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1、June 12, 2003,73-347 Game Theory-Lecture 17,1,Dynamic Games of Complete Information,Dynamic Games of Complete and Imperfect Information,June 12, 2003,73-347 Game Theory-Lecture 17,2,Outline of dynamic games of complete information,Dynamic games of complete information Extensive-form representation D
2、ynamic games of complete and perfect information Game tree Subgame-perfect Nash equilibrium Backward induction Applications Dynamic games of complete and imperfect information More applications Repeated games,June 12, 2003,73-347 Game Theory-Lecture 17,3,Todays Agenda,Review of previous class Bank r
3、uns (2.2.B of Gibbons) Tariffs and imperfect international competition (2.2.C of Gibbons),June 12, 2003,73-347 Game Theory-Lecture 17,4,Dynamic (or sequential-move) games of complete information,A set of players Who moves when and what action choices are available? What do players know when they mov
4、e? Players payoffs are determined by their choices. All these are common knowledge among the players.,June 12, 2003,73-347 Game Theory-Lecture 17,5,Dynamic games of complete information,Perfect information A player knows Who has made What choices when she has an opportunity to make a choice Imperfec
5、t information A player may not know exactly Who has made What choices when she has an opportunity to make a choice.,June 12, 2003,73-347 Game Theory-Lecture 17,6,Information set,Gibbons definition: An information set for a player is a collection of nodes satisfying: the player has the move at every
6、node in the information set, and when the play of the game reaches a node in the information set, the player with the move does not know which node in the information set has (or has not) been reached. All the nodes in an information set belong to the same player The player must have the same set of
7、 feasible actions at each node in the information set.,June 12, 2003,73-347 Game Theory-Lecture 17,7,Information set: illustration,an information set for player 3 containing three nodes,an information set for player 3 containing a single node,two information sets for player 2 each containing a singl
8、e node,June 12, 2003,73-347 Game Theory-Lecture 17,8,Perfect information and imperfect information,A dynamic game in which every information set contains exactly one node is called a game of perfect information. A dynamic game in which some information sets contain more than one node is called a gam
9、e of imperfect information.,June 12, 2003,73-347 Game Theory-Lecture 17,9,Subgame-perfect Nash equilibrium,A Nash equilibrium of a dynamic game is subgame-perfect if the strategies of the Nash equilibrium constitute or induce a Nash equilibrium in every subgame of the game. A subgame of a game tree
10、begins at a singleton information set (an information set containing a single node), and includes all the nodes and edges following the singleton information set, and does not cut any information set; that is, if a node of an information set belongs to this subgame then all the nodes of the informat
11、ion set also belong to the subgame.,June 12, 2003,73-347 Game Theory-Lecture 17,10,Find subgame perfect Nash equilibria: backward induction,a subgame,a subgame,Starting with those smallest subgames Then move backward until the root is reached,One subgame-perfect Nash equilibrium ( IR, AR ),June 12,
12、2003,73-347 Game Theory-Lecture 17,11,Find subgame perfect Nash equilibria: backward induction,a subgame,a subgame,Starting with those smallest subgames Then move backward until the root is reached,Another subgame-perfect Nash equilibrium ( ED, BD ),June 12, 2003,73-347 Game Theory-Lecture 17,12,Fin
13、d subgame perfect Nash equilibria: backward induction,What is the subgame perfect Nash equilibrium?,June 12, 2003,73-347 Game Theory-Lecture 17,13,Bank runs (2.2.B of Gibbons),Two investors, 1 and 2, have each deposited D with a bank. The bank has invested these deposits in a long-term project. If t
14、he bank liquidates its investment before the project matures, a total of 2r can be recovered, where D r D/2. If banks investment matures, the project will pay out a total of 2R, where RD. Two dates at which the investors can make withdrawals from the bank.,June 12, 2003,73-347 Game Theory-Lecture 17
15、,14,Bank runs: timing of the game,The timing of this game is as follows Date 1 (before the banks investment matures) Two investors play a simultaneous move game If both make withdrawals then each receives r and the game ends If only one makes a withdrawal then she receives D, the other receives 2r-D
16、, and the game ends If neither makes a withdrawal then the project matures and the game continues to Date 2. Date 2 (after the banks investment matures) Two investors play a simultaneous move game If both make withdrawals then each receives R and the game ends If only one makes a withdrawal then she receives 2R-D, the other receives D, and the game ends If neither makes a withdrawal then the bank returns R to each investor and the game ends.,June 12, 2003,73-347
