1 Introduction

A growing number of experimental studies focus on issues related to probabilistic losses associated with the provision or maintenance of public goods. Motivated by issues of climate change, several previous studies have examined the behavioral response to variations in exogenous probabilities of group losses (Milinski et al. Reference Milinski, Sommerfeld, Krambeck, Reed and Marotzke2008, Reference Milinski, Röhl and Marotzke2011; Barrett and Dannenberg Reference Barrett and Dannenberg2012). Other contributions, more relevant to this study, examine endogenous probabilistic losses. In particular, Dickinson (Reference Dickinson1998) and Gangadharan and Nemes (Reference Gangadharan and Nemes2009) examine provision-point public goods settings where the probability of provision of a public good increases in contributions. In addition, Walker and Gardner (Reference Walker and Gardner1992) and Blanco et al. (Reference Blanco, Lopez and Walker2016b) explore the relevance of endogenous probabilistic losses in appropriation game settings.

An important example of the relevance of endogenous probabilistic losses in appropriation settings relates to the provision or maintenance of ecosystem services, which have the characteristics of public goods in that they yield positive externalities to a population. In this context, probabilistic loss externalities are relevant in a wide collection of settings (see TEEB 2010) where increasing pressure by resource users results in an increased likelihood that a major ecosystem disturbance occurs and compromises the capacity of the ecosystem to generate ecosystem services or even to survive. The vulnerability of ecosystems to appropriation pressures is dependent upon a number of factors, including the geographical location, the ecosystem network, and the level of biodiversity. Similar issues of vulnerability and probabilistic losses apply to the provision of public goods, for example mitigation investments to dampen the effects of climate change.

In the context of an appropriation setting, this experimental study examines how subjects respond to changes in the magnitude of an endogenous probabilistic loss of a public good, where the probability of occurrence of the loss decreases with greater cooperation. Specifically, we examine loss parameters that entail losing 10, 50 or 90% of the value of the public good maintained through cooperation. The study makes several important contributions to the social dilemma literature. First, while previous experimental research has compared the response of subjects to endogenous probabilities of public good provision to settings with exogenous probabilities (Dickinson Reference Dickinson1998; Gangadharan and Nemes Reference Gangadharan and Nemes2009) and to settings without a probabilistic component (Gangadharan and Nemes Reference Gangadharan and Nemes2009; Blanco et al. Reference Blanco, Lopez and Walker2016b), the experiments reported here are the first to address the response of subjects to manipulations in the magnitude of the endogenous probabilistic component. Thus, we provide the first results on the quantitative response to the magnitude of endogenous losses rather than just to the existence of the endogenous component.

Secondly, to our knowledge, no previous social dilemma study has identified the asymmetry in behavior we observe between subjects who are optimistic about group performance and those who are pessimistic. On average, we show that own cooperation increases with forecasts of total group cooperation (and the corresponding marginal incentives to cooperate), and the treatment effects show that the quantitative response is greater the larger the magnitude of the loss parameter. Moreover, our results provide a novel insight by discovering significant differences in the response to variations in the loss parameter depending on whether subjects are pessimistic or optimistic about group behavior. Pessimistic subjects reduce cooperation the higher the magnitude of the loss parameter and their decisions are tied systematically to changes in the marginal incentives that correspond to their expectations of others’ appropriation. Optimistic subjects are more cooperative, but their decisions are not systematically tied to changes in marginal incentives that correspond to changes in expectations of others’ appropriation.

These novel results add to the emergent experimental literature that explores individual differences in the responses to marginal incentives and reciprocity (see for example, Brandts and Schram Reference Brandts and Schram2001; Fischbacher et al. Reference Fischbacher, Gächter and Fehr2001; Goeree et al. Reference Goeree, Holt and Laury2002; Brandts et al. Reference Brandts, Saijo and Schram2004; Blanco et al. Reference Blanco, Haller, Lopez and Walker2016a, Reference Blanco, Lopez and Walkerb). These studies, like ours, address decisions in a menu game setting where subjects report cooperation levels for variations in marginal incentives or others’ cooperation. This attribute of the design allows us to examine within subject decisions in regard to how they respond to changes in game parameters. Moreover, like most of these studies (Fischbacher et al. Reference Fischbacher, Gächter and Fehr2001; Goeree et al. Reference Goeree, Holt and Laury2002; Blanco et al. Reference Blanco, Haller, Lopez and Walker2016a, Reference Blanco, Lopez and Walkerb), we report one-shot decisions. This allows us to abstract from group dynamics related to strategic play across decision rounds, and thus avoid the complexity of modeling subject’s responses to the dynamics of a repeated game. This type of setting is reminiscent of individual decisions in large group settings where there is limited or no knowledge of decisions by others and where group dynamics play little role in decision making.

In the games examined, we use a “take some” frameFootnote ¹ where appropriation leads to (1) deterministic losses, by reducing the value of a shared group resource, and to (2) endogenous probabilistic losses, where greater appropriation increases the probability that the shared resource faces an additional loss in value.Footnote ²

Given the existence of an endogenous probabilistic loss, the expected value of the shared resource to an individual and the expected harm to others from appropriation is endogenously defined by the first order beliefs of others’ appropriation and exogenously defined through parameter variations in the magnitude of the probabilistic loss. By eliciting subjects’ expectations of other’s behavior, the experimental design allows for estimation of the relationship between changes in expected marginal incentives and appropriation decisions. In addition, for control purposes, the experimental design includes a “benchmark game” without the possibility of a probabilistic loss and fixed marginal incentives to appropriate. Using subjects’ decision in this game as a measure for “baseline cooperation”, we are able to examine individual responses to the addition of a probabilistic loss and to the changes in the magnitude of that loss.Footnote ³

The studies cited above that focus on endogenous probabilistic losses vary in regard to how they approach the issue that expected marginal incentives change as group behavior changes. More specifically, Walker and Gardner (Reference Walker and Gardner1992) focus on game continuation, and not on individual subject responses to expectations of marginal incentives. Both Dickinson (Reference Dickinson1998) and Gangadharan and Nemes (Reference Gangadharan and Nemes2009) focus on expected per capita return of contributions, based on one-period lagged behavior, implicitly assuming that expectations of current round group contributions are based on behavior in the previous round. Blanco et al. (Reference Blanco, Lopez and Walker2016b) introduces the deterministic and probabilistic degradation games used in this study and investigate the response to variations in subjects’ private benefits in settings without probabilistic losses and in settings with a probabilistic loss of 50% of the shared resource. This previous paper explicitly links expected value of marginal net benefits to subjects’ forecast of other group members’ appropriation. In contrast to the present study, that study does not examine the response to changes in the magnitude of the probabilistic loss nor does it examine differences in individual responses for optimistic and pessimistic subjects on group performance, as defined herein.Footnote ⁴

2 Decision settings and parameters

The experimental design included four one-shot decisions from a menu of games (part A), an incentivized first-order belief-elicitation task related to each of the games (part B), a risk aversion task (part C) and a dictator donation to charities (part D). In part A, incentives in all games are measured in Experimental Currency Units (ECUs). In these games, groups of n = 4 individuals face allocation decisions between a “Group Fund” and an “Individual Fund.” Each four member group begins with a Group Fund endowment of w = 100 tokens, where every token left in the Group Fund has a value of g = 2 ECUs. Each individual begins the game with 0 tokens allocated to their Individual Fund. Individuals privately decide how many tokens to move from the Group Fund which are then placed in their Individual Fund, with a maximum appropriation limit of e = 25 tokens per individual. Each token an individual i moves from the Group Fund, in a given treatment condition j, yields a private benefit increasing the value of his/her Individual Fund by h = 1 ECU. Each token left in the Group Fund has a value of g/n = 0.5 ECUs for every member of the group and thus appropriation generates a deterministic degradation to the group of g. Concurrently, appropriation generates a probabilistic degradation, implemented as a hazard rate that depends on the aggregate number of tokens appropriated from the Group Fund. Subjects confront a fractional loss L of the total value remaining in the Group Fund after all decisions are final. The endogenous probability of this loss occurring is $\left(p{\sum }_{i=1}^n{z}_i\right)$ , where p = 0.01 the fractional increase in the probability associated with each token appropriated from the Group Fund. The feasible range of values of $p\in \left(0,1\right)$ and $L\in \left(0,1\right)$ .

Letting $z_{ij}$ denote the amount individual i appropriates from the Group Fund in treatment j, Eq. (1) presents the payoff to individual i in ECUs. The probabilistic degradation externality is described in the last component of Eq. (1).

(1) ${\pi }_i^j=h{z}_{ij}+\frac{g}{n}w-\frac{g}{n}Z-\left(\frac{g}{n}\right)\left(L_j·\left(p\text{Z}\right)\left(w-\text{Z}\right)\right)$

where $Z={\sum }_{i=1}^n{z}_{ij}$ , and $p\text{Z}\le 1$ . Ceteris paribus, the experimental design varies L across games in Part A, with L = 0.10, 0.50 and 0.90, and p = 0.01 in all cases.Footnote ⁵ We refer to these treatment conditions as L10, L50 and L90, and the benchmark game, where L = 0 as L0. A total of 111 subjects participated in these sessions.Footnote ⁶

The instructions for each game in Part A, as well as quizzes to check subjects’ understanding of the games, were presented sequentially (see the Electronic Supplementary Material). As in Brandts and Schram (Reference Brandts and Schram2001), it was the subjects’ choice to determine the order in which he/she made decisions in the games of part A. Importantly, at any point during decision-making in part A, subjects had the opportunity to review and change any of the choices they had already made. After all participants had time to finalize their decisions, the experimenter announced the end of part A, after which no one was allowed to change their decisions.

Part B was an incentivized belief elicitation task following Croson (Reference Croson2007), in which subjects were asked to report a forecast of the average per-person appropriation level for the other members of their group for each of the four games in part A. Subjects learned of the details of part B only after completing part A, with no feedback of results from part A. While making their forecasts, subjects could refer to a copy of their own decision-making sheet from part A.

Part C was a risk elicitation task that was a modified version from Dohmen et al. (Reference Dohmen, Falk, Huffman and Sunde2010), with the stake sizes used in Balafoutas et al. (Reference Balafoutas, Kerschbamer and Sutter2012). In this task subjects had to choose between a certain payment or a lottery yielding 5 Euros or 0 Euros, each with a 50% probability. Subjects made a total of 10 decisions, where the amount they received in the certain payment increased from 0.5 to 5 Euros in 50 cent intervals.

Part D was a dictator task with charities as recipients, where subjects had to allocate 3 Euros between themselves and one (or several) of eight charities offered to them. The decision sheet included a list of the charities as well as a short description of their mission. In order to circumvent the issue that some subjects might prefer to donate to one of the charities following the experiment, subjects were informed that the experimenter would increase the amount a subject allocated to the charities by 25%.

After finishing part D, subjects answered a short questionnaire. Payments were based on one of the games in each of the parts A, B, and C, and the amount of money subjects kept for themselves in part D. All drawings used for determining the games for computing experiment earnings were made in public. Subjects were paid in private in cash.Footnote ⁷

3 Expected marginal incentives

Based on the payoff functions given in Eq. (1), the marginal net benefit ( $MN{B}_i^j$ ) of appropriation for individual i in treatment j is:

(2) $MN{B}_i^j=\frac{\partial {\pi }_i^j}{\partial z_i}=h-\frac{g}{n}-\frac{g}{n}{L}_{j}p·\left(w-2\text{Z}\right)$

where $p\text{Z}\le 1$ . Notice, with p = 0.01 in all games, the probabilistic nature of the game implies that the magnitude of $MN{B}_i^j$ depends on aggregate group appropriation Z, and the parameter L. Thus, based on differences in first order beliefs of others’ behavior, subjects facing the same parameter values will face different marginal incentives to appropriate.Footnote ⁸ Table 1 displays the specific functional relation between $MN{B}_i^j$ and aggregate group appropriation for each of the treatment conditions j = L0, L10, L50 and L90. Figure 1 displays the value of $MN{B}_i^j$ at each possible level of group appropriation, as well as illustrating how it changes across treatment conditions.

Table 1 Decision settings: parameters and marginal net benefits

Decision setting	L	p	Marginal net benefit functions
L0 (benchmark game)	0	0.01	$MN{B}_i^{L0}=0.5$
L10	0.1	0.01	$MN{B}_i^{L10}=0.45+0.001·\text{Z}$
L50	0.5	0.01	$MN{B}_i^{L50}=0.25+0.005·\text{Z}$
L90	0.9	0.01	$MN{B}_i^{L90}=0.05+0.009·\text{Z}$

Parameters n = 4, w = 100, e = 25, h = 1 are constant in all games

Fig. 1 Marginal net benefits as a function of aggregate group appropriation

Of course, as noted above, some subjects may also respond to incentives beyond their own pecuniary return. As discussed, appropriation by subject i creates a deterministic and probabilistic negative externality on other group members. More specifically, the marginal harm to each other group member −i from appropriation by subject i, $M{H}_{-i}^j$ , is based on the last two components in Eq. (2):

(3) $M{H}_{-i}^j=\frac{\partial {\pi }_{-i}^j}{\partial z_i}=\frac{\left(n-1\right)g}{n}+\frac{\left(n-1\right)g}{n}{L}_{j}p·\left(w-2\text{Z}\right)$

5 Discussion of results and conclusions

This study examines how subjects respond to changes in the vulnerability of a shared resource to appropriation pressures. Experimentally we vary the magnitude of a probabilistic loss of a group fund, where the probability of occurrence of the loss increases in appropriation. Thus, this study adds to the experimental literature that examines subjects’ responses to endogenous probabilistic losses in social dilemmas (Walker and Gardner Reference Walker and Gardner1992; Dickinson Reference Dickinson1998; Gangadharan and Nemes Reference Gangadharan and Nemes2009; Blanco et al. Reference Blanco, Lopez and Walker2016b). We contribute to this literature by providing the first results on the quantitative response to variations in the magnitude of endogenous losses rather than just to the existence of the endogenous component.

We find that average group cooperation increases as the loss parameter (L) increases; with important heterogeneities in individual behavior. Controlling for individual decisions in the benchmark game without probabilistic losses, we observe a behavioral difference between those subjects who forecast lower levels of group cooperation versus those who forecast higher levels of group cooperation. In particular, subjects who are pessimistic regarding others’ appropriation appropriate at higher levels on average (84% of their appropriation capacity), and respond systematically and significantly to changes in their expectations of others’ appropriation. Subjects who are optimistic about other’s appropriation appropriate at lower levels on average (16% of their appropriation capacity). However, the latter group does not make appropriation decisions that are as systematically linked to changes in expectations of group appropriation.

More generally, these novel results add to the emergent literature that explores individual differences in the responses to marginal incentives and reciprocity. In particular, we show the relevance of threshold expectations on group cooperation in defining subjects’ responsiveness (or lack of it) to others’ behavior. These findings extend the results reported in Goeree et al. (Reference Goeree, Holt and Laury2002), for provision decisions in a public good setting with deterministic marginal benefits. In that study, the authors observe, on average, a positive relationship between public good contributions and the internal return to the contributor, as well as the return to other group members. However, the effect of the internal return is larger and more systematic, as their model estimates of individual’s altruism toward others suggests considerable heterogeneity in responses.

What are possible explanations for the behavioral differences we observe between optimistic and pessimistic subjects in this study? Suppose subjects’ behavior focuses primarily on individual expected marginal incentives. In the context of our decision setting, pessimistic expectations of higher group appropriation imply subjects perceive, relative to the benchmark condition, a higher expected marginal private return from appropriation and a lower expected marginal harm to the group. In this sense, if our subjects’ individual response to incentives is consistent with that observed in Goeree et al. (Reference Goeree, Holt and Laury2002), across the group of pessimistic subjects, the increase in appropriation relative to their baseline appropriation can be expected to make appropriation decisions that correlate more systematically with their expectations of others’ appropriation. And, this is in fact what we observe for the pessimistic subjects.

However, optimistic expectations of lower group appropriation imply subjects perceive, relative to the benchmark condition, a lower expected marginal private return from appropriation and a higher expected marginal harm to the group. In line with the results from Goeree et al. (Reference Goeree, Holt and Laury2002), we find that the optimistic subjects lower appropriation overall. However, in terms of statistical significance, they do not respond as systematically to changes in expected marginal incentives. One could alternatively suppose that optimistic subjects’ decisions are influenced more strongly by additional motives such as warm-glow from the act of cooperating or fairness heuristics. These additional motivations could lead to decisions by optimistic subjects that are less sensitive to changes in pecuniary incentives inferred from changes in expectations of others’ behavior. In the context of our experiment, such motivations would be compatible with the evidence that the optimistic group made more cooperative decisions in the benchmark game, which is what we observe.

In summary, in very different decision settings, one where marginal incentives are deterministic and one where they are endogenous and dependent on expectations of others’ behavior, both Goeree et al. (Reference Goeree, Holt and Laury2002) and this study find evidence that decisions are consistent with a more systematic response to changes in private marginal incentives relative to the impact on other group members. In our study, this heterogeneity in behavior is linked to threshold expectations of whether subjects are more (or less) pessimistic about the actions of other group members.

The policy implications of these results can be illustrated by the motivating example of the maintenance of ecosystem services provided in the introduction. As this study demonstrates, individuals may have heterogeneous responses to the potential of increasingly severe endogenous destruction of a resource. While some individuals might be willing to make necessary sacrifices in resource use by limiting their appropriation (despite pecuniary incentives), others might perceive conservation objectives to be unrealistic (or unfeasible) and thus engage in highly extractive strategies leading to a race-to-the-bottom. Which of these strategies individuals undertake might be (at least partially) influenced by their first order beliefs of others’ behavior.