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Collins SWARMFEST2015 Strategic coalition formation

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my slides from SWARMFEST 2015 which show my beginnings into place a heuristic core into agent-based simulation

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Collins SWARMFEST2015 Strategic coalition formation

  1. 1. Strategic Coalition Formation in Agent-based Modeling and Simulation Andrew J. Collins, Ph.D. Erika Frydenlund, Ph.D. Terra L. Elzie R. Michael Robinson, Ph.D. Swarmfest 2015 Conference July 10-12, 2015 Columbia, SC
  2. 2. Overview • Motivation • Cooperative Game Theory • Model • Results • Conclusion 2
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  4. 4. 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) 4
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  6. 6. • 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
  7. 7. • 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
  8. 8. 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: iS xi  v(S)
  9. 9. 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 9
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  11. 11. 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 11
  12. 12. 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 …. 12 A
  13. 13. 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) 13 A
  14. 14. 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 14
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  16. 16. 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 16
  17. 17. 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… 17
  18. 18. 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 18
  19. 19. 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” 19
  20. 20. Heterogeneity • Similar story for heterogeneous case 20
  21. 21. Future directions 21
  22. 22. 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 22
  23. 23. Contact Information: Andrew Collins ajcollin@odu.edu Virginia Modeling, Analysis and Simulation Center Old Dominion University Norfolk, Virginia 23
  24. 24. 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. 24
  25. 25. 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 25
  26. 26. • 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) 𝑖=1 𝑛 𝑥𝑖 = 𝑣 𝑁 (Efficient) 2) 𝑥𝑖 ≥ 𝑣 𝑖 ∀𝑖 ∈ 𝑁 (individually rational)
  27. 27. 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 27
  28. 28. 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 28

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