Including economics, language, cultural influence, collective actions, ecosystems and predator / prey dynamics, social dilemmas – such as conflict resolution / negotiation, coalition formation, elections, co-operation and evolution of trust, one shot games, or else repeated interactions – these offer chance to study such things such as reputation, trust, and evolving / adaptive strategies.
In a normal form game, the best move is always known. Extended form games describe the long list of all the possible moves
For more on this categorisation see the paper “The Economics of Social Networks” (2007) by Matthew O. Jackson
Itis necessary for everyone to decide at the same time whether they will go to the bar or not.
Reinforcement learning,Not useful predictors get moved down in priority and ignored
A model of co-evolution captures this interplay between games on the network, and structural changes of the network that support / constrain the behaviour.
The models shown so far have represented the breadth of applications in which game theory can model social network formation.
e.g. a model purporting to show evolution of co-operation only worked with synchronous updates. Hence, the findings model were very fragile to changing some simple assumptions.
By defining a set of actions and outcomes as equations it allows exploration of the problem space2.Repeated games on a network, allow for network topology and / or strategy evolution. (e.g. evolution of co-operation, trust or community structure / clustering.
Game theory social networks cmccabe-12
Game Theory & Social Network Models Connor McCabe PhD Candidate in Web Science University of Southampton email@example.com Agents, Interaction &May 2012 Complexity (AIC) Group
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 for investigating social scenarios.
Game theory: basic concepts & models• Game theory (GT) is used to model situations in which multiple participants (players) interact or affect each other’s outcomes.• The origins of GT are from the field of economics, (and remains most active in that area) although it has been applied elsewhere in fields including sociology, psychology and complexity science.
Payoff Matrices • Normal Form Extended Form Player 1 A B A 1,1 0,0Player 2 B 0,0 1,1 Payoff Matrix for a co-ordination game
Exempli Gratia (e.g): Prisoner’s Dilemma Player 1 Cooperate Defect Cooperate R=1, R=1 S=10, T=0 Reward for Sucker’s mutual payoff, and Player 2 cooperation temptation to defect = Defect T=0, S=10 P=5, P=5 Temptation to Punishment defect and for mutual sucker’s payoff defectionFor PD, T(temptation) > R(reward) > P(punishment) > (S)suckerSee http://plato.stanford.edu/entries/prisoner-dilemma/ for full description
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 an overall societal welfare corresponding to any network that might arise, and• (ii) links are formed by the agents themselves, and the resulting networks can be predicted through notions of equilibrium or dynamic processes
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 many participants with conflicting constraints. – Won’t 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 the network the player can gain most from co-ordinating their actions 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.
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) anti–coordination (x=0, y=1).Players are assigned to play different type of games with others with equalprobability.Uij =
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 player’s games with their network contacts.
Core Mechanisms and Results• Players flip their current strategy if doing so means they can co-ordinate with most of their ‘friends’, and to anti- coordinate with ‘non-friends’, in order to received a positive payoff from these different social ties.• Then, a dynamic social structure is modelled, by varying weighting assigned to each connection as agents learn who they most often co-ordinate with, (and who they don’t).• Ties now represent continuous values between 0 and 1, strongly weighted connections represent the interactions (games).
1. Non-Habitual Agentst=0 R L Player 1 L Player 2 R Player 2 L R Player 1 L 0 5 R 5 0 True Utility Coordination game (+5 utility for being the same) AntiCoordination game (+5 utility for being different)
1. Non-Habitual Agentst=0 R Utility=5 L Utility=10 L Utility=10 R System Utility = 30 Utility=5 Coordination game (+5 utility for being the same) AntiCoordination game (+5 utility for being different)
1. Non-Habitual Agentst=1 R L Utility=5 Utility=10 L Utility=5 L Utility=15 R System Utility = 40 Utility=10 Coordination game (+5 utility for being the same) AntiCoordination game (+5 utility for being different)
1. Non-Habitual Agentst=2 R L Utility=5 Utility=15 R L Utility=5 Utility=10 L Utility=15 R System Utility = 55 Utility=15 Coordination game (+5 utility for being the same) AntiCoordination game (+5 utility for being different)
1. Non-Habitual Agentst=4 R L Utility=5 Utility=15 R L Utility=5 Utility=10 L Utility=15 R System Utility = 55 Utility=15 Coordination game (+5 utility for being the same) AntiCoordination game (+5 utility for being different)
1. Non-Habitual Agentst = 1000… R L Utility=5 Utility=15 R L Utility=5 Utility=10 L Utility=15 R Utility=10 End of relaxation Coordination game (+5 utility for being the same) AntiCoordination game (+5 utility for being different)
Core mechanisms & results• Now we add a preference matrix Pij so that agents perceive satisfying some connections and sacrificing others.• The preference matrix contains a value for each player pairing; value is initially set to zero, and is adjusted each time step
2. Habitual Agents R R Player 1 R Player 2 Player 2 L R LPlayer 1 Player 2 L 0 0 L R Player 1 R 0 0 L 5 0 Perception Transformation R 0 5 True Utility
2. Habitual Agents R R Player 1 R Player 2 Player 2 L R LPlayer 1 Player 2 L -0.1 0.1 L R Player 1 R 0.1 -0.1 L 5 0 Player 2 Perception Transformation R 0 5 L R Player 1 True Utility L 4.9 0.1 R 0.1 4.9 Perceived Utility
2. Habitual Agents R R Player 1 R Player 2 L• Habitual agents use perceived utilityto make strategy decisions Player 2 L R Player 1 L 4.9 0.1 R 0.1 4.9 Perceived Utility
2. Habitual Agents R R Player 1 R Player 2 L• Habitual agents use perceived utilityto make strategy decisions Player 2 L R• But system utility is always measured Player 1using true utility L 4.9 0.1 R 0.1 4.9 Perceived Utility
Research case 2: El Farrol bar model• The El Farol bar model involves N people (N=100), each have to decide each evening, at same time, whether they want to go out to a bar, or else stay in.• If less than 60% of the population go to the bar, theyll all have a better time than if they stayed at home.• If more than 60% of the population go to the bar, theyll all have a worse time than if they stayed at home.• This model represents a case of inductive reasoning, since deterministic / pure strategies are guaranteed to fail.
Core mechanisms and results• The actors make decisions based on probability of certain outcomes occurring.• Assume 100 actors each can individual form predictors / hypothesis, of the past d week’s 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 week’s – a rounded average of last four weeks 
Core mechanisms and results• Actors decide to go or stay based on most accurate predictor they have found so far (active predictor)• Once decisions are made, the actor updates the accuracies of their predictors.• Good predictors are kept, while those found evaluated as not presently useful are not selected. A whole ecology containing the active predictors of actors emerges.
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.
Core Mechanisms and Results• Permit each player to use a mixed strategy, where a choice is made with a particular probability.• For the El Farol Bar problem there exists a Nash equilibrium 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
Final note on mixed strategies• Following a pure strategy, will enable other players to guess your move
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!
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 randomising their moves.• To do well in these games involves players finding the optimal probability with which to choose each strategy.
Research case 3: Co-evolution ofcooperation• A model of co-evolution of a social network captures the interplay between dynamics (games) on the network, and structural changes of the network that influence the dynamics(games).• Van Segbroek et. al’s (2010) model of prisoner’s dilemma considers how players strategies change and evolve alongside which games are being played between whom.• In this study they varied the payoff matrices between the different linking strategies updatesVan Segbroeck S et. al. (2010).Coevolution of Cooperation, Response to Adverse Social Ties and Network Structure. Games. 1(3):317-337.
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 represents fast linking dynamic,and W>> 1 a slow linking dynamic.For values upwards of W=0.1, fixation ofcooperation is certain
Core mechanisms & resultsHow does the network of players evolve to co-operation?• Heterogenous actors with different link strategies: – Slow cooperators (SC’s), and defectors (SD’s) whose adverse interactions last longer before they switch – Fast coperators (FC’s) and defectors (FD’s)• Actors can change both their strategy of co-operate or defect, and their link strategy to fast or slow.
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) affect evolution of the network? (graph b)
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%), and most actors switched to slow defecters (SD)• Increasing M had a positive effect on increasing selection of cooperation (59.8% of time was spent co-operating).
Discussion• The simplicity of game theory, using strategies and payoffs, can become very analytical when considering a large number of players representing a social network. This is where computer simulation can aid.• In practice, there are some decisions to be made when designing a game theoretic model of a network, one key issues which we now discuss: interactions over time
Interactions over time• Repeated games are very sensitive to the order in which players make their choices.• One of careful considerations is whether all the players make their decision at the same time (synchronous) or not (asynchronous).• Synchronous player updates often create coupled dynamics, – such as how coupled oscillators sync their rotations / frequency over time.
Interactions over time• Dynamics may disappear entirely with asynchronous updates. E.g. a model of local co-operation was not stable when 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):7715–7718.
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
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 the‘blogosphere.’
Further extensions for social models• Bounded rationality for humans (limits on cognitive processing, imperfect information) – Recognise costs of gathering and processing information – More realistic, multi-valued utility function• Most simple game theoretic models involve agents changing strategy on a fixed network structure.• Some, however, are complex adaptive system models, in which both the agent strategies and network structure co- evolve.
Summary• In this talk we discussed 3 game theoretic models involving simulated social networks and their results involving 1) Satisfycing / Structural balance • Aim to satisfy relations that often paid off in the past 2) Mixed strategies in a social decision problem • Use heuristics to make a best guess, and keep a record of which guesses were most often correct 3) 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 practitioners need to deal with in applying game theory in social network modelling
References• Arthur, 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 Press• Simon, H. A. (1956). Rational choice and the structure of the environment. Psychological Review, Vol. 63 No. 2, 129-138• Huberman, B. A. and Glance, N. S. (1993). Evolutionary games and computer simulations. Proceedings of the National Academy of Sciences USA, 90(16):7715–7718.• 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
References (continued)• The two ‘magics’ of web science: www.w3.org/2007/Talks/0509-www-keynote- tbl/ (Accessed on 29/04/2012)• Prisoner’s dilemma <http://plato.stanford.edu/entries/prisoner- 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 Competition and Collaboration, Princeton Studies in Complexity (1997)
Question for discussionWhat sort of useful role (or not) can game theory provide as atool to investigate the theory and practice of Web Science?
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