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19. Game Theory

# 19. Game Theory - Game Theory and Applications Professor...

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Game Theory and Applications Professor John Diamond ECON 370: Microeconomic Theory Lecture 19

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2 Game Theory: Introduction Game theory A means of modeling strategic behavior Agents act to maximize own welfare Agents understand their actions affect actions of other agents
3 Game Theory: Applications Game theory has been applied to analyze Oligopolies Cartels: OPEC Tax competition across jurisdictions, countries Military strategies Externalities: using common resources like fishery

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4 Game Theory: Overview A game consists of a set of players (often two) a set of strategies for each player the payoffs to each player for all combinations of possible strategy choices by the players Simplest possible game Two-player game Each chooses between only two strategies
5 Two-Player Game: Example Players are A and B A has two strategies: “Up” and “Down” B has two strategies: “Left” and “Right” Payoff matrix - table showing payoffs to A and B for each of four possible strategy combinations Payoff: (A,B)

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6 L R U D (3,9) (0,0) (1,8) (2,1) Two-Player Game: Matrix Payoff matrix for game Player B Player A Player A’s payoff is shown first Player B’s payoff is shown second
7 If A plays U p and B plays R ight then A’s payoff is 1 B’s payoff is 8 Player B Player A L R U D (3,9) (0,0) ( 1 , 8 ) (2,1) Two-Player Game: Matrix

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8 L R U D (3,9) (0,0) (1,8) (2,1) Player B Player A A play is a pair such as (U,R) where 1st element is strategy chosen by A 2nd element is strategy chosen by B Two-Player Game: Matrix
9 Player B Player A If B plays Right, A will play Down (2 v. 1) So (U,R) is not a likely play Is (U,R) a likely play? L R U D (3,9) (0,0) ( 1 , 8 ) (2,1) Two-Player Game: Matrix

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10 Player B Player A If B plays Right then A’s best reply is Down If A plays Down then B’s best reply is Right So (D,R) is a likely play Is (D,R) a likely play? L R U D (3,9) (0,0) (1,8) ( 2 , 1 ) Two-Player Game: Matrix
11 Player B Player A If A plays Down then B’s best reply is Right So (D,L) is not a likely play L R U D (3,9) ( 0 , 0 ) (1,8) (2,1) Two-Player Game: Matrix Is (D,L) a likely play?

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12 Player B Player A If A plays Up then B’s best reply is Left If B plays Left then A’s best reply is Up So (U,L) is a likely play Is (U,L) a likely play? L R U D ( 3 , 9 ) (0,0) (1,8) (2,1) Two-Player Game: Matrix
13 Nash Equilibrium: Introduction Nash equilibrium a play where each strategy is a best response to the other Can be multiple Nash equilibria Example has two Nash equilibria (U,L) and (D,R)

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Nash Equilibrium: Matrix Player B Player A ( U , L ) and ( D , R ) are both Nash equilibria ( U , L ) is Pareto preferred to ( D , R ) Is (U,L) the only (likely) equilibrium? L
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19. Game Theory - Game Theory and Applications Professor...

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