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 Emiliano Lorini Fr´ ed´ eric Moisan

IRIT, CNRS, Toulouse, France

IRIT, July 2013

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Problems of Cooperation: Prisoner’s Dilemma

(2, 2) (1, 1) (3, 0) (0, 3) C C D D

Alice Bob

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

(2, 2) (1, 1) (0, 0) (0, 0) C C D D

Alice Bob

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, {Si|i ∈ Agt}, {Ui|i ∈ Agt} where: Agt = {1, . . . , n} is the set of agents; Si defines the set of strategies for agent i; Ui :

i∈Agt Si → R is a total payoff function mapping every

strategy profile to some real number for some agent i.

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Outline

1

Theories of team reasoning

2

Theory of social ties

3

Conclusion

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Outline

1

Theories of team reasoning

2

Theory of social ties

3

Conclusion

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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)

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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 UJ to be maximized. It is common knowledge among all members of J that solution s uniquely maximizes UJ. Then I should choose my strategy in s.

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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

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Games with group payoffs

Definition (Game with Group Payoffs)

G = Agt, {Si|i ∈ Agt}, {UJ|J ∈ 2Agt∗} where: Agt = {1, . . . , n} is the set of agents; Si defines the set of strategies for agent i; UJ :

i∈Agt Si → 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)

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Examples of group payoff functions UJ

Pure utilitarianism: UJ(s) =

i∈J Ui(s)

Rawls’ maximin principle: UJ(s) = mini∈J Ui(s)

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Bacharach’s theory of team reasoning

Definition (Unreliable Team Interaction)

An unreliable team interaction structure is a tuple UTI = Agt, {Si|i ∈ Agt}, {UJ|J ∈ 2Agt∗}, {Ωi|i ∈ Agt} where: Agt, {Si|i ∈ Agt}, {UJ|J ∈ 2Agt∗} is a strategic game with group payoffs; Ωi is a probability distribution on Group(i). For J1, . . . , Jn ∈ Groups, Ω(J1, . . . , Jn) = Ω1(J1) × . . . × Ωn(Jn) is the probability that “agent 1 identifies with group J1 and... and agent n identifies with group Jn”

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Bacharach’s theory of team reasoning

Team protocol α: function mapping every agent i ∈ Agt and every team J ∈ Group(i) to a strategy si ∈ Si

◮ 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 si ∈ Si

◮ α(i, J) is agent i’s strategy when identifying with the group J ◮ the set of all protocols is denoted by ∆

αJ =

i∈J α(i, J) denotes the strategy of group J specified by

protocol α

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Bacharach’s theory of team reasoning

Team protocol α: function mapping every agent i ∈ Agt and every team J ∈ Group(i) to a strategy si ∈ Si

◮ α(i, J) is agent i’s strategy when identifying with the group J ◮ the set of all protocols is denoted by ∆

αJ =

i∈J α(i, J) denotes the strategy of group J specified by

protocol α αJ ∙ β−J denotes the protocol such that:

1

for all i ∈ J, αJ ∙ β−J(i, J) = α(i, J), and

2

for all H ∈ 2Agt∗ such that H = J and for all i ∈ H, αJ ∙ β−J(i, H) = β(i, H)

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Bacharach’s theory of team reasoning

Expected value of protocol α ∈ ∆ for group J ∈ 2Agt∗: EVJ(α) =

• J1,...,Jn∈Groups

Ω(J1, . . . , Jn) ∙ UJ(α(1, J1), . . . , α(n, Jn))

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Bacharach’s theory of team reasoning

Expected value of protocol α ∈ ∆ for group J ∈ 2Agt∗: EVJ(α) =

• J1,...,Jn∈Groups

Ω(J1, . . . , Jn) ∙ UJ(α(1, J1), . . . , α(n, Jn))

Definition (UTI Equilibrium)

A protocol α is an UTI equilibrium if and only if: ∀J ∈ 2Agt∗, ∀β ∈ ∆, EVJ(βJ ∙ α−J) ≤ EVJ(αJ ∙ 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 = |2Agt∗|

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Prisoner’s dilemma and team reasoning

(2, 2) (1, 1) (3, 0) (0, 3) C C D D

Alice Bob

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 = Ωa({a, b}) = Ωb({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

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Hi-Lo and team reasoning

(2, 2) (1, 1) (0, 0) (0, 0) C C D D

Alice Bob

Let β be the following protocol: β(a, {a}) = β(b, {b}) = β(a, {a, b}) = β(b, {a, b}) = C Then, if ω = Ωa({a, b}) = Ωb({a, b}): β is the unique UTI equilibrium when ω > 2

3 with both utilitarian

group payoff and group payoff based on Rawls’ maxmin

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Limitations of Bacharach’s theory

Complexity of computing an equilibrium solution Interpretation of exogenous probability distribution Ωi

◮ 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)!

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Limitation of all theories of team reasoning

A counter-example to all theories of team reasoning:

(8, 0) (5, 7)

A B C

(7, 4) Bob Alice

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!)

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Limitation of all theories of team reasoning

A counter-example to all theories of team reasoning:

(8, 0) (5, 7)

A B C

(7, 4) Bob Alice

Fact

For every probability distribution ΩBob, 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.

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Outline

1

Theories of team reasoning

2

Theory of social ties

3

Conclusion

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What are social ties?

Common sense

◮ ⇒ friends, married couples, family relatives, colleagues,

classmates, etc. . .

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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, . . .

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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

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What are social ties?

How to quantify the social tie between i and j? Quantity and importance of shared social features that define both i and j’s social identities

◮ How many social features i and j share? ◮ Are shared social features as important for i as for j?

Quantity and quality of past interactions between i and j

◮ How often i and j had meaningful interactions with each other? ◮ e.g., exchanging ideas, opinions, sharing positive emotions, . . . 22/ 35

Modeling social ties

Definition (Social Ties Game)

A social ties game is a tuple ST = Agt, {Si|i ∈ Agt}, {UJ|J ∈ 2Agt∗}, {ki|i ∈ Agt} where: Agt, {Si|i ∈ Agt}, {UJ|J ∈ 2Agt∗} is a strategic game with group payoffs; every ki is a total function ki : Group(i) → [0, 1], such that:

C1 for every i ∈ Agt,

J∈Group(i) ki(J) = 1

C2 for all i, j ∈ J, ki(J) = kj(J)

Intuitions:

◮ ki(J) = agent i’s social tie with group J ◮ C1 ⇒ a distribution of social ties ◮ C2 ⇒ social ties are bilateral 23/ 35

How utility is affected by social ties

Definition (Social Ties Utility)

The social ties utility function of strategy profile s for agent i in the social ties game ST is defined by: UST

i

(s) =

• J⊆Agt\{i}

ki(J ∪ {i}) ∙ max

s′

J∈SJ

UJ∪{i}(s−J, s′

J)

General idea: ⇒ In the presence of a strong social tie between agent i and group J ∈ Group(i), i is strongly motivated to maximize the collective payoff of group J, assuming that the other agents in J are also motivated to maximize the collective payoff of group J

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Illustration: Social Ties in 2-player Games

Let Agt = {i, j} then for every s ∈ S: UST

i

(s) = (1 − kij) ∙ Ui(s) + kij ∙ max

s′

j ∈Sj

U{i,j}(si, s′

j)

(where kij = ki({i, j})) If kij = 1: i maximizes collective payoff assuming that j does the same i does not face a strategic problem anymore: the utility of strategy profile s for i becomes independent from j’s part in s If kij = 0: i maximizes individual payoff

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Strategic game induced by social tie game

Definition (Induced Strategic Game)

Let ST = Agt, {Si|i ∈ Agt}, {UJ|J ∈ 2Agt∗}, {ki|i ∈ Agt} be a social ties game. The strategic game induced by ST is the tuple GST = Agt, {Si|i ∈ Agt}, {U′

i |i ∈ Agt} where, for all i ∈ Agt and for all

s ∈ S: U′

i (s) = UST i

(s)

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Illustration: Social Ties in 2-player Games

⇒ Transformation of utilities with group payoff based on Rawls’ maxmin and kab = kba = 1

(2, 2) (1, 1) (3, 0) (0, 3) C C D D

Alice Bob

(2, 2) (1, 1) (1, 2) (2, 1) C C D D

Alice Bob

⇒

(2, 2) (1, 1) (0, 0) (0, 0) C C D D

Alice Bob

(2, 2) (1, 1) (1, 2) (2, 1) C C D D

Alice Bob

⇒

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Prisoner’s dilemma and social ties

(2, 2) (1, 1) (3, 0) (0, 3) C C D D

Alice Bob

If kab = kba > 1

2 then (C, C) is the unique Nash equilibrium in the

induced strategic game using either the utilitarian notion of group payoff or the notion of group payoff based on Rawls’ maxmin

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Hi-Lo and social ties

(2, 2) (1, 1) (0, 0) (0, 0) C C D D

Alice Bob

Utilitarian notion of group payoff: (C, C) is the unique Nash equilibrium in the induced strategic game if kab = kba > 1

3

Notion of group payoff based on Rawls’ maxmin: (C, C) is the unique Nash equilibrium if kab = kba > 1

2

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Comparison with Bacharach’s team reasoning

⇒ Main difference: Bacharach’s theory assumes that an agent in a given strategic setting reasons either in the I-mode or in the We-mode (no possibility of being at the same time in I-mode and in We-mode) Our theory assumes that an agent can be partially tied with a given group (notion of partial identification with a group)

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Comparison with Bacharach’s team reasoning (8, 0) (5, 7)

A B C

(7, 4) Bob Alice

⇒ Differently from Bacharach’s theory, we can predict the outcome C

Fact

If kab = kba = 0.5 then the unique Nash equilibrium in the induced strategic game is C, using either the utilitarian notion of group payoff or the notion based on Rawls’ maximin principle.

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Comparison with Bacharach’s team reasoning

Consider binary versions of strategic games:

◮ Binary UTI structure: no uncertainty about which group each agent

identifies with

◮ Binary social ties game: every agent can only identify with a unique

group

In binary version of 2-player games:

◮ the two theories make the same predictions!

In binary version of n-player games:

◮ the two theories make the same predictions if the game has no

conflicting collective goals!

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Outline

1

Theories of team reasoning

2

Theory of social ties

3

Conclusion

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Conclusion

Bacharach’s theory of team reasoning:

◮ explains cooperation in PD and coordination in Hi-Lo ◮ beyond classical decision theory ◮ no notion of partial group identification

Our theory of social ties:

◮ explains cooperation in PD and coordination in Hi-Lo ◮ close in spirit to theories of social preferences (utility

transformation)

◮ a type of Tuomela’s pro-group I-mode ◮ can model partial group identification and graded social ties 34/ 35

Ongoing and future work

Interpreting social ties towards groups in terms of social ties between individuals Experimental validation of our theory of social ties Social ties in extensive games Epistemic foundation for social ties Comparison between our theory of social ties and Binmore’s theory of empathetic preferences [Binmore, 1994,2005] Comparison between our theory of social ties and Alger & Weibull’s theory of homo moralis [Alger and Weibull, 2012]

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