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.