Microeconomics: Game Theory P . v. Mouche Wageningen University Spring 2020
Game Theory First watch the following video up to time 14:15: https://www.youtube.com/watch?v=pC--lK8KNwo
Game Theory (ctd.) So in this video You encountered various notions, in particular the (for this chapter very important) notion of bimatrix. Below we shall, among other things, reconsider them. After studying these slides, You should be able to determine for a bimatrix game the strictly dominant strategies; the Nash equilibria; the Pareto efficient strategy profiles; whether the game is a prisoner’s dilemma.
What is game theory? Traditional game theory deals with mathematical mod- els of conflict and cooperation in the real world between at least two rational intelligent players. ’Traditional’ because of rationality assumption. Player: concerns individuals, organisations, countries, animals, computers, ... . Further on we only deal for simplicity with two players: player 1 and player 2, or white and black, ... . In some games a device like a die decides who is which player. Situations with only one player are studied in classical optimisation theory.
Game theory Applications. Economics: Nobel prices in 1994 for Nash, Harsanyi and Selten, in 2005 for Aumann and in 2007 for Meyerson and Maskin. Sociology, psychology, antropology, politocology. Military strategy. Biology. Design of computer games and robots. Game theory provides a language that is very appropriate for conceptual thinking. Many game theoretical concepts can be understood without higher mathematics.
Concrete economical games: oligopolies Cournot duopoly: two producers are the players. π i : profit function of producer i . c i : cost function of producer i . p inverse market demand function. π i ( x 1 , x 2 ) = p ( x 1 + x 2 ) x i − c i ( x i ) . Example: 10 π 1 ( x 1 , x 2 ) = x 1 − 5 x 1 , 3 + x 1 + x 2 10 π 2 ( x 1 , x 2 ) = x 2 − 5 x 2 . 3 + x 1 + x 2 Oligopolies are our next topic this week!
Bimatrix game Before we continue, first some remarks on bimatrix games. Consider, for example, the bimatrix game 3 ; 3 2 ; 2 7 ; − 1 − 3 ; 1 . 1 ; 2 127 ; − 1 , 45 This is a 3 × 2-bimatrix game, i.e. it has 3 rows and 2 columns. Player 1 chooses a row: row 1, row 2 or row 3; so player 1 has 3 strategies. Player 2 chooses a column: column 1 or column 2; so player 2 has 2 strategies. At the strategy profile ( 3 , 2 ) , i.e. row 3 and column 2, player 1 has payoff 127 and player 2 has payoff − 1 , 45.
Rationality Rationality’ and ’Intelligence’ are completely different concepts. Rationality concerns a consistence condition for the choices a player makes in order to fulfill his goal as good as possible. A player that behaves according to that condition is called rational. Especially, if there is more than one player, this notion becomes problematic For example, what would You as player 1 play in the following bimatrix game: � 300 ; 400 � 600 ; 250 ? 200 ; 600 450 ; 500 Remember (from the video): In a bimatrix game player 1 chooses a row, player 2 a column. In the corresponding cell the first (second) number is the payoff to the first (second) player.
Outcomes and payoffs A game can have different outcomes. Each outcome has its own payoffs for each of the players. For instance: There is a winner and a looser. There is a winner, a looser or it is draw. Interpretation of payoff: ‘satisfaction’ at end of game. Nature of payoff: money, honour, activity, nothing at all, utility, real number, ... .
Concrete parlor game: tic-tac-toe 1 2 3 Notations: 4 5 6 7 8 9 Player 1: X. Player 2: O. Many outcomes (more than three). Three types of payoffs: player 1 wins, draw, player 1 looses. Payoffs (example): winner obtains 13 euro from looser. When draw, then each player cleans the shoes of the other. Example of a play of this game:
Tic-tac-toe (cont.) X X X X O O X X O X X O X X O O O X O X O So: player 2 is the winner.
Bimatrix game A bimatrix game is a special case of a game in strategic form (also sometimes called ‘normal form’), like the above oligopoly game, where there are two players. 2 players: player 1 chooses a row and player 2 a column. Choices are made simultaneously and independently. Each player has a finite number of strategies. This situation can be represented by a bimatrix.
Bimatrix (ct.) Many games can be represented in a natural way as a bimatrix game. For example: 0 ; 0 − 1 ; 1 1 ; − 1 1 ; − 1 0 ; 0 − 1 ; 1 − 1 ; 1 1 ; − 1 0 ; 0 Stone-paper-scissors. The above game is an example of zero sum game, i.e. a game where the sum of the payoffs at each cell of the bimatrix is zero.
Fundamental notions Strategy of a player: completely elaborated plan of playing. Strategy profile: for each player a strategy. Strictly dominant strategy of a player: the best strategy of that player independently of strategies of the other players. Strictly dominant equilibrium: strategy profile where each player has a strictly dominant strategy. Nash equilibrium: strategy profile such that no player wants to deviate from it.
Fundamental notions (ctd.) (Weakly) Pareto efficient strategy profile: a strategy profile for which there is no other strategy profile in which each player is better off. A strategy profile is fully cooperative if the sum of the payoffs in this strategy profile is maximal. Prisoner’s dilemma game: a game with a strictly dominant equilibrium that is Pareto inefficient. Important remark: there also exists another notion of Pareto-efficiency, the so-called strong Pareto efficiency. We shall not deal with this notion and only consider the above more simple one.
Example Example 1. 2 ; 4 1 ; 4 4 ; 3 3 ; 0 1 ; 1 1 ; 2 5 ; 2 6 ; 1 . 1 ; 2 0 ; 5 3 ; 4 7 ; 3 0 ; 6 0 ; 4 3 ; 4 1 ; 5 No strictly dominant strategies, thus no strictly dominant equilibrium. The game has the following Nash equilibria: ( 1 , 1 ) , ( 1 , 2 ) , ( 2 , 2 ) and ( 2 , 3 ) (i.e. row 2 and column 3).
Example Example 6 ; 1 3 ; 1 1 ; 5 No player has a strictly dominant 2 ; 4 4 ; 2 2 ; 3 2. 5 ; 1 6 ; 1 5 ; 2 strategy, thus the game does not have a strictly dominant equilibrium. The game has one Nash equilibrium: ( 3 , 3 ) .
Examples Example 3 ; − 1 3 ; 1 6 ; 0 3. 1 ; 0 3 ; 1 6 ; 0 . No player has a strictly dominant 2 ; 2 4 ; 1 8 ; 2 strategy and ( 3 , 3 ) is the only Nash equilibrium. The game is not a prisoner’s dilemma. � 1 ; 0 � 3 ; 1 6 ; 0 4. . Only player 1 has a strictly dominant 2 ; 1 4 ; 1 8 ; 1 strategy: the second one. There are three Nash equilibria: ( 2 , 1 ) , ( 2 , 2 ) , ( 2 , 3 ) . The game is not a prisoner’s dilemma.
Examples Example 6 ; 1 3 ; 1 1 ; 5 5. 2 ; 4 4 ; 2 2 ; 3 . No player has a strictly dominant 5 ; 1 6 ; 1 5 ; 2 strategy and ( 3 , 3 ) is the only Nash equilibrium. The game is not a prisoner’s dilemma. � 1 ; 0 � 1 ; 4 0 ; 2 6. . No player has a strictly dominant 0 ; 6 0 ; 2 2 ; 0 strategy and ( 1 , 2 ) is the only Nash equilibrium. The game is not a prisoner’s dilemma.
Examples Example � � 7. 1 ; 0 1 ; 2 0 ; 4 . The first player has a strictly dominant strategy: the first one. The second player has a strictly dominant strategy: the third one. ( 1 , 3 ) is the only Nash equilibrium. The game is not a prisoner’s dilemma. � 1 ; 0 � − 1 ; 4 0 ; 2 8. . No player has a strictly dominant 0 ; 6 0 ; 2 0 ; 3 strategy and there is no Nash equilibrium. The game is not a prisoner’s dilemma.
Examples Example � − 1 ; − 1 � 2 ; 0 9. . Both players have a strictly dominant 0 ; 2 3 ; 3 strategy: their second one. ( 2 , 2 ) is the only Nash equilibrium. This Nash equilibrium is Pareto efficient. The game is not a prisoner’s dilemma game. � � 2 ; 2 − 1 ; 3 10. . Both players have a strictly dominant 3 ; − 1 0 ; 0 strategy: their second one. ( 2 , 2 ) is the only Nash equilibrium. This Nash equilibrium is Pareto inefficient. The game is a prisoner’s dilemma game.
Examples Example − 1 ; 0 − 1 ; 1 0 ; 0 2 ; − 2 − 3 ; 3 − 1 ; 3 11. . Only player 1 has a strictly 4 ; − 3 5 ; − 5 1 ; − 7 3 ; − 3 3 ; − 5 − 6 ; 8 dominant strategy: his third one. Only ( 3 , 1 ) is a Nash equilibrium. The game is not a prisoner’s dilemma game.
John Nash Enjoy, if You like, the following video about John Nash: http://topdocumentaryfilms.com/ a-brilliant-madness-john-Nash
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