Endowment Inequality in Common Pool Resource Games: An Experimental Analysis

Abstract

This work addresses whether heterogeneity in player endowments influences investment decisions in common pool resource (CPR) games, shedding light on the relationship between inequality and economic decision making. We explore two theoretical avenues from behavioral economics—linear other-regarding preferences and inequity aversion—and examine the predictions of each with a laboratory experiment. Our experimental results roundly reject the majority of these explanations: in treatments with endowment inequality, high endowment individuals invest more in the common pool resource than low endowment individuals. We discuss these results in the context of the literature on psychological entitlement and positional preferences.

1. Introduction

Economists have long studied the impact of imperfect property rights in environments where individual incentives run counter to those of society as a whole. Among these, the common pool resource (CPR) features individual resource utilization that generates negative spillovers in the resource value. As such, self-interested individuals are incentivized to access the resource at levels that degrade it below the socially optimal level. The pioneering work in this area by Ostrom et al. ((1994), henceforth OGW) developed both a theoretical and empirical foundation for the study of common pool resources, including fisheries, forests, and irrigation systems. Their benchmark theoretical model results in a Nash equilibrium where individuals extract the resource at levels above the socially optimal level, while experiments confirm that relative to this social optimally level, individuals tend to extract at Nash equilibrium levels.

An area which remains underexplored in the CPR literature is the influence of heterogeneity of extraction opportunities—endowments in the CPR setting—among individual agents1. The standard CPR model2 (of OGW) predicts that changes in individual endowments have no impact on Nash equilibrium extraction levels. This result arises because while changes in endowments impact individual payoffs, they do not influence individual incentives to extract, which are based on the marginal value and marginal cost of extraction. In an early test, Walker et al. (1990) study the impact of symmetric changes in individual endowments in an experimental setting, increasing endowments by 2.5 times for each individual, and find that increasing all subjects’ extraction opportunities equally does not significantly alter equilibrium play relative to the Nash prediction.

This work re-examines the question by introducing heterogeneous endowments in a laboratory CPR environment. Alongside symmetric high endowment and symmetric low endowment treatments, in which each participant has the same endowment, we implement an asymmetric endowment treatment where participants are randomly assigned either a high endowment or low endowment role. As in the standard CPR setting, asymmetric endowments do not influence equilibrium or socially optimal contribution levels directly. Therefore, the standard model makes a clear prediction: endowments should not impact individuals’ extraction decisions.

While this question has not been addressed directly in CPR settings, experimental studies featuring heterogeneity in related settings have yielded mixed results. Buckley and Croson (2006) introduce endowment heterogeneity in public goods games, and do not find that it significantly changes individual contribution levels, while Heap et al. (2016) find that high endowment individuals contribute less to a public good when there is endowment inequality. Cherry et al. (2005) find overall public good contribution levels decline in the presence of endowment inequality. More generally, heterogeneity along other dimensions in public goods games has been explored in Chan et al. (1999), McGinty and Milam (2013), Casari and Luini (2009), and others.

Several experimental studies touch on adjacent forms of inequality in CPR contexts. Lecoutere et al. (2015) run a behavioral experiment in rural Tanzania and find the impact of both social status3 and gender on the usage of common watersheds. In particular, they find that while men and women with low social status distribute water equally, men with high social status keep greater shares of water for themselves, while women with high social status distribute water more altruistically. A CPR experiment in Namibian and South African farming communities conducted by Hayo and Vollan (2012) finds impacts of several sociodemographic characteristics on cooperative behavior in standard CPR games; they find less cooperative behavior from male participants, and that an unequal distribution of wealth leads to lower levels of cooperation.

Much of the experimental CPR literature has incorporated behavioral elements—such as other-regarding preferences (ORP)—to explore extraction decisions. Cox et al. (2009) contrast common property rights with private rights regarding agent trust, finding a significant but minor rise in trust with common rights. The influences of linear ORP are more directly modeled, though not the primary focus, in Casari and Plott (2003). They find that agents with spiteful tendencies increase extraction and those with altruistic tendencies decrease extraction of the CPR. Falk et al. (2001) apply a model of reciprocal preferences to the standard CPR model, pointing out that such ORP can explain the observed improvements in efficiency with the introduction of communication or informal sanctions. Velez et al. (2009) use models of altruism, reciprocity, inequity aversion, and conformity to analyze field data from CPR experiments conducted in Colombia; they find the strongest evidence for a preference for conformity which leads to more cooperative behavior.

This paper is unique in its use of behavioral game theory to explain departures from the prediction of the standard CPR model in the presence of heterogeneous endowments. Building on the work of others, we examine versions of the CPR model which incorporate either inequity aversion (Fehr & Schmidt, 1999) or linear other-regarding preferences. In particular, since the focus of this work is on inequality of endowments, we build a linear ORP model which allows for any individual to experience an other-regarding preference for individuals within their same-endowment group which differs from their other-regarding preference for individuals outside of their same-endowment group. For example, this two-dimensional linear ORP allows high endowment individuals to simultaneously feel altruistic toward low endowment individuals and neutral (or even spiteful) toward fellow high endowment individuals, or low endowment individuals to feel spiteful exclusively toward high endowment individuals.

Inequity aversion and linear other-regarding preferences in the asymmetric endowment CPR environment generate a number of testable hypotheses regarding behavioral predictions. These are identified formally below. Generally speaking, altruistic tendencies toward others tend to decrease one’s investment in the common property resource while spite towards others increases it. The presence of inequity aversion suggests high endowment individuals should invest less, while low endowment individuals invest more. We evaluate these predictions using laboratory experiments which include a symmetric low endowment treatment, symmetric high endowment treatment, and an asymmetric endowment treatment—the main focus of this work.

Experimental results from both symmetric endowment treatments mirror individual investment behavior generally observed in the CPR literature. Results from the asymmetric endowment treatment, however, unambiguously reject this pattern of behavior. In particular, we find that in asymmetric endowment treatments, high endowment individuals invest above equilibrium levels, while low endowment individuals invest below equilibrium levels.

This pattern of investment behavior rejects the prediction of the standard CPR model, is not consistent with inequity aversion, and is only consistent with linear other-regarding preferences if high endowment individuals exhibit spite. A robustness check against the symmetric high endowment treatment indicates it is the introduction of asymmetry, and not high endowments alone, which drives behavior in the asymmetric endowment treatment. A critical dimension of our results is that unlike much of the related literature, our experiments isolate the impact of asymmetric endowments independent of any institutional context. The divergence in investment decisions emerges solely from the introduction of asymmetric endowments.

Our behavioral models are therefore unable to explain the individual investment pattern observed in the asymmetric endowment treatment data. We therefore propose alternate hypotheses from two studies. The first, on psychological entitlement, suggests some individuals with more resources feel deserving of a larger slice of the pie, which would incentivize this pattern of behavior. The second, positional preferences, suggests individuals often pursue not only high payoffs but payoffs which are relatively larger than others. In this context, high endowment individuals would be incentivized to stake out a larger share of the CPR and widen the payoff gap between themselves and low endowment individuals. While the experimental design in this manuscript is not suited to examining these hypotheses directly, both psychological entitlement and positional preferences present opportunities for future research.

In Section 2, we lay out common pool resource models which introduce two-dimensional linear other-regarding preferences and inequity aversion. We discuss experimental design and offer experimental hypotheses in Section 3. We present and analyze experimental data in Section 4, and conclude with a discussion of key results and alternate hypotheses in Section 5.

2. Model

2.1. Standard CPR Model

We employ a common pool resource model which mirrors Ostrom et al. (1994), Casari and Plott (2003), and others in the CPR literature. A group is composed of n individuals, where each individual i commits a level of effort to extract from or “invest in” the CPR, denoted $x_i\in [0,{\omega }_i]$. ${\omega }_i>0$ gives the endowment or maximum investment potential for individual i. Any units not utilized in the CPR are invested in an outside alternative which generates a constant per-unit return. The total amount of investment in the CPR is $X={\sum }_{i=1}^n{x}_i$ and $X_{-i}={\sum }_{j\ne i}{x}_j$. The value of the CPR is determined by a concave function, $F\left(X\right)$, increasing for low values of total investment and decreasing for high values of total investment. To be consistent with the literature, we use a negative quadratic CPR value function, $F\left(X\right)=aX-b{X}^2$, with $a>0$ and $b>0$. In a standard CPR game, each individual earns a payoff ${\pi }_i$ composed of their return from both investments:

$${\pi }_i(x_i;X_{-i})=\alpha ({\omega }_i-x_i)+{\displaystyle \frac{x_i}{X}}[aX-b{X}^2]$$

(1)

where $\alpha >0$ is the constant return earned per unit invested in the outside alternative. When ${\omega }_i={\omega }_j$, $\forall i$, j; this is a symmetric endowment CPR model. A symmetric model without behavioral considerations is a standard CPR model in the OGW sense.

The standard CPR model is by definition a symmetric game. This gives each individual’s best response function as

$$B{R}_i\left(X_{-i}\right)={\displaystyle \frac{a-\alpha }{2b}}-{\displaystyle \frac{1}{2}}{X}_{-i}$$

(2)

The symmetric Nash equilibrium level of investment is

$$x_i^{*}={\displaystyle \frac{a-\alpha }{b}}{\displaystyle \frac{1}{n+1}}$$

(3)

while the symmetric Nash equilibrium level of total investment is

$$X^{*}={\displaystyle \frac{a-\alpha }{b}}{\displaystyle \frac{n}{n+1}}$$

(4)

2.2. Behavioral Considerations I: Linear Other-Regarding Preferences

We now introduce two potential behavioral factors to individual payoffs: linear other-regarding preferences and inequity aversion. Since both approaches allow individuals to consider the payoffs of others in the group, we will denote individual payoffs by $U_i\left(\pi \right)$, where $\pi =({\pi }_1,{\pi }_2,\dots ,{\pi }_n)$ is the vector of payoffs for each individual in the group4. In a CPR game without any behavioral considerations, for example, $U_i\left(\pi \right)=U_i\left({\pi }_i\right)$; in the standard CPR game, $U_i\left(\pi \right)=U_i\left({\pi }_i\right)={\pi }_i$. When an individual weighs the impact of the sum of others’ payoffs, we denote this sum by ${\Pi }_{-i}={\sum }_{j\ne i}{\pi }_j$.

A simple linear other-regarding preferences model (or simple LORP model) grants each individual utility from their own payoff, plus additional utility based on the sum of all others’ payoffs:

$$U_i\left(\pi \right)=U_i({\pi }_i,{\Pi }_{-i})={\pi }_i+{\gamma }^i{\Pi }_{-i}$$

(5)

The LORP coefficient for each individual is represented by $-1\le {\gamma }^i\le 1$. An altruistic individual (someone who receives positive utility from others’ payoffs) would exhibit ${\gamma }^i>0$, while an individual who receives negative utility from others’ payoffs would exhibit ${\gamma }^i<0$. The LORP coefficient is “simple” in two senses: (1) individuals treat all others’ identically, and cannot be altruistic toward some but not all others; (2) individuals weight all others’ identically, and cannot incorporate weaker or stronger other-regarding preferences toward different individuals. Notice as well that this model embeds the standard CPR model as a special case when ${\gamma }^i=0$, $\forall i$.

In the simple LORP model, individual best response functions are given as

$$B{R}_i\left(X_{-i}\right)={\displaystyle \frac{a-\alpha }{2b}}-{\displaystyle \frac{1}{2}}(1+{\gamma }^i)X_{-i}$$

(6)

If ${\gamma }^i={\gamma }^j=\gamma$, $\forall i,j$, then the symmetric Nash equilibrium level of investment is

$$x_i^{*}={\displaystyle \frac{a-\alpha }{b}}{\displaystyle \frac{1}{(n+1)+\gamma (n-1)}}$$

(7)

If, for example, all individuals are identically altruistic, then equilibrium levels of investment are lower than in the standard CPR model.

Our primary question of study involves introducing an asymmetric distribution of endowments in the CPR environment. The asymmetry is introduced by splitting the group of n individuals into a subgroup of high endowment individuals and a subgroup of low endowment individuals. Since it may be relevant to allow high endowment individuals and low endowment individuals to express different other-regarding preferences, we develop a general linear other-regarding preferences model which can be applied to subgroups of two types.

Split the n individuals in the group into two “types”: i types and j types. Without loss of generality, let i types be high endowment players and let j types be low endowment players.5 Let k denote the number of i types in the group; $n-k$ therefore denotes the number of j types. In the two-type LORP model, each i type chooses $x_i$ to maximize $U_i={\pi }_i+{\gamma }^i{\Pi }_{-i}$, while each j type chooses $x_j$ to maximize $U_j={\pi }_j+{\gamma }^j{\Pi }_{-j}$. If individuals are symmetric within types—meaning all i types behave identically and all j types behave identically—then the best response for an i type individual is given by

$$B{R}_i\left(X_{-i}\right)={\displaystyle \frac{a-\alpha }{2b}}-{\displaystyle \frac{1}{2}}(1+{\gamma }^i)X_{-i}$$

(8)

but $X_{-i}$ is now composed of the $k-1$ remaining i types plus all $n-k$ of the j types. Similarly, for any j type individual,

$$B{R}_j\left(X_{-j}\right)={\displaystyle \frac{a-\alpha }{2b}}-{\displaystyle \frac{1}{2}}(1+{\gamma }^j)X_{-j}$$

(9)

where $X_{-j}$ is composed of all k of the i types plus the remaining $n-k-1$j types.6 Nash equilibrium levels of investment for each type are then given as

$$x_i^{*}={\displaystyle \frac{a-\alpha }{b}}\left({\displaystyle \frac{1+{\gamma }^j(n-k-1)-{\gamma }^i(n-k)}{R}}\right)$$

(12)

and

$$x_j^{*}={\displaystyle \frac{a-\alpha }{b}}\left({\displaystyle \frac{1+{\gamma }^i(k-1)-{\gamma }^{j}k}{R}}\right)$$

(13)

where $R=(k+1+{\gamma }^i(k-1))(n-k+1+{\gamma }^j(n-k-1))-k(n-k)(1+{\gamma }^i)(1+{\gamma }^j)$. This model can therefore analyze the impact of other-regarding preferences across subgroups on equilibrium investment levels; if, for example, ${\gamma }^i>0$ and ${\gamma }^j=0$, then the combination of altruistic high endowment i types and neutral low endowment j types leads the altruistic types to invest less than the neutral types.7

The two-type LORP model allows two types of linear other-regarding preferences by breaking the n individuals into groups, but it does not allow the individuals to weight different individuals differently. What if, for example, a low endowment individual felt altruistic or neutral toward other low endowment individuals but spiteful toward high endowment individuals? To address this concern, we utilize a two-dimensional (or $2\times 2$) linear other-regarding preferences model.

Assume the n total individuals are split between ki types and $n-k$j types. The $2\times 2$ model will permit i types to have two LORP coefficients: one regarding fellow i types, and one regarding j types. An i type chooses $x_i$ to maximize their payoff:

$$U_i\left(\pi \right)={\pi }_i+{\gamma }_i^i{\Pi }_{-i}^i+{\gamma }_j^i{\Pi }_{-j}^i$$

(14)

where ${\gamma }_i^i$ represents an i type’s other-regarding preference toward other i types; ${\gamma }_j^i$ represents an i type’s other-regarding preference toward j types; ${\Pi }_{-i}^i$ gives the sum of the payoffs to all other i types as an i type; ${\Pi }_{-j}^i$ gives the sum of the payoffs to all j types.8 In a similar fashion, a j type chooses $x_j$ to maximize their payoff:

$$U_j\left(\pi \right)={\pi }_j+{\gamma }_i^j{\Pi }_{-i}^j+{\gamma }_j^j{\Pi }_{-j}^j$$

(15)

Best response functions here look familiar:

$$B{R}_i(X_{-i}^i,X_{-j}^i)={\displaystyle \frac{a-\alpha }{2b}}-{\displaystyle \frac{1}{2}}(1+{\gamma }_i^i)X_{-i}^i-{\displaystyle \frac{1}{2}}(1+{\gamma }_j^i)X_{-j}^i$$

(16)

and

$$B{R}_j(X_{-i}^j,X_{-j}^j)={\displaystyle \frac{a-\alpha }{2b}}-{\displaystyle \frac{1}{2}}(1+{\gamma }_i^j)X_{-i}^j-{\displaystyle \frac{1}{2}}(1+{\gamma }_j^j)X_{-j}^j$$

(17)

This generates Nash equilibrium investments for each type as

$$x_i^{*}={\displaystyle \frac{a-\alpha }{b}}\left({\displaystyle \frac{1+{\gamma }_j^j(n-k-1)-{\gamma }_j^i(n-k)}{R^{\prime }}}\right)$$

(18)

and

$$x_j^{*}={\displaystyle \frac{a-\alpha }{b}}\left({\displaystyle \frac{1+{\gamma }_i^i(k-1)-{\gamma }_i^{j}k}{R^{\prime }}}\right)$$

(19)

where $R^{\prime }=(k+1+{\gamma }_i^i(k-1))(n-k+1+{\gamma }_j^j(n-k-1))-k(n-k)(1+{\gamma }_j^i)(1+{\gamma }_i^j)$. With just one type and ${\gamma }_i^i={\gamma }_j^i={\gamma }^i$ for all i individuals, this model nests the simple LORP model. Interestingly, the relative level of equilibrium investment for an i type depends on exactly two LORP parameters: (1) if they feel altruistic toward j types, ${\gamma }_j^i>0$ and $x_i^{*}$ decreases; (2) if the j types feel altruistic toward j types, ${\gamma }_j^j>0$ and $x_i^{*}$ increases.

2.3. Behavioral Considerations II: Inequity Aversion

Inequity aversion provides an additional framework from which our question can be addressed. In a standard CPR model, initial endowments and Nash equilibrium are symmetric, and players earn identical payoffs at the NE. Without behavioral considerations, introducing asymmetric endowments does not change NE play, but it does create inequality in payoffs. Therefore, in the context of asymmetric endowments, players may reasonably anticipate an inequality in payoffs. Our inequity aversion model allows players to anticipate and respond to this anticipated inequality. See Falk et al. (2001) for a detailed analysis of inequity aversion in CPR settings.

In the inequity aversion model, we again distinguish between two types, denoted by i and j. Assume play is symmetric within types; that is, all i types behave identically and all j types behave identically. ${\pi }_i$ still denotes the direct payoff from the CPR to an i type. For an i type with inequity aversion, their payoff is

$$U_i\left(\pi \right)={\pi }_i-A_i{\displaystyle \frac{1}{n-1}}\sum _{j\ne i}max\{{\pi }_j-{\pi }_i,0\}-B_i{\displaystyle \frac{1}{n-1}}\sum _{j\ne i}max\{{\pi }_i-{\pi }_j,0\}$$

(20)

where $0\le B_i<1$ and $A_i\ge B_i$. The coefficient $A_i$ represents the weight on the negative utility received from having lower payoffs than other group members. The coefficient $B_i$ represents the weight on the negative utility from having higher payoffs than other group members. First, notice that if $A_i=B_i=0$, the model reduces to the standard CPR result. Moreover, if all individuals have the same payoff at the NE, then $\forall i$, $U_i={\pi }_i$ and the result is that of the standard CPR model. Here, beliefs play a critical role: it is sufficient when generating the standard CPR result to have all individuals believe that payoffs will be identical for everyone.

WLOG, we let i types be the high endowment types, and let j types be the low endowment types. We propose that in the asymmetric endowment case, individuals have the belief that ${\pi }_i>{\pi }_j$.9 Under this belief, the $B_j$ term drops out of $U_j$ for every j type and the $A_i$ term drops out of $U_i$ for every i type. No high endowment players, for example, will experience negative utility from having less than any of the others.

Once again, let there be k individuals who are i types, and $n-k$ individuals who are j types. Given each individual’s choice of $x_i$ or $x_j$, best response functions can be derived for the representative i type and j type, respectively, as

$$B{R}_i\left(X_{-i}\right)={\displaystyle \frac{a-\alpha }{2b}}-{\displaystyle \frac{1}{2}}{X}_{-i}-{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{B_i\frac{k}{n-1}}{1-B_i\frac{k}{n-1}}}\right)x_j$$

(21)

and

$$B{R}_j\left(X_{-j}\right)={\displaystyle \frac{a-\alpha }{2b}}-{\displaystyle \frac{1}{2}}{X}_{-j}+{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{A_j\frac{n-k}{n-1}}{1+A_j\frac{n-k}{n-1}}}\right)x_i$$

(22)

The first two terms in the inequity aversion model best responses are identical to the standard CPR model. The third term in each represents the additional response under inequity aversion. A (low endowment) j type, for $A_j>0$, is incentivized to respond with higher investment, while a (high endowment) i type, for $B_i>0$, is incentivized to respond with lower investment. The magnitude of these additional responses depends on the inequity coefficients ($A_j$ and $B_i$) and the number of each type (k). Since a CPR game is inherently a game with strategic substitutes10, an interpretation of these additional incentives reads that while every individual maintains a downward-sloping best response, under inequity aversion, i types exhibit weaker strategic substitutes (have flatter best responses) while j types exhibit stronger strategic substitutes (have steeper best responses.)

To simplify, let $A^{\prime }=A_j{\displaystyle \frac{n-k}{n-1}}$ and $B^{\prime }=B_i{\displaystyle \frac{k}{n-1}}$. Notice that $A^{\prime }>0$; $0<{\displaystyle \frac{A^{\prime }}{1+A^{\prime }}}<1$; $0\le B^{\prime }<1$; and, ${\displaystyle \frac{B^{\prime }}{1-B^{\prime }}}>0$. Solving for Nash equilibrium investment levels in the inequity aversion model yields

$$x_j^{*}={\displaystyle \frac{a-\alpha }{b}}\left({\displaystyle \frac{1+\frac{A^{\prime }}{1+A^{\prime }}}{S}}\right)$$

(23)

and

$$x_i^{*}={\displaystyle \frac{a-\alpha }{b}}\left({\displaystyle \frac{1-\frac{B^{\prime }}{1-B^{\prime }}}{S}}\right)$$

(24)

where $S=(k+1)(n-k+1)-(k+{\displaystyle \frac{B^{\prime }}{1-B^{\prime }}})(n-k-{\displaystyle \frac{A^{\prime }}{1+A^{\prime }}})$. This NE backs out to the standard CPR when $A_j=B_i=0$. Importantly, at the inequity aversion NE, $x_j^{*}>x_i^{*}$ unambiguously.

3. Materials and Methods

3.1. Experimental Design

Our approach to investigating the behavioral influence of endowment asymmetries in this CPR environment builds upon that established by OGW and extended by many others. We implement the above CPR game in the laboratory, with parameters consistent with OGW: $\alpha =5$, $a=23$, $b=0.25$, and $n=8$.11

Subject endowments $\omega$ are the primary treatment variable, as described below. Subjects were endowed with experimental tokens and provided two options in which to invest these tokens: an investment in a constant marginal value asset, Fund A, and an investment representing the CPR resource, Fund B.12 Investments translated participant tokens into points, which were converted into monetary (USD) payouts at a rate of USD 0.07 per point; participants were informed of the rate verbally and in writing on the board in the lab. From their individual endowment of tokens $\omega$, subjects decided how many tokens to invest in Fund A and how many tokens to invest in Fund B. Subjects were informed of the relationship between total investment X and Fund B valuation in both tabular form and with a graphical representation.13 Given these parameters, the standard Nash equilibrium investment is 8 tokens for each individual, which generates 64 total tokens invested in Fund B at the standard NE. The socially optimal level of total investment in Fund B is 36 tokens in the standard model.

Interaction between subjects occurred via networked computers. The experiment was coded using oTree, an open source experimental platform created by Chen et al. (2016). Overall, 64 undergraduate students at the University of Puget Sound participated in the experiment. Students were recruited from a wide range of majors to prevent overrepresentation from economics or related fields. Each laboratory session featured 20 rounds per treatment and a within-subjects design. Following each round, subjects were informed of the Fund B contribution decisions of each subject within their group, the total value of Fund B, the per-unit return from Fund B, and their earnings for the round in points. To avoid wealth effects, laboratory earnings above the USD 5 show-up fee were determined by random selection of a single paid round per treatment at the end of the laboratory session. Across all sessions, median participant earnings were USD 28.75, while average earnings were USD 29.07. Following each session, subjects completed a qualitative survey to gauge their perceptions of the session.

Participants were randomly assigned into groups of eight; group memberships remained the same throughout the experiment. Via the within-subjects design, each group played all three treatments, executed in randomized order: $SY{M}_{LOW}$ is a symmetric, low endowment (${\omega }_i=10$, $\forall i=1,\dots ,n$) treatment; $SY{M}_{HIGH}$ is a symmetric, high endowment ($\omega =25$, $\forall i=1,\dots ,n$) treatment; and, $ASYM$ is the asymmetric treatment with 4 members of each group receiving the low ($\omega =10$) and high ($\omega =25$) endowments, respectively.14 Information regarding the allocation of endowments within groups was common but group member identities were private.

Table 1. Summary statistics from experimental sessions. The variables Asymmetric, Symmetric_Low, and Symmetric_High represent dummy variables for each treatment, while the Asymmetric_Low and Asymmetric_High represent subgroups within the asymmetric treatment. The middle set of variables covers measures of subject investment decisions and subject payoffs. Treatment_round measures the round number within each treatment. The Treat_order variables are dummy variables for the randomized treatment order in which each session was run.

	n	Mean	St. Dev.	Min	Max
Asymmetric	3840	0.33	0.47	0.00	1.00
Symmetric_Low	3840	0.33	0.47	0.00	1.00
Symmetric_High	3840	0.33	0.47	0.00	1.00
Asymmetric_Low	3840	0.17	0.37	0.00	1.00
Asymmetric_High	3840	0.17	0.37	0.00	1.00
Individual_investment	3840	8.00	5.57	0.00	25.00
Group_total_investment	3840	64.01	13.63	21.00	115.47
Player_payoff	3840	148.72	149.35	−65.25	767.75
Player_endowment	3840	17.50	7.50	10.00	25.00
Treatment_round	3840	10.50	5.77	1.00	20.00
Treat_order_TR1	3840	0.33	0.47	0.00	1.00
Treat_order_TR2	3840	0.33	0.47	0.00	1.00
Treat_order_TR3	3840	0.33	0.47	0.00	1.00

presents summary statistics regarding experimental sessions.

3.2. Model Hypotheses

As a benchmark, we include the standard CPR model—without any behavioral considerations:

Hypothesis 1 (H1). 

Under the standard CPR model, in the $ASYM$ treatment, there should be no significant difference in investment levels between low endowment types and high endowment types.

The inequity aversion model makes a clear prediction on individual contribution levels:

Hypothesis 2 (H2). 

Under inequity aversion, in the $ASYM$ treatment, low endowment types should invest more in the common pool resource than high endowment types. As shown above, this is derived from participant beliefs and corresponding equilibrium investment levels.

The two-dimensional linear other-regarding preferences model makes an array of predictions depending on the parametrization of the LORP values. Several examples are shown in

Table 2. Nash equilibrium individual investment values for parameterizations of the linear other-regarding preference model.

Hypotheses	$({\mathit{\gamma }}_{\mathit{i}}^{\mathit{i}},{\mathit{\gamma }}_{\mathit{j}}^{\mathit{i}};{\mathit{\gamma }}_{\mathit{i}}^{\mathit{j}},{\mathit{\gamma }}_{\mathit{j}}^{\mathit{j}})$	${\mathit{x}}_{\mathit{i}}^{*}$	${\mathit{x}}_{\mathit{j}}^{*}$
H1	$(0,0,0,0)$	8	8
	$(0.1,0.1,0,0)$	5.28	10
H3	$(0,0,-0.1,-0.1)$	6.4	10
	$(0.1,0.1,-0.1,-0.1)$	5.28	10
	$(0.1,-0.1,-0.1,0.1)$	8.08	8.08
H4	$(0.1,0,0,0.1)$	7.74	7.74
	$(0,-0.1,-0.1,0)$	8.37	8.37
H5	$(-0.1,-0.1,0,0)$	11.07	5.54
	$(0,-0.1,0,0)$	9.51	6.79

. For example, if high endowment types are altruistic and low endowment types are neutral, then the LORP model predicts low endowment types investing more ($x_j^{*}=10$) than high endowment types ($x_i^{*}=5.28$). If individuals are altruistic within groups, then both types invest less than the standard CPR model prediction ($x_i^{*}=x_j^{*}=7.74$) and equilibrium group level of investment decreases.15

Given the numerous possibilities, we highlight several hypotheses:

Hypothesis 3 (H3). 

If high endowment types are altruistic, low endowment types are spiteful, or both, in the $ASYM$ treatment, low endowment types should invest more in the common pool resource than high endowment types. Under this hypothesis, low types invest their entire endowment of 10 tokens.16

Hypothesis 4 (H4). 

If either type is altruistic within their own group, spiteful toward those outside of their own group, or both, in the $ASYM$ treatment, there should be no significant difference in investment levels between low endowment types and high endowment types.

Hypothesis 5 (H5). 

If high endowment types are uniquely spiteful, in the $ASYM$ treatment, high endowment types should invest more in the common pool resource than low endowment types.17

4. Results

Our CPR models offer clear predictions which we can examine with the laboratory data. First, we present a comparison of group-level CPR investment across the three treatments. Figure 1 illustrates mean aggregate investments by treatment across rounds, as well as the standard Nash equilibrium investment level (64) and the level which maximizes the social resource value (36). From this, we can see that aggregate behavior of the groups of eight approaches the standard Nash level, particularly as rounds progress, but does not converge smoothly. We formally tested the hypothesis of median equality of aggregate CPR investment with the standard Nash equilibrium prediction. The results support the above impression. Wilcoxon signed-rank tests for each treatment reject the null hypothesis of equality to Nash for each treatment at a better than one-tenth of the one percent level of significance ($p<0.001$).

As the main focus of this work is the examination of individual CPR investment levels, we test hypotheses H1 through H5 regarding individual behavior with endowment asymmetry to check for the presence of inequity aversion or two-dimensional linear other-regarding preferences. To this end, it is useful to illustrate the individual investment behavior across treatments. In Figure 2, we see the mean levels of investment (across all sessions) by individual subjects for each round of the treatments with symmetric low and symmetric high endowments, as well as the standard Nash equilibrium investment level.18 These baseline treatments reflect a strong tendency toward equilibrium behavior on average, a trend which strengthens in later rounds of the treatments; this tendency is broadly consistent with the foundational work of OGW.

Figure 3 presents the primary contribution of this project: mean individual investment levels for the asymmetric treatment. There is a remarkably consistent pattern of behavior which emerges when subjects are presented with endowment asymmetry: relative to the standard Nash prediction, the average investment by high endowment types rises and investment by low endowment types falls. Further evidence appears in

Table 3. Mean Fund B individual investment levels across all rounds in each treatment. $ASY{M}_{LOW}$ and $ASY{M}_{HIGH}$ capture mean individual investment levels of respective endowment types in the asymmetric treatment. Standard deviation in parentheses.

Treatment	Sess 1	Sess 2	Sess 3	Sess 4	Sess 5	All
$SY{M}_{LOW}$	7.75	7.54	6.87	7.20	6.35	7.02
	(2.71)	(2.25)	(2.79)	(2.42)	(2.76)	(2.66)
$SY{M}_{HIGH}$	8.96	8.39	9.07	7.94	8.91	8.65
	(8.65)	(6.49)	(8.05)	(6.06)	(6.04)	(7.01)
$ASYM$	8.76	8.43	7.99	8.38	8.39	8.34
	(7.69)	(5.06)	(5.79)	(6.30)	(5.12)	(5.95)
$ASY{M}_{LOW}$	4.99	5.54	5.20	6.58	5.69	5.68
	(3.52)	(2.90)	(2.89)	(2.95)	(3.09)	(3.08)
$ASY{M}_{HIGH}$	12.52	11.33	10.79	10.18	11.09	10.99
	(8.84)	(5.12)	(6.58)	(8.02)	(5.32)	(6.87)

, which shows mean individual investment behavior by treatment for each of the five experimental sessions, as well as all-session averages. On average across all rounds, individuals with high endowments consistently invested more in Fund B than those with low endowments in asymmetric endowment treatments.

A formal analysis of the data reinforces the evidence from asymmetric endowment treatments. We consider each of the hypotheses regarding ORP in turn below. Both Hypothesis 2, arising from inequity averse preferences, and Hypothesis 3, regarding a range of LORP values, suggest a pattern of reduced investment by high endowment and increased low endowment type investment. The pattern observed in the data is the inverse of what Hypotheses 2 and 3 suggest. Hypothesis 4, suggesting that the effects of certain LORP essentially cancel each other out, predicts behavior by all types which closely resembles the standard Nash levels of investment. This is also not borne out by the data. Hence, neither inequity aversion nor any of the more intuitive patterns of LORP (representing altruism or spite) appear to describe observed behavior.

The only prediction which the data supports is Hypothesis 5: in the asymmetric endowment treatment, high endowment agents are spiteful and low endowment agents are neutral. In the context of the LORP model, spite captures high endowment players’ negative payoff from others’ positive payoffs. Initially, this explanation is not particularly convincing: one might expect spite to manifest either in all endowment types or in those disadvantaged by asymmetry, not exclusively in high endowment types. What is it about having a high endowment that would uniquely trigger this spitefulness?

If high endowment alone were enough to induce spiteful preferences, it would be natural to expect high endowment to correlate with high investment in the CPR in both the symmetric (high endowment) and asymmetric endowment treatments. Given our within-subject design, we are able to observe decisions by the same individuals in each of the symmetric endowment and asymmetric endowment roles utilizing a Wilcoxon signed-rank test, with observations paired by subject and treatment round. We tested those assigned high endowments in the asymmetric treatment against their symmetric high endowment investments and those assigned low endowments in the asymmetric treatment against their symmetric low endowment investments. These one-sided tests provide evidence that the introduction of asymmetry is responsible for higher investment levels by high endowment types and lower investment levels by low endowment types.

Based on this evidence, therefore, Hypothesis 5 suggests high endowment alone does not generate the observed behavior. Rather, it results from the combination of high endowment and the presence of asymmetry to elicit a kind of context-induced spite from high endowment agents. This explanation may not be intuitive, but it cannot be explicitly ruled out. While we share two alternatives in the discussion in Section 5 which are also consistent with our primary results, Hypothesis 5 nevertheless provides one explanation for the unique impact of asymmetry on CPR investment behavior.

A more robust analysis of the full data set further supports the evidence above. We conduct a set of panel data regressions with subject fixed effects and clustered standard errors at the subject group level, controlling for period and treatment order effects.19 A first set of regressions evaluates individual investment levels across treatments. In the second, we evaluated group-level investments. These models are presented in

Table 4. Panel regression with individual fixed effects: individual CPR investment levels with group clustered standard errors.

	1	2	3	4
$SY{M}_{HIGH}$	$1.57$ **		$1.57$ **
	$\left(0.30\right)$		$\left(0.30\right)$
$SY{M}_{LOW}$		$-1.57$ **		$-1.57$ **
		(0.30)		$\left(0.30\right)$
$ASYM$	$1.25$ ***	−0.31
	$\left(0.23\right)$	(0.28)
$ASY{M}_{HIGH}$			$3.75$ ***	$2.17$ **
			$\left(0.47\right)$	$\left(0.61\right)$
$ASY{M}_{LOW}$			$-1.25$ *	$-2.82$ ***
			$\left(0.42\right)$	$\left(0.24\right)$
$Period$	$0.00$	0.00	$0.00$	$0.00$
	$\left(0.01\right)$	(0.01)	$\left(0.01\right)$	$\left(0.01\right)$
$TreatmentOrder1st$	$-0.27$	-0.27	$-0.27$	$-0.27$
	$\left(0.28\right)$	$\left(0.28\right)$	(0.28)	$\left(0.28\right)$
$TreatmentOrder2nd$	$0.10$	$0.10$	$0.10$	$0.10$
	$\left(0.28\right)$	$\left(0.28\right)$	$\left(0.28\right)$	$\left(0.28\right)$
Num. obs.	3840	3840	3840	3840
Num. groups	64	64	64	64
Adj. R2	$0.29$	0.29	$0.34$	$0.34$

*** $p<0.001$; ** $p<0.01$; * $p<0.05$.

and

Table 5. Panel regression with group fixed effects: group CPR investment levels.

	5	6
$SY{M}_{HIGH}$	$12.61$ ***
	$\left(2.28\right)$
$SY{M}_{LOW}$		$-12.61$ ***
		$\left(2.28\right)$
$ASYM$	$10.07$ ***	$-2.55$
	$\left(1.74\right)$	$\left(2.15\right)$
$Period$	$0.02$	$0.02$
	$\left(0.10\right)$	$\left(0.10\right)$
$TreatmentOrder1st$	$-2.17$	$-2.17$
	$\left(2.10\right)$	$\left(2.10\right)$
$TreatmentOrder2nd$	$0.76$	$0.76$
	$\left(2.10\right)$	$\left(2.10\right)$
Num. obs.	480	480
Adj. R2	$0.16$	$0.16$

*** $p<0.001$; ** $p<0.01$; * $p<0.05$.

, respectively.

In Table 4, models 1 and 2 examine the impact of the introduction of asymmetry, considering the individual investment levels in the asymmetric treatment overall ($ASYM$), without distinguishing the separate endowment levels within. In model 1, with the symmetric low dummy variable ($SY{M}_{LOW}$) excluded, both the asymmetry and the uniform high endowment ($SY{M}_{HIGH}$) treatments demonstrate significantly higher investments. In order to examine whether investments were higher overall under $ASYM$ vs. $SY{M}_{HIGH}$, we duplicated the analysis with $SY{M}_{HIGH}$ excluded. This is performed in model 2, where we see that the average investments under $ASYM$ are shown as lower, though not significant, consistent with the preliminary evidence in Table 3.

Models 3 and 4 provide direct support for the primary contributions of this work, evaluating the distinct behavioral changes under asymmetry for those assigned the low ($ASY{M}_{LOW}$) and high ($ASY{M}_{HIGH}$) endowments. These results directly align with both the casual impressions and the non-parametric evidence above, with each reflecting a significant response to asymmetry relative to the excluded $SY{M}_{LOW}$ treatment consistent with that discussed above. To complete the analysis, we repeated the model with the $SY{M}_{HIGH}$ dummy excluded in model 4, which indicates that $ASY{M}_{HIGH}$ investment levels are also significantly higher than $SY{M}_{HIGH}$. Across all four models, no additional controls show significance.

We conducted a final set of panel regressions in order to better explore the possible impact of asymmetry at the subject group level. These models, presented in Table 5, consider the group aggregate investment level across each of the treatments, with group fixed effects, controlling for period and treatment order. From model 5, with $SY{M}_{LOW}$ excluded, we see that group investment levels are significantly higher under $SY{M}_{HIGH}$ and $ASYM$, as suggested by Figure 1. When $SY{M}_{HIGH}$ is excluded under model 6, we see that $ASYM$ is not significantly different. Once again, no additional controls show significance.

5. Discussion

Upon examination of individual CPR investment behavior under asymmetric endowments, our laboratory results clearly show a consistent pattern of investment behavior: high endowment individuals invest more in the common pool resource than low endowment individuals. Our analysis therefore roundly rejects hypotheses of the standard CPR model, inequity aversion model, and most models of linear other-regarding preferences. This suggests two primary consequences: (1) the standard CPR model, in which endowments do not have an impact on equilibrium investment levels, is insufficient; and (2) the impact of asymmetric endowments on investment levels does not necessarily align with intuitive hypotheses. It presents a bit of a puzzle.

It is tempting to offer risk aversion as a possible explanation for this pattern of behavior. However, if individuals’ risk preferences are consistent throughout the experiment, we would not observe differential behavior in the asymmetric treatment relative to the symmetric treatments. Additionally, individual investment levels in the asymmetric treatment are consistent across treatment orders, which rules out early-treatment risk aversion (cautious play at the onset of the experiment, for example) or late-treatment risk aversion as explanatory factors.

Although our experimental design is not particularly well-suited to address this puzzle, we propose two hypotheses which show potential for future research. As an initial explanation for these results, the literature on psychological entitlement may shed some light. Campbell et al. (2004) “conceptualize psychological entitlement as a stable and pervasive sense that one deserves more and is entitled to more than others. This sense of entitlement will also be reflected in desired or actual behaviors.” While there is a related body of work which aims to elicit decision makers’ preferences for income distributions in social dilemmas such as CPR games20, one study21 conducted by Campbell et al. (2004) is particularly relevant. Researchers conducted a CPR experiment which asked participants to make decisions about harvesting trees from a shared forest. Participant scores on several scales, including a measure for “greed” and the Personal Entitlement Scale (PES), were recorded. Experimental results showed that individuals with higher entitlement scores (1) were correlated with higher measures of “greed”; and (2) desired to harvest marginally more trees in the first round of the experiment.

Our experimental design was not constructed with entitlement in mind, so we cannot provide post-experiment surveys or measures of participants’ PES scores to address this hypothesis directly. In their absence, if we interpret psychological entitlement as motivating higher marginal resource extraction for high endowment types, then the data from our experiments gives some cursory evidence for this behavior. If, however, we interpret this entitlement as high endowment types’ perception that they deserve higher payoffs, entitlement may not tell the full story.

Table 6. Mean individual payoff levels across all rounds in asymmetric treatments. $ASY{M}_{LOW}$ and $ASY{M}_{HIGH}$ capture payoffs for high endowment and low endowment types, respectively.

Endowment Type	Expected NE Payoff	Actual ASYM Treatment Mean Payoff
$ASY{M}_{HIGH}$	141	139.2
$ASY{M}_{LOW}$	66	57.34

below shows that as a result of the common pattern in asymmetric treatments across all experimental sessions showing increased investment by high endowment types and decreased investment by low types, the net effect on total payoffs is a very modest payoff decrease for high endowment types but a greater (13%) payoff decrease for low endowment types. If high types raise their investment and this is accommodated by low types as they best respond to this increase, it serves to drive up the difference in payoffs at little absolute cost to the high endowment types.

This would suggest high endowment types may have a preference not for higher absolute payoffs but higher payoffs relative to others. One possible other-regarding preference which could explain this pattern of behavior is positional preferences, in which agents are not only interested in their own payoff but also where they stand relative to others, with a positive preference for a greater difference in their favor. Frank and Hutchens (1993) find evidence of this in labor markets. A broad literature utilizing experimental survey data finds evidence of preferences for greater relative payoffs or consumption (Carlsson et al., 2007; Easterlin, 2001; Ferrer-i-Carbonell, 2005; Luttmer, 2005). In our experiments, even if high endowment types cannot increase their payoffs above the NE payoff in absolute terms, they may receive some positive utility from generating a larger gap between their payoffs and the payoffs of low endowment individuals. Even if the low endowment types have a similar positional preference, they have little opportunity to act on it. Adjusting their investment from the Nash level of 8 to their full endowment of 10 simply exacerbates their negative positional situation, reducing payoffs by a greater amount than that imposed on the high endowment types. The advantageous position of high endowment types, therefore, means they can absorb the cost of overinvestment while benefitting from stretching the payoff gap.

The behavior observed across all treatments is consistent with positional preferences. Even if all players exhibit positional preferences, only high endowment types have the resources to invest more in the common pool resource and position themselves with relatively higher payoffs. As with the entitlement hypothesis, our experimental design is not well-suited to testing a positional preferences hypothesis explicitly, but the clear pattern in response to endowment asymmetry exhibited in our experimental results certainly invites further exploration. While there is not much literature assessing positional preferences in lab settings like ours, experimental designs which explicitly target a positional hypothesis represent a promising area for future work motivated by our data.