Communications of the ACM

The skill parameters reflect the abilities of players: the larger the value of a player’s skill parameter, the more proficient is the player. If the production cost is according to a linear function, we can normalize the payoff functions such that a player’s skill parameter can be interpreted as the reciprocal of his or her production cost per unit effort. The game as defined here allows us to study equilibrium outcomes under different prize allocation mechanisms, such as assigning fixed shares of a prize budget in decreasing order of invested efforts, or splitting a prize budget among players in proportion to their effort investments. The prize allocation can be interpreted as the winning probabilities for an indivisible prize item, or as the shares of an infinitely divisible prize. The game allows us to study equilibrium outcomes for different types of production costs. For example, it is common to consider linear production costs, we refer to as constant marginal production costs, under which the production cost per unit effort is constant; in particular, we refer to unit marginal production cost when the production cost per unit effort is of unit value. We may also consider production costs with either decreasing or increasing marginal costs. It is noteworthy that the game as defined here formally corresponds to an auction, where efforts, skills, and production costs are in correspondence with bids, valuations, and payments, respectively We say a game is with complete information if the players have perfect information about each other’s skill parameters. A game with complete information can be used as a model of a contest when the players are informed about who is going to participate in the contest and about the skills of the participants. For example, a situation like this can be found in competition-based software development platforms such as TopCoder, where a contest takes place after a registration phase, which reveals identities of participants. A game is said to be with incomplete information if the value of each player’s skill parameter is his or her private information. In a game with incomplete information, skill parameters are assumed to be random variables according to a prior distribution, which is a common knowledge.

The design of skill-rating methods is based on statistical models of ranking outcomes developed from 1920s onward. More recent developments include skill-rating methods that allow for contests among two or more teams of players, which are common in online gaming and online labour platforms. New results have been recently developed in the area of statistical inference for statistical models of ranking data, including new characterizations of the accuracy of various skill parameter estimators and new iterative methods for skill parameter estimation.

In this article, we survey some main results of contest theory. Specifically, we discuss basic game theory models of contests that are found in online services. We explain the conditions under which to optimally allocate prizes to maximize a given objective, such as the total effort or the maximum individual effort, in a strategic equilibrium. We will focus on games in which players make simultaneous effort investments; the games that involve some aspect of sequential play are only briefly discussed.

We consider both games that model a single contest (see Figure 1) and games that model a system of two or more simultaneous contests (Figure 2). Simultaneous contests are common in the context of online crowdsourcing platforms. We explain basic principles of popular skill rating systems and point out some new results in this area. We conclude with an outlook on future research directions.

This article complements existing surveys on the game-theoretic aspects in contest theory, for example, Corchon, 9 Konrad, 25 and Nitzan. 32 We provide an overview of some of the topics covered in the book by Vojnović, 42 where the reader may find a more extensive coverage of references.

Strategic Game Models of Contests The standard game theory framework for studying contests is based on the assumption that agents are rational and strategic players who invest effort with a selfish goal to maximize their individual payoffs. The payoff of a player combines the utility of winning a prize and the cost of production. Specifically, we consider a normal-form game that models a contest, defined by: