Academic journal article Economic Inquiry

Two-Player Asymmetric Contests with Ratio-Form Contest Success Functions

Academic journal article Economic Inquiry

Two-Player Asymmetric Contests with Ratio-Form Contest Success Functions

Article excerpt

I examine players' equilibrium effort levels in two-player asymmetric contests with ratio-form contest success functions. I first characterize the Nash equilibrium of the simultaneous-move game. I show that the equilibrium effort ratio is equal to the valuation ratio, and that the prize dissipation ratios for the players are the same. I also show that the prize dissipation ratio for each player is less than or equal to the minimum of the players' probabilities of winning at the Nash equilibrium and thus never exceeds a half. Then I examine how the equilibrium effort ratio, the prize dissipation ratios, and the players' equilibrium effort levels, respond when the players' valuations for the prize or their abilities change. (JEL D72, C72)


A contest is a situation in which players compete with one another by expending irreversible effort to win a prize. Typical examples are various types of rent-seeking contests: competition among firms to win a monopoly rent secured under government protection or by a government procurement contract, competition between domestic and foreign firms to obtain governmental trade policies favorable to them, competition among firms to acquire a rent generated by rights of ownership to an import quota, and competition among firms to capture rents created by governmental decisions to establish tariffs or other trade barriers. Other examples of contests include auctions, patent races, research and development (R&D) competition among firms, litigation, competition for jobs among job candidates, competition among candidates to win promotion to higher ranks, election campaigns between political candidates, and competition between local governments to invite business firms, government institutions, or government-owned corporations into their districts.

Naturally, due to their prevalence and importance in economies, such contests have been studied by many economists: Loury (1979), Lee and Wilde (1980), Tullock (1980), Rogerson (1982), Rosen (1986), Appelbaum and Katz (1987), Dixit (1987), Hillman and Riley (1989), Hirshleifer (1989), Ellingsen (1991), Nitzan (1991, 1994), Krishna and Morgan (1997), Che and Gale (1998), Hurley and Shogren (1998), and Konrad (2000), to name a few. In this vast literature on the theory of contests, one of the main issues is: How much effort do the players exert in pursuit of the prize? Indeed, it is of great interest because the players' effort levels determine the profitability of the players' engaging in the contest and, in some cases, they are revenues collected by the contest organizer or bribes given to government officials. Furthermore, they account for other important outcomes of the contest. For example, in a rent-seeking contest, efforts expended by the players are viewed as social costs due to rent-seeking activities, so that total effort level is a measure of economic efficiency. In an R&D contest, effort levels expended by the players--these are R&D expenditures of the firms--determine the expected date of invention.

This article also addresses the issue: How much effort do the players exert in pursuit of the prize? But it differs from previous research by dealing with contests with ratio-form contest success functions.1 Specifically, the novelty of this article is to consider two-player asymmetric contests in which each player's probability of winning is a function of the ratio of the two players' effort levels.

Two-player contests with ratio-form contest success functions or two-player contests that can be best modeled with ratio-form contest success functions are easily observed in the real world. Examples include various types of two-player rent-seeking contests, litigation between a plaintiff and a defendant, election campaigns between two parties or candidates, and R&D competition between two firms. Consider, for example, a rent-seeking contest in which two firms, potential monopolists, compete with each other to win a government monopoly franchise contract. …

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