Incentives and Behavior Prof. Dr. Heiner Schumacher KU Leuven 1. Game Theory I Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 1. Game Theory I 1 / 27
Introduction Game Theory is a mathematical language that is used in social sciences to analyze strategic interaction. We have a situation of strategic interaction when the behavior of two or more parties in‡uences each party’s well-being. There is an in…nite number of examples for strategic interactions: competition between …rms, the interaction between …rms and customers, auctions, bargaining, etc. This chapter introduces the basic notation and analyzes static games under complete information. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 1. Game Theory I 2 / 27
Introduction Overview Fundamental Concepts Nash Equilibrium Mixed Strategies Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 1. Game Theory I 3 / 27
Fundamental Concepts De…nition of a game: A number of players i 2 N = f 1 , ..., n g . A number of possible strategies for each player: s i 2 S i . A strategy pro…le is a vector s = ( s 1 , s 2 , ..., s n ) 2 S = S 1 � S 2 � ... � S n . The vector s � i denotes the vector of strategies of all players except player i , i.e., s � i = ( s 1 , ..., s i � 1 , s i + 1 , ..., s n ) . A utility function that determines the players’ payo¤s as a function of the chosen strategies, u i : S ! R (player i ’s Bernoulli utility function). A n � player game is a list G = f N , S 1 , ..., S n ; u 1 , ..., u n g which speci…es the set of players, the strategies available to each player and the utilities of each player. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 1. Game Theory I 4 / 27
Fundamental Concepts Players need not to act simultaneously. Each player only cares for her own payo¤. Each player knows the whole structure of the game. In a game with complete information each player’s payo¤ function is common knowledge among all players. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 1. Game Theory I 5 / 27
Fundamental Concepts When there are only two players, we can display the game in a payo¤ matrix. For example, consider the following game with two players S 1, S 2 and two strategies for each player. S 1/ S 2 Left Right Up 1,3 0,1 Down 2,1 1,0 Interpretation: If S 1 chooses “Down” and S 2 chooses “Right”, then S 1 receives the payo¤ 1 and S 2 the payo¤ 0. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 1. Game Theory I 6 / 27
Fundamental Concepts Dominant Strategies Consider the game on the last slide. What is the optimal strategy for player S 1? If S 2 chooses “Left”, then “Down” is better than “Up”. If S 2 chooses “Right”, then “Down” is better than “Up”. It is therefore a dominant strategy for S 1 to play “Down”. What would be the optimal strategy for S 2? If S 1 chooses “Up”, then “Left” is better than “Right”. If S 1 chooses “Down”, then “Left” is better than “Right”. It is therefore a dominant strategy for S 2 to play “Left”. A dominant strategy is a strategy which is always optimal, that is, regardless of the opponents’ strategies. In the example above, we get an equilibrium in dominant strategies. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 1. Game Theory I 7 / 27
Fundamental Concepts Iterated elimination of strictly dominated strategies (IESDS) In most games, there are no dominant strategies (and therefore no equilibrium in dominant strategies). How can we analyze such games? Consider the following one. S 1/ S 2 Left Middle Right Up 1,0 1,2 0,1 Down 0,3 0,1 2,0 Idea: We drop strictly dominated strategies from the payo¤ matrix. A strategy s 0 i is strictly dominated by strategy s i if strategy s i yields a strictly larger payo¤ than strategy s 0 i for all possible choices of the opponents: u i ( s 0 i , s � i ) < u i ( s i , s � i ) 8 s � i 2 S � i . Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 1. Game Theory I 8 / 27
Fundamental Concepts We apply the method of “iterated elimination of strictly dominated strategies” to the game on the previous slide. “Right” is strictly dominated by “Middle”. Thus, a rational S 2 will never choose “Right”. If S 1 knows that S 2 is rational, she can eliminate “Right” from the payo¤ matrix. In this case, “Down” is strictly dominated by “Up”. If S 2 knows that S 1 is rational and that S 1 knows that S 2 is rational, she can eliminate “Down” from the payo¤ matrix. In this case, “Left” is strictly dominated by “Middle”. Conclusion: The outcome of the game is “Up, Middle”. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 1. Game Theory I 9 / 27
Fundamental Concepts IESDS requires that the rationality of the players is common knowledge: All players are rational; All players know that all the players are rational; All the players know that all the players know that all the players are rational; asf. The order of elimination does not matter (you may try to show this as an exercise). Do not eliminate weakly dominated strategies (they are not ruled out by common knowledge of rationality). Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 1. Game Theory I 10 / 27
Nash Equilibrium The previous solution concepts are not helpful for most games of interest. Consider the following example. S 1/ S 2 Left Middle Right Up 3,3 2,4 3,2 Middle 4,2 6,6 0,8 Down 2,3 8,0 4,4 In this game, there are neither dominant, nor strictly dominated strategies. We therefore search for the best responses . The strategy s i is a best response to s � i if u i ( s i , s � i ) � u i ( s 0 i , s � i ) 8 s 0 i 2 S i . Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 1. Game Theory I 11 / 27
Nash Equilibrium Mark for each strategy of S 2 the best response of S 1 by underlining the corresponding payo¤ of player S 1. Then mark for each strategy of S 1 the best response of S 2 by underlining the corresponding payo¤ of player S 2. If there are more than one maximal payo¤s in a column (row), mark all of them. We have a Nash equilibrium if there is a cell where both payo¤s are underlined. The corresponding strategies are “mutual best responses”. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 1. Game Theory I 12 / 27
Nash Equilibrium The strategy pro…le ( s � 1 , ..., s � n ) is a Nash equilibrium if, for each player i and every strategy s i of player i , s � i is a best response to s � � i , i.e. for every player i and s i u i ( s � i , s � � i ) � u i ( s i , s � � i ) 8 s i 2 S i . Note that only a weak inequality is required. In a Nash equilibrium, no player has an incentive to unilaterally deviate (i.e., deviate from the predicted outcome when the other players play according to the prediction). Note: When he chooses his strategy, he does not know his opponent’s strategy. He only plays a best response against the opponent’s expected behavior. In an equilibrium, expectations must be correct. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 1. Game Theory I 13 / 27
Nash Equilibrium We can interpret the Nash equilibrium as follows: Suppose the players meet before the game and discuss what strategies they should play. They can only agree on strategy pro…les that are self-enforcing : each player’s strategy must be a best-response to the strategies of the other players. If players do not meet before playing the game, then in equilibrium they must have beliefs about the opponents’ strategy that are mutually consistent . Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 1. Game Theory I 14 / 27
Nash Equilibrium Why should we expect that a Nash equilibrium is played? Rational analysis of the game (which outcome makes sense?). Recommendation of a third party (examples: tra¢c signals, social norms). Communication before the game is played. Learning through trial and error. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 1. Game Theory I 15 / 27
Nash Equilibrium Multiple equilibria In many games, there are several Nash equilibria. Consider, for example, the following game (which is called “the battle of the sexes”): “She”/“He” ballet boxing ballet 2,1 0,0 boxing 0,0 1,2 Show that both “ballet, ballet” and “boxing, boxing” are Nash equilibria! Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 1. Game Theory I 16 / 27
Nash Equilibrium When there are several Nash equilibria, it is not clear which one will be played (or whether a Nash equilibrium will be played at all). There are some concepts that try to solve the problem. Focal Points (Schelling) Pareto optimality Coordination by communication Equilibrium re…nements Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 1. Game Theory I 17 / 27
Nash Equilibrium Focal points Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 1. Game Theory I 18 / 27
Nash Equilibrium Focal points are properties of the environment that can be used to anticipate the opponents’ behavior. They demonstrate that people can coordinate without communication. Schelling’s example goes as follows: Tomorrow you have to meet a stranger in New York. Where and when do you meet him? Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 1. Game Theory I 19 / 27
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