly true in a small network of
8 agents, but it also can be the case between di erent subnetworks of larger networks.
However, the relative e ectiveness of strategies should be universal regardless of the
structure of the agent network, even though their absolute payo s may di er.
Absolute payo s are expected to vary widely across di erent, small, agent networks. Even
though two networks can face the same parameters in the meeting scheduling problem of
distributing jMjmeetings over jAjagents, a well-connected network would have more agents
sharing these meetings evenly, while a sparse network with some lone agents would
CHAPTER 5. EXPERIMENTAL RESULT S 67
have a fraction of the agents shouldering most of the burden. Further, the lone agent who
does not have meetings would arti cially have a utility of 0; the average utility of a network
relates inversely to the proportion of lone agents. We need a robustness check that is
agnostic to these di erences in network structure, but measures the e ectiveness of relative
payo s of various strategies.
We conduct the following robustness check. First, we randomly generate 100 agent
networks using the same initial parameters that have been tted to the LiveJournal network.
Then, we run 100 trials, each using a distinct agent network, and compare the computed
payo matrix to another that has been computed from data using only one agent network. In
completing this computation, we need to guarantee that it be feasible to generate meeting
scheduling problems over every agent network, in particular for sparse networks, so we
slightly varied the parameters to decrease the level of constraints.
Instead of comparing absolute di erences in the payo s of various strategies between the
two experiments, we analyze the di erences between the means of pairs of strategies in
each entry of the payo matrix. Even though network structures may and do vary, if
strategy Ais always more e ective than strategy Bby the same margin, then we can
conclude that the result in the e cacy of strategy Ais robust. We show the results in Figure
5.5. Each color corresponds to di erences between a pair of strategies in the two trials. The
three boxes on the left refer to trials using 100 networks, while the three boxes on the right
refer to trials using only 1 network. The large size of the boxes on the left side substantiate
the claim that running experiments over randomly generated instances of the agent
network would have introduced a signi cant amount of noise. We conducted a paired
sample t-test for each pair, obtaining p-values of 0.1773, 0.8692, and 0.3299, respectively,
all of which are not su cient to reject the null hypothesis that there are di erences between
each pair of di erences.
-0.010 -0.005 0.000 0.005 0.010
Differences between Expected Payoffs
CHAPTER 5. EXPERIMENTAL RESULT S 68
100 networks (1) ..(2) ..(3) 1 network (1) ..(2) ..(3)
Pairs of Strategies
Figure 5.5: Two experiments of 100 trials each using 100 randomly generated agent
networks and 1 network, respectively. They were otherwise identical, with parameters jTj=
16;jAj= 8;jMj= 16 over 50 periods and with S= fthreshold, threshold-wealth, threshold-sel shg.
The boxplot shows the distribution of relative e ectiveness of pairs of strategies within each
entry of the payo matrix, represented by the di erences in respective expected payo s, i.e.
(1) corresponds with the di erences between payo s of threshold-wealth and threshold
strategies, and so on. The distributions vary more for the experiment using 100 agent
networks, but the di erences in means are not signi cant.
5.3 Results
In this section, we rst present results on the auction mechanism, then the doodle
mechanism, and nally we compare the two.
5.3.1 Auction: Preliminary Results and New Strategies
In order to make meaningful comparisons between the di erent bidding strategies, we limit
the strategy pro les to consist of one strategy of each type: only depend on oneself,
speculate on others, or a hybrid of the two. Table 5.2 presents the results from experiments
over all sets of strategies in these limited combinations. For each set, we calculate the
payo matrix, then iteratively seek the proportions of each strategy played in symmetric
mixed Nash equilibrium (henceforth abbreviated \NE").
CHAPTER 5. EXPERIMENTAL RESULT S 69
Symmetric Mixed NE Sets of
Permissible Strategies Proportions Expected Payo s
truthful popular hybrid 1 0.0664 0.9336 0 0.7918 truthful popular hybrid 2 0.0563
0.1805 0.7632 0.7948 truthful wealth hybrid 1 0.4223 0.5764 0.0013 0.7909 truthful
wealth hybrid 2 0 0 1 0.7956* truthful wealth contrarian hybrid 1 0.7119 0.0613
0.2268 0.7924 truthful wealth contrarian hybrid 2 0.0633 0.0003 0.9364 0.7955
threshold popular hybrid 1 1 0 0 0.8046* threshold popular hybrid 2 1 0 0 0.8045*
sel sh popular hybrid 1 1 0 0 0.7930* sel sh popular hybrid 2 0.4106 0 0.5894
0.7935* sel sh wealth hybrid 1 1 0 0 0.7932* sel sh wealth hybrid 2 0.1565 0 0.8435
0.7955 sel sh wealth hybrid 2 0.9307 0 0.0693 0.7948 sel sh wealth contrarian hybrid
1 0.9566 0.0434 0 0.7951 sel sh wealth contrarian hybrid 2 0.2233 0.0033 0.7733
0.7954 sel sh wealth contrarian hybrid 2 0.7738 0.1944 0.0318 0.7946 aggressive
popular hybrid 1 0.7734 0.1767 0.0498 0.7922 aggressive popular hybrid 2 0.1331 0
0.8669 0.7951 aggressive wealth hybrid 1 0.0037 0.5001 0.4962 0.7919 aggressive
wealth hybrid 2 0 0 1 0.7954 aggressive wealth contrarian hybrid 1 1 0 0 0.7919
aggressive wealth contrarian hybrid 2 0.6051 0.0006 0.3943 0.7935
Table 5.2: Symmetric mixed NE found using the auction model and actual utility for various
strategy pro les. Expected payo s in the last column are given by equation (4.1). * indicates
that replicator dynamics algorithm has not completely converged, but as the cycles are
miniscule and there seems to be an indisputable winner in these pro les, I rounded the
results.
We should note that some strategy pro les have multiple symmetric mixed NE that appear
repeatedly after more trials, such as S = fsel sh, wealth contrarian, hybrid 2g. This particular
case is perhaps due to the fact that the hybrid 2 strategy, depending on the agent’s wealth,
toggles between the sel sh and the wealth contrarian strategies, so it is unsurprising that it
appears to be almost symmetrical with the sel sh strategy as they may act similarly.
The anomaly to notice in this table is that the threshold strategy is always the only
CHAPTER 5. EXPERIMENTAL RESULT S 70
strategy played in equilibrium if it has been included in the set of permissible strategies.
Further, its expected payo s average 0.8046 while all other expected payo s lie between
0.7909 and 0.7956. This suggests that current strategies can be improved by taking on
features of the threshold strategy.
I introduce a set of threshold- strategies, all of which maintain their original characteristics
but mimic the threshold strategy by placing 0 bids for assignments with values below a
threshold ci, as given in Equation (3.4). The overall e ect is to amplify the admissible bids
such that they are only placed on the most important meetings, and to discard entirely bids
for all other meetings. For instance, the threshold-aggressive bidder iwould still bid
aggressively but only over the set of assignments where pi(m) ci. I run simulations over the
following strategies: threshold, threshold-sel sh, threshold-aggressive, threshold-wealth, and
threshold-wealth-contrarian. The threshold-popular strategy was initially considered, but
performed strictly worse than the others so it was subsequently dropped. The original
threshold strategy becomes analogous to the truthful strategy in this set because it does not
distort the preferences for assignments that are above the threshold. Table 5.3 presents the
results for this set of threshold- strategies.
At rst glance, the mixed equilibria almost always includes the threshold strategy.
However, it would be naive to conclude at this point that agents gravitate towards a particular
mix of strategies in equilibrium. We need to rst understand how to interpret the simpler case
of pure strategy NE, then develop a metric for evaluating the e ectiveness of each strategy
relative to others, and nally draw conclusions for the mixed strategy cases.
5.3.2 Interpreting Pure Equilibria
In order to interpret the presented symmetric mixed NE, we need to rst establish that each
entry represents a stable point from which no agent in the population can improve their
outcome by deviating to a di erent strategy. This is challenging to con rm with mixed
strategy NE, so we shed light using the simpler case of pure strategy NE.
CHAPTER 5. EXPERIMENTAL RESULT S 71
Symmetric Mixed NE Sets of
Permissible Strategies Proportions Expected Payo s
threshold -aggressive -contrarian 0.6731 0 0.3269 0.8057 threshold -aggressive
-sel sh 0.0459 0.7427 0.2114 0.8049 threshold -aggressive -sel sh 1 0 0 0.8052
threshold -aggressive -wealth 0.1829 0.7440 0.0731 0.8049 threshold -sel sh
-contrarian 0 1 0 0.8045* threshold -wealth -contrarian 0.3171 0.5098 0.1730
0.8046 threshold -wealth -contrarian 0.3875 0.0698 0.5427 0.8050 threshold
-wealth -sel sh 0.3952 0.2931 0.3116 0.8046 -aggressive -contrarian -sel sh
0.2022 0 0.7978 0.8046* -aggressive -contrarian -sel sh 1 0 0 0.8052* -aggressive
-contrarian -wealth 0.8353 0.1647 0 0.8050* -aggressive -sel sh -wealth 0.6563
0.2248 0.1189 0.8047 -contrarian -sel sh -wealth 0 0.2868 0.7132 0.8044*
-contrarian -sel sh -wealth 0.6253 0 0.3747 0.8046*
Table 5.3: Symmetric mixed Nash equilibria found using the auction model and the set of
threshold- strategies. Since the context is clear, the strategies are abbreviated to only the
parts after \threshold"; further, \threshold-wealth-contrarian" is abbreviated to \contrarian."
A pure strategy Nash equilibrium is a Nash equilibrium in which only one strategy in the
permissible set is played. For example, given S= fthreshold, threshold-sel sh,
thresholdwealth-contrariang, replicator dynamics indicate that in equilibrium, players would
all play the pure strategy threshold-sel sh. We consider an agent’s expected payo s if he
was to switch to playing another strategy, in this case, either threshold or
threshold-wealthcontrarian. Table 5.7 presents the relevant rows of the payo matrix. When
all agents are playing the threshold-sel sh strategy, the expected payo is 0.8045. If an
agent deviates to the threshold strategy, he can expect to have a lower payo of 0.7921,
which he would not prefer. If an agent deviates to the threshold-contrarian strategy, he can
expect to have an even lower payo of 0.7872, which he also would not prefer. This simple
test veri es that in this case, no agent is willing to deviate from the pure strategy equilibria.
CHAPTER 5. EXPERIMENTAL RESULT S 72
# Agents Playing Expected Payo s threshold -sel sh -contrarian threshold
-sel sh -contrarian p-values
Equilibrium 0 8 0 0 0.8045 0 NA Deviation 1 1 7 0 0.7921 0.8075 0 0.0660 Deviation 2 0 7 1 0
0.8052 0.7872 0.0348
Table 5.4: Relevant rows of the payo matrix for S = fthreshold, threshold-sel sh,
threshold-wealth-contrariang. P-values are given by the t-test for di erence in mean expected
payo s between each pair of strategies; they are signi cant at = 0:10.
5.3.3 Navigating Di erent Mixed Nash Equilibria
Because our equilibria are computed over games with restricted sets of strategies, we cannot
generalize to the ultimate equilibrium behavior: what would agents play if they could choose
any strategy? A way to navigate the set of symmetric mixed NE is to start from one NE,
introduce another strategy not included in the mix by having one agent play the new strategy
while others still play the original mixed strategy. The \invasion" is successful if the expected
payo of the \invading" strategy is higher than that of the original equilibrium, in which case
we can compute the mixed NE including the invading strategy. Another advantage of this
approach is that the successfulness of a given strategy in invading mixed equilibria of other
strategies signi es its relative e ectiveness.
In order to conclude whether a strategy can successfully invade an equilibrium of x
strategies, we require the expected payo s information for all strategy pro les of the x+ 1
strategies. Given our data of payo matrices consisting of 3 strategies in each set, we can
study invasions into 2-strategy mixed NE. This process generalizes to larger sets of
strategies if given the requisite payo matrices.
From a payo matrix, we rst calculate the mixed strategy NE and expected payo s for
each of the three sets of two strategies over the entries where the respective third strategy
does not appear. Then, we introduce the unused strategy into the population by letting one
agent play the invading strategy, while seven agents keep their mix of two strategies at NE.
We compute the expected payo of the deviating agent using equation (4.1). If it is lower
CHAPTER 5. EXPERIMENTAL RESULT S 73
than the equilibrium payo of the two original strategies, then no agent would deviate by
playing this strategy and the 2-strategy equilibrium would be maintained. If it is higher, then
this strategy has successfully invaded the population.
Over every set of strategies previously considered in Table 5.3, we evaluate the
successfulness of each component strategy in invading a mix of the other two strategies in
equilibrium. Table 5.7 summarizes the results. Our trials suggest that the aggressive strategy
is the most successful, at 100%, while the threshold strategy is a close second at 83%. On
the other hand, the sel sh and speculating on wealth strategies are each successful only
once, at 17%. This result is promising, indicating that agents often bene t by using the
aggressive or threshold strategies, but are usually harmed if they adopt the sel sh or
speculating on wealth strategies.
Invading Strategy # Successful Invasions # Attempted Invasions Success Rate
-aggressive 6 6 1
threshold 5 6 0.8333 -contrarian 2 6 0.3333 -sel sh 1 6 0.1667 -wealth 1 6 0.1667
Table 5.5: Summary of performance of di erent strategies in invading 2-strategy equilibria
using the auction model.
5.3.4 Doodle: Sel sh Strategy Dominates
The set of applicable strategies in the doodle model is di erent because of the trivial notion
of \wealth." In particular, speculating on wealth reduces to the truthful strategy, while the
hybrid strategies that depend on wealth do not correlate to any realistic scenarios. The total
wealth contrarian strategy still applies, but here it morphs into a strategy that speculates on
the number of attendees at each meeting. Further, since preferences cannot be
over-expressed in the doodle model, certain strategies have slightly di erent consequences
on poll responses as opposed to on bids. For instance, the aggressive strategy maps to a
CHAPTER 5. EXPERIMENTAL RESULT S 74
generous or exible strategy, because the aggressive agent agrees to attend meetings at
more times. With the remaining strategies|truthful, threshold, sel sh, aggressive, popular, and
contrarian|we run experiments with various combinations and present results in Table 5.6.
Symmetric Mixed NE Sets of
Permissible Strategies Proportions Expected Payo s
contrarian popular truthful 1 0 0 0.7273 contrarian popular aggressive 1 0
0 0.7274 sel sh popular truthful 1 0 0 0.7273* sel sh popular contrarian 1
0 0 0.7276* sel sh popular aggressive 1 0 0 0.7271 sel sh threshold
popular 1 0 0 0.7270* sel sh threshold popular 0.5386 0.4614 0 0.7111*
sel sh threshold aggressive 0.3680 0.6320 0 0.7035* threshold popular
contrarian 0.8860 0.1140 0 0.6899* threshold popular truthful 0.8157
0.1842 0 0.6945* threshold popular aggressive 0.9165 0.0835 0 0.6878*
Table 5.6: Symmetric mixed Nash equilibria using the doodle model for various strategy
pro les.
The sel sh strategy emerges as the only strategy played in equilibrium. By observation,
the truthful strategy is never included in any NE while the sel sh strategy is always included,
if it is part of the initial strategy pro le. We show this more rigorously. Starting from an
arbitrary mixed NE that does not include the sel sh strategy, such as playing the threshold
strategy with 89% probability and the popular strategy with 11% probability, we consider one
agent’s outlook if he was to deviate. However, none of the strategies besides the sel sh
strategy can remain in the population because they do not appear in any mixed NE. Once an
agent switches to the sel sh strategy, the expected payo rises from 0.69 to 0.71 or 0.73,
depending on the particular equilibrium, and the sel sh strategy takes over to become a pure
strategy NE. The result is the same if we start with all agents in the population playing the
contrarian strategy.
Table 5.7 summarizes the successfulness of each strategy in invading mixes of two other
strategies in equilibrium. The superiority of the sel sh strategy is further substantiated;
CHAPTER 5. EXPERIMENTAL RESULT S 75
an agent can improve his utility 100% of the time by behaving sel shly. The contrarian
strategy follows at 75%, but the threshold strategy is at 0%. Interestingly, strategies are
almost completely reversed in order of e ectiveness in the auction and doodle models.
Invading Strategy # Successful Invasions # Attempted Invasions Success Rate sel sh
551
contrarian 3 4 0.75 aggressive 2 4 0.5 popular 4 9 0.444 truthful 1 3 0.333
threshold 0 5 0
Table 5.7: Summary of performance of di erent strategies in invading 2-strategy equilibria
using the doodle model.
Chapter 6
Discussion
In this chapter, I discuss the results from our experiments and o er explanations for what we
observe. Next, I acknowledge major critique of the work and present open questions for
future research.
6.1 Auction vs. Doodle
We observe two salient di erences between the auction and doodle models:
Agents can expect to improve utility by 11% in the auction model over the pollbased,
doodle model. All else being equal, expected payo s in equilibrium is 0.81 under auction
and 0.73 under doodle, in terms that are relative to their own notions of ideal outcomes.
Agents bene t the most from employing aggressive and threshold strategies in the
auction model, as opposed to sel sh and speculative strategies in the doodle model.
We o er some explanations and propose a few theories as to why. Given a meeting mthat
agent iwants to attend, recall that his true preferences are
represented by the set of v(m;t), while his reported preferences are expressed in the form of
bii(m;t) under auction and li(m;t) under doodle. Also recall that the auction mechanism
76
CHAPTER 6. DISCUSSION 77
chooses the assignment with the highest overall bids by solving the maximization problem presented in Equation
(3.1):
Xj2 bj(m;t):
argmax A
m;t
The doodle mechanism chooses the assignment with the highest percentage of bids, thus maximizing a slightly
di erent quantity, as presented in Equation (3.3):
P j2A (m;t)
lj
;
d(m)
argmax
m;t
where d(m) = degree of meeting m2G
M.
The presence of d(m) in the second equation creates an inverse relationship betwe
the importance of an a rmative poll response from agent iand the number of attendees
the meeting. This perhaps explains why speculating against wealth is relatively succes
in invading an equilibrium in the doodle model, but not as e ective in the auction mode
Each agent solves the problem of maximizing the value of the chosen assignment
given others’ expressed preferences. For ease of notation, let us represent poll respon
as binary \bids," li(m;t) : bi(m;t) 2f0;1g, and
vi(m;t) = arg max
Oi(m;t) = Xj2A j6=ibj(m;t)
represent the collective the reported preferences of all other agents. Then, the agent’s
maximization problem becomes:
max(bi(m;t) + Oi(m;t)):
t
arg max
vi(m;t)
bi(m;t)
We consider two extreme cases of an agent’s state of knowledge about others’ preferences to draw conclusions
about his behavior across the spectrum. In one extreme, if an agent is able to accurately predict Oi(m;t), then he
would act the most sel shly in both the auction
CHAPTER 6. DISCUSSION 78
and the doodle models, i.e. bidding a positive amount only at the time that solves his
maximization problem, and 0 elsewhere. The sel sh strategy is the most rational choice for
the omniscient agent.
However, it is di cult to have perfect knowledge of other people’s bids, in practice and in
execution|it would require everyone solving the same equation simultaneously. In the Doodle
application, since everyone can see all previously submitted poll responses, the last person
to answer the poll e ectively has perfect knowledge, but not everyone can be the last person.
In the other extreme, if an agent does not have any knowledge of Oi(m;t), then he would
continue to act sel shly under the doodle model, but be more generous under the auction
model. The case is trivial if other agents’ preferences positively correspond to the agent’s
own preferences, because the favored meeting time will be chosen as long as the agent bids
rationally, i.e. bid a greater amount for more favored time slots than for less favored ones.
However, rationality diverges in the two models if other agents’ preferences negatively
correspond to the agent’s own preferences. In the doodle model, agents cannot inuence a
less favored time slot from being chosen before a more favored one. If he places a bid, or
bids, on less favored time slot, then he will only amplify the misalignment. So his dominant
strategy is to only say \yes" so long as others’ preferences agree with his own, but as soon
as others’ preferences disagree, he should never say \yes" thereafter. This implies that if
others highly disagree over their preferences, then the agent has no choice but to say \yes"
only on his favorite time slot, or as many favorite time slots as he is indi erent between. It
becomes a dilemma because the agent cannot indicate in any way his preferences between
time slots where he has replied, \no," so should one of them be chosen, the choice would not
have any input from him.
In the auction model, as long as an agents bids rationally, his bids are would in e ect
\smooth" over the parts of the curve where others’ preferences negatively correlate with his
own. Bidding truthfully, or proportional to truthful preferences, would not incur any
CHAPTER 6. DISCUSSION 79
additional cost to the agent in the sense that he receives in value proportional to the bid he
has paid. If he has exaggerated his bid over an assignment and it is chosen, then he would
be burdened to pay an excessive amount compared to the value that he gains. If given the
means, bidding more generously is better than purely being sel sh because the agent is still
able to voice his opinion on the time slots that he does not prefer as much as his others,
while still able to distinguish between various tiers of preferences.
Most real world scenarios are in-between the two extremes: agents often have imperfect
knowledge of Oi(m;t). These two scenarios shows that expressed preferences in the doodle
model depend heavily on the accuracy with which one can predict others’ responses, thus
acting sel shly is safe and rational. But in the auction, a good strategy is bidding over more
time slots regardless of whether people agree or disagree on preferences, though a better
knowledge of Oi(m;t) will allow the agent to use his wealth more wisely. The agent has no
choice but to bid conservatively in the doodle mechanism, but has
a choice to bid conservatively or more liberally in the auction mechanism. In the latter
scenario, the agent is able to make better choices to inuence the meetings that he really
cares about|as opposed to depending his bids on arbitrary factors such as the number of
attendees present|and let others take the lead in deciding for meetings that he do not care as
much. Thus, this agent is likely to be happier with his overall schedule.
6.2 Critique
We acknowledge and address major critiques in our work.
Better prediction of others’ preferences may change the relative e ectiveness of bidding
strategies.
This points to the crux of strategies and why they matter. We have studied a few
speculative strategies that modify reported preferences based on the individuals’ relative
importance and exibility, represented in the form of popularity and virtual wealth.
CHAPTER 6. DISCUSSION 80
The common rationale underlying both is that more popular and busier people are often
more constrained by time, so one should be more exible in order ensure that a meeting
will occur. Results show that these strategies do not perform well in the auction
mechanism but are more useful in the doodle mechanism, which suggest that the
auction mechanism may be more impervious to speculative manipulation. However,
since these strategies only modify relative bids between meetings, there is merit to
exploring more strategies that are time slot-speci c or can learn from experience.
The model for the meeting scheduling problem makes simplifying assumptions.
We had assumed that there are no externalities between projects in the sense that the
values to completing multiple projects are absolutely additive. This is perhaps not
reasonable. Certain projects may be substitutable, and some projects may enhance, or
detract from, the value of others. For instance, one may wish to either play laser tag
with friends or attend a party with colleagues, but not both. Learning yoga along with
taking a class on healthy diet may nicely complement each other, and an individual
values the combined health bene ts as a positive externality. If these externalities
cannot be captured within the construct of a project, then the model can bene t from
additional expressiveness that incorporates relationships between projects and
accounts for them in evaluating an individual’s true preferences. Other possible
extensions include generalizing our auction framework to extend the value revision
process between bids to account for these externalities, or extending the auction
process to understand more expressive preferences.
Is there an e ective \doodle strategy" in the mix?
While the e ectiveness of various strategies in the doodle model depends heavily on the
accuracy of predicting the preferences of others, better strategies also seem to only
incentivize more sel shness under doodle. While we recognize that rationality vary by
CHAPTER 6. DISCUSSION 81
context, and perhaps another strategy is more optimal for doodle, it is unclear whether
individuals would be happier with their outcomes as a result of better predictive ability.
6.3 Open Questions and Future Research
Our work opens two intriguing directions for future investigation: more realism on one hand,
and theoretical proof on the other.
Due to the applicable nature of the research question, many individuals face situations
where they need to coordinate their schedules with others’, and for a few, it is a way of life. A
realistic model that can extend and apply to these situations would improve productivity,
reduce frustration over scheduling as well as the amount of time required. Towards this end
of making an application that people can use, we can pose various questions and propose
extensions to improve the realism of the model. Some include:
How can we account for the psychology of agents? Agents may wish to bid in a way to
earn people’s trust, perhaps in the form of \karma points." Agents may even act in ways
that appear to be irrational. Can the system capture more complicated reasons people
would want to conceal their preferences? For instance, if one was unwilling to attend a
meeting requested by his boss but does not want his boss to know.
Encoding the relative importance of guests at a meeting, perhaps by variations in initial
distribution of wealth. If someone is absolutely necessary for a meeting to proceed, then
this person would have a di erent strategy: bid less and be more selective about times.
How can we improve the expressiveness of the model to extend dependency
relationships between projects?
How can we allow projects to be generated dynamically? A model where projects are
CHAPTER 6. DISCUSSION 82
generated dynamically can approximate real-world scenarios where people organically
schedule meetings with others.
How can the ndings be generalized to an actual scheduling application: people over a
network input commitments, dependencies, priorities and the app would produce a uid
and optimal schedule?
Another direction is to generalize the results and demonstrate their robustness in a
variety of settings. In moving towards a theoretical proof of the properties required for a
strategyproof and optimal system to schedule meetings, we can pose many questions
regarding rationality of strategies and the impact of mechanism design choices in equilibrium
outcomes of strategies. Some include:
How can strategies be improved? Some possibilities include developing learning
algorithms that can remember from past experiences with similar meetings, similar
people, or similar time slots. Also, if it is possible to predict when people will disagree the
most about meeting times, we can develop smarter strategies, i.e. if an agent is indi erent
between timeslots of a meeting, then he can bid low amounts across the board. The
doodle application partially has this property in the sense that the polls are visible to
everyone.
How do di erences in network structures a ect outcomes and equilibrium strategies? Can
we rule out these network e ects, if they exist? Investigate networks of larger size, varying
structures, and of subnetworks within networks.
How can one change the auction design to make sure that one’s bids would only depend
on his preferences, and not at all with others’ bids? What is necessary?
Chapter 7
Conclusion
We have established that the problem of coordinating schedules among self-interested
individuals is a salient problem for which there exist multiple solutions. However, these
solutions, such as e-mails, Microsoft Outlook, or Doodle, require trade-o s between privacy
and e ciency and do not guarantee the best schedule. Further, and this is perhaps most
vexing from a user’s standpoint, they are optimal only for niche subproblems, but prove
cumbersome when applied outside. A more natural interface is one in which users merely
need to provide an ordering of their commitments (with deadlines) by priority. The challenge
then lies in best using this information for multiple users at once to generate the ideal
schedule providing the maximal utility to all users.
We draw on ideas from current economic theory to propose and evaluate the e cacy of
scheduling meetings using an automated auction system with a virtual currency. Here, bids
are placed for meeting times in accordance with the priorities provided, until a complete
schedule is reached. We simulate the model over a realistic set of potential users so that our
conclusions accurately reect the viability of this approach. Further, we show that the
schedule generated is more utile and incentivizes people to represent priorities more
truthfully, so that there is little bene t in attempting to game the system, which is important for
a natural interface.
83
CHAPTER 7. CONCLUSION 84
One contribution of this work is to implement a faithful model simulating the commitments
and priorities of people over a network. In order to produce relevant results, the model must
be based on insights common to many large organizations, so that the potential results of the
model are realized when applied to actual groups of people. The model proposed is exible
and generalizable to larger instances, more complex settings, and open to various
extensions. The crux of the model lies in a heuristic function which creates an ordering of
preferences from priorities and deadlines of commitments, which relates the user’s input to
our auction mechanism. Further, we recognize the potential to re ne results by nessing this
function.
The main contribution of this work is to demonstrate the theoretical potential of using an
auction-based mechanism to move towards an optimal and strategyproof way to schedule
meetings between people. Simulating the auction mechanism over a set of ordered
commitments generates schedules which are 11% more utile than those generated by a
poll-based mechanism using the same input. This increase in utility is due to the auction
mechanism’s ability to understand and fully exploit the expressiveness of the preferences
provided, in the form of bids.
Further, we show that the most e ective bidding strategies do not require knowledge of
other agents’ information in the auction mechanism. Two variations of the truthful strategy,
the threshold and aggressive strategies, represent their preferences proportional to values,
though ampli ed. On the other hand, the sel sh strategy disproportionately weights the top
priority meeting at the cost of the others. In the doodle mechanism, this is manifest as only
voting for the meeting times one wants as opposed to those where one is available. We nd
that the sel sh strategy is almost always harmful in the auction mechanism, whereas it is the
most e ective in the doodle mechanism. Thus, the former favors a relative notion of
truthfulness, while the latter favors sel shness and manipulation by speculating on outside
information.
This thesis demonstrates that coordinating meetings among collaborating entities using
CHAPTER 7. CONCLUSION 85
the auction mechanism can nd schedules e ectively incorporating the preferences of all
these entities. Furthermore, the conceptual framework of the auction mechanism lends
itself well to an intuitive user interface which merely requires an ordering of commitments.
Both the theoretical and practical aspects imply that such a tool would be transformative in
many aspects of our busy social and professional lives.
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