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Public Choice

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01-02-2019

Sabotage in team contests

Authors: Serhat Doğan, Kerim Keskin, Çağrı Sağlam

Published in: Public Choice | Issue 3-4/2019

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Abstract

In the contest literature, sabotage is defined as a deliberate and costly activity that damages the opponent’s likelihood of winning the contest. Most of the existing results suggest that, anticipating a possible sabotage, contestants would be discouraged from exerting high efforts. In this paper we investigate the act of sabotage in a team contest wherein team members exert costly efforts as a contribution to their team’s aggregate effort, which in turn determines the contest’s outcome. For the baseline model with no sabotage, there exists a corner equilibrium implying a free-rider problem in each team. As for the model with sabotage, our characterization of Nash equilibrium reveals two important results: (i) a unique interior equilibrium exists so that the free-rider problem no longer is a concern and (ii) the discouragement effect of sabotage vanishes for some players. On top of those conclusions, we investigate the team owner’s problems of prize allocation and team formation with the objective being to maximize his team’s winning probability.

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For more real-life examples, see a recent survey by Chowdhury and Gürtler (2015).

A result that is verified observationally by a number of studies (e.g., Garicano and Palacios-Huerta 2014; Harbring and Irlenbusch 2005, 2011; Vandegrift and Yavas 2010).

Treating a group of attackers/defenders as a single decision-maker, we label them as a team member. The same is true for the alternative interpretations that follow.

Sabotage activity is said to be directed if a player is facing multiple opponents and is able to choose the victim of his sabotage. Here we restrict the possible directions for sabotage, arguing that the attackers/defenders in a football team are facing the defenders/attackers in the opposing team. For the interested reader, we analyze the case of directed sabotage in Appendix 2 and characterize the conditions under which the model reduces to our original model with directionally restricted sabotage.

Indeed, if one considers our baseline model in the context of public good provision, such results date back to Olson (1965).

Following the war interpretation, the destruction of a rival’s weaponry or resources can be labeled as a sabotage act. Or, following the election interpretation, a possible sabotage activity would be negative campaigning.

From another perspective, we study a team contest played on a small network with (i) four nodes representing the team members and (ii) three types of links representing their interactions (see Fig. 1). In particular, a1 has a friendship link with d1; a2 has a friendship link with d2; a1 and d1 have competition links with both a2 and d2; a1 has a sabotage link with d2; and d1 has a sabotage link with a2.

There is another well-known solution to complete free riding in team contests, even when sabotage is not available. If players have strictly convex cost functions, then it is possible to construct an equilibrium in which both team members exert positive productive efforts (see Esteban and Ray 2001). Accordingly, it can be argued that the introduction of sabotage plays a role analogous to that played by a strictly convex cost function.

An oversimplified model refers to a model that disregards the possibility of sabotage although the real-life scenario to be explained includes a sabotage act. Apparently, such an oversimplified model might make significantly different predictions.

We thank an anonymous reviewer for bringing this point to our attention.

Suppose that the effectiveness parameters could differ across teams and assume without loss of generality that $\mu _1^d > \mu _2^d$. That assumption implies that team 1 cannot hire a defender with a sabotage cost lower than that of the defenders in team 2. This conclusion surely sounds odd. Here we simply assume that $\mu ^j$ is a property of a player, but not a team.

The setup eliminates the strategic interaction between team owners. Independent of what the owner of team $-i$ does, the owner of team i would always choose the same values of effectiveness parameters for his team’s attackers and defenders.

In the baseline model with no sabotage, the free riding result makes the team owner’s problems trivial, since the team owner would concentrate on the reward for the most motivated member of his team.

Notice that if all winning prizes are multiplied by the same scalar, then the equilibrium values for $E_1$ and $E_2$ remain unchanged. Accordingly, for any quadruple of winning prizes, a scalar can be found above which the respective winning prizes lead to positive sabotage efforts for all players.

Note that

$$\begin{aligned} \begin{aligned} \left( \frac{f(x)}{f(x)+g(x)}\right) ' = \frac{f'(x)(f(x) + g(x)) - f(x)(f(x)'+g(x)')}{(f(x) + g(x))^2} = \frac{f'(x)g(x) - f(x) g'(x)}{(f(x) + g(x))^2}, \end{aligned} \end{aligned}$$

so that in order for this derivative to be equal to zero, it must be that

$$\begin{aligned} f'(x)g(x) = f(x) g'(x). \end{aligned}$$

We are referring to the football game or war interpretations here. If we consider the election interpretation, we would expect that $\mu _1^{aa} < \mu _1^{da}$, since player a in team 1 is now closer to player a in team 2 than player d in team 1 is.

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Title: Sabotage in team contests
Authors: Serhat Doğan
Kerim Keskin
Çağrı Sağlam
Publication date: 01-02-2019
Publisher: Springer US
Published in: Public Choice / Issue 3-4/2019
Print ISSN: 0048-5829
Electronic ISSN: 1573-7101
DOI: https://doi.org/10.1007/s11127-019-00643-1

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