Facilitating Pareto-Optimum Coordination by Subsidies in

Deterministic and Stochastic Settings

CRED Lab Meeting

Sept. 16th, 2010

Introduction: Social Reinforcement

• Agents in an interactive game or a social network reinforce each other’s decisions– Schelling’s (1978) tipping example on racial

composition changes – Leibenstein’s (1950) “bandwagon effects”

• Often results in a coordination game with multiple Pareto-ranked Nash Equilibria.

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Games with Pareto-ranked Equilibria

• Garbage Disposal

Games with Pareto-ranked Equilibria

• VHS vs. Betamax

Games with Pareto-ranked Equilibria in Stochastic Settings

• Wildfire Defensive Space

• Computer Virus

Advance in Theory

• Heal and Kunreuther (2009)– Model social reinforcement in a game theoretical

framework– With increasing differences, changes in a subset can

shift the equilibrium– Public policy can be used to encourage such changes

• Zhuang et al. (2007) – A dynamic theoretical model of interdependent

security – Subsidy is more efficient if allocated to those agents

least likely to improve security on their own.

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Current Research

• Purpose: investigate the relationship between theory and reality

• Method: A subset of players are partially subsidized for choosing the Pareto preferable option.

• Predictions: – On the one hand, the theory predicts that subsidies

promote the Pareto optimum; – On the other hand, at the behavior level, the

motivation crowding theory suggests that external monetary intervention may undermine intrinsic motivations (Frey and Jegen, 2001 ; Meier (2007) ) .

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Deterministic Coordination Games in the Lab

• Three types of Coordination games (Camerer 2003)• Order-statistic Game

– Van Huyck et al. (1990); Review paper: Devetag (2007)– A player's payoff is increasing in order statistics (median or

minimum of all choices) and decreasing in the distance of her choice from the order statistic.

– A payoff dominant strategy and a security dominant strategy.

• Extensive evidence on coordination failure• Various Methods to Improve Coordination

• Finer grade and longer game• Loss aversion• Communication

Critical Mass Game

• In a critical mass game, players usually make binary decisions (X or Y), and once a threshold of players choose X (Y), all other players will follow X(Y). – The urban segregation and faculty seminar

participation in Schelling (1978) – The threshold public good (Van de Kragt et al., 1983;

Isaac et al., 1989). • Full information about other player’s historical

decisions and increasing returns above the critical mass are found to encourage players to reach the efficient equilibrium (Devetag 2003).

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Coordination in Stochastic Settings

• The interdependent security : the decisions of the agents depend on both their expectation of others’ actions and their own risk preference.

• Hess et al. (2007) find that coordination failure is common in an interdependent security game in which players simultaneously make decisions to coordinate investment to reduce the probability of losses, but coordination is improved when decisions are made in sequence, or when the security interdependency is small relative to the overall risk.

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Experimental Design

• 2X2 between-subject design: – Stochastic vs. Deterministic Game– Subsidy(cost of A=10) vs. Baseline (cost of A=32)

• 6 players as a group making simultaneous decisions; each has an initial wealth of 2000

• 2 random players in each round pay 10 instead of 32 in the subsidy condition

• Repeat 20 rounds• After-game surveys for risk preference,

decision motivations, demographics, etc.

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Baseline Conditions

• Stochastic-Baseline • Table 1: Probabilities of Losing 100 Talers in the Stochastic Game

• Deterministic-Baseline • Table 2: Possible Losses in the Deterministic Game

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  Number of Other Players Who Choose Option A

0 1 2 3 4 5Your Choice

Option A (cost= 32) 67% 59% 49% 36% 20% 0%

Option B (cost= 0) 80% 75% 69% 61% 52% 40%

  Number of Other Players Who Choose Option A

0 1 2 3 4 5

Your ChoiceOption A (cost= 32) 67 59 49 36 20 0

Option B (cost= 0) 80 75 69 61 52 40

(Expected) Loss in the Coordination Games

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Testing the Subsidy Carry-over Effect

• Running a second session in each condition– At the beginning of Session 2, players’ wealth

level is restored to 2000 Talers.

– The same 6 players play the same type of game (stochastic or deterministic) for another 20 rounds, with the subsidy removed or added, depending on which condition they were in the first session.

– The players are not aware of the existence of the second session until all finished the first one.

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The Complete Design

• 288 people (48 6-person groups)

• 82% of participants’ age was between 18 and 25. 62% of them were females.

Table 3: the Complete Design

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Name DB1-DS2 DS1-DB2 SB1-SS2 SS1-SB2# of 6-Person Groups

13 13 10 12

  Description Percentage Description Percentage Description Percentage Description Percentage

Session 1Deterministic-

BaselineDeterministic –

SubsidyStochastic-

BaselineStochastic-

Subsidy

Session 2Deterministic-

SubsidyDeterministic-

Baseline-Stochastic-

SubsidyStochastic-

Baseline

Hypotheses

• The General Subsidy Effect Hypothesis– H1: Players are more likely to choose

Strategy A with subsidy than without subsidy in both the stochastic and deterministic game.

– H1 will be tested using the between-subject data from 48 groups in Session 1 only.

• The Subsidy Carry-over Effect Hypothesis – H2: The higher cooperation rate in Session 1

brought by subsidy sustains after subsidy is removed in Session 2.

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Means in all Sessions

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Name DB1-DS2 DS1-DB2 SB1-SS2 SS1-SB2# of 6-

Person

Groups

13 13 10 12

 Descripti

onPercent

ageDescripti

onPercent

ageDescrip

tionPercent

ageDescripti

onPercent

age

Session 1

Deterministic-

Baseline0.64

Deterministic –

Subsidy0.74

Stochastic-

Baseline

0.71Stochastic- Subsidy

0.79

Session 2

Deterministic-

Subsidy0.79

Deterministic-

Baseline-0.68

Stochastic-

Subsidy0.76

Stochastic-

Baseline0.79

Results in Session 1

• Confirmed H1: Players were more likely to choose Strategy A with subsidy than without subsidy (p<0.01), after controlling for the period effect and individual subject differences.

• Social Welfare(summed payoff minus subsidy cost) was 7% higher in the subsidy conditions than in the baseline conditions.

• No interaction between game type and subsidy

Table 4. Random Individual Logit model for Choosing Strategy A in Session 1

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Variable Coefficient Standard Error z value Pr(>|z|)Dependent Variable        

Choosing A        Independent Variables        

Constant 1.23 0.26 4.68 0.00Stochastic Game 0.57 0.31 1.86 0.06

Subsidy 0.98 0.31 3.19 0.001Fixed Effects        

Period -0.01 0.006 -2.06 0.04Rho 5.76 2.40    Log likelihood -2422Sample size 5760

Figure 3: Cooperation Rate in Groups in

Period 16-20 in Session 1

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Group Behavior in Session 1

• Subsidy improved the cooperation in both games• Both Nash equilibria were observed

– But some groups never reached either equilibrium – No groups were trapped in All-B with subsidy – The subsidized players chose A 91% of the time.

• Subsidy tipped some groups toward All-A– The expectation of unsubsidized players in the

subsidy condition is significantly higher than that of the players in the baseline condition (p<0.01)

– They are also more likely to choose A (p=0.05, one-tail z test).

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Group Differences in the Subsidy Effect

• Stochastic Game:• Baseline: only 2 out of 10 groups had a cooperation rate over 0.83 (5

person)• Subsidy: 6 out of 12 groups successfully reached the efficient

equilibrium. • Deterministic Game

• Baseline: A dichotomous distribution -- 7 groups reached the efficient equilibrium, 4 groups clustered at the inefficient equilibrium where 1 or less player chooses A, and only 2 groups settled in the middle.

• Subsidy: the majority of the groups had 2-4 players choosing A, with no group falling into the inefficient equilibrium, and slightly less groups reaching the efficient equilibrium than in the baseline condition.

• To summarize, subsidy improves coordination by tipping half groups towards the efficient equilibrium in the stochastic game, and by diverting one third of groups away from the inefficient equilibrium.

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Lower Tipping Point in the Stochastic Game

• Theoretical tipping point: 4 in both games• But, in the stochastic game, decisions depend on both

expectation and risk preference– Explains why so many divided groups in the stochastic-

baseline condition: risk aversion drives players to choose A to reduce risk even when expecting less than 4 persons to choose A

– Most participants were risk averse (survey data) might have a lower tipping point than 4 Subsidy can more easily increased players’ expectation to above the tipping point in the stochastic game than in the deterministic game subsidy tipped half groups towards the efficient equilibrium in the stochastic game, but not in the deterministic game

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The Subsidy Carry-Over Effect

• Data from Session 2 : – 13 groups in DS1-DB2 , 12 groups in SS1-SB2

• A significant interaction between the game type (stochastic vs. deterministic) and the subsidy carry-over effect (p<0.01). – Participants in the deterministic game were

significantly less likely to choose A after the subsidy was removed than with subsidy (p<0.01),

– but those in the stochastic game sustained the same level of the coordination without subsidy as with the subsidy (p>0.10).

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Figure 4: Cooperation Rates in Period 16-20 in the Subsidy Condition

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Crowding Reasoning in the Post-Game Survey

• In the deterministic game, 43% of players believed that paying a lower cost was the only reason to choose A, and that others would choose A only when subsidized.

• In the stochastic game, only 22% of players viewed paying a lower cost as the only reason for choosing A: 78% of the players simply regarded A as a safer option, and assumed that others also preferred reduced risk, once the subsidy helped the group reach a higher number of players choosing A.

• In summary, subsidy seems to crowd out other possible reasons for cooperation in the deterministic setting, but safety is the principal reason for coordination on A in the stochastic setting. As a result, the subsidy effect carries over in the stochastic setting, but not in the deterministic one.

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Conclusions

• Partially subsidizing 2 out of 6 players in a laboratory coordination game usually produced better coordination and higher total payoff:– Subsidy improves coordination by tipping half groups

towards the efficient equilibrium in the stochastic game, and by diverting one third of groups away from the inefficient equilibrium;

• After removal of the subsidy, high coordination continued in most groups with stochastic payoffs but declined with deterministic ones– Temporary subsidies may promote lasting coordination in

risk reduction, whereas in a deterministic setting, subsidy may be counterproductive, because it crowds out other bases for coordination.

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Implications and Extensions

• Public Policy Implications– A limited budget might best be used to support

temporary subsidies in stochastic settings, spread among many groups.

– In deterministic settings subsidies might need to be maintained indefinitely and might crowd out cooperation based on social expectation.

• Possible Extensions– Communication and cheap talk– Natural vs. imposed subsidy– # of subsidized players

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