Game Theory and Evolution of
cooperation
Gilberto Câmara, Earth System Science Center, INPE
Licence: Creative Commons By Attribution Non Commercial Share Alike
http://creativecommons.org/licenses/by-nc-sa/2.5/
Acknowledgments for using previous material
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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 Theory
GT 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 Theory
Explanation: 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 players
Strategies: all actions available to all players
Payoffs: 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 both
Two possible strategies: A party cooperates when he performs
value-increasing promises, and defects when he breaches
Modeling choice in non-cooperative games
Player 2
Cooperate
Cooperate
Player 1
Defect
Defect
Player 1
Both cooperate cooperates,
Player 2 defects
Player 1
defects, Player
2 cooperates
Both defect
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 nada
Se os dois jogarem “sete”, não ganham nada.
Prisoners’ Dilemma
Two 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 prison
2. 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”
“Rebel without a cause”
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.
The hawk-dove game (== chicken game)
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.
Maynard Smith and Price, "The logic of animal conflict“ (Nature, 1973 )
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
C
D
C
10,10
0,6
D
6,0
5,5
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..."
Generalizing...
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
C
D
C
b–c
-c
D
b
0
you
Payoff matrix
other
Terminology
Player 2
T = Temptation to defect
R = Reward for mutual cooperation
P = Punishment for mutual defection
S = Sucker's payoff
Generalizing...
Payoff matrix
R: mutual cooperation
other
P: mutual defection
C
D
C
R(1)
S(-c)
D
T(b)
P(0)
S : sucker’s payoff
T : temptation to defect
you
Taking R = 1 and P = 0
Generalizing...
Payoff matrix
R: mutual cooperation
opponent
P: mutual defection
C
D
C
1
S
D
T
0
S : sucker’s payoff
T : temptation to defect
you
Taking R = 1 and P = 0
Different ordering -> Different tensions
greed
C D
C R S
D T P
fear
Chicken game
T >R > S > P
Stag-hunt game
R>T > P > S
Prisoner’s dilemma
T >R > P > S
(Macy&Flache, PNAS 2002)
Different ordering -> Different tensions
greed
C D
C R S
D T P
fear
Chicken game
T >1 > S > 0
Stag-hunt game
1>T > 0 > S
Prisoner’s dilemma
T >1 > 0 > S
(Macy&Flache, PNAS 2002)
Spatial Prisioner´s Dillema
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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’ Dilemma
Two 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 prison
2. 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
You are the herdsman. What are your options? Do you
cooperate or defect?
C
D
C
1
-1/3
D
4/3
1/3
you
Payoff matrix
other
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
Four different strategies for repeated prisioner´s dillema
source: Novak, May and Sigmund (Scientific American, 1995)
Repeated prisioner´s dillema
Evolution of prisioner´s dillema comparing different strategies
source: Novak, May and Sigmund (Scientific American,
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