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Pamela Schmitt United States Naval Academy

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Game Theory REVIEW payoff matrix REVIEW definition and determination of dominant strategies NASH EQUILIBRIA with dominant strategies NASH EQUILIBRIA without dominant strategies (cover and underline best response method) Applications: Oligopolies, Prisoner dilemma (suboptimal outcomes), Battle of the Sexes, Chicken, Hotelling's Beach

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The Payoff Matrix: Dominant Strategy Equilibrium LeftRight Top4,7 5, 8 Bottom 2, 13, 6 Danny Lily

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The “row” player Lily Lily has two strategies “Top” and “Bottom” LeftRight Top 4, 7 5, 8 Bottom 2, 13, 6 Danny “row” Lily

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The “column” player Danny Danny has two strategies “Left” and “Right” LeftRight Top 4, 7 5, 8 Bottom 2, 13, 6 “column” Danny Lily

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The Payoff Matrix The first number in each cell is the payoff the row player (Lily) receives if both players choose the action that leads to that cell. Similarly, the second number in each cell is the payoff the column player (Danny) receives.

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If Lily chooses “Top”: Lily earns 4 if Danny chooses “Left” and 5 if Danny chooses “Right” LeftRight Top 4, 7 5, 8 Bottom 2, 13, 6 Danny Lily

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If Lily chooses “Bottom”: Lily earns 2 if Danny chooses “Left” and 3 if Danny chooses “Right” LeftRight Top 4, 7 5, 8 Bottom 2, 13, 6 Danny Lily

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Dominant strategies A dominant strategy is the best strategy regardless of what the other player chooses. If both players have a dominant strategy, the outcome is a dominant strategy equilibrium. All dominant strategy equilibrium are Nash Equilibrium.

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Lily has a dominant strategy: choosing Top always leads to a higher payoff regardless of what Danny chooses: 4>2 and 5>3 LeftRight Top 4, 7 5, 8 Bottom 2, 13, 6 Danny Lily

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Dominant strategies But not all Nash Equilibrium are dominant strategy equilibrium. A Nash Equilibrium is the outcome in which neither player has a desire to choose a different strategy given the choice of the other player. (mutual best responses)

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Let’s try it with AP questions

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Note: Neither has a dominant strategy.

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But, we can now answer (a): if Red Shop chooses “South” Blue Mart chooses “North” (1 pt). b/c 4000>1000 (1 pt.)

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And for (b): “South” is not a dominant strategy for Red Shop chooses (1 pt.) If Blue Mart chooses south, Red Shop is better off choosing north. (Red Shop’s best response depends on Blue Mart’s move.) (1 pt.)

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Part (c): the highest combined payoff are at (S,N): (5,000 +4,000) > 6,5000 > 2,7000,> 2,500. (1 pt.) Stating that Red Shop chooses south and Blue Mart chooses north

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Part (d) redraw such that +$2,000 are added to “South” payoffs NorthSouth North 900, ,5500 South 7000, , 3000 Blue Mart Red Shop

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When teaching game theory, I prefer to have students start with their own intuition. Ariel Rubinstein has an online resource that allows teachers to use simple games (and more complex ones!) to build this intuition. This is following Rubinstein, A. (1999). “Experience from a Course in Game Theory: Pre- and Post-class Problem Sets as a Didactic Device” Games and Economic Behavior 28, 155 – 170.

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Basic Attacker/Defender GameTwo Person/Binary Decision Game of Strategy Multi-Site Attacker/Defender Game New! Defaults Implement a Simple Profiling Game CentipedeAlternating Two-person "Pass or Take" Game Coordination Minimum-Effort Game, with Incentive Pay Options Guessing GameWith Incentive to Guess Others' Decisions 2x2 Matrix GamePrisoner's Dilemma, Battle of Sexes, etc. Asymmetric Matrix Game "Large" Setup, e.g. Coordination with 7 Effort Choices Symmetric Matrix GameNxN Matrix Game with Symmetric Payoffs Security Coordination GameCoordination of Security Investment Decisions Traveler's DilemmaSocial Dilemma with No Dominant Strategy 2-Stage GameGeneric Two-Stage Extensive-Form Game View ResultsView Results of Any Prior Setup Vecon Lab Games

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