Sequential round-robin tournaments with multiple prizes

Abstract

We examine the fairness and intensity of sequential round-robin tournaments with multiple prizes. With three symmetric players and two prizes, the tournament is completely fair if and only if the second prize is valued half of the first prize, regardless of whether matches are organized as Tullock contests or as allpay auctions. For second prizes different from half of the first prize, three-player tournaments with matches organized as Tullock contests are usually fairer than tournaments with matches organized as all-pay auctions. However, unless the second prize is very small, they are less intense in the sense that players exert less ex-ante expected aggregate effort per unit of prize money. Moreover, we specify how the relative size of the second prize influences the extent and the direction of discrimination as well as the intensity of three-player tournaments. Finally, we show that there is no prize structure for which sequential round-robin tournaments with four symmetric players are completely fair in general.

Optimal and Fair Prizing in Sequential Round-Robin Tournaments: Experimental Evidence

“…This result is surprising because previous experimental studies of contest environments, in which multiple prizes are predicted to elicit more aggregate effort than a single prize, mostly support these predictions (see, e.g., Lim et al, 2009;Müller and Schotter, 2010;Freeman and Gelber, 2010). In line with the theory, aggregate effort is lower in the 100%-tournament than in the 50%-tournament, which can be explained by lean-back effects (Laica et al, 2021): if the second prize equals the first prize, a player who has won her first match will lose much of her incentives to provide additional effort in her second match.…”

“…Krumer et al (2020) illustrate that in round-robin tournaments with four players, matches organized as all-pay auctions, and two identical prizes a player may even have adverse ex-interim incentives in the sense that he may prefer losing over winning some match depending on the course of the tournament. Laica et al (2021) extend the analysis to sequential round-robin tournaments with an arbitrary number of heterogeneous players, matches organized as general Tullock contests, and multiple arbitrary rank-dependent prizes. They show that a tournament with three players is fair if and only if the second prize equals half of the first prize; 4 this prize structure also maximizes expected aggregate effort if matches are organized as all-pay auctions.…”

“…By contrast, with more than three players, no prize structure exists for which a tournament with a fully sequential exogenous match schedule is fair. 5 Our experimental design allows us to test the main theoretical predictions by Laica et al (2021) for roundrobin tournaments with three players.…”

“…a Nash equilibrium (in mixed strategies) exists and has the following properties (Krumer et al, , 2019Laica et al, 2021): 9 For i, j ∈ {1, 2, 3} with i = j and w k i − k i ≤ w k j − k j , the expected equilibrium efforts are…”

“…The reason is a discouragement effect of trailing players that implies a strategic momentum (see, e.g., Mago et al, 2013) and has been identified in many forms of dynamic contests (Konrad, 2009, Chapter 8). Allowing for multiple prizes, Laica et al (2021) show that no rank-dependent prize structure exists for which sequential round-robin tournaments with more than three players are fair, and round-robin tournaments with three players are fair if and only if the second prize equals half of the first prize. Only in this case, the participant's intertemporal effort decisions are not distorted either: in each single match, they only depend on the players' characteristics but not on the position of the match in the schedule of the tournament.…”