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Lecture_25_Game_Applications

Lecture_25_Game_Applications - Lecture 25 Game Theory...

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Lecture 25: Game Theory Applications c ° 2008 Je/rey A. Miron Outline 1. Introduction 2. Best Response Curves 3. Mixed Strategies 4. Games of Coordination 5. Games of Competition 6. Games of Coexistence 7. Games of Commitment 8. Bargaining 9. The Ultimatum Game 1 Introduction The °rst lecture on game theory described a number of important concepts and illustrated these with a few examples. We now examine four important issues in game theory and see how they work in various strategic interactions. These four issues are cooperation, competition, coexistence, and commitment. As a °rst step, we will develop two tools for analyzing games. 1

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2 Best Response Curves Consider a two-person game, and put yourself in the position of one of the players. For any choice the other player can make, your best response is the choice that maximizes your payo/. If several choices maximize your payo/, then your best response is the set of all such choices. We can illustrate this concept in the game below: 2
Table: A Simple Game Row Column Top Bottom Left Right 2, 1 0, 0 0, 0 1, 2 For this game we can write down the best response functions: Table: Best Response Functions Column±s choice: Left Right Row±s best response: Top Bottom Row±s choice: Top Bottom Column±s best response: Left Right Now consider a general two-person game in which Row has choices r 1 ; :::; r R and Column has choices c 1 ; :::; c C : For each choice r that Row makes, let b c ( r ) be the best response for Column, and for each choice c that Column makes, let b r ( c ) be the best response for Row. Then a Nash equilibrium is a pair of strategies ( r ° ; c ° ) such that c ° = b c ( r ° ) r ° = b r ( c ° ) 3

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Thus, the choices are mutually consistent. If Row expects Column to play Left, then Row will choose Top. If Column expects Row to play Top, Column will want to play Left. The beliefs and actions of the players are mutually consistent in a NE. In some cases one of the players may be indi/erent between several responses. That is why we only require the c ° be one of Column±s best responses, and similarly for Row. If a unique best response exists for each choice, then the best response curves can be represented as best response functions. This way of looking at NE makes clear it is a generalization of the Cournot case. There, the choice variable is the amount of output, which is a continuous variable. The Cournot equilbrium has the property that each °rm chooses its pro°t- maximizing output, given the choice of the other °rm. The Bertrand equilibrium is also a NE, but in prices rather than quantities. This discussion shows that best response curves generalize our earlier models and are a simple way to solve for NE. 3 Mixed Strategies Let±s examine the use of best response functions to analyze the game below: 4