Unraveling Public Good Games

Abstract

This paper provides experimental evidence on how players predict end-game effects in a linear public good game. Our regression analysis yields a measure of the relative importance of priors and signals on subjects' beliefs on contributions and allows us to conclude that, first, the weight of the signal is relatively unimportant, while priors have a large weight and, second, priors are the same for all periods. Hence, subjects do not expect end-game effects and there is very little updating of beliefs. We argue that the sustainability of cooperation is related to this pattern of belief formation.

1. Motivation

Previous experimental research on public good games has shown that contributions are relatively high in one-shot games (40%–60% of endowment) and they fall over time in finitely repeated public good games (see Davis and Holt [1], Isaac, McCue and Plott [2], Kim and Walker [3], Ledyard [4]). Deviations from the free-riding zero contribution outcome and the decline over time have been rationalized through social preferences, learning effects, strategic considerations or conditional cooperation.¹ Binmore [14] offers an explanation based on social norms. In the case of inexperienced laboratory subjects, the framing of the game triggers a social norm coming from an indefinitely repeated game. The experience acquired in the game through trial and error adjustments changes behavior and may explain the decline in contributions.

Cooperation may survive in an infinitely repeated game, but even in a finite game, if there is a small probability that some subjects are not fully rational, rational subjects may react by contributing in the early periods and stop contributing toward the end of the game (see Kreps et al. [15]). Players may not want to trigger a break in cooperation when the others are contributing, but, of course, this argument is no longer valid as the end of the game approaches and, in particular, when lowering the contribution in the last period cannot trigger any retaliation. This argument can be extended. If players were aware that there would be no consequence from lowering contributions in the last period, and they thought others were likewise aware, they might also realize that lowering contributions in the previous to last period would not trigger any retaliation either. This unraveling would make the finite game equivalent to a one-shot game but it requires common knowledge of rationality. The question is therefore whether subjects do indeed solve the game by backward induction. There is some evidence that subjects find it difficult to apply this type of reasoning. Palacios and Volij [16] find that agents used to the backward induction arguments (chess players) applied it when playing the centipede game, while subjects who were more unfamiliar with it (students) did not use it to the same extent. Using backward induction seems to require some learning. The usual laboratory experiment repetitions of the PGG will not provide that learning as subjects face the end of the game only once. Johnson, Camerer, Sen and Rymon [17] have shown that subjects taught to use backward induction made equilibrium offers in an alternative offer bargaining game when playing with robots; however, when they played with untrained subjects they behaved differently, although closer to the equilibrium offers than prior to training. They conclude that both social preferences and a limited use of backward induction play a role in the discrepancy between the experimental outcome and the equilibrium prediction.

Problems with backward induction are not the only cognitive difficulties faced by players. Understanding the incentives in the one-shot game may also be an issue. Most papers have focused on this last type of limited cognition and on how learning through repetitions of the one-shot PGG mitigate its effects (see Anderson et al. [18], Andreoni [5,6], Brandts and Schram [11], Goeree et al. [19], Houser and Kurzban [7], Palfrey and Prisbey [20,21]). However, little attention has been paid to another source of cognitive limitation in PGG: the fact that subjects are not used to applying backward induction arguments in finite games, nor do they believe that other subjects will use this type of reasoning. To analyze this problem, we focus on the end-game effect² in PGG and subjects' beliefs in this effect.³ Our work confirms the difficulties related to backward induction arguments in finite PGG; a majority of subjects do not predict any end-game effect at all, even when beliefs are elicited after playing the game.

After providing subjects experience with a 5-period repeated public good game and information on the resulting average contribution to the public good within their own group for each period (referred to as signals), we ask them to make predictions of contributions (referred to as guesses or ex-post beliefs) of all the groups that participate in the same 5-period game. We model ex-post beliefs as a linear combination of prior beliefs and the signals observed during the game. Our purpose is to determine how individuals process these two sources of information to establish their beliefs on the behavior of others. Our results suggest that the signal has a low weight in determining ex-post beliefs and, even though subjects experienced an end-game effect, this effect is absent from their ex-post beliefs.

The rest of the paper is organized as follows. Section 2 describes the game and Section 3 the experimental design and procedures. Section 4 presents our main results on average behavior and beliefs and in Section 5, we analyze individual behavior. Section 6 concludes.

2. Description of the Game

Players participate in a public good game in groups of n players. The game is repeated T times. At each stage of the game, player i receives an endowment w and has to decide how to distribute it between a contribution to a public good, c_i, and private consumption, w − c_i. The amount contributed by the n members of a group is multiplied by β, β < n, and equally shared by the n members. Denoting m = β/n, m < 1, the monetary payoff at each stage as a function of the contribution to the public good is: ${\mathit{\pi }}_i=w-c_i+m{\sum }_{i=1}^n{c}_i$.

If subjects maximize their monetary payoff, contributions in the subgame perfect equilibrium are zero. However, experimental research has shown that some subjects behave as conditional cooperators, that is, they are willing to contribute more to the public good game the more others contribute, while others behave as free riders. Assume some players find it optimal to contribute a proportion x ≤ 1 of the average contribution c̅ (conditional cooperators) and the rest are free riders (x = 0),⁴ then for any c that players may initially hold as common belief on the average, contributions should collapse to zero since the belief c̅ and the expected behavior of players (contribute at most c̅) generate a lower estimated average which in turn lowers expected optimal contributions and the estimated average and so on. That is, if players unravel the game their expected average and the optimal contributions would be zero⁵ even if they have social preferences and conditionally cooperate.

Players may not have common beliefs on the average contribution (for example at the first period), but if they all use the observed average in the previous period as their belief on c̅ the same argument applies. In the laboratory, convergence to zero contribution takes time so players do not unravel the game or they do not expect others to do so. The question is then how subjects form expectations about the contribution of others.

The previous argument suggests that the sustainability of cooperation may be more related to how players form beliefs regarding others' behavior than to social preferences so more research is needed on how these expectations are formed. Players may use their priors or the previous period average as an estimate of c̅ and they may also take into account that contributions will be declining over time.

We model belief formation in repeated interactions as follows. At the first stage of the game, subjects maximize their preferences using priors as the estimate of the average behavior and after playing the game, they observe the average contribution and use it as a signal s_i for the average behavior for next period. Players update beliefs using all the information available, priors and the signal, and generate posterior beliefs g_i that will be used as the estimate of average behavior at the next stage of the game ⁶:

$$g_i=\left(1-\mathit{\alpha }\right)p_i+\mathit{\alpha }{s}_i$$

Our empirical research is focused on how players update their beliefs after observing other players' contribution. We also check whether subjects expect an end-game effect.

3. Experimental Design

The experiment was carried out in a single session at Universidad de Granada on 31 May, 2007. Participants were first-year undergraduate students in Economics. The total number of participants was 48 divided into 12 groups. Students were told that they would perform several tasks (See the Instructions in the Appendix 2).

For the first task, subjects played a linear public good game (PGG) in each group for five periods. Subjects were informed that they would be playing with the same partners for the five periods. In each period, the subjects were given an endowment of 100 2-eurocent coins. They were asked to make a decision on how much to allocate to a private account and how much to allocate to a public account. Contributions were expressed in number of coins; thus, they were integer numbers between 0 and 100, c_it ∈ [0,100]. Participants were informed that they could keep any money allocated to the private account for themselves, and this would be independently of the other subjects' actions, while all the money allocated to the public account (the sum of the money allocated by the four members of the group) would be multiplied by 1.5 and then divided equally among the four members. Each participant earned the sum of payoffs obtained in the five periods.

After each period, each subject received feedback privately on his own payoff, π_it. Prior to the new period starting, they were given a new endowment of 100 2-eurocent coins. Figure 1 summarizes the timing of the experiment.

After making decisions on contributions to the public account for 5 periods, and getting feedback on their payoffs, subjects started Task 2.

In Task 2, they were asked about their beliefs regarding the average contributions to the public account (in number of coins) of the 48 participants and for each of the five periods (g_it). We used an incentive scheme according to their errors, ε_it = int c_i − g_it (being int c_i the integer of the observed average and t = 1, …, 5). More precisely,⁷

• If | ɛ_it | > 10 participant i did not receive anything;

• If 5<| ɛ_it| ≤ 10 participant i received 1 euro;

• If 0 < | ɛ_it | ≤ 5 participant i received 2 euros;

• Finally, if ε_it = 0 participant i would receive 20 euros.

Participants were told that only one of the periods chosen at random would determine their payoff for Task 2.

We did not perform a belief elicitation step before Task 1 to avoid any possible effects on contributions.⁸ Since our main interest was to determine how priors are affected by the experience of playing the game and whether the priors or the posterior beliefs incorporate any end-game effect, we chose a design with a low number of periods. This design makes the end-effect very important in the game experience and increases the chances of observing it in the forecasts. Subjects also had enough time to think about what they would do; participants were given a few minutes after each decision, then the feedback about payoffs was received and then the following period would start. In Task 2, subjects had the feedback received in the five periods at their disposal, so that any possible differences in recall between subjects could not introduce any noise in the results. Note that subjects do not observe contributions but may infer the level of group contributions very easily as $\frac{\mathit{\text{feedback}}\cdot 4}{1.5}$. Even if they did not explicitly calculate contributions, it should be noted that our purpose was not to determine whether subjects are accurate in their predictions, but rather only if they could predict the end-game effect. If they observed a decline in profits from the public account, this could only come from a decline in contributions. We did not provide data on contributions to avoid the implicit suggestion that they should use the average of the group in their predictions. This design allows us to measure the relative importance of priors and the subjects' experience in Task 1 (group signals) in the subjects' forecasts in Task 2. The complete experiment lasted about an hour and subjects earned, on average, 13.47 euros.

4. Average Behavior

We first compared actions and elicited beliefs. We checked whether subjects, who had played the PGG for five periods and had received feedback about their own payoff after each period, could accurately predict the mean contribution of the population and to what extent they could predict any end-game effects. Since forecasts were elicited in Task 2, they will be called posterior beliefs.

Figure 2 shows both the average posterior beliefs (over the whole population) and average actions in each period in the 4-player public good game.⁹

The average of contributions in the first three periods is 35.3, which is not very different from the average forecasts, 33.4. In the last two periods, however, there is a discrepancy between average contributions (18.1) and beliefs (29.3), suggesting that the end-game effect observed in contributions in the last two periods was not predicted in Task 2.

Observe that whereas subjects changed their behavior in period 4, this change was not incorporated into posterior beliefs for that period and, despite a small decrease in guesses at the last period (weakly significant), subjects overstated the value of the participants' contribution at the end of the game.

To explore differences between actions and beliefs in period t, we define the error or discrepancy between them as e_it, e_it = c_t − g_it (t = 1, 2, …, 5).

Table 1. Belief accuracy (n = 48).

errors	mean	median	st. dev.
e1	4.02	4.01	19.68
e2	0.97	5.39	18.29
e3	0.81	4.39	17.29
e4	−13.04	−10.63	18.17
e5	−9.37	−5.63	17.01

summarizes these discrepancies.

Recall that positive values indicate low guesses. The mean difference between actions and beliefs was relatively small for the first three periods. However, the average difference increased in periods 4 and 5 and became negative. Subjects did not predict end-game effects, and contributions and guesses diverged. We checked whether c_t and g_t are drawn from the same population using paired non-parametric test. The Wilcoxon test compares c_t and g_t for each round. Z₁ = −0.50 (p-value = 0.61); Z₂ = −0.11 (0.91); Z₃ = −0.28 (0.77); Z₄ = −3.83 (0.00) and Z₅ = −2.96 (0.00). The sign test yields similar results: S₁ = 0.00 (p-value = 1.00); S₂ = 0.00 (1.00); S₃ = −0.43 (0.66); S₄ = −3.89 (0.00) and S₅ = −3.20 (0.00).

Hence, subjects' average beliefs matched average actions fairly well for the first three rounds (despite the high variance of belief accuracy) but failed to do so in T and T-1. In period 4, when the end-game effect was first observed, and in period 5, the differences between the two are statistically significant. Therefore, subjects did not incorporate the decline in contributions to their beliefs, although the small decline in guesses at the last period is weakly significant (see Figure 2 and

Table 2. Evolution of c_t and g_t (n = 48). Non-parametric test for 2-paired samples. Wilcoxon and Sign tests. H₀: data are drawn from the same sample, Significant coefficients are reported in bold; p-v = p-value.

	Wilcoxon		Sign			Wilcoxon		Sign
Δc	Z	p-v	S	p-v	Δg	Z	p-v	S	p-v
c1, c2	−0.02	0.98	0.00	1.00	g1, g2	−0.25	0.80	−0.45	0.65
c2, c3	−0.19	0.84	0.00	1.00	g2, g3	1.30	0.19	−1.22	0.22
c3, c4	−2.34	0.01	−1.83	0.06	g3, g4	−0.15	0.88	0.00	1.00
c4, c5	−0.52	0.60	−0.50	0.61	g4, g5	−1.69	0.09	−1.73	0.08

).

Regarding average behavior, we may conclude that:

• Result 1(a) There was an end-game effect at period T-1.

• 1(b) On average, players did not incorporate the strong decline in contributions to their posteriors beliefs.

Result 1 refers to average behavior. However, different types of players may follow different patterns.¹⁰ We will address this issue in the next section.

5. Individual Results

Figure 2 shows the extent of the end-game phenomenon in aggregate beliefs and contributions. We will now analyze individual behavior and beliefs to explore the question in greater depth. Figure 3 shows the histogram for the discrepancies between contributions and guesses for each period. Non-negative errors are represented in the shaded area. Individuals within the unshaded area overestimated mean contributions, that is, they were optimistic.

The percentage of subjects over- and underestimating contributions remained balanced for periods 1 to 3. However, the percentage of subjects with optimistic predictions increased notably after round 4 and broke the balance.

Figure 2 shows that there was an end-game effect at period 4, but the mean predictions did not incorporate it. We may conclude from Figure 3 that not only the mean but a relevant percentage of the individuals did not predict this decline.

To improve our understanding of these phenomena we will now focus on the period when they lowered contributions and the period when they believed the end game phenomenon would occur. We define a decrease in contributions (in rows) as lowering the contribution to a value (i) lower than 2/3 of the previous value and (ii) lower than 2/3 of the average of the own contribution in previous periods, provided the decrease is maintained up to the last period.¹¹ Identical criterion is used for guesses.

• contributions: 25% (12 out of 48) of subjects decreased their contribution in period 4, 12.5% defected at period 5, but a high percentage of subjects (23%) did not decrease their contribution as the end of the game approached.

• beliefs: 25 out of 48 subjects (52%) did not predict any end-game effect; 10 subjects (21%) believed that the end-game effect would occur at the last period and only one made the right prediction (decline at period 4).

This means that 73% (35 out 48) of the players either predicted the decrease in contributions later than the period in which the decrease took place or they did not predict it at all. This is remarkable since at the time of the prediction they had already seen the outcome of the five periods of the contribution game in their own group of four subjects (although the prediction referred to the average of all participants). Subjects had the opportunity to update their beliefs with the observed behavior in their group, in case they had not predicted ex ante the end-game effect.

• Result (2) Half of the subjects did not incorporate the observed end-game effects into their posterior beliefs.

We will now try to rationalize this result by looking at how posterior beliefs are formed. Beliefs were elicited after playing the PGG so that they must be a combination of ex-ante beliefs (priors) and the signals observed throughout the game. Subjects did not observe other players contributions, but they did observe the part of the payoff that comes from their group contributions to the public account.¹² We define this value as the signal¹³ observed by individual i in group z at each period t:

$$S_{\mathit{\text{it}}}=\frac{1.5\text{∑}_{i\in t}{c}_{\mathit{\text{it}}}}{4}$$

We model ex-post beliefs as a weighted average of prior beliefs and the signal observed in the game for each individual i; the weights represent the relative importance of each source of information:

$$g_{\mathit{\text{it}}}=\left(1-\mathit{\alpha }\right)p_{\mathit{\text{it}}}+\mathit{\alpha }{s}_{\mathit{\text{it}}}$$

where prior beliefs of individual i, p_it. might vary across periods.

As we observe s_it and g_it we can obtain an estimation of α. The assumption is that the weight given to the signal and that given to the priors are the same for all individuals. We estimate the following panel regression with fixed effects:

$$g_{\mathit{\text{it}}}=\underset{\left(1-\mathit{\alpha }\right)p_{\mathit{\text{it}}}}{\underbrace{[{\gamma }_0+{\gamma }_i\text{+}\sum _{t=2}^5{\beta }_t{d}_t]}}+\mathit{\alpha }{s}_{\mathit{\text{it}}}+u_{\mathit{\text{it}}}$$

(1)

where γ₀ is the constant, γ_i are individual fixed effects (reflecting subjects' heterogeneity), and d_t are period dummies allowing priors to be different across periods; the estimated parameters β̂_t will be an indication of how individuals predict the end-game effect (if they do, parameters β̂₄ and β̂₅ will be negative and significant), and u_it is the error term.

Table 3. Regression results. Beliefs, g_it.

beliefs (git)	(1)	(2)	(3)	(4)
signal (sit)	0.11 *** (0.04)	0.14 *** (0.03)	0.12 *** (0.04)	0.12 *** (0.04)
constant	28.80 *** (3.11)	25.95 *** (1.68)	29.28 *** (4.01)	31.09 *** (5.38)
d2	−0.21 (2.38)		−0.16 (2.38)	−0.15 (2.38)
d3	−3.38 (2.42)		−3.27 (2.42)	−3.25 (2.42)
d4	−0.41 (2.74)		−0.11 (2.72)	−0.06 (2.72)
d5	−4.49 (2.76)		−4.18 (2.74)	−4.14 (2.73)
Free-rider			−2.94 (4.53)
End-gamer				−3.83 (5.06)
n = 240	F = 4.15	F = 14.98	F = 23.36	F = 23.46
	p-value = 0.00	p-value = 0.00	p-value = 0.00	p-value = 0.00

^(***)significant at 1%; (st. errors).

shows the regression results. The period dummies are not significant. The implication is that when we separate the effect of the priors and that of the signal, priors are not time dependent, i.e., on average subjects did not predict ex ante a decline in contributions over time.

Eliminating the time dummies from the regression yields the coefficients shown on the second column of Table 3, regression (2). The estimated weight of the signal, α̂, is 14% and the weight of the priors, 1- α̂, is six times larger.¹⁴

We have also added two dummies to check if different types of players have a different behavior concerning beliefs regarding the endgame. Surprisingly, the beliefs of subjects who decreased their contributions at the end of the game (endgamer) do not follow a different pattern. The dummy for free riders (subjects who gave less than 5 in at least three periods) is not significant either.

In regression (2) we may obtain a measure of each individual prior beliefs weighted by (1 − α̂) through the predicted constant and fixed effects: γ̂₀ + γ̂_i (see Equation (1)). We calculate the main statistics for the (predicted) prior beliefs: the mean (std. dev.) is 25.97 (14.47) and the max (min) is 69.3 (−0.8).¹⁵ Therefore, we observe a large heterogeneity in priors across subjects.

Summing up our results in this section,

• Result 3(a) Priors are constant across periods, that is, subjects did not predict ex ante any end-game effects. There is a large heterogeneity in priors.

• 3(b) In the formation of posterior beliefs, the weight given to the signal (–α̂) is relatively low: 10%–15%. Priors are given a much larger weight.

To check the robustness of this result, we considered two alternative signals that the subject could use to update his priors: own contributions or the payoff he received in each period. However, these signals turned out not to be significant.¹⁶ In sum, subjects do not use their own contribution or the payoff as signals to form their posterior beliefs, but the average contribution of their group (the relevant signal is 1.5 times the average contribution of the group, the payoff from the public good).

There were two groups which did not experience any endgame effect. In terms of beliefs, these two groups did not behave differently from the others and therefore having experienced the endgame effect does not seem to affect beliefs. Looking at the endgame effect in beliefs group by group, only in one of the groups the average guess went below the 2/3 rule at the end; there was no endgame effect in guesses in the other 11 groups.

The low weight given to the signal is consistent with the fact that although individuals experienced an end-game effect, they did not guess it after the game. Other papers have found evidence in the same direction: subjects barely update their beliefs (see Kovarik [26]). Given the low weight given to signals, we should not expect large learning effects from repetitions of a finite PGG.¹⁷

6. Conclusions

In the experimental literature on PGG, repetition of the one-shot game has been shown to decrease contributions. Repetition introduces learning effects, strategic considerations and the possibility of punishment for the unfair behavior of others¹⁸ that could be related to the decrease in contributions.

We contribute to this literature on experimental public good games with the idea that the subjects' abilities to unravel the game (or their beliefs on the ability of others to do so) may be an important factor behind the experimental results. We performed this analysis by asking subjects about their beliefs regarding average contributions for each period. The belief elicitation was conducted after the PGG to avoid any interference with contributions.

Our regression analysis allowed us to measure the relative importance of priors and signals on subjects' belief formation. Our main results are that priors are constant for all periods and they have a significant weight compared to the signals observed throughout the game.

Our analysis suggests that, prior to playing the game, subjects do not expect backward induction, not even in the last few periods, and their updating using the observed signals is slow. Therefore, the posteriors beliefs do not incorporate the end-game effect.

Previous papers have studied the reasons behind contributions: conditional cooperation (Fischbacher and Gächter [33]), kindness, altruism or warm-glow vs. errors or confusion (see Croson [32-34]; Andreoni [6]; Houser and Kurzban [7] among others). Our paper focuses on a different kind of confusion: people are not able to predict end-game effects. However, this confusion is not inconsistent with individuals endowed with other-regarding preferences and, more precisely, with subjects who consider that other players could have social preferences. Moreover, we have shown that the rate of decline in contributions and the sustainability of cooperation may be related to how players form expectations.