Beliefs and Public Good Provision with Anonymous Contributors

• Date: 01 October 2016
• Author: JOSE A. RODRIGUES‐NETO,WILFREDO LEIVA MALDONADO
• DOI: http://doi.org/10.1111/jpet.12161
• Published date: 01 October 2016

BELIEFS AND PUBLIC GOOD PROVISION WITH ANONYMOUS

CONTRIBUTORS

WILFREDO LEIVA MALDONADO

Catholic University of Bras´

ılia

JOSE A. RODRIGUES-NETO

Australian National University

Abstract

We analyze a static game of public good contributions where finitely

many anonymous players have heterogeneous preferences about the

public good and heterogeneous beliefs about the distribution of pref-

erences. In the unique symmetric equilibrium, the only individuals who

make positive contributions are those who most value the public good

and who are also the most pessimistic; that is, according to their beliefs,

the proportion of players who most like the public good is smaller than

it would be according to any other possible belief. We predict whether

the aggregate contribution is larger or smaller than it would be in an

analogous game with complete information and heterogeneous prefer-

ences, by comparing the beliefs of contributors with the true distribu-

tion of preferences. A trade-off between preferences and beliefs arises

if there is no individual who simultaneously has the highest preference

type and the most pessimistic belief. In this case, there is a symmetric

equilibrium, and multiple symmetric equilibria occur only if there are

more than two preference types.

1. Introduction

Paul Samuelson pioneered the analysis of public goods provision.1More recent litera-

ture on the provision of public goods under incomplete information usually assumes

that incompleteness arises from either ignorance about players’ preferences, for in-

stance Bac(1996), Menezes, Monteiro, and Temimi (2001), and Bag and Roy (2008),

1See Samuelson (1954, 1955), as well as Olson (1965).

Wilfredo Leiva Maldonado, Catholic University of Braslia, Brazil (wilfredo@pos.ucb.br). Jose A.

Rodrigues-Neto, Australian National University, Canberra, Australia (jarnwi@yahoo.com).

Wilfredo Leiva Maldonado acknowledges the financial support of CNPq (Brazil) through Grant

303420/2012-0. We would like to thank James Taylor and seminar participants at the Australian Na-

tional University for comments, and Merrilyn L´

arusson for the English editing. We would also like to

thank three excellent reviewers for their many corrections and suggestions. All remaining errors are

our own responsibility.

Received February 20, 2015; Accepted March 3, 2015.

C2016 Wiley Periodicals, Inc.

Journal of Public Economic Theory, 18 (5), 2016, pp. 691–708.

691

692 Journal of Public Economic Theory

or from ignorance about peers’ wealth, as in Andreoni (1988), Gradstein (1992), and

Gradstein, Nitzan, and Slutsky (1994). However, in some situations in practice, it is un-

likely that each individual knows the identity of all her peers. Even when each player

knows the identities of all players, we may not wish to assume that each player is capa-

ble of forming a joint distribution about the profiles of types of other players. It seems

more reasonable that each player simply has her own perception of the distribution of

relevant characteristics of the population.

Hellwig (2011) introduces a general mathematical framework to work with incom-

plete information games in a large population of anonymous players.2This is a con-

venient modeling setup in games where the relevant variable for a player is only the

aggregate action of the others. Examples of this framework are voting games, Cournot

competition and public contribution games.

Our work analyzes a static game of individual contributions to a continuous public

good by a finite population of heterogeneous anonymous players. The heterogeneity

comes from two sources: (1) the preferences for, or values of, the public good—what

we refer to as the player’s type or preference type; and (2) the beliefs about the distri-

bution of types among the population. Unlike most of the literature, we do not assume

that each player knows the correct joint distribution of her peers’ (preference) types.

Instead, each player has a belief about how the population is divided by types. Each

individual has quasi-linear utility in her private good consumption and chooses how

much of her initial endowment to consume or to give to the pool of contributions to

the public good.

In this setup, we prove that there is always a symmetric equilibrium; that is, a Nash

equilibrium where all players with the same characteristic (same type and belief) make

identical contributions. In such a symmetric equilibrium, only players with the highest

preference type and the most pessimistic belief (i.e., the one with the lowest proportion

of the highest type players) make positive contributions, provided that there is at least

one such player. In this situation, the total contribution to the public good could be

greater, smaller, or equal to the total contribution under an analogous game of com-

plete information with heterogeneous preferences. It is greater if and only if the players

that actually contribute are pessimistic relative to the true distribution of preferences,

which means that they believe that the number of the highest type players is smaller

than it actually is. This result allows us to evaluate how the classical free-rider problem

depends on the heterogeneity of beliefs and on the preferences present in the popula-

tion. If there are many possible types and players, and only few of them have the most

pessimistic belief, then the free-rider problem is severe. This is because only a relatively

small minority of players contributes and these contributions are relatively large. More-

over, contributors may be better off than noncontributors of different types. This is true

because contributors benefit more from the same amount of public good than players

with a different preference type.

If there is no individual who simultaneously has the highest preference type and

the most pessimistic belief, a compensatory interaction between preferences and be-

liefs arises. Depending on the parameters, there may be symmetric equilibria where all

2As Hellwig (2011) explains, “This formulation of incomplete information departs from the personal-

istic approach where each agent is assumed to receive a signal about the underlying state of the world

and uses the signal to form probabilistic beliefs about the state of the world, about the signals received

by the other agents and about the other agents’ beliefs that are induced by their signals . .. here, agents

do not form beliefs about any particular other agents. They only form beliefs about the cross-section

distribution of the other agents’ characteristics.”