1. Gilberto Câmara, Earth System Science Center, INPE
Game Theory
Gilberto Câmara, Earth System Science Center, INPE
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2. 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)

3. 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

4. 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?

5. 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

6. 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

7. 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

8. Modeling choice in non-cooperative games
Player 2
Cooperate
Defect
Both cooperate
Player 1 cooperates, Player 2 defects
Player 1 defects, Player 2 cooperates
Both defect
Player 1

9. 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.

10. Prisoners’ Dilemma
Two suspects are caught and put in different rooms (no communication). They are offered the following deal:
If both of you confess, you will both get 3 years in prison
If you confesses whereas the other does not, you will get 1 year and the other gets 5 years in prison .
If neither of you confess, you both will get 2 years in prison.

11. 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.

12. 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 )

13. The hawk-dove game (== chicken game)
Encyclopedia Britannica

14. 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.

15. The stag-hunt game: conflict between safety and social cooperation
10,10
0,6
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..."

16. Generalizing... D C b – c -c b Payoff matrix
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 D C
b – c -c b
To efficiently study several social tension we can start from the benefit cost dilemma presented in the beginning. If we consider that each player plays simulataneously as a donor and receptor we reach to the following payoff matrix. We have 4 parameters for each situation. The reward for mutual cooperation, the punishment for mutual defection, the sucker’s payoff (when an individual is cheated and finally, the payoff received by a defector interacting with a cooperator.
you
Payoff matrix
other

17. Terminology Player 2 T = Temptation to defect
R = Reward for mutual cooperation
P = Punishment for mutual defection
S = Sucker's payoff
Prisioner’s dillema: T > R > P > S

18. Generalizing... Payoff matrix D C R(1) S(-c) P(0) T(b)
R: mutual cooperation
P: mutual defection
S : sucker’s payoff
T : temptation to defect
other D C
R(1)
S(-c)
P(0)
T(b)
To efficiently study several social tension we can start from the benefit cost dilemma presented in the beginning. If we consider that each player plays simulataneously as a donor and receptor we reach to the following payoff matrix. We have 4 parameters for each situation. The reward for mutual cooperation, the punishment for mutual defection, the sucker’s payoff (when an individual is cheated and finally, the payoff received by a defector interacting with a cooperator.
you
Taking R = 1 and P = 0

19. Generalizing... Payoff matrix D C 1 S T R: mutual cooperation
P: mutual defection
S : sucker’s payoff
T : temptation to defect
opponent D C 1 S T
To efficiently study several social tension we can start from the benefit cost dilemma presented in the beginning. If we consider that each player plays simulataneously as a donor and receptor we reach to the following payoff matrix. We have 4 parameters for each situation. The reward for mutual cooperation, the punishment for mutual defection, the sucker’s payoff (when an individual is cheated and finally, the payoff received by a defector interacting with a cooperator.
you
Taking R = 1 and P = 0

20. Different ordering -> Different tensions
greed fear
Chicken game
Stag-hunt game
Prisoner’s dilemma
T > 1 > S > 0 S P D C R T
1 > T > 0 > S
T > 1 > 0 > S
So, we can obtain the chicken game or hawk-dove game proposed by Maynard-smith; the stag-hunt game proposed recently by brian skyrms and the famous prisoner’s dilemma. Note, the rational choices change dependenging on the game: Contrary to the PD, in the SG is always better to choose a strategy which is opposite from your opponente while in the SH you should mimic your opponent (that’s why is also called a coordination game). From this, different tension raises. in the SG, people defect because the gains received by deceiving someone are huge. Agents defect because they are greedy to receive the benefits. In the SH game agents defect because they are afraid of being cheated. Unilateral cooperation is the worst case possible. The PD mixes the 2 tensions.
ll this dilemmas represent a certain tension. When Temptation is big than individuals cooperate because they are greedy to receive high benefits without providing any help. In the stag-hunt game the opposite happens. People defect because they are afraid of being cheated. The sucker’s payoff is a quite bad outcome. The prisoner’s dilemma merges these 2 tensions being the worst regime for the emergence of cooperation.
(Macy&Flache, PNAS 2002)

21. 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.

22. 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.