Nash equilibrium, which admits an explicit characterization, established in DiPalantino and Vojnović. 11 In this equilibrium, there is a segregation of players in different skill levels, such that the players of the same skill level choose contests according to identical mixed strategies. A player of a higher skill level chooses a contest to participate from a smaller set of contests that offer highest prizes. A higher expected participation is attracted by contests that offer high prizes, according to a relation that exhibits diminishing returns with respect to the values of the prizes.

Another type of production costs is when each player is endowed with an effort budget that he or she can split arbitrarily over available contests. This game is closely related to so-called Colonel Blotto game: there are two colonels and two or more battlefields; each colonel is endowed with a number of troops that are simultaneously deployed over the battlefields; a battlefield is won by the colonel who places a larger number of troops on this battlefield, and the game is won by the colonel who wins more battlefields. A continuous Colonel Blotto game assumes that each colonel is endowed with an infinitely divisible amount of army force.

The game with players endowed with effort budgets has a rich set of equilibrium properties. There are game instances with a continuum of mixed-strategy Nash equilibria. For example, this is the case for the game with two players that have non-identical effort budgets and two or more standard all-pay contests that offer identical prizes. When players have identical effort budgets, the game has both pure and mixed-strategy Nash equilibria in which each player invests all his or her effort in one contest, provided that the number of players is sufficiently large. In the limit of many players, the equilibrium participation of players across different contests is proportional to the values of prizes.

Games that model simultaneous contests with players endowed with effort budgets have also been studied for other prize allocation mechanisms, including proportional allocation and equal-share allocation.

The game that models simultaneous
contests with proportional alloca-
tion and players endowed with effort
budgets is not guaranteed to have a
pure-strategy Nash equilibrium. A suf-
ficient condition for the existence of a
pure-strategy Nash equilibrium is that
each contest has at least two players
with strictly positive skill parameters.
The social efficiency in a pure-strategy
Nash equilibrium can be arbitrarily
low in a worst case.

Sharing of the utility of production. There have been various studies of production systems where agents invest effort in one or more activities, which results in a utility of production that is shared among contributors according to a utility sharing mechanism. Some online services rely on user contributions and award credits to incentiv-ize contributions. For example, some online services rely on user-generated content, such as questions and answers in online Q&A services, and award credits in terms of attention or reputation points, which are commensurate to user contributions. Sharing the utility of production has been also studied in the context of cognitive labor and allocation of scientific credit, for example, Kitcher23 and Kleinberg and Oren. 24

A central question here is about the social efficiency of production in strategic equilibrium outcomes. Several factors can contribute to social inefficiency of production, including the choice of the utility sharing mechanism, the nature of the utility of production functions, and the nature of production cost functions. Special attention has been paid to local utility sharing mechanisms, which specify the shares of the utility of production associated with a project exclusively based on the effort investments in this project, and not on the effort investments in other projects. It is of interest to understand social efficiency of simple local utility sharing mechanisms, for example, allocating a priori fixed shares of the utility of production in decreasing order of individual contributions or allocating in proportion to individual contributions.

The nature of the utility of produc-
tion is a critical factor for the social
efficiency of equilibrium outcomes. If
the utility of production is allowed to
be according to a non-monotonic func-
tion of effort investments, then there
are game instances for which the utility
of production in a pure-strategy Nash
equilibrium is an arbitrarily small
fraction of the optimum; for example,
this can be for a single project game
with proportional allocation. This is
an instance of a general phenomenon
known as the tragedy of the commons, 19
referring to an inefficient use of con-
gestible resources that arises from
non-cooperative behavior of selfish
agents. The nature of the production
cost functions is also a critical factor.
If, in a single project game with pro-
portional allocation, the utility of pro-
duction is a monotone function, but
players incur unit marginal production
costs, then a similar inefficiency of pro-
duction can arise.

Are there conditions for the games under consideration under which equilibrium is guaranteed to exist and all equilibria are approximately socially efficient? Here we may settle for the utility of production to be at least a constant factor of the optimum value. Such conditions have been identified by Vetta41 for the class of games referred to as monotone valid utility games. A game is said to be a monotone valid utility game if the players’ payoffs are according to utility functions whose sum is less than or equal to the value of a social utility function, and the following two conditions hold. The game is required to satisfy a monotonicity condition, which restricts to social utility functions whose value cannot increase by some player opting out from participation. The game is also required to satisfy a marginal contribution condition, which restricts each player’s utility to be at least as large as his or her marginal contribution to the social utility. In the context of games that model simultaneous projects, whether or not the marginal contribution condition holds depends on the nature of the utility of production functions and the utility sharing mechanism. For example, the marginal contribution condition holds if the project utility functions are increasing functions with diminishing returns in the total effort invested in a project, and the utility sharing is according to proportional allocation. For monotone valid utility games, the utility of production in any pure-strategy Nash