More Experiments on Game Theory Syngjoo Choi Spring 2010 Experimental Economics (ECON3020) Game theory 2 Spring2010 1/28 Playing Unpredictably In many situations there is a strategic advantage associated with being unpredictable. Experimental Economics (ECON3020) Game theory 2 Spring2010 2/28 Playing Unpredictably In many situations there is a strategic advantage associated with being unpredictable. Experimental Economics (ECON3020) Game theory 2 Spring2010 2/28 Mixed (or Randomized) Strategy In the penalty kick, Left (L) and Right (R) are the pure strategies of the kicker and the goalkeeper. A mixed strategy refers to a probabilistic mixture over pure strategies in which no single pure strategy is played all the time. e.g., kicking left with half of the time and right with the other half of the time. A penalty kick in a soccer game is one example of games of pure con‡icit, essentially called zero-sum games in which one player’s winning is the other’sloss. Experimental Economics (ECON3020) Game theory 2 Spring2010 3/28 The use of pure strategies will result in a loss. What about each player playing heads half of the time? Matching Pennies Games In a matching pennies game, each player uncovers a penny showing either heads or tails. One player takes both coins if the pennies match; otherwise, the other takes both coins. Left Right Top 1, 1, 1 1 Bottom 1, 1, 1 1 Does there exist a Nash equilibrium involving the use of pure strategies? Experimental Economics (ECON3020) Game theory 2 Spring2010 4/28 Matching Pennies Games In a matching pennies game, each player uncovers a penny showing either heads or tails. One player takes both coins if the pennies match; otherwise, the other takes both coins. Left Right Top 1, 1, 1 1 Bottom 1, 1, 1 1 Does there exist a Nash equilibrium involving the use of pure strategies? The use of pure strategies will result in a loss. What about each player playing heads half of the time? Experimental Economics (ECON3020) Game theory 2 Spring2010 4/28 Mixed-Strategy Equilibrium In the matching pennies game, both players playing heads with half probability is the unique Nash equilibrium. On one hand, this strategy is used to not be predictable. On the other hand, given the other’sstrategy, there is no (either pure or mixed) strategy giving higher payo¤s than playing head with half probability. One way of seeing this equilibrium is drawing each player’sbest response: Suppose player 2 chooses (Right) with probability p. Then, player 1’spayo¤ di¤erence betwee (Top) and (Bottom) is (1 2p) ( 1 + 2p) = 2 (1 2p). If p< 1/2, it is optimal for player 1 to choose (Top) with probability 1. Similarly for other cases. Experimental Economics (ECON3020) Game theory 2 Spring2010 5/28 Experimental Economics (ECON3020) Game theory 2 Spring2010 6/28 Classroom Experiment Now let’splay a game in a classroom experiment. Go to http://veconlab.econ.virginia.edu/login.htm. Session name is sjc8. Experimental Economics (ECON3020) Game theory 2 Spring2010 7/28 Does there exist a pure-strategy Nash equilibrium? (Intuitive guess) Given the change of payo¤s, who should still play each pure strategy with half probability in an equilibrium? How did you play? Asymmetric Matching Pennies Game The game you just played is an asymmetric matching pennies game: Left Right Top 360, 36, 36 72 Bottom 36, 72, 72 36 Experimental Economics (ECON3020) Game theory 2 Spring2010 8/28 (Intuitive guess) Given the change of payo¤s, who should still play each pure strategy with half probability in an equilibrium? How did you play? Asymmetric Matching Pennies Game The game you just played is an asymmetric matching pennies game: Left Right Top 360, 36, 36 72 Bottom 36, 72, 72 36 Does there exist a pure-strategy Nash equilibrium? Experimental Economics (ECON3020) Game theory 2 Spring2010 8/28 Asymmetric Matching Pennies Game The game you just played is an asymmetric matching pennies game: Left Right Top 360, 36, 36 72 Bottom 36, 72, 72 36 Does there exist a pure-strategy Nash equilibrium? (Intuitive guess) Given the change of payo¤s, who should still play each pure strategy with half probability in an equilibrium? How did you play? Experimental Economics (ECON3020) Game theory 2 Spring2010 8/28 Suppose player 2 chooses (Right) with probability p. Player 1’sexpected payo¤ for (Top) = 360 (1 p) + 36p = 360 324p Player 1’sexpected payo¤ for (Bottom) = 36 (1 p) + 72p = 36 + 36p. Thus, the payo¤ di¤erence between (Top) and (Bottom) is 324 360p. If p < 0.9, player 1 will choose (Top) with probability 1. What about player 1’sstrategy? Suppose player 1 chooses (Bottom) with probability q. The payo¤ di¤erence for player 2 between (Right) and (Left) is (72 36q) (36 + 36q) = 36 (1 2q). Player 1 should still choose (Top) and (Bottom) with equal probability!!! Instead, player 2 increases the probability of choosing (Right) up to 0.9 probability. Mixed-strategy Equilibrium Let’sderive the mixed-strategy equilibrium: Experimental Economics (ECON3020) Game theory 2 Spring2010 9/28 Player 1’sexpected payo¤ for (Top) = 360 (1 p) + 36p = 360 324p Player 1’sexpected payo¤ for (Bottom) = 36 (1 p) + 72p = 36 + 36p. Thus, the payo¤ di¤erence between (Top) and (Bottom) is 324 360p. If p < 0.9, player 1 will choose (Top) with probability 1. What about player 1’sstrategy? Suppose player 1 chooses (Bottom) with probability q. The payo¤ di¤erence for player 2 between (Right) and (Left) is (72 36q) (36 + 36q) = 36 (1 2q). Player 1 should still choose (Top) and (Bottom) with equal probability!!! Instead, player 2 increases the probability of choosing (Right) up to 0.9 probability. Mixed-strategy Equilibrium Let’sderive the mixed-strategy equilibrium: Suppose player 2 chooses (Right) with probability p. Experimental Economics (ECON3020) Game theory 2 Spring2010 9/28 Player 1’sexpected payo¤ for (Bottom) = 36 (1 p) + 72p = 36 + 36p. Thus, the payo¤ di¤erence between (Top) and (Bottom) is 324 360p. If p < 0.9, player 1 will choose (Top) with probability 1. What about player 1’sstrategy? Suppose player 1 chooses (Bottom) with probability q. The payo¤ di¤erence for player 2 between (Right) and (Left) is (72 36q) (36 + 36q) = 36 (1 2q). Player 1 should still choose (Top) and (Bottom) with equal probability!!! Instead, player 2 increases the probability of choosing (Right) up to 0.9 probability. Mixed-strategy Equilibrium Let’sderive the mixed-strategy equilibrium: Suppose player 2 chooses (Right) with probability p. Player 1’sexpected payo¤ for (Top) = 360 (1 p) + 36p = 360 324p Experimental Economics (ECON3020) Game theory 2 Spring2010 9/28 Thus, the payo¤ di¤erence between (Top) and (Bottom) is 324 360p. If p < 0.9, player 1 will choose (Top) with probability 1. What about player 1’sstrategy? Suppose player 1 chooses (Bottom) with probability q. The payo¤ di¤erence for player 2 between (Right) and (Left) is (72 36q) (36 + 36q) = 36 (1 2q). Player 1 should still choose (Top) and (Bottom) with equal probability!!! Instead, player 2 increases the probability of choosing (Right) up to 0.9 probability. Mixed-strategy Equilibrium Let’sderive the mixed-strategy equilibrium: Suppose player 2 chooses (Right) with probability p. Player 1’sexpected payo¤ for (Top) = 360 (1 p) + 36p = 360 324p Player 1’sexpected payo¤ for (Bottom) = 36 (1 p) + 72p = 36 + 36p. Experimental Economics (ECON3020) Game theory 2 Spring2010 9/28 What about player 1’sstrategy? Suppose player 1 chooses (Bottom) with probability q. The payo¤ di¤erence for player 2 between (Right) and (Left) is (72 36q) (36 + 36q) = 36 (1 2q). Player 1 should still choose (Top) and (Bottom) with equal probability!!! Instead, player 2 increases the probability of choosing (Right) up to 0.9 probability. Mixed-strategy Equilibrium Let’sderive the mixed-strategy equilibrium: Suppose player 2 chooses (Right) with probability p. Player 1’sexpected payo¤ for (Top) = 360 (1 p) + 36p = 360 324p Player 1’sexpected payo¤ for (Bottom) = 36 (1 p) + 72p = 36 + 36p. Thus, the payo¤ di¤erence between (Top) and (Bottom) is 324 360p. If p < 0.9, player 1 will choose (Top) with probability 1. Experimental Economics (ECON3020) Game theory 2 Spring2010 9/28 The payo¤ di¤erence for player 2 between (Right) and (Left) is (72 36q) (36 + 36q) = 36 (1 2q). Player 1 should still choose (Top) and (Bottom) with equal probability!!! Instead, player 2 increases the probability of choosing (Right) up to 0.9 probability. Mixed-strategy Equilibrium Let’sderive the mixed-strategy equilibrium: Suppose player 2 chooses (Right) with probability p. Player 1’sexpected payo¤ for (Top) = 360 (1 p) + 36p = 360 324p Player 1’sexpected payo¤ for (Bottom) = 36 (1 p) + 72p = 36 + 36p. Thus, the payo¤ di¤erence between (Top) and (Bottom) is 324 360p. If p < 0.9, player 1 will choose (Top) with probability 1. What about player 1’sstrategy? Suppose player 1 chooses (Bottom) with probability q. Experimental Economics (ECON3020) Game theory 2 Spring2010 9/28 Player 1 should still choose (Top) and (Bottom) with equal probability!!! Instead, player 2 increases the probability of choosing (Right) up to 0.9 probability. Mixed-strategy Equilibrium Let’sderive the mixed-strategy equilibrium: Suppose player 2 chooses (Right) with probability p. Player 1’sexpected payo¤ for (Top) = 360 (1 p) + 36p = 360 324p Player 1’sexpected payo¤ for (Bottom) = 36 (1 p) + 72p = 36 + 36p. Thus, the payo¤ di¤erence between (Top) and (Bottom) is 324 360p. If p < 0.9, player 1 will choose (Top) with probability 1. What about player 1’sstrategy? Suppose player 1 chooses (Bottom) with probability q.