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Game theory

Game theory . By Joe Taylor. Outline. Game Theory definition and basic facts How to play a Game The Nash Equilibrium Defined Mixed Strategy Equilibrium Defined with examples. Game Theory Basics. In all societies, there is interaction between two separate parties

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Game theory

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  1. Game theory By Joe Taylor

  2. Outline • Game Theory definition and basic facts • How to play a Game • The Nash Equilibrium Defined • Mixed Strategy Equilibrium Defined with examples

  3. Game Theory Basics • In all societies, there is interaction between two separate parties • These interactions can be cooperative, like business partners working on a project • Or they can be competitive, like multiple firms fighting for a market share or several coworkers fighting for a promotion • In either case, interdependence applies

  4. Game Theory Defined • There are 5 elements of a game: • A list of players • A complete description of what the players can do(strategies) • A description of what the players know (payoffs) • A specification of how the players’ actions lead to outcomes (information) • A specification of the players’ preferences over outcomes (objective)

  5. Strategy Behind the Game • A strategy is a complete contingent plan for a player • The goal of the game is to maximize your own payoff. There is no “feeling sorry” for the other player because they may not get a high payoff • Try to choose the best strategy profile given you and the other player(s) known all the payoffs for each player for each strategy combination

  6. Types of Games Extended Form Normal Form 2 1 C D

  7. How to Play • When playing a game, each player attempts to best guess what the other player will choose, and through that they attempt to maximize their own profit. 2 1 C D A B

  8. Dominance • An important concept in Game Theory is Dominance. • A strategy “B’ is dominated if there is another strategy “A” that will provide a higher payoff for the player, regardless of the other player’s choices • Iterative Dominance

  9. Dominance Example 2 1 L C R U M D

  10. Dominance Example 2 1 L C R U M D

  11. Best Response • To maximize your own profit, the objective is to find a best response to any of the other players actions • Do this by seeing if any strategies are dominated by the other choice In this game, we can see that it does not matter if player 2 chooses C or D, because player 1 will always be able to maximize payoff by choosing B. Therefore, we can say that strategy A is dominated by B Once A is gone, player 2 chooses his best payoff, which is D. Strategy D is the Best Response to Player 1’s actions. 2 1 C D A B

  12. Nash Equilibrium • a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his or her own strategy unilaterally. • So, each player lands on a strategy combination where they have no incentive to change their position in the future, therefore each player is choosing the best response to the other player’s choice.

  13. Nash Equilibrium Example Can you identify the best responses for each strategy for each player? 2 1 a b c w x y

  14. Nash Equilibrium Example • Here, (w,b) and (y,c) are Nash Equilibria 2 1 a b c w x y

  15. Mixed Strategy Equilibria • Very similar to a pure strategy Nash Equilibrium, but in this case the player chooses percentages for each strategy that will give him the highest payoff, on average. • Each player’s total strategies will add up to 1, so they will have a probability of p for strategy A and (1-p) for strategy B, given there are only two strategies.

  16. Football Opera Game • In this game, Alice wants to go to the opera, but Bob wants to go to the game. However, both would be better off being together rather than alone. p 1-p Bob Alice opera football opera football q 1-q

  17. Football Opera Game Cont. • To find a mixed strategy equilibrium, we must find a probability p for which Bob is indifferent between opera and football, and do the same for Alice: q 1-q Bob Alice opera football opera football p 1-p

  18. Graph of Best Responses q 1 1/5 0 0 4/5 1 p So there are 3 Nash Equilibria in this game, one where both choose opera with probability 1, one where both choose opera with probability 0 (both choose football), and one where Alice chooses opera with probability 4/5 and Bob chooses opera with probability 1/5.

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