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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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.

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}

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

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 ^{2}^{3}

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.

Players participate in a public good game in groups of _{i}_{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 ^{4}^{5}

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

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

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 _{i}_{i}^{6}

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.

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

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, _{it}

After each period, each subject received feedback privately on his own payoff, _{it}

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 (_{it}_{it}_{i}_{it}_{i}^{7}

If _{it} |

If _{it}|

If _{it} |

Finally, if _{it}

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}

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.

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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 _{it}_{it}_{t}_{it}

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 _{t}_{t}_{t}_{t}_{1}_{2}_{3}_{4}_{5}_{1}_{2}_{3}_{4}_{5}

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

Regarding average behavior, we may conclude that:

Result 1 refers to average behavior. However, different types of players may follow different patterns.^{10}

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.

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 (^{11}

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.

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}^{13}

We model ex-post beliefs as a weighted average of prior beliefs and the signal observed in the game for each individual _{it}

As we observe _{it}_{it}_{0} is the constant, _{i}_{t}_{t}_{4} and _{5} will be negative and significant), and _{it}

Eliminating the time dummies from the regression yields the coefficients shown on the second column of ^{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 (

In regression (2) we may obtain a measure of each individual prior beliefs weighted by (1 − _{0} + _{i}^{15}

Summing up our results in this section,

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}

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 [^{17}

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 others^{18}

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 [

Timing of the experiment. Task 1, 5 periods of contributions with feedback; Task 2, belief elicitation.

Contributions (task 1) and beliefs (task 2).

Histograms for discrepancies between contributions and guesses. Non-negative errors are represented in the shaded area. The unshaded area indicates overestimation of mean contributions.

Belief accuracy (

e_{1} |
4.02 | 4.01 | 19.68 |

e_{2} |
0.97 | 5.39 | 18.29 |

e_{3} |
0.81 | 4.39 | 17.29 |

e_{4} |
−13.04 | −10.63 | 18.17 |

e_{5} |
−9.37 | −5.63 | 17.01 |

Evolution of _{t}_{t}_{0}: data are drawn from the same sample, Significant coefficients are reported in bold; p-v = p-value.

Δc | Z | p-v | S | p-v | Δg | Z | p-v | p-v | |

c_{1}, c_{2} |
−0.02 | 0.98 | 0.00 | 1.00 | g_{1}, g_{2} |
−0.25 | 0.80 | −0.45 | 0.65 |

c_{2}, c_{3} |
−0.19 | 0.84 | 0.00 | 1.00 | g_{2}, g_{3} |
1.30 | 0.19 | −1.22 | 0.22 |

c_{3}, c_{4} |
− |
0.01 | − |
0.06 | g_{3}, g_{4} |
−0.15 | 0.88 | 0.00 | 1.00 |

c_{4}, c_{5} |
−0.52 | 0.60 | −0.50 | 0.61 | g_{4}, g_{5} |
− |
0.09 | − |
0.08 |

Regression results. Beliefs, _{it}

_{it} |
||||
---|---|---|---|---|

signal (_{it} |
0.11 |
0.14 |
0.12 |
0.12 |

constant | 28.80 |
25.95 |
29.28 |
31.09 |

d_{2} |
−0.21 (2.38) | −0.16 (2.38) | −0.15 (2.38) | |

d_{3} |
−3.38 (2.42) | −3.27 (2.42) | −3.25 (2.42) | |

d_{4} |
−0.41 (2.74) | −0.11 (2.72) | −0.06 (2.72) | |

d_{5} |
−4.49 (2.76) | −4.18 (2.74) | −4.14 (2.73) | |

| ||||

Free-rider | −2.94 (4.53) | |||

End-gamer | −3.83 (5.06) | |||

| ||||

significant at 1%; (st. errors).

We gratefully acknowledge comments and suggestions from three anonymous referees. We also thank comments from Jordi Brandts, Werner Guth, Vitoria Levatti, Maite Martínez-Granado and participants at the Max Planck Institute Jena seminar and Alhambra Experimental Workshop. María Paz Espinosa acknowledges financial aid from MICINN (ECO2009-09120) and Gobierno Vasco, DEUI (IT-313-07); Pablo Brañas-Garza from MICINN (ECO2010-17049) and Junta de Andalucía (P07-SEJ-2547).

Descriptive statistics, contributions and beliefs

| ||||
---|---|---|---|---|

35.27 | 19.98 | 0 | 78 | |

34.42 | 18.23 | 0 | 80 | |

30.58 | 17.90 | 0 | 78 | |

31.42 | 18.17 | 3 | 78 | |

27.25 | 17.02 | 0 | 78 | |

| ||||

| ||||

t = 1 | 39.29 | 36.53 | 0 | 100 |

t = 2 | 35.40 | 33.78 | 0 | 100 |

t = 3 | 31.40 | 33.83 | 0 | 100 |

t = 4 | 18.38 | 24.07 | 0 | 100 |

t = 5 | 17.87 | 28.74 | 0 | 100 |

Granada, May 31, 2007

In this experiment you will perform several tasks.

Task 1 consists of 5 independent rounds. You will be a member of the same 4-member group during the 5 rounds.

At the

Your only decision is how much you want to keep for yourself (Private Account) and how much to assign to a Public Account in your group. Any amount not assigned to the Public Account goes to your Private Account.

The amount you will get from the Private Account is equal to the amount of money you assigned to it and this is independent of the decisions of the other participants.

The amount you will get from the Public Account in your group depends on the sum of the amounts of money assigned to it by all members of your group (that is, the amount you have decided to assign to it plus the amounts that the other 3 members of your group have decided to assign to the Public Account). This sum is multiplied by 1.5 and then divided in 4 parts. Each of these four equal parts goes to a member of the group.

Summing up, the amount of money you win in each round is calculated as follows:

Before we start Task 1, remember that you have to decide how to distribute the money between your PRIVATE Account and the PUBLIC Account of your group.

You will play 5 rounds. Remember that in each round your will have 100 coins of two euro cents (that is, 2 Euros).

We will now start Task 1:

Write in the first row of the form provided (Round 1) how much money you put into the PUBLIC Account.

The amount you write must be between 0 and 100.

After you make your decision (you will have to wait a few minutes) you will be informed of the amount of money you have won in the round (we will fill out the cell on the right, “The amount you win”).

Round 1

Contribution to the PUBLIC Account | The amount you win | |
---|---|---|

Round 1 | xxxxxx | (we fill out this cell) |

Round 2 | ||

Round 3 | ||

Round 4 | ||

Round 5 |

We will now start a second round. You have 100 coins to assign to the Public Account or to the Private Account.

Please write in the second row of the form provided (Round 2) the amount you want to put into the Public Account in this second round.

As before, after a few minutes we will inform you of the amount of money you have won in this second round.

Round 2

Contribution to the PUBLIC Account | The amount you win | |
---|---|---|

Round 1 | ||

Round 2 | XXXXXX | (we fill out this cell) |

Round 3 | ||

Round 4 | ||

Round B |

We will now start a third round. You have 100 coins to assign to the Public Account or ……

Round 3

Contribution to the PUBLIC Account | The amount you win | |
---|---|---|

Round 1 | ||

Round 2 | ||

Round 3 | xxxxxx | (we fill out this cell) |

Round 4 | ||

Round 5 |

We will now start a fourth round. You have 100 coins to assign to the Public Account or ……

Round 4

Contribution to the PUBLIC Account | The amount you win | |
---|---|---|

Round 1 | ||

Round 2 | ||

Round 3 | ||

Round 4 | XXXXXX | (we fill out this cell) |

Round 5 |

Now we start a fifth round. You have 100 coins to assign to the Public Account or …..

Round 5

Contribution to the PUBLIC Account | The amount you win | |
---|---|---|

Round 1 | ||

Round 2 | ||

Round 3 | ||

Round 4 | ||

Round 5 | XXXXXX | (we fill out this cell) |

The five rounds are over.

All the money you have won IS YOURS. We will now start Task 2 and you may earn more money.

Your task is to find out the average contribution to the Public Account of ALL THE PARTICIPANTS in this experiment (including yourself) in each of the rounds.

We want you to guess the average amount that the participants have put into the Public Account in each round. Given that in each round the contribution could be a number between 0 and 100, your guess should also be in that interval.

We will now explain how you can earn money in this task.

If the value you guess is:

If your guess is between 5 and 10 above or below of the actual average, you win 1 euro.

If your guess is between 0 and 5 above or below of the actual average, you win 2 euros.

If your guess is equal to the average contribution (an integer between 0 and 100) you will win 20 euros!!!

otherwise, you do not win anything.

Try to make a good guess in each round because we are going to pay you according to your guess in only ONE of the rounds CHOSEN AT RANDOM.

Work out the average in each round and write the number on the form provided (Task 2).

Remember that you should write down 5 numbers, one for each round. Remember also that each number must be between 0 and 100.

See Andreoni [

Several papers have dealt with the question of end-game effects. Gonzalez

Several papers explore beliefs and elicitation mechanisms in PGG (see for instance Gätcher and Renner [

For example, if players maximize (Bolton and Ockenfels [_{i}_{i}_{i}_{i}

This is similar to a guessing game.

This expression for the updating of beliefs is justified as follows: If the individual has _{1},…, _{k}

Alternative reward functions include quadratic and linear scoring rules and other procedures that correct for risk attitudes (Karni [

The evidence on whether belief elicitation may affect contribution is mixed. See for example Gächter and Renner [

The descriptive statistics of contributions and beliefs can be found in the

Previous work on PGG has shown evidence of subjects' heterogeneity. For instance, Fischbacher

The actual decrease in the average contribution in period 4 was from (39.3; 35.4; and 31.4) to (18.4; 17.9) which fulfills this criterion. Small changes in the threshold do not change results (choosing 0.6 or 0.7 leaves results almost unchanged).

Concerning accuracy when beliefs are elicited period by period or at the end of the game, the average absolute estimation error in Croson [

An alternative signal could be the subjects' payoffs (private + public account). We also used this variable as the signal (see footnote 16).

If we normalize signals to reflect average contribution of the group:

The unweighted values are 30.20 and 80.58, respectively.

Using the individual payoffs as signals yields a coefficient 0.02 (_{it}

This is also consistent with the low speed of learning observed in the centipede game (see Palacios and Volij [

See Andreoni [