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Partial equilibrium q analysis: y Some game theory Lectures in Microeconomic Theory Fall 2010, Part 19 07.07.2010 G.B. Asheim, ECON4230-35, #19 1 Game theory studies multi-person decision problems, and analyzes agents that are rational (have well-defined preferences) reason strategically (take into account their knowledge and beliefs about what others do) Applications Industrial organization (incl. oligopoly theory) g g andd auction c theoryy Bargaining Labor market and financial economics Macro economics International economics 07.07.2010 G.B. Asheim, ECON4230-35, #19 2 1 Game matrix The players choose strategies. Each player’s payoff depends on the strategy profile. 2 Demand 1 Demand 3 1 for self for the other Demand 1 for self 1, 1 Demand 3 for the other 0, 4 4, 0 Prisoners’ dilemma If the players do not choose strictly dominated strategies, both will demand 1 for themselves. 07.07.2010 3, 3 G.B. Asheim, ECON4230-35, #19 3 Solving games through iterated elimination of strongly dominated strategies r T 1,, 1 1,, 0 B 0, 0 0, 1 The game is solved through h h iiter. elimi. li i of str. dom. strat. 1 does not choose a str. dom. strategy 1 chooses T 2 believes that 1 does not choose a str. dom. strat. & 2 does not choose a str. dom. strat. 2 chooses r Battle of the No strategy is str. T 2, 1 0, 0 sexes B 0, 0 1, 2 07.07.2010 G.B. Asheim, ECON4230-35, #19 dominated. for any of the players! 4 2 Nash equilibrium Are there strategies for the two players so that no player will regret his own choice when hen being told of the other pla player’s er’s choice? If yes, then such a strategy profile is a Nash equilibrium. r Battle of the T 2, 1 0, 0 sexes B 0, 0 1, 2 (T, ) and (B, r) are Nash equilibria. 07.07.2010 G.B. Asheim, ECON4230-35, #19 5 The strategic form specifies Players: {1, ... , i, ... , n} For each p player, a strategy set: Si For each player, a payoff function: ui G ( S1 ,, Sn ; u1 ,, un ) Payoff for each player i : ui ( s1 , , sn ) ui ( si , si ) where, for all j , s j S j , and where we write si ( s1 , , si 1 , si1 , , sn ) 07.07.2010 G.B. Asheim, ECON4230-35, #19 6 3 Definitions Definition : si is strongly dominated by si if ui ( si, si ) ui ( si, si ) for all si . Iterated elimination of str. dom. strategies requires that it is commonly believed that players are rational. Definition : ( s1 , , sn ) is a Nash equilibrium for G ( S1 , , S n ; u1 , , u n ) if, for each player i , it holds that ui ( si , si ) ui ( si , si ) for all si . Nash equilibrium requires that players are rational and have correct conjectures conc. the opponents’ choices. 07.07.2010 G.B. Asheim, ECON4230-35, #19 7 Examples C L U 1, 0 1, 2 D 0, 3 R 0, 1 0, 1 2, 0 Iterated elimination of strongly dominated strategies works well in some examples. C R L T 0, 4 4, 0 5, 3 In other games, no strategy is str. dom. M 4, 0 0, 4 5, 3 Even in games w/only one Nash equil., incorr. conjectures can be held. B 3, 5 3, 5 6, 6 07.07.2010 G.B. Asheim, ECON4230-35, #19 8 4 Some games have no Nash equilibrium Monitoring (of traffic) I L l Illegal Legal Ill l Monitoring - 1, 0 - 1, - 4 No monitoring 0, 0 - 3, 1 1 3 Player 2’s 2 s best response fn Player 1’s best response fn Such games have a L Nash equilibrium in 1 M N 5 mixed strategies. Interpretation as a steady state. 07.07.2010 G.B. Asheim, ECON4230-35, #19 9 Some games have multiple equilibria Will the players coordinate, and if yes, on which equilibrium? A game of trust Each of you is to choose 1 or 2 indep. of each other. A player who chooses 1, receives 100 kr from UofO. A player who chooses 2, receives 200 kr from UofO if allll others h also l choose h 2 but 2, b must pay 500 kr k if at least one other player chooses 1. Write 1 or 2 on a piece of paper! 07.07.2010 G.B. Asheim, ECON4230-35, #19 10 5 Multiple equilibria (cont.) A game of trust 2 1 Choose 1 Choose 2 100, 110 Choose 1 100, 100 100,-500 Will a guarantee help? Choose 2 -110, 500,100 100 200, 200 Conclusion: C l i S Some games (e.g., ( the h battle b l off the h sexes and the game of trust) have multiple Nash equilibria. Will the players coordinate on a particular Nash equil.? 07.07.2010 G.B. Asheim, ECON4230-35, #19 11 Multiple equilibria (cont.) In some games the Nash equilibria require that the players choose differently. Chicken 2 1 Not R&D R&D Not R & D 2, 2 R&D 3 0 3, 0, 3 - 1, - 1 Will the players coordinate on a particular Nash equil.? 07.07.2010 G.B. Asheim, ECON4230-35, #19 12 6