Private provision of discrete public goods: the correlated cost case

Abstract

This paper explores the effects of correlated costs between players in an incomplete information game in the context of the private provision of discrete public goods. In such an incomplete information game, without correlation between players, equilibria in cut-point strategies (in other words, monotone strategies) always exist. A cut-point strategy prescribes to contribute a positive amount up to a certain cost level (‘the cut-point’) and to contribute zero above that critical level. However, when the players’ costs of contributing are correlated, an equilibrium in cut-point strategies may fail to exist because new incentives to free ride arise. A sufficient condition for the existence of an equilibrium in cut-point strategies is provided, and equilibria in non-monotone strategies are studied. The case without correlation is analyzed as a special case.

Appendix

1.1 Proof of Proposition 1

When (12) is satisfied, the slope of $G{ \Pr }(n = k - 1{\mid }c_{i} )$ with respect to $c_{i}$ is less than 1. Thus, $\widehat{{c_{i} }}$ is a unique solution to (11). Player $i$, whose disutility is $\widehat{{c_{i} }}$, is indifferent between contributing and not contributing. From (8), player $i$’s best action is to contribute at a sufficiently small $c_{i}$ and to not contribute at a sufficiently large $c_{i}$. Thus, around $\widehat{{c_{i} }}$, player $i$ changes his/her action from contributing to not contributing. The cut-point strategy is the best response to other players’ cut-point strategies, and an equilibrium in the single cut-point strategies exists. Therefore, it is sufficient to ensure that the maximum slope of $G{ \Pr }(n = k - 1{\mid }c_{i} )$ is below 1.

1.2 Necessary and sufficient condition for the existence of an equilibrium in cut-point strategies

Here, we study a more detailed condition of Proposition 1. An equilibrium fails to exist in the single cut-point strategies if and only if (11) has multiple solutions. In this case, there is at least one intersection with the slope of the left-hand side of (11) is greater than or equal to 1. Therefore, an equilibrium in the single cut-point strategies exists if and only if the slope of the left-hand side of (11) is below 1 at any intersection $\widehat{{c_{i} }}$. The following is a necessary and sufficient condition for the existence of equilibrium in the single cut-point strategies:

$$ \begin{aligned} \left. {\frac{{\partial H(c_{i} , \hat{c})}}{{\partial c_{i} }}} \right|_{{at c_{i} = \widehat{{c_{i} }}}} = & \frac{{H\left( {\widehat{{c_{i} }}, \hat{c}} \right)\left\{ {k - 1 - (M - 1) \varPhi \left( {\gamma \left( {\hat{c} - \frac{{\alpha \theta + \beta \widehat{{c_{i} }}}}{\alpha + \beta }} \right)} \right)} \right\}\frac{ - \,\gamma \beta }{\alpha + \beta }}}{{\left\{ {\varPhi \left( {\gamma \left( {\hat{c} - \frac{{\alpha \theta + \beta \widehat{{c_{i} }}}}{\alpha + \beta }} \right)} \right)} \right\}\left\{ {1 - \varPhi \left( {\gamma \left( {\hat{c} - \frac{{\alpha \theta + \beta \widehat{{c_{i} }}}}{\alpha + \beta }} \right)} \right)} \right\}}} \\ & \phi \left( {\gamma \left( {\hat{c} - \frac{{\alpha \theta + \beta \widehat{{c_{i} }}}}{\alpha + \beta }} \right)} \right) < 1\,for\,any\,\widehat{{c_{i} }}, \\ & where H(c_{i} ,\hat{c}) \equiv G\left( {\begin{array}{*{20}c} {M - 1} \\ {k - 1} \\ \end{array} } \right)\left\{ {\varPhi \left( {\gamma \left( {\hat{c} - \frac{{\alpha \theta + \beta c_{i} }}{\alpha + \beta }} \right)} \right)} \right\}^{k - 1}\\ & \left\{ {1 - \varPhi \left( {\gamma \left( {\hat{c} - \frac{{\alpha \theta + \beta c_{i} }}{\alpha + \beta }} \right)} \right)} \right\}^{M - k} . \\ \end{aligned} $$

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In addition to the arguments in Proposition 1, the following results are found. When $\theta$ is sufficiently large or small, $H\left( {\widehat{{c_{i} }}, \hat{c}} \right)/\left[ {\left\{ {\varPhi \left( {} \right)} \right\}\left\{ {1 - \varPhi \left( {} \right)} \right\}}] \right.$ becomes close to 0. Therefore, when $\theta$ is sufficiently large or small, an equilibrium in the single cut-point strategies exists.

Moreover, there is a unique equilibrium if and only if the slope of the left-hand side of (15) is below 1 at any intersection $\hat{c}^{*}$. A necessary and sufficient condition for the uniqueness of the equilibrium is:

$$ \begin{aligned}\left. {\frac{{\partial H\left( {\hat{c},\hat{c}} \right)}}{{\partial \hat{c}}}} \right|_{{at\,\hat{c} = \hat{c}^{*} }} &= \frac{{H\left( {\hat{c}^{*} ,\hat{c}^{*} } \right)\left\{ {k - 1 - (M - 1)\varPhi \left( {\gamma \alpha \frac{{\hat{c}^{*} - \theta }}{\alpha + \beta }} \right)} \right\}\frac{ - \,\gamma \beta }{\alpha + \beta }}}{{\left\{ {\varPhi \left( {\gamma \alpha \frac{{\hat{c}^{*} - \theta }}{\alpha + \beta }} \right)} \right\}\left\{ {1 - \varPhi \left( {\gamma \alpha \frac{{\hat{c}^{*} - \theta }}{\alpha + \beta }} \right)} \right\}}}\phi \left( {\gamma \alpha \frac{{\hat{c}^{*} - \theta }}{\alpha + \beta }} \right)\\ &\quad < 1\,for\,any\,\hat{c}^{*} .\end{aligned} $$

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1.3 Proof of Proposition 2

If (15) has a unique solution, this solution gives a unique equilibrium. Therefore, similar to Proposition 1, it is sufficient to ensure that the maximum slope of the left-hand side of (15) is below 1.

1.4 With general distribution

The arguments for the single cut-point strategy are repeated here for the case with a general distribution function. As in former sections, symmetric Bayesian Nash equilibria are studied. Disutility $c_{i}$ consists of the common term $c$ and the characteristic term $d_{i}$:

$$ c_{i} = c + d_{i} , $$

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where $c$ is common across players and follows a continuous distribution. $d_{i}$ is independent across players and follows a continuous distribution. From these expressions, a posterior distribution of $c_{j}$ for player $i$ whose disutility is $c_{i}$ is acquired, and its probability function is defined as:

$$ {\text{P}}(c_{j} {\mid }c_{i} ). $$

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Player $i$’s net expected utility from contributing is:

$$ \begin{aligned} & {\text{L}}(\hat{c},c_{i} ) - c_{i} , \\ & where\,{\text{L}}(\hat{c},c_{i} ) \equiv {\text{G}}\left( {\begin{array}{*{20}c} {{\text{M}} - 1} \\ {{\text{k}} - 1} \\ \end{array} } \right)\{ {\text{P}}(\hat{c}{\mid }c_{i} )\}^{k - 1} \{ 1 - {\text{P}}(\hat{c}{\mid }c_{i} )\}^{M - k} . \\ \end{aligned} $$

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At the cut-point for player $i\left( {\widehat{{c_{i} }}} \right)$, the following is satisfied:

$$ {\text{L}}\left( {\hat{c},\widehat{{c_{i} }}} \right) = \widehat{{c_{i} }}. $$

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We can restate the condition regarding the existence of an equilibrium in the single cut-point strategies as:

$$ \left. {\frac{{\partial {\text{L}}\left( {\hat{c},c_{i} } \right)}}{{\partial c_{i} }}} \right|_{{at\,c_{i} = \widehat{{c_{i} }}}} = \frac{{{\text{L}}\left( {\hat{c},\widehat{{c_{i} }}} \right)\left\{ {k - 1 - (M - 1){\text{P}}\left( {\hat{c}{\mid }\widehat{{c_{i} }}} \right)} \right\}}}{{\left\{ {{\text{P}}(\hat{c}{\mid }\widehat{{c_{i} }})} \right\}\left\{ {1 - {\text{P}}(\hat{c}{\mid }\widehat{{c_{i} }})} \right\}}}\left. {\frac{{\partial {\text{P}}(\hat{c}{\mid }c_{i} )}}{{\partial c_{i} }}} \right|_{{at\,c_{i} = \widehat{{c_{i} }}}} < 1\,for\,any\,\widehat{{c_{i} }}. $$

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This condition is necessary and sufficient for the existence of an equilibrium.

Symmetric equilibria are found from:

$$ {\text{L}}\left( {\hat{c}^{*} ,\hat{c}^{*} } \right) = \hat{c}^{*} . $$

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$\hat{c}^{*}$ is an equilibrium cut-point. The condition that ensures the uniqueness of the equilibrium is:

$$ \left. {\frac{{\partial {\text{L}}(\hat{c},\hat{c})}}{{\partial \hat{c}}}} \right|_{{at\,\hat{c} = \hat{c}^{*} }} = \frac{{{\text{L}}(\hat{c},\hat{c})\left\{ {k - 1 - (M - 1){\text{P}}(\hat{c}{\mid }\hat{c})} \right\}}}{{\left\{ {{\text{P}}(\hat{c}{\mid }\hat{c})} \right\}\left\{ {1 - {\text{P}}(\hat{c}{\mid }\hat{c})} \right\}}}\left. {\frac{{\partial {\text{P}}(\hat{c}{\mid }\hat{c})}}{{\partial \hat{c}}}} \right|_{{at\,\hat{c} = \hat{c}^{*} }} < 1\,for\,any\,\hat{c}^{*} . $$

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This condition is necessary and sufficient for the uniqueness of the equilibrium.