tional allocation. 14 For the game that
models the contest with proportional
allocation, the total effort in the pure-strategy Nash equilibrium cannot be
increased by excluding some of the
players from competition.
Simultaneous contests. In the context of online services, a contest is
often run simultaneously with other
contests. For example, in competition-based crowdsourcing services, there
are typically many open contests at any
given time. Similarly, in online labor
marketplaces, there are usually many
open jobs at any given time. Multiple
open contests provide players with
alternative options to invest efforts,
which can have a significant effect on
the effort invested in any given contest.
A player can invest effort only in a limited number of contests over a period
of time, or he or she has a limited effort budget to invest over available contests. A worker may only be able to produce a high-quality work by focusing to
a small number of projects at any given
time, or he or she may only be able to
devote a limited number of work hours
per week. Game theory provides us
with a framework to study the relation
between the values of prizes offered by
different contests and the effort investments across different contests in a
strategic equilibrium.
We consider games that model si-
multaneous standard all-pay contests
that offer prizes of arbitrary values.
Such games have been studied for dif-
ferent types of production costs. We
first consider the case where produc-
tion costs are such it is feasible for
each player to participate in at most
one contest, in which he or she incurs
a unit marginal production cost. In
such a game, strategic decision mak-
ing of a player consists of two compo-
nents: choosing in which contest to
invest effort, and deciding how much
effort to invest in the chosen contest.
This strategic decision making is in-
formed by the available information,
which consists of the values of prizes
offered by different contests and the
prior information about the skills of
players. We consider the game with
incomplete information, where the
skill parameters of players are inde-
pendent and identically distributed
according to a prior distribution.
This game has a symmetric Bayes-
the prize allocated to a highest-effort
player. For more details about smooth
allocations, see, for example, Corchon
and Dahm10 and Vojnović. 42
One may ask how do equilibrium
outcomes in the game that models
the standard all-pay contest compare
with those in the game with a smooth
prize allocation, say, according to proportional allocation. A first notable
difference is that unlike the game that
models the standard all-pay contest,
the game with proportional allocation
has a pure-strategy Nash equilibrium,
which is unique.
The total effort in any pure-strategy
Nash equilibrium is guaranteed to be
at least v2/2. The total effort increases
in the highest-skill parameter v1 and it
can be larger than v2. This is in contrast
to the game that models the standard
all-pay contest, where the total effort in
any mixed-strategy Nash equilibrium
is at most v2. One may ask whether
there exists a smooth allocation of
prizes that guarantees the total effort
to be within a constant factor of v1 in
any pure-strategy Nash equilibrium.
The answer is negative. 41 This gives us
a useful insight that randomized prize
allocations can achieve a larger total effort, but there are fundamental limits
that cannot be surpassed.
The maximum individual effort can
be an arbitrarily small fraction of the
total effort; for example, this is so for
the simple game instance with equally
skilled players by taking the number
of players to be sufficiently large. This
is in contrast to the game that models
the standard all-pay contest where we
noted that in any mixed-strategy Nash
equilibrium, the expected maximum
individual effort is at least 1/2 of the expected total effort.
The social welfare in any pure-strategy
Nash equilibrium of the game with
proportional allocation is always at
least 3/4 of the optimum value, a result
by Johari and Tsitsiklis. 21 It has been
shown the game with proportional allocation is a smooth game (for example, see Roughgarden34), which implies
that the expected social welfare is at
least 1/2 of the optimum value in any
mixed-strategy Nash equilibrium.
Unlike the game that models the
standard all-pay contest, the exclusion
principle does not hold for the game
that models the contest with propor-
An interesting
question to ask is
how should a prize
budget be allocated
to maximize a
given objective
in equilibrium,
without making
a commitment
to allocate the
entire prize budget
to players, no
matter what effort
investments they
make.