Uncertainty In Coordination Games Christos A. Ioannou
Motivation ◮ Common Knowledge Coordination games emphasize that equilibrium cannot be pinned down uniquely because beliefs are indeterminate. ◮ Global Coordination games assume that agents face idiosyncratic uncertainty about economic fundamentals (see Morris and Shin (1998), Heinemann (2000) and Heinemann and Illing (2002)). ◮ Recent experimental literature finds mixed results (see Heinemann, Nagel, and Ockenfels (2004), and Cabrales, Nagel, and Armenter (2007)).
Objective ◮ Myerson (2000) models the number of actual players in the game as a Poisson random variable. 1. Does uncertainty about the number of actual players have an impact on subjects’ behavior? 2. Is such behavior consistent with the prediction of Poisson Coordination games? ◮ We provide the first experimental investigation of Poisson Coordination games.
Findings 1. Uncertainty about the number of actual players may influence subjects’ behavior. 2. Subjects’ behavior in Poisson Coordination games is consistent with the theoretical prediction. 3. Robustness checks confirm that the aforementioned results are insensitive to smaller and larger sample sizes.
Literature Review
Heinemann, Nagel, Ockenfels (2004) Their design imitates the speculative attack model of Morris and Shin (1998). It consists of: ◮ 15 participants per session, ◮ Y ∼ unif (10 , 90), ◮ x i ∼ unif ( Y − 10 , Y + 10), ◮ 2 stages of 8 rounds with 10 independent decisions per round, ◮ two alternatives: A and B, ◮ for A, the payoff was a sure gain of T, ◮ for B, the payoff was Y, subject to the threshold α ( Y ) = 15 · (80 − Y ) being met, Z where T=20 or T=50, and Z=100 or Z=60.
Heinemann, Nagel, Ockenfels (2004) ◮ Subjects use threshold strategies. ◮ Thresholds in Global games are above and, sometimes, significantly different from the ones in Common Knowledge games. Payoff-Dominant Global Game Solution Risk-Dominant Maximin
Heinemann, Nagel, Ockenfels (2004) ◮ Subjects use threshold strategies. ◮ Thresholds in Global games are above and, sometimes, significantly different from the ones in Common Knowledge games. Global Games Payoff-Dominant Global Game Solution Risk-Dominant Maximin
Heinemann, Nagel, Ockenfels (2004) ◮ Subjects use threshold strategies. ◮ Thresholds in Global games are above and, sometimes, significantly different from the ones in Common Knowledge games. Common Knowledge Games Global Games Payoff-Dominant Global Game Solution Risk-Dominant Maximin
Cabrales, Nagel, Armenter (2007) Their 2 × 2 game has a unique strategy profile that survives the iterative deletion of strictly dominated strategies. Their experimental design consists of: ◮ 16 participants per session, ◮ 15 or 50 rounds with random matching, ◮ a discrete state space with 5 possible states and signals, ◮ a Global game solution which coincides on average with the risk-dominant equilibrium outcome.
Cabrales, Nagel, Armenter (2007) ◮ In Global games with long play, behavior converges towards the Global game solution. Perhaps because it coincides with the risk-dominant equilibrium? ◮ In Common Knowledge and Global games with short play, some participants settle on the payoff-dominant equilibrium, others on the Global game solution, and others in between.
Theoretical Predictions
Primitives ◮ N > 1 is the number of players, who decide whether to register to buy a cash reward. ◮ T is the registration fee. ◮ Y / 2 is the cash reward gross of the fee, with Y ∈ [ Y min , Y max ]. ◮ The cash amount is awarded if the number of registered players is at least as high as threshold α ( Y ). Therefore, after letting ν be the number of other players who register, the payoff of each player is 0 if he does not register , − T if he registers and ν < α ( Y ) − 1 , Y / 2 − T if he registers and ν ≥ α ( Y ) − 1 .
The minimum number of registrations required for the cash amount to be awarded is set as α ( Y ) = C − Y D with C > 0 , D > 0 and C − Y max ≤ 1 . D For Y ≥ Y ≡ α − 1 (1) a single registration is enough for the cash amount to be awarded, while for Y < Y more than one registrations will be needed. We assume that 2 T < Y . Let Y denote the supremum of all levels of economic fundamentals for which it is not profitable to register given that all other N − 1 players register. We assume that 2 T > Y min .
Common Knowledge Games ◮ Parameters Y , α ( Y ) , Y / 2, and N are common knowledge. ◮ 0 registrations is the unique equilibrium for Y < Y . ◮ N registrations is the unique equilibrium for Y ≥ Y . ◮ In the “grey area,” there is multiplicity of equilibria.
Poisson Games ◮ Parameters Y , Y / 2, and α ( Y ) are common knowledge. ◮ The number of players being a Poisson random variable with mean n is common knowledge. ◮ The predictions for economic fundamentals such that Y < 2 T or Y ≥ Y coincide with the corresponding predictions of Common Knowledge games. ◮ For economic fundamentals within the remaining area, the unique equilibrium is where no player registers if and only if 1 − F ( ⌈ α ( Y ) ⌉ − 2 | n ) < 2 T / Y , where F ( · | n ) is the cumulative distribution function with parameter n , and the symbolic function ⌈·⌉ rounds-up the fraction to the nearest integer from above. When the above inequality does not hold then multiplicity of equilibria is predicted instead.
Experimental Design
Experimental Design Table: Characteristics of the Experimental Sessions Common Knowledge Games # of Subj. Mean Threshold Fee ( £ ) Amount ( £ ) Acronym 34 - 16 9 12.50 CK169 34 - 15 10 12.50 CK1510 34 - 16 9 12.50 CK169R Poisson Games # of Subj. Mean Threshold Fee ( £ ) Amount ( £ ) Acronym 40 17 16 9 12.50 P169 44 17 15 10 12.50 P1510 45 17 16 9 12.50 P169R ◮ The experiments were conducted over the Internet. ◮ We conducted 2 sessions per treatment. ◮ The theoretical prediction based on the chosen parameters is that all players do not register in the Poisson games.
Poisson Games ◮ Stage 1: Computer draw to determine the number of participants in Stage 2. The Poisson process was based on a mean n = 17. 26 8 9 25 24 10 11 23 22 12 21 13 20 14 19 15 18 16 17 ◮ Stage 2: Subjects had the option of buying a cash amount of £ 12.50 at a fee of £ 9/ £ 10 subject to a threshold of 16 / 15 being met. The fee was subtracted immediately from their endowment conditional on registering.
Common Knowledge Games ◮ The number of participants N was 17. ◮ Subjects had the option of buying a cash amount of £ 12.50 at a fee of £ 9/ £ 10 subject to a threshold of 16 / 15 being met. The fee was subtracted immediately from their endowment conditional on registering.
Parameter Choices ◮ Y / 2 = £ 12.50, ◮ N = 17, ◮ α ( Y ) = 16 or α ( Y ) = 15, ◮ T = £ 9 or T = £ 10, and ◮ e = £ 12 or e = £ 15.
General Hypotheses 1. Subjects’ behavior is statistically similar across the Common Knowledge and Poisson games when controlling for the parameter choices of each pairwise comparison. 2. Subjects in the Poisson games will choose to forego registering to buy the cash amount in accordance with the prediction of the Poisson games for the parameters specified. 3. Subjects in the Common Knowledge games will either all coordinate on registering to buy the cash amount or all coordinate on foregoing to resister to buy the cash amount in accordance with the prediction of the Common Knowledge games for the parameters specified.
Results
Single Shot Experiments Table: Descriptive Statistics Common Knowledge Games Registered Not Registered Amount Acronym Freq. % Freq. % Awarded? CK169 16 47.1 18 52.9 No CK1510 18 52.9 16 47.1 No Poisson Games Registered Not Registered Amount Acronym Freq. % Freq. % Awarded? P169 2 5.0 38 95.0 No P1510 2 4.6 42 95.5 No Total 38 114
Single Shot Experiments Table: Differences in Subjects’ Behavior Across Game Types Alternative hypothesis: decision i � = decision j p -values Common Knowledge games vs Poisson games CK169 & P169 0.000 CK1510 & P1510 0.000
Single Shot Experiments Table: Theory and Subjects’ Behavior Panel A Alternative hypothesis: decision i � = 1 p -values CK169 0.000 CK1510 0.000 P169 0.152 P1510 0.153 Panel B Alternative hypothesis: decision i � = 0 p -values CK169 0.000 CK1510 0.000
Single Shot Experiments Table: Marginal Effects Dependent variable: decision Regressor dy / dx CK1510 -0.059 (0.121) P169 0.421*** (0.092) P1510 0.425*** (0.091)
Repeated Experiments 1 .9 Proportion .8 .7 .6 .5 0 5 10 15 20 Period P169R CK169R
Robustness Analysis
Robustness Analysis Table: Characteristics of the Robustness Sessions Common Knowledge Games # of Subj. Mean Threshold Fee ( £ ) Amount ( £ ) Acronym 16 - 4 10 12.50 CK410 38 - 18 9 12.50 CK189 38 - 17 10 12.50 CK1710 Poisson Games # of Subj. Mean Threshold Fee ( £ ) Amount ( £ ) Acronym 16 4 4 10 12.50 P410 48 19 18 9 12.50 P189 46 19 17 10 12.50 P1710
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