Evolutionarily Stable Strategies in Sports Contests

Abstract

Most articles on sports economics presume the well-known Nash equilibrium concept. In this article, however, we apply evolutionary game theory in a sports-contest model. If clubs follow evolutionarily stable strategies (ESS), then ESS generate greater investments and smaller profits than predicted by Nash’s strategies, independent of whether a club is win-maximizing or profit-maximizing. Overdissipation of the rent is possible for Nash strategies and for ESS.

Keywords

contest evolutionary stable strategies utility maximization team sports league

Introduction

Most research in the field of sports economics applies traditional game theory with the Nash equilibrium concept to analyze the interaction of clubs. It is assumed that club owners are fully rational and maximize their payoffs with an adequate choice of their investment level. Following Nash strategies (NS), clubs inherently maximize their absolute payoffs. Evolutionary game theory uses a different approach as originally introduced by Smith (1982, 1974) and Smith and Price (1973) in the context of animal conflicts. These articles analyze the strategies and survival of different types in a population. Adopting evolutionary game theory to an economic environment, Schaffer (1989) argued that profit-maximizing firms are not the best survivor, because they are not immune to the spiteful behavior of their competitors.

In this article, we introduce evolutionary game theory in sports contests. In particular, the special characteristics of sports contests (i.e., win vs. profit maximization and competitive balance) are analyzed applying evolutionary game theory. Moreover, differences to the outcome under the standard Nash equilibrium concept are discussed.

The origin of evolutionary game theory stems from the biology literature. In this context, strategies are evolutionarily stable if no player invades the population. Based on Schaffer (1988), Leininger (2003) defined an evolutionarily stable strategy (ESS) for a finite population of $n$ individuals as follows:

Definition 1: Let a strategy $x$ be adapted by all players $i$ , $i = 1, . . ., n$ .

A strategy $\overset{ˉ}{x} \neq x$ can invade $x$ , if the payoff for a single player using $\overset{ˉ}{x}$ (against $(n - 1)$ players using $x$ ) is strictly higher than the payoff of a player using $x$ (against $(n - 2)$ other players using $x$ and the deviationist using $\overset{ˉ}{x}$ ).

A strategy $x^{E S S}$ is evolutionarily stable, if it cannot be invaded by any other strategy.

The main difference from the Nash strategy is that players may increase their investments even if their absolute payoffs decrease, provided that the other players’ payoffs decrease even more. ESS implies that players consider relative payoffs and not their absolute payoffs, in contrast to NS.

Hehenkamp, Leininger, and Possajennikov (2004) and Leininger (2003) introduced and advocated the concept of evolutionary games in contest theory. Leininger (2003), for instance, argued:

[ … ] contest or more general conflict-theory portrays and emphasizes competition through struggle and contention in order to allocate economic goods and is thus opposed to the market-oriented exchange paradigma. Bringing contests and evolutionary analysis – both concepts that entered economics via biology – together in a (micro-)economic setting may therefore offer some promising perspectives.

Thus, if evolutionary game theory applies to general contests as a result of the characteristics of conflicts, then this theory should be especially suitable for sports contests. The concept of ESS understands participants’ behavior as the result of a dynamic process in which participants do not have to be fully rational. Agents will survive in the long run if their payoffs are relatively greater than their opponents. Thus, participants may have an incentive to increase their efforts to harm their competitors. Leininger (2003) proved that the outcome with ESS always differs from the outcome with NS. However, for an infinite population, the two concepts coincide.¹ Furthermore, Leininger demonstrated that an ESS contest can be transformed and interpreted as a contest in which participants try to beat the average competitor.

Hehenkamp et al. (2004) find a simple rule for ESS regarding the possibility of overdissipation of the rent depending on the sensitivity of the winning probability. They revealed that overdissipation holds for a high sensitivity of the winning probability, full rent dissipation holds for a medium sensitivity of the winning probability, and underdissipation holds for a low sensitivity of the winning probability. Moreover, they are able to exclude full dissipation or overdissipation for NS in their model.

Let me state that the main objective of this article is not to redefine ESS in economic theory or contest theory. My intention is to introduce ESS in sports economics. Evolutionary game theory seems to be a promising concept analyzing sports contests. To do that, we will consider two important peculiarities in sports economics that influence the outcome of the contest. First, a club’s objective, that is, the payoff, does not necessarily have to be profits. Clubs possibly maximize their winning probability, their profits, or a mix of the two possibilities (see e.g., Dietl, Grossmann, & Lang, 2011). Second, different from the assumptions used in general contest theory, clubs’ revenues are influenced by competitive balance. Large inequalities in team strength negatively affect (gate and broadcasting) revenues, owing to a lack of suspense provided by the game (see e.g., Neale 1964; Rottenberg 1956; Szymanski, 2003).

The main findings of this article are as follows: (i) Clubs’ investments with NS are always smaller than investments with ESS, independent of whether a club is win maximizing or profit maximizing, (ii) however, in contrast to Hehenkamp et al. (2004), overdissipation of the rent is possible for NS as well as for ESS. The possibility for overdissipation increases if the weight of the win maximization is sufficiently great and simultaneously, if competitive balance has a minor influence on clubs’ revenues. Similar to Hehenkamp et al. (2004), a higher sensitivity of the winning probability increases the possibility of overdissipation, (iii) profits are greater for NS than ESS. This means that ESS provide an additional explanation for small or even negative profits in the major European soccer leagues.

The remainder of this article is structured as follows. The following section presents the model. Then, we derive clubs' investment behavior with NS and ESS. Afterwards, the outcomes of the two concepts are compared and we discuss how ESS can explain the high investments in European soccer leagues. Finally, we sum up the main findings in the last section.

Model

In this contest, $n \geq 2$ symmetric clubs compete to win a contest. Club $i,$ $i \in {1, . . ., n}$ , invests $x_{i}$ in order to increase the probability $p_{i}$ of winning the contest. Club $i$ ’s probability of winning depends on relative investments and is determined according to the Tullock contest success function (CSF):²

p_{i} (x_{1}, . . ., x_{n}) = \frac{x_{i}^{r}}{\sum_{j = 1}^{n} x_{j}^{r}} .

Parameter $r$ determines the so-called discriminatory power of the CSF, that is, the sensitivity of the probability function with respect to investments. A high $r$ means that a small difference of effort between contestants has a great effect on the differences in the winning probabilities. Otherwise, a low $r$ implies that a small difference in effort has a minor impact on the winning probabilities.

Club $i$ ’s revenue is a concave function with respect to the CSF and increasing in the market potential $m_{i} \in R^{+}$ :

R_{i} (m_{i}, x_{1}, . . ., x_{n}) = m_{i} p_{i} - \frac{b}{2} p_{i}^{2} .

The parameter $b \in R_{0}^{+}$ determines the effect of the competitive balance on club revenues. If $b = 0$ , competitive balance has no effect on revenue. Club $i$ ’s expected profit $π_{i}$ is therefore given by

π_{i} (x_{1}, . . ., x_{n}) = R_{i} (m_{i}, x_{1}, . . ., x_{n}) - c x_{i} .

In the sports economic literature, it is often assumed that clubs’ market potentials differ (e.g., Grossmann & Dietl, 2009; Grossmann, Dietl, & Trinkner, 2008; Lang, Grossmann, & Theiler, 2011; Szymanski 2003; Szymanski & Késenne 2004). It is a commonly accepted form to introduce asymmetry and discuss the ability of different mechanisms (such as salary caps, revenue sharing, and luxury taxes) to influence competitive balance in sports leagues. In this article, however, the primary aim is to analyze individual and aggregate investments under different equilibrium concepts. The main research question is under which equilibrium concept do clubs invest more. Therefore, we assumed identical market sizes and set $m_{i} = m$ $\forall$ $i = 1, . . ., n$ to isolate the effect of the equilibrium concept.³ Consequently, the contest is fully symmetrical.

As widely discussed in the sports economic literature, club owners may not only be interested in profit but also in the win percentage (Dietl, Grossmann, and Lang, 2011; Késenne 1996, 2000, 2006). Like Dietl, Grossmann, and Lang (2011), we assumed that club owner $i$ maximizes his expected utility $u_{i} (π_{i}, p_{i})$ , which contains his expected profit as well as his win probability.⁴ The weight attached to profit is $α \in (0, 1]$ , whereas the weight attached to the win probability is $1 - α$ .⁵ Therefore, the club owner $i$ ’s objective function is

u_{i} (x_{1}, . . ., x_{n}) = α π_{i} + (1 - α) p_{i} = α [R_{i} (m, x_{1}, . . ., x_{n}) - c x_{i}] + (1 - α) p_{i} .

In order to measure how much of the possible revenue (or more general rents) are dissipated, we use the following approach. Because each club has a market size of $m$ and acquires it with probability $p_{i}$ , the maximum total expected rents in the league are $\sum_{i = 1}^{n} p_{i} m = m$ . The ratio of the sum of investments $\sum_{i = 1}^{n} x_{i}$ in equilibrium and the maximum total expected rents in the league $m$ measures the extent of “rent-dissipation.” Therefore, rent-dissipation $D$ is defined as⁶

D \equiv \frac{\sum_{i = 1}^{n} x_{i}}{m} .

Within this context, it will be interesting to analyze whether the rent is overdissipated $(D > 1),$ fully dissipated $(D = 1)$ , or underdissipated $(D < 1)$ for the different equilibrium concepts.

Solution

Nash Strategies

Club $i$ chooses its investment according to NS, and therefore maximizes its expected utility $u_{i}$ with respect to $x_{i}$ . The following Proposition sums up the main findings.

Proposition 1: Under NS,

club $i$ invests $x_{i}^{*} = \frac{r (n - 1)}{α c n^{2}} (α (m - \frac{b}{n}) + (1 - α)) .$

club $i$ ’s investment $x_{i}^{*}$ is increasing in $r$ and $m$ , decreasing in $c, b,$ and $α,$ and ambiguous in $n .$

overdissipation of the rent is possible for small values of $c, b,$ and $α,$ and for large values of $n$ and $r$ .

Proof. See Appendix ▪

(i) It is easy to see that the condition $α (m - \frac{b}{n}) + (1 - α) > 0$ is required to guarantee positive investments. Henceforth, we assume that this condition holds. (ii) A higher sensitivity of the probability function increases marginal revenue of an investment and, therefore, club $i$ ’s investment increases in equilibrium. The same intuition holds for the market potential $m$ . On the other hand, higher marginal costs decrease investments. If competitive balance has a large impact on revenues, that is, for a high $b$ , clubs decrease their investments. It is also intuitive that clubs’ investment are larger if they care more about win maximizing, that is, for a lower value of $α$ .

Result (iii) is interesting because Hehenkamp et al. (2004) can rule out overdissipation for NS in their model with profit maximization and without competitive balance. However, we find that with the introduction of a high weight of win maximization in combination with a small effect of competitive balance on revenues, overdissipation of the rent is possible. The intuition for this result is that a higher weight of win maximization as well as a lower influence of competitive balance on revenues fosters incentive to invest. However, even if the weight for win maximization is high, a high effect of competitive balance on revenue could overcompensate incentives to invest, so that underdissipation results. Therefore, the combination of win maximization and competitive balance are an important source for the extent of rent dissipation. In addition, for many clubs, a high sensitivity of the probability function or low-marginal costs increase investments and therefore foster overdissipation.

Evolutionarily Stable Strategies

If clubs follow ESS, then we get the following Proposition.

Proposition 2: Under ESS,

club i's investment is $x_{i}^{E S S} = \frac{r}{c n^{2}} (n (m - 1 + \frac{1}{α}) - b) .$

club $i$ ’s investment $x_{i}^{E S S}$ is increasing in $r$ and $m$ , decreasing in $c, b,$ and $α$ , and ambiguous in $n$ .

overdissipation of the rent $(D > 1)$ is possible for small values of $c, b,$ and $α$ , and for large values of $n$ and $r$ .

Proof. See Appendix ▪

(i) It is easy to see that the condition $n (m - 1 + \frac{1}{α}) - b > 0$ is required to obtain positive investment $x_{i}^{E S S}$ .⁷ (ii) A higher $r$ and $m$ , or a lower $c, b,$ and $α$ , imply greater club investments. Result (iii) is interesting because Hehenkamp et al. (2004) reveal in their model, with profit maximization and without any effects of competitive balance on revenues, that overdissipation for ESS depends only on $r ⋛ 1$ . According to result (iii), there could be two additional sources for overdissipation for sports leagues. Similar to the case with NS, a combination of a great weight on win maximization (i.e., a low value of $α$ ) and a low effect of competitive balance on revenues, that is, a low value of $b$ , fosters overdissipation for ESS. In addition, for many clubs, a high sensitivity of the probability function or low marginal costs increases investments and therefore induces overdissipation.

Comparison

In the last section, we derived a clubs’ behavior within the different concepts of NS and ESS. In the following two subsections, we compare the outcomes of NS with ESS. First, we compare investments and then profits.

Investment

It is interesting to analyze whether NS or the ESS produces larger investments. The following Proposition sums up the main findings.

Proposition 3:

Club $i$ ’s investment is lower with NS than ESS.

The ratio of ESS investment to NS investment $x_{i}^{E S S} / x_{i}^{*}$ equals $n / (n - 1)$ .

Proof. See Appendix ▪

(i) This result implies that the concept of ESS provides a new explanation for the large investment in sports leagues. Provided that clubs have a positive weight on profits, ESS investments are larger than NS investments. As $α$ converges to $0$ , investments converge to infinity for NS as well as for ESS. In this extreme case, clubs only care about their win probability and invest as much as possible. Implicitly, it is assumed that clubs can borrow as much as they want in this situation.⁸ (ii) It is interesting that the relative investments between ESS and NS $x_{i}^{E S S} / x_{i}^{*}$ only depend on the number of participating clubs. Neither the weight of win maximization nor the sensitivity of the probability function influences this ratio. The result implies that more clubs in the league decrease relative investments between ESS and NS with a decreasing rate. As the number of clubs converges to infinity, the ratio $x_{i}^{E S S} / x_{i}^{*}$ converges to $1$ .

Profit

Even if clubs consider their utility, we focus on clubs’ profit in this subsection. The reason is that we aim to discuss the source of low or even negative profits of many clubs in the major European soccer leagues. The following Proposition sums up the main findings.

Proposition 4 : Club $i$ ’s profit is greater for NS than ESS.

Proof. See Appendix ▪

According to this result, the concept of ESS provides a new explanation for the small profits in sports leagues. Clubs invest more with ESS than NS, so that this destructive competition leads to relatively smaller profits. If $α$ converges to $0$ , a clubs’ profit converges to $- \infty$ , independent of NS or ESS. The intuition for this result is that clubs have an incentive to invest as much as possible in this situation because they care only about win maximization.

Overinvestment in the European Soccer Leagues

It is a matter of fact that many soccer clubs in the major European soccer leagues have accumulated vast debts in recent years. The net debt in the English Premier league was £3.1 billion at the end of 2007/2008. In this season, the French League (Ligue 1) was confronted by operating losses caused by major wage increases (Deloitte, 2009). Furthermore, the Italian League (Serie A) generated losses, even though the league was the fastest growing one—revenues increased by 34% in 1 year—of the five biggest European leagues. In recent years, some clubs even went bankrupt (e.g., AC Fiorentina [Serie A] in 2002, AC Parma [Serie A] in 2004, Servette Genf [Swiss Football League] in 2005).

The economic literature provides several explanations for this development. Ruinous rat races between clubs, league structure (open league vs. closed league), and inadequate club constitutions have been identified as possible sources for overinvestment. Whitney (1993) argued that clubs’ wish for stars leads to destructive competition and suboptimal welfare because some participants drop out of the market even if leaving is inefficient. Dietl, Franck, and Lang (2008) argued that overinvestment originates from the ruinous interaction between clubs in competition, that is, the negative externalities of an investment on the opponent. They theoretically derive that the league structure—open league or closed league—also affects the extent of revenue dissipation. A system with promotion and relegation (i.e., an open league) fosters overinvestment. On the other hand, Dietl and Franck (2000) emphasized that inadequate club constitutions generate incentives for overinvestment. Traditional clubs are often constituted as an organization without residual claimant. Club managers try rather to maximize the probability of winning the championship instead of maximizing profits, caused by less control.⁹ However, this argument can only partly explain overinvestment, because we also observe large debts for clubs as “for-profit” corporations. In this article, we provide another theoretical channel that can be responsible for overinvestment. Ideas from evolutionary game theory are applied to a general model in sports economics, and the results are compared with the outcome applying traditional game theory. We reveal that the concept of ESS can explain the race to the bottom as observed for some clubs in professional soccer leagues.

Conclusion

Sports economic models are often analyzed with traditional game theory. The Nash equilibrium concept proves to be a useful tool for analyzing contestants’ behavior. Nonetheless, it is based on strong assumptions regarding contestants’ rationality. Evolutionary game theory is an approach that focuses rather on the outcome in a dynamic process. As the notion “evolutionary” suggests, types in a population survive who develop the best strategies for relative payoffs. It is quite intuitive that contestants in sports also think in relative terms. Usually, the absolute result is less important than the relative result. For instance, a marathon runner is primarily interested in being the first at the finish line and not beating a specific time limit.

In this article, we analyzed a sports contest in which $n$ clubs compete in a championship. Club owners’ objective function includes a mix of win maximization as well as profit maximization. We demonstrated that club investments are greater and profits smaller if clubs pursue ESSs rather than NS. Hence, ESS can better explain than NS why many clubs in the European soccer leagues are submerged in debt in recent years. Furthermore, we conclude that overdissipation of the rent is possible for NS as well as for ESS. The possibility for overdissipation increases if the weight of win maximization is sufficiently great and, simultaneously, competitive balance has a minor influence on clubs’ revenues. Moreover, a greater sensitivity of the winning probability increases the possibility of overdissipation.

As evolutionary game theory has not been applied to sports contests so far, this study can be seen as a first step in this field of research. We would like to invoke further research on this topic in future. The effect of asymmetry with ESS could be an especially interesting issue to analyze, for instance, effects of different redistributive mechanisms (such as revenue sharing) on competitive balance.

Footnotes

Appendix

Declaration of Conflicting Interests

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

Funding

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

Notes

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