10032021  Original Paper  Issue 2/2021 Open Access
Bygones in a public project
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 Social Choice and Welfare > Issue 2/2021
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1 Introduction
Contributions to a public good that are made in incremental amounts over a period of time are very prevalent in real life, including donations to charities and even countries’ contributions to mitigating climate change. For example, in the context of the Paris Agreement, countries take stock of their contributions every five years and set new target contributions for the next five years. This stocktaking every cycle of five years is equivalent to a round of play in which information over the incremental contributions of the coplayers becomes available. The targets for the next cycle of 5 years are nonbinging pledges, while the actual contribution efforts are yet to be made during the next round of five years. Therefore, the setting of the Paris Agreement can be seen as a dynamic game in which contributions are made in increments to reach a threshold of contributions, i.e. the level of emissions that will keep the temperature below the famous two degrees celsius target.
This dynamic contribution setting allows players to condition their contributions on the contributions made by other players in the past. This is in contrast to the static setting in which the contributor makes a single contribution decision simultaneously with other players’ contributions. The experimental literature on dynamic contributions games suggests, directly or indirectly, that total contributions in a dynamic setting exceed the contributions in the static one (Duffy et al.
2007). One explanation for this is provided by Schelling (
1960), who argues that incremental contributions allow coplayers to test each others’ trustworthiness for a small price. This hypothesis has been experimentally tested by Duffy et al. (
2007) who find that contributions made dynamically without feedback between the contribution rounds are statistically equal to those made in a dynamic setting with feedback; however, they are significantly higher than those made in a static setting. This finding is puzzling because the dynamic contribution setting without feedback is theoretically equivalent to the static setting and, therefore, the contributions in these settings should not differ from one another.
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What is, nevertheless, the common feature between the dynamic settings with feedback and the dynamic setting without feedback is the sunk character of the contributions made in previous rounds of play. In the static setting, on the other hand, there is no opportunity to sink contributions. If some contributions are already sunk, a rational decision maker disregards them, such that current contributions are independent of past contributions. This, in turn, leads to more subsequent contributions especially when a target overall contribution is set, as is the case with a threshold public good. Unfortunately, Duffy et al. (
2007) only report how individuals condition their contributions on coplayers’ past contributions, but not how they condition their contributions on their own past contributions. Therefore, it is not possible to make conclusions regarding individual responses to sunk contributions.
In this paper I test whether it is the individual rationality rather than, or in addition to, the conditionality on others’ behavior that explains the higher production of the public good in the dynamic setting compared to the static one.
^{1} In particular, I ask if the sunk individual contributions can explain the better performance of the dynamic setting in collecting contributions to goods that are provided only if a threshold is met. For this I set up an experiment in which subjects make contributions over two stages to a threshold public good. The experimental treatments differ with respect to the action space available in the second stage. In one treatment subjects have the option to withdraw the contributions made in the first stage, in part or entirely. This amounts to the possibility of making negative contributions in the second stage and it means that the firststage contributions are not sunk. In the second experimental treatment subjects cannot withdraw their firststage contributions and can only make null or strictly positive contributions in the second stage. Hence, the firststage contributions are sunk.
Because the purpose is to test whether the sunk character of the first stage individual contributions induces higher subsequent contributions, equal conditions should be ensured in the first stage. This is achieved by providing subjects with the same information before the firststage contributions are decided. In particular, in the
noinformation condition the participants know that the option to withdraw the firststage contributions occurs with 50% probability and that it is equally likely that they cannot withdraw these contributions in the second stage. As a control I include an
information condition in which participants know before making their firststage contributions whether these can be withdrawn or not in the second stage. Thus, the experimental design keeps the dynamic character of the contribution decisions, which was made responsible for the higher contributions by the previous literature, but it isolates the effect of the sunk or nonsunk character of these contributions.
The results show that the secondstage contributions are, indeed, higher when the firststage contributions are sunk as compared to when they are not sunk, in both informational conditions. Moreover, the difference in the secondstage contributions between the sunk and the nonsunk treatments increases with the amount contributed in the first stage for both informational conditions and more so for the noinformation condition. This is evidence that players respond positively to their own sunk contributions. Thus, these results support the hypothesis that the opportunity to sink contributions in the dynamic setting explains, at least partially, its better performance relative to the static one. However, in this experiment I do not find evidence of sunk contributions improving the provision of the public good. In particular, I fail to find statistical evidence that the total group contributions or the project’s success rates differ across treatments. However, this result does not invalidate the fact that individual player’s response to own sunk contributions may explain the better performance of the dynamic setting. Recall that in the setting of this experiment both treatments have a dynamic nature and although the static setting and the nosunk treatments are theoretically equivalent, there may be behavioral differences which are not the objective of this paper.
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Finally, contrary to Duffy et al. (
2007), I do not find evidence of subjects conditioning their contributions on the previous contributions of their coplayers. However, I find that a player’s contribution in the second stage increases in her expectations about her coplayers contributions in the same stage. This effect is stronger and statistically significant in the “no information” condition, showing that the beliefs about other players’ contributions substitutes for some of the uncertainty present in this informational condition.
The remainder of the paper is organized as follows as follows. The next section discusses the related literature. Section
3 outlines the experimental design, the theoretical predictions and the hypotheses to be tested. In Sect.
4 present the results, using both nonparametric tests and regression analysis. Section
5 concludes and discusses the limitations of the study.
2 Related literature
As explained above, this paper is motivated by the finding of Duffy et al. (
2007) who set up an experiment to test two hypotheses that explain why the dynamic publicgood game provides higher contributions than the static one. The two hypotheses are the “smallpriceof trust” hypothesis (Schelling
1960, pp. 45–46) and the condition that a jump in payoff should exist at the completion point (Marx and Matthews
2000). While the authors find that the dynamic game leads to higher contributions than the static one, they also find that none of the above hypotheses explains this result. Moreover, a dynamic game without feedback about the group’s contribution between the contribution rounds, yields higher contributions than the static one, but similar to the dynamic contributions with feedback. Similar conclusions emerge from Dorsey (
1992) when comparing their realtime contribution environment with a provision point with the static one of Isaac et al. (
1989). Using a provision point mechanism, in which the public good is binary and the contributions are allornothing, Goren et al. (
2003,
2004) also conclude that a realtime protocol of play provides higher contributions than the static play. In a realtime provision environment players can make contributions at any time during a time window. Hence, not only are the contributions dynamic, but the order and timing of the contributions are determined endogenously by the players themselves. Thus, this is different from the dynamic setting used in the current paper.
An important feature relevant for the current study is that the abovementioned papers study dynamic contributions with and without the possibility to withdraw previouslymade contributions. Specifically, Dorsey (
1992) compares these contribution institutions both for a linear payoff function and for a payoff function with a jump at the provision point. The author finds no significant difference between the two institutions for neither of the payoff functions, although contributions are higher when there is no possibility to withdraw (in the language of the current paper, the previous contributions are sunk). By contrast, the realtime reversible contributions in Goren et al. (
2004) reach the provision threshold significantly less often than in Goren et al. (
2003) in which alreadymade contributions remain sunk (Goren et al.
2004). However, the difference with Dorsey (
1992) and with the current study is that the contribution decisions are constrained to be binary, i.e. contribute all endowment or nothing. Battaglini et al. (
2016) also study experimentally the effect of the reversibility versus irreversibility of contributions to a dynamic nonlinear public good. They find that the setup with irreversible contributions leads to a higher production of the public good than the reversiblecontributions setup, although contributions decline over time in both environments. Similar conclusions are reached by Kurzban et al. (
2001) who use an environment with realtime contributions to a linear public good game.
These papers, however, differ in at least three main respects from the current study. First, in Dorsey (
1992) and Goren et al. (
2003,
2004) contributions are made in continuous rather than in discrete time. While the continuous time setting may be more realistic, the discrete time protocol implemented in the current study eliminates issues related to reaction to lastsecond decisions and attention to status updates, allowing to identify the pure effect of sunk contributions. Second, in Goren et al. (
2003,
2004) contributions are binary, thus considerably limiting the actions space of the players. Using continuous contributions instead, allows studying the effect of the previous sunk contributions on the subsequent ones, both in the direction of increased and decreased contributions. Third, the games played in Battaglini et al. (
2016) and Kurzban et al. (
2001) differ considerably from the game of the current paper, of which the type of the public good and the payoff functions, hence the equilibria, are the most notable differences. The choice of a threshold public good game in this paper is explained in Sect.
3.
The literature that studies the effect of seed money, i.e. money that are already collected before asking contributions from donors, also bears similarities with the current paper, to the extend that the firststage contributions can be regarded as seed money from the perspective of the second stage. For example, testing a previously advanced theory, List and LuckingReiley (
2002) conduct a threshold public good field experiment in which their three treatments differ with respect to the percentage of the threshold that represents the seed. Indeed, they find that contributions and participation to the donation campaign increase monotonically in the amount of seed money. Bracha et al. (
2011), on the other hand, find that seed money provided by a firstmover in a sequential contribution game does not have a positive effect on individual contributions, but it increases the likelihood of provision if the threshold is sufficiency high. Donazzan et al. (
2016) also find a positive effect of seed money on participation, but find no effect of seed money on the average donation size, similar with Verhaert and den Poel (
2012) and Rondeau and List (
2008). Nevertheless, the channels through which seed money affects contributions, most notably through reciprocity and trust, differ from the ones investigated in the current paper. This is because seed money is a prior contribution provided by other donors and not by the solicitee herself. By contrast, the experiment of this paper investigates the effect of the contributions made by a donor in the past on her own current contributions.
Finally, the literature that investigated sequential contributions environments in which players contribute to the public good in turns is also somewhat related to the current study. This literature has studied the effect of the information about the history of play available to the players (e.g. Erev and Rapoport
1990; Steiger and Zultan
2014), the role of refunds in a threshold public good game (Coats et al.
2009) or the role of players’ asymmetry (Gächter et al.
2010). It should be noted however, that while the sequential setting has a dynamic character in the sense that players have the chance to observe the behavior of the coplayers before making their decisions, this is different from the notion of dynamism used in this paper. In the dynamic game used in this experiment players make sequential decisions in stages, but in each stage all players make simultaneous decisions.
3 Experimental design
Groups of
n symmetric players contribute simultaneously over two stages to a threshold public good, with
T denoting the provision point. Each player has the same endowment of play
w, from which she can make contributions to the threshold public good. The good is provided if the sum of all players’ contributions over the two stages exceeds the provision point, in which case every player receives a bonus
B, regardless of her contribution. There is no refund of the contributions made over the two stages if the threshold is not reached or if it is overreached.
The choice of a threshold public good for this experiment has a twofold motivation. First, it allows for comparisons with the previous literature that motivates the current research and which also uses threshold public good games to compare dynamic and static contributions settings. Second, the nature of the research question makes the threshold public good game the appropriate game. This is because it allows for a straightforward derivation of player’s best response in the second stage and thus, for clear predictions of behavior both with and without sunk contributions, as it will become clear below. Moreover, while the previous literature has employed several rounds of contributions (e.g. in Duffy et al. (
2007) the dynamic setting treatment has four stages of contributions), for the purpose of the current paper, in which the response to own sunk contributions is searched for, employing only two rounds of contributions has the advantage of making the actual response to own past contributions more clearcut, by reducing the multiplicity of equilibria. A further advantage is reflected in the data analysis because it allows the tested hypotheses to be directly derived from the theoretical predictions, as it is shown in Sect.
3.1.
The experimental treatments differ in two dimensions. First, in order to identify the effect of the sunk contributions, the groups differ in whether the firststage contribution is sunk or not when players decide on their secondstage contribution. I label these treatments
Sunk and
NoSunk, respectively. Thus, in the
Sunk treatment players can only make weakly positive contributions in the second stage, while in the
NoSunk treatment they can also withdraw fully or partially the contributions made in the first stage by making negative contributions. Second, in order to isolate the pure sunkcontribution effect, the firststage contributions should be identical between the two conditions. This is achieved by providing the same information before the firststage contributions, i.e. players know their secondstage contribution options of withdrawing the firststage contribution or not only after the firststage contributions are completed. Before they make their firststage contributions, players know the two possible secondstage options and that each of them has equal chance of being realized. This is the noinformation condition which will be labeled
NoInfo. As a control, in the information condition, labeled
Info, players know their secondstage options with certainty before making their firststage contributions. This yields the twobytwo design summarized in Table
1.
Table 1
Summary of the experimental treatments
Treatment

Information condition



Information

No information


Firststage contributions are sunk

InfoSunk

NoInfoSunk

Firststage contributions are
not sunk

InfoNoSunk

NoInfoNoSunk

In what follows I continue to refer to the differences regarding the firststage contributions being sunk or not as
treatments and at the differences along the exante information as
(informational) conditions. In all treatments and conditions, players are informed about each players’ individual contributions, as well as about the total group contributions after the first stage. In fact, the history of the firststage contributions is available to the players while making their secondstage contributions.
3.1 Theoretical analysis and parameters
Let
\(g_i^t\) be the contribution of player
\(i\in \{1,\dots ,n\}\) in stage
\(t\in \{1,2\}.\) Then, the individual payoff of player
i is given by:
where
\(G_{i}\) is the total contribution to the public good over the two stages by all players except for player
i. Hence,
\(G_i+G_{i}\) is the total contribution of the group to the public good. Finally, function
is the individual return to the public good.
$$\begin{aligned} U_i(G_i,\,G_{i},\,T)=wG_i+r(G_i+G_{i}),\,\, \text {with}\,\, G_i=g_i^1+g_i^2, \end{aligned}$$
(1)
$$\begin{aligned} r(G_i+G_{i})=\left\{ \begin{matrix} B, &{} \quad \text {if} &{} G_i+G_{i}\ge T \\ 0, &{} \quad \text {if} &{} G_i+G_{i}< T \\ \end{matrix} \right. \end{aligned}$$
(2)
I further focus on subgame perfect Nash equilibria. In the static version of the game, the best response of player
i to the total contribution by the other players is given by:
Let us now analyze the bestresponse function of player
i to the contributions of the other players in the second stage, both when players can withdraw and when they cannot withdraw the firststage contributions. I denote by
\(G^t=\sum _{i=1}^{n}g_i^t\) the total group contribution made in stage
\(t=1,\,2\) and by
\(G^t_{i}=\sum _{j=1, j\ne i}^{n}g_j^t\) the total contribution made in stage
t by all players expect for player
i.
$$\begin{aligned} g_i(G_{i})=\left\{ \begin{matrix} TG_{i}, &{} \quad \text {if} &{} TG_{i}\le \min \{B, w\} \\ 0, &{} &{} \text {otherwise}\\ \end{matrix} \right. . \end{aligned}$$
The
NoSunk treatment, in which players can withdraw their firststage contributions, is equivalent with the static game. Therefore, player
i completes the projects only if her total contribution over the two stages does not exceed the value of the project
B, and withdraws her firststage contribution otherwise. Formally, accounting for the budget constraint, her bestresponse function for the contribution in the second stage is given by:
Then, the conditions for an efficient equilibrium, i.e. in which no player contributes more than the value of the project and the threshold is exactly met, are:
In the
Sunk treatment, regardless of the informational condition, the individual firststage contribution is sunk and, therefore, it does not matter for the contribution decision in the second stage. In this case, player
i completes the project in the second stage if the remaining difference to the threshold, given the total contribution of the coplayers, does not exceed the value of the project
B and satisfies the budget constraint. Hence, player
\(i's\) bestresponse function reads:
Thus, the secondstage contribution depends on the firststage contribution only if the budget constraint binds. If the available funds are sufficient to complete the project, then the player does not take into account the contribution made in the first stage. It follows that the conditions for an efficiency equilibrium in this case are:
Comparing (
3) with (
4) it is easy to see that, for any level of the firststage contribution, the secondstage upper limit of the bestresponse contribution of player
i is always higher in the
Sunk treatment than in the
NoSunk treatment, provided the endowment is sufficiently high, i.e.
\(w>B\). If
\(w\le B\), the two treatments allow for the same level of contributions in the secondstage, given the same level of the firststage contributions. In sum, ceteris paribus, the upper bound of the bestresponse contribution is weakly higher in the
Sunk treatment than in the
NoSunk treatment.
^{2}
$$\begin{aligned} g_i^2(G_{i}^2)=\left\{ \begin{matrix} TG^1G_{i}^2, &{} \, \text {if} &{} {TG^1G_{i}^2 \le \min \{B  g_i^1, w  g_i^1\}}\\  g_i^1, &{}&{} \text {otherwise} \\ \end{matrix} \right. . \end{aligned}$$
(3)
(i)
\(\sum _{i=1}^{n}g_i^1+\sum _{i=1}^{n}g_i^2=T \text { and }\)
(ii)
\(g_i^1\le B, \text { with } g_i^2\le \min \{B  g_i^1, w  g_i^1\}, \forall i\)
$$\begin{aligned} g_i^2(G_{i}^2)=\left\{ \begin{matrix} TG^1G_{i}^2, &{} \, \text {if} &{} {TG^1G_{i}^2 \le \min \{B, wg_i^1\} }\\ 0, &{}&{} \text {otherwise} \\ \end{matrix} \right. \end{aligned}$$
(4)
(i)
\(\sum _{i=1}^{n}g_i^1+\sum _{i=1}^{n}g_i^2=T \text { and }\)
(ii)
\(g_i^1\le B, \text { with } g_i^2\le \min \{B, wg_i^1\}, \forall i\)
Because theoretically the players are symmetric, i.e. they have the same endowments, the same value for the public good and the same payoff function, it is worth at this point to mention the symmetric efficient subgame perfect equilibria. These are the equilibria for which the public good is exactly provided and the cost of the public good is shared among the group members, i.e.
\(g_i^1+g_i^2=g_j^1+g_j^2={T}/{n},\,\forall i\ne j.\) There is an infinity of these equilibria and they differ with respect to the profile of the individual contributions over the two stages.
The experimental parameters are borrowed from Croson and Marks (
2000) are:
\(n=5,\,B=50,\,w=55\) and
\(T=125.\) Note that these parameters ensure that the project is feasible, i.e. the total endowment of the players exceeds the threshold
\(T\le nw\) and that efficient equilibria exist, i.e. the total benefit from the project exceeds the provision point
\(nB\ge T.\) Note also that
\(w>B\) such that in an efficient equilibrium the secondstage upper limit of the bestresponse contribution of player
i is strictly higher in the
Sunk treatment than in the
NoSunk treatment for the same level of the firststage contribution.
^{3} For an illustration, Fig.
1 shows the bestresponse functions derived in Eqs. (
3) and (
4) for the above parameters and the history of play
\(g_i^1=10\) and
\(G_{i}^1=40.\)
×
We are now ready to formulate the experimental hypotheses. As demonstrated by the bestresponse functions, contributions in the second stage should be larger in the
Sunk treatment than in the
NoSunk treatment. However, because the exante information plays a role for the decision of the first stage contributions, the expectation is that they will differ significantly between the two treatments in the
Info condition, but not in the
NoInfo condition. Therefore, the following two hypotheses result:
Hypothesis 1
Holding the firststage individual contributions constant, in the
Info condition the secondstage individual contributions are larger in the
Sunk than in the
NoSunk treatment.
Hypothesis 2
In the
NoInfo condition, the secondstage individual contributions are larger in the
Sunk than in the
NoSunk treatment.
If Hypothesis
2 is confirmed, then the rate of project completion should be higher in the
NoInfoSunk treatment than in the
NoInfoNoSunk treatment. Note also that, while the payoff structure is the same in the
Sunk treatment as in the
NoSunk treatment, the confirmation of Hypotheses
1 and
2 solely depends on subject’s rationality regarding sunk contributions, which remains to be observed in the experiment.
3.2 Procedure
The experimental sessions were conducted in a computer laboratory at the University of Kassel, in Germany. The participants were randomly selected from the pool of volunteers recruited from the general student population, with no prior experience of participating in economic experiments. The recruitment software was ORSEE (Greiner
2004) and the experiment was programmed in zTree (Fischbacher
2007). No subject took part in more than one session and subjects were randomly assigned to treatments, i.e. a betweensubject design was implemented.
After being seated at the computer stations, the instructions were read aloud to ensure common knowledge.
^{4} The participants were randomly assigned to groups of five which stayed fixed throughout the experiment. The randomization was executed by the experimental software. In a single session was conducted either the
InfoSunk treatment, the
InfoNoSunk treatment or the
NoInfo condition in which groups were randomly assigned to the
Sunk and
NoSunk treatment. Thus, in the
NoInfo condition the randomization for the two treatments was done at session level.
Each experimental session consisted of two parts. While the introduction informed the participants that the experiment consisted of two parts, they received the instructions for the second part only after the first part was completed. The first part elicited subjects’ risk preferences according to Holt and Laury (
2002) menu of lottery choices. The riskpreference elicitation task was played for real stakes such that the earnings in this part of the experiment were added to the final earnings of the participants. However, the outcome of the lottery and the earnings from this task were announced to the participants only at the end of the experiment, after the main experimental task was completed. Therefore, this task could not affect the decisions made in the main experimental task.
The second part of the experiment consisted of the main threshold public good game, which was played in one single round. Before the participants played the oneshot threshold public good game, they answered control questions to ensure a good understanding of the rules of the game and to ensure that the participants were aware of the available strategies and their implications. Any questions that occurred during this time were answered privately. After all participants answered the control questions correctly, they tried out the game in four trial rounds played against the computer. The instructions made it clear that in the trial rounds the computer played the role of their coplayers, i.e. dummy players. The contributions of the computer playing on behalf of the other four coplayers were the same for all participants, in all treatments and were set before the experiment. In particular, in order to experience all possible scenarios in the game and to understand their pivotal role in reaching the threshold, the four scenarios implemented in the trial rounds involved two rounds with low contributions by the dummy players and two rounds with high contributions by the dummy players. Hence, the choice of the scenarios, with both low and high contributions by the coplayers ensures that the trial rounds do not have any influence on the actual behavior of the players in the subsequent oneshot game. However, even if there is suspicion of the trial rounds influencing the subsequent behavior in the game, this influence is the same across all treatments.
^{5} Only when the trial rounds were completed, did the experiment proceed with the actual game. During the game, contributions and earnings were displayed in Taler and it was common knowledge that the final payments would be calculated by applying an exchange rate of 20 Euro cents per Taler.
Expectations about coplayers’ contributions were elicited as follows. Before deciding on their firststage contributions, subjects in all treatments were asked to give their best guess of whether their group would complete the project or not. After submitting their firststage contributions, but before receiving feedback about the other group members’ contributions, the subjects gave their guesses about the average contribution of their coplayers in this stage and over the two stages together. After completing their secondstage contribution decisions, the subjects were asked again to give their guesses about the secondstage average contributions of their coplayers. In fact, the questions about beliefs came as a surprise immediately after the contribution screen and just before the screen that displayed the group’s contribution in each stage. Guesses about the completion of the project were not incentivized. Guesses about others’ contributions were incentivized such that, with a tolerance of one Taler, each correct guess was rewarded with 5 Taler. However, earnings and thus the outcome of the beliefs elicitation tasks were only displayed at the end of the experiment, together with the total earnings from the main experimental task and from the riskpreference elicitation task. The reader may be concerned about the riskpreference and beliefs elicitation tasks creating a hedging problem and thus affecting the decisions in the main experiment (see, Blanco et al.
2010). Note that the small stakes of these tasks relative to the stakes of the main experimental task, as well as the very small tolerance range in the belief elicitation task, minimize concerns about hedging incentives.
^{6}
In total, 240 students took part in the experiment. This means a total of 48 groups distributed as follows: 10 groups in the
InfoSunk treatment, 9 groups in the
InfoNoSunk, 13 in the
NoInfoSunk and 16 in
NoInfoNoSunk.
^{7} At the end of the experiment, the participants were paid their earnings privately, in cash. The total earnings over the two parts of the experiment ranged from 3 to 23 Euro with an average of 13 Euro.
4 Results
The results are structured as follows. First I analyze the individual contributions in the two stages of the game separately and then I discuss the group outcome over the two stages. At each stage I use nonparametric and regression analysis, controlling for the expectations about coplayers’ contributions and, in the second stage, for coplayers’ firststage contributions.
Before presenting the results, Table
2 shows, for each of the four treatments, the summary statistics with respect to the observable demographic variables collected via the final questionnaire of the experiment, together with the measure of the risk aversion. The last column in the table shows the
p values of the Kruskal–Wallis equalityofpopulations rank test, which tests whether at least two of the four treatment groups differ significantly from each other. As this test shows, there are no significant differences across the treatments with regard to these observables.
Table 2
Means of observables by treatment
Treatment

InfoNoSunk

InfoSunk

NoInfoNoSunk

NoInfoSunk

p value


N

45

50

80

65


Male

0.49 (0.075)

0.48 (0.071)

0.55 (0.056)

0.60 (0.061)

0.5408

Age

25.69 (0.675)

26.08 (0.603)

25.61 (0.493)

25.98 (0.492)

0.7440

HL safe choices

5.47 (0.197)

5.28 (0.297)

5.54 (0.173)

5.80 (0.224)

0.6632

In the analysis that follows I will report
p values from the Wilcoxon ranksum (Mann–Whitney), unless otherwise specified.
4.1 Firststage behavior
×
In the analysis of the firststage behavior, the firststage contributions in the
NoInfo condition are pooled together. This can be done for two reasons. First, as shown in Table
2, there are no statistically significant differences in subjects’ observables across the treatment groups. Second, in the first stage of contributions there are no differences in subjects’ experience across the two conditions of this treatment. Thus, the bars in Fig.
2 show the individual mean contributions in the first stage for the two treatments in the
Info condition and for the
NoInfo treatment. The errors bars represent the
\(95\%\) confidence interval of the mean contributions. As this figure shows, the firststage contributions are significantly lower in the
InfoSunk treatment than in the
InfoNoSunk treatment (
\(p=0.000\)), reflecting the noncommitment feature of the
NoSunk treatment. In fact, the firststage contributions in
InfoNoSunk are not statistically different from the fairshare of 25 Taler (Wilcoxon signedrank test
\(p=0.571\)). This shows that in riskfree conditions, people naturally direct their behavior towards fairness.
Table 3
Individual contributions in Stage 1 (OLS regression)
Dependent variable: Individual Contributions in Stage 1



(1)

(2)

(3)


InfoSunk

− 2.354

− 3.795

− 2.433

(1.371)*

(1.091)***

(1.099)**


InfoNoSunk

8.877

7.633

5.736

(1.273)***

(1.457)***

(1.508)***


Guess completion

11.114

6.795


(1.208)***

(1.651)***


Guess Stage 1

0.444


(0.114)***


Guess Total

0.184


(0.080)**


Constant

15.434

7.540

− 0.191

(0.780)***

(1.078)***

(1.361)


Adjusted
\(R^2\)

0.16

0.41

0.53

N

240

240

240

The regressions in Table
3 explain the individual firststage contributions as a function of the treatment conditions, in which the
Sunk and the
NoSunk treatments of the
NoInfo condition are pooled together and form the reference group. As regression (1) of this table shows, the contributions in the
NoInfo condition, including both the
Sunk and the
NoSunk treatment, are larger than in the
InfoSunk, but they are significantly smaller than in the
InfoNoSunk treatment. These results are intuitive and they reflect the firststage uncertainty of the
NoInfo condition in which the subjects do not know whether in the second stage their firststage contributions become sunk or not. Regressions (2) and (3) in Table
3 include further controls. Regression (2) shows that in the reference group, i.e. the
NoInfo condition, believing that her group will complete the project prompts a player to contribute on average 11 Taler more than a player who does not expect a project completion. This is both economically and statistically significant. Recall that these beliefs were elicited before the player made her firststage contribution. The last column of the table includes player’ beliefs about coplayers contributions in the first stage and in total over both stages. The results show that both beliefs affect positively and significantly players’ contributions in the first stage. Note that these beliefs were elicited after the firststage contributions were made, but before the contributions of the coplayers were revealed.
4.2 Secondstage behavior
I now turn to analyzing the secondstage contributions which are the focus of the research hypotheses of this study. The bars in Fig.
3 show the distributions and the smooth lines show the kernel density estimations of the second stage contributions by treatment. As seen in Sect.
4.1, in the
InfoNoSunk treatment subjects contribute on average their fairshare in the first stage. However, they withdraw some of these contributions in the second stage (see the negative contributions in Fig.
3) such that on average they are lower than the positive contributions in the
InfoSunk.
The unconditional treatment effect shown in regression (1) of Table
4 confirms this result, i.e. the subjects in the
Sunk treatment contributed significantly more in the second stage than the subjects in the
NoSunk treatment. However, since this test does not condition on the firststage contributions, it provides only a partial support for Hypothesis
1.
×
Similarly, regression (4) of Table
4 shows that in the
NoInfo conditions, the secondstage contributions in the
Sunk are significantly higher than in the
NoSunk treatment. This provides support for Hypothesis
2, showing that subjects rationally ignore the sunk contributions made in the first stage.
^{8} This effect does not seem to be explained by a larger crowding of zero contributions in the
Sunk treatment, as the lowest possible contribution in this treatment, compared to the
NoSunk treatment or, in general, by the larger action space of the latter treatment compared to the former. First, as Fig.
3 shows, there are no differences in the frequencies of zero contributions between the
NoInfoNoSunk and the
NoInfoSunk treatments (40% in
NoInfoSunk versus 35% in
NoInfoNoSunk,
\(p=0.5371\)). Second, only 15% of the contributions made in the second stage in the
NoInfoNoSunk treatment were negative. Third, while the frequencies of strictly positive contributions in the two treatments are not statistically different (
\(p=0.2308\)), they are higher in the
NoInfoSunk treatment than in
NoInfoNoSunk treatment (
\(p=0.0586\)).
In the remaining regressions of Table
4 I control for the contributions made in the first stage.
^{9} Furthermore, for the ease of interpretation, the variable for the firststage contributions (
mStage1) is centered around the average contribution within each group, i.e. it is expressed as the difference of player
i’s contribution from the average contributions of his/her group in the first stage, i.e.
\(g_i^1\frac{1}{n}\sum _{j=1}^{n}{g_j^1}\). In addition, I control for the contributions made by the coplayers in the first stage and the beliefs about the average contributions made by the coplayers in the second stage. Finally, the specifications in columns (3) and (5) allow for the
Sunk treatment to vary with the contributions make in the first stage by interacting it with the dummy for the
Sunk treatment of each informational condition.
Table 4
Treatment effects (OLS regression)
Dependent variable: Individual contributions in stage 2



Info

NoInfo


(1)

(2)

(3)

(4)

(5)

(6)


Sunk

10.080

0.299

0.240

4.400

3.765

3.703

(2.577)***

(3.487)

(3.426)

(1.870)**

(1.850)*

(1.853)*


mStage 1

− 0.827

− 1.054

− 0.237

− 0.422


(0.135)***

(0.140)***

(0.109)**

(0.115)***


Sunk
\(\times\) mStage 1

0.347

0.375


(0.202)

(0.147)**


Others Stage 1

− 0.189

− 0.189

0.032

0.035


(0.096)*

(0.096)*

(0.065)

(0.065)


Beliefs Stage 2

0.240

0.250

0.437

0.479


(0.114)**

(0.115)**

(0.137)***

(0.128)***


Constant

− 0.400

16.275

16.233

3.200

− 3.040

− 3.638

(2.433)

(8.668)*

(8.760)*

(1.390)**

(5.459)

(5.350)


Adjusted
\(R^2\)

0.202

0.419

0.421

0.045

0.225

0.248

N

95

95

95

145

145

145

Regardless of the specification, the secondstage contributions in the
NoSunk treatments are decreasing in the firststage contributions in both informational conditions (see the coefficient of
mStage1), albeit at a lower rate in the
NoInfo condition. Specifically, the more a player contributed above her group’s average in the first stage, the less she contributed in the second stage. This is explained by the fact that due to the nonsunk character of the firststage contributions, the contributions in this treatment are essentially made in one stage (static game). Therefore, higher contributions in the first stage bring lower contributions in the second stage and viceversa.
The effect of controlling for the firststage contributions is that for both informational conditions the magnitude of the unconditional treatment effect diminishes, and significantly so for the
Info condition for which it also becomes statistically insignificant. For the
NoInfo condition, however, the drop in magnitude is negligible and the coefficient remains statistically significant. Specifically, subjects that contributed close to the group’s average contribution in the first stage, contributed on average about 4 Taler more in the second stage if they were in the
Sunk treatment as compared to those in the
NoSunk treatment.
×
Moreover, as the interaction terms in columns (3) and (6) show, the difference between the
Sunk and the
NoSunk treatments increases the further away we move above the group’s mean firststage contribution, but this is statistically significant only for the
NoInfo condition. Moreover, the difference between the two treatments is larger in the
NoInfo condition than in the
Info condition.
^{10} This means that, ceteris paribus, the more a player contributed in the first stage, the larger is her secondstage contribution in the
Sunk treatment compared to the
NoSunk treatment. To visualize these effects, Fig.
4 shows the linear predictions of the secondstage contributions as a function of the deviations of the firststage contributions from their mean, at the average values of the covariates.
^{11} At all levels of the firststage contributions that are above the average group’s contribution, the contributions in the
Sunk treatment are higher than in the
NoSunk treatment. This can be interpreted as high contributors, i.e. those who contribute over the group’s average, having a rational behavior towards completing the project after learning that the firststage contributions are irreversible. This finding indicates that the reaction of the contributors who withdraw their contributions in final stages in response to the behavior of freeriders is not the only explanation for the differences between a setting with revocable and one with irrevocable contributions as in Goren et al. (
2004). Instead, the individual response of the contributors to their own previous sunk contributions adds to this explanation.
Remarkably, in the
NoInfo condition, the contributions made by the coplayers in the first stage fail to predict the contributions made by a player in the second stage, i.e. the coefficients on
Others Stage 1 are not statistically significant and are economically null. These coefficients are negative, but only marginally significant in the
Info condition indicating a tendency to freeride on the coplayers that signal to be high contributors through their firststage behavior. This is unlike the evidence reported by Duffy et al. (
2007) who, in a setting with complete information and irreversible contributions, found that players respond positively to previous positive contributions of their coplayers. What seems to matter more, however, is the expectations about the coplayers’ contributions in the second stage. This increases significantly one’s contribution in the second stage and this effect is stronger, both economically and statistically, in the
NoInfo than in the
Info condition.
^{12}
4.3 Group outcome
At group level, the firststage contributions in the
NoSunk treatment are significantly higher than in the
Sunk treatment, for each of the two informational conditions (
\(p=0.0351\) for
NoInfo and
\(p=0.0002\) for
Info). While in the
Info condition, this difference may be explained by the perfect knowledge regarding the secondstage options, there is no explanation for why this difference exists in the
NoInfo condition. The direction of the comparisons is reversed for the secondstage contributions such that they are lower in the
NoSunk treatment than in the
Sunk treatment (
\(p=0.0392\) for
NoInfo and
\(p=0.0006\) for
Info). As a result, there are no statistically significant differences in the total group contributions over the two stages between the
Sunk and the
NoSunk treatments for either of the two informational conditions. In fact, only
\(40\%\) of all groups reached total group contributions of 125 or higher, i.e. completed or overcompleted the project. In the
NoInfoSunk treatment
\(50\%\) of the groups completed the project, while in the
InfoNoSunk treatment
\(55\%\) of the groups reached the provision threshold. This share amounts to
\(31\%\) in both treatments of the
NoInfo condition.
×
This can be grasped from Fig.
5 in which the left picture shows the cumulative distribution and the right picture shows the estimated probability distribution of the total group contributions for each treatment. In both pictures the vertical dashed line shows the provision point of 125 Taler. The black lines correspond to the
Info condition and the gray lines correspond to the
NoInfo condition. In both informational conditions the continuous lines show the probability distribution for the
Sunk treatment and the dashed lines show the same for the
NoSunk treatment.
Remarkable from Fig.
5 is the change in the steepness of the cumulative density functions around the level of 100 Taler of contribution, indicating that most groups reached at least this contribution level. Indeed, the majority of the groups, i.e.
\(71\%\), reached total contributions of at least 100 Taler. This is a special contribution point because it means that, on average, one out of the five group members did not contribute her fair share of 25 Taler, leading to a miscoordination problem. This type of miscoordination seems to have been more sever in the
NoInfo condition as confirmed by the peak of the probability distributions around the 100 Taler contribution point.
^{13} Although these distributions are not statistically different from each other, neither within nor across the informational conditions, the
Sunk treatments of each condition exhibit higher frequencies in the regions of high contributions than do the respective
NoSunk treatments, i.e. in the region of 100 Taler in the
NoInfo condition and 125 Taler in the
Info condition.
^{14} This result is an indication of the fact that making contributions sunk from one stage to another can improve the overall provision of the good, though the current experiment fails to find a statistically significant effect of sunk contributions on total group contributions.
5 Discussion and conclusion
This paper investigated players’ responses to own past contributions in a threshold publicgood game in which contributions are spread over two stages, i.e. a dynamic contribution setting. Unconditional tests show that previouslysunk contributions induce higher subsequent contributions, even when it is known in advance of the firststage contributions that they will become sunk in the second stage. Furthermore, players for which past contributions are sunk contribute subsequently more to the public good than those whose past contributions are not sunk, and this is more so the higher their past contributions. In the twostage contribution game used in this paper, this difference is larger for those players who do not know exante whether their firststage contributions become sunk or not in the second stage.
Note that in Duffy et al. (
2007), Goren et al. (
2003,
2004) and Dorsey (
1992) players know in advance of the first contribution stage whether their past contributions become sunk or not in the second stage. This means that in these papers only the
Info treatment is implemented. Another important difference with the previous studies is that in the experiment reported here there were only two opportunities to give, while in the previous studies there are multiple such opportunities. In only two opportunities to give, players have also fewer opportunities to test others’ willingness to give. Therefore, when the firststage contributions are not sunk and this is known exante, subjects in this experiment test the trustworthiness of their coplayers “all the way”, i.e. on average, they make the fairshare contribution right in the first stage. This creates a significant gap in the firststage contributions between the two sunk conditions of the fullinformation treatment.
This study failed to find that the total group contributions in the dynamic setting with sunk past contributions are higher than the contributions made in a dynamic setting that is theoretically equivalent to a static one, i.e. in which past contributions are not sunk. This finding is in line with Duffy et al. (
2007) in which the contributions made in a dynamic game that is theoretically equivalent to a static game are no different from those made in a pure dynamic game. However, the purpose of the current study was not to show that sunk contributions lead to overall higher group contributions. Instead, the experiment was set to show that the individual responses to own sunk contributions may explain the better performance of the dynamic play, in which past contributions are sunk, as compared to the static play, in which there is no opportunity to sink contributions. Moreover, as already mentioned, the experiment reported here had only two opportunities to give. If each opportunity makes it more likely for players to fructify their next opportunity to give, then more rounds of contributions would bring about higher completion rates. Indeed, Choi et al. (
2008) find that extending the horizon of contributions improves the provision rates, and significantly so the higher the provision threshold. Additionally, increased overall contributions could also result from providing interstage feedback only about the total group contributions, as in Duffy et al. (
2007), and not about individual contributions. This may have the effect of focusing players’ attention on the group’s goal and minimizing antisocial reactions to individual freeriders.
Finally, the low performance at group level across all treatments and conditions is likely due to player’ failure to coordinate on one of the multiple completion equilibria. While repeated play would have provided information on how players learn to coordinate, this endeavor was out of the scope of this paper. Since, the response to sunk contributions is an individual behavior, this study was concerned with understanding the individual response to own past contributions and not on how players coordinate in a social context. Therefore, one round of play provides the crudest form of this response, unaltered by a learning effect. While, these are all interesting avenues for understating the effect of individual biases on social outcomes, I leave them for future research.
Acknowledgments
This work has been supported by the European Union (EU) Horizon 2020 Program, action ERC2014STG, Project HUCO, grant number 636746.
Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit
http://creativecommons.org/licenses/by/4.0/.
Appendix 1: Experimental instruction
These are the experimental instructions for the
NoInfo treatment (translated from the German language). The instructions for
InfoSunk and
InfoNoSunk are very similar.
Welcome to the experiment and thank you for your participation!
General instructions
In this experiment you can earn money. Your payment will be determined by the course of the game, which means that it depends on your own decisions and on the decisions of your coplayers. Now carefully read the following rules of the game. If you have questions, please raise your hand. We will come to you and answer your questions.
All decisions in this experiment are anonymous. For the payment of your earnings you will have to sign a receipt. The receipts are needed only for billing and accounting. However, under no circumstance will we connect your names with the decisions in the experiment.
Important rules:
1.
From our side: NO DECEPTION. We promise that this experiment will be conducted exactly as described in these instructions. This is the rule for all experiments that are conducted in KLab. We can publish our results only when we follow this rule consistently and under all circumstances.
2.
From your side: NO COMMUNICATION. Please do not communicate with other participants during the experiment and take your decisions individually. Your mobile phones and other communication devices must be switched off during the entire experiment. The experiment will run at computers. You are only allowed to use those features of the computer that are needed for the course of the experiment. Communication with other participants, the use of mobile phones and the use of other functions of the computer than those required for the experiment lead to exclusion from the experiment and the loss of all earnings.
Part 1
The experiment consists of two parts. These are the instructions for Part 1. You will receive the instructions for Part 2 as soon as Part 1 is concluded. The two parts of the experiment are independent from each other. Your earning from both parts will be paid to you in cash right at the end of the experiment. The payment will made in private such that no other participant can see how much you have earned. After the experiment you will be asked to answer a short questionnaire.
Your earnings in Part 1 depend only on your individual decisions. The decision table below shows ten decisions. Every decision is a choice between “Option A” and “Option B”. You will make the ten decisions at the computer and document them by clicking either on “Option A” or on “Option B”. Before you made the ten decisions, please take a look at how exactly these decisions influence your earnings.
Decision

Option A

Option B


1

With 1/10 Probability: 2.00 Euro;

With 1/10 Probability: 3.85 Euro;

With 9/10 Probability: 1.60 Euro

With 9/10 Probability: 0.10 Euro


2

With 2/10 Probability: 2.00 Euro;

With 2/10 Probability: 3.85 Euro;

With 8/10 Probability: 1.60 Euro

With 8/10 Probability: 0.10 Euro


3

With 3/10 Probability: 2.00 Euro;

With 3/10 Probability: 3.85 Euro;

With 7/10 Probability: 1.60 Euro

With 7/10 Probability: 0.10 Euro


4

With 4/10 Probability: 2.00 Euro;

With 4/10 Probability: 3.85 Euro;

With 6/10 Probability: 1.60 Euro

With 6/10 Probability: 0.10 Euro


5

With 5/10 Probability: 2.00 Euro;

With 5/10 Probability: 3.85 Euro;

With 5/10 Probability: 1.60 Euro

With 5/10 Probability: 0.10 Euro


6

With 6/10 Probability: 2.00 Euro;

With 6/10 Probability: 3.85 Euro;

With 4/10 Probability: 1.60 Euro

With 4/10 Probability: 0.10 Euro


7

With 7/10 Probability: 2.00 Euro;

With 7/10 Probability: 3.85 Euro;

With 3/10 Probability: 1.60 Euro

With 3/10 Probability: 0.10 Euro


8

With 8/10 Probability: 2.00 Euro;

With 8/10 Probability: 3.85 Euro;

With 2/10 Probability: 1.60 Euro

With 2/10 Probability: 0.10 Euro


9

With 9/10 Probability: 2.00 Euro;

With 9/10 Probability: 3.85 Euro;

With 1/10 Probability: 1.60 Euro

With 1/10 Probability: 0.10 Euro


10

With 10/10 Probability: 2.00 Euro;

With 10/10 Probability: 3.85 Euro;

With 0/10 Probability: 1.60 Euro

With 0/10 Probability: 0.10 Euro

At the end of the experiment the computer will randomly generate two numbers between 1 and 10. The first number determines which of the ten decisions will be used for payment. The second number determines the earnings from the option you chose: A or B. That means that although you make ten decisions, only one will be used for payment. Every single decision has the same chance of being drawn. Since you do not know in advance which decision will be drawn, you should make every decision as if it determines your payment.
Please look at decision 1 in the table. Option A provides a gain of 2.00 Euro if the random number is 1 and a gain of 1.60 Euro if the random number is 2–10. Option B provides a gain of 3.85 Euro if the random number is 1 and a gain of 0.10 Euro if the random number is 2–10. The other decisions are similar, whereby the chances for a higher gain rises as you go further down in the table. For the decision number 10 in the last raw of the table the second random number is not needed since the probability for the highest gain is one. Thus, the decision is between 2.00 Euro in Option A and 3.85 Euro in Option B.
In summary, you will make ten decisions of which only one will be used for payment at the end. For each decision in the table, you must choose between Option A and Option B. When you are done, click “Next”.
You will learn the two computer generated random numbers and your gain in Part 1 only after completing Part 2.
Part 2
In Part 2 you are a member of a group of
five players. That means that apart from you there are four other players in your group. Each player faces exactly the same decision problem. Each player receives 55 Taler. You will see these Taler in your
private account displayed on the topright corner of your computer screen. In the game, you will decide whether you want to contribute your Taler to a
common project. It is possible to contribute any integer amount from 0 to 55 Taler. The Taler that you do not contribute to the common project remain in your private account.
If the group contributes in total
125 Taler or more to the common project, then every player in the group receives a
bonus of 50 Taler. This bonus is then added to the Taler remaining in the private account, regardless of how much a player contributed to the common project. Your profit in this case is:
If the group contributes in total
124 Taler or less to the common project, then there is no bonus. Your profit in this case is:
Below we describe a few numerical examples.
$$\begin{aligned} {{\varvec{Your\, profit\, = \,Remaining\, Taler\, in\, private\, account\, +\, 50\, Taler}}} \end{aligned}$$
$$\begin{aligned} {{\varvec{Your\, profit\, =\, Remaining\, Taler\, in\, private\, account}}} \end{aligned}$$
Example 1: Suppose that the other four players in your group contribute in total 80 Taler to the common project. If you contribute 10 Taler, your payoff is 45 Taler (= 55 − 10). If you contribute 25 Taler your payoff is 30 Taler (= 55 − 25). If you contribute 45 Taler your payoff is 60 Taler (= 55 − 45 + 50).
Example 2: Suppose that the other four players in your group contribute in total 100 Taler to the common project. If you contribute 10 Taler, your payoff is 45 Taler (= 55 − 10). If you contribute 25 Taler your payoff is 80 Taler (= 55 − 25 + 50). If you contribute 45 Taler your payoff is 60 Taler (= 55 − 45 + 50).
Example 3: Suppose that the other four players in your group contribute in total 120 Taler to the common project. If you contribute 10 Taler, your payoff is 95 Taler (= 55 − 10 + 50). If you contribute 25 Taler your payoff is 80 Taler (= 55 − 25 + 50). If you contribute 45 Taler your payoff is 60 Taler (= 55 − 45 + 50).
Please notice the following important rules of the game. The game will be played over
two stages. Your contribution to the common project is
the sum of your contribution in Stage 1 and your contribution in Stage 2. If, for example, you contribute 5 Taler in Stage 1 and 10 Taler in Stage 2, then your contribution to the common project is 15 Taler.
In Stage 1 you can contribute to the common project any integer amount from 0 to 55 Taler. After all players have chosen their contributions in Stage 1, the individual contributions of all players in Stage 1 as well as the sum of all contributions in Stage 1 will be shown on the computer screen. The own contribution for each player will be shown in boldface.
In Stage 2 there are two options. The computer decides randomly which of the two options realizes for your group, while each option has equal chance. In Option 1 it is only possible that you leave the current contribution unchanged or you contribute more Taler to the common project. In Option 2 it is possible that you leave the current contribution unchanged, that you contribute more Taler to the common project or that you withdraw from your contribution make in Stage 1. If you want to contribute more Taler, any contribution from 0 to the remaining funds in your private account is possible. You will see the funds remaining in your private account displayed in the upperright corner of the computer screen. If you want to withdraw (and this is possible) from the contribution made in Stage 1, you must enter a negative value as your contribution in Stage 2. For example, if you want to withdraw 5 Taler, then you must enter − 5 in the entry field. You can only withdraw maximum as many Taler as you contributed in Stage 1. The withdrawn Taler will be added back to your private account.
On the computer screen it will be shown which of the two options in Stage 2 is realized for your group. Please note that in Stage 1 you do not know yet which of the two options will be realized. You will only learns this at the beginning of Stage 2. The realization of the option is valid for the whole group. This means that either all players in your group can withdraw the previously made contributions or nobody in your group can do this.
After all players have made their contributions in Stage 2, the individual contributions of all players in both stages as well as the sum of all contributions in both stages are displayed on the screen. The own contribution will be shown in boldface for each player. Note that the group as a whole must contribute at least 125 Taler to the common project over both stages such that the bonus is realized for all players.
The Taler earned in the experiment will be converted in Euro. You will receive 1 Euro for every 5 Taler (0.20 Euro per Taler). If, for example, you earn in total 60 Taler, then you receive 12 Euro.
Before the game in Part 2 begins, we would like to ask you to answer a few control questions at the computer. This way we can make sure that all participants have understood the rules of the game. If you have questions, please raise your hand. We come to you and answer the questions.
After answering the control questions, there will be
four trial rounds, in which you can try out the game. The trial rounds are
not relevant for your payment. Please note that in these trial rounds your four coplayers will be played by the computer. Therefore, you cannot draw
any conclusions about the behavior of your future coplayers.
Appendix 2
See Table
5 and Figs.
6,
7.
Table 5
The difference in the treatment effects across the informational conditions (OLS differenceindifferences regression)
Dependent variable: individual contributions in Stage 2



(1)

(2)


Info

− 0.225

− 0.182

(2.023)

(2.042)


Sunk

2.742

2.689

(1.938)

(1.941)


Info
\(\times\) Sunk

1.730

1.667

(2.540)

(2.549)


mStage 1

− 0.446

− 0.599

(0.105)***

(0.109)***


Sunk
\(\times\) mStage 1

0.412


(0.141)***


Info
\(\times\) Sunk
\(\times\) mStage 1

− 0.419


(0.154)***


Others Stage 1

− 0.086

− 0.084

(0.069)

(0.070)


Beliefs Stage 2

0.329

0.356

(0.094)***

(0.092)***


Constant

5.811

5.524

(5.362)

(5.394)


Adjusted
\(R^2\)

0.273

0.290

N

240

240

×
×
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Footnotes
1
Assuming that subjects have the intention to participate in providing the public good and, thus, make early contributions, the failure of continuing to participate is a manifestation of the fallacy of sunk costs. If the fallacy is observed, the situation is similar to the one in which a competitive firm exits the market prematurely, when the price is above the average variable cost, but below the average total cost. In the short run the fixed costs, which are included in the total costs, are sunk.
2
Note that within the same treatment there are no differences with respect to the set of equilibria across informational conditions.
3
An example of a symmetric subgame perfect efficient equilibrium is in this case
\((g_i^1=0; g_i^2=25),\forall i=1,\dots 5.\)
4
The experimental instructions for the
NoInfoSunk and the
NoInfoNoSunk treatments are reproduced in Appendix
1 (translation from German). The instructions for the other treatments are very similar and they are available from the author upon request.
5
The contributions of the coplayers in the trial rounds where the following: Trial round 1 (low)—Stage 1: 0; 5; 0; 4. Stage 2: 2; 1; 8; 1. Trial round 2 (low)—Stage 1: 2; 3; 4; 1. Stage 2: 1; 2; 2; 4. Trial round 3 (high)—Stage 1: 10; 9; 25; 20. Stage 2: 6; 15; 8; 12. Trial round 4 (high)—Stage 1: 13; 12; 16; 10. Stage 2: 11; 13; 10; 15.
7
I aimed at 10 groups per treatment. This was not realized due to noshows in one of the
InfoNoSunk sessions and to the randomization in the
NoInfo condition, which was done by the experimental software at session level. For the latter reason I had to run more sessions than planned in order to equalize the sample sizes across the treatments of this condition.
9
As the scatter plots in Fig.
6 suggest, the relationship between the first and the secondstage contributions is linear.
10
In fact, as the regression in column (2) of Tables
5 in Appendix
2 shows, the treatment effect in the
NoInfo condition increases significantly relative to the
Info condition, the further away we move above the group’s mean firststage contribution. This is given by statistically significant negative coefficient of the interaction term between the informational condition, the treatment dummy and the distance from the group’s mean firststage contribution.
11
The figure is based on the models with interaction terms in columns (2) and (4), since these models provided the highest explanatory power according to the adjusted
\(R^2.\) The shaded areas represent the
\(95\%\) confidence intervals of the linear predictions. For each condition, the darkgray area pertains to the
Sunk treatment, while the lightgray area refers to the
NoSunk treatment.
12
Although the subjects face uncertainty about the secondstage options, individual firststage contributions in the
NoInfo condition were not found to correlate with the measure of the risk aversion used in this paper (
\(corr=0.1411\),
\(p=0.0904\)). Moreover, the number of safe choices in the riskpreference elicitation task where not statistically significant in any of the specifications presented in Table
4 and, therefore, omitted.
14
In the
NoInfo condition
\(69\%\) of the groups reached 100 Taler in the
Sunk treatment and
\(56\%\) of the groups reached this level in the
NoSunk treatment.