When the Group Matters A Game-Theoretic Analysis of Team Reasoning - PowerPoint PPT Presentation

When the Group Matters A Game-Theoretic Analysis of Team Reasoning and Social Ties Fr´ ed´ eric Moisan Emiliano Lorini IRIT, CNRS, Toulouse, France IRIT, July 2013 1/ 35

Problems of Cooperation: Prisoner’s Dilemma Bob D C (2 , 2) (0 , 3) C Alice (3 , 0) (1 , 1) D What should/will each player do? ◮ ⇒ Normative: they should choose ( D , D ) (Nash equilibrium) ◮ ⇒ In real-life: 30-50% of people choose C 2/ 35

Problems of Coordination: Hi-Lo matching game Bob D C (2 , 2) (0 , 0) C Alice (0 , 0) (1 , 1) D What should/will each player do? ◮ ⇒ Normative: both ( C , C ) and ( D , D ) are Nash equilibria ◮ ⇒ Real life: most people choose ( C , C ) 3/ 35

Formalizing strategic interaction Definition (Standard Strategic Game) G = � Agt , { S i | i ∈ Agt } , { U i | i ∈ Agt }� where: Agt = { 1 , . . . , n } is the set of agents; S i defines the set of strategies for agent i ; U i : � i ∈ Agt S i → R is a total payoff function mapping every strategy profile to some real number for some agent i . 4/ 35

Outline Theories of team reasoning 1 Theory of social ties 2 Conclusion 3 5/ 35

Outline Theories of team reasoning 1 Theory of social ties 2 Conclusion 3 6/ 35

Tuomela’s theory of team reasoning Tuomela’s I-mode / we-mode distinction [Tuomela,2010]: I-mode = the agent conceives the situation as a classical decision making problem (maximization of expected utility) ◮ plain I-mode = make a decision with the individual goal to maximize the individual payoff ◮ pro-group I-mode = make a decision with the individual goal to maximize the group payoff We-mode = the agent identifies himself as a member of the team and frames the situation as a problem for the team (beyond classical decision theory) 7/ 35

Sugden’s theory of team reasoning According to [Sugden,2003,2007]: Statement (Simple team reasoning) If I believe that: I am a member of group J. It is common knowledge among all members of J that we all identify with J. It is common knowledge among all members of J that we all want group payoff U J to be maximized. It is common knowledge among all members of J that solution s uniquely maximizes U J . Then I should choose my strategy in s. 8/ 35

Bacharach’s theory of team reasoning Concept of unreliable team interaction [Bacharach,1999]: A game theoretic model of team reasoning People frame the problem as a I or we problem: a psychological factor, prior to any rational choice Agents “know” their own type of reasoning (e.g., I-mode / we-mode ) Agents can be uncertain about others’ types of reasoning 9/ 35

Games with group payoffs Definition (Game with Group Payoffs) G = � Agt , { S i | i ∈ Agt } , { U J | J ∈ 2 Agt ∗ }� where: Agt = { 1 , . . . , n } is the set of agents; S i defines the set of strategies for agent i ; U J : � i ∈ Agt S i → R is a total payoff function mapping every strategy profile to some real number for some team J . For all i ∈ Agt , Group ( i ) = { J ⊆ Agt | i ∈ J } ⇒ Group ( i ) = “the set of groups i may identify with” Groups = � i ∈ Agt Group ( i ) 10/ 35

Examples of group payoff functions U J Pure utilitarianism : U J ( s ) = � i ∈ J U i ( s ) Rawls’ maximin principle : U J ( s ) = min i ∈ J U i ( s ) 11/ 35

Bacharach’s theory of team reasoning Definition (Unreliable Team Interaction) An unreliable team interaction structure is a tuple UTI = � Agt , { S i | i ∈ Agt } , { U J | J ∈ 2 Agt ∗ } , { Ω i | i ∈ Agt }� where: � Agt , { S i | i ∈ Agt } , { U J | J ∈ 2 Agt ∗ }� is a strategic game with group payoffs; Ω i is a probability distribution on Group ( i ) . For � J 1 , . . . , J n � ∈ Groups , Ω( � J 1 , . . . , J n � ) = Ω 1 ( J 1 ) × . . . × Ω n ( J n ) is the probability that “agent 1 identifies with group J 1 and ... and agent n identifies with group J n ” 12/ 35

Bacharach’s theory of team reasoning Team protocol α : function mapping every agent i ∈ Agt and every team J ∈ Group ( i ) to a strategy s i ∈ S i ◮ α ( i , J ) is agent i ’s strategy when identifying with the group J ◮ the set of all protocols is denoted by ∆ 13/ 35

Bacharach’s theory of team reasoning Team protocol α : function mapping every agent i ∈ Agt and every team J ∈ Group ( i ) to a strategy s i ∈ S i ◮ α ( i , J ) is agent i ’s strategy when identifying with the group J ◮ the set of all protocols is denoted by ∆ α J = � i ∈ J α ( i , J ) denotes the strategy of group J specified by protocol α 13/ 35

Bacharach’s theory of team reasoning Team protocol α : function mapping every agent i ∈ Agt and every team J ∈ Group ( i ) to a strategy s i ∈ S i ◮ α ( i , J ) is agent i ’s strategy when identifying with the group J ◮ the set of all protocols is denoted by ∆ α J = � i ∈ J α ( i , J ) denotes the strategy of group J specified by protocol α α J ∙ β − J denotes the protocol such that: for all i ∈ J , α J ∙ β − J ( i , J ) = α ( i , J ) , and 1 for all H ∈ 2 Agt ∗ such that H � = J and for all i ∈ H , 2 α J ∙ β − J ( i , H ) = β ( i , H ) 13/ 35

Bacharach’s theory of team reasoning Expected value of protocol α ∈ ∆ for group J ∈ 2 Agt ∗ : � EV J ( α ) = Ω( � J 1 , . . . , J n � ) ∙ U J ( α ( 1 , J 1 ) , . . . , α ( n , J n )) � J 1 ,..., J n �∈ Groups 14/ 35

Bacharach’s theory of team reasoning Expected value of protocol α ∈ ∆ for group J ∈ 2 Agt ∗ : � EV J ( α ) = Ω( � J 1 , . . . , J n � ) ∙ U J ( α ( 1 , J 1 ) , . . . , α ( n , J n )) � J 1 ,..., J n �∈ Groups Definition (UTI Equilibrium) A protocol α is an UTI equilibrium if and only if: ∀ J ∈ 2 Agt ∗ , ∀ β ∈ ∆ , EV J ( β J ∙ α − J ) ≤ EV J ( α J ∙ α − J ) Equilibrium solution ⇔ no individual AND no team can increase expected value by unilaterally deviating ⇒ Equivalent to finding a Nash equilibrium in a transformed n -player game with n = | 2 Agt ∗ | 14/ 35

Prisoner’s dilemma and team reasoning Bob D C (2 , 2) (0 , 3) C Alice (3 , 0) (1 , 1) D Let α be the following protocol: α ( a , { a } ) = α ( b , { b } ) = D and α ( a , { a , b } ) = α ( b , { a , b } ) = C Then, if ω = Ω a ( { a , b } ) = Ω b ( { a , b } ) : Notion of group payoff based on Rawls’ maxmin : α is the unique UTI equilibrium when ω > 2 3 Utilitarian notion of group payoff : α is the unique UTI equilibrium for all values of ω 15/ 35

Hi-Lo and team reasoning Bob D C (2 , 2) (0 , 0) C Alice (0 , 0) (1 , 1) D Let β be the following protocol: β ( a , { a } ) = β ( b , { b } ) = β ( a , { a , b } ) = β ( b , { a , b } ) = C Then, if ω = Ω a ( { a , b } ) = Ω b ( { a , b } ) : β is the unique UTI equilibrium when ω > 2 3 with both utilitarian group payoff and group payoff based on Rawls’ maxmin 16/ 35

Limitations of Bacharach’s theory Complexity of computing an equilibrium solution Interpretation of exogenous probability distribution Ω i ◮ e.g., intrinsic to game structure? subjective probabilities? Only binary types of reasoning ( I-mode / we-mode ) ◮ ⇒ No gradual group identification (at best vacillations between modes)! 17/ 35

Limitation of all theories of team reasoning A counter-example to all theories of team reasoning: Alice (8 , 0) A (5 , 7) Bob B (7 , 4) C In I-mode ⇒ Bob will play A In we-mode ⇒ Bob will play B ◮ Using either the utilitarian notion of group payoff or the notion based on Rawls’ maximin principle ⇒ Bob will never play C (counter-intuitive!) 18/ 35

Limitation of all theories of team reasoning A counter-example to all theories of team reasoning: Alice (8 , 0) A Bob (5 , 7) B (7 , 4) C Fact For every probability distribution Ω Bob , if α is a UTI equilibrium then α ( Bob , { Bob } ) = A and α ( Bob , { Bob , Alice } ) = B using either the utilitarian notion of group payoff or the notion based on Rawls’ maximin principle. 19/ 35

Outline Theories of team reasoning 1 Theory of social ties 2 Conclusion 3 20/ 35

What are social ties? Common sense ◮ ⇒ friends, married couples, family relatives, colleagues, classmates, etc. . . 21/ 35

What are social ties? Common sense ◮ ⇒ friends, married couples, family relatives, colleagues, classmates, etc. . . Psychological foundation (Social Identity Theory, [Tajfel & Turner,1979]) ◮ Social features that define one’s social identity ◮ e.g., to identify as a student of Toulouse university, a supporter of Barcelona’s soccer team, a Democrat, . . . 21/ 35

What are social ties? Common sense ◮ ⇒ friends, married couples, family relatives, colleagues, classmates, etc. . . Psychological foundation (Social Identity Theory, [Tajfel & Turner,1979]) ◮ Social features that define one’s social identity ◮ e.g., to identify as a student of Toulouse university, a supporter of Barcelona’s soccer team, a Democrat, . . . Epistemic condition ◮ Minimal criterion : a social tie between i and j ⇔ i and j commonly believe that they share the same social features defining their social identities 21/ 35