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