A game with incomplete information allows us to model uncertainty about skills of competitors in a contest; in the context of online services, such an uncertainty may arise because it may not be a priori known who is going to participate in a contest.

The strategic effort investment by a player can be according to a pure strategy, specifying a value of the effort investment, or according to a mixed strategy, specifying a probability distribution over pure strategies. An investment of efforts by players is a pure-strategy Nash equilibrium if no player can increase his or her payoff by a unilateral deviation. Similarly, a set of mixed strategies is a mixed-strategy Nash equilibrium if no player can increase his or her expected payoff by a unilateral deviation. A Bayes-Nash equilibrium is a mapping of an individual’s skill to a value of effort such that no player can increase his or her expected payoff by a unilateral deviation.

The utility of production is typically studied with respect to the following two metrics: the total effort and the maximum individual effort. The total effort has been studied extensively because it corresponds to the revenue accrued in an all-pay auction, and the total outlay accrued in a rent-seeking contest. 26, 40 The maximum individual effort has been studied motivated by applications in contests, such as in crowdsourcing services, where a contest owner makes use only of the best submitted solution. The utility of production has also been studied from a societal perspective, defined by a social welfare function, which is commonly defined as the sum of payoffs of all the parties involved (players and the contest owner). For example, when players incur unit marginal production costs and the payoff to the contest owner is the total effort invested by the players, social welfare corresponds to the total valuation of prizes by those who win them. Social welfare in an equilibrium can be smaller than optimal value; in some instances, optimum social welfare is achieved only if a given prize budget is fully assigned to highest-skill players, while in equilibrium a lower-skill player can have a strictly positive winning probability.

Single contest. We now consider a
normal-form game that models a sin-
gle contest among two or more players,
for different prize allocation mecha-
nisms and production cost functions.
A model of a single contest allows us to
study situations in which players have
no outside options such as investing
effort in an alternative contest; we will
later discuss games that model simul-
taneous contests, which provide play-
ers with such outside options.

Standard all-pay contest. A classic game that models a contest, we refer to as the standard all-pay contest, assumes a prize allocation mechanism that allocates entire prize budget to a highest-effort player with random tie break, and unit marginal production costs. This game corresponds to the well-known game that models an all-pay auction, studied in auction theory. The given prize allocation mechanism is commonly referred to as perfect discrimination, because it assumes perfect identification of a highest-effort player, achieved by some flawless mechanism for comparison of individual efforts.

We first discuss Nash equilibrium outcomes in the game with complete information that models the standard all-pay contest. This game does not have a pure-strategy Nash equilibrium. It can be easily verified that for any given effort investments, there is always a player who has a beneficial unilateral deviation. On the other hand, the game always has one or more mixed-strategy Nash equilibria, which were first fully characterized by Baye, Kovenock, and de Vries. 2

The game has a unique mixed-strategy Nash equilibrium only in some special cases, such as in a two-player contest, or in a contest with three or more players but where two players have individual skills larger than that of any other player. In general, the game has a continuum of mixed-strategy Nash equilibria. This may be considered a drawback because it implies a lack of predictive power. The mixed-strategy Nash equilibria are payoff equivalent: whenever a game has two or more mixed-strategy Nash equilibria, the expected payoffs in these equilibria are equivalent. In general, the equilibrium outcomes are not equivalent with respect to either the expected total effort or the expected maximum individual effort. It is noteworthy that

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.