How to Analyse Social Network? : Part 2 Game Theory Thank you for all referred contexts and figures

Introduction Methods to address SNA Tasks  Traditional Approaches: Graph Theory  Such as Centrality Measures Optimization Techniques  Such as Genetic Algorithms ….  Recent Advances Data Mining Techniques Game Theory 2 Source: Source:

Introduction Methods to address SNA Tasks  Recent Advances Data Mining Techniques  Process of analyzing data from different perspectives and summarizing it into useful information Game Theory 3 Source:

Introduction General Issue:  Economists and game theorists have been interested in understanding how individuals (people) or institutions (businesses, corporation, and countries) behave in different economic situations. 4

Introduction General Issue:  Classical game theory predicts how rational agents behave in strategic settings Advertising Business interactions Job market 5

Introduction In many network settings, the behavior of the system is driven by the actions of a large number of autonomous individuals (or agents)  Research collaborations among both organizations and researchers  Telecommunication networks (Service Providers)  Online social communities such as Facebook Individuals are always self-interested and optimize their respective objectives 6

Introduction Social connectedness is the measure of how people come together and interact.  The connectedness of a complex social system really means two things: Structure of interconnecting links Interdependence in the behaviors of the individuals who inhabit the system  Game Theory 7 Source:

Introduction What does the Game Theory mean?  Game theory is the formal study of decision- making where several players (individuals or groups) must make choices that potentially affect the interests of the other players. Study of conflict and cooperation A game with only one player is usually called a decision problem. 8 Source:

9 Game Theory Game Theory aims to help us understand situations in which decision-makers interact.  Social connections by means of acquaintanceship, friendship, or levels of influence that can factor in decision-making are modeled with an undirected graph (network), where each vertex (node) represents an individual and an edge (link) denotes potential social ties.  Social Network: Interaction between people as a competitive activity Firms competing for business Animals fighting over prey Bidders competing in an auction  Game-theoretic modeling starts with an idea related to some aspect of the interaction of decision-makers.

Game Theory To Understand “Game Theory”  Players: Decision Makers  Payoff: Utility or Desirability of an outcome to a player  Nash Equilibrium: Strategic equilibrium (Lists of Strategies) 10 Who gets benefits, Who loses benefits!!

Strategic Games Strategic games:  A model of interacting decision-makers.  Each player has a set of possible actions. The model captures interaction between the players by allowing each player to be affected by the actions of all players, not only her own action.  Each player has preferences about the action profile— the list of all the players’ actions. 11

Strategic Games Consists of  a set of players  for each player, a set of actions  for each player, preferences over the set of action profiles. Examples:  Players may be firms, the actions prices, and the preferences a reflection of the firms’ profits.  Players may be animals fighting over some prey, the actions concession times, and the preferences a reflection of whether an animal wins or loses. 12

Prisoner's Dilemma One of the most well-known strategic games is the Prisoner’s Dilemma.  Two suspects in a major crime are held in separate cells.  There is enough evidence to convict each of them of a minor offense, but not enough evidence to convict either of them of the major crime unless one of them acts as an informer against the other (finks). 13

Prisoner's Dilemma Prisoner’s Dilemma.  If they both stay quiet, each will be convicted of the minor offense and spend one year in prison.  If one and only one of them finks, she will be freed and used as a witness against the other, who will spend four years in prison.  If they both fink, each will spend three years in prison. 14

Prisoner's Dilemma Players: The two suspects. Actions: Each player’s set of actions is {Quiet, Fink}. Preferences:  Suspect 1’s ordering of the action profiles, from best to worst, is (Fink, Quiet) (she finks and suspect 2 remains quiet, so she is freed), (Quiet, Quiet) (she gets one year in prison), (Fink, Fink) (she gets three years in prison), (Quiet, Fink) (she gets four years in prison).  Suspect 2’s ordering is (Quiet, Fink), (Quiet, Quiet), (Fink, Fink), (Fink, Quiet). 15

Prisoner's Dilemma Utility Function (from best to worst )  Suspect 1: u1(Fink, Quiet) > u1(Quiet, Quiet) > u1(Fink, Fink) > u1(Quiet, Fink).  Suspect 2: u2(Quiet, Fink) > u2(Quiet, Quiet) > u2(Fink, Fink) > u2(Fink, Quiet) If u1(Fink, Quiet) = 3, u1(Quiet, Quiet) = 2, u1(Fink, Fink) = 1, and u1(Quiet, Fink) = 0. If u2(Quiet, Fink) = 3, u2(Quiet, Quiet) = 2, u2(Fink, Fink) = 1, and u2(Fink, Quiet) = 0. 16

Prisoner's Dilemma The Prisoner’s Dilemma models a situation in which there are gains from cooperation  each player prefers that both players choose Quiet than they both choose Fink  but each player has an incentive to “free ride” (choose Fink) whatever the other player does. 17

Prisoner’s Dilemma The Prisoner’s Dilemma is a game in strategic form between two players.  Each player has two strategies, called “cooperate” and “defect,” which are labeled C and D for player I (suspect 1) and c and d for player II (suspect 2) 18

Prisoner’s Dilemma Player I chooses a row, either C or D, and simultaneously player II chooses one of the columns c or d. The strategy combination (C; c) has payoff 2 for each player, and the combination (D; d) gives each player payoff 1. The combination (C; d) results in payoff 0 for player I and 3 for player II, When (D; c) is played, player I gets 3 and player II gets 0. 19

Prisoner’s Dilemma “Defect” is a strategy that dominates “cooperate.”  Strategy D of player I dominates C since if player II chooses c, then player I’s payoff is 3 when choosing D and 2 when choosing C  If player II chooses d, then player I receives 1 for D as opposed to 0 for C. The unique outcome in this game, as recommended to utility-maximizing players, is therefore (D; d) with payoffs (1,1). 20

Prisoner's Dilemma 21 Prisoner A= Player I Prisoner B= Player II Confess=Cooperate Remain Silent=Defect

Prisoner’s Dilemma Applying Prisoner’s Dilemma in Social Network  Each node plays one of two strategies, cooperation or defection,  Each time step nodes decide whether to switch to a new strategy or keep playing the same.  All nodes connected to a node i, form its neighborhood.  To compute the payoff of a node one needs to account for all pair interactions (cooperator-cooperator, cooperator-defector, defector-cooperator and defector-defector) happening in the node's neighborhood. 22

Duopoly Examples: Two firms produce the same good, for which each firm charges either a low price or a high price.  Each firm wants to achieve the highest possible profit.  If both firms choose High then each earns a profit of $1000.  If one firm chooses High and the other chooses Low then the firm choosing High obtains no customers and makes a loss of $200, whereas the firm choosing Low earns a profit of $1200 (its unit profit is low, but its volume is high).  If both firms choose Low then each earns a profit of $

Bach or Stravinsky? (BoS) Two people wish to go out together. Two concerts are available: one of music by Bach, and one of music by Stravinsky. One person prefers Bach and the other prefers Stravinsky. If they go to different concerts, each of them is equally unhappy listening to the music of either composer. 24

Nash Equilibrium In a game, the best action for any given player depends on the other players’ actions.  When choosing an action, a player must have in mind the actions the other players will choose. That is, she/he must form a belief about the other players’ actions. 25

Reference David Easley and Jon Kleinberg, Networks, Crowds, and Markets: Reasoning About a Highly Connected World, Cambridge University Press,