Are sequential round‐robin tournaments discriminatory?

• Date: 01 February 2019
• Published date: 01 February 2019
• Author: Marco Sahm
• DOI: http://doi.org/10.1111/jpet.12308

S

44 © 2018 Wiley Periodicals, Inc. wileyonlinelibrary.com/journal/jpet Journal of Public Economic Theory. 2019;21:44–61.

Received:13 September 2017 Accepted:13 April 2018

DOI:10.1111/jpet.12308

ARTICLE

Are sequential round-robin tournaments

discriminatory?

Marco Sahm1,2

1Universityof Bamberg, Department of Eco-

nomics,Feldkirchenstraße 21, D–96052 Bam-

berg,Germany

2CESifo,Poschingerstraße 5, D–81679 Munich,

Germany

Correspondence

MarcoSahm, Department of Economics,

Universityof Bamberg, Feldkirchen-

straße21, D–96052 Bamberg, Germany.

Email:Marco.Sahm@uni-bamberg.de

I examine sequential round-robin tournaments with three and four

symmetric players. Each player is matched once with each other

playerand each match is organized as a Tullock contest. A single prize

is allocated to the player with the most matches won. I show that,

depending on the position of their matches in the sequence of the

tournament, the players'ex ante winning probabilities and expected

payoffs differ. Compared to tournaments with matches organized

as all-pay auctions, however, discrimination is weaker and partially

reversed. Moreover, aggregate effort in tournaments with suitable

Tullockcontests is larger than in tournaments with all-pay auctions.

The fairness of round-robin tournaments may be improved by an

endogenous sequence of matches or the requirement that playersfix

their effort ex ante.

1INTRODUCTION

Around-robin tournament, also referred to as an all-play-all tournament, is a form of competition in which each par-

ticipant is matched with each other participant in a pairwise contest, the participants are ranked according to the

number of matches won, and awarded prizes according to this ranking. Round-robin tournaments are widely used

as an organizational form of competition, particularly in sports. Most sports leagues around the world are—at least

partially—organized as some version of a round-robin tournament. Prominent examples of double round-robin tour-

naments include the major European football leagues in England, Spain, Germany, and Italy.Moreover, smaller pure

round-robin tournaments are embedded in early stages of many multistage sports tournaments; e.g., the first round

(group stage) of the Fédération Internationale de Football Association (FIFA) World Cup (since 1950) and Union of

European Football Associations European Championship (since 1980) with four teams per group, or the second round

of the FIFAWorld Cup (1982) with three teams per group.1In tournaments with three teams, the natural schedule of

the matches is sequential. And even in tournaments with four teams, the matches are mostly scheduled sequentially

due to technological constraints (better accessibility) or economic considerations (higher profitability).2

1InJanuary 2017, FIFA decided that, from 2026 on, it will extend its World Cup to 48 teams and organize the first round in form of round-robin tournaments

withthree teams per group.

2Withfour teams, as inthe examples mentioned above, often only the last two matches are played simultaneously in order to avoid certain teams havingthe

advantageof knowing the results of other teams.

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One reason for its high popularity may be the common wisdom according to which “a round-robin tournament is

the fairest way to determine the champion among a known and fixed number of participants. Each participant, player

or team, has equal chances against all other opposites” (Wikipedia3, 2016). This paper addresses the question whether

this common wisdom is true. Are round-robin tournaments really fair? Or is their sequential structure inherently dis-

criminatory?

Despite the practical and scientific relevanceof round-robin tournaments, the economic analysis of sequential con-

tests has mainly focused on other forms of dynamic battles like elimination tournaments or races.4An exception is

the article by Krumer, Megidish, and Sela (2017a) who investigatethe winning probabilities and expected payoffs in

sequential round-robin tournaments with three and four symmetric players and a single prize.5The different pair-

wise matches take place one after the other and the player with the most victories receives the prize. The authors

assume that each single match is organized as an all-pay auction (APA)and find substantial discrimination by the order

of matches in the subgame perfect equilibrium of such APA tournaments. For example,they show that in the three-

player tournament, the player who competes in the first and last match has a much higher winning probability and

expected payoff than the two other players.Indeed, Krumer and Lechner (2017) provide some empirical evidence for

such discrimination in round-robin sports tournaments.

Thesefindings are in stark contrast to the common wisdom of fairness.6The intuition is that the sequential structure

implies an asymmetry in intermediate scores that discourages trailing participants from providingequivalent effort as

they (anticipate having to) face some competitor with a head start who thus has a higher ex interim winning probabil-

ity.7In the model of Krumer et al. (2017a), this discouragementeffect is particularly strong because the contest suc-

cess function of an APA that determines the players'winning probabilities in each single match of the tournament is

fully discriminatory in the sense that the player with the highest effort wins for sure. This raises the question whether

the sequential round-robin tournament will be fairer if its matches are organized as contests with less discriminatory

contest success functions.

In this paper,I address the fairness issue under the assumption that the matches of the sequential round-robin tour-

nament are organized as Tullock contests. I find that such Tullocktournaments are discriminatory as well. Quantita-

tively,however, in Tullock tournaments with sufficiently low discriminatory powers (i.e., exponents),the extent of dis-

crimination, as measured by the relative standard deviations of the players'exante winning probabilities or expected

payoffs, is smaller than in APA tournaments. Intuitively, the discouragement of trailing participants will be mitigated

if the respective contests are less discriminatory because the winning probability and expected payoff of a player is

positive evenif he invests less than his opponent.

Moreover, the analysis shows that the discriminatory power of the contest success function impacts not only the

extentbut also the direction of discrimination. For example, in the three-player tournament with lottery contests (LCs),

the player who competes in the first and last match has the lowest winning probability and expected payoff—thedia-

metric opposite result to what Krumer et al. (2017a) find for tournaments with APAs.

Besides aiming at a fair tournament, the designer of a (sports) contest may be interested in (expected) aggregate

effort because more intense competition attracts more attention from viewers and sponsors. As is well known from

the literature on rent dissipation, a static winner-take-all contest between a finite number of symmetric players will

be more intense if it is organized as an APA than as a LC. However, the opposite is true if players are sufficiently

3Accessedat https://en.wikipedia.org/wiki/Round-robin_tournament on 16/12/2016

4Therelated literature is summarized, e.g., by Konrad (2009, chapter 8). More recent contributions include, i.e., Ghosh and Stong (2018) and Walker,Wooders,

andAmir (2011).

5Three recent working papers discuss the role of multiple prizes in round-robin tournaments (Dagaev & Zubanov,2017; Krumer, Megidish, & Sela, 2017b;

Laica,Lauber, & Sahm, 2017).

6Here, fairness is considered from a horizontal perspective:a tournament will be called (horizontally) fair,if ex ante symmetric players have equal ex ante

winning probabilities and expectedpayoffs. In reality, players are usually not symmetric. One might thus additionally require some kind of vertical fairness:a

playerwho is stronger (due to higher ability or higher valuation of the prize) should have a higher ex ante winning probability and expected payoff. The analysis

of Krumer,Megidish, and Sela (2017c) suggests that most results on discrimination in round-robin tournaments with symmetric playerswill also hold with

asymmetricplayers.

7Similardiscouragement effects occur in most forms of dynamic competition (Konrad, 2009, Chapter 8).