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