Externalities in appropriation: responses to probabilistic losses

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 \( \left( {p\sum\nolimits_{i = 1}^{n} {z_{i} } } \right) \), 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 \( L \in \left[ {0,1} \right] \).

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 (2001), 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 (2007), 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. (2010), with the stake sizes used in Balafoutas et al. (2012). 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.⁷

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

$$ MNB_{i}^{j} = \frac{{\partial \pi_{i}^{j} }}{{\partial z_{i} }} = h - \frac{g}{n} - \frac{g}{n}L_{j} p \cdot \left( {w - 2{\text{Z}}} \right) $$

(2)

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 \( MNB_{i}^{j} \) 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.⁸ Table 1 displays the specific functional relation between \( MNB_{i}^{j} \) and aggregate group appropriation for each of the treatment conditions j = L0, L10, L50 and L90. Figure 1 displays the value of \( MNB_{i}^{j} \) 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	\( MNB_{i}^{L0} = 0.5 \)
L10	0.1	0.01	\( MNB_{i}^{L10} = 0.45 + 0.001\cdot{\text{Z}} \)
L50	0.5	0.01	\( MNB_{i}^{L50} = 0.25 + 0.005\cdot{\text{Z}} \)
L90	0.9	0.01	\( MNB_{i}^{L90} = 0.05 + 0.009\cdot{\text{Z}} \)

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

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Note that, for any value of L in the range [0, 1], the unique Nash equilibrium for self-interested payoff-maximizing agents is to appropriate at capacity. This follows from the observation that, given a maximum group capacity to appropriate of 100, the \( MNB_{i}^{j} \) is positive for any value of L in the range [0, 1].

However, a broad range of previous research on social dilemma settings has shown that subjects make decisions that reflect complex and diverse motivations beyond simple self-income maximization (see research summarized in Camerer 2003; Camerer and Fehr 2006; Ostrom and Walker 2003). Some but not all of these motivations support models where subjects respond systematically to the private benefits of their actions (internal returns) and the magnitude of externalities imposed on others (external return) (see for example Goeree et al. 2002). The literature also provides support for models where decision makers follow other regarding preferences that are not sensitive to changes in magnitudes of externalities imposed on others, such as the concept of “warm glow” as introduced by Andreoni (1990), and examined in Palfrey and Prisbrey (1997) among others.

If subjects were to respond solely to marginal net benefits, how would we expect behavior to change with respect to the benchmark game in the treatment conditions? Referring back to Fig. 1, first note that as compared to the constant \( MNB_{i}^{L0} = 0.5 \) in the L0 game, the expected marginal net benefit increases (decreases) for expectations of group appropriation above (below) a critical threshold of 50 tokens in all treatment conditions. Further, note that the range of values of \( MNB_{i}^{j} \) as a function of group appropriation (vertical axis) increases with the size of L. In particular, \( MNB_{i}^{L0} = 0.5 \), \( MNB_{i}^{L10} \in \left[ {0.45, 0.55} \right] \), \( MNB_{i}^{L50} \in \left[ {0.25, 0.75} \right] \), \( MNB_{i}^{L90} \in \left[ {0.05, 0.95} \right] \). Thus, across treatment conditions, the influence of first order beliefs of group appropriation on the expected magnitude of \( MNB_{i}^{j} \) increases with increases in L.

As shown, \( MH_{ - i}^{j} \) is inversely related to \( MNB_{i}^{j} \), increasing in L, and decreasing in Z. Similarly, as discussed for \( MNB_{i}^{j} \), the directional response to treatment conditions resulting from \( MH_{ - i}^{j} \) is affected by the critical threshold of first order beliefs of group appropriation of 50 tokens. In sum, while higher values of Z increase the magnitude of pecuniary benefits \( MNB_{i}^{j} \), higher values of Z decrease the magnitude of the marginal damage associated with appropriation \( MH_{ - i}^{j} \).⁹

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 1992; Dickinson 1998; Gangadharan and Nemes 2009; Blanco et al. 2016b). 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. (2002), 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. (2002), 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. (2002), 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. (2002) 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.