1. Game Theory & Social NetworkModels Connor McCabe PhD Candidate in Web Science University of Southampton cm7e09@ecs.soton.ac.uk Agents, Interaction &May 2012Complexity (AIC) Group

2. OverviewTalk: Game Theory & Social Network Models Introduction of basic concepts & models Examples of social network models using game theory Discussion of using game theory as a method forinvestigating social scenarios. 3. Game theory: basic concepts & models Game theory (GT) is used to model situations in whichmultiple participants (players) interact or affect each othersoutcomes. The origins of GT are from the field of economics, (andremains most active in that area) although it has beenapplied elsewhere in fields including sociology, psychologyand complexity science. 4. Payoff Matrices Normal FormExtended Form Player 1 AB A 1,10,0Player 2 B 0,01,1 Payoff Matrix for a co-ordination game 5. Exempli Gratia (e.g): Prisoners Dilemma Player 1 Cooperate DefectCooperateR=1, R=1S=10, T=0 Reward forSuckers mutualpayoff, andPlayer 2 cooperation temptation to defect =Defect T=0, S=10 P=5, P=5 Temptation to Punishment defect andfor mutual suckers payoff defectionFor PD, T(temptation) > R(reward) > P(punishment) > (S)suckerSee http://plato.stanford.edu/entries/prisoner-dilemma/ for full description 6. Models of network formationThere are 2 key aspects of game theoretic approach tomodelling network formation: (i) agents get some utility from the network, and there is anoverall societal welfare corresponding to any network thatmight arise, and (ii) links are formed by the agents themselves, and theresulting networks can be predicted through notions ofequilibrium or dynamic processes 7. Research case 1: Satifysing What is Satisfycing? (Satisfy + Sacrifice) Similar to the idea of structural balance (see chapter 5,Easley & Kleinberg, Networks, Crowds, and Markets, 2010) An example is the co-ordination game, played among manyparticipants with conflicting constraints. Wont be able to co-ordinate with everyone most likely(because players have different friends / strategies) Hence, the problem is then to identify the subset of thenetwork the player can gain most from co-ordinating theiractions with. The example that we discuss here is Davies et al. (2011)Adam P. Davies et al. (2011) "if you cant be with the one you love, love the one youre with"Artif. Life 17, 3 167-181. 8. Core mechanisms & resultsN=100 actors (players)Uij = symmetric payoff matrix, which defines for actors i and j either :(i)a coordination game ( x =1, y =0), or(ii) anticoordination (x=0, y=1).Players are assigned to play different type of games with others with equalprobability.Uij = 9. Core mechanisms & results Adding up the payoffs for a single player i,e.g. Ui(t) = sum(1 + 0 + 1 + 1 ) for games with player j = (1, 2, 3 n) and the whole social network G(t):e.g. G(t) = sum(50 + 49 + 53 + 40 .) representing combined outcome forevery players games with their network contacts. 10. Core Mechanisms and Results Players flip their current strategy if doing so means theycan co-ordinate with most of their friends, and to anti-coordinate with non-friends, in order to received a positivepayoff from these different social ties. Then, a dynamic social structure is modelled, by varyingweighting assigned to each connection as agents learn whothey most often co-ordinate with, (and who they dont). Ties now represent continuous values between 0 and 1,strongly weighted connections represent the interactions(games). 11. 1. Non-Habitual Agentst=0 RLPlayer 1 L Player 2RPlayer 2 L R Player 1L05R50True UtilityCoordination game (+5 utility for being the same)AntiCoordination game (+5 utility for being different) 12. 1. Non-Habitual Agentst=0RUtility=5 L Utility=10 L Utility=10RSystem Utility = 30Utility=5Coordination game (+5 utility for being the same)AntiCoordination game (+5 utility for being different) 13. 1. Non-Habitual Agentst=1 RLUtility=5Utility=10 L Utility=5 L Utility=15RSystem Utility = 40Utility=10Coordination game (+5 utility for being the same)AntiCoordination game (+5 utility for being different) 14. 1. Non-Habitual Agentst=2 RLUtility=5Utility=15RL Utility=5 Utility=10 L Utility=15RSystem Utility = 55Utility=15Coordination game (+5 utility for being the same)AntiCoordination game (+5 utility for being different) 15. 1. Non-Habitual Agentst=4 RLUtility=5Utility=15RL Utility=5 Utility=10 L Utility=15RSystem Utility = 55Utility=15Coordination game (+5 utility for being the same)AntiCoordination game (+5 utility for being different) 16. 1. Non-Habitual Agentst = 1000 RLUtility=5Utility=15RL Utility=5 Utility=10 L Utility=15R Utility=10End of relaxationCoordination game (+5 utility for being the same)AntiCoordination game (+5 utility for being different) 17. Core mechanisms & results Now we add a preference matrix Pij so that agentsperceive satisfying some connections and sacrificingothers. The preference matrix contains a value for each playerpairing; value is initially set to zero, and is adjusted eachtime step 18. 2. Habitual AgentsRR Player 1RPlayer 2 Player 2LR LPlayer 1Player 2 L0 0 L R Player 1 R0 0L5 0 Perception TransformationR0 5True Utility 19. 2. Habitual Agents R R Player 1RPlayer 2Player 2LRLPlayer 1 Player 2 L-0.1 0.1L RPlayer 1 R0.1-0.1 L5 0Player 2 Perception Transformation R0 5 L RPlayer 1 True UtilityL4.9 0.1 R0.1 4.9 Perceived Utility 20. 2. Habitual AgentsR R Player 1RPlayer 2L Habitual agents use perceived utilityto make strategy decisions Player 2L RPlayer 1 L4.9 0.1 R0.1 4.9 Perceived Utility 21. 2. Habitual AgentsR R Player 1RPlayer 2L Habitual agents use perceived utilityto make strategy decisions Player 2L R But system utility is always measuredPlayer 1using true utility L4.9 0.1 R0.1 4.9 Perceived Utility 22. Core Mechanisms and Results 23. Research case 2: El Farrol bar model The El Farol bar model involves N people (N=100), eachhave to decide each evening, at same time, whether theywant to go out to a bar, or else stay in. If less than 60% of the population go to the bar, theyll allhave a better time than if they stayed at home. If more than 60% of the population go to the bar, theyll allhave a worse time than if they stayed at home. This model represents a case of inductive reasoning, sincedeterministic / pure strategies are guaranteed to fail. 24. Core mechanisms and results The actors make decisions based on probability of certainoutcomes occurring. Assume 100 actors each can individual form predictors /hypothesis, of the past d weeks attendance figures. If for example, recent attendance might be: 44 78 56 15 23 67 84 34 45 76 40 56 22 35 Predictors of attendance might be: same as last weeks[35] a rounded average of last four weeks [49] 25. Core mechanisms and results Actors decide to go or stay based on most accuratepredictor they have found so far (active predictor) Once decisions are made, the actor updates the accuraciesof their predictors. Good predictors are kept, while those found evaluated asnot presently useful are not selected. A whole ecologycontaining the active predictors of actors emerges. 26. Results of the model Bar attendance in the first 100 weeks. Notice how there are no persistent cycles, Interesting, mean attendance always converges to 60 This is because the predictors self-organize into a pattern / equilibrium. 27. Core Mechanisms and Results Permit each player to use a mixed strategy, where a choiceis made with a particular probability. For the El Farol Bar problem there exists a Nashequilibrium where a mixed strategy involves each player deciding to go to the bar with a certain probability that is a function of the number of players, and the relative utility of going to a crowded or an uncrowded bar compared to staying home 28. Final note on mixed strategies Following a pure strategy, will enable other players to guess your move 29. Final note on mixed strategies Following a pure strategy, will enable other players to guess your move.Lisa: Look, theres only one way to settle this.Rock-paper-scissors.Lisas brain: Poor predictable Bart. Always takes `rock.Barts brain: Good old `rock. Nothing beats that!Bart: Rock!Lisa: Paper.Bart: Doh! 30. Final note on mixed strategies Following a pure strategy, will enable other players to guess your move.Lisa: Look, theres only one way to settle this.Rock-paper-scissors.Lisas brain: Poor predictable Bart. Always takes `rock.Barts brain: Good old `rock. Nothing beats that!Bart: Rock!Lisa: Paper.Bart: Doh! Hence the need for mixed strategies involving players randomisingtheir moves. To do well in these games involves players finding the optimalprobability with which to choose each strategy. 31. Research case 3: Co-evolution ofcooperation A model of co-evolution of a social network captures theinterplay between dynamics (games) on the network, andstructural changes of the network that influence thedynamics(games). Van Segbroek et. als (2010) model of prisoners dilemmaconsiders how players strategies change and evolvealongside which games are being played between whom. In this study they varied the payoff matrices between thedifferent linking strategies updatesVan Segbroeck S et. al. (2010).Coevolution of Cooperation, Response to Adverse Social Ties and Network Structure. Games. 1(3):317-337. 32. Core mechanisms & resultsTime scale Ta denotes evolution of the network structure,and Ts denotes strategy evolutionThe impact of network dynamics on the strategy dynamicsdepends on the ratio: W = Ts / TaWhere W 1 a slow linking dynamic.For values upwards of W=0.1, fixation ofcooperation is certain 33. Core mechanisms & resultsHow does the network of players evolve to co-operation? Heterogenous actors with different link strategies: Slow cooperators (SCs), and defectors (SDs) whose adverseinteractions last longer before they switch Fast coperators (FCs) and defectors (FDs) Actors can change both their strategy of co-operate or defect, and theirlink strategy to fast or slow. 34. Core mechanisms & results How does the temptation payoff (T) affect the stability of co-operation(graph a) How does speed at which links are adjusted between others (Y) affectevolution of the network? (graph b) 35. Core mechanisms & results Let M represent the number (types) of linking strategies. When M = 2, time spent in co-operation was lower (only 7.2%), andmost actors switched to slow defecters (SD) Increasing M had a positive effect on increasing selection ofcooperation (59.8% of time was spent co-operating). 36. Discussion The simplicity of game theory, using strategies and payoffs,can become very analytical when considering a largenumber of players representing a social network. This iswhere computer simulation can aid. In practice, there are some decisions to be made whendesigning a game theoretic model of a network, one keyissues which we now discuss: interactions over time 37. Interactions over time Repeated games are very sensitive to the order in whichplayers make their choices. One of careful considerations is whether all the playersmake their decision at the same time (synchronous) or not(asynchronous). Synchronous player updates often create coupleddynamics, such as how coupled oscillators sync their rotations / frequency over time. 38. Interactions over time Dynamics may disappear entirely with asynchronousupdates. E.g. a model of local co-operation was not stablewhen asynchronous updates were used. Source: Huberman, B. A. and Glance, N. S. (1993). Evolutionary games and computer simulations. Proceedings of the National Academy of Sciences USA, 90(16):77157718. 39. Game theory as a research method For social science and Web Science: Provides a means to describe exactly a set of actions and outcomes for social interactions. Amenable to simulation modelling Offers a link to investigate micro-macro behaviour Evolutionary game theory involving repeated games on a network are useful to model evolution of social systems / networks 40. Investigating macro from the micro-levelGame theoretic models encouragesfinding the simple micro rules thathelp understand the evolution ofcomplex macro phenomena, like theWeb, and emergent systems on itsuch as Web 2.0 and theblogosphere. 41. Further extensions for social models Bounded rationality for humans (limits on cognitiveprocessing, imperfect information) Recognise costs of gathering and processing information More realistic, multi-valued utility function Most simple game theoretic models involve agentschanging strategy on a fixed network structure. Some, however, are complex adaptive system models, inwhich both the agent strategies and network structure co-evolve. 42. Summary In this talk we discussed 3 game theoretic models involving simulatedsocial networks and their results involving1) Satisfycing / Structural balance Aim to satisfy relations that often paid off in the past2) Mixed strategies in a social decision problem Use heuristics to make a best guess, and keep a record ofwhich guesses were most often correct3) Evolution of co-operation in dynamic social network Responding promptly to adverse social ties promotes evolution of co-operation The discussion also addressed some of the assumptions practitionersneed to deal with in applying game theory in social network modelling 43. ReferencesArthur, W. B. (1994). Inductive Reasoning and Bounded Rationality (The El Farol Problem). Amer. Econ. Review (Papers and Proceedings), 84(406).Davies, A. P. et al. (2011) if you cant be with the one you love, love the one youre with: How individual habituation of agent interactions improves global utility. Artif. Life 17, 3 167-181.Easley., D. and Kleinberg., J. Networks, Crowds, and Markets (2010) Cambridge University PressSimon, H. A. (1956). Rational choice and the structure of the environment. Psychological Review, Vol. 63 No. 2, 129-138Huberman, B. A. and Glance, N. S. (1993). Evolutionary games and computer simulations. Proceedings of the National Academy of Sciences USA, 90(16):77157718.Van Segbroeck S et. al. (2010). Coevolution of Cooperation, Response to Adverse Social Ties and Network Structure. Games. 1(3):317-337.Zu Erbach-Schoenberg, Elisabeth, McCabe, Connor and Bullock, Seth (2011) On the interaction of adaptive timescales on networks. ECAL 2011, Paris, 08 - 12 Aug 2011 44. References (continued) The two magics of web science: www.w3.org/2007/Talks/0509-www-keynote-tbl/ (Accessed on 29/04/2012) Prisoners dilemma (Accessed on 29/04/2012)Other useful sources M. O. Jackson, Social and Economic Networks (2008) R. A. Axelrod, Complexity of Cooperation: Agent-Based Models of Competitionand Collaboration, Princeton Studies in Complexity (1997) 45. Question for discussionWhat sort of useful role (or not) can game theory provide as atool to investigate the theory and practice of Web Science? 46. Question for discussionWhat sort of useful role (or not) can game theory provide as atool to investigate the theory and practice of Web Science?Perhaps it can be viewed as:-too simplistic? toy models?-non realistic? mostly utilises selfish, maximising behaviour+ good way to look at link micro-level and macro-level+ useful for analysis and prediction (sometimes) of outcomesof social scenarios