ByDieter Balkenborgand Todd KaplanThe Watchman and the Thief -An Experiment on the Comparative Statics in Games
IntroductionMixed strategy: Randomizing between different optionsCentral role of mixed strategy in gamesMixed strategy as a tool for preventing to be outguessed by the opponentBart Simpson in episode 9F16 "Good ol' rock. Nuthin' beats that!. Lisa: Poor Bart. He is so predictable.
von Neumann on game theoryJacob 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.
Mixed Strategies are a way of remaining unpredictablePenalty Kick:
92.91Kick RDive L58.394.9769.92Dive RKick L
R = strong side of kickerNash 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)
Q: How does increase in punishment affect equilibrium behaviour?WATCHMAN61564THIEF7WATCHMAN564564THIEF474High Punishment:Low Punishment:45The Watchman and the ThiefWatchWatchRestHomeStealRestHomeSteal????
A: Watchman gets more lazy, theft remains as before.WATCHMAN61564THIEFRestWatchSteal7WATCHMAN564564THIEFRestHomeSteal474High Punishment:Low Punishment:45WatchHome
Game Structures: The Watchman and the Thief00-P10-FR-1WATCHMANTHIEFNot watchingWatchingNot stealingStealingP >0, punishment for the thief.F >0, fine for the watchman.R >0, reward for the watchman.
Mixed Strategy Nash EquilibriaMixed Nash equilibria determined by the condition to make the opponent indifferentHow a player randomizes depends only on payoff of his opponent Counterintuitive ImplicationsRelevance for economics: Relevant economic implications (Dasgupta, Stiglitz 1980, Kaplan, Luski, Wettstein 2003) (More firms implies less innovation)
Main design featurePopulation design (Nagel, Zamir 2001):10 subjects are row players (thieves)10 are column players (watchmen)In each period every thief plays against every watchman and vice versa (10 plays)Placement random
QuestionsQualitative predictions of theory correct?Own Payoff effect to be expected. Will they be wielded out in market-like matching environment?Relevance of security strategies?
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
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 gamesEmpirical: Walker and Wooders (2001)tennis Ciappori, Levitt, Groseclose (2003)soccer Palacios-Huerta (2003)soccer Volij & Palacios-Huerta (2006)soccer
OutlineExperimental DesignAggregate ResultsOwn payoff EffectQuantal Response and Risk AversionThe learning cycleIndividual BehaviourHeterogeneityMovers and ShakersConclusions
Experimental Design:Computerized Experiment at FEELE (the Finance and Economics Experimental Laboratory at Exeter University).Programmed by Tim Miller in z-tree.A session lasted approx. 1:30 h, on average a student earned 18.120 subjects
Written instructions, summaryTest questions2 trial rounds50 paid roundsQuestionnaire, paymentNo storyEither row- or column player throughoutEach player makes 10 decisions per period
Six Sessions:H1: High penaltyH2: High penalty, row interchangedH3: High penalty, column interchangedL4: Low penaltyL5: Low penalty, row interchangedL6: Low penalty, column interchanged(Data adjusted here)
Row players decision screenScreen shot:
Screen shot:Column players decision screen
Row players waiting screenScreen shot:
Row players result screen Screen shot:
Column players result screenScreen shot:
Results 1, Watchmen (col)The average proportion of watching (R) is lower in the high-punishment treatments.The observation is robust over time.High Punishment: Too close to 50%?
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.
Results 3The 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.
Chart1
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period50503period505111period50512period50510period505112period505113
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Timeseries
50503
505111
50512
50510
505112
50513
Timeseries
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50503Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10505111Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 1050512Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 1050510Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10505112Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 1050513Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10
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Data 050503
Thief
50503Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10Sum505111Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10Sum50512Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10Sum50510Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10Sum505112Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10Sum50513Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10Sum
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50503Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10Sum505111Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10Sum50512Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10Sum50510Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10Sum505112Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10Sum50513Th 1Th 2Th 3Th 4Th 5Th 6Th 7Th 8Th 9Th 10Sum
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