GameTheory

GameTheory - All games in the game theory framework have...

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All games in the game theory framework have four common features: - Rules - Strategies - Payoffs - Outcome Rules – the rules of the game define how the game will be played. It usually consists of a time frame, a place where the game is played, and any other information that is necessary for the game. Strategies – all possible actions that a player (or agent) can take in a game Payoffs – payoffs for each player and each strategy are given in a payoff matrix. A payoff matrix is a table that shows the payoffs for every possible action by each for every possible action by another player. Outcome – An outcome for a game happens when, given the strategies of all other players, a player has no incentive to deviate from a strategy. When this is true for all players in the game, this is referred to as a Nash equilibrium. A Nash Equilibrium is a strategy for each player such that there is no incentive for any player to change strategies. If there are two players A and B , a Nash Equilibrium occurs when player A makes his best possible choice given the choices of player B , and at the same time player B makes the best possible choice, given the choices for player A . Let’s consider a very simply game It is the day before an exam. You and your roommate are both in the same class and face the exam the next day. You both have two decision, study or not study. Let’s identify the features of the game: - Rules – two players make one decision tonight, choosing their action at the same time. Player can talk. - Strategies – Study or not study - Payoffs – the blue triangles represent the payoff for you, the yellow represent your roommates. No Study Study Study No Study Roommate You A A B D F F D B
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- Outcome The way we are going to find the Nash EQ outcome is to pick a particular strategy for both people (say no study – no study) and see if you or your roommate has an incentive to deviate from that choice. Let’s look at your decision. If you choice to study rather than not study, you can improve your grade from an F to a B. Thus No study, no study is not an equilibrium. Let’s look at the study, no study box. Does any player have an incentive to move to the other strategy. If you change strategies and choose not to study, your grade falls – so you cannot be made better off. However, if your roommate choices to study, he can improve his grade from a D to an A, so he would choose to study. Thus study, no study is not an EQ. Now let’s see if study, study is a Nash EQ. If you choose to not study, while your roommate studies, you will get a D, which is worse than an A, so you don’t do that. Your roommate has the same decision, and chooses to study. Since both you and your roommate have no incentive to deviate the strategy from study-study, study, study is a nash eq. Now it is possible to have more than one Nash Equilibrium in a game – however there is only
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This note was uploaded on 02/06/2012 for the course ECON 251 taught by Professor Blanchard during the Summer '08 term at Purdue.

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GameTheory - All games in the game theory framework have...

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