Strategic Coalition Formation in Agent-based Modeling and Simulation

Andrew J. Collins, Ph.D.Erika Frydenlund, Ph.D.Terra L. ElzieR. Michael Robinson, Ph.D.

Swarmfest 2015 ConferenceJuly 10-12, 2015Columbia, SC

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Overview

•Motivation•Cooperative Game Theory•Model•Results•Conclusion

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Motivation

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Group Formation• Assume people tend to move and interact in

groups▫ What is the impact of this?

i.e. evacuation behavior Move towards danger to pick up kids

• Group formation has been well studied▫ Social Network Analysis (SNA) (Watts, 2004)▫ Formation based on:

Popularity, physical location (neighbors), or homophily (Wang and Collins, 2014).

• But what about strategic group formation?

• Wish to incorporate strategic group formation in Agent-based modeling (ABM)

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Cooperative Game Theory

•Not your common game theory▫Core, Shapley Value, nucleolus, ….

• Which coalitions form? • Who cooperates with whom?• How do coalitions share rewards between

members?

Cooperative Game Theory

•Coalitions S {1,2,...,n} = N▫Worst that can happen is remaining N-S forms own coalition that tries to minimise S's payoff i.e. 2-player zero-sum game forms

•Characteristic function “v(S)” gives a value that reflects this worst scenario

Characteristic Function

The Core

xi is reward that ‘i’ gets

x =(x1,x2,.........xn) is in the core if and only if

• x1+x2+.........+xn= v(1,2,..,n)

• S: iS xi v(S)

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How implement Core into ABMS?

•Options:▫Randomly form different collections of

coalitions▫Exhaustively test to see if in the core▫BUT would need to test all subgroups within

a coalition Group of 50 has 1015 subgroups

•Heuristic approach:▫Monte Carlo selection of subgroups

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Model

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Game

• Game consists of stationary agents interacting on a Von Neumann Grid

• Each link is worth 1.

▫ Split depends on strength of agents

• If the agent is:

▫ Stronger, it gets one

▫ Weaker, it gets zero

▫ Equal or in the same coalition, it gets 0.5

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Coalition Effects

• Members in an agent A’s coalition can add to your strength if they are neighbors of the agent A’s opponent

• For example, the black agent has support from 2 other agents; the blue agent has the support of only one

• We are interested in the characteristic value so ….

A

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Coalition Effects

• … it is assumed that all your opponents will join forces to beat you which means▫ Assume that red and

yellow are blue• In which case, agent A

loses and would get a zero• This process is repeated

for all four of A’s neighbors and summed

• All a coalition agent’s values are summed to determine its characteristic value, v(S)

A

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Algorithm

1. Select random agent and associated coalition.2. Randomly determine subgroup containing

selected agent and determine value of subgroup If value greater than in coalition, then subgroup

detaches from the main coalition

3. Determine if coalition is better off without agent If so, then agent gets kicked out

4. Determine if the agent’s current coalition benefits from joining another random local coalition Other coalition picked that is connected to current

agent

5. Repeat

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Results

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Homogeneous case

• Repeat the runs 100 times• Always resulted in

dominant group being formed.

• Most dominated groups would have a v(S) = 0▫ However, special

circumstance could result in differences

• Sometimes groups look larger because the same color has been used for multiple groups

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Heterogeneity

• Heterogeneous case was not so straight-forward

• Sometimes would get a super-group which would split▫ Never saw a super-group

splitting in the homogeneous case

• Did see some minor stability…

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Heterogeneity

• The blue group will never split or join another group as they both get 2.5 out of a max 3!

• Would need to get 3 (the max) to make it beneficial to split or join another group

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Does the stability exist?

• Two groups▫ Checkerboard

• No wrap around• Values

▫ 40 get 1 or 1.5 (edges)▫ 81 get 2

• No subset benefits from deviation

• Never saw this being converged to “Unstable”

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Heterogeneity

• Similar story for heterogeneous case

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ConclusionsFuture directions

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Conclusion

Expectations Future Directions

• Was hoping to see scale-free behavior or two competing large groups

• Warm-up states where interesting▫“Dead Fish Fallacy”

• Emergent behavior▫Globalization with

subjugation

• Add side payments

• Prove that process converges to:▫1) Imputation▫2) Core

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Questions?Contact Information: Andrew Collins ajcollin@odu.edu

Virginia Modeling, Analysis and Simulation CenterOld Dominion UniversityNorfolk, Virginia

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References• AXELROD, R. 1997. The complexity of cooperation: Agent-based models of competition and

collaboration, Princeton, Princeton University Press.• COLLINS, A. J., ELZIE, T., FRYDENLUND, E. & ROBINSON, R. M. 2014. Do Groups Matter? An Agent-

based Modeling Approach to Pedestrian Egress. Transportation Research Procedia, 2, 430-435.• ELZIE, T., FRYDENLUND, E., COLLINS, A. J. & MICHAEL, R. R. 2014. How Individual and Group

Dynamics Affect Decision Making. Journal of Emergency Management, 13, 109-120.• EPSTEIN, J. M. 1999. Agent‐based computational models and generative social science. Complexity,

4, 41-60.• EPSTEIN, J. M. 2014. Agent_Zero: Toward Neurocognitive Foundations for Generative Social Science,

Princeton University Press.• GILLIES, D. B. 1959. Solutions to general non-zero-sum games. Contributions to the Theory of

Games, 4, 47-85.• MILLER, J. H. & PAGE, S. E. 2007. Complex Adaptive Systems: An Introduction to Computational

Models of Social Life, Princeton, Princeton University Press.• SCHMEIDLER, D. 1969. The nucleolus of a characteristic function game. SIAM Journal on applied

mathematics, 17, 1163-1170.• SHAPLEY, L. 1953. A Value of n-person Games. In: KUHN, H. W. & TUCKER, A. W. (eds.) Contributions

to the Theory of Games. Princeton: Princeton University Press.• SHEHORY, O. & KRAUS, S. 1998. Methods for task allocation via agent coalition formation. Artificial

Intelligence, 101, 165-200.• WANG, X. & COLLINS, A. J. Popularity or Proclivity? Revisiting Agent Heterogeneity in Network

Formation. 2014 Winter Simulation Conference, December 7-10 2014 Savannah, GA.• WATTS, D. J. 2004. The “New” Science of Networks. Annual Review of Sociology, 30, 243-270.

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Cooperative Game Theory

• a.k.a. N-person Game Theory

• Forget the Nash Equilibrium Based around maximin solution

▫Characteristic functions▫ Imputations▫Core▫ Shapley Value (1953)▫Nucleous (Schmidler 1969)

• Side-payments▫With or without▫With or without enforcement

• Imputation is a "reasonable" share out of rewards.• An imputation in a n-person game with characteristic

function v is a set of rewards x1,x2,...,xn, where:

• The first condition is a Pareto optimality condition that ensures the players get the same out of the game as if they all cooperated.

• The second condition assumes a player’s reward is as good as non-cooperation.

Imputation

1) (Efficient)

2) (individually rational)

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Details

•Assume no side-payments allowed•v({i}) = 0 (homogenous case)•What to find imputation and core▫(need sum xi = V(N))

•How?▫Iteratively test coalition subgroups to see if

would benefit from splitting▫Iteratively test feasible joining of groups to see

if would benefit from join into super-coalition

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Results

• Placed agents on 11 x 11 grid▫Strict borders

•Run model with homogeneous case▫Every agent had a strength of one

•Heterogeneous case▫U[0,1] strength

•Result where all imputation▫ xi= 162+ 27 +4 = V(N)▫Possible to not be

Consider 121 random coalitions of {i} each xi= 0