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. 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. Overview • Motivation • Cooperative Game Theory • Model • Results • Conclusion 2
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- 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. • 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. • 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. 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)
- 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. 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. 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. 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. 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. 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. 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. 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. 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. Heterogeneity • Similar story for heterogeneous case 20
- 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. Contact Information: Andrew Collins ajcollin@odu.edu Virginia Modeling, Analysis and Simulation Center Old Dominion University Norfolk, Virginia 23
- 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. 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. • 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. 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. 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
Editor's Notes
- Hello Andy from VMASC Overview Motivation - Group mix things up and are studied but not strategic Different normal gt Value is what get if every one out for your coalition No new coalition could do better Normally would get Exhaustive or analytic - Think of countries interacting Help neighbors Worst case scenarios of neighbors Split, kick out or join - Tend to get one group Heterogeneous not stable Some stability in heterogeneous case Some stability Heterogneous version - Thought get scale free
- Game Theory provides a mechanism for introducing sophisticated behavior into a model that involves multiple autonomous decision-makers like agent-based modeling. Unsurprisingly, Game Theory and ABM have a long history of being combined with many leading ABM researchers coming from a Game Theory background (Axelrod, 1997, Epstein, 1999). Surprisingly, the agent-based community has almost completely ignored one form of Game Theory-- the n-person or cooperative game-- though it has been considered by the related computer science multi-agent community (Shehory and Kraus, 1998). N-person game theory differs from traditional game theory in that it is focuses on coalition formation, which means that the Nash Equilibrium concept must be replaced with other solution concepts, e.g., the core (Gillies, 1959), the nucleous (Schmeidler, 1969) or the Shapley valve (Shapley, 1953). One reason for the disregard for n-person game theory is due to the computational complexity of solving these types of games. Therefore, scientists usually solve n-person games analytically.
- Wanted to study ERIKA’S COMMENTS: I think this slide could be slimmed down. Like, the first bullet doesn't need the subbullets-just say thaT part. Is SNA research on group formation or group dynamics? Does the second bullet related to SNA, or group formation more generally? The last two bullets are the same-- Combine and then use spoken words to clarify. Also "with" not "in" ABM
- ERIKA’S COMMENTS: Is the “sensible solution criterion” supposed to be singular? There’s only one criterion with which to judge the sensible solution? Just checking…
- Needs to be strictly super-additive (V(S U T) ≥ V(S) + V(T)) otherwise trivial. ERIKA: First subbullet: “N-S forms” what? Second bullet, first subbullet: “this maximum value” –what is “this” referring to? Bullet #1? Should it be “a”? Does the #2 bullet second subbullet need to be a sub-subbullet? Can you demote that one and still get the same meaning? If not, I think the first word is “giving” not “gives” If bullet three is still referring to the Characteristic Function, shouldn’t it be a subbullet of #2? What’s V(T)?
- The core of a game is the set of imputations which are not dominated by any coalition. Useful result in finding the core: Good news: The core is easy to calculate for a small game. Bad news: The core need not exist. If x =(x1,x2,.........xn), and y = (y1,y2,.........yn) are imputations x1+x2+.......+xn = v(12......n) = y1+y2+.......+yn One cannot have xi > yi for everyone. Maybe can have xi > yi for everyone in a coalition S with enough strength to get x If x and y are imputations x dominates y over S if i) xi > yi for all players i in S ii) iS xi v(S) ( S has strength to get x) x dominates y if it dominates on some coalition S. So could get x dominating y and y dominating x
- 2^N subgroups ERIKA: There’s a word missing in #1 subbullet 2: “see if in” –missing “group” or some word there…
- ERIKA: I reformatted this slide a little…
- ERIKA: Is bullet one “bet you” or “beat you”? I didn’t change it, but I think bullet four should say: “All a coalition’s agent values”
- ERIKA: Does “Dead Fish Fallacy” need to be a subbullet? “Subjection” to what? I reformatted this slide because I think two ideas were getting jumbled—conclusions and future work
- Shapley value: find split independent of when joined Nucleus - Idea: Find the distribution (imputation) where the most unhappy coalition under it is happier than the most unhappy coalition under any other imputation.
- Are Pareto optimal Assumptions Rewards are infinitely divisible Rewards can be transferred between players - Side Payments ERIKA: What does “share out” mean? Is it different than just “share of rewards”? Second bullet “where” characteristic function v? Also, do you need so many dots between x2 and xn? Is that the standard? Third bullet “the same out of the game as if they all co-operated” the same what? Does this mean the same payout in a completely non-cooperative game? Or the same individually? (pardon my GT ignorance)
- ERIKA: The third bullet does read right—I know what you’re saying, but I think this is phrased weird. Fourth bullet: I think it would read better “to see if EACH would benefit from splitting” Same in the second subbullet.
- ERIKA: Bullet 4 second subbullet “Possible to not be” what?