Nash equilibrium Non-standard preferences Experimental design Results Other projects Nash Equilibrium in Tullock Contests Aidas Masiliunas 1 1 Aix-Marseille School of Economics Controversies in Game Theory III, ETH Zurich 2 June, 2016
Nash equilibrium Non-standard preferences Experimental design Results Other projects Rent-seeking (Tullock) contest Two players compete for a prize (16 ECU) by making costly investments ( x 1 , x 2 ≤ 16) Higher investments increase the probability to win the prize x i Probability that player i receives the prize: x i + x j
Nash equilibrium Non-standard preferences Experimental design Results Other projects Rent-seeking (Tullock) contest Two players compete for a prize (16 ECU) by making costly investments ( x 1 , x 2 ≤ 16) Higher investments increase the probability to win the prize x i Probability that player i receives the prize: x i + x j Applications: Competition for monopoly rents Investments in R&D Competition for a promotion/bonus Political contests
Nash equilibrium Non-standard preferences Experimental design Results Other projects Theory x i E ( π ) = x i + x j · 16 + 16 − x i BR i ( x j ) : x ∗ i = � 16 x j − x j RNNE : x ∗ i = 4, dominance solvable in three steps. Standard preferences 16 15 14 12 Best Response 10 10 9 8 7 6 5 5 4 3 2 1 1 2 3 4 5 5 6 7 8 9 10 10 11 12 13 14 15 15 16 Other plays
Nash equilibrium Non-standard preferences Experimental design Results Other projects Explanatory power of Nash equilibrium in experiments 7.04% of choices are exactly Nash 60.19% of choices are strictly dominated Investments are spread across the whole strategy space Experience does not help Less stability compared to auctions
Nash equilibrium Non-standard preferences Experimental design Results Other projects Comparative statics of Nash equilibrium An alternative to point predictions is comparative statics Is behaviour sensitive to changes in the Nash prediction?
Nash equilibrium Non-standard preferences Experimental design Results Other projects Comparative statics of Nash equilibrium An alternative to point predictions is comparative statics Is behaviour sensitive to changes in the Nash prediction? Players Nash Mean investment 2 250 325 3 222 283 4 188 302 5 160 322 9 99 326 Source: Lim, Matros & Turocy, 2014
Nash equilibrium Non-standard preferences Experimental design Results Other projects Why should players choose Nash equilibrium? Interpretation #1: Nash equilibrium is the unique action profile that can be justified by common knowledge of rationality. Rationality = maximization of expected payoff given some belief.
Nash equilibrium Non-standard preferences Experimental design Results Other projects Rationalizable strategies x i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 BR ( x i ) 3 4 4 4 4 4 4 3 3 3 2 2 1 1 1 1
Nash equilibrium Non-standard preferences Experimental design Results Other projects Rationalizable strategies x i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 BR ( x i ) 3 4 4 4 4 4 4 3 3 3 2 2 1 1 1 1 Rationality Rationalizable: 3, 4, 2, 1
Nash equilibrium Non-standard preferences Experimental design Results Other projects Rationalizable strategies x i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 BR ( x i ) 3 4 4 4 4 4 4 3 3 3 2 2 1 1 1 1 BR ( BR ( x i )) 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 Rationality Rationalizable: 3, 4, 2, 1 Rationality + belief that the opponent is rational Rationalizable: 3, 4
Nash equilibrium Non-standard preferences Experimental design Results Other projects Rationalizable strategies x i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 BR ( x i ) 3 4 4 4 4 4 4 3 3 3 2 2 1 1 1 1 BR ( BR ( x i )) 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 BR ( BR ( BR ( x i ))) 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 Rationality Rationalizable: 3, 4, 2, 1 Rationality + belief that the opponent is rational Rationalizable: 3, 4 Rationality + belief that the opponent is rational + belief that the opponent believes in my rationality Rationalizable: 4
Nash equilibrium Non-standard preferences Experimental design Results Other projects Rationalizable strategies x i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 BR ( x i ) 3 4 4 4 4 4 4 3 3 3 2 2 1 1 1 1 BR ( BR ( x i )) 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 BR ( BR ( BR ( x i ))) 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 Rationality Rationalizable: 3, 4, 2, 1 Rationality + belief that the opponent is rational Rationalizable: 3, 4 Rationality + belief that the opponent is rational + belief that the opponent believes in my rationality Rationalizable: 4 Epistemic definition of Nash equilibrium: common belief in rationality + simple belief hierarchy
Nash equilibrium Non-standard preferences Experimental design Results Other projects Why should players choose Nash equilibrium? Nash equilibrium is the unique action profile that cannot be ruled out by common knowledge of rationality. Players care about expected payoffs 1 Players have the ability to calculate expected payoffs and 2 identify dominated strategies Players believe that other players satisfy 1-2, and believe that 3 they believe that they satisfy 1-2...
Nash equilibrium Non-standard preferences Experimental design Results Other projects Why should players choose Nash equilibrium? Nash equilibrium is the unique action profile that cannot be ruled out by common knowledge of rationality. Players care about expected payoffs 1 Players have the ability to calculate expected payoffs and 2 identify dominated strategies Players believe that other players satisfy 1-2, and believe that 3 they believe that they satisfy 1-2... Nash equilibrium is the rest point of various learning dynamics Belief-based learning, e.g. Cournot best-response, fictitious play Assumption 3 is not necessary
Nash equilibrium Non-standard preferences Experimental design Results Other projects Why should players choose Nash equilibrium? Nash equilibrium is the unique action profile that cannot be ruled out by common knowledge of rationality. Players care about expected payoffs 1 Players have the ability to calculate expected payoffs and 2 identify dominated strategies Players believe that other players satisfy 1-2, and believe that 3 they believe that they satisfy 1-2... Nash equilibrium is the rest point of various learning dynamics Belief-based learning, e.g. Cournot best-response, fictitious play Assumption 3 is not necessary Payoff-based learning, e.g. reinforcement learning Players must be willing to explore, remember past payoffs, receive accurate feedback.
Nash equilibrium Non-standard preferences Experimental design Results Other projects Which assumptions are violated?
Nash equilibrium Non-standard preferences Experimental design Results Other projects Preference-based explanations: joy of winning Participants receive non-monetary utility from winning (Parco et al, 2005, Sheremeta, 2011) or lose utility after losing (Delgado et al., 2008). Sheremeta (2011) elicits joy of winning by implementing a contest where prize has no value. Joy of winning with w=3 Joy of winning with w = 8 16 16 15 15 14 14 12 12 Best Response 10 10 Best Response 10 10 9 9 8 8 7 7 6 6 5 5 5 5 4 4 3 3 2 2 1 1 1 2 3 4 5 5 6 7 8 9 10 10 11 12 13 14 15 15 16 1 2 3 4 5 5 6 7 8 9 10 10 11 12 13 14 15 15 16 Other plays Other plays
Nash equilibrium Non-standard preferences Experimental design Results Other projects Preference-based explanations: risk preferences CRRA untility function: u ( π i ) = π 1 − ρ i 1 − ρ Risk aversion if ρ = 0 . 5, risk seeking if ρ = − 0 . 5 Risk aversion Risk seeking 16 16 15 15 14 14 12 12 Best Response 10 10 Best Response 10 10 9 9 8 8 7 7 6 6 5 5 5 5 4 4 3 3 2 2 1 1 1 2 3 4 5 5 6 7 8 9 10 10 11 12 13 14 15 15 16 1 2 3 4 5 5 6 7 8 9 10 10 11 12 13 14 15 15 16 Other plays Other plays
Nash equilibrium Non-standard preferences Experimental design Results Other projects Preference-based explanations: social preferences Fehr & Schmidt (1999) inequality aversion: � π i − α ( π j − π i ) if π i ≤ π j u ( π i , π j ) = π i − β ( π i − π j ) if π i > π j Fehr and Schmidt (1999) inequality aversion a=0, b=0 15 15 a=0.5, b=0 a=1, b=0 13 11 Best Response 10 9 8 7 6 5 5 4 3 2 1 1 2 3 4 5 5 6 7 8 9 10 10 12 14 15 16 Other plays
Nash equilibrium Non-standard preferences Experimental design Results Other projects All preferences from Sheremeta (2015)
Nash equilibrium Non-standard preferences Experimental design Results Other projects ”Behavioral Variation in Tullock Contests”, joint with F. Mengel and Ph. Reiss Deviations from NE could be a result of bounded rationality Players optimize given the feedback in previous rounds. Noisy feedback prevents players from discovering optimal actions
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