The “sports wars”: A contest theory approach to cities hosting game

Abstract

The competition among cities to host a sports team or a large-scale international sports event is modeled as a winner-pay contest with an entry fee. In the first stage, each contestant (city) decides whether to pay the entry fee (infrastructure required by the team, for example), which allows it to participate in the second stage, that is, the actual contest. We show that the contest organizer’s choice of the optimal entry fee does not depend on the number of contestants. Furthermore, in some cases, the result is a form of zero-sum game, in which the sports team or event organizer is the main beneficiary. The findings shed light on this type of competition and under what conditions on the entry fee a city might benefit from hosting a sports team or large-scale international sports event.

Keywords

Competition entry fee event organisation risk aversion winner-pay contest zero-sum game

Introduction

Cities have fought with one another over economic resources and development opportunities as long as they have existed. Sports is a modern-day arena for that competition. Specifically, cities invest significant efforts to attract major league teams or large-scale international sports events with the goal of stimulating their economic development.^1,2 In the US, for example, cities compete for the right to host major league baseball, football, basketball, and hockey teams,^1,3 and there are similar examples in European countries.⁴ There is also global competition to host the quadrennial Olympic games.^5,6 In order to host sports teams, simply offering incentives is usually not enough. Cities also need to invest in their sports infrastructure.^3,4 In the case of the Olympic Games, cities must first invest heavily in preparing a detailed plan, which may or may not bear fruit only 4 to 7 years later; and that is just the beginning. Once a city has won the right to host the games, it must make massive investments in sports facilities and its supporting infrastructure, including roads, accommodations, and transportation, while at the same time significantly expanding its payrolls.^7,8

A great deal of investment is required to host a major league sports team, and for many cities, the cost can be prohibitive. They must heavily invest in their sports infrastructure simply in order to present their proposals to the teams. This pre-investment can be viewed as an entry fee to enter the competition. For example, the incentive package offered by Norfolk to attract the NBA’s Charlotte Hornets in 2001 included a proposed $215-million 18,500-seat arena. To host the Olympics, a city must offer sufficiently attractive municipal infrastructure and it must incur pre-investment costs in the form of a detailed plan and investment in basic sports facilities. For example, Chicago spent at least $70 million on its unsuccessful application to host the 2016 Games.⁹ While the cost of investment in sports facilities is incurred only in the case of winning, the cost of the required pre-investment and the detailed plan can be viewed as the entry fee to a competition. Robert Baade, a sports economics professor and author, once said:¹⁰ “There are no guarantees that if you build it they will come. ”

Is it even worthwhile to fight in the “sports wars”? According to most of the sports literature, hosting a sports team or sports event has no impact on a city’s development; on the contrary, in some cases, it even have a negative effect.^1,3,8,11,12 Few studies shown that winning this type of competition might be beneficial to a city.^3,4 The model presented below will employ the contest theory approach in order to examine this issue.

As previously mentioned, the cost of an offer to attract a sports team is incurred only in the case of winning the competition, while the infrastructure investment must be made before the competition for hosting begins. Furthermore, the investment in infrastructure is a basic requirement to attract a team and is known to the competitors before the competition even begins. Most importantly, it does not have any effect on the probability of winning. It is essentially the “ticket” to enter the competition, and the sports team will profit from that “ticket” whichever city wins. The same applies to competition over hosting an event such as the Olympics. We can therefore model this type of competition as a winner-pay contest with an entry fee. Only the winner incurs the cost of its offer, while the entry fee, namely the investment in the basic infrastructure required by the sports team or the sports event organizer, is paid up front.¹

Winner-pay contests^13,14 with a Tullock success function (also known as winner-pay Tullock contests) have been widely studied in a symmetric setting.^15–18 Yates¹³ studies a more general form of the success function in pure strategy and finds that Nash equilibrium exists and is unique in two-contestant cases with a more general success function and asymmetry. The few related studies have employed the standard all-pay contest (auction) or the standard all-pay Tullock contest with complete information and an entry fee. The first to do so were Kaplan and Sela¹⁹ who examined the asymmetric and symmetric cases of an all-pay auction when the entry fee is private information. They showed that the entry fee can lead to an inefficient contest when low-ability contestants have a higher chance of winning than high-ability ones. They also showed that if the goal is an efficient contest, then the entry fee should be paid by the winner. Jia and Sun²⁰ considered the use of a commonly known entry fee in a standard symmetric Tullock contest with two stages where the decision to pay the entry fee constitutes the first stage and the exertion of effort constitutes the second. They derive the optimal entry fee as a function of population size and the parameters of the contest organizer. The entry fee effect was also studied by Duffy et al.²¹ in the context of a one-stage standard (all-pay) symmetric Tullock contest. They showed that an entry fee increases the contest designer’s revenue. Duffy et al.²¹ also conducted an experimental study of their contest model and found mixed support for their theoretical predictions. Specifically, they were mismatched in terms of optimal entry fees, over-bidding by the contestants, under-participation with a low entry fee, and over-participation with a high entry fee.

We study the effect of introducing an entry fee in the winner-pay contest with complete information and a sufficiently general success function. Each contestant must pay the entry fee up front, which constitutes the first stage of the contest. The entry fee is considered to be common knowledge, that is, it is determined by the organizer (team) and is therefore publicly observable (while in “The entry fees are private information” section, it is assumed to be private information and, therefore, not publicly observable). The contestants compete in the second stage. We show that in the case of a common-knowledge entry fee, if the contest organizer chooses the optimal entry fee, then his expected revenue does not depend on the number of contestants. Furthermore, the case can be viewed as a kind of zero-sum game. In other words, the contest organizer or designer gets all of the payoff and the contestants get none. In the case of a private-information entry fee, the contestants will have a positive expected payoff. The findings provide additional evidence on the question of whether it is profitable for a city to host a sports team or sports event.

The rest of the article is organized as follows: “The model” section introduces the model. The “Results” section describes the effect of an entry fee on effort exerted and on the designer’s revenue. The “Extensions” section considers extensions of the basic model and the “Conclusion and discussion” section concludes the article.

The model

Consider $n \geq 2$ risk-neutral contestants (cities) competing in a winner-pay contest for a single common-knowledge prize $V$ (i.e. the value of hosting a team).² Each contestant must pay an entry fee $e$ (the cost of infrastructure investment and a detailed plan) which is also common knowledge. In the first stage, each contestant $i,$ $i \in {1, \dots, n}$ chooses whether to pay the entry fee, which determines whether he can enter the contest. In the second stage, each contestant simultaneously chooses his non-negative effort $x_{i}$ and pays the cost of exerting that effort, that is, $c (x_{i}) = x_{i}$ if he wins $.$

The probability of contestant $i$ winning the contest when there are $k \leq n$ contestants who paid the fee is $q_{i} (x_{1}, \dots, x_{k}) = q_{i}^{k} (x_{i}, x_{- i})$ . We assume that $q_{i}^{k} (x_{i}, x_{- i})$ is continuous, concave, increasing in $x_{i},$ and decreasing in $x_{- i}$ and that $q_{i}^{k} (t x_{i}, t x_{- i}) = q_{i}^{k} (x_{i}, x_{- i})$ for all $t > 0$ (homogeneous of degree zero).³ This form of contest success function is the generalization to $n$ contestants of the contest success function studied by Yates.¹³,⁴ Additionally, we deviate from the models of Jia and Sun²⁰ and Duffy et al.²¹ in two main ways. First, they consider all-pay contests while the current model employs a winner-pay contest. Second, they use the Tullock success function, while the current model employs a more general success function.

Results

Backward induction from stage $2$ is used to solve for the subgame perfect equilibrium for $k + 1$ contestants:

x_{i}^{*} = max_{x_{i}} (V - x_{i}) q_{i}^{k + 1} (x_{i}, x_{- i}) .

The first-order conditions in the symmetric case (

x_{1}^{*} = x_{2}^{*} = \dots = x_{k + 1}^{*} = x^{*})

are therefore given by

\frac{\partial q^{k + 1} (x^{*})}{\partial x} (V - x^{*}) - q^{k + 1} (x^{*}) = 0.

(1)

Since

q^{k + 1}

is homogeneous of degree zero according to Simon and Blume,²²

\frac{\partial q^{k + 1} (x^{*})}{\partial x}

is homogeneous of degree minus one. Thus, (1) can be written as follows:

\frac{1}{x^{*}} \frac{\partial q^{k + 1} (1)}{\partial x} (V - x^{*}) - q^{k + 1} (1) = 0,

(2)

and by rewriting (2), we obtain

x^{*} = (\frac{\frac{\partial q^{k + 1} (1)}{\partial x}}{\frac{\partial q^{k + 1} (1)}{\partial x} + q^{k + 1} (1)}) V .

(3)

Since

q_{i}^{k} (x_{i}, x_{- i})

is a concave, it is easy to verify that the second-order conditions are satisfied. Notice that in the symmetric case of

q_{i}^{k + 1} (x^{*}) = \frac{1}{k + 1}

, we take as given that the symmetric subgame has a Nash equilibrium in which contestants enter the contest with identical probability

p \in [0, 1] .

Thus, the utility of contestant

i^{'}

s decision to enter the contest when there are

k

contestants and effort is given by (3) is as follows:

\begin{aligned} U_{k + 1} & = (V - x_{i}) q_{i}^{k + 1} (x_{i}, x_{- i}) - e \\ = (\frac{q^{k + 1} (1)}{\frac{\partial q^{k +!} (1)}{\partial x} + q^{k + 1} (1)}) V q_{i}^{k + 1} (x^{*}) - e \\ = (\frac{q^{k + 1} (1)}{\frac{\partial q^{k + 1} (1)}{\partial x} + q^{k + 1} (1)}) \frac{V}{k + 1} \\ - e, k \in {1, \dots, n - 1} \end{aligned}

(4)

Since in equilibrium the entry fee is set such that the contestant is indifferent to entering the contest, the probability of entering is determined as follows:

\begin{aligned} \sum_{k = 0}^{n - 1} (\begin{matrix} n - 1 \\ k \end{matrix}) p^{k} (1 - p)^{n - k - 1} (\frac{q^{k + 1} (1)}{\frac{\partial q^{k + 1} (1)}{\partial x} + q^{k + 1} (1)}) \frac{V}{k + 1} \\ = \frac{1}{n p} \sum_{k = 0}^{n - 1} (\begin{matrix} n \\ k + 1 \end{matrix}) p^{k + 1} (1 - p)^{n - k - 1} (k + 1) \\ (\frac{q^{k + 1} (1)}{\frac{\partial q^{k + 1} (1)}{\partial x} + q^{k + 1} (1)}) \frac{V}{k + 1} = \frac{1}{n p} \sum_{k = 1}^{n} (\begin{matrix} n \\ k \end{matrix}) p^{k} (1 - p)^{n - k} \\ (\frac{q^{k} (1)}{\frac{\partial q^{k} (1)}{\partial x} + q^{k} (1)}) V = e . \end{aligned}

(5)

The organizer’s expected payoff in this contest is

\begin{aligned} R_{n} = & \sum_{k = 1}^{n} (\begin{matrix} n \\ k \end{matrix}) p^{k} (1 - p)^{n - k} [max x + k e] \\ = \sum_{k = 1}^{n} (\begin{matrix} n \\ k \end{matrix}) p^{k} (1 - p)^{n - k} [(\frac{\frac{\partial q^{k} (1)}{\partial x}}{\frac{\partial q^{k} (1)}{\partial x} + q^{k} (1)}) V + k e] \\ = \sum_{k = 1}^{n} (\begin{matrix} n \\ k \end{matrix}) p^{k} (1 - p)^{n - k} [(\frac{\frac{\partial q^{k} (1)}{\partial x}}{\frac{\partial q^{k} (1)}{\partial x} + q^{k} (1)}) V] + n p e . \end{aligned}

In other words, the expectation is constructed from the expected number of participants multiplied by the winner’s effort plus the participants’ entry fees. By (5), the organizer’s revenue is given by

\begin{aligned} R_{n} = \sum_{k = 1}^{n} (\begin{matrix} n \\ k \end{matrix}) p^{k} (1 - p)^{n - k} (\frac{\frac{\partial q^{k} (1)}{\partial x}}{\frac{\partial q^{k} (1)}{\partial x} + q^{k} (1)}) V \\ + \sum_{k = 1}^{n} (\begin{matrix} n \\ k \end{matrix}) p^{k} (1 - p)^{n - k} (\frac{q^{k} (1)}{\frac{\partial q^{k} (1)}{\partial x} + q^{k} (1)}) V \\ = \sum_{k = 1}^{n} (\begin{matrix} n \\ k \end{matrix}) p^{k} (1 - p)^{n - k} V = [1 - (1 - p)^{n}] V . \end{aligned}

(6)

This leads to the following proposition:

Proposition 1

The unique symmetric subgame perfect equilibrium satisfies $p^{*} = 1$ and $e^{*} = (\frac{q^{n} (1, 1)}{\frac{\partial q^{n} (1, 1)}{\partial x} + q^{n} (1, 1)}) \frac{V}{n}$ .

Proof: Notice that the probability of entering the contest is determined by the entry fee. Thus, the optimal probability will determine the optimal entry fee. Differentiating the organizer’s payoff (6) yields

\frac{\partial R_{n}}{\partial p} = n (1 - p)^{n - 1} V \geq 0.

Thus, the corner solution is

p^{*} = 1

and substituting it in (5) yields the result. □

By Proposition 1 and (6), we get

R_{n} = V .

(7)

In other words, the entry fee complements the effort exerted, bringing it up to

V .

This full rent dissipation result is in line with Duffy et al.²¹ for a standard Tullock contest with an entry fee. Furthermore, the contestants’ payoff is zero according to Proposition 1 and (4). Thus, the contestants are indifferent to participating in the contest, and therefore it can be viewed as a kind of zero-sum game. Note that the expected revenue in this contest is equal to that in the standard symmetric Tullock contest with complete winner reimbursement, which was studied by Matros and Armanios.¹⁷ Furthermore, the expected revenue is identical to that in the winner-pay contest when the yardstick approach is the cost determinant.²³ Furthermore,

V

is the highest possible revenue that can be obtained in the standard symmetric winner-pay Tullock contest.¹⁷ Notice that by (4) and (6), we get

R_{n} < V

and

U_{n} > 0

in the case of a zero entry fee

.

In other words, in the absence of an entry fee we do not arrive at a zero-sum game. Notice that we are focusing on a symmetric equilibrium, in which all contestants are active. In this form of the game, an asymmetric equilibrium, in which some of the contestants will be inactive, always exists. We focus only on a symmetric equilibrium because a symmetric setting is the most plausible scenario.^24,25 Furthermore, there is another possible equilibrium when

e^{*} = V,

in which case only one city enters the competition, and no additional effort is exerted. This result is not realistic in the sense that the incentives offered to the teams (or infrastructure additions in the case of the Olympics) are mandatory. The result in Proposition 1 captures a more realistic scenario in which optimal entry fees and efforts are determined according to the number of contestants.

Corollary 1

The organizer’s revenue in a winner-pay contest with an entry fee does not depend on the number of contestants.

Proof: Strictly by (7). □

Because the entry fee depends on the number of contestants, it is added to the prize value. This is because when the number of contestants is high, the fee is low, and vice versa, and they are added together to obtain $V .$

Corollary 2

Let $q_{i}^{k} (x_{i}, x_{- i})$ be the Tullock success function. The organizer’s revenue in a winner-pay contest with an entry fee is then higher than without.

Proof: According to Matros and Armanios¹⁷ and Minchuk,¹⁸ the organizer’s revenue in a winner-pay contest satisfies $R < V$ for any $n,$ yielding the result. □

Notice that in some cases a city gains benefit from the infrastructure simply because it would have been built even without the contest, thereby lowering the true cost to the city since the same infrastructure is being used, even in part, as the contest entry fee. In this situation, the cost of the entry fee $e$ will be $α e$ where $0 < α < 1,$ and the main result remains unchanged. The main difference is that the entry fee becomes: $e^{*} = \frac{1}{α} (\frac{q^{n - 1} (1, 1)}{\frac{\partial q^{n - 1} (1, 1)}{\partial x} + q^{n - 1} (1, 1)}) \frac{V}{n},$ which now includes the city’s future gains. Nonetheless, all other results remain unchanged.

Extensions

In this section, the results are extended in four directions: the designer benefits only from the winner’s entry fee, the entry fee is privately known, risk-averse contestants, and asymmetric prizes.

The organizer benefits only from the entry fee paid by the winner

In this section, it is assumed that the contest designer (the team) benefits only from the highest-effort entry fee. Since the cities are competing for a specific sports team, the team benefits only from the infrastructure investment made by the winning city but not that made by the others. Thus, the organizer’s revenue is given by:

\begin{aligned} R_{1} & = \sum_{k = 1}^{n} (\begin{matrix} n \\ k \end{matrix}) p^{k} (1 - p)^{n - k} [max x^{*} + e] \\ = \sum_{k = 1}^{n} (\begin{matrix} n \\ k \end{matrix}) p^{k} (1 - p)^{n - k} [(\frac{\frac{\partial q^{k} (1)}{\partial x}}{\frac{\partial q^{k} (1)}{\partial x} + q^{k} (1)}) V + e] \\ = \sum_{k = 1}^{n} (\begin{matrix} n \\ k \end{matrix}) p^{k} (1 - p)^{n - k} [(\frac{\frac{\partial q^{k} (1)}{\partial x}}{\frac{\partial q^{k} (1)}{\partial x} + q^{k} (1)}) V] \\ + [1 - (1 - p)^{n}] e . \end{aligned}

By (5)

\begin{aligned} R_{1} & = \sum_{k = 1}^{n} (\begin{matrix} n \\ k \end{matrix}) p^{k} (1 - p)^{n - k} [(\frac{\frac{\partial q^{k} (1)}{\partial x}}{\frac{\partial q^{k} (1)}{\partial x} + q^{k} (1)}) V] \\ + \frac{[1 - (1 - p)^{n}]}{n p} \sum_{k = 1}^{n} (\begin{matrix} n \\ k \end{matrix}) p^{k} (1 - p)^{n - k} \\ [(\frac{q^{k} (1)}{\frac{\partial q^{k} (1)}{\partial x} + q^{k} (1)}) V] . \end{aligned}

(8)

Differentiating by

p

, we get

\begin{aligned} \frac{\partial R_{1}}{\partial p} & = \frac{1}{p} \sum_{k = 1}^{n} (\begin{matrix} n \\ k \end{matrix}) [k - n p] p^{k} (1 - p)^{n - k - 1} \\ [(\frac{\frac{\partial q^{k} (1)}{\partial x}}{\frac{\partial q^{k} (1)}{\partial x} + q^{k} (1)}) V] \\ + \frac{[(1 - p)^{n} + n p (1 - p)^{n - 1} - 1]}{n p^{2}} \\ \sum_{k = 1}^{n} (\begin{matrix} n \\ k \end{matrix}) p^{k} (1 - p)^{n - k} [(\frac{q^{k} (1)}{\frac{\partial q^{k} (1)}{\partial x} + q^{k} (1)}) V] \\ + . \frac{[1 - (1 - p)^{n}]}{n p^{2}} \sum_{k = 1}^{n} (\begin{matrix} n \\ k \end{matrix}) [k - n p] p^{k} (1 - p)^{n - k - 1} \\ [(\frac{q^{k} (1)}{\frac{\partial q^{k} (1)}{\partial x} + q^{k} (1)}) V] . \end{aligned}

In the case of

p = 1

\frac{\partial R_{1}}{\partial p} |_{p = 1} = - n (\frac{\frac{\partial q^{n - 1} (1)}{\partial x}}{\frac{\partial q^{n - 1} (1)}{\partial x} + q^{n - 1} (1)}) V - \frac{n + 1}{n} (\frac{q^{n} (1)}{\frac{\partial q^{n} (1)}{\partial x} + q^{n} (1)}) V < 0.

(9)

This leads to the following proposition:

Proposition 2

If the organizer benefits only from the winner’s entry fee, then the symmetric subgame perfect equilibrium satisfies $p^{*} \in (0, 1)$ .

Proof: By (9), we have $\frac{\partial R_{1}}{\partial p} |_{p = 1} < 0$ , and, therefore, $p^{*} \in (0, 1)$ , yielding the result. □

Thus, since the sports team benefits only from the winner’s entry fee (infrastructure), it has an incentive to limit the number of competing cities. Thus, since $p^{*} \in (0, 1)$ and by (5), it might be that $e^{*} \geq (\frac{q^{n} (1, 1)}{\frac{\partial q^{n} (1, 1)}{\partial x} + q^{n} (1, 1)}) \frac{V}{n}$ with $k^{*} < n$ participants. Furthermore, we can rewrite (8) as follows:

\begin{aligned} R_{1} & = \sum_{k = 1}^{n} (\begin{matrix} n \\ k \end{matrix}) p^{k} (1 - p)^{n - k} V + \frac{[1 - (1 - p)^{n} - n p]}{n p} \\ \sum_{k = 1}^{n} (\begin{matrix} n \\ k \end{matrix}) p^{k} (1 - p)^{n - k} [(\frac{q^{k} (1)}{\frac{\partial q^{k} (1)}{\partial x} + q^{k} (1)}) V] \end{aligned}

and by (5)

\begin{aligned} R_{1} & = [1 - (1 - p)^{n}] V + [1 - (1 - p)^{n} - n p] e \\ \leq [1 - (1 - p)^{n}] V = R_{n} . \end{aligned}

Thus, we get

R_{1} < V

for

p^{*} \in (0, 1)

and it is again easy to verify that the contestants’ (cites’) expected payoffs remain unchanged, as in the previous case. In other words, this is not a zero-sum game. It is also easy to show that this result continues to hold if the contest designer gains no benefit from the entry fee (in which case the derivative of the designer’s payoff is

- n (\frac{\frac{\partial q^{n - 1} (1)}{\partial x}}{\frac{\partial q^{n - 1} (1)}{\partial x} + q^{n - 1} (1)}) V

p = 1)

The entry fees are private information

Following the approach of Kaplan and Sela,¹⁹ we assume that the entry fees are private information. First, the organizer chooses a uniform entry requirement.⁵ Each contestant $i,$ $i \in 1, \dots, n$ then realizes the cost of meeting the requirement $e_{i},$ which is known only to him. Following that, he chooses whether or not to pay the entry fee (i.e. whether or not to enter the contest). Finally, each contestant simultaneously chooses a non-negative level of effort. The only common knowledge in this case is the distribution of entry fees. The entry fees are drawn independently from a cumulative distribution function $F$ on support $[0, \bar{e}] .$

The assumption that the entry fees are private information leads to the following proposition:

Proposition 3

Let the entry fees be private information. Then there exists an equilibrium cutoff $\hat{e}$ that satisfies $p^{*} = {\begin{matrix} 1 & e \leq \hat{e} \\ 0 & e > \hat{e} \end{matrix} |$ and $\hat{e} = \sum_{k = 1}^{n} (\begin{matrix} n \\ k \end{matrix}) F (\hat{e})^{k - 1} (1 - F (\hat{e}))^{n - k} (\frac{q^{k} (1)}{\frac{\partial q^{k} (1)}{\partial x} + q^{k} (1)}) \frac{V}{n}$ .

Proof: The proof is similar to that of Proposition 1 by Kaplan and Sela.¹⁹ The main difference is that the utility function in the second stage is as in (4) and in the first stage a contestant’s decision to enter is made according to (5), where $e$ is substituted by $\hat{e}$ and $p$ by $F (\hat{e})$ . □

The contestant’s expected utility function is given by

\begin{aligned} E (U_{n}) & = \int_{0}^{\hat{e}} (\sum_{k = 1}^{n} (\begin{matrix} n \\ k \end{matrix}) F (\hat{e})^{k - 1} (1 - F (\hat{e}))^{n - k} \\ (\frac{q^{k} (1)}{\frac{\partial q^{k} (1)}{\partial x} + q^{k} (1)}) \frac{V}{n} - e) d F (e) \\ = \int_{0}^{\hat{e}} (\hat{e} - e) d F (e) > 0. \end{aligned}

Thus, for any entry requirement (including the optimal one), it is beneficial for the contestants (the cities) if their entry fees are private information.

Risk aversion

Now assume that the contestants are risk-averse. By means of backward induction from stage $2$ to solve for the subgame perfect equilibrium with $k + 1$ contestants, we get

x_{i, a v e r}^{*} = max_{x_{i}} u (V - x_{i, a v e r}) q_{i}^{k + 1} (x_{i, a v e r}, x_{- i, a v e r}),

(10)

where

u (0) = 0, u^{'} > 0

and

u^{″} < 0

(strictly concave)

.

Thus, the first-order conditions in the symmetric case are

\frac{1}{x_{a v e r}^{*}} \frac{\partial q^{k + 1} (1)}{\partial x} u (V - x_{a v e r}^{*}) - q^{k + 1} (1) u^{'} (V - x_{a v e r}^{*}) = 0.

Rearranging yields

x_{a v e r}^{*} = \frac{\frac{\partial q^{k + 1} (1)}{\partial x}}{q^{k + 1} (1)} \frac{u (V - x_{a v e r}^{*})}{u^{'} (V - x_{a v e r}^{*})} .

(11)

This leads to the following corollary:

Corollary 3

The efforts in a winner-pay contest with risk-averse contestants are greater than in the case of risk-neutral contestants.

Proof: Since $u$ is strictly concave, we have $\frac{u (V - x_{a v e r}^{*})}{u^{'} (V - x_{a v e r}^{*})} > V - x_{a v e r}^{*} .$ According to (11),

x_{a v e r}^{*} > \frac{\frac{\partial q^{k + 1} (1)}{\partial x}}{q^{k + 1} (1)} (V - x_{a v e r}^{*}) .

Rearranging

x_{a v e r}^{*} > (\frac{\frac{\partial q^{k + 1} (1)}{\partial x}}{\frac{\partial q^{k + 1} (1)}{\partial x} + q^{k + 1} (1)}) V,

which yields the result. □

From the corollary above, we see that in the case of risk aversion, the cities will pay more if they win than in the risk-neutral case.⁶ This result also holds in the symmetric auctions case with incomplete information.²⁶ The higher level of effort has two effects: an increase in the probability of winning and a decrease in payoffs. In contests, the second effect usually predominates.²⁷ We assume that since the cities are already indifferent as to whether to participate or not in the risk-neutral case (zero expected payoff), the same result will obtain under the assumption of risk aversion.

Asymmetry

In this subsection, we address the effect of asymmetry in valuations. Assume that there are two contestants and that $V_{1} > V_{2.}$ In other words, the first contestant is “stronger” than the second. We also make additional assumptions on the probability function $q_{i} (x_{1}, x_{2}),$ such that $q_{i} (0, 0) = \frac{1}{2}$ , $q_{i} (1, 0) > \frac{1}{2}$ and $q_{i} (0, 1) < \frac{1}{2}$ . Thus, in the second stage, we obtain the solution of the model studied by Yates,¹³ according to which an equilibrium exists (no closed-form solution) such that $x_{1}^{*} > x_{2}^{*} > 0$ and $V_{1} - x_{1}^{*}$ $> V_{2} - x_{2}^{*}$ . In other words, according to (4) and with identical entry fees in equilibrium, we get $U_{1} \geq U_{2} \geq 0.$ Thus, in the case of asymmetry in valuations, at least the “stronger” city may profit from the competition.⁷

Conclusion

A winner-pay contest with $n$ symmetric contestants (cities) and an entry fee was considered. The optimal entry fee and the optimal investment required of a contestant in order to participate in the contest were derived. It was also found that the optimal expected revenue of the team or event organizer does not depend on the number of contestants and is higher than in the corresponding contest without an entry fee and with a Tullock success function. Furthermore, the contestants have zero payoffs, such that the contest is a kind of zero-sum game. Finally, if there is no entry fee or the entry fee is private information, then the contestants (cities) may profit from the contest. Interestingly, a sports team or international sports event organizer can adjust their infrastructure requirements so as to be the only beneficiary from the competition.

The results are in line with those reported in the literature which show that hosting a sports team or sports event is not profitable for a city but is beneficial for the team or the event organizer.^{1,4,6,8,28,29} Although there are examples in which hosting events or sports teams has been profitable, they usually involved a jurisdiction larger than a city or cities that were already highly attractive to tourists.^1,2,8 Nonetheless, cities and their citizens appear to be willing to incur the cost of hosting a sports event or a sports team.^30,31 The question is how high a cost.

The analysis indicates that hosting a team or sports event might not be worthwhile for a city, as a result of the entry fee (infrastructure demands or a detailed plan) which can be manipulated by the team or event organizer according to the number of contestants (the cities) or some other economic parameters for their own benefit. If there is no entry fee, the entry fee is private information, or the entry fee is not adjustable (not optimal), then the city might be able to profit from hosting a team or sports event. If the contest designer wishes to maximize his profit and reach its upper bound of profit, he can get it under the current model. However, if he has a different goal, say, efficiency, then the contest model in the asymmetric case might be inefficient, and another mechanism, such as bargaining, might be the preferred solution. Finally, the results assume rational behavior and complete information regarding the prize. A possible direction for future research would be to relax one or all of these assumptions.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

ORCID iD

Yizhaq Minchuk

Notes

References

Nunn

Rosentraub

. Sports wars. J Sport Soc Issues 1997; 21: 65–82.

Boer

IJW

Koning

Mierau

. ex ante and ex post willingness to pay for hosting a large international sport event. J Sports Econom 2019; 20: 159–176.

Rosentraub

Swindell

. Negotiating games: cities, sports, and the winner’s curse. J Sport Manag 2002; 16: 18–35.

Gratton

Shibli

Coleman

. Sport and economic regeneration in cities. Urban Stud 2005; 42: 985–999.

Short

. Globalization, cities and the summer olympics. City 2008; 12: 321–340.

Nitch

Wendland

. The IOC’s midas touch: summer olympics and city growth. Urban Stud 2017; 54: 971–983.

Billings

Holladay

. Should cities go for the cold? The long-term impacts of hosting the olympics. Econ Inq 2012; 50: 754–772.

Baade

Maytheson

. Going for the gold: the economics of the olympics. J Econ Perspect 2016; 30: 201–218.

Zimbalist

. Circus Maximus: The Economic Gamble Behind Hosting the Olympics and the World Cup. Washington: Brookings Institution Press, 2015.

10.

Bozik

. What does it take to attract a pro sports team? DailyPress, https://www.dailypress.com/2014/01/13/what-does-it-take-to-attract-a-pro-sports-team (2014, accessed 27 April 2022).

11.

Andranovich

Burbank

. Olympic cities: lessons learned from mega-event politics. J Urban Aff 2001; 23: 113–131.

12.

Bradbury

. Does hosting a professional sports team benefit the local community? Evidence from property assessments. Econ Gov 2022; 23: 219–252.

13.

Yates

. Winner-pay contests. Public Choice 2011; 142: 93–106.

14.

Minchuk

. Transfer market – a contest theory approach. memo 2022b. DOI: https://doi.org/10.13140/RG.2.2.19606.52809.

15.

Skaperdas

Gan

. Risk aversion in contests. Econ J 1995; 105: 951–962.

16.

Wärneryd

. In defense of lawyers: moral hazard as an aid to cooperation. Games Econ Behav 2000; 33: 145–158.

17.

Matros

Armanious

. Tullock contests with reimbursements. Public Choice 2009; 141: 49–63.

18.

Minchuk

. Winner – pay contests with a no-winner possibility. Manage Decis Econ 2022a; 43: 1874–1879.

19.

Kaplan

Sela

. Effective contests. Econ Lett 2010; 106: 38–41.

20.

Jia

Sun

C-J

. The optimal entry fee-prize ratio in Tullock contest. J Math Econ 2021; 96: 1–11.

21.

Duffy

Matros

Valencia

. Contests with entry fees: theory and evidence. Rev Econ Des 2023; 27: 725–761.

22.

Simon

Blume

. Mathematics for Economists. New York: Norton, 1994.

23.

Minchuk

. Yardstick approach in a Tullock contest. Managerial and Decision Economics 2024. DOI: doi.orgq10.1002/mde.4103.

24.

Barut

Kovenock

Noussair

. A comparison of multiple-unit all-pay and winner-pay auctions under incomplete information. Int Econ Rev 2002; 43: 675–708.

25.

Jiao

. Contests with endogenous entry. Int J Game Theory 2015; 44: 387–424.

26.

Krishna

. Auction Theory. Academic Press: New York, 2010.

27.

Cornes

Hartley

. Risk aversion in symmetric and asymmetric contests. Econ Theory 2012; 51: 247–275.

28.

Johnson

Groothuis

Whitehead

. the value of public goods generated by a major league sports team: the CVM approach. J Sports Econom 2001; 2: 6–21.

29.

Kuper

Szymanski

. Soccernomics: Why England Loses, Why Germany and Brazil Win, and Why the US, Japan, Australia, Turkey – and Even Iraq – Are Destined to Become the Kings of the World’s Most Popular Sport: Nation Books, 2009.

30.

Castellanos

García

Sánchez

. The willingness to pay to keep a football club in a city: How important are the methodological issues? J Sports Econom 2011; 12: 464–486.

31.

Walton

Longo

Dawson

. A contingent valuation of the 2012 London Olympic Games: a regional perspective. J Sports Econom 2012; 9: 304–317.

32.

Lagerlöf

NMJ

. Hybrid all-pay and winner-pay contests. Am Econ J: Microecon 2020; 12: 144–169.

33.

Treich

. Risk-aversion and prudence in rent-seeking games. Public Choice 2010; 145: 339–349.