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1 Chapter 14 – Game Theory 14.1 Nash Equilibrium 14.2 Repeated Prisoners’ Dilemma 14.3 Sequential-Move Games and Strategic Moves

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2 Game Theory and Life You are on a first date with the love of your dreams. You can propose 2 activities: 1)Safe activity (Coffee) 2)Exciting Activity (Waterpark) Your date could either want a safe activity or an exciting activity. There are different results if your ideas match up or clash:

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3 Mr/Miss Right Mr/Miss Right You Chapter Fourteen First Date Game What is the outcome of this game? Payoff format is (Left, Top)

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4 Chapter Fourteen Game Theory Components Players: agents participating in the game (You and Your Date Strategies: Actions that each player may take under any possible circumstance (Coffee, Waterpark) Outcomes: The various possible results of the game (four, each represented by one cell of the payoff matrix) Payoffs: The benefit that each player gets from each possible outcome of the game (the profits entered in each cell of the payoff matrix)

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5 Best Responses In all game theory games, players choose strategies without knowing with certainty what the opposing player will do. Players construct BEST RESPONSES -optimal actions given all possible actions of other players

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6 Mr/Miss Right Mr/Miss Right You Chapter Fourteen First Date Game Best Responses If you know your date will pick coffee, you should pick coffee, since 10 > -5 If you know your date will pick waterpark, you should pick waterpark, since 20 > 0

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7 Mr/Miss Right Mr/Miss Right You Chapter Fourteen First Date Game Best Responses If your date knows you will pick coffee, they should pick coffee, since 10 > -5 If your date knows you will pick waterpark, they should pick waterpark, since 20 > 0 Note that this game is SYMMETRICAL

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8 Chapter Fourteen Nash Equilibrium Definition: A Nash Equilibrium occurs when each player chooses a strategy that gives him/her the highest payoff, given the strategy chosen by the other player(s) in the game. ("rational self-interest") Nash Equilibria occur when best responses line up The Date Game: Nash equilibria: Each proposes coffee or each proposes waterpark. Definition: A Nash Equilibrium occurs when each player chooses a strategy that gives him/her the highest payoff, given the strategy chosen by the other player(s) in the game. ("rational self-interest") Nash Equilibria occur when best responses line up The Date Game: Nash equilibria: Each proposes coffee or each proposes waterpark.

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9 Game Theory A special kind of Best Response: Strategy that is best no matter what the other player does. DOMINANT STRATEGY

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10 Advertising A’s profit= $50 000 A’s loss = $25 000 A’s profit= $75 000 A’s profit = $10 000 B’s profit = $50 000 B’s profit = $75 000 B’s loss = $25 000 B’s profit = $10 000 Don’t advertise Advertise B’s STRATEGY A’s STRATEGY Don’t advertise Advertise

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11 Dominant Strategy A’s profit= $50 000 A’s loss = $25 000 A’s profit= $75 000 A’s profit = $10 000 B’s profit = $50 000 B’s profit = $75 000 B’s loss = $25 000 B’s profit = $10 000 Don’t advertise Advertise B’s dominant strategy is advertise A’s dominant strategy is advertise Don’t advertise Advertise

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12 Prisoner’s Dilemma This is an example of a prisoner’s dilemma type of game. –There is dominant strategy. –The dominant strategy does not result in the best outcome for either player. –It is hard to cooperate even when it would be beneficial for both players to do so –Cooperation between players is difficult to maintain because cooperation is individually irrational. eg., The dominant strategy: advertise

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13 Classic Prisoners’ Dilemma Rocky’s strategies ConfessDeny Ginger’s strategies Confess 5 years Prison 5 years Prison 7 years Prison Go free 1 year Prison 1 year Prison 7 years Prison Go free Deny Dominant strategy: confess, even though they would both be better off if they both kept their mouths shut.

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Dominant Strategy Equilibrium Definition: A Dominant Strategy Equilibrium occurs when each player uses a dominant strategy. Honda Toyota

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15 Chapter Fourteen Dominated Strategy Definition: A player has a dominated strategy when the player has another strategy that gives it a higher payoff no matter what the other player does. Honda Toyota

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16 Chapter Fourteen Dominant or Dominated Strategy Why look for dominant or dominated strategies? A dominant strategy equilibrium is particularly compelling as a "likely" outcome Similarly, because dominated strategies are unlikely to be played, these strategies can be eliminated from consideration in more complex games. This can make solving the game easier. Why look for dominant or dominated strategies? A dominant strategy equilibrium is particularly compelling as a "likely" outcome Similarly, because dominated strategies are unlikely to be played, these strategies can be eliminated from consideration in more complex games. This can make solving the game easier.

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17 Honda Dominated Strategy Toyota "Build Large" is dominated for each player By eliminating the dominated strategies, we can reduce the game matrix.

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18 Chapter Fourteen Finding Nash Equilibrium Cases 1)Nash Equilibrium where Dominant Strategies overlap 2)Nash Equilibrium with one Dominant Strategy 3)Nash Equilibrium by eliminating Dominated Strategy 4)Nash Equilibrium through Best Responses 1)Nash Equilibrium where Dominant Strategies overlap 2)Nash Equilibrium with one Dominant Strategy 3)Nash Equilibrium by eliminating Dominated Strategy 4)Nash Equilibrium through Best Responses

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19 Student Nash Equilibrium – Dominant Overlap Professor

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20 Student Nash Equilibrium – One Dominant Professor

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21 Student Nash Equilibrium – Eliminate Dominated Professor

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22 Student Nash Equilibrium – Best Responses Professor

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23 Nash Equilibrium However it is found, a Nash Equilibrium ALWAYS occurs where Best Responses line up If Multiple Nash Equilibria exist, we can’t conclude WHICH outcome will occur, only the possible outcomes that can occur Also, it is often APPEARS that no Nash Equilibria exist:

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24 Barney No Nash Equilibrium Fred

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25 Mixed Strategies Pure Strategy – A specific choice of a strategy from the player’s possible strategies in a game. (ie: Rock) Mixed Strategy – A choice among two or more pure strategies according to pre-specified probabilities. (ie: Rock, Paper or Scissors each 1/3 rd of the time) If Pure Strategies can’t produce a Nash Equilibrium, Mixed Strategies can: If both players randomize each choice 1/3 rd of the time, nether have an incentive to deviate.

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Non-cooperative Game Theory Notes by Alberto Bressan.

Non-cooperative Game Theory Notes by Alberto Bressan.

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