Game Theory

1. Game Theory

2. Contents Introduction Normal and extensive forms Dominated strategies and Nash equilibria Mixed strategies and refinements Some important games and game iteration Evolutionary game theory Dynamics and basic results Spatial effects Discussion and conclusions

3. Introduction A universal language to treat with behavioral sciences in a unified manner A toolbox to solve complicated problems� � without a lot of math A way to research the world An study of emergency, transformation, stabilization and diffusion of �strategies� Adventure and fantasy!

4. Introduction Ernst Zermelo (1913): Chess John von Neumann y Oskar Morgenstern (1944): The theory of games and economic behavior

5. Introduction Ernst Zermelo (1913): Chess John von Neumann y Oskar Morgenstern (1944): The theory of games and economic behavior John Nash (1950)�: �Solution�

6. Introduction Ernst Zermelo (1913): Chess John von Neumann y Oskar Morgenstern (1944): The theory of games and economic behavior John Nash (1950)�: �Solution� John Maynard Smith y George Price (1973): Evolution (biological)

7. Introduction Ernst Zermelo (1913): Chess John von Neumann y Oskar Morgenstern (1944): The theory of games and economic behavior John Nash (1950)�: �Solution� John Maynard Smith y George Price (1973): Evolution (biological) William Hamilton y Robert Axelrod (1981): Human cooperation John Harsanyi� y Reinhard Selten� (1988): Equilibrium problem

8. Extensive and normal forms

9. Big Monkey Wait Climb Little Monkey CC (Climb no matter what Big Monkey does) WW (Wait no matter what Big Monkey does) WC (Do exactly as Big Monkey) CW (Do the opposite to Big Monkey) Extensive and normal forms

10. Extensive and normal forms

11. Extensive and normal forms

12. Extensive and normal forms

13. Extensive and normal forms

14. Extensive and normal forms

15. Concepts Perfect information Perfect rationality N players, symmetry/asymmetry Dominated strategies For player i, si dominates s�i if, for any choice of the rest of players, the payoff obtained with si is larger than that gained from s�i Weakly dominated strategies If elimination of dominated strategies leaves a unique one for each player, the result is a Nash equilibrium

16. Concepts Nash equilibrium A set of strategies (one per player) from which no player benefits by changing unilaterally A set of strategies such that each one of them is a best response to the joint strategies of the rest Some times weakly dominated strategies Pure strategy equilibria Mixed strategy equilibria Randomization (Populations)

17. s:=(p1,�,pn) are the probabilities of the strategies of one player (s1,�, sN) is a set of strategies If pi y pj are nonzero in s, then the payoffs for si y sj played against the rest is the same Way to find equilibria in mixed strategies Nash Theorem

20. Refinements A unique Nash equilibrium is only guaranteed in zero-sum games In general we expect more than one Nash equilbrium. �How should we decide which one is the solution? Refinements of Nash equlibria: criteria to choose among the possible ones Incredible threats: Subgame perfection Pareto-dominance vs risk-dominance �Trembling hands� Motivation for evolutionary game theory

22. Unique Nash equilibrium Dilemma: the best thing is not to cooperate Communication among players Paradigm in the study of human cooperation (mainly in iterated form) Symmetrical

23. Two Nash equlibria (C,C) is Pareto dominant (D,D) is risk dominant Experiments: different behaviors

24. Two undecidable Nash equilibria Choice of payoffs to represent real situations is arbitrary A mixed equilibria Coordination game

25. No Nash equilibria in pure strategies Mixed strategy equilibria are difficult to justify in applications

27. W. Brian Arthur, 1992

28. The tragedy of the commons, G. Hardin (1965)

30. (G�th, Schmittberger & Schwarze, 1982)

32. Two prisoners play the prisoner�s dilemma an unknown number of times R. Axelrod and W. D. Hamilton, Science 211, 1390 (1981) Basic strategies: All C All D Tit-for-Tat (TFT)

33. Strategy tournament: TFT winner In general, the best strategies are �nice�, �punishing� and �forgiving� Other important strategies: TF2T Pavlov Applications: trench war during WWI

34. If the game is played a predetermined number of times, the predicted equilibria are problematic Usually, discount factors are introduced With an infinite number of iterations, the �Folk theorem� guarantees a continuum of Nash equlibria

35. Evolutionary game theory John Maynard Smith (1982): �Evolution and the theory of games� Biology meets economics Three main concepts shift as compared to classical game theory: Strategy Equilibrium Interaction among players

36. Evolutionary game theory Classical theory: players have strategy sets from where to choose their actions Biology: species have strategy sets from which every individual inherits one Society: the set of alternative cultures can be identified with the strategy set, which individuals inherit or choose

37. Evolutionary game theory Classical theory: Nash equilibrium Biology: evolutively stable strategy (ESS) Society: similar concept We move from trying to explain the actions of individuals to model the changes and diffusion of behaviors in biology or in the society

38. Evolutionary game theory Classical theory: one-shot games and iterated games Biology: random and repeated pairing of individuals, with strategies based on their genome and not on the past Society: applied better as the world becomes more and more interconnected

39. Evolutionary game theory Strategies {s1,�,sn} Payoff for the player using si vs another one using sj: pij (and pji for its opponent) Game does not depend on being player 1 o 2: symmetrical Game matrix A=(pij) At every time t=1,2,�, agents in a large population are paired and play the game. There are as many types of agents as strategies

40. Evolutionary game theory

41. Evolutionary game theory

42. Evolutionary game theory

43. Evolutionary game theory s is an ESS if it cannot be invaded by any mutant (introduced in small quantities) In terms of Nash equilibria: we can see it as a population of equal agents all playing the mixed strategy s In biological terms, we have a population of agents, each one with a pure strategy, in the proportion given by s

44. Evolutionary game theory

45. Dynamics and basic results We have discussed �invasion� or �displacement� of some strategies by others We have not specified the dynamics of such process In game theory there are no Newton�s laws or Hamilton�s equation: we have to pose a dynamics depending on the process we want to model There are many possible dynamics

46. Dynamics and basic results

47. Dynamics and basic results

48. Dynamics and basic results

50. Everybody begins with a randomly chosen strategy Everybody plays against everybody else Infinite population Payoffs add up Total payoff determines the number of copies: Selection Copies inherit approximately their parent�s strategy: Mutation (Notice the relationship with genetic algorithms) Talk about darwin�s theory of evolution- the frequencies of genes and increase over time if they are associated with features which lead to the production of more offspring. So the proportion of the given feature within the population will increase over time, eg. finches with a gene for sharp beaks in an environment where such a peak is very important for accessing food. If the feature is behavioural, such as for instance the propensity to back down in a conflict (cf. hawk-dove game) then whether the gene is selected for depends on the make up of the population. Evolutionary game theory models this kind off evolutionary process. It can be used to show that the proportion of hawks in a population of hawks and doves will tend to fitness gain for winning territory/ fitness loss for getting injured (in our example 1/5). Here it implies that heavily in armed species, such as stags, which can potentially inflict mortal wounds on one another, very few individuals will escalate a conflict. Paradoxically in species of doves who under normal circumstances can�t do each other much damage, escalation is much more likely. Indeed when confined to small cages doves will often peck each other to death.Talk about darwin�s theory of evolution- the frequencies of genes and increase over time if they are associated with features which lead to the production of more offspring. So the proportion of the given feature within the population will increase over time, eg. finches with a gene for sharp beaks in an environment where such a peak is very important for accessing food. If the feature is behavioural, such as for instance the propensity to back down in a conflict (cf. hawk-dove game) then whether the gene is selected for depends on the make up of the population. Evolutionary game theory models this kind off evolutionary process. It can be used to show that the proportion of hawks in a population of hawks and doves will tend to fitness gain for winning territory/ fitness loss for getting injured (in our example 1/5). Here it implies that heavily in armed species, such as stags, which can potentially inflict mortal wounds on one another, very few individuals will escalate a conflict. Paradoxically in species of doves who under normal circumstances can�t do each other much damage, escalation is much more likely. Indeed when confined to small cages doves will often peck each other to death.

51. Dynamics and basic results Strategies {s1,�,sn} pi(t): proportion of players with strategy i at time t Payoff for si: pi[P(t)]:= pit, P(t)=(p1,�,pn) At every t, we order p1t= p2t =�= pnt In dt, an agent using si changes to sj with probability (learning)

52. Dynamics and basic results

53. Dynamics and basic results Replicator dynamics is not a best response dynamics The sum of pit is always 1 Equation can be derived in other contexts Fundamental theorem of natural selection (Fisher, 1930):

54. Dynamics and basic results Theorems: In general, dominated strategies do not survive Every Nash equilibrium is a fixed point Every stable fixed point is a Nash equilibrium Every ESS is an asymptotically stable fixed point. If it uses all the strategies, stability becomes global

55. Spatial effects So far, space has not played any role, and every player interacts with every other one If agents are spatially distributed, interactions go local In spatial models, there is nothing similar to the replicator equation Numerical simulations (agent based modelling) Equilibria change drastically

56. Spatial effects Example: M. Nowak and R. May, Nature 359, 826 (1992)

57. Spatial effects Example: M. Nowak and R. May, Nature 359, 826 (1992)

58. Spatial effects Example: M. Nowak y R. May, Nature 359, 826 (1992)

59. Spatial effects Example: M. Nowak y R. May, Nature 359, 826 (1992)

60. Spatial effects

61. Spatial effects

62. Discussion and conclusions We have done a quick tour over the very basic concepts of Game Theory Equilibrium, in its different flavors, is the fundamental concept: Nash equilibrium Evolutionary stable strategy (ESS) Stable fixed point of the dynamics The equilibrium selection problem remains open (and possibly it has no solution)

63. Discussion and conclusions We have not analyzed in depth very many concepts: Equilibrium refinements Asymmetric games n player games Continuous strategies Different dynamics Discrete dynamics Finite size/population effects Games and networks

64. Discussion and conclusions Game Theory as a tool to model any kind of systems where we have interacting agents: Biological Economical Social Modelling can be done at different levels Key: a clear identification/specification of the game and its dynamics