Rank-order allocation of prizes. We
now consider a more general situation
where a prize budget can be arbitrarily split among two or more position
prizes, which are assigned to players
in decreasing order of effort, subject to
the constraint that any position prize
is at least as large as any lower position prize (see Figure 3). For example,
a prize budget may be split between
two position prizes such that 2/3 of the
prize budget is allocated to first place
prize and the remaining part is allocated to second place prize; a common
way of splitting a prize budget in TopCoder contests.
We consider the question of how
should a prize budget be split among
position prizes to maximize a given
objective in equilibrium. Clearly,
the answer depends on the choice of
the objective, equilibrium concept,
heterogeneity of skills, and production costs. Suppose the objective is
to maximize the expected total effort
in equilibrium of the game with incomplete information, where players
have identical prior distributions of
skills and unit marginal production
costs. Under these assumptions, it
is optimal to allocate the entire prize
budget to first place prize, which was
shown by Moldovanu and Sela. 28 Under the same assumptions, allocating
the entire prize budget to the first-place prize is also optimal for the
objective of maximizing the expected
maximum individual effort, which
was shown by Chawla, Hartline, and
Sivan. 7 These results hold even more
generally for any production cost
function with decreasing marginal
costs. In contrast, for production cost
functions with increasing marginal
costs, it may be optimal to split a prize
budget among two or more position
prizes. The optimality of allocating
entire prize budget to first place prize
holds also for the game with complete information, under the assumption that players have identical skills
and decreasing marginal production
costs, as shown by Glazer and Hassin17
and Ghosh and McAfee. 16
The assumption that in the game
with incomplete information the skills
of players have identical prior distri-
butions is critical for the optimality of
allocating entire prize budget to first
place prize. Similarly, the assumption
there always exists a mixed-strategy
Nash equilibrium in which all but two
highest-skill players invest zero effort.
The expected total effort in this equi-
librium is at least as large as in any
other equilibrium.
We now discuss some properties
that hold in any mixed-strategy Nash
equilibrium. Without loss of generality, assume that players’ identities are
in decreasing order of their skill parameters. The expected total effort is
of value between v2/2 and v2. Interestingly, the expected maximum individual effort is always at least half of the
expected total effort. This provides a
theoretical support for the efficiency
of competition-based crowdsourcing
services in which a contest owner solicits solutions from multiple workers, but makes use only of the best
submitted solution. Intuitively, one
would expect that such a production
system is bound to be highly inefficient because much of the invested
work ends up being wasted. However,
by this result, inefficiency can only be
to a limited extent in any mixed-strategy Nash equilibrium. With regard to
social welfare, there can be some efficiency loss in equilibrium, because
a player whose skill is not the highest
may have a strictly positive winning
probability. However, this can only be
up to a limited extent in any mixed-strategy Nash equilibrium: the expected social welfare is always at least 4/5
of the optimum social welfare.
Another noteworthy property is the
so-called exclusion principle, which
refers to the existence of game instances for which the expected total
effort in equilibrium can be increased
by excluding some players from the
competition. In particular, for some
game instances, it can be beneficial
to exclude the highest-skill player.
Intuitively, such exclusion may result
in a more intense competition among
players with more balanced skills,
and, as a result, yield a higher expected total effort.
We now move on to discuss the
game with incomplete information
that models the standard all-pay contest. We restrict our discussion to prior
distributions according to which skills
of players are independent and identically distributed random variables. The
game has a unique symmetric Bayes-Nash equilibrium, in which players
play identical strategies. The expected
total effort in this equilibrium is equal
to the expected value of the second-highest skill of a player. Interestingly,
the expected maximum individual effort is at least half of the expected total effort in any symmetric Bayes-Nash
equilibrium, which was established by
Chawla, Hartline, and Sivan. 7 This is
exactly the same relation we previously
noted to hold between the expected total effort and the expected maximum
individual effort in any mixed-strategy
Nash equilibrium of the game with
complete information.
Figure 3. Rank order allocation of prizes: Allocation of fixed shares w1 ≥ w2 ≥ … ≥ wn ≥0
of a prize budget in decreasing order of effort.
2-nd prize P r i z
e
Prize allocation: 1-st prize
Effo
r
t
Players
prize budget
( 1) ( 2) (n)
n-th prize