that in the game with complete information the skills of players are identical is critical for the optimality of
allocating entire prize budget to first
place prize. If the skills of players have
non-identical prior distributions, then
there exist game instances such that it
is profitable to split the prize budget
over two or more position prizes, which
is shown by the following example.
Three players, two prizes example.
Consider a game where a unit prize
budget is split between two position
prizes such that ½ ≤ α ≤ 1 is allocated
to the first place prize and the remaining part is allocated to the second place
prize. Assume there are three players: a
high-skill player with the skill parameter of value v> 1 and two low-skill players whose skill parameters are of value
1. Assume that each player incurs unit
marginal production cost. This game
has a mixed-strategy Nash equilibrium
such that the two low-skill players play
symmetric strategies. This equilibrium
is such that in the limit of asymptotically large skill of the high-skill player,
the mixed strategy of the high-skill
player converges to a uniform distribution on [ 1 – α,α], and that of low-skill
players converges to a uniform distribution on [0, 1 – α]. In this limit, the
expected effort of the high-skill player
is ½, and that of each low-skill player is
( 1 – α)/2. This adds up to the expected
total effort of value ³/2 – α. Therefore,
we observe the more balanced the split
of the prize budget between the two
position prizes, the larger the expected
total effort.
An interesting question to ask is
how should a prize budget be allo-
cated to maximize a given objective in
equilibrium, without making a com-
mitment to allocate the entire prize
budget to players, no matter what
effort investments they make. This
question has been resolved for the
game with incomplete information
and the objective of maximizing the
expected total effort by the celebrated
work of Myerson. 28 In particular, if
the skill parameters are independent
and identically distributed according
to a prior distribution that satisfies a
certain regularity condition, it is opti-
mal to award the entire prize budget
to a highest-effort player subject to
his or her effort being larger than or
equal to a minimum required effort,
and withhold the prize by the contest
owner, otherwise. Chawla, Hartline,
and Sivan7 have recently established
similar characterization of the opti-
mum prize allocation for the objec-
tive of maximizing the expected maxi-
mum individual effort.
Smooth allocation of prizes. Now
consider prize allocation mechanisms
that have a positive bias to awarding
players who invest high effort, but do
not guarantee that the prize is allo-
cated to a highest-effort player. Such
prize allocation mechanisms can arise
due to various factors. One factor is
the stochasticity of production, where
individual production outputs are ran-
dom variables, positively correlated
with invested efforts. Another factor is
allocation of prizes based on a ranking
of players derived from noisy observa-
tions of individual production outputs.
Such prize allocation mechanisms are
referred to be with imperfect discrimi-
nation. The stochasticity of production
may result in prize allocation accord-
ing to a smooth function of invested
efforts, for all vectors of efforts except
for some corner cases such as when all
players invest zero efforts.
An example of a smooth allocation
of prizes is proportional allocation
that splits a prize budget among play-
ers in proportion to invested efforts,
conditional on at least one player in-
vesting a strictly positive effort; oth-
erwise, the prize is evenly split among
players (Figure 4). A smooth prize al-
location may be enforced by the de-
sign of a resource allocation mecha-
nism. For example, proportional
allocation has been used for alloca-
tion of computing resources37 and
network bandwidth. 22 Such resources
typically consist of a large number of
small units and, thus, for any practi-
cal purposes, can be regarded as infi-
nitely divisible resources.
A more general class of smooth allocations is defined by allocating in
proportion to an increasing positive-valued function of invested effort, referred to as a general logit allocation.
A special case is allocation in proportion to a power function of invested
effort, with a positive exponent parameter r. This is commonly referred to
as Tullock allocation, which has been
studied extensively in the literature on
rent-seeking contests. 40 Proportional
allocation is a special case of a Tullock allocation for the value of parameter r equal to 1. The larger the value
of parameter r, the larger the share of
Figure 4. Proportional allocation.
Prize 1st 2nd 3rd nth
123 n
Effort
Rank