Chapter Twenty-Eight,Game Theory,Game Theory,Game theory models strategic behavior by agents who understand that their actions affect the actions of other agents.,Some Applications of Game Theory,The study of oligopolies (industries containing only a few firms) The study of cartels; e.g. OPEC The study of externalities; e.g. using a common resource such as a fishery. The study of military strategies.,What is a Game?,A game consists of a set of players a set of strategies for each player the payoffs to each player for every possible list of strategy choices by the players.,Two-Player Games,A game with just two players is a two-player game. We will study only games in which there are two players, each of whom can choose between only two strategies.,An Example of a Two-Player Game,The players are called A and B. Player A has two strategies, called “Up” and “Down”. Player B has two strategies, called “Left” and “Right”. The table showing the payoffs to both players for each of the four possible strategy combinations is the games payoff matrix.,An Example of a Two-Player Game,This is the games payoff matrix.,Player B,Player A,Player As payoff is shown first. Player Bs payoff is shown second.,An Example of a Two-Player Game,E.g. if A plays Up and B plays Right then As payoff is 1 and Bs payoff is 8.,This is the games payoff matrix.,Player B,Player A,L,R,U,D,(3,9),(0,0),(1,8),(2,1),An Example of a Two-Player Game,And if A plays Down and B plays Right then As payoff is 2 and Bs payoff is 1.,This is the games payoff matrix.,Player B,Player A,L,R,U,D,(3,9),(0,0),(1,8),(2,1),An Example of a Two-Player Game,Player B,Player A,A play of the game is a pair such as (U,R) where the 1st element is the strategy chosen by Player A and the 2nd is the strategy chosen by Player B.,An Example of a Two-Player Game,What plays are we likely to see for this game?,Player B,Player A,An Example of a Two-Player Game,Player B,Player A,Is (U,R) a likely play?,L,R,U,D,(3,9),(0,0),(1,8),(2,1),An Example of a Two-Player Game,Player B,Player A,If B plays Right then As best reply is Down since this improves As payoff from 1 to 2. So (U,R) is not a likely play.,Is (U,R) a likely play?,L,R,U,D,(3,9),(0,0),(1,8),(2,1),An Example of a Two-Player Game,Player B,Player A,Is (D,R) a likely play?,L,R,U,D,(3,9),(0,0),(1,8),(2,1),An Example of a Two-Player Game,Player B,Player A,Is (D,R) a likely play?,If B plays Right then As best reply is Down.,L,R,U,D,(3,9),(0,0),(1,8),(2,1),An Example of a Two-Player Game,Player B,Player A,If B plays Right then As best reply is Down. If A plays Down then Bs best reply is Right. So (D,R) is a likely play.,Is (D,R) a likely play?,L,R,U,D,(3,9),(0,0),(1,8),(2,1),An Example of a Two-Player Game,Player B,Player A,Is (D,L) a likely play?,L,R,U,D,(3,9),(0,0),(1,8),(2,1),An Example of a Two-Player Game,Player B,Player A,If A plays Down then Bs best reply is Right, so (D,L) is not a likely play.,Is (D,L) a likely play?,L,R,U,D,(3,9),(0,0),(1,8),(2,1),An Example of a Two-Player Game,Player B,Player A,Is (U,L) a likely play?,L,R,U,D,(3,9),(0,0),(1,8),(2,1),An Example of a Two-Player Game,Player B,Player A,If A plays Up then Bs best reply is Left.,Is (U,L) a likely play?,L,R,U,D,(3,9),(0,0),(1,8),(2,1),An Example of a Two-Player Game,Player B,Player A,If A plays Up then Bs best reply is Left. If B plays Left then As best reply is Up. So (U,L) is a likely play.,Is (U,L) a likely play?,L,R,U,D,(3,9),(0,0),(1,8),(2,1),Nash Equilibrium,A play of the game where each strategy is a best reply to the other is a Nash equilibrium. Our example has two Nash equilibria; (U,L) and (D,R).,An Example of a Two-Player Game,Player B,Player A,(U,L) and (D,R) are both Nash equilibria for the game.,L,R,U,D,(3,9),(0,0),(1,8),(2,1),An Example of a Two-Player Game,Player B,Player A,(U,L) and (D,R) are both Nash equilibria for the game. But which will we see? Notice that (U,L) is preferred to (D,R) by both players. Must we then see (U,L) only?,L,R,U,D,(3,9),(0,0),(1,8),(2,1),The Prisoners Dilemma,To see if Pareto-preferred outcomes must be what we see in the play of a game, consider a famous second example of a two-player game called the Prisoners Dilemma.,The Prisoners Dilemma,What plays are we likely to see for this game?,Clyde,Bonnie,(-5,-5),(-30,-1),(-1,-30),(-10,-10),S,C,S,C,The Prisoners Dilemma,If Bonnie plays Silence then Clydes best reply is Confess.,Clyde,Bonnie,(-5,-5),(-30,-1),(-1,-30),(-10,-10),S,C,S,C,The Prisoners Dilemma,If Bonnie plays Silence then Clydes best reply is Confess. If Bonnie plays Confess then Clydes best reply is Confess.,Clyde,Bonnie,(-5,-5),(-30,-1),(-1,-30),(-10,-10),S,C,S,C,The Prisoners Dilemma,So no matter what Bonnie plays, Clydes best reply is always Confess. Confess is a dominant strategy for Clyde.,Clyde,Bonnie,(-5,-5),(-30,-1),(-1,-30),(-10,-10),S,C,S,C,The Prisoners Dilemma,Similarly, no matter what Clyde plays, Bonnies best reply