What is a game?

Game: a contest between players with rules to determine a winner. Strategy: a long term plan of action designed to achieve a particular goal, most often “to win”.

There can be many possible strategies: PLAYER 1 PLAYER 2 S2S1 S3S2 S3 S4

Strategies can change over time: PLAYER 1 PLAYER 2 S2S1 S3S2 S3 S4

To understand games, one must be able to anticipate game dynamics, on the assumption that all players act rationally and are trying to win. Question: Is the a strategy such that, once adopted, there will be no further incentive for any player to change to another strategy? Supposedly, if all players have found such a strategy, all players will continue to play their strategies, and the game will have predictable outcomes (either deterministic or stochastic). This is a kind of “equilibrium solution” to a game.

John Forbes Nash Born 1928 In West Virginia Boy genius Ph.D. Princeton: Nash Equilibrium Professor, Massachusetts Institute of Technology Diagnosed with paranoid schizophrenia in 1959 Overcame schizophrenia in the 1980s and resumed his work at Princeton Nobel Price in Economics 1994

NASH EQUILIBRIUM A set of strategies enacted by players, in which no player has anything to gain by changing only his or her own strategy unilaterally. For example, two players A and B are in Nash equilibrium if A is making the best decision A can, taking into account B's decision, and B is making the best decision B can, taking into account A's decision. It came as a shock to economic theory, that the rational choices made by all the players in a non-cooperative game (in the Nash equilibrium) do not necessarily give the best payoffs for all the players.

Rules of the game = Payoff Matrix: Prisoner B stays silentPrisoner B betrays Prisoner A stays silent Each serves 6 monthsPrisoner A serves ten years, Prisoner B goes free Prisoner A betrays Prisoner B serves ten years, Prisoner A goes free Each prisoner serves 5 years The prisoner’s dilemma:

The weak Nash equilibrium: This is a “weak” solution because there may be several strategy sets fulfilling the requirement that no better strategy can be found. p A,p B is a weak Nash Equilibrium if

The strict Nash equilibrium: This is a “strong” solution because it maximizes the payoff for all players and implies stability of the equilibrium: Players are locked into their strategies, because all other choices reduce the payoff. p A,p B is a strict Nash Equilibrium if

Prisoner B stays silentPrisoner B betrays Prisoner A stays silent Each serves 6 monthsBoth prisoners serve 10 years. Prisoner A betrays Both prisoners serve 10 years. Example of weak and strict Nash equilibria: (betray,betray) and (stay silent, stay silent) are N.E. (betray,betray) is a weak N.E., because payoffs cannot be improved by unilateral switching. (stay silent, stay silent) is a strict N.E., because switching strategies can only make things worse.

Some more definitions: Pure strategy:only one action can be taken (e.g. betray or stay silent once) Mixed strategy:several actions can be taken with specific probabilities (e.g. betray 50% of the time, stay silent 50% of the time) Symmetric game:The game rules are the same for all players (prisoner’s dilemma). Thus, in a N. E. a strategy that is the best reply to itself. Asymmetric game:Different rules apply to different players (poker).

Biological/evolutionary interpretation: (introduced by Maynard Smith and Price) An evolutionarily stable strategy (ESS) is a strategy which, if adopted by a population cannot be invaded by any competing alternative strategy. 1)A strategy is conceptualized a specific phenotype within a population containing several more rivaling phenotypes (a symmetric game). 2)Many play the game repeatedly in pairwise contests. 3)Payoff = fitness. Phenotypes that achieve higher fitness than competing phenotypes increase in frequency = Evolution. 4)Evolutionary Endpoint: change in phenotype frequency stops when all remaining phenotypes (perhaps just one) have equal fitness.

Definition of the Evolutionarily Stable Strategy (ESS): Strategy (I,I) is ESS if either: E(I,I) > E(J,I)for all I ≠ J or: E(I,I) = E(J,I)andE(I,J) > E(J,J) An ESS is also an evolutionarily stable state if the population has the genetic apparatus to return to the ESS when it is perturbed away from it. (strict N.E.) (Maynard Smith’s second condition)

Evolution as dynamical system: the hawk-dove game: A resource contest with the following rules: if resource is won without fight: fitness gain = V if the resource is won after a fight: fitness gain = V-C hawkdove hawk½(V-C)V dove0½ V The frequency of pairwise encounters is determined by the frequency of hawks and doves in the population.

Hawks have frequency p, doves have frequency (1-p) Hawk fitness depends on the probability of competing with another hawk or a dove: f hawk = f 0,H + 0.5*(V-C)*p + V*(1-p) The same holds for doves: f dove = f 0,D + 0.5*V*(1-p) In the ESS (the equilibrium frequencies of hawks and doves) the average payoff of contest is equal for hawks and doves.

Iterated games: What if a game is played repeatedly among the same players? possibility of learning possibility of trust-building Strategies consist of the following considerations: How to open the game (first move). How to respond to the opponents actions. How to end the game on the last move (if moves are finite).