Evolutionary game theory and cognitionakazna/EGTCog20101109.pdf · 2010. 11. 21. · Two player...

Evolutionary game theory and cognition

Artem Kaznatcheev

University of Waterloo

November 9, 2010

Artem Kaznatcheev (University of Waterloo) Evolutionary game theory and cognition November 9, 2010 0 / 17

Two player games

I A game between two players (Alice and Bob) is represented by amatrix G of pairs.

Example ((3, 1) (2, 3)

(−1, 2) (3,−1)

I If Alice plays strategy i and Bob plays strategy j then (a, b) := Gij isthe outcome, where a corresponds to the change in Alice’s utility andb to Bob’s.

Two player games

I A game between two players (Alice and Bob) is represented by amatrix G of pairs.

Example ((3, 1) (2, 3)

(−1, 2) (3,−1)

)I If Alice plays strategy i and Bob plays strategy j then (a, b) := Gij is

the outcome, where a corresponds to the change in Alice’s utility andb to Bob’s.

Zero-sum games

Definition

A game G is a zero-sum game if for each (a, b) := Gij we have a + b = 0.

Example ((1,−1) (−1, 1)(−1, 1) (1,−1)

I Zero-sum games are the epitome of competition. Any gain for Alice isa loss for Bob, and vice-versa.

Zero-sum games

Definition

Example ((1,−1) (−1, 1)(−1, 1) (1,−1)

Zero-sum games

Definition

Example ((1,−1) (−1, 1)(−1, 1) (1,−1)

Coordination games

Definition

A two-strategy game G is a coordination game if we have

((a1, b1) (c2, d1)(c1, d2) (a2, b2)

)And a1 > c1, a2 > c2, b1 > d1, b2 > d2.

Examples((1, 1) (−1,−1)

(−1,−1) (1, 1)

((2, 1) (0, 0)(0, 0) (1, 2)

((4, 4) (0, 2)(2, 0) (3, 3)

I The diagonals are always better for both players, they just have tofigure out how to pick the same strategy.

I Captures the idea of win-win, lose-lose situations.

Coordination games

Definition

((a1, b1) (c2, d1)(c1, d2) (a2, b2)

)And a1 > c1, a2 > c2, b1 > d1, b2 > d2.

Examples((1, 1) (−1,−1)

(−1,−1) (1, 1)

((2, 1) (0, 0)(0, 0) (1, 2)

((4, 4) (0, 2)(2, 0) (3, 3)

Coordination games

Definition

((a1, b1) (c2, d1)(c1, d2) (a2, b2)

)And a1 > c1, a2 > c2, b1 > d1, b2 > d2.

Examples((1, 1) (−1,−1)

(−1,−1) (1, 1)

((2, 1) (0, 0)(0, 0) (1, 2)

((4, 4) (0, 2)(2, 0) (3, 3)

What do these two types of games tell us?

I Zero-sum and coordination games are mutually exclusive: there is nogame that is both zero-sum and a coordination game.

I Upside: zero-sum and coordination provide a good duality betweenimpossibility of cooperation and obvious cooperation.

I Downside: both types of games are really boring. The mostinteresting games (from a mathematical and modeling point of view)are neither zero-sum nor coordination.

I Being non-zero-sum does not ensure cooperation.

Prisoner’s dilemma

((b − c , b − c) (−c, b)

(b,−c) (0, 0)

)I b is the benefit of receiving and c is the cost of giving.

I Strategy 1 is called cooperate or C and strategy 2 is called defect orD.

I The rational strategy (or Nash equilibrium) is mutual defection.

I The best for the players taken together (or Pareto optimum) ismutual cooperation.

((b − c , b − c) (−c, b)

(b,−c) (0, 0)

((b − c , b − c) (−c, b)

(b,−c) (0, 0)

Symmetric games

Definition

G is a symmetric game if for all strategies p and q we have:

fst(G (p, q)) = snd(G (q, p))

I Neither player is preferred or treated differently: every player isidentical with respect to the game rules.

I The representation of a game does not require writing those confusingpairs

I For example, all 2 player 2 strategy symmetric games can be writtenin the form: (

(R,R) (S ,T )(T ,S) (P,P)

(R ST P

Symmetric games

Definition

(R,R) (S ,T )(T ,S) (P,P)

(R ST P

Symmetric games

Definition

(R,R) (S ,T )(T , S) (P,P)

(R ST P

)Artem Kaznatcheev (University of Waterloo) Evolutionary game theory and cognition November 9, 2010 6 / 17

Cooperate-Defect games

Definition

A symmetric two-strategy game is a cooperate-defect game if the two purestrategies p and q have G (p, p) > G (q, q). In this case, we call pcooperation and q defection.

I This captures the interesting two player games.

I Allows us to reduce the general game to two parameters by removingconstant offset and picking our units:(

R ST P

(1 UV 0

Definition

R ST P

(1 UV 0

Definition

R ST P

(1 UV 0

U-V plane

(1 UV 0

Nash equilibrium

Definition

A strategy pair (p, q) is a Nash equilibrium of a game G if for all otherstrategies r we have:

fst(G (p, q)) ≥ fst(G (r , q))

andsnd(G (p, q)) ≥ snd(G (p, r))

I Informally: neither Alice nor Bob can improve their payoff byunilateral change of strategy.

I If we only allow pure strategies then replace G (i , j) by Gij

I If we allow mixed strategies, then every game has at least one Nashequilibrium

Nash equilibrium

Definition

Nash equilibrium

Definition

Pareto optimum

Definition

A strategy pair (p, q) is a Pareto optimum of a game G is there is noother strategy pair (p′, q′) such that G (p′, q′) > G (p, q)

I Informally: there is no other strategy such that both Alice and Bobget a better payoff.

I Every game has at least one Pareto optimum

Example ((2, 1) (0, 0)(0, 0) (1, 2)

((2,−3) (−1, 1)(0, 0) (−2, 2)

Pareto optimum

Definition

Example ((2, 1) (0, 0)(0, 0) (1, 2)

((2,−3) (−1, 1)(0, 0) (−2, 2)

Pareto optimum

Definition

Example ((2, 1) (0, 0)(0, 0) (1, 2)

((2,−3) (−1, 1)(0, 0) (−2, 2)

Cognitive demands of rationality

I Alice needs to be aware of her own utility function

I To check if she is currently in Nash equilibrium (at least from herperspective) Alice needs to be able to simulate the game in her mind(thus she must understand the interaction)

I To find a Nash equilibrium Alice needs to be able to simulate thegame and she must be able to place herself in Bob’s shoes.

I Do we even expect humans to be able to do all of this?

I Let’s bound rationality and see what happens!

Evolutionary game theory

I Strategy is a genetic trait and immutable by the agent.

I All cognition is stripped away

I Game payoffs change the fitness of the agent.

I Agents reproductive rate increases with higher fitness.

I Simplest model of biological evolution.

I Also applicable outside of biology.

I What happens to rationality?

Evolutionary stable strategy

Definition

A strategy s is an evolutionary stable strategy for a game G if for all otherstrategies r we have (a) fst(G (s, s)) > fst(G (r , s)), or (b)fst(G (s, s)) = fst(G (r , s)) and fst(G (s, r)) > fst(G (r , r)).

I Consider a population all with strategy s, a mutant with strategy rcannot invade the population only if one of the following conditionsholds:

I r has a higher fitness than s in a population of all s.I r has the same fitness when interacting with s and the same or greater

fitness when interacting with other r .

I Compare this to the Nash equilibrium conditions.

I The conditions are almost identical: we can think of the evolutionaryprocess as a rational process (entity?)!.

Definition

I r has a higher fitness than s in a population of all s.

I r has the same fitness when interacting with s and the same or greaterfitness when interacting with other r .

Definition

Wait a second: what about cooperation?

I The ESS predicts mutual defection in the Prisoner’s dilemma, but weobserve cooperation through out nature.

I The assumptions of the ESS:I Random interactions (inviscid population)I No repeated interactionsI Zero cognition in individual agents

I Various augmentations of the model create fascinating results, amongthem: cooperation.

I The assumptions of the ESS:

I Random interactions (inviscid population)I No repeated interactionsI Zero cognition in individual agents

I The assumptions of the ESS:I Random interactions (inviscid population)

I No repeated interactionsI Zero cognition in individual agents

I The assumptions of the ESS:I Random interactions (inviscid population)I No repeated interactions

I Zero cognition in individual agents

Cognitively relevant augmentations

I Kin selection:

the ability to recognize your children, siblings andparents

I Direct reciprocity (reciprocal altruism): the ability to rememberprevious interactions

I Indirect reciprocity: the ability to track social constructs likereputation

I Tag-based conditional strategies

I Kin selection: the ability to recognize your children, siblings andparents

I Direct reciprocity (reciprocal altruism):

the ability to rememberprevious interactions

I Indirect reciprocity:

the ability to track social constructs likereputation

Ethnocentrism in spatial models

Thank you!For more info feel free to contact me at:[email protected]

Some fun resources:

1. Robert Wright: ”The evolution of compassion”http://www.ted.com/talks/lang/eng/robert_wright_the_

evolution_of_compassion.html

2. Howard Rheingold: ”On collaboration”http://www.ted.com/talks/lang/eng/howard_rheingold_on_

collaboration.html

3. Jonathan Haidt: ”On the moral roots of liberals and conservatives”http://www.ted.com/talks/jonathan_haidt_on_the_moral_

mind.html

4. Artem Kaznatcheev: ”Evolving Cooperation”http://www.youtube.com/watch?v=bRuE3oP-JT8

5. Stanford Encyclopedia of Philosophy: ”Evolutionary Game Theory”http://plato.stanford.edu/entries/game-evolutionary/

Thank you!For more info feel free to contact me at:[email protected]

Some fun resources:

1. Robert Wright: ”The evolution of compassion”http://www.ted.com/talks/lang/eng/robert_wright_the_

evolution_of_compassion.html

2. Howard Rheingold: ”On collaboration”http://www.ted.com/talks/lang/eng/howard_rheingold_on_

collaboration.html

3. Jonathan Haidt: ”On the moral roots of liberals and conservatives”http://www.ted.com/talks/jonathan_haidt_on_the_moral_

mind.html

4. Artem Kaznatcheev: ”Evolving Cooperation”http://www.youtube.com/watch?v=bRuE3oP-JT8

5. Stanford Encyclopedia of Philosophy: ”Evolutionary Game Theory”http://plato.stanford.edu/entries/game-evolutionary/

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