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The Watchman and the Thief - An Experiment on the Comparative Statics in Games By Dieter Balkenborg and Todd Kaplan

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Introduction Mixed strategy: Randomizing between different options Mixed strategy: Randomizing between different options Central role of mixed strategy in games Central role of mixed strategy in games Mixed strategy as a tool for preventing to be outguessed by the opponent Mixed strategy as a tool for preventing to be outguessed by the opponent Bart Simpson in episode 9F16 "Good ol' rock. Nuthin' beats that!“. Lisa: “Poor Bart. He is so predictable.” Bart Simpson in episode 9F16 "Good ol' rock. Nuthin' beats that!“. Lisa: “Poor Bart. He is so predictable.”

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von Neumann on game theory Jacob Bronowski recalls in “The Ascent of Man”, BBC, 1973, a conversation with John von Neumann: … I (JB) naturally said to him, since I am an enthusiastic chess player “You mean the theory of games like chess”. “No, no,” he (vN) said. “Chess is not a game. Chess is a well-defined form of computation. You may not be able to work out the answers, but in theory there must be a solution, a right procedure in every position. Now real games,” he said, “are not like that at all. Real life is not like that. Real life consists of bluffing, of little tactics of deception, of asking yourself what is the other man going to think I mean to do. And that is what games are about in my theory.”

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Mixed Strategies are a way of remaining unpredictable Penalty Kick: Kick R Dive L Dive R Kick L “R” = strong side of kicker Nash prediction for (Kicker, Goalie)=(41.99L+58.01R, 38.54L+61.46R) Actual Data =(42.31L+57.69R, 39.98L+60.02R) Palacios-Huerta (2003), Volij & Palacios-Huerta (2006)

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Q: How does increase in punishment affect equilibrium behaviour? WATCHMAN THIEF 7 WATCHMAN THIEF High Punishment: Low Punishment: 4 5 The Watchman and the Thief Watch Watch Rest Home Steal Rest Home Steal ? ? ??

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A: Watchman gets more lazy, theft remains as before. WATCHMAN THIEF Rest Watch Steal 7 WATCHMAN THIEF Rest Home Steal High Punishment: Low Punishment: 4 5 Watch Home

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The Watchman and the Thief Game Structures: The Watchman and the Thief 0 0 -P 1 0 -FR WATCHMAN THIEF Not watching Watching Not stealing Stealing P >0, punishment for the thief. F >0, fine for the watchman. R >0, reward for the watchman.

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Mixed Strategy Nash Equilibria Mixed Nash equilibria determined by the condition to make the opponent indifferent Mixed Nash equilibria determined by the condition to make the opponent indifferent How a player randomizes depends only on payoff of his opponent How a player randomizes depends only on payoff of his opponent Counterintuitive Implications Counterintuitive Implications Relevance for economics: Relevant economic implications (Dasgupta, Stiglitz 1980, Kaplan, Luski, Wettstein 2003) (More firms implies less innovation) Relevance for economics: Relevant economic implications (Dasgupta, Stiglitz 1980, Kaplan, Luski, Wettstein 2003) (More firms implies less innovation)

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Main design feature Population design (Nagel, Zamir 2001): Population design (Nagel, Zamir 2001): 10 subjects are row players (thieves) 10 subjects are row players (thieves) 10 are column players (watchmen) 10 are column players (watchmen) In each period every thief plays against every watchman and vice versa (10 plays) In each period every thief plays against every watchman and vice versa (10 plays) Placement random Placement random

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Questions Qualitative predictions of theory correct? Qualitative predictions of theory correct? “Own Payoff effect” to be expected. Will they be wielded out in market-like matching environment? “Own Payoff effect” to be expected. Will they be wielded out in market-like matching environment? Relevance of security strategies? Relevance of security strategies?

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Literature: Experiments on normal form games with a unique mixed equilibrium Atkinson and Suppes (1963)2x2, 0-sum O'Neill (1987)4x4, 0-sum Malawski (1989)2x2, 3x3 game Brown & Rosenthal (1990)4x4, 0-sum Rapoport & Boebel (1992)5x5 Mookerherjee & Sopher (1994)matching pennies Bloomfield (1995)2x2, r. dev, Ochs (1995) 2x2, compet., r.d., r.m. Mookerherjee & Sopher (1997)4x4 games, const sum Erev & Roth (1998)many data sets McKelvey, Palfrey & Weber (1999)2x2 Fang-Fang Tang (1999, 2001, 2003) 3x3 Rapoport & Almadoss (2000)investment game

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Literature (cont): Goree, Holt, Palfrey (2000) 2x2, 0-sum Bracht (2000)2x2, 0-sum Bracht & Ichimura (2002)2x2, 0-sum Binmore, Swierpinski & Proulx2x2, 3x3, 4x4, 0-sum Shachat (2002)4x4, 5x5 games, 0-sum Shachat & Walker 2x2 games Rosenthal, Shachat & Walker (2001)2x2 games, 0-sum Nagel & Zamir (2000)2x2 games Selten & Chmura (2005) 2x2 games Empirical: Walker and Wooders (2001)tennis Ciappori, Levitt, Groseclose (2003)soccer Palacios-Huerta (2003)soccer Volij & Palacios-Huerta (2006)soccer

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Outline Experimental Design Experimental Design Aggregate Results Aggregate Results Own payoff Effect Own payoff Effect Quantal Response and Risk Aversion Quantal Response and Risk Aversion The learning cycle The learning cycle Individual Behaviour Individual Behaviour Heterogeneity Heterogeneity Movers and Shakers Movers and Shakers Conclusions Conclusions

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Experimental Design: Computerized Experiment at FEELE (the Finance and Economics Experimental Laboratory at Exeter University). Computerized Experiment at FEELE (the Finance and Economics Experimental Laboratory at Exeter University). Programmed by Tim Miller in z-tree. Programmed by Tim Miller in z-tree. A session lasted approx. 1:30 h, on average a student earned £18. A session lasted approx. 1:30 h, on average a student earned £ subjects 120 subjects

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Written instructions, summary Written instructions, summary Test questions Test questions 2 trial rounds 2 trial rounds 50 paid rounds 50 paid rounds Questionnaire, payment Questionnaire, payment No “story” No “story” Either row- or column player throughout Either row- or column player throughout Each player makes 10 decisions per period Each player makes 10 decisions per period

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Six Sessions: 1. H1: High penalty 2. H2: High penalty, row interchanged 3. H3: High penalty, column interchanged 4. L4: Low penalty 5. L5: Low penalty, row interchanged 6. L6: Low penalty, column interchanged (Data adjusted here)

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Row players’ decision screen Screen shot:

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Column players’ decision screen

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Row players’ “waiting” screen Screen shot:

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Row players’ result screen Screen shot:

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Column players’ result screen Screen shot:

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Results 1, Watchmen (col) The average proportion of watching (R) The average proportion of watching (R) is lower in the high-punishment treatments. is lower in the high-punishment treatments. The observation is robust over time. The observation is robust over time. High Punishment: Too close to 50%? High Punishment: Too close to 50%? H1H2H3 L4L5L6 "1-50" 0.28 * * * "25-45" 0.28 * * *

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Results 2, Thieves (row) average proportion of stealing (B) (Own-payoff effect): In the high-penalty sessions the average proportion of stealing is 10-15% below equilibrium while it is close to equilibrium in the other session. (Own-payoff effect): In the high-penalty sessions the average proportion of stealing is 10-15% below equilibrium while it is close to equilibrium in the other session. H1H2H3 L4L5L6 "1-50" 0.42 * 0.36 * 0.42 * * "25-45" 0.41 * 0.36 * 0.44 * 0.43 * *

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Results 3 The noise in the aggregate per-period data is large. The data fit roughly in a circle of diameter 0.5. The noise in the aggregate per-period data is large. The data fit roughly in a circle of diameter 0.5. There is no convergence to equilibrium. There is no convergence to equilibrium.

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q 1-p Session H1

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q 1-p Session H2

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q 1-p Session H3

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Session L q 1-p

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q 1-p Session L5

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q 1-p Session L6

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Quantal Response with Risk Aversion

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PENALTY=1; (HIGH) WATCH=.28, STEAL=.42 r= , = WATCH=.32, STEAL=.36 r= , = WATCH=.34, STEAL=.42 r= , = PENALTY=4; (LOW) WATCH=.57, STEAL=.48 NO CONVERGENCE. WATCH=.65, STEAL=.52 r= , = WATCH=.51, STEAL=.52 r= , = PENALTY HIGH WATCH=.28, STEAL=.42 r= , =

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Results 4 Fraction of watching. The trial rounds matter. The trial rounds matter. The adjustment seems to happen in the first 5 rounds (including trial rounds). The adjustment seems to happen in the first 5 rounds (including trial rounds). H1H2H3 L4L5L

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Results 5 The ten-period moving averages lie in a circle of radius 0.1. The ten-period moving averages lie in a circle of radius 0.1. Session 6:

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Results 6 The moving averages data points seem to spin counter-clockwise, although not necessarily around the equilibrium. This is indicated in the following graphs. The moving averages data points seem to spin counter-clockwise, although not necessarily around the equilibrium. This is indicated in the following graphs.

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Session H1 Direction of movement Equilibrium

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Session H2 Direction of movement Equilibrium

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Session H3 Direction of movement Equilibrium

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Session L4 Direction of movement Equilibrium

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Direction of movement Equilibrium Session L5

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Direction of movement Equilibrium Session L6

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Test 1 We run a OLS regression on the moving average data using a linear difference equation as the model. We run a OLS regression on the moving average data using a linear difference equation as the model. Does the matrix describe a rotation counter- clockwise? Does the matrix describe a rotation counter- clockwise?

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Session H1 Graphical Analysis: Direction of movement Equilibrium Graph exhibiting the DE fitted to the original MA data.The movements are always anticlockwise and converge.

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Session H2 Graphical Analysis: Direction of movement Equilibrium Graph exhibiting the DE fitted to the original MA data.The movements are always anticlockwise and converge.

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Session L3 Graphical Analysis: Direction of movement Equilibrium Graph exhibiting the DE fitted to the original MA data.The movements are always anticlockwise and converge.

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Session L4 Graphical Analysis: Direction of movement Equilibrium Graph exhibiting the DE fitted to the original MA data.The movements are always anticlockwise and converge.

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Direction of movement Equilibrium Session L5 Graphical Analysis: Graph exhibiting the DE fitted to the original MA data.The movements are always anticlockwise and converge.

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Graphical Analysis: Direction of movement Equilibrium Session L6 Graph exhibiting the DE fitted to the original MA data.The movements are always anticlockwise and converge.

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Experiment-5 The one-period prediction are quite close. 1 stage Predicted True Values

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Test 2 For the smoothed (or original) time series add the angular movements from period to period. For the smoothed (or original) time series add the angular movements from period to period. H1H2H3 L4L5L The total angle is always positive The total angle is always positive

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Results 7 Behaviour of subjects in the same role is very heterogeneous. Behaviour of subjects in the same role is very heterogeneous. Many subjects do either not use best replies or are constantly indifferent. Many subjects do either not use best replies or are constantly indifferent. However, pure maximin players are rare. However, pure maximin players are rare. Thieves, then watchmen Thieves, then watchmen Count bottom and right Count bottom and right

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H1 L4 H2L5 H3 L6

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H1 L4 H2 L5 H3L6

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Results 8 The distribution of aggregate choices tends to have a typical shape, with modes at the ends and a peak around the equilibrium distribution. It is distinctly different from a distribution as on the right. The distribution of aggregate choices tends to have a typical shape, with modes at the ends and a peak around the equilibrium distribution. It is distinctly different from a distribution as on the right.

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H1 L4 H2 L5 H3L6

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H1 L4 H2 L5 H3L6

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Results 9 Those who played the strategy that was overall the best strategy tended to win the most. Those who played the strategy that was overall the best strategy tended to win the most.

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THIEVES’ DECISIONS L 4 L 6 L 5 H 1 H 3 H 2

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EXPERIMENT -4 EXPERIMENT -6 EXPERIMENT -5 EXPERIMENT -1 EXPERIMENT -3 EXPERIMENT -2 WATCHMENS’ DECISIONS

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Conclusions Population game like a market? Population game like a market? Do anomalies get wielded out? Do anomalies get wielded out? Heterogeneity in behaviour. Heterogeneity in behaviour. Still, aggregate behaviour gets close enough to equilibrium for the comparative statics to be true. Still, aggregate behaviour gets close enough to equilibrium for the comparative statics to be true. What explains the own-payoff effect? What explains the own-payoff effect? Repetition with 2 players? Repetition with 2 players? A good class room experiment? A good class room experiment?

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