Cooperation and Coordination in Threshold Public Goods Games with Asymmetric Players

Abstract

In this paper, we study cooperation and coordination in a threshold public goods game with asymmetric players where players have different endowments $e_i$, productivities $p_i$, and rewards $r_i$. In general, this game has a defective Nash equilibrium (NE), in which no one contributes, and multiple cooperative NEs, in which the group’s collective contribution equals the threshold. We then study how multiple dimensions of inequality influence people’s cooperation and coordination. We show that heterogeneity in $e_i{p}_i$ can promote cooperation in the sense that the existence condition of the defective NE becomes stricter. Furthermore, players with higher $e_i{p}_i$ are likely to contribute more at a cooperative NE in terms of collective contribution (i.e., absolute contribution multiplied by productivity).

1. Introduction

Exploring cooperation and coordination stands out as a fundamental application of game theory. This field seeks to elucidate which mechanisms drive cooperative behavior and how individuals coordinate towards a more efficient equilibrium. Much of the existing work in this field considers homogeneous populations. However, inequality is ubiquitous in humans. Individuals often vary in their endowments, productivities, shares of rewards, and social hierarchy positions. Asymmetric games raise many exciting questions that are difficult to tackle within the standard setup of symmetric games. For example, what is the impact of various forms of inequality on cooperation and coordination? Does endowment inequality make cooperation and coordination more difficult than productivity inequality? How do individuals coordinate when they differ in multiple dimensions?

The existing literature suggests that the effect of inequality on cooperation is nontrivial. A considerable number of studies based on public goods games has found that endowment inequality tends to reduce cooperation [1,2,3] but that asymmetric productivities and the sharing of public goods tend to positively influence contributions or maintain a neutral impact [4,5,6,7]. In addition, in cases of both endowment and productivity heterogeneity, optimal cooperation is achieved when individuals with a higher productivity also possess greater endowments [8].

In contrast, the interaction between inequality and coordination has attracted less attention. In this paper, we discuss this issue within the framework of a simple coordination game, a threshold public goods game (TPGG). In a TPGG, individuals derive benefits from contributing to a public good only when their total collective contributions (i.e., absolute contribution multiplied by productivity) reach a certain threshold [9,10,11]. In general, a threshold public goods game tends to have many equilibria, including a defective equilibrium in which no contributions are made and a set of cooperative equilibria in which the group’s collective contribution exactly meets the threshold [9,11]. A cooperative equilibrium is more efficient than a defective equilibrium; therefore, everyone has the motivation to achieve the threshold. However, different players may prefer different cooperative equilibria. They may prefer other group members to contribute more, allowing them to contribute less, thereby leading to a nontrivial coordination problem.

When the TPGG is symmetric, i.e., all players have the same endowment, productivity, and share of the reward, a natural solution is that all the players contribute the same amount of their endowment to the public good. However, when the TPGG is asymmetric, experimental studies have demonstrated a negative effect of endowment inequality on coordination1: while rich subjects often make higher contributions compared to the symmetric case, the much less contributions from the poor lead to an overall decrease in the total collective contribution in the asymmetric scenario [11,12,13]. In contrast, it seems that productivity inequality does not have a significant effect [11]. The effect of inequality on cooperation and coordination has also been investigated in a variation of the TPGG, the climate game (also known as the collective-risk dilemma game) [14]. Experiments have found that achieving coordination is hindered more by endowment inequality than by productivity inequality [15,16,17,18]. Furthermore, the effects of multiple inequalities on coordination are generally more complex [19].

Most of the existing studies on asymmetric TPGGs have considered two types of players with only one source of asymmetry: they either differ in endowments or in productivities2. It is not clear how multiple dimensions of inequality influence people’s cooperation and coordination in large groups. In this paper, we consider an $n$-player asymmetric TPGG where players have different endowments, productivities, and rewards. We focus on the following two questions:

• What is the impact of various forms of inequality on cooperation?
• How do people coordinate when group members differ among multiple dimensions?

We investigate the impact of various forms of inequality on cooperation by examining the existence conditions of Nash equilibria. Furthermore, to analyze the coordination among players with multidimensional differences, we quantify the size of cooperative NE subset in which a specific player makes the largest relative, absolute, or collective contribution.

Section 2 provides a detailed description of the asymmetric TPGG model. In Section 3, two theorems are presented to address the two key questions. Section 4 conducts a numerical analysis, illustrating the theoretical findings with concrete examples. Conclusive remarks are given in Section 5.

2. Model

Let us consider a threshold public goods game (TPGG) with $n$ asymmetric players. Each player $i$ (with $i=1,\dots , n$) starts with some endowment, meaning $e_i>0$. Then, each player independently decides which fraction $x_i$ of their endowment to contribute to the public good. Each player’s contribution is multiplied by a productivity factor $p_i$. We refer to $x_i$, ${e_{i}x}_i$, and ${e_i{p}_{i}x}_i$ as player $i$’s relative contribution, absolute contribution, and collective contribution, respectively [11]. If the total collective contribution of the $n$ players reaches a predefined threshold $\theta$, then each player $i$ obtains a reward $r_i$ along with their remaining endowment. Otherwise, the players only receive their remaining endowment. As a result, the players’ payoffs are as follows:

$$f_i\left(\mathit{x}\right)=\left\{\begin{array}{cc}{e}_i\left(1-x_i\right)+r_i,& \mathrm{i}\mathrm{f} {\sum }_{i=1}^n{e}_i{p}_i{x}_i\ge \theta ,\\ e_i\left(1-x_i\right),& \mathrm{i}\mathrm{f} {\sum }_{i=1}^n{e}_i{p}_i{x}_i<\theta .\end{array}\right.$$

This game can have various kinds of heterogeneities with respect to endowment $e_i$, productivity $p_i$, and reward $r_i$. To compare the impact of various forms of inequality, we assume ${\sum }_{i=1}^n{e}_i{p}_i=G$ and ${\sum }_{i=1}^n{r}_i=R$, i.e., the maximum amount of group collective contribution and the total amount of reward are fixed. Furthermore, we assume $r_i\ge e_i$ for all $i=1,2,\dots ,n$, i.e., the social welfare is improved if the group reaches the threshold3.

3. Results

In general, an asymmetric TPGG has two types of NE. First, there is a set of cooperative NEs in which the total collective contribution exactly meets the threshold. In addition, there may exist a defective NE in which all the players choose not to contribute. Theorem 1 provides the existence conditions for the two types of NE (see Appendix A for the proof).

Theorem 1.

The defective NE, $\mathit{x} \mathbf{=} \mathbf{0}$, exists if and only if, for any player $i$, (1) $\theta >e_i{p}_i$ or (2) $\theta =e_i{p}_i$ and $r_i=e_i$. In contrast, a cooperative NE (i.e., at least one player contributes to the public good) exists if and only if $0<\theta \le G$. In addition, the set of cooperative NEs has the form $\left\{\mathit{x}|{\sum }_{i=1}^n{e}_i{p}_i{x}_i=\theta \right\}$.

Theorem 1 reveals the impact of various inequalities on cooperation from the perspective of NE. It indicates that individual differences have no effect on the existence of a cooperative NE. The cooperative NE set is not empty if and only if the maximum amount of group collective contribution is not less than the threshold. We then analyze the effect of heterogeneity on the defective NE by assuming $r_i>e_i$ (i.e., the critical case $r_i=e_i$ is excluded). For the symmetric case, the existence condition of the defective NE can be simplified as $\theta >\frac{G}{n}$. For the asymmetric case, the defective NE exists if and only if $\theta >e_i{p}_i$ for all players. Without a loss of generality, we consider that players differ in $e_i{p}_i$, with $e_1{p}_1>e_2{p}_2\dots >e_n{p}_n$ (i.e., player 1 is the most ‘able’ player). In this case, the defective NE exists if and only if $\theta >e_1{p}_1$. Let us note that $e_1{p}_1>\frac{{\sum }_{i=1}^n{e}_i{p}_i}{n}=\frac{G}{n}$, and the existence condition of the defective NE becomes stricter than that in the symmetric case. This implies that heterogeneity in $e_i{p}_i$ has a positive impact on cooperation. In particular, defection ceases to be an NE when $e_1{p}_1\ge \theta$ because, in this case, player 1 is motivated to contribute even if all the other players defect.

Theorem 1 also shows that, when $0<\theta <G$, the game has an infinite number of cooperative NEs. In all these equilibria, the group collective contribution reaches the threshold (i.e., ${\sum }_{k=1}^n{e}_k{p}_k{x}_k=\theta$); yet, they vary in how the contributions are distributed among the players. We then analyze which player contributes more to a cooperative NE in the presence of asymmetries. We note that there are three measures of individual contribution, namely, relative contribution, absolute contribution, and collective contribution. We denote the subsets of the cooperative NE set where player $i$ has the highest relative contribution, absolute contribution, and collective contribution by $S_i^1$, $S_i^2$, and $S_i^3$, respectively, and we denote their sizes (under the Hausdorff measure) by $\left|S_i^1\right|$, $\left|S_i^2\right|$, and $\left|S_i^3\right|$, respectively. Intuitively, $\left|S_i^k\right|>\left|S_j^k\right|$ for all $j\ne i$ ($k=1,2,3$) means that player $i$ is more likely to contribute most at a randomly chosen cooperative NE in terms of the relative, absolute, or collective contribution4. Theorem 2 compares the sizes of the different subsets (see Appendix B for the proof).

Theorem 2.

Let us suppose that $e_1{p}_1>e_2{p}_2\dots >e_n{p}_n$.

(1)

For relative contribution, $\left|S_1^1\right|<\dots <\left|S_n^1\right|$ if $0<\theta <e_n{p}_n,$ and $\left|S_1^1\right|>\dots >\left|S_n^1\right|$ if ${\sum }_{i=1}^{n-1}{e}_i{p}_i<\theta <G$.

(2)

For absolute contribution, when there is endowment heterogeneity, i.e., $e_1>\dots >e_n$ and $p_1=\dots =p_n$, $\left|S_1^2\right|\ge \dots \ge \left|S_n^2\right|$ for all $0<\theta <G$. When there is productivity heterogeneity, i.e., $e_1=\dots =e_n$ and $p_1>\dots >p_n$, $\left|S_1^2\right|<\dots <\left|S_n^2\right|$ if $0<\theta <e_n{p}_n,$ and $\left|S_1^2\right|>\dots >\left|S_n^2\right|$ if ${\sum }_{i=1}^{n-1}{e}_i{p}_i<\theta <G$.

(3)

For collective contribution, $\left|S_1^3\right|\ge \dots \ge \left|S_n^3\right|$ for all $0<\theta <G$. Furthermore, $\left|S_1^3\right|=\dots =\left|S_n^3\right|$ if $0<\theta <e_n{p}_n$.

Theorem 2 summarizes the contribution patterns in cooperative NEs when group members differ across multiple dimensions. First, ‘able’ players (i.e., players with larger $e_i{p}_i$) are likely to make higher collective contributions in the sense that $\left|S_i^3\right|\ge \left|S_j^3\right|$ if $e_i{p}_i>e_j{p}_j$. In addition, if the threshold is sufficiently low such that any single player is able to achieve it, then all players are equally likely to make the highest collective contribution in a cooperative NE, irrespective of their abilities. Second, ‘able’ players are likely to have higher relative contributions only for large threshold, where successfully reaching the threshold requires the joint efforts of all players. However, for sufficiently low threshold, ‘able’ players tend to make smaller relative contributions. Finally, the results for absolute contribution depend on both $e_i$ and $p_i$. When there is endowment inequality, the players with higher endowments are likely to contribute more for all thresholds. However, when there is productivity inequality, the more productive players are likely to contribute more only for large threshold.

4. Numerical Analysis

Theorems 1 and 2 qualitatively analyze the effect of various forms of inequality on cooperation and coordination. In particular, Theorem 2 compares the relative sizes of $S_i^k$ for low thresholds (i.e., $0<\theta <e_n{p}_n$) and high thresholds (i.e., ${\sum }_{i=1}^{n-1}{e}_i{p}_i<\theta <G$). In order to enhance the comprehension of the two theorems, we calculate the Nash equilibria and absolute sizes of the three types of subsets $S_i^1$, $S_i^2$, and $S_i^3$ for two-player threshold public goods games with all $0<\theta <G$. Inspired by [8], we consider five scenarios, namely, full equality, endowment inequality, productivity inequality, aligned inequality, and misaligned inequality. Full equality corresponds to a homogeneous scenario, whereas in the endowment inequality and productivity inequality scenarios players only differ in a single dimension. In the last two scenarios, players differ in two dimensions: in the aligned inequality scenario, both advantages concentrate in one player, with the high-endowment player being more productive, while in the misaligned inequality scenario these two advantages distribute across two players, meaning the high-endowment player is less productive. Furthermore, $e_1{p}_1=e_2{p}_2=30$ in the full equality and misaligned inequality scenarios; $e_1{p}_1=40,e_2{p}_2=20$ in the endowment inequality and productivity inequality scenarios, and $e_1{p}_1=48,e_2{p}_2=12$ in the aligned inequality scenario. Detailed parameter settings for $e_i$ and $p_j$ can be found in the first column of

Table 1. Nash equilibria and sizes of $S_i^k$ in two-player threshold public goods games. Column 1: Five game scenarios are considered, namely, full equality, endowment inequality, productivity inequality, aligned inequality, and misaligned inequality. Column 2: For each scenario, the threshold $\theta$ is categorized into three ranges according to Theorem 2. Column 3: The cooperative NE set is non-empty for all parameter combinations, and the defective NE exists only for high thresholds. Columns 4–6: Absolute sizes of the three sets $S_i^1$, $S_i^2$, and $S_i^3$. The size relationship between $\left|S_1^k\right|$ and $\left|S_2^k\right|$ is marked using different colors. Red: $\left|S_1^k\right|>\left|S_2^k\right|$. Black: $\left|S_1^k\right|=\left|S_2^k\right|$. Blue: $\left|S_1^k\right|<\left|S_2^k\right|$. Grey: the relationship between $\left|S_1^k\right|$ and $\left|S_2^k\right|$ depends on $\theta$. Three cells are highlighted. In these cells, $\left|S_1^3\right|>\left|S_2^3\right|$ for almost all $\theta$. The exceptions are $\theta =20$ in the first two cells and $\theta =12$ in the last cell.

$\mathbf{(}{\mathit{e}}_1\mathbf{,}{\mathit{e}}_2\mathbf{,}{\mathit{p}}_1\mathbf{,}{\mathit{p}}_2\mathbf{)}$	Threshold	NE	$\mathbf{\left(}\mathbf{\right|}{\mathit{S}}_1^1\mathbf{|}\mathbf{,}\mathbf{|}{\mathit{S}}_2^1\mathbf{\left|}\mathbf{\right)}$	$\mathbf{\left(}\mathbf{\right|}{\mathit{S}}_1^2\mathbf{|}\mathbf{,}\mathbf{|}{\mathit{S}}_2^2\mathbf{\left|}\mathbf{\right)}$	$\mathbf{\left(}\mathbf{\right|}{\mathit{S}}_1^3\mathbf{|}\mathbf{,}\mathbf{|}{\mathit{S}}_2^3\mathbf{\left|}\mathbf{\right)}$
Full equality $(30,30,1,1)$	$0<\theta <30$	$x_1+x_2=\frac{\theta }{30}$	$\left(\frac{\sqrt{2}\theta }{60}, \frac{\sqrt{2}\theta }{60}\right)$	$\left(\frac{\sqrt{2}\theta }{60}, \frac{\sqrt{2}\theta }{60}\right)$	$\left(\frac{\sqrt{2}\theta }{60}, \frac{\sqrt{2}\theta }{60}\right)$
	$\theta =30$	$x_1+x_2=1$	$\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$	$\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$	$\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$
	$30<\theta <60$	$x_1+x_2=\frac{\theta }{30};$ $x_1=x_2=0$	$\left(\sqrt{2}-\frac{\sqrt{2}\theta }{60}, \sqrt{2}-\frac{\sqrt{2}\theta }{60}\right)$	$\left(\sqrt{2}-\frac{\sqrt{2}\theta }{60}, \sqrt{2}-\frac{\sqrt{2}\theta }{60}\right)$	$\left(\sqrt{2}-\frac{\sqrt{2}\theta }{60}, \sqrt{2}-\frac{\sqrt{2}\theta }{60}\right)$
Endowment inequality $(40,20,1,1)$	$0<\theta <20$	$2x_1+x_2=\frac{\theta }{20}$	$\left(\frac{\sqrt{5}\theta }{120}, \frac{\sqrt{5}\theta }{60}\right)$	$\left(\frac{\sqrt{5}\theta }{80}, \frac{\sqrt{5}\theta }{80}\right)$	$\left(\frac{\sqrt{5}\theta }{80}, \frac{\sqrt{5}\theta }{80}\right)$
	$20\le \theta \le 40$	$2x_1+x_2=\frac{\theta }{20}$	$\left(\frac{\sqrt{5}\theta }{120}, \frac{\sqrt{5}}{2}-\frac{\sqrt{5}\theta }{120}\right)$	$\left(\frac{\sqrt{5}\theta }{80}, \frac{\sqrt{5}}{2}-\frac{\sqrt{5}\theta }{80}\right)$	$\left(\frac{\sqrt{5}\theta }{80}, \frac{\sqrt{5}}{2}-\frac{\sqrt{5}\theta }{80}\right)$
	$40<\theta <60$	$2x_1+x_2=\frac{\theta }{20};$ $x_1=x_2=0$	$\left(\sqrt{5}-\frac{\sqrt{5}\theta }{60}, \frac{\sqrt{5}}{2}-\frac{\sqrt{5}\theta }{120}\right)$	$\left(\frac{3\sqrt{5}}{2}-\frac{\sqrt{5}\theta }{40}, 0\right)$	$\left(\frac{3\sqrt{5}}{2}-\frac{\sqrt{5}\theta }{40}, 0\right)$
Productivity inequality $(20,20,2,1)$	$0<\theta <20$	$2x_1+x_2=\frac{\theta }{20}$	$\left(\frac{\sqrt{5}\theta }{120}, \frac{\sqrt{5}\theta }{60}\right)$	$\left(\frac{\sqrt{5}\theta }{120}, \frac{\sqrt{5}\theta }{60}\right)$	$\left(\frac{\sqrt{5}\theta }{80}, \frac{\sqrt{5}\theta }{80}\right)$
	$20\le \theta \le 40$	$2x_1+x_2=\frac{\theta }{20}$	$\left(\frac{\sqrt{5}\theta }{120}, \frac{\sqrt{5}}{2}-\frac{\sqrt{5}\theta }{120}\right)$	$\left(\frac{\sqrt{5}\theta }{120}, \frac{\sqrt{5}}{2}-\frac{\sqrt{5}\theta }{120}\right)$	$\left(\frac{\sqrt{5}\theta }{80}, \frac{\sqrt{5}}{2}-\frac{\sqrt{5}\theta }{80}\right)$
	$40<\theta <60$	$2x_1+x_2=\frac{\theta }{20};$ $x_1=x_2=0$	$\left(\sqrt{5}-\frac{\sqrt{5}\theta }{60}, \frac{\sqrt{5}}{2}-\frac{\sqrt{5}\theta }{120}\right)$	$\left(\sqrt{5}-\frac{\sqrt{5}\theta }{60}, \frac{\sqrt{5}}{2}-\frac{\sqrt{5}\theta }{120}\right)$	$\left(\frac{3\sqrt{5}}{2}-\frac{\sqrt{5}\theta }{40}, 0\right)$
Aligned inequality $(24,12,2,1)$	$0<\theta <12$	$4x_1+x_2=\frac{\theta }{12}$	$\left(\frac{\sqrt{17}\theta }{240}, \frac{\sqrt{17}\theta }{60}\right)$	$\left(\frac{\sqrt{17}\theta }{144}, \frac{\sqrt{17}\theta }{72}\right)$	$\left(\frac{\sqrt{17}\theta }{96}, \frac{\sqrt{17}\theta }{96}\right)$
	$12\le \theta \le 48$	$4x_1+x_2=\frac{\theta }{12}$	$\left(\frac{\sqrt{17}\theta }{240}, \frac{\sqrt{17}}{4}-\frac{\sqrt{17}\theta }{240}\right)$	$\left(\begin{array}{c}\min \left\{\frac{\sqrt{17}\theta }{144},\frac{\sqrt{17}}{4}\right\},\\ \max \left\{\frac{\sqrt{17}}{4}-\frac{\sqrt{17}\theta }{144},0\right\}\end{array}\right)$	$\left(\begin{array}{c}\min \left\{\frac{\sqrt{17}\theta }{96},\frac{\sqrt{17}}{4}\right\},\\ \max \left\{\frac{\sqrt{17}}{4}-\frac{\sqrt{17}\theta }{96},0\right\}\end{array}\right)$
	$48<\theta <60$	$4x_1+x_2=\frac{\theta }{12};$ $x_1=x_2=0$	$\left(\sqrt{17}-\frac{\sqrt{17}\theta }{60}, \frac{\sqrt{17}}{4}-\frac{\sqrt{17}\theta }{240}\right)$	$\left(\frac{5\sqrt{17}}{4}-\frac{\sqrt{17}\theta }{48}, 0\right)$	$\left(\frac{5\sqrt{17}}{4}-\frac{\sqrt{17}\theta }{48}, 0\right)$
Misaligned inequality $(30,15,1,2)$	$0<\theta <30$	$x_1+x_2=\frac{\theta }{30}$	$\left(\frac{\sqrt{2}\theta }{60}, \frac{\sqrt{2}\theta }{60}\right)$	$\left(\frac{\sqrt{2}\theta }{45}, \frac{\sqrt{2}\theta }{90}\right)$	$\left(\frac{\sqrt{2}\theta }{60}, \frac{\sqrt{2}\theta }{60}\right)$
	$\theta =30$	$x_1+x_2=\frac{\theta }{30}$	$\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$	$\left(\frac{2\sqrt{2}}{3}, \frac{\sqrt{2}}{3}\right)$	$\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$
	$30<\theta <60$	$x_1+x_2=\frac{\theta }{30};$ $x_1=x_2=0$	$\left(\sqrt{2}-\frac{\sqrt{2}\theta }{60}, \sqrt{2}-\frac{\sqrt{2}\theta }{60}\right)$	$\left(\begin{array}{c}\max \left\{\sqrt{2}-\frac{\sqrt{2}\theta }{90},2\sqrt{2}-\frac{\sqrt{2}\theta }{30}\right\},\\ \max \left\{\sqrt{2}-\frac{\sqrt{2}\theta }{45},0\right\}\end{array}\right)$	$\left(\sqrt{2}-\frac{\sqrt{2}\theta }{60}, \sqrt{2}-\frac{\sqrt{2}\theta }{60}\right)$

($G=60$ and $r_1=r_2=45$ in all five scenarios). For each scenario, the threshold $\theta$ is categorized into three ranges according to Theorem 2, i.e., low thresholds $0<\theta <e_2{p}_2$, intermediate thresholds $e_2{p}_2\le \theta \le e_1{p}_1$, and high thresholds $e_1{p}_1<\theta <G$.

We first study the effect of inequality on cooperation. Theorem 1 says that heterogeneity in $e_i{p}_i$ can promote cooperation in the sense that the existence condition of the defective NE becomes stricter. From Table 1, the existence conditions of the defective NE in the full equality, endowment inequality, productivity inequality, aligned inequality, and misaligned inequality scenarios are $\theta >30$, $\theta >40$, $\theta >40$, $\theta >48$, and $\theta >30$, respectively. This shows clearly that the existence condition of the defective NE is harder to satisfy when heterogeneity in $e_i{p}_i$ increases.

We then analyze the set of cooperative NEs. Figure 1 shows the set of cooperative NEs for different game scenarios and thresholds $\theta$. For a two-player threshold public goods game, the set of cooperative NEs is a line segment in the $x_1$-$x_2$ plane with slope $-\frac{e_1{p}_1}{e_2{p}_2}$. Furthermore, the length of the line segment (which measures the size of the cooperative NE set) does not change monotonically in the threshold: it first increases and then decreases as $\theta$ increases from $0$ to $G$.

Columns 4–6 in Table 1 show the (absolute) sizes of the three sets $S_i^1$, $S_i^2$, and $S_i^3$ for different game scenarios and ranges of $\theta$. Overall, the difference between the sizes of the two cooperative NEs sets, $\left|S_1^k\right|-\left|S_2^k\right|$, expands as $\theta$ increases for all$k=1, 2, 3$. In other words, higher thresholds result in an increased probability that the more ‘able’ player makes a greater contribution at a randomly chosen cooperative NE. However, the size relationship between $\left|S_1^k\right|$ and $\left|S_2^k\right|$ depends crucially on $e_i$, $p_i$, $\theta$, and the contribution type.

For relative contribution, as pointed out in Theorem 2(1), in game scenarios with heterogeneity in $e_i{p}_i$ (i.e., the endowment inequality, productivity inequality, and aligned inequality scenarios), the size of $\left|S_1^1\right|$ is less than $\left|S_2^1\right|$ for low thresholds $0<\theta <e_2{p}_2$, whereas the opposite is true for high threshold $e_1{p}_1<\theta <G$. In addition, for intermediate thresholds $e_2{p}_2\le \theta \le e_1{p}_1$, the size of $\left|S_1^1\right|$ (or $\left|S_2^1\right|$) is increasing (or decreasing) in $\theta$. In contrast, in game scenarios without heterogeneity in $e_i{p}_i$ (i.e., the full equality and misaligned inequality scenarios), $\left|S_1^1\right|=\left|S_2^1\right|$ for all $\theta$. For collective contribution, Theorem 2(3) predicts $\left|S_1^3\right|=\left|S_2^3\right|$ for low thresholds $0<\theta <e_2{p}_2$. This is indeed the case. Furthermore, we observe $\left|S_1^3\right|>\left|S_2^3\right|$ for the intermediate and high thresholds $e_2{p}_2<\theta <G$ in the three game scenarios with heterogeneity in $e_i{p}_i$. This observation is slightly stronger than the prediction of Theorem 2(3) (which says that $\left|S_1^3\right|\ge \left|S_2^3\right|$). Finally, the numerical analysis confirms that the results for the absolute contribution are sensitive to $e_i$ and $p_i$. In particular, $\left|S_1^2\right|=\left|S_2^2\right|$ in the full equality scenario; $\left|S_1^2\right|\ge \left|S_2^2\right|$ for all $\theta$ in the endowment inequality and misaligned inequality scenarios, and $\left|S_1^2\right|<\left|S_2^2\right|$ for low thresholds and $\left|S_1^2\right|>\left|S_2^2\right|$ for high thresholds in the productivity inequality and aligned inequality scenarios.

5. Concluding Remarks

In this paper, we consider an $n$-player asymmetric TPGG, where players have different endowments, productivities, and rewards. This game can have a defective NE and multiple cooperative NEs. We show that heterogeneity in $e_i{p}_i$ can promote cooperation in the sense that the existence condition of the defective NE becomes stricter. Furthermore, players with higher $e_i{p}_i$ are likely to contribute more at a cooperative NE in terms of collective contribution, but they do not necessarily have a higher relative contribution or absolute contribution. This result is consistent with a recent experimental study on a two-player asymmetric TPGG [11]. In the study [11], subjects either differ in their endowments or in their productivities. In most of the successful groups (i.e., the group has reached the threshold), the collective contribution of the rich individual or the more productive individual is higher, but the two group members tend to have similar relative contributions in the endowment inequality scenario, while having similar absolute contributions in the productivity inequality scenario. In sum, our study highlights the nontrivial effects of inequality on cooperation and coordination in a TPGG.

To deepen our understanding on human behaviors in TPGGs, a possible direction for future research is to empirically validate the theoretical findings for scenarios where players differ in two or more dimensions (e.g., the aligned and misaligned inequality scenarios discussed in Section 4). Another direction for a future investigation could focus on the coordination problem in an asymmetric climate game. A recent study showed that cooperation can be an equilibrium outcome if and only if the weighted average of climate risk of all countries reaches or exceeds the coefficient of emission reduction target [20]. However, the climate game can have multiple cooperative equilibria, in which countries have different responsibilities for carbon emission reduction at different equilibria. Thus, a natural question would be the following: how do countries with different endowments, climate risks, and emission reduction costs coordinate? Finally, the reward value for each player in our model is considered to be fixed. It would be interesting to investigate the TPGG model within a cooperative game framework, where rewards are assigned, according to the players’ values, to each coalition. Specifically, one could calculate the stable reward allocation schemes in asymmetric TPGGs based on the Shapley value, CIS value, and nucleolus and compare the fairness and efficiency of these allocation schemes.