Behavioral Graph Coloring
“An Experimental Study of the Coloring Problem
on Human Subject Networks”
[Science 313, August 2006]
Michael Kearns
Computer and Information Science
University of Pennsylvania
Collaborators:
Siddharth Suri
Nick Montfort
Special Thanks: Colin Camerer, Duncan Watts, Huanlei Ni
Background and Motivation
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Network Science: Structure, Dynamics and Behavior
– sociology, economics, computer science, biology…
– network universals and generative models
– empirical studies: network is given, hard to explore alternatives
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Navigation and the Six Degrees
– Travers & Milgram  Watts, Kleinberg
– distributed all-pairs shortest paths
– what about other problems?
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Behavioral Economics and Game Theory
– human rationality in the lab
– typically subjects in pairs
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This Work:
– human subject experiments in distributed graph coloring
– controlled variation of network structure (and other variables)
(Behavioral) Graph Coloring
solved
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not solved
Undirected graph; imagine a person “playing” each vertex
Finite vocabulary of colors; each person picks a color
Goal: no pair connected by an edge have the same color
Computationally well-understood and challenging…
– no efficient centralized algorithm known (exponential scaling)
– strong evidence for computational intractability (NP-hard)
– even extremely weak approximations are just as hard
• …Yet simple and locally verifiable
The Experiments: Overview
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Designed and built a system for distributed graph coloring
Designed specific sequence of experiments
Obtained human subjects review (IRB) approval
Recruited human subjects (n = 38, two sessions)
Ran experiments!
Analyzed findings
Experimental Design Variables
• Network Structure
– six different topologies
– inspired by recent generative models
• Information View
– three different views
• Incentive Scheme
– two different mechanisms
• Design space: 6 x 3 x 2 = 36 combinations
• Ran all 36 of them (+2)
Research Questions
• Can large groups of people solve these problems at all?
• What role does network structure play?
– information view, incentives?
• What behavioral heuristics do individuals adopt?
• Can we do collective modeling and prediction?
– some interesting machine learning challenges
Choices of Network Structure
Small
Worlds
Family
Simple Cycle
5-Chord Cycle
Preferential Attachment,
Leader Cycle
n=2
20-Chord Cycle
Preferential Attachment,
n=3
Choices of Information Views
Choices of Incentive Schemes
• Collective incentives:
– all 38 participants paid if and only if entire graph is properly colored
– payment: $5 per person for each properly colored graph
– a “team” mechanism
• Individual incentives
– each participant paid if they have no conflicts at the end of an experiment
– payment: $5 per person per graph
– a “selfish” mechanism
• Minimum payout per subject per session: $0
• Maximum: 19*5 = $95
The Experiments: Some Details
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5 minute (300 second) time limit for each experiment
Population demographics: Penn CSE 112 students
Handout and intro lecture to establish understanding
Intro and exit surveys
No communication allowed except through system
Experiments performed Jan 24 & 25, 2006
– Spring 2005: CSE 112 paper & pencil face-to-face experiments
– Sep 2005: system launch, first controlled experiments
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Jan 24 session: collective incentives; Jan 25 session: individual incentives
Randomized order of 18 experiments within each session
First experiment repeated as last to give 19 total per session
The Results:
Overview
• 31 of 38 experiments solved
• mean completion time of solved = 82s
•median = 44s
• exceeded subject expectations (52 of 76)
Effects of Network Structure
Colors
required
Min.
degree
Max.
degree
Avg.
degree
S.D.
Avg.
distance
Simple
cycle
2
2
2
2
0
9.76
144.17
5/6
378
5-chord
cycle
2
2
4
2.26
0.60
5.63
121.14
7/7
687
20-chord
cycle
2
2
7
3.05
1.01
3.34
65.67
6/6
8265
Leader
cycle
2
3
19
3.84
3.62
2.31
40.86
7/7
8797
Pref. att.,
newlinks=2
3
2
13
3.84
2.44
2.63
219.67
2/6
1744
Pref. att.,
newlinks=3
4
3
22
5.68
4.22
2.08
154.83
4/6
4703
Avg. duration &
fraction solved
• smaller diameter  better performance
• preferential attachment much harder than cycle-based
• distributed heuristic gives reverse ordering
Distributed
heuristic
Small
Worlds
Family
Simple Cycle
5-Chord Cycle
Preferential Attachment,
Leader Cycle
n=2
20-Chord Cycle
Preferential Attachment,
n=3
Effects of Information View
Effects of Incentive Scheme
Towards Behavioral Modeling
Algorithmic Introspection
Prioritize color matches to high degree nodes. That is, I tried to arrange it so that the high degree
nodes had to change colors the least often. So if I was connected to a very high degree node I
would always change to avoid a conflict,
vice comments)
versa, if I was higher degree than the others I
(Sep and
2005
was connected to I would usually stay put and avoid changing colors. [many similar comments]
Strategies in the local view: I would wait a little before changing my color to be sure that the nodes
in my neighborhood were certain to stay with their color. I would sometimes toggle my colors
impatiently (to get the attention of other nodes) if we were stuck in an unresolved graph and no one
was changing their color.
Strategies in the global view: I would look outside my local area to find spots of conflict that were
affecting the choices around me. I would be more patient in choices because I could see what was
going on beyond the neighborhood. I tried to solve my color before my neighbors did.
I tried to turn myself the color that would have the least conflict with my neighbors (if the choices
were green, blue, red and my neighbors were 2 red, 3 green, 1 blue I would turn blue). I also tried
to get people to change colors by "signaling" that I was in conflict by changing back and forth.
If we seemed to have reached a period of stasis in our progress, I would change color
and create conflicts in my area in an attempt to find new solutions to the problem.
When I had two or three neighbors all of whom had the same color, I would go back and forth
between the two unused colors in order to inform my neighbors that they could use either one if
they had to.
(Sep 2005 data)
signaling behaviors
Machine Learning for the Collective
(work in progress)
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Natural approach to develop a model of individual behavior:
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Some model details:
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aggregate all subject data to learn a single model
gradient descent on log-loss
Standard ML evaluation: log-loss on the test data
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weight vector for each action
take inner product with feature values
run through sigmoid squashing function
normalize output values to obtain conditional distribution
Some learning details:
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treat each subject & time step as an example
develop a set of state features believed to be salient (neighbor conflicts, degrees, history,…)
transform data to <features,action> where action is new color or no change
learn a conditional model: Pr[action|features]
still care about this, but…
New and interesting additional evaluation: collective behavior
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run 38 copies of the model in simulation on graphs
can the learned model explain/reconstruct the ordering of the human subjects?
makes collective predictions as well
Cycle-Based Model: Training
sumdeg, opp
constant (bias)
own degree > max same
own degree > max opp
fraction same
fraction opp
own degree
maxdeg, same
maxdeg, opp
sumdeg, same
#same
#opposite
Cycle-Based Model: Weights
Cycle-Based Model: Collective Behavior
(over 96 trials)
mean
standard
soln time deviation
Simple cycle
14559
13661
5-chord cycle
1730
1683
20-chord cycle
112
69
Leader cycle
220
254
Summary
• Human groups can solve rather complex coloring problems
– including from very limited, local information
• Network structure has clear effects
– within cycle-based family, solution time decreases with diameter
– preferential attachment appears considerably harder
• More info helpful for cycle-based, harmful for preferential attachment
• Individuals adopt sensible and natural heuristics
– inverse dependence of activity on degree
– signaling behaviors
– injection of “randomization” to escape local minima
Future Work
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More human subject experiments!
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wider variety of graph topologies
larger subject pools
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controlled vs. web-based
approximations and the “behavioral price of anarchy”
imposed vs. “natural” network structure
richer communication channels
other collective problems (independent set, consensus vs. differentiation,…)
etc. etc. etc.
Currently designing and developing portable Java-based system
Contact:
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email mkearns@cis.upenn.edu
web www.cis.upenn.edu/~mkearns