Payment schemes in infinite-horizon experimental games

Abstract

We consider payment schemes in experiments that model infinite-horizon games by using random termination. We compare paying subjects cumulatively for all periods of the game; with paying subjects for the last period only; with paying for one of the periods, chosen randomly. Theoretically, assuming expected utility maximization and risk neutrality, both the cumulative and the last period payment schemes induce preferences that are equivalent to maximizing the discounted sum of utilities. The last period payment is also robust under different attitudes toward risk. In comparison, paying subjects for one of the periods chosen randomly creates a present-period bias. We further provide experimental evidence from infinitely repeated prisoners’ dilemma games that supports the above theoretical predictions.

Appendices

Appendix A: Discount factors in random continuation games in periods beyond period 1

Consider how the relative weights between the current and the future change under random pay as the game progresses beyond period 1. In period 2, using manipulations similar to those for (3), we obtain that the expected payoff is:

(9)

Here, $\delta^{rt}_{\tau}$ denotes the weight put on period τ under random pay when the game is in period t. Observe that $\delta^{r2}_{1}=\delta^{r2}_{2}$ in period 2, whereas we had, from (3), $\delta^{r1}_{1}>\delta^{r1}_{2}$ in period 1.

In general, assume the game has progressed to period t≥2. We obtain that the expected payoff in period t is:

(10)

where $\delta^{rt}_{s}=\frac{1-p}{p^{t}}\{-\log(1-p)-\sum_{\tau=2}^{t} \frac{p^{\tau-1}}{\tau-1}\}$ is the weight put on each past period, s<t, and on the current period, s=t; and $\delta^{rt}_{s}=\frac{1-p}{p^{t}}\{-\log(1-p)-\sum_{\tau=2}^{s} \frac{p^{\tau-1}}{\tau-1}\}$ is the payoff weight put on a future period s>t. This implies that the relative weights put on the current period t and the future periods s>t, $\delta^{rt}_{t}/ (\sum_{s=t+1}^{\infty} \delta^{rt}_{s})$, change as t increases.

Appendix B: Incentives to cooperate in later periods

Do relative incentives to cooperate and defect change as the game progresses beyond the first period? As noted before, under cumulative pay, the gains and losses from defection do not change in periods beyond t=1. Under random pay, the relative gains and losses from cooperation and defection may change in later periods due to the changes in relative weights put on the present and the future. Comparing gains from cooperation and defection under random pay in period t≥2, from (10), the weight put on the current period t is $\delta^{rt}_{t}$, and the future payoff weights are $\sum_{\tau =t+1}^{\infty} \delta^{rt}_{\tau}$. Hence, under the Nash reversion, the players in the PD game will have incentives to cooperate in period t>1 if

$$\delta^{rt}_t (b-a) \leq (a-c) \sum _{\tau =t+1}^{\infty} \delta^{rt}_\tau , $$

or

$$ \frac{\delta^{rt}_t}{\sum_{\tau =t+1}^{\infty} \delta^{rt}_\tau } \leq \frac{a-c}{b-a} , $$

(11)

a condition less demanding than the requirement for cooperation in period t=1. Figure 4 presents a numerical simulation of the current to future payoff weight ratios under continuation probability p=3/4.

Fig. 4

Ratios of the current payoff weight to the future payoff weights, p=3/4

The figure indicates that, for periods t>1, the random payment scheme continues to induce the present-period bias in behavior compared to the cumulative payment scheme, as $\frac{\delta^{rt}_{t}}{\sum_{\tau =t+1}^{\infty} \delta^{rt}_{\tau}} > \frac{1-p}{p}$. However, this bias decreases and $\frac{\delta^{rt}_{t}}{\sum_{\tau =t+1}^{\infty} \delta^{rt}_{\tau}}$ approaches $\frac{1-p}{p}$ from above as t grows. This suggests that incentives to cooperate increase as the game progresses under random pay, but they are never as strong as the incentives to cooperate under cumulative pay.