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