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Static ggames: Nash equilibrium q Lectures in Game Theory Fall 2011, Part 3 24.07.2011 G.B. Asheim, ECON3/4200-3 1 Rationalizability is appropriate … in a situation which does not recur often. with no communication or outside coordination when players are strategically sophisticated. What if … the game recurs often (even though the opponents t change h from f time ti to t time). ti ) the players can communicate. there is outside coordination. 24.07.2011 G.B. Asheim, ECON3/4200-3 2 1 Two concepts Consider a set of strategy profiles X X 1 X n , where X i Si for all i. Best response property (weak congruity): The set X contains only best responses. responses X has the best response property if, for each i and each si X i , there is i X i s.t. si BRi ( i ). Best response completeness: Th set X contains The i allll b best responses X is best response complete if, for each i and each i X i , BRi ( i ) X i . 24.07.2011 G.B. Asheim, ECON3/4200-3 3 Nash equilibrium Are there strategies for the two players so that no player will regret his own choice when being told of the other player’s p y choice? If yyes,, then such a strategy profile is a Nash equilibrium. Some examples: Definition : ( s1 , , sn ) is a Nash equilibrium if it, for each player i , holds that ui ( si , si ) ui ( si, si ) for all si. I.e., for each player i , si BR ( si ). If, for each player i , {si } BR ( si ), then ( s1 , , sn ) is a strict Nash equilibrium . 24.07.2011 G.B. Asheim, ECON3/4200-3 4 2 Observations Result : If ( s1 , , sn ) is a Nash equilibrium, then ( s1 , , sn ) survives iterated strict eliminatio n. Result : If ( s1 , , sn ) is the only strategy profile which survives iterated strict eliminatio n,, then ( s1 , , sn ) is a Nash equilibrium. 24.07.2011 G.B. Asheim, ECON3/4200-3 5 Some games have no Nash equilibrium Some examples: Such games have a Nash equilibrium in mixed strategies. Interpretation as a steady state. Definition : Consider a strategy profile ( 1 , , n ), where i Si for each player i. The profile ( 1 , , n ) is a mixed - strategy Nash equilibrium if it, for each player i , holds that ui ( i , i ) ui ( si , i ) for all si . Result (Nash, 1950) : Every finite game has a mixed - strategy Nash equilibriu m. 24.07.2011 G.B. Asheim, ECON3/4200-3 6 3 Steady-state interpretation of Nash equilibr. The game is a model designed to explain some regularity in a family of similar situations. Each participant "knows" the equilibrium and tests the optimality of his behavior given this knowledge, which he has acquired from long experience. The interpretation requires that the players meet different opponents each time. In games with multiple equilibria, will the players coordinate, di and d if yes, on which hi h equilibrium? ilib i ? Some examples: Third tension: Coordination on an inefficient NE. 24.07.2011 G.B. Asheim, ECON3/4200-3 7 Can Nash equilibrium be used as a solution concept if the game is only played once? Yes, if each player can predict what each opponent will do. For each p player, y , onlyy one strategy gy survives iterative elimination of strictly dominated strategies. Through communication before the game starts, the players make a self-enforcing agreement (coordinate on an equilibrium). Given a common background, the players are able to co-ordinate on an equilibrium without communication before the game starts (Schelling, 1960, focal point). A unique Nash equilibrium is not sufficient. 24.07.2011 G.B. Asheim, ECON3/4200-3 8 4 Behavioral game theory Standard game theory provides discipline for our analysis of the relation between the outcome of strategic interaction and our assumptions p about behavior. But does the theory accurately describe and predict real behavior? To test game theory, one can … — Gather data about behavior in real strategic situations. — Perform laboratory experiments with monetary payoffs. Behavioral game theory seeks to learn about real behavior through laboratory experiments. Problems: — Lab. settings may not resemble real strategic settings. — May be difficult to control the subjects’ payoffs. 24.07.2011 G.B. Asheim, ECON3/4200-3 9 5