Uncertainty In Coordination Games Christos A. Ioannou Motivation - - PowerPoint PPT Presentation

Christos A. Ioannou

Uncertainty In Coordination Games

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), ◮ xi ∼ 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 )

Z

being met, 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.

Global Games Maximin Common Knowledge Games Payoff-Dominant Global Game Solution Risk-Dominant

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,

• thers 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 ∈ [Ymin, Ymax].

◮ 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 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 − Ymax D ≤ 1. 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 2T < 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 2T > Ymin.

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 < 2T

• r 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) < 2T/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

10 11

12 13

15 16 14 17 18 19

20 21

22 23

24 25

◮ 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: decisioni = decisionj 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: decisioni = 1 p-values CK169 0.000 CK1510 0.000 P169 0.152 P1510 0.153 Panel B Alternative hypothesis: decisioni = 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

.5 .6 .7 .8 .9 1 Proportion 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

Single Shot Experiments

Table: Descriptive Statistics of Smaller & Larger Sample Sizes

Common Knowledge Games Registered Not Registered Amount Acronym Freq. % Freq. % Awarded? CK410 7 43.8 9 56.3 No CK189 16 42.1 22 57.9 No CK1710 18 47.4 20 52.6 No Poisson Games Registered Not Registered Amount Acronym Freq. % Freq. % Awarded? P410 1 6.3 15 93.8 No P189 3 6.3 45 93.8 No P1710 2 4.4 44 95.7 No Total 47 155

Single Shot Experiments

Table: Robustness Analysis for Small Samples

Panel A Alternative hypothesis: decisioni = decisionj p-value Common Knowledge games vs Poisson games CK410 & P410 0.019 Panel B Alternative hypothesis: decisioni = 1 p-value Poisson games P410 0.500

Single Shot Experiments

Table: Robustness Analysis for Large Samples Panel A Alternative hypothesis: decisioni = decisionj p-values Common Knowledge games vs Poisson games CK189 & P189 0.000 CK1710 & P1710 0.000 Panel B Alternative hypothesis: decisioni = 1 p-values Poisson games P189 0.128 P1710 0.153 Panel C Dependent variable: decision Regressor dy/dx P189 0.359*** (0.087) P1710 0.378*** (0.086)

Discussion

Comparative Statics

Table: Characteristics of the Comparative Statics Sessions

Panel A # of Subj. # of Ses. Mean Threshold Fee (£) Amount (£) Acronym 21 1 17 13 9 12.50 P139 14 1 17 14 9 12.50 P149 16 1 17 15 9 12.50 P159 40 2 17 16 9 12.50 P169 23 1 17 17 9 12.50 P179 15 1 17 18 9 12.50 P189 Panel B Registered Not Registered Amount Acronym Freq. % Freq. % Awarded? P139 9 42.9 12 57.1 No P149 4 28.6 10 71.4 No P159 3 18.8 13 81.2 No P169 2 5.0 38 95.0 No P179 1 4.3 22 95.7 No P189 0.0 15 100.0 No

Single Shot Experiments

Global Games

◮ In Global games, there is a unique symmetric Bayesian Nash

Equilibrium (BNE), where all players register if and only if their signal is higher than x∗, where x∗ is defined by 1 2εY

x∗+εY

• x∗−εY

Y 2 [1 −

⌈a(Y )−2⌉

• j=0

Bin(j, N − 1, p(Y , x∗))]dY = T. The symbolic function ⌈·⌉ rounds-up the fraction to the nearest integer from above, and Bin(·) is the binomial distribution where p(Y , x∗) = Y + εY − x∗ 2εY .

◮ However, the above result relies heavily on the assumption

that the state and signal are continuous random variables. If,

• n the other hand, these are discrete random variables, then

there may not be a unique symmetric BNE.

Global Games

◮ In the experiments, we chose Y , εY and xY such that, in the

symmetric BNE, the theoretical prediction prescribes that all players do not register.

Table: Characteristics of the Global Sessions

Panel A # of Subj. # of Ses. Threshold Fee (£) Amount (£) Acronym 34 2 22 −

• Y

4

• = 16

9

Y 2 = 12.50

G169 34 2 21 −

• Y

4

• = 15

10

Y 2 = 12.50

G1510 Panel B Registered Not Registered Amount Acronym Freq. % Freq. % Awarded? G169 14 41.2 20 58.8 No G1510 16 47.1 18 52.9 No

Concluding Remarks

Summary

We design online experiments that attempt to capture “large” games between players to study population uncertainty in Coordination games.

• 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 the aforementioned results with

smaller and larger sample sizes. Policy Implications ...

Future Direction

◮ Provision of a unified theory of explaining behavior across

various treatments (see Heinemann, Nagel, and Ockenfels (2009) and Kneeland (2012)).

◮ Do our results carry over to Voting games and Discrete Public

Goods games (see Bouton and Castanheira (2012) and Makris (2009))?

Poisson Games

Poisson Games

Poisson Games

Global Games

Global Games

Global Games

Global Games

n 11 12 13 14 15 16 17 18 19 x = 0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 2 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 3 0.004 0.002 0.001 0.000 0.000 0.000 0.000 0.000 0.000 4 0.015 0.007 0.003 0.001 0.000 0.000 0.000 0.000 0.000 5 0.037 0.020 0.010 0.005 0.002 0.001 0.000 0.000 0.000 6 0.078 0.045 0.025 0.014 0.007 0.004 0.002 0.001 0.000 7 0.143 0.089 0.054 0.031 0.018 0.010 0.005 0.002 0.001 8 0.232 0.155 0.099 0.062 0.037 0.022 0.012 0.007 0.003 9 0.340 0.242 0.165 0.109 0.069 0.043 0.026 0.015 0.008 10 0.459 0.347 0.251 0.175 0.118 0.077 0.049 0.030 0.018 11 0.579 0.461 0.353 0.260 0.184 0.127 0.084 0.054 0.034 12 0.688 0.576 0.463 0.358 0.267 0.193 0.135 0.091 0.060 13 0.781 0.681 0.573 0.464 0.363 0.274 0.200 0.142 0.098 14 0.854 0.772 0.675 0.570 0.465 0.367 0.280 0.208 0.149 15 0.907 0.844 0.763 0.669 0.568 0.466 0.371 0.286 0.214 16 0.944 0.898 0.835 0.755 0.664 0.566 0.467 0.375 0.292 17 0.967 0.937 0.890 0.827 0.748 0.659 0.564 0.468 0.378 18 0.982 0.962 0.930 0.882 0.819 0.742 0.655 0.562 0.469 19 0.990 0.978 0.957 0.923 0.875 0.812 0.736 0.650 0.560 20 0.995 0.988 0.975 0.952 0.917 0.868 0.805 0.730 0.647 21 0.997 0.993 0.985 0.971 0.946 0.910 0.861 0.799 0.725 22 0.999 0.997 0.992 0.983 0.967 0.941 0.904 0.855 0.793 23 0.999 0.998 0.996 0.990 0.980 0.963 0.936 0.898 0.849 24 0.999 0.999 0.998 0.995 0.988 0.977 0.959 0.931 0.893 25 0.999 0.999 0.999 0.997 0.993 0.986 0.974 0.955 0.926 26 1.000 0.999 0.999 0.998 0.996 0.992 0.984 0.971 0.951 27 1.000 0.999 0.999 0.999 0.998 0.995 0.991 0.982 0.968 28 1.000 1.000 0.999 0.999 0.999 0.997 0.995 0.989 0.980 29 1.000 1.000 1.000 0.999 0.999 0.998 0.997 0.994 0.988 30 1.000 1.000 1.000 0.999 0.999 0.999 0.998 0.996 0.993 31 1.000 1.000 1.000 1.000 0.999 0.999 0.999 0.998 0.996 32 1.000 1.000 1.000 1.000 1.000 0.999 0.999 0.999 0.997 33 1.000 1.000 1.000 1.000 1.000 0.999 0.999 0.999 0.998 34 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.999 0.999 35 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.999 36 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 0.999 37 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.999 38 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

Poisson Cumulative Distribution Table

n 11 12 13 14 15 16 17 18 19 x = 0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 2 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 3 0.004 0.002 0.001 0.000 0.000 0.000 0.000 0.000 0.000 4 0.015 0.007 0.003 0.001 0.000 0.000 0.000 0.000 0.000 5 0.037 0.020 0.010 0.005 0.002 0.001 0.000 0.000 0.000 6 0.078 0.045 0.025 0.014 0.007 0.004 0.002 0.001 0.000 7 0.143 0.089 0.054 0.031 0.018 0.010 0.005 0.002 0.001 8 0.232 0.155 0.099 0.062 0.037 0.022 0.012 0.007 0.003 9 0.242 0.165 0.109 0.069 0.043 0.026 0.015 0.008 10 0.251 0.175 0.118 0.077 0.049 0.030 0.018 11 0.260 0.184 0.127 0.084 0.054 0.034 12 0.267 0.193 0.135 0.091 0.060 13 0.274 0.200 0.142 0.098 14 0.280 0.208 0.149 15 0.286 0.214 16 0.292

Poisson Cumulative Distribution Table

n 11 12 13 14 15 16 17 18 19 x = 0 1 2 3 4 5 6 0.078 7 0.143 0.089 8 0.232 0.155 0.099 9 0.242 0.165 0.109 10 0.251 0.175 0.118 11 0.260 0.184 0.127 12 0.267 0.193 0.135 13 0.274 0.200 0.142 14 0.280 0.208 0.149 15 0.286 0.214 16 0.292

Poisson Cumulative Distribution Table

n 11 12 13 14 15 16 17 18 19 x = 0 1 2 3 4 5 6 0.078 7 0.143 0.089 8 0.232 0.155 0.099 9 0.242 0.165 0.109 10 0.251 0.175 0.118 11 0.260 0.184 0.127 12 0.267 0.193 0.135 13 0.274 0.200 0.142 14 0.280 0.208 0.149 15 0.286 0.214 16 0.292