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

2Source: http://www.cse.iitm.ac.in/snaworkshop/presentations/Ramasuri_Narayanam_Game_Theoretic_Models_for_Social_Network_Analysis_I.pdf

Source:http://psychgames.weebly.com/game-theory.html

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

3Source: http://www.cse.iitm.ac.in/snaworkshop/presentations/Ramasuri_Narayanam_Game_Theoretic_Models_for_Social_Network_Analysis_I.pdf

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.

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Introduction

General Issue: Classical game theory predicts how rational

agents behave in strategic settings Advertising Business interactions Job market

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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

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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

7Source:http://psychgames.weebly.com/game-theory.html

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.

8Source: http://professional-paper-writing-service.blogspot.com/2013/05/leadership-decision-making-and-problem.html

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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)

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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.

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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.

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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).

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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.

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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).

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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.

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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.

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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)

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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.

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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).

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Prisoner's Dilemma

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Prisoner A= Player IPrisoner B= Player II

Confess=CooperateRemain 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.

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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 $600.

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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.

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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.

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Reference

David Easley and Jon Kleinberg, Networks, Crowds, and Markets: Reasoning About a Highly Connected World, Cambridge University Press, 2010.

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