Optimizing Scrip Systems:
Efficiency, Crashes, Hoarders, and
Altruists
By Ian A. Kash, Eric J. Friedman, Joseph Y. Halpern
Presentation by
Avner May
12/10/08
Overview
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Introduction
Review of Earlier Model
Basic Results about Equilibrium in System
Examining Distribution of Money in System
Optimizing System Performance Through Money
Supply
Effects of Altruists and Hoarders on System
Conclusion
Intro: Brief History
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NGO’s issue their own currency
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Examples
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These are known as “scrip”
In company towns where government currency
was scarce
To prevent robbery in DC buses
Online Systems: Everquest, Second Life
Creates market for exchange of
goods/services that would be impractical or
undesirable with real currency
Introduction
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Motivating Example: Capital Hill Baby Sitting Coop
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Effects of too much or too little money on effectiveness
of system
Optimal Money Supply?
Note: fixed price
Homogeneous vs. Heterogeneous Population
Altruists and Hoarders
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Gnutella: 1% users satisfy 50% of requests
Effect on System?
Model
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Very similar to previous model
Agent Types:
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Tuple: t = (αt, βt, γt, δt, ρt) in T (finite set)
αt: Cost of satisfying request
βt: Probability that agent can satisfy request
γt: Utility an agent gains for having a request satisfied
δt: Rate at which agent discounts utility
ρt: Represents an agent’s relative request rate
Agent population entirely determined by (T,f,n)
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T: finite set of types
f: vector of ft, fraction of agents of type t
n: # of agents
Model (cont.)
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Infinite number of rounds
Each round:
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An agent is picked to request service (prob. proportional to
ρt)
If chosen agent has money, every other agent is able to
satisfy request with certain probability (βt: independent of
round). Else round ends
If at least one agent is willing and able to satisfy request,
transaction occurs
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Requester (type t) gets γt utils, pays $1 (normalized scrip cost)
Volunteer (type t’) pays cost of αt’ utils, receives $
n rounds per unit time (time b/w rounds is 1/n)
Model (cont.)
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Payoff Heterogeneity
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The results in this paper only explicitly apply to “payoff
heterogeneous” populations
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Population of types T that differ only in α, γ, δ (utility
variables)
Don’t differ in probability of being able to satisfy request (β),
or in frequency of requests (ρ)
Authors don’t believe it would fundamentally change results
to extend it to an arbitrary heterogeneous population
Recap: This paper extends the results from previous
paper to payoff heterogeneous populations
Discussion: Model
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Any thoughts?
Are the assumptions too strong?
Payoff-Heterogeneity Assumption?
Agent types remain constant over time?
Choice of volunteer is independent over
time?
Threshold Strategies?
Theoretical Results
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Many of the results in this paper are direct
analogs of the theorems we saw in the
previous paper
Theorem 3.1
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General idea: Given a payoff-heterogeneous system, it
will approach the max-entropy monetary distribution
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More specifically, if there are enough agents in the system, after
sufficient time, it will be arbitrarily close to the max-entropy
distributions with arbitrarily high probability
Note: This is assuming a given distribution of strategies, and the
final monetary distribution maximizes entropy subject to the
constraints of the strategy distribution
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Theoretical Results (cont)
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Theorem 3.2
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Given a payoff-heterogeneous system, if each agent of
type t plays a threshold strategy, then every agent has a
best-response which is a threshold strategy
Lemma 3.2
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The best-response function for agents of type t is nondecreasing
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Note: input to best-response function is vector k of threshold
values for agents of each type
Intuition: If all agents increase their threshold value, an agent
will earn money less often while wanting to spend money at
least as often. Thus, they will increase their threshold value.
Theoretical Results (cont)
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Theorem 3.3
 Given a payoff-heterogeneous system, their exists a non-trivial
Nash equilibrium where all agents of type t play the same threshold
strategy Sk.
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Sketch of proof
 By Lemma 3.2, for each type, BRt(k) is non-decreasing (k is vector)
 Thus, BR(k) is non-decreasing function
 So, Tarski’s fixed point theorem guarantees the existence of a
greatest and least fixed point
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Least fixed point: trivial equilibrium (all types play S0)
Greatest fixed point: Non-trivial equilibrium
Tarski’s Fixed Point Theorem
 Given any monotone increasing function on a complete lattice L (f:
LL) , their exist a greatest and a least fixed point.
Theoretical Results (cont)
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What is a fixed point?
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Vector k s.t. BR(k) = k
This means that given k, the best-response strategy profile
is precisely k
Thus, this is an equilibrium
Computing the fixed point efficiently
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Best-reply dynamics
Begin with strategy profile (∞, ∞, …, ∞)
Iteratively compute best-reply strategy profile
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BR(BR(….(BR((∞, ∞, …, ∞))
This process converges to greatest fixed point
Note: The greatest fixed point might not be the one
which maximizes social welfare
Theoretical Results (cont)
EXAMPLE
Using Best-Reply Dynamics
Identifying User Strategies
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In section 3 of paper, used max-entropy to
get explicit formula for monetary distribution
given strategies of agents (fraction of agents
playing each strategy)
πM
Now, we would like to try to determine the
strategies of the agents given a certain
monetary distribution
πM
Identifying User Strategies (cont)
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Given a monetary distribution M, an “explanation” of M to
be a distribution π over strategies such that the maximum
entropy distribution which results from π is M. (i.e., π
“explains” M if π  M)
Lemma 4.1
 If M is a fully supported distribution of money with finite
support, then there exist an infinite number of
explanations of M
The paper explains an algorithm for finding the “minimal
explanation,” that in which fewest # of strategies are played.
From minimal explanation, can further make inferences
about fraction of population of each type, and strategy they
are playing
Discussion
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At this point, the authors motivate why the minimal
explanation is useful
They think that there will be clusters of agents of similar
types, and within a cluster agents will play similar
strategies.
Does this make sense?
What if small differences in type make large differences in
strategy?
Can people be expected to compute their best response
functions accurately?
What if there are not clusters, but rather a uniform
distribution of agents of different types?
Optimizing the Money Supply
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In sections 3 & 4, we assumed money supply was
fixed, and considered how money is distributed
among agents of different types
We will now examine what happens to the
distribution of money when the money supply is
altered
We want to determine the money supply which will
optimize the performance of the system (maximize
social welfare)
Optimizing the Money Supply
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Result: Increasing the
amount of money
improves performance up
to the point where the
system crashes.
Thus, optimizing the
system is simply a matter
of finding the point at
which the system would
experience monetary
crash
Optimizing Money Supply
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But how do we find this optimal point?
Observation: We would like to find money supply
which minimizes fraction M0 of people without any
money
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Why?
Expected utility gain in each round
 (1-M0)(γt-αt) * PROB(at least one agent volunteers)
Expected total utility (assuming homogeneous populations)
 (1-M0)(γt-αt)/(1-δt)
Analysis is very similar for heterogeneous population
Optimizing Money Supply
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Theorem 5.1
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Increasing average money decreases fraction of people
with no money (assuming there is no monetary crash)
Thus, as long as non-trivial equilibria exist, the more money
the better
Corollary 5.1
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There exists a finite threshold m*, such that:
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If average money/person < m*, then non-trivial equilibria exist
If average money/person > m*, then system crashes (no nontrivial equilibrium exists)
Optimizing Money Supply
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But how do we find m*???
Given that you know T and f, you can use
techniques from section 3 (best-reply
dynamics) to see if non-trivial equilibria exists
Binary search: Given T,f. Vary m
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But how do we know T,f?
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m: average money/person
You can use results from section 4 to solve for T,f.
Reasonable?
Altruists and Hoarders
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Altruists: Provide services at no cost
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Hoarders: Always volunteer to satisfy
requests, but never make requests
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Presumably gain some utility by satisfying a
request
Can be modeled as playing S∞
Intuition
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Having altruists is like increasing money supply
Having hoarders is like decreasing money supply
Altruists and Hoarders
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Theorem 6.1
 Increasing fraction of agents which are altruists increases social
welfare up to the point where the system crashes
Theorem 6.2
 Increasing fraction of agents which are hoarders decreases
social welfare
System designer can see actions of these agents and modify
money supply accordingly to prevent sub-optimal outcomes
Additional property: Hoarders stabilize system by helping to
prevent monetary crash
 Theoretically, they are always willing to work for $1, regardless of
how depreciated the dollar is
Discussion
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Floating prices? Auctions with scrips?
Other types of irrational behavior?
Would these theoretical results hold-up well
in real-world scrip systems?
What next?
Any other thoughts