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Field
Intervention (Hidden)
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Before
We propose a laboratory experimental design that allows for varying discount rates. We do this in a deliberately simple manner: (1) subjects never face more than two possible values of δ, (2) they always know the current value of δ, and (3) they know the stochastic process governing changes to δ. Presumably none of these features are replicated in most relevant field settings, but we view this as a necessary first step in exploring the effect of varying δ.
All subjects play IRPD games with perfect monitoring using the payoff matrix shown below. Treatments vary the value(s) of δ and whether the value of δ is fixed or possibly changes at the beginning of a new stage game. We use a between subjects design, so all subjects participate in a single treatment, with random assignment of subjects to treatments. Our design contains the following treatments.
PD Payoff Matrix
C D
C 156 5
D 259 110
Baseline (Low and High): The value of δ is fixed at 0.75 throughout a session in Baseline Low and 0.90 in a session of Baseline High. This implies that the expected length of a supergame is only four stage games in Baseline Low versus ten stage games in Baseline High. Combining δ with the payoff matrix, BAD = 0.75 in Baseline Low versus 0.25 in Baseline High. Based on past results, cooperation rates are expected to be low in Baseline Low and substantially higher in Baseline High.
Switching: At the beginning of each supergame, an initial continuation probability is randomly selected, either δ = 0.75 or δ = 0.90. Each starting value is equally likely to be selected. At the end of each stage game when δ = 0.75, there is a 20% chance that δ switches to δ = 0.90 for the current stage game. Likewise, if δ = 0.90, there is a 20% chance it switches to δ = 0.75 for the next stage game. It is common knowledge when the value of δ changes.
Control (Low and High): The dynamic incentives in Baseline Low and the switching treatments are not the same even in stage games where δ = 0.75 in both cases. In the Baseline Low treatment, the subjects know that this low continuation probability will hold in all future stage games, but in the switching treatments they know that δ might switch. This implies that the expected length of the supergame is less in Baseline Low than in the switching treatments when δ = 0.75. As a result, we do not expect cooperation rates to be the same between Baseline Low and Switching Low even if random switching has no effect beyond altering the dynamic incentives. Similar logic applies to the comparison of Baseline High with Switching High.
We therefore added two additional control treatments, Control Low and Control High. We calculate the value of BAD for the initial stage game of Switching with δ = 0.75, accounting for the possibility of switching, and then calibrate the fixed value of δ (rounded to the closest 100th) that gives the same value of BAD. The resulting value of δ = 0.80, yielding BAD = 0.56 as in the switching treatments for stage games where δ = 0.75. Performing the same exercise for Switching with δ = 0.90 yields a fixed value of δ = 0.85 and BAD = 0.40. The value of δ is, therefore, fixed at 0.80 throughout a session in Control Low and 0.85 in a session of Control High.
Subjects in all treatments will play ten supergames using a strangers matching, with payment made for one randomly selected supergame. Payoffs will be calibrated to achieve an average payoff of 12 – 15 dollars per hour. Random seeds for the probability of continuation and the probability of switching δ are matched across treatments. This does not generate identical supergame lengths since the continuation probabilities vary, but does imply a similar pattern in terms of when relatively long and short supergames will occur within a session. We plan for five sessions, split into two matching groups, of each treatment. We plan to recruit at least 20 subjects per session. This gives us a minimum of 100 subjects and ten matching groups per treatment. We plan to double the sample for the Switching treatment; this treatment is of greater interest than the others, plus we want the extra data to ease fitting of a structural model.
Edit: 3/3/20
After an initial pilot session, we have decided to change the payoff table to the following:
C D
C 140 10
D 240 100
We have also changed the value of the continuation probabilities to 0.8 and 0.95 for the low and high values respectively. These changes were made for two reasons.
1) The pilot indicated lower than expected cooperation rates. This would make detection of treatment effects quite difficult. The changes lower the value of BAD, making cooperation more likely.
2) We rescaled the payoffs so we could pay in pennies rather than having to convert ECUs to dollars. This was done to reduce subject confusion.
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After
We propose a laboratory experimental design that allows for varying discount rates. We do this in a deliberately simple manner: (1) subjects never face more than two possible values of δ, (2) they always know the current value of δ, and (3) they know the stochastic process governing changes to δ. Presumably none of these features are replicated in most relevant field settings, but we view this as a necessary first step in exploring the effect of varying δ.
All subjects play IRPD games with perfect monitoring using the payoff matrix shown below. Treatments vary the value(s) of δ and whether the value of δ is fixed or possibly changes at the beginning of a new stage game. We use a between subjects design, so all subjects participate in a single treatment, with random assignment of subjects to treatments. Our design contains the following treatments.
PD Payoff Matrix
C D
C 156 5
D 259 110
Baseline (Low and High): The value of δ is fixed at 0.75 throughout a session in Baseline Low and 0.90 in a session of Baseline High. This implies that the expected length of a supergame is only four stage games in Baseline Low versus ten stage games in Baseline High. Combining δ with the payoff matrix, BAD = 0.75 in Baseline Low versus 0.25 in Baseline High. Based on past results, cooperation rates are expected to be low in Baseline Low and substantially higher in Baseline High.
Switching: At the beginning of each supergame, an initial continuation probability is randomly selected, either δ = 0.75 or δ = 0.90. Each starting value is equally likely to be selected. At the end of each stage game when δ = 0.75, there is a 20% chance that δ switches to δ = 0.90 for the current stage game. Likewise, if δ = 0.90, there is a 20% chance it switches to δ = 0.75 for the next stage game. It is common knowledge when the value of δ changes.
Control (Low and High): The dynamic incentives in Baseline Low and the switching treatments are not the same even in stage games where δ = 0.75 in both cases. In the Baseline Low treatment, the subjects know that this low continuation probability will hold in all future stage games, but in the switching treatments they know that δ might switch. This implies that the expected length of the supergame is less in Baseline Low than in the switching treatments when δ = 0.75. As a result, we do not expect cooperation rates to be the same between Baseline Low and Switching Low even if random switching has no effect beyond altering the dynamic incentives. Similar logic applies to the comparison of Baseline High with Switching High.
We therefore added two additional control treatments, Control Low and Control High. We calculate the value of BAD for the initial stage game of Switching with δ = 0.75, accounting for the possibility of switching, and then calibrate the fixed value of δ (rounded to the closest 100th) that gives the same value of BAD. The resulting value of δ = 0.80, yielding BAD = 0.56 as in the switching treatments for stage games where δ = 0.75. Performing the same exercise for Switching with δ = 0.90 yields a fixed value of δ = 0.85 and BAD = 0.40. The value of δ is, therefore, fixed at 0.80 throughout a session in Control Low and 0.85 in a session of Control High.
Subjects in all treatments will play ten supergames using a strangers matching, with payment made for one randomly selected supergame. Payoffs will be calibrated to achieve an average payoff of 12 – 15 dollars per hour. Random seeds for the probability of continuation and the probability of switching δ are matched across treatments. This does not generate identical supergame lengths since the continuation probabilities vary, but does imply a similar pattern in terms of when relatively long and short supergames will occur within a session. We plan for five sessions, split into two matching groups, of each treatment. We plan to recruit at least 20 subjects per session. This gives us a minimum of 100 subjects and ten matching groups per treatment. We plan to double the sample for the Switching treatment; this treatment is of greater interest than the others, plus we want the extra data to ease fitting of a structural model.
Edit: 3/3/20
After an initial pilot session, we have decided to change the payoff table to the following:
C D
C 140 10
D 240 100
We have also changed the value of the continuation probabilities to 0.8 and 0.95 for the low and high values respectively. These changes were made for two reasons.
1) The pilot indicated lower than expected cooperation rates. This would make detection of treatment effects quite difficult. The changes lower the value of BAD, making cooperation more likely.
2) We rescaled the payoffs so we could pay in pennies rather than having to convert ECUs to dollars. This was done to reduce subject confusion.
Edit 5/19/20
We've had a long delay due to coronavirus, but are ready to start running sessions in a serious way. After finishing piloting, we have made a final tweak to the payoff table to make cooperation a bit easier.
C D
C 145 30
D 255 100
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