Game Theory and Evolution of cooperation

Gilberto Câmara, Earth System Science Center, INPE

Licence: Creative Commons ���� By Attribution ���� Non Commercial ���� Share Alikehttp://creativecommons.org/licenses/by-nc-sa/2.5/

Acknowledgments for using previous material

Martin Nowak (Harvard University, USA) Francisco C. Santos (Université Libre de Bruxelles, Belgium) Craig Callender (Philosophy, Univ California San Diego, USA) Ana Aguiar (INPE, Brazil) Tiago Carneiro (Federal University of Ouro Preto, Brazil) Guy Brasseur (NCAR, USA)

Game TheoryGT is an analytical tool for social sciences that is used to model

strategic interactions or conflict situations.Strategic interaction: When actions of a player influence payoffs to

other players

Game TheoryExplanation: What is the game to be played?Prediction: What outcome will prevail?Advice or prescription: Which strategies are likely to yield good results

in which situations?

Where can we use Game Theory?

Any situation that requires us to anticipate our rival’s response to our action is a potential context for GT.

Economics, Political science, Biology

What is a Normal Form Game?Players: list of playersStrategies: all actions available to all playersPayoffs: a payoff assigned to every contingency (every possible

strategy profile as the outcome of the game)

John Kennedy and Nikita Khrushchev

Modeling two-party games

Payoffs for each player depend on actions of bothTwo possible strategies: A party cooperates when he performs

value-increasing promises, and defects when he breaches

Cooperate Defect

Cooperate Both cooperatePlayer 1 cooperates, Player 2 defects

DefectPlayer 1 defects, Player 2 cooperates

Both defect

Player 2

Player 1

Modeling choice in non-cooperative games

Silvio Santos e o jogo do “Sete e Meio”

Dois jogadores se enfrentam na TV.Se dois jogarem “meio”, cada um ganha R$ 14 mil.Se um jogar “sete” e o outro “meio”, o primeiro ganha R$ 112

mil e outro não ganha nadaSe os dois jogarem “sete”, não ganham nada.

Prisoners’ DilemmaTwo suspects are caught and put in different rooms (no

communication). They are offered the following deal:1. If both of you confess, you will both get 3 years in prison2. If you confesses whereas the other does not, you will get 1

year and the other gets 5 years in prison .3. If neither of you confess, you both will get 2 years in prison.

The “chicken game”

Two persons drive their cars towards a cliff. They must stop or both may die in the fall. The one that stops first will be called a "chicken," meaning a coward.

“Rebel without a cause”

The hawk-dove game (== chicken game)

Maynard Smith and Price, "The logic of animal conflict“ (Nature, 1973 )

Two individuals compete for a resource (In biological terms, its value increases in the Darwinian fitness of the individual who obtains the resource)

Hawk Initiate aggressive behaviour, not stopping until injured or until one's opponent backs down.

Dove Retreat immediately if one's opponent initiates aggressive behaviour.

The hawk-dove game (== chicken game)

Encyclopedia Britannica

The stag-hunt game: conflict between safety and social cooperation

Two hunters want to kill a stag. Success is uncertain and, if it comes, require the efforts of both. On the other hand, either hunter can forsake his partner and catch a hare with a good chance of success.

The stag-hunt game: conflict between safety and social cooperation

Rousseau, in A Discourse on Inequality:“If it was a matter of hunting a deer, everyone well realized that he must remain faithful to his post; but if a hare happened to pass within reach of one of them, we cannot doubt that he would have gone off in pursuit of it without scruple..."

C D

C 10,10 0,6

D 6,0 5,5

Generalizing...

Payoff matrix

DC

C

D

b – c -c

0 b

other

you

Cooperation requires at least two individuals:A: the one providing cooperation (DONOR)B: the one benefiting from cooperation (RECEIVER)

Donor has a cost c to cooperate and confers a benefit b to other player

Player 2

Terminology

T = Temptation to defect

R = Reward for mutual cooperation

P = Punishment for mutual defection

S = Sucker's payoff

T = Temptation to defect

R = Reward for mutual cooperation

P = Punishment for mutual defection

S = Sucker's payoff

Generalizing...

Payoff matrix R: mutual cooperation

P: mutual defection

S : sucker’s payoff

T : temptation to defect

DC

C

D

R(1) S(-c)

P(0) T(b)

other

you

Taking R = 1 and P = 0

Generalizing...

Payoff matrix R: mutual cooperation

P: mutual defection

S : sucker’s payoff

T : temptation to defect

DC

C

D

1 S

0 T

opponent

you

Taking R = 1 and P = 0

Different ordering -> Different tensions

Chicken game

Stag-hunt game

Prisoner’s dilemma

SP

DCCD R

TT > R > P > S

R > T > P > S

T > R > S > P greed fear

(Macy&Flache, PNAS 2002)

Different ordering -> Different tensions

Chicken game

Stag-hunt game

Prisoner’s dilemma

SP

DCCD R

TT > 1 > 0 > S

1 > T > 0 > S

T > 1 > S > 0 greed fear

(Macy&Flache, PNAS 2002)

Spatial Prisioner´s Dillema

Nowak and May considered a large lattice with each cell occupied by one player. The players engage in one round of the Prisoner’s Dilemma game against each of their neighbors.

Afterward, the next generation is formed: each cell is taken over by a copy of the highest-scoring strategy within the neighborhood.

Tragedy of the Commons (Hardin, 1968)

Assume a common-property resource (exclusion is difficult and joint use involves subtractability) with no property rights. (Pasture open to all)

Each herdsman tries to keep as many sheep as possible on the commons. Each tries to maximize gain.

Add those sheep!

The rational herdsman concludes that he should add another sheep. And another…And another…And so does each herdsman

“Ruin is the destination toward which all men rush, each pursuing his own best interest…”

Prisoners’ DilemmaTwo suspects are caught and put in different rooms (no

communication). They are offered the following deal:1. If both of you confess, you will both get 3 years in prison2. If you confesses whereas the other does not, you will get 1

year and the other gets 5 years in prison .3. If neither of you confess, you both will get 2 years in prison.

Prisioner´s Dillema as a Model for the Tragedy of the Commons

1. Suppose the commons can support 2 sheep at no cost and that each additional sheep put in the commons has a cost of 1/3 of its price due to overgrazing.

2. Assume two herdsman with one sheep on the commons each.

3. If a herdsman puts another sheep in the commons, he receives all the proceeds from the sale of each additional animal. His temptation is 4/3 and the sucker´s payoff for the other herdsman is -1/3.

Prisioner´s Dillema as a Model for the Tragedy of the Commons

Payoff matrix

DC

C

D

1 -1/3

1/3 4/3

other

you

You are the herdsman. What are your options? Do you cooperate or defect?

Tragedy of the Commons?

Everybody’s property is nobody’s property (Hardin)

Preconditions for the tragedy of the commons

Lack of restraint on pursuits of self-interest

Consequences are externalities (I don’t have to pay)

Externalities in the global commons

Activity of one person has an impact on the well-being of another.

Positive externalities (or external benefits): Benefits realized by those who didn’t pay for them.

Negative externalities (or external costs): Costs borne by those who didn’t generate them. Byproducts that harm others.

SUVs in USA Climate Change in Africa

Is the tragedy of the commons inevitable?

Experiments show that cooperation emerges if virtuous interactions exist

source: Novak, May and Sigmund (Scientific American, 1995)

Repeated prisioner´s dillema

source: Novak, May and Sigmund (Scientific American, 1995)

Four different strategies for repeated prisioner´s dillema

Repeated prisioner´s dillema

source: Novak, May and Sigmund (Scientific American, 1995)

Evolution of prisioner´s dillema comparing different strategies

How can cooperation happen?Nowak MA (2006). “Five rules for the evolution of cooperation” Science 314:1560-1563(most highly cited multidisciplinary paper – ISI, 1st quarter 2010)

"I would lay down my life for two brothers or eight cousins“ (J.B.S. Haldane)

Five rules for evolution of cooperation

b = benefit for the recepient c= cost for the donor