Elimination contests with collusive team players

• Published date: 01 February 2023
• Author: Bo Chen,Shanlin Jin
• Date: 01 February 2023
• DOI: http://doi.org/10.1111/jpet.12600

Received: 7 March 2022

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Accepted: 21 May 2022

DOI: 10.1111/jpet.12600

ORIGINAL ARTICLE

Elimination contests with collusive team

players

Bo Chen

1,2

|Shanlin Jin

1

Department of Economics, Southern

Methodist University, Dallas, Texas, USA

2

Institute for Advanced Studies in

Finance and Economics, Hubei

University of Economics, Wuhan, China

Correspondence

Bo Chen, Department of Economics,

Southern Methodist University, Dallas,

TX, USA.

Email: bochen@smu.edu

Funding information

National Natural Science Foundation of

China, Grant/Award Number: 71973040

Abstract

We consider a standard two‐stage elimination (Tullock)

contest where multiple (team) players can perfectly

and publicly collude with each other throughout. We

analyze and compare equilibrium outcomes under

various seedings where the collusive players meet or

are separated in the group stage. We identify the impact

of collusion on the contest organizer and non‐collusive

players, as well as the organizer's optimal seeding.

We find that collusion, while always undermining

fairness of the competition, can hurt or benefit the

organizer, depending on the discriminatory powers of

the two stages. We also discuss issues such as sequential

group‐stage competitions, comparison between the

elimination contest and the corresponding one‐shot

contest, secret collusion, and large discriminatory

powers.

1|INTRODUCTION

Collusion in sports, or match fixing to restrict competition, has been a long‐standing and

widespread problem in professional sports. Indeed, match fixing events, which undermine the

integrity and fairness of sports, have occurred in various professional sports in the past.

1

A

famous incident, called the “disgrace of Gijón”in the 1982 FIFA World Cup, occurred in a

group‐stage football match in Gijón, Spain, where West Germany and Austria were accused of

fixing the match so that both would progress to and face more ideal opponents in the next

J Public Econ Theory. 2023;25:61–89. wileyonlinelibrary.com/journal/jpet © 2022 Wiley Periodicals LLC.

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1

A Wikipedia page https://en.wikipedia.org/wiki/List_of_match-fixing_incidents lists numerous match‐fixing inci-

dents, ranging (alphabetically) from American football to volleyball.

round, at the expense of Algeria.

2,3

Electronic sports (eSports), which are organized multi‐

player video game competitions, offer another setting where match‐fixing problems are

“incredibly widespread.”Unlike traditional sports, eSports and its research (on proper training

of eSports athletes and composition of teams) are still at an infant stage and professional

eSports players are much less known or recognized, according to a 2018 Guardian article.

4

One

particular phenomenon of eSports though, according to the Guardian article, is that its hyper‐

digital nature makes the athletes more integrated and enables them to easily communicate and

coordinate their actions.

In this paper, we analyze collusion among players using a standard elimination contest with

multiple stages. Elimination contests, while perhaps arising more prominently in traditional

sports and eSports contests, have also been widely used in other real‐life competitions such as

internal job promotions in organizations, political competitions where candidates compete and

exert efforts and resources in multiple stages, science or research competitions with a

preliminary stage and a final stage, as illustrated in Moldovanu and Sela (2006).

To be concrete, we consider a stylized four‐player elimination contest, where the (ex ante

identical) players are arranged into pairwise competitions in a group stage, and the two

resulting group‐stage winners then compete to win a prize in a final stage. The two stages are

modeled as Tullock contests. While such two‐stage elimination contests are familiar in the

contest literature, our setting differs from previous research in two respects. First, we allow for

the discriminatory powers of the two stages to be different. Such a flexible feature captures real‐

life elimination contests where competition can vary from one round to another, such as the

format of the competition, the winning criterion of each round, the judging panel of each

round, etc.

5

Second, there is explicit collusion among a set of players in our contest. Formally, it

is publicly known that two players out of the four, called collusive agents, are team players who

can coordinate their play and aim to maximize their joint payoff throughout the contest.

The existence of public collusion introduces (hidden) heterogeneity into our otherwise

symmetric setting and also prompts the remaining players to strategically respond to it.

Consequently, how to seed the four players in the group stage has direct implications on their

subsequent interactions. The majority of our analysis focuses on investigating equilibrium

outcomes and the impact of collusion under two group‐stage seedings, one where the collusive

agents meet in the group stage, and the other where the collusive agents are separated in the

group stage. More explicitly, our analysis answers the following questions: If the seeding of the

group stage can be arranged, should the organizer separate the collusive agents or let them

compete against each other in the group stage? How does each arrangement affect the

2

After 1982, FIFA changed the World Cup group stage format so that all the teams in a group would play final matches

simultaneously. We discuss the impact of making the group stage competitions sequential rather than simultaneous in

Section 4.1.

3

Match fixing incidents have also occurred in badminton competitions in the Summer Olympics, where players from

the same country colluded in semi‐finals to increase the probability of winning the gold medal in finals. See https://en.

wikipedia.org/wiki/List_of_match-fixing_incidents#Badminton.

4

See https://www.theguardian.com/games/2018/jul/31/its-incredibly-widespread-why-esports-has-a-match-fixing-

problem.

5

For example, the format of the postseason of a sports league (such as NBA, NFL, NCAA) is typically different from its

previous regular season. In addition, popular TV shows such as American Idol and Golden Balls also exhibit different

rules or formats in different rounds. We emphasize, however, that for our analysis to be applicable, such changes in

competition or discriminatory powers should affect symmetric players symmetrically, rather than reflect bias against

some player, so as to be consistent with the interpretation of discriminatory power in the literature.

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CHEN AND JIN