# The Impact of Organizer Market Structure on Participant Entry Behavior in a Multi-Tournament Environment

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## Abstract

**:**

## 1. Introduction

## 2. Overview of Model

- Stage 1.
- prize levels for each event are set,9
- Stage 2.
- tournament participants/entrants decide which events to enter,
- Stage 3.
- competition takes place in each tournament (by way of participants/entrants exerting effort) and prizes are awarded.

## 3. Decisions of Tournament Participants

#### 3.1. Tournament Level Competition

#### 3.2. Entry Decision of Tournament Participants

## 4. Monopsonist Tournament Organizer

#### 4.1. Optimal Prizes to Realize Separating Composition

**Lemma 1.**

**Proof of Lemma 1.**

#### 4.2. Optimal Prizes to Realize Pooling Composition

**Lemma 2.**

**Proof of Lemma 2.**

#### 4.3. Separating Composition or Pooling Composition?

**Theorem 1.**

**Proof of Theorem 1.**

- $\delta $ were larger (so the difference in abilities between H and L would be greater, and agents would exert less effort in an event with a field of $(H,L)$);
- ${V}_{\left\{H,H\right\}}$ were larger, which would imply that ${V}_{\left\{H,H\right\}}-{V}_{\left\{H,L\right\}}$ would be larger (so that the marginal benefit to the organizer of having a second high ability agent in a particular tournament would be greater);
- ${V}_{\left\{L,L\right\}}$ were larger, which would imply that ${V}_{\left\{H,L\right\}}-{V}_{\left\{L,L\right\}}$ would be smaller (so that the marginal benefit to the organizer of having a first high ability agent in a particular tournament would be smaller);
- ${V}_{\left\{H,L\right\}}$ were smaller, which would imply that ${V}_{\left\{H,H\right\}}-{V}_{\left\{H,L\right\}}$ would be larger (so that the marginal benefit to the organizer of having a second high ability agent in a particular tournament would be greater) and ${V}_{\left\{H,L\right\}}-{V}_{\left\{L,L\right\}}$ would be smaller (so that the marginal benefit to the organizer of having a first high ability agent in a particular tournament would be smaller);
- r were larger (so that the organizer valued effort to a greater degree).

**Corollary 1.**

**Proof of Corollary 1.**

**Corollary 2.**

**Proof of Corollary 2.**

## 5. Competing Tournament Organizers

**Proposition 1.**

**Proof of Proposition 1.**

## 6. Impact of Organizer Market Structure

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1

## Appendix B. Numerical Results

(a). $\mathit{r}=1$, $\mathit{\delta}=0.60$, ${\mathit{V}}_{\mathit{LL}}=1$. | ||||||
---|---|---|---|---|---|---|

${\mathit{V}}_{\mathit{HH}}$ | ||||||

${\mathit{V}}_{\mathit{HL}}$ | 1.2 | 2.4 | 3.6 | 4.8 | 6.0 | 7.2 |

1.1 | (CS,0.9679) | (LP,1.4805) | (LP,1.8438) | (LP,2.1992) | (LP,2.5501) | (LP,2.8982) |

2.2 | (CS,0.9650) | (LP,1.6332) | (LP,2.1992) | (LP,2.5501) | (LP,2.8982) | |

3.3 | (CS,0.9638) | (CS,0.7495) | (LP,2.3026) | (LP,2.8982) | ||

4.4 | (CS,0.9631) | (CS,0.8119) | (LP,1.9178) | |||

5.5 | (CS,0.9626) | (CS,0.8458) | ||||

6.6 | (CS,0.9623) | |||||

(LP) | (1,1.0937] | (1,1.7688] | (1,2.4719] | (1,3.1836] | (1,3.8999] | (1,4.6193] |

(b).$\mathbf{r}=\mathbf{1}$, $\mathbf{\delta}=\mathbf{0}.\mathbf{75}$, ${\mathit{V}}_{\mathit{LL}}=\mathbf{1}$. | ||||||

${\mathit{V}}_{\mathit{HH}}$ | ||||||

${\mathit{V}}_{\mathit{HL}}$ | 1.2 | 2.4 | 3.6 | 4.8 | 6.0 | 7.2 |

1.1 | (CS,0.9756) | (LP,1.8443) | (LP,2.4968) | (LP,3.1365) | (LP,3.7691) | (LP,4.3972) |

2.2 | (CS,0.9739) | (CS,0.7988) | (LP,1.7255) | (LP,2.7922) | (LP,3.8589) | |

3.3 | (CS,0.9733) | (CS,0.8572) | (CS,0.7234) | (LP,1.5390) | ||

4.4 | (CS,0.9730) | (CS,0.8861) | (CS,0.7894) | |||

5.5 | (CS,0.9728) | (CS,0.9033) | ||||

6.6 | (CS,0.9726) | |||||

(LP) | (1,1.0791] | (1,1.5030] | (1,1.9543] | (1,2.4139] | (1,2.8780] | (1,3.3451] |

(c).$\mathbf{r}=\mathbf{1}$, $\mathbf{\delta}=\mathbf{0}.\mathbf{90}$, ${\mathit{V}}_{\mathit{LL}}=\mathbf{1}$. | ||||||

${\mathit{V}}_{\mathit{HH}}$ | ||||||

${\mathit{V}}_{\mathit{HL}}$ | 1.2 | 2.4 | 3.6 | 4.8 | 6.0 | 7.2 |

1.1 | (LP,1.1067) | (LP,2.0220) | (LP,2.8261) | (LP,3.6158) | (LP,4.3978) | (LP,5.1746) |

2.2 | (CS,0.9666) | (CS,0.8245) | (CS,0.6730) | (LP,2.1223) | (LP,3.1890) | |

3.3 | (CS,0.9697) | (CS,0.8757) | (CS,0.7782) | (CS,0.6747) | ||

4.4 | (CS,0.9713) | (CS,0.9012) | (CS,0.8293) | |||

5.5 | (CS,0.9723) | (CS,0.9165) | ||||

6.6 | (CS,0.9730) | |||||

(LP) | (1,1.1309] | (1,1.4678] | (1,1.8108] | (1,2.1555] | (1,2.5012] | (1,2.8475] |

(a). $\mathit{r}=1$, $\mathit{\delta}=0.60$, ${\mathit{V}}_{\mathit{LL}}=1$. | ||||||
---|---|---|---|---|---|---|

${\mathit{V}}_{\mathit{HH}}$ | ||||||

${\mathit{V}}_{\mathit{HL}}$ | 1.2 | 2.4 | 3.6 | 4.8 | 6.0 | 7.2 |

1.1 | (MP,CS,0.9645) | (MP,LP,1.0269) | (MP,LP,1.0799) | (MP,LP,1.1215) | (MP,LP,1.1546) | (MP,LP,1.1814) |

2.2 | (MS,CS,0.9736) | (MP,LP,1.1007) | (MP,LP,1.1215) | (MP,LP,1.1546) | (MP,LP,1.1814) | |

3.3 | (MS,CS,0.9819) | (MS,CS,1.0659) | (MP,LP,1.1753) | (MP,LP,1.1814) | ||

4.4 | (MS,CS,0.9880) | (MS,CS,1.0571) | (MS,LP,1.3488) | |||

5.5 | (MS,CS,0.9926) | (MS,CS,1.0519) | ||||

6.6 | (MS,CS,0.9963) | |||||

(MP,CS) | $[1.0938,1.1039]$ | $\left\{\varnothing \right\}$ | $\left\{\varnothing \right\}$ | $\left\{\varnothing \right\}$ | $\left\{\varnothing \right\}$ | $\left\{\varnothing \right\}$ |

(MS,LP) | $\left\{\varnothing \right\}$ | [1.7040,1.7688] | [2.3040,2.4719] | [2.9040,3.1836] | [3.5040,3.8999] | [4.1040,4.6193] |

(b).$\mathbf{r}=\mathbf{1}$, $\mathbf{\delta}=\mathbf{0}.\mathbf{75}$, ${\mathit{V}}_{\mathit{LL}}=\mathbf{1}$. | ||||||

${\mathit{V}}_{\mathit{HH}}$ | ||||||

${\mathit{V}}_{\mathit{HL}}$ | 1.2 | 2.4 | 3.6 | 4.8 | 6.0 | 7.2 |

1.1 | (MP,CS,0.9988) | (MP,LP,0.9930) | (MP,LP,1.0217) | (MP,LP,1.0451) | (MP,LP,1.0640) | (MP,LP,1.0794) |

2.2 | (MS,CS,0.9799) | (MP,CS,1.0185) | (MP,LP,1.1637) | (MP,LP,1.1335) | (MP,LP,1.1127) | |

3.3 | (MS,CS,0.9826) | (MS,CS,0.9772) | (MP,CS,1.0577) | (MP,LP,1.2942) | ||

4.4 | (MS,CS,0.9848) | (MS,CS,0.9833) | (MS,CS,0.9954) | |||

5.5 | (MS,CS,0.9866) | (MS,CS,0.9870) | ||||

6.6 | (MS,CS,0.9881) | |||||

(MP,CS) | [1.0792,1.1264] | [1.5031,1.7264] | [1.9544,2.3264] | [2.4140,2.9264] | [2.8781,3.5264] | [3.3452,4.1264] |

(MS,LP) | $\left\{\varnothing \right\}$ | $\left\{\varnothing \right\}$ | $\left\{\varnothing \right\}$ | $\left\{\varnothing \right\}$ | $\left\{\varnothing \right\}$ | $\left\{\varnothing \right\}$ |

(c).$\mathbf{r}=\mathbf{1}$, $\mathbf{\delta}=\mathbf{0}.\mathbf{90}$, ${\mathit{V}}_{\mathit{LL}}=\mathbf{1}$. | ||||||

${\mathit{V}}_{\mathit{HH}}$ | ||||||

${\mathit{V}}_{\mathit{HL}}$ | 1.2 | 2.4 | 3.6 | 4.8 | 6.0 | 7.2 |

1.1 | (MP,LP,0.9665) | (MP,LP,0.9791) | (MP,LP,0.9966) | (MP,LP,1.0120) | (MP,LP,1.0247) | (MP,LP,1.0351) |

2.2 | (MS,CS,0.9777) | (MP,CS,1.0304) | (MP,CS,1.2685) | (MP,LP,1.1869) | (MP,LP,1.1566) | |

3.3 | (MS,CS,0.9819) | (MS,CS,0.9654) | (MP,CS,1.0424) | (MP,CS,1.2103) | ||

4.4 | (MS,CS,0.9844) | (MS,CS,0.9731) | (MS,CS,0.9691) | |||

5.5 | (MS,CS,0.9861) | (MS,CS,0.9779) | ||||

6.6 | (MS,CS,0.9874) | |||||

(MP,CS) | [1.1310,1.1705] | [1.4679,1.7705] | [1.8109,2.3705] | [2.1556,2.9705] | [2.5013,3.5705] | [2.8476,4.1705] |

(MS,LP) | $\left\{\varnothing \right\}$ | $\left\{\varnothing \right\}$ | $\left\{\varnothing \right\}$ | $\left\{\varnothing \right\}$ | $\left\{\varnothing \right\}$ | $\left\{\varnothing \right\}$ |

## Notes

1 | The abbreviation PGA stands for Professional Golfers Association. See https://www.pgatour.com/media-guide/brief-tour-history.html (last accessed on 30 December 2022) for a brief history. |

2 | LIV is not an abbreviation, but rather Roman numerals for 54 (which is both the number of holes played in each LIV Golf tournament and the total score that a golfer would record for a round on an 18 hole, par 72 course by getting a birdie (one stroke better than par) on each hole). See https://www.livgolf.com/ (last accessed on 30 December 2022) for more information about LIV Golf. |

3 | See https://www.golfmonthly.com/news/how-much-are-liv-players-being-paid (last accessed on 30 December 2022). For some perspective on these amounts, the leading money winner on the PGA TOUR for the 2021-22 season, Scottie Scheffler, earned just over $14 million for the year: https://www.pgatour.com/stats/stat.109.y2022.html (last accessed on 30 December 2022). |

4 | In reality, the PGA TOUR has always faced some competition for players over its entire existence, most notably from what had been know as the European Tour (now the DP World Tour; https://www.europeantour.com/dpworld-tour/, last accessed on 30 December 2022). However, in practice, the PGA TOUR has always treated the European Tour as a “partner” as opposed to an “adversary,” in stark contrast to the way it has treated LIV Golf. For example, members of the PGA TOUR have always had to apply for a “conflicting event release” if they wanted to compete in an event on a non-PGA TOUR circuit. Such requests have always been granted, almost without exception, when PGA TOUR members have wanted to play in European Tour events. In contrast, before the first LIV Golf event was played, the PGA TOUR made it clear that they would take a hard stance and revoke the membership of any PGA TOUR player who chose to compete on the LIV Golf circuit. |

5 | The “Official World Golf Ranking” (https://www.owgr.com/, last accessed on 30 December 2022) is updated every week to provide an ordinal ranking of golfers based upon performance in sanctioned events over the most recent two years. In the 521 rankings released between 7 January 2001 and 26 December 2010: Tiger was ranked first 480 times (i.e., over 92% of the time); Phil was ranked second 265 times (i.e., over 50% of the time). In each of the 265 weeks that Phil was ranked second, Tiger was ranked first (thus, over this decade Tiger was ranked first and Phil was ranked second over half the time). |

6 | The modern day “major champsionships” consist of four tournaments per year: The Masters, PGA Championship, U.S. Open, and British Open. Dustin Johnson, Martin Kaymer, and Brooks Koepka have each won multiple majors. |

7 | Numerical results suggest that there is instead a first mover advantage when competition leads to a pooling composition. |

8 | The ability of agents is assumed to be common knowledge. |

9 | With a monopsonist organizer, a pair of prizes will be announced by the single organizer at this stage. With competing organizers, first “Organizer l” (i.e., the “leader”) announces a prize, after which “Organizer f” (i.e., the “follower”) announces a prize. |

10 | Many features of this model are similar to that analyzed by [6]. Their focus is on the entry decision of tournament participants of three different skill levels over two tournaments, with exogenously set prizes. The focus here is on the strategic, endogenous choice of tournament prizes. To conduct this analysis, we presently assume a less general contest success function, tournament participants of only two ability levels (not three ability levels), and a second place prize of zero (assumptions which allow the choice of the organizer to be examined with greater ease). |

11 | |

12 | Henceforth $\delta >\frac{1}{2}$ will specifically refer to the probability with which an agent of type H will be the winner in a tournament with a field of $(H,L)$. |

13 | As previously noted, [6] provides a detailed analysis of the specific factors influencing such a tournament entry decisions in a framework more general than the one considered here. The restrictions imposed here allow the tournament prizes to be endogenously determined while keeping the model tractable. |

14 | In order to identify a unique choice by each agent at each decision node, it is assumed that if an agent has the same expected payoff in each tournament he will enter Event 1. Thus, for the present decision, if $\frac{1}{4}{p}_{1}={\delta}^{2}{p}_{2}$ we assume ${H}_{ii}$ enters Event 1. |

15 | For this mixed strategy equilibrium, the two separating compositions, the pooling composition in which both enter Event 1, and the pooling composition in which both enter Event 2 would each arise with positive probability. |

16 | In Theorem 1, and hereafter, $\beta \left({V}_{\left\{H,H\right\}},{V}_{\left\{H,L\right\}},{V}_{\left\{L,L\right\}},r\right)$ is written as just $\beta $. |

17 | We assume that the organizer will implement the pooling composition if ${\gamma}_{MP}^{\ast}={\gamma}_{MS}^{\ast}$. |

18 | We assume that the marginal value of having an additional high ability agent in any field is always non-negative (i.e., $\left({V}_{\left\{H,L\right\}}-{V}_{\left\{L,L\right\}}\right)\ge 0$ and $\left({V}_{\left\{H,H\right\}}-{V}_{\left\{H,L\right\}}\right)\ge 0$). |

19 | Recall, $g\left(\delta \right)>0$ for all $\delta \in \left(\frac{1}{2},1\right)$. Thus, if $\beta <0$, then $g\left(\delta \right)\ge \beta $ for all values of r. Therefore, a change in the value of r can possibly alter the relation between $g\left(\delta \right)$ and $\beta $ only when $\beta >0$ (in which case $\beta $ is decreasing in r). As a result, the only possible impact of a change in r on the relation between $g\left(\delta \right)$ and $\beta $ is the following: if initially $g\left(\delta \right)<\beta $, then a larger value of r may lead to $g\left(\delta \right)\ge \beta $ instead of $g\left(\delta \right)<\beta $. |

20 | A similar technique has been used in the study of opinion dynamics (c.f. [18]). |

21 | Numerical results were obtained for many more parameter values than those reported in Appendix B. All of the insights discussed in this subsection and the following section hold true for these non-reported results as well. |

22 | The larger value states the largest ${V}_{HL}$ rounded to four decimal places for which $LP$ arises. For example, from the last column of Table (1a) we have that for $r=1$, $\delta =0.60$, ${V}_{LL}=1$, and ${V}_{HH}=7.2$: $LP$ arises for ${V}_{HL}\le 4.6193$, while $CS$ arises for ${V}_{HL}\ge 4.6194$. |

23 | Though not reported, results were obtained for different values of r, to see how the equilibrium depends upon this parameter. Generally, as r increases, the maximum ${V}_{HL}$ leading to $LP$ appears to be “u-shaped.” For example, with $\delta =0.75$, ${V}_{LL}=1$, and ${V}_{HH}=4.8$, the largest ${V}_{HL}$ leading to $LP$: decreases from $2.5183$ to $2.4139$ as r increases from $0.2$ to 1, but then increases from $2.4139$ to $2.6141$ as r increases from 1 to 5. |

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## Share and Cite

**MDPI and ACS Style**

Mathews, T.; Namoro, S.D.; Boudreau, J.W.
The Impact of Organizer Market Structure on Participant Entry Behavior in a Multi-Tournament Environment. *Games* **2023**, *14*, 4.
https://doi.org/10.3390/g14010004

**AMA Style**

Mathews T, Namoro SD, Boudreau JW.
The Impact of Organizer Market Structure on Participant Entry Behavior in a Multi-Tournament Environment. *Games*. 2023; 14(1):4.
https://doi.org/10.3390/g14010004

**Chicago/Turabian Style**

Mathews, Timothy, Soiliou Daw Namoro, and James W. Boudreau.
2023. "The Impact of Organizer Market Structure on Participant Entry Behavior in a Multi-Tournament Environment" *Games* 14, no. 1: 4.
https://doi.org/10.3390/g14010004