The Allocation of a Prize (R)

37 Pages Posted: 4 Apr 2012

See all articles by Pradeep K. Dubey

Pradeep K. Dubey

SUNY Stony Brook - Center for Game Theory in Economics

Siddhartha Sahi

Rutgers, The State University of New Jersey - New Brunswick/Piscataway

Date Written: April 3, 2012

Abstract

Consider agents who undertake costly effort to produce stochastic outputs observable by a principal. The principal can award a prize deterministically to the agent with the highest output, or to all of them with probabilities that are proportional to their outputs. We show that, if there is sufficient diversity in agents' skills relative to the noise on output, then the proportional prize will, in a precise sense, elicit more output on average, than the deterministic prize. Indeed, assuming agents know each others' skills (the complete information case), this result holds when any Nash equilibrium selection, under the proportional prize, is compared with any individually rational selection under the deterministic prize. When there is incomplete information, the result is still true but now we must restrict to Nash selections for both prizes.

We also compute the optimal scheme, from among a natural class of probabilistic schemes, for awarding the prize; namely that which elicits maximal effort from the agents for the least prize. In general the optimal scheme is a monotonic step function which lies "between" the proportional and deterministic schemes. When the competition is over small fractional increments, as happens in the presence of strong contestants whose base levels of production are high, the optimal scheme awards the prize according to the "log of the odds," with odds based upon the proportional prize.

JEL Classification: C70, C72, C79, D44, D63, D82

Suggested Citation

Dubey, Pradeep K. and Sahi, Siddhartha, The Allocation of a Prize (R) (April 3, 2012). Cowles Foundation Discussion Paper No. 1858, Available at SSRN: https://ssrn.com/abstract=2033815 or http://dx.doi.org/10.2139/ssrn.2033815

Pradeep K. Dubey (Contact Author)

SUNY Stony Brook - Center for Game Theory in Economics ( email )

Stony Brook, NY 11794
United States
631-632-7555 (Phone)
631-632-7516 (Fax)

Siddhartha Sahi

Rutgers, The State University of New Jersey - New Brunswick/Piscataway ( email )

94 Rockafeller Road
New Brunswick, NJ 08901
United States

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