Strategic Form Games with Complete Information I

Transcription

1 Strategic Form Games with Complete Information I Georg Nöldeke Wirtschaftswissenschaftliche Fakultät, Universität Basel Advanced Microeconomics, HS 11 Lecture 1 1/22

2 Strategic Form Games A strategic form game is specified by defining the players: i = 1,...,N. a strategy set S i for each player, specifying the strategies s i player i might choose. a payoff function for each player, specifying the player s utility u i (s 1,...,s n ) for every possible strategy profile s = (s 1,...,s n ) S 1... S n := S. Definition (Strategic Form Game) An N-person strategic form game is given by G = (S 1,,S N ;u 1,,u N ), where for each i the set S i is non-empty and u i : S R. Advanced Microeconomics, HS 11 Lecture 1 2/22

3 Strategic Form Games Note: When analyzing a strategic form game, it is assumed that strategies are chosen simultaneously... but this does not mean that players have to act simultaneously. Indeed, as we will see later strategic form games can be used to analyze situations in which players move sequentially. Definition (Finite Game) An N-person strategic form game is called finite if for all i the strategy set S i contains a finite number of elements. Many economic applications consider infinite games. Advanced Microeconomics, HS 11 Lecture 1 3/22

4 Strategic Form Games Finite 2-player strategic form games can be represented by a bi-matrix. Every row of the bi-matrix represents a strategy of player 1. Every column of the bi-matrix represents a strategy of player 2. Every cell thus represents a strategy profile. In every cell there are two numbers, representing the payoffs the players obtain from the corresponding strategy profile. By convention the first number is player 1 s payoff, the second number is player 2 s payoff. Advanced Microeconomics, HS 11 Lecture 1 4/22

5 Some Simple Examples C D C 2,2 0,3 D 3,0 1,1 Prisoner s Dilemma Advanced Microeconomics, HS 11 Lecture 1 5/22

6 Some Simple Examples A B A 2,1 0,0 B 0,0 1,2 Battle of the Sexes Advanced Microeconomics, HS 11 Lecture 1 6/22

7 Some Simple Examples T S T 3,3 2,5 S 5,2 0,0 Chicken Advanced Microeconomics, HS 11 Lecture 1 7/22

8 Some Simple Examples H T H 1, 1 1,1 T 1,1 1, 1 Matching Pennies Advanced Microeconomics, HS 11 Lecture 1 8/22

9 Strict Domination Write s = (s i,s i ) to denote the strategy profile in which player i plays s i and s i specifies strategies for the remaining players. Definition (Strict Domination) 1. A strategy ŝ i S i strictly dominates s i S i if u i (ŝ i,s i ) > u i ( s i,s i ) holds for all s i S i = j i S j. 2. A strategy s i S i is strictly dominated in S if there is some strategy in S i strictly dominating it. 3. A strategy s i S i is strictly dominant if it strictly dominates all other strategies of player i. Advanced Microeconomics, HS 11 Lecture 1 9/22

10 Strict Domination Example: Prisoner s Dilemma Strategy D strictly dominates strategy C for both players. Hence, for both players C is strictly dominated. Eliminating strictly dominated strategies thus identifies (D,D) as the solution of the game. Question: Why eliminate strictly dominated strategies? Advanced Microeconomics, HS 11 Lecture 1 10/22

11 Weak Domination Definition (Weak Domination) 1. A strategy ŝ i S i weakly dominates s i S i if u i (ŝ i,s i ) u i ( s i,s i ) holds for all s i S i = j i S j and there exists at least one s i such that the above inequality is strict. 2. A strategy s i S i is weakly dominated in S if there is some strategy in S i weakly dominating it. 3. A strategy ŝ i S i is weakly dominant if it weakly dominates all other strategies of player i. Advanced Microeconomics, HS 11 Lecture 1 11/22

12 Weak Domination Example: L R U 3,1 0,0 D 2,2 0,2 U weakly dominates D and L weakly dominates R. Hence, U and L are weakly dominant. Eliminating weakly dominated strategies thus identifies (U,D) as the solution of the game. Observe: There are no strictly dominates strategies. Question: Why eliminate weakly dominated strategies? Advanced Microeconomics, HS 11 Lecture 1 12/22

13 Iterated Dominance Consider the following procedure: Step 0 Let S 0 i = S i for all players i. Step 1 Eliminate all strictly dominated strategies from S 0 i for all i. Let S 1 i denote the set of remaining strategies for player i. Step 2 Eliminate all strategies from S 1 i that are strictly dominated in S 1 for all i. Let S 2 i denote the set of remaining strategies for player i. Step n Eliminate all strategies from Si n 1 that are strictly dominated in S n 1 for all i. Let Si n denote the set of remaining strategies for player i. Advanced Microeconomics, HS 11 Lecture 1 13/22

14 Iterated Dominance Definition (Iteratively Strictly Undominated Strategies) A strategy s i for player i is iteratively strictly undominated if s i S n i for all n 1. In a finite game iteratively strictly undominated strategies exist. the iterative procedure stops after a finite number of rounds. the order of iteration does not matter. Question: What motivates the iterative elimination of strictly dominated strategies? Advanced Microeconomics, HS 11 Lecture 1 14/22

15 Iterated Dominance Replacing strictly dominated by weakly dominated in the above procedure defines strategy sets W 0 i,,w n i,. i,w 1 Definition (Iteratively Weakly Undominated Strategies) A strategy s i for player i is iteratively weakly undominated if s i W n i for all n 1. In a finite game iteratively weakly undominated strategies exist. the iterative procedure stops after a finite number of rounds. the order of iteration may matter. Question: What motivates the iterative elimination of weakly dominated strategies? Advanced Microeconomics, HS 11 Lecture 1 15/22

16 Iterated Dominance Guess the Average: An example for the iterated elimination of weakly dominated strategies N 2 players simultaneously choose an integer between 1 and 100. The player closest to 1/3 of the average of the guesses wins CHF100.; if there are ties the prize is split evenly. Payoffs are given by the expected amount of money a player wins. All strategies with s i > 33 are weakly dominated. Once these strategies are eliminated, all strategies with s i > 11 are weakly dominated. For all players the only iteratively weakly undominated strategy is s i = 1. Questions: Is this what players in this game actually do? Is this what you would do? Advanced Microeconomics, HS 11 Lecture 1 16/22

17 Pure Strategy Nash Equilibrium John Nash Winner of the Nobel Prize in Economics 1994 Photo: Elke Wetzig wikipedia/commons/9/91/john_ f_nash_ _3.jpg Advanced Microeconomics, HS 11 Lecture 1 17/22

18 Pure Strategy Nash Equilibrium Definition (Pure Strategy Nash Equilibrium) Given a strategic form game G = (S 1,...,S N ;u 1,...,u N ) a strategy profile ŝ S is a pure strategy Nash equilibrium of G if for all players i u i (ŝ) u i (s i,ŝ i ) holds for all s i S i. A pure strategy Nash equilibrium describes behavior in which each individual 1. correctly anticipates the behavior of other individuals 2. has no incentive to change his own behavior. Advanced Microeconomics, HS 11 Lecture 1 18/22

19 Pure Strategy Nash Equilibrium Definition (Best Response) Let G = (S 1,...,S N ;u 1,...,u N ) be a strategic form game. A strategy ŝ i S i is a best response against s i S i if holds for all s i S i. u i (s i,s i ) u i (s i,s i ) Write BR i (s i ) for the set of best responses against s i. The mapping BR i : S i S i is known as player i s best response correspondence. Note: If s i is strictly dominated then it is never a best response Advanced Microeconomics, HS 11 Lecture 1 19/22

20 Pure Strategy Nash Equilibrium The following observations are useful in finding (all) pure strategy Nash equilibria of a game: Observation A strategy profile ŝ is a pure strategy Nash equilibrium if and only if for all players i the strategy ŝ i is a best response against ŝ i, that is: ŝ i BR i (ŝ i ). Observation If ŝ is a pure strategy Nash equilibrium then for all players ŝ i is (iteratively) strictly undominated. Hence, the set of pure strategy Nash equilibria is not affected by the (iterated) elimination of strictly dominated strategies. Advanced Microeconomics, HS 11 Lecture 1 20/22

21 Pure Strategy Nash Equilibrium Some games have multiple pure strategy Nash equilibria. Examples: Battle of the Sexes. Chicken. Other games have no pure strategy Nash equilibrium. Example: Matching Pennies. A game may have a unique pure strategy Nash equilibrium without having strictly dominated strategies. Example: L C R T 0,0 1, 1 1,1 M 0,0 1,1 1, 1 B 5,5 0,0 0,0 In this game there are no strictly dominated strategies. The unique Nash equilibrium is (B, L). Advanced Microeconomics, HS 11 Lecture 1 21/22

22 Pure Strategy Nash Equilibrium Two Methods for Finding all Pure Strategy Nash Equilibria: 1. Check for all possible strategy profiles whether or not there is some player who could strictly increase his payoff by switching to a different strategy. If there is no such player, the strategy profile is a pure strategy Nash equilibrium. Otherwise it is not. 2. Determine the best response correspondence for each player and identify those strategy profiles satisfying the condition s i BR i (s i ) for all i. Before using either of the above, it may be advisable to eliminate strictly dominated strategies (and to do so iteratively). In economic applications, it is usually advisable to develop some intuition before applying brute force to find pure strategy Nash equilibria. Advanced Microeconomics, HS 11 Lecture 1 22/22