EC487 Advanced Microeconomics, Part I: Lecture 6 Leonardo Felli 32L.LG.04 3 November, 2017
Game Theory ◮ It is the analysis of the strategic interaction among agents. ◮ This is a situation in which each agent when deciding how to behave explicitly takes into account the decision of the other agents that interact with him. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 2 / 54
Example: Entry Game ◮ Two individuals have to decide whether to sell newspapers at a given exit of the underground. ◮ They take this decision without observing the decision taken by the other individual. ◮ If only one individual decides to locate herself at the exit she will make the highest level of profits since she will serve all clients. Let this profit be £ 300. ◮ If both individuals decide to locate themselves at the exit then clients are equally shared (we assume newspaper prices are pre-set). Each individual’s profit is £ 150. ◮ Finally if an individual does not locate herself at the exit than she makes zero profits. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 3 / 54
Example (cont’d) ◮ We can describe the situation with the following table: 1 \ 2 E NE E 150 , 150 300 , 0 NE 0 , 300 0 , 0 ◮ Rows denote individual 1’s decisions. ◮ Columns denote individual 2’s decisions. ◮ The first number of each ordered pair denotes individual 1’s profit, while the second number denotes individual 2’s profit. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 4 / 54
Example (cont’d) ◮ Notice that predicting the outcome of this situation is fairly easy provided that we assume that both individuals wants to maximize profits , in other words they are rational . ◮ The predicted outcome is that both individuals locate themselves at the exit ( E , E ). 1 \ 2 E NE E 150 , 150 300 , 0 NE 0 , 300 0 , 0 ◮ Notice that this conclusion can be reached without requiring each individual to make a prediction on the behavior of the other individual. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 5 / 54
Battle of Sexes ◮ This is not true in general. ◮ Consider for example the following situation known as battle of sexes : 1 \ 2 B S B 1 , 2 0 , 0 S 0 , 0 2 , 1 ◮ In this case each individual needs to make a prediction on the behavior of the other individual. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 6 / 54
Coordination Game ◮ Consider the following simple coordination game (no conflict of interest). ◮ There is still a need for predictions: 1 \ 2 M C M 2 , 2 0 , 0 C 0 , 0 1 , 1 ◮ Notice that we will be more confident in our prediction if the individuals involved encounter this situation more than once. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 7 / 54
Jargon and Definitions ◮ The strategic situations we described above are known as games. ◮ A simple static game or game in normal (strategic) form (no time dimension) comprises three elements: 1. Set of players , economic agents: N = { 1 , . . . , I } 2. For each player i ∈ N an action space, or a pure strategy space denoted A i . This is the set of choices available to each player: A 1 = { locate at the exit, do not locate at the exit } . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 8 / 54
Definitions ◮ Denote: a i ∈ A i player i ’s strategy choice; ◮ Then a − i = ( a 1 , . . . , a i − 1 , a i +1 , . . . , a I ) is the strategy profile of every player but player i . ◮ Therefore a = ( a i , a − i ) ∈ A 1 × . . . × A I = A . ◮ Finite games are games with finite strategy spaces (a finite number of strategies). Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 9 / 54
Definitions (cont’d) 3. Finally define for each player i ∈ N a payoff function associated with his strategy choice a i and the other players’ strategy choice a − i : u i ( a 1 , . . . , a I ) = u i ( a i , a − i ) = u i ( a ) . ◮ The payoffs u i ( · ) is taken to be the utility representation of player i ’s preferences. ◮ The objective of game theoretic analysis is to give predictions on the behavior of agents in strategic situations. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 10 / 54
Rationality: ◮ What assumptions do we need on the players’ behavior to deliver these predictions? ◮ First assumption rationality (maximization of utility or payoff). ◮ In our example above rationality and knowledge of own payoff is enough to deliver a prediction: 1 \ 2 E NE E 150 , 150 300 , 0 NE 0 , 300 0 , 0 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 11 / 54
Prisoners’ dilemma ◮ An other classic example of a situation in which rationality and knowledge of own payoff is enough to deliver a prediction is the the prisoners’ dilemma game . ◮ This is characterized by the following normal form: 1 \ 2 C NC C 0 , 0 4 , − 1 NC − 1 , 4 3 , 3 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 12 / 54
Prisoners’ dilemma (cont’d) ◮ The three elements of the game are: ◮ N = { 1 , 2 } , ◮ A i = { C , NC } , ◮ u 1 ( C , C ) = u 2 ( C , C ) = 0, u 1 ( NC , C ) = u 2 ( C , NC ) = − 1, u 1 ( C , NC ) = u 2 ( NC , C ) = 4, u 1 ( NC , NC ) = u 2 ( NC , NC ) = 3. 1 \ 2 C NC C 0 , 0 4 , − 1 NC − 1 , 4 3 , 3 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 13 / 54
Prisoners’ dilemma (cont’d) Consider: 1 \ 2 C NC C 0 , 0 4 , − 1 NC − 1 , 4 3 , 3 ◮ Each player will choose the strategy C independently of the action chosen by the other player. ◮ The predicted outcome is therefore ( C , C ). This is clearly the inefficient outcome, it is Pareto dominated by ( NC , NC ). ◮ The only information needed to make a prediction is the fact that players are rational and they know their own payoffs. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 14 / 54
Knowledge of Rationality ◮ Consider now the following modification of the previous game: 1 \ 2 L C R T 0 , 0 4 , − 1 1 , − 1 M − 1 , 4 3 , 3 3 , 2 B − 1 , 2 0 , 1 4 , 1 ◮ In this case we need some extra assumptions to make a prediction on the outcome of the game. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 15 / 54
Knowledge of Rationality (cont’d) ◮ Indeed: 1 \ 2 L C R T 0 , 0 4 , − 1 1 , − 1 M − 1 , 4 3 , 3 3 , 2 B − 1 , 2 0 , 1 4 , 1 ◮ L dominates C and R for player 2; ◮ if player 1 knows that player 2 is rational then he will focus only on the first column; Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 16 / 54
Knowledge of Rationality (cont’d) ◮ Therefore: 1 \ 2 L T 0 , 0 M − 1 , 4 B − 1 , 2 ◮ In the first column T dominates M and B . ◮ The prediction is therefore ( T , L ). Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 17 / 54
Relevant Assumptions: ◮ The information needed to make a prediction is then: ◮ both players are rational; ◮ both players know their own and the other player’s payoff; ◮ player 1 knows that player 2 is rational. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 18 / 54
Common Knowledge of Rationality ◮ Consider now the following game: 1 \ 2 L C R T 1 , 0 1 , 2 0 , 1 B 0 , 3 0 , 1 2 , 0 ◮ Player 2 will never play R since R is a strictly dominated strategy and both players are rational and know each other payoffs. ◮ Since player 1 knows that player 2 is rational he also knows that R will never be played. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 19 / 54
Common Knowledge of Rationality (cont’d) ◮ Notice now that: 1 \ 2 L C T 1 , 0 1 , 2 B 0 , 3 0 , 1 ◮ For player 2 none of the remaining strategies is strictly dominated: ◮ if player 2 believes that player 1 will play B then 2 will choose L ; ◮ while if player 2 believes that player 1 will play T then 2 will choose C . Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 20 / 54
Common Knowledge of Rationality (cont’d) ◮ However if we now assume that: player 2 knows that player 1 knows that player 2 is rational ◮ then player 2 knows that player 1 realizes that he will never play R so for all intents and purposes the game is: 1 \ 2 L C T 1 , 0 1 , 2 B 0 , 3 0 , 1 ◮ In this new game player 1’s strategy B is strictly dominated, therefore 1 will never choose it. ◮ Therefore since player 2 knows that player 1 is rational the predicted outcome will be ( T , C ). Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 21 / 54
Common Knowledge of Rationality (cont’d) ◮ The assumptions needed to make this prediction are then: ◮ that both players are rational; ◮ that both players know their own and the other player’s payoff; ◮ that both players know that the other player is rational; ◮ that player 2 knows that player 1 knows that player 2 is rational. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 3 November, 2017 22 / 54
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