Eponine Lupo

 Game Theory is a mathematical theory that deals with models of conflict and cooperation.  It is a precise and logical description of a strategic setting  It can be applied to many social sciences, evolutionary biology, and has many applications in economics.  Game Theory is often used in more complex situations where chance and a player’s choice are not the only factors that are contributing to the outcome.  Ex. Oil deposits

 Games—situations where the outcome is determined by the strategy of each player  Strategy—a complete contingent plan outlining all the actions a player will do under all possible circumstances  Key assumption: players are rational with complete information and want to maximize their payoffs

 Classic Games  Matching Pennies  Coordination  Battle of the Sexes  Prisoner’s Dilemma  Normal Form  Extensive Form  Strategies—pure strategy set  Solution  Nash Equilibrium (D,D)

 A probability distribution over the pure strategies for a player  Must add up to 1 or 100%  Infinite number of mixed strategies  Choose a mixed strategy to keep opponents guessing  Use a mixed strategy if the game is not solvable using pure strategies (no cominant or efficient strategies)

 Dominance—Prisoner’s Dilemma  S 1 is dominated by S 1 1 if S 1 1 gives Player 1 better payoffs than S 1, no matter what the other players do.  Compares 1 strategy to another of a single player  Iterated Dominance—Pigs  Efficiency—Pareto Coordination  S is more efficient than S 1 if everyone prefers S to S 1  Compares 2 strategy combinations involving all players  S is efficient if there is nothing that’s more efficient than S.  Best Response  S 1 is a Best Response to S 2 if S 1 gives player 1 the highest payoff given player 2 is playing S 2

 Named after John Nash  American mathematician  Subject of A Beautiful Mind  Definition: A strategy profile is a Nash equilibrium if and only if each player’s prescribed strategy is a best response to the strategies of the others.  No player can do better by unilaterally changing his or her strategy  Equilibrium that is reached even if it is not the best joint outcome

 Pure and Mixed Strategy N.E.  Some games do not have a pure strategy N.E.  One always exists in a mixed form  All finite games have at least one N.E.  A N.E. will/must be played in the last stage  In a Mixed N.E., each player chooses his probability mixture to maximize his value conditional on the other player’s selected probability mixture.

 Matching Pennies—mixed strategy only  (.5,.5)X(.5,.5)  Coordination  Prisoner’s Dilemma

 Find the Dominant strategies  Find the Best Responses for each player  Find the pure strategy N.E.

 Find the mixed strategy N.E. for 2X2 games  Find more than 1 mixed strategy NE  2 player games with more than 2 strategies  3 player games