Note“Small, yet Beautiful”: Reconsidering the optimal design of multi-winner contests☆
Introduction
Contests, in which players exert costly and irreversible efforts to win a prize, are ubiquitous in day-to-day life. In cases such as war, terrorism or territorial conflicts, contests are not designed by an organizer. However, there are very many situations including sports, patent race, promotion tournament, crowdsourcing, legal battle etc. in which an organizer organizes the contest, and contest design issues become highly important. The topic of optimal contest design, hence, has been an active area of research. In the literature one of the most frequently attempted questions is how to maximize the total effort exerted in a contest. For a contest with noisy outcome it is a further important question whether arranging a grand contest elicits a higher equilibrium effort than arranging several sub-contests.
For a single-winner setting, Moldovanu and Sela, 2001, Moldovanu and Sela, 2006 show that under certain conditions a grand contest indeed elicits higher effort. Adding an important contribution to this area, Fu and Lu (2009) characterize the optimal structure for multi-winner contests.1 They employ a nested winner-selection procedure as in Clark and Riis (1996) and show that a grand contest elicits greater equilibrium efforts than what a collection of mutually exclusive and exhaustive sub-contests does. This is an important finding since this extends the single-winner contest results of Moldovanu and Sela, 2001, Moldovanu and Sela, 2006 into multi-winner settings, and provides with clear policy design prescriptions.
In the specific mechanism employed above, the winners are selected sequentially. Players simultaneously exert their effort, and K winners are selected by K consecutive draws. Once a winner is selected through a Tullock (1980) contest success function, he/she is immediately removed from the pool of candidates up for the next draw. This procedure is repeated until all the prizes are exhausted.
In the field, however, the winner-selection procedure in a multi-winner contest is not always the one suggested above. Clark and Riis (1996) mention that when “the imperfectly discriminating rent-seeking contest [...] ha(s) several winners, there is no unique method for selecting those winners”. Indeed, the very first winner-selection mechanism suggested in the multi-winner contest literature is by Berry (1993), who considers a one-shot winner-selection mechanism. Under this, the players exert effort and the set of winners are taken out simultaneously. The probability of a player to win one of K prizes is the sum of efforts exerted by any combination of a K-player group that includes that specific player, divided by the sum of efforts exerted by any combination of a K-player group. There are different instances in which either a simultaneous (Berry, 1993) or a sequential (Clark and Riis, 1996) winner-selection mechanism is employed in the field.
There are both pros and cons of employing the simultaneous mechanism. Clark and Riis (1996) show that with this mechanism the very first prize is allocated according to the effort outlays whereas all the other prizes are implicitly allocated randomly – allowing for an incentive to free-ride. Chowdhury and Kim (2014), on the other hand, find an equivalence of the simultaneous mechanism to a mechanism in which the losers are sequentially taken out – essentially providing a microfoundation for the contest success function arising out of the simultaneous mechanism.2 Since the loser-elimination mechanism is well implemented in real life, this helps one to reformulate those real life situations as well. Hence, it is important to understand whether the answer to the original question (of comparing grand contest with sub-contests) depends on the particular winner-selection mechanism implemented.
In this study we reconsider such comparison of a grand contest with a collection of mutually exclusive and exhaustive sub-contests from a design point of view. We employ the simultaneous winner-selection mechanism (Berry, 1993) and find that the result of Fu and Lu (2009) gets reversed, i.e., a collection of sub-contests elicit a higher level of equilibrium effort than what a grand multi-winner contest does. In such a situation we characterize the optimal allocation of players and prizes for the case with identical prizes. We further show that with the sequential loser-elimination mechanism (Chowdhury and Kim, 2014) the Fu and Lu (2009) result is again reversed. We then characterize the optimal contest structure when the number of sub-contests is limited to two and no prize can be wasted.
Section snippets
Contest design under simultaneous winner-selection
Consider N identical players competing for K indivisible prizes with . The common values of the prizes are , and a player can win at most one prize. Without any loss of generality, assume that . The contest designer can run the grand contest by putting all the players and the prizes together or run M small contests by dividing the contestants and the prizes into mutually exclusive groups. Let be the number of contestants in group g and be
Further analysis: limit in the number of contests
Thus far, we have assumed that there is no additional cost for the designer to organize more contests, and showed that dividing a grand contest into smaller ones can increase the total effort. Let us now suppose that there exist operational costs for running these contests, which increases in the number of groups. Because of the costs, one cannot run more than two contests, i.e., .6
Discussion
In this study we reconsider the design of multi-winner contests. Fu and Lu (2009) employ a sequential winner-selection mechanism (Clark and Riis, 1996) and find that a grand contest always elicits higher equilibrium effort than a collection of sub-contests. We show that the result is completely reversed if a simultaneous winner-selection or a sequential loser-elimination mechanism is implemented. This result is obtained because the simultaneous winner-selection mechanism (Berry, 1993) and the
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We appreciate the useful comments of Qiang Fu, Jingfeng Lu and the seminar participants at the University of East Anglia. Any remaining errors are our own.