Bidding for a group-specific public-good prize

Abstract

We examine the equilibrium effort levels of individual players and groups in a contest in which two groups compete with each other to win a group-specific public-good prize, the players choose their effort levels simultaneously and independently, and the winning group is determined by the selection rule of all-pay auctions. We first prove nonexistence of a pure-strategy Nash equilibrium, and then construct a mixed-strategy Nash equilibrium. At the Nash equilibrium, the only active player in each group is a player whose valuation for the prize is the highest in that group; all the other players expend zero effort; and the equilibrium effort levels depend solely on two values — the highest valuation for the prize in each group.

Introduction

A contest is a situation in which individual players or groups compete with one another by expending irreversible effort to win a prize. In a group contest, if a prize is a public good within a group, we call it a group-specific public-good prize. Contests involving group-specific public-good prizes are easily observed. Consider, for example, a situation in which the government first decides whether to regulate a monopoly and then decides which firm to be the monopolist. In each stage, there is competition. In the first stage, firms — potential monopolists — lobby for the unregulated monopoly while consumer groups lobby for the regulated monopoly.¹ In the second stage, after knowing the government’s decision on the form of the monopoly, the firms compete against each other to win the monopoly. The prize for each firm in the first-stage competition is a group-specific public good: if a firm wins its prize, the unregulated monopoly, then all of the firms enjoy being candidates for the unregulated monopoly. The prize for each consumer group is also a group-specific public good: if a consumer group wins its prize, the regulated monopoly, then all consumers enjoy the lower price.

Other examples of contests with group-specific public-good prizes include competition between domestic and foreign firms to obtain governmental trade policies favorable to them, R&D competition between consortiums, election campaigns between political parties, and competition between local governments to invite business firms into their districts.

The purpose of this paper is to examine the equilibrium effort levels of individual players and groups in such contests. To do so, we consider the following contest. Two groups compete with each other to win a group-specific public-good prize. The individual players choose their effort levels simultaneously and independently. Each player’s effort is irreversible. The winning group is determined by the selection rule of all-pay auctions — a group which expends more effort (or submits a higher group-bid) than its rival wins the prize with certainty. Each player’s valuation for the prize is publicly known and the valuations may differ across the individual players.

We first show that no pure-strategy Nash equilibrium exists. Then, constructing a mixed-strategy Nash equilibrium, we show that the equilibrium effort levels of individual players and groups depend solely on two values — the highest valuation for the prize in each group. This implies that the equilibrium effort levels are independent of the number of players, the sum of valuations, and the distribution of valuations in each group, as long as changes in these do not change the highest valuation for that group. We also show that, at the mixed-strategy Nash equilibrium, there are only two active players, one for each group; each active player is one who has the highest valuation for the prize in his group; and all the other players except these two expend zero effort.² We argue that the free-rider problem occurs at the equilibrium since, in each group, a highest-valuation player obtains the greatest gross marginal payoff, while all members including him experience the same marginal cost.

Katz et al., 1990, Ursprung, 1990, Baik, 1993, Riaz et al., 1995 and Baik and Shogren (1998) also study contests with group-specific public-good prizes. Among them, Baik (1993) and Baik and Shogren (1998) are closely related to this paper. The main difference is that their rules of selecting the winning group differ from the rule in this paper — their probability-of-winning functions are continuous while our function is discontinuous. According to our selection rule, the group which expends the largest effort wins the prize with certainty. Interestingly, however, the main result in Baik (1993) and Baik and Shogren (1998) is similar to ours: the equilibrium effort levels of individual players and groups depend solely on two values — the highest valuation for the prize in each group.

Hillman and Riley (1989) and Baye et al. (1996) consider a contest in which many individual players compete with one another by expending irreversible effort to win a private-good prize, and the winner is determined by the selection rule of all-pay auctions — a player who submits the highest individual bid wins the prize with certainty. They show that no pure-strategy Nash equilibrium exists.³ They also show that, if top two players (according to valuation for the prize) have greater valuations than the third, then there exists a unique mixed-strategy Nash equilibrium at which only the top two players are active and all the other players expend zero effort (or bid zero) with probability one. Note that we also obtain the two-active-player result. In this paper, however, the two active players are not top two players (according to valuation) in the contest. They consist of a highest-valuation player in each group.

This paper is related to the literature on the private (also called, voluntary) provision of public goods — the literature which deals with situations in which public goods are financed by voluntary contributions of individuals (see, for example, Olson, 1965, Olson and Zeckhauser, 1966, Bergstrom et al., 1986, Gradstein et al., 1994, Varian, 1994, Vicary, 1997, Boadway and Hayashi, 1999). Indeed, in this paper, the players in each group play a game of the private provision of a public good — they choose their effort levels noncooperatively to win their public-good prize. We show that, as is often the case with the private-public-goods-provision literature, the free-rider problem arises. The free-rider problem in this paper is the severest form in that, in each group, all the players except a highest-valuation player are free riders.

This paper is also related to the literature on mechanism design, particularly, designing mechanisms for the provision of public goods. Papers in this literature include Groves and Ledyard, 1977, Jackson and Moulin, 1992, Kleindorfer and Sertel, 1994, Bag and Winter, 1999, Deb and Razzolini, 1999 and Saijo and Yamato (1999). Based on our model, one could propose the ‘all-pay-auction mechanism’ for the provision of group-specific public goods. According to the mechanism, the government auctions off a group-specific public good;⁴ all individual players in participating groups or communities submit their bids noncooperatively; the government collects all the bids and provides the public good for the group which submits the highest group-bid for the public good; and the collected money goes to the general fund of government revenues. The main idea of this proposed mechanism is to finance a group-specific public good partly or fully by auctioning it off.

The paper proceeds as follows. Section 2 develops the model. Section 3 first proves the nonexistence of a pure-strategy Nash equilibrium, and then finds a mixed-strategy Nash equilibrium at which there is just one active player in each group. Section 4 presents two modified models in which there are many active players in equilibrium. Finally, Section 5 offers our conclusions.