Voting for the distribution rule in a Public Good Game with heterogeneous endowments

Abstract

This paper analyzes the impact of inequality in the distribution of endowments on cooperation. We conduct a lab experiment using a dynamic Public Good Game to test this relation. We introduce the possibility of choosing among three different redistribution rules: Equidistribution, Proportional to contribution and Progressive to endowment. This novelty in a dynamic environment allows us to analyze how the inequality within groups changes according to individual choices and to investigate if players show inequity averse preferences. Results show that inequality has a negative impact on individual contribution. Players act in order to reduce the initial exogenous inequality. Indeed, in the Treatment with the highest level of inequality, agents vote for reducing the endowment heterogeneity. Moreover, individual contribution is strongly influenced by others’ contributions.

Appendices

Appendix 1: Figures—individual behavior

See Figs. 8, 9, 10

Fig. 8

Average individual contributions per period (Treatment 1)

Fig. 9

Average individual contributions per period (Treatment 2)

Fig. 10

Average individual contributions per period (Treatment 3)

Appendix 2: Theoretical solution of the game

1.1 Equidistribution

In the case of the Equidistribution each player takes the same amount which is independent from her level of contribution. The coefficient $\alpha$ is given by:

$$\begin{aligned} \alpha = \frac{1}{n} \end{aligned}$$

and $\delta = 2$.

The net payoff corresponding to the strategy cooperate is given by:

$$\begin{aligned} \pi _C = \alpha \delta \sum _{i=1}^{5} g_i - g_i \end{aligned}$$

The net payoff corresponding to the strategy not cooperate is given by:

$$\begin{aligned} \pi _{NC} = \alpha \delta \sum _{i=1}^{4} g_i \end{aligned}$$

In this case the dominant strategy is to not cooperate. In fact each player cooperates if $\pi _C > \pi _{NC}$, that is:

$$\begin{aligned} \alpha \delta \sum _{i=1}^{5} g_i - g_i > \alpha \delta \sum _{i=1}^{4} g_i \end{aligned}$$

Since $\sum _{i=1}^{5} g_i - \sum _{i=1}^{4} g_i = g_i$, the equilibrium solution is to cooperate if

$$\begin{aligned} \alpha > \frac{1}{\delta } \end{aligned}$$

this condition is never satisfied and so the equilibrium solution is to not cooperate.

1.2 Proportional to contribution

In this case players receive from the Public Good an amount proportional to their own contribution. Indeed, the coefficient $\alpha$ is determined after the collection of the individual contributions and it is equal to:

$$\begin{aligned} \alpha = \frac{g_i}{\sum _{i=1}^{n} g_i} \end{aligned}$$

and $\delta = 2$. The individual net payoff related to the strategy cooperate is given by:

$$\begin{aligned} \pi _C = \alpha \delta \sum _{i=1}^{5} g_i - g_i \end{aligned}$$

while in the case of not cooperate players obtain zero, i.e.

$$\begin{aligned} \pi _{NC} = 0 \end{aligned}$$

In this case the equilibrium strategy is to cooperate if and only if $\pi _C > \pi _{NC}$, that is

$$\begin{aligned} \alpha \delta \sum _{i=1}^{5} g_i - g_i > 0 \end{aligned}$$

This implies that, the optimal solution of the game is to contribute if

$$\begin{aligned} \alpha > \frac{g_i}{\delta \sum _{i=1}^{n} g_i } \end{aligned}$$

In the extreme case in which $\sum _{i=1}^{n} g_i = g_i$, player i has the convenience to cooperate if $\alpha > 1/2$. In the case in which there only one player who contributes, the coefficient $\alpha$ is always equal to 0.5. In this case, the individual i is indifferent to the two strategies.

According to our parametrization, and considering that the value of $\alpha$ is endogenously derived after the contribution stage, this condition holds for each value of $g_i$, in fact:

$$\begin{aligned} \alpha = \frac{g_i}{\sum _{i=1}^{n} g_i} > \frac{1}{2} \frac{g_i}{\sum _{i=1}^{n} g_i} \end{aligned}$$

This means that the best solution is to cooperate.

1.3 Progressive

According to this rule, the highest share of the Public Good goes to players with the lowest endowment. This share does not depend on the individual contribution neither of the rich, nor of that of the poor. The coefficient $\alpha$ is given by:

$$\begin{aligned} \alpha = \frac{1}{n-1} \left( 1 - \frac{d_i }{ \sum _{i=1}^{n} d_i } \right) \end{aligned}$$

with $delta = 2$. The net payoff related to the strategy cooperate is:

$$\begin{aligned} \pi _C = \alpha \delta \sum _{i=1}^{5} g_i - g_i \end{aligned}$$

and the net payoff associated to the strategy not cooperate is:

$$\begin{aligned} \pi _{NC} = \alpha \delta \sum _{i=1}^{4} g_i \end{aligned}$$

According to this specification, the optimal strategy is to cooperate if and only if:

$$\begin{aligned} \alpha \delta \sum _{i=1}^{5} g_i - g_i > \alpha \delta \sum _{i=1}^{4} g_i \end{aligned}$$

Given that $\sum _{i=1}^{5} g_i - \sum _{i=1}^{4} g_i = g_i$, the inequality is true if

$$\begin{aligned} \alpha > \frac{1}{\delta } \end{aligned}$$

According to our parametrization, this condition never holds. In fact:

$$\begin{aligned} 0 < \frac{d_i }{\sum _{i=1}^{n} d_i } < 1 \quad \forall d \end{aligned}$$

As a consequence:

$$\begin{aligned} 0 < \left( 1 - \frac{d_i }{\sum _{i=1}{n} d_i } \right) < 1 \end{aligned}$$

and

$$\begin{aligned} \frac{1}{n-1} \left( 1 - \frac{d_i }{ \sum _{i=1}^{n} d_i} \right) < \frac{1}{2} \quad \forall n, d \end{aligned}$$

This means that the optimal solution of the game is to not cooperate.