Appendix 1: Theoretical model
We introduce the possibility of self-restriction into the model of strategic effort choices in round-robin tournaments with three heterogeneous players. The players may restrict their resources in the first match to increase the probability of having unrestricted resources in their second match. Whether a player uses this option in equilibrium depends on the extent of heterogeneity between that player and the opponents in the first and second match.
Assumptions
The structure of the tournament
A round-robin tournament, also referred to as an all-play-all tournament, matches each participant with each other participant in a pairwise contest, ranks them according to the number of matches won, and awards prizes according to this ranking. For simplicity, we focus on round-robin tournaments with three risk-neutral players and consider an exogenous sequence in which player 1 is matched with player 2 in the first match, player 1 is matched with player 3 in the second match, and player 2 is matched with player 3 in the third match (Krumer et al., 2017; Sahm, 2019).Footnote 26
If a player earns two victories, he is ranked first, and the player with one victory is ranked second; if there is a tie because each player has won one match, the ranks are assigned randomly with equal probabilities of 1/3 for each player and rank. The player ranked first (last) receives the first (last) prize, the value of which is identical for all players and normalised to 1 (0). The value of the second prize is assumed to be identical for all players as well and fixed at 1/2, or half of the first prize. This prize structure (is the only one that) ensures a fair tournament, in the sense that all potential differences in players’ equilibrium winning probabilities and expected payoffs result from differences in their abilities, not from their position in the sequence of matches (Laica et al. 2017).
The structure of the resulting sequential game with its \(2^3=8\) potential courses is depicted in Fig. 3. The seven nodes \(k \in \{A,\ldots ,F\}\) represent all combinations for which the ranking of the tournament has not been fully determined when the respective match starts.
Tournament matches
Each match of the tournament is organised as a lottery contest between two potentially heterogeneous players, A and B, with linear costs of effort.Footnote 27 Specifically, player A’s probability of winning match \(k \in \{A,\ldots ,F\}\) is
$$\begin{aligned} p^k_A = \left\{ \begin{array}{cc} 1/2 &{} \text {if }\quad x^k_A=x^k_B=0,\\ \frac{\theta ^k_Ax^k_A}{\theta ^k_Ax^k_A + \theta ^k_Bx^k_B} &{} \text {else,} \end{array} \right. , \end{aligned}$$
where \(\theta ^k_i\) describes ability, and \(x^k_i\) denotes the effort of player \(i \in \{A,B\}\) in match k (Leininger 1993; Baik 1994).
Player A chooses \(x^k_A \ge 0\) to maximise the expected payoff
$$\begin{aligned} E^k_A=p^k_A(w^k_A-x^k_A) + \left( 1-p^k_A\right) (\ell ^k_A-x^k_A), \end{aligned}$$
(3)
where \(w^k_i\) denotes player i’s expected continuation payoff from winning match k, and \(\ell ^k_i\) denotes the expected continuation payoff from losing match k, with \(w^k_i \ge \ell ^k_i \ge 0\) for \(i \in \{A,B\}\). For \(w^k_A=\ell ^k_A\), the optimal choice is \(x^k_A=0\) for any \(x^k_B \ge 0\). If \(x^k_A=0\) and \(w^k_B > \ell ^k_B\), player B has no best response unless there is a smallest monetary unit \(\varepsilon >0\), in which case the best response is \(x^k_B=\varepsilon\). As \(\varepsilon \rightarrow 0\), in the limit, \(x^k_B \rightarrow 0\), and \(p^k_B \rightarrow 1\). Otherwise, there is a unique Nash equilibrium in pure strategies (e.g., Cornes & Hartley, 2005). The equilibrium effort levels can be derived from the following necessary conditions:
$$\begin{aligned} \frac{\partial E^k_i}{\partial x^k_i}=\frac{\theta ^k_i\theta ^k_jx^k_j}{(\theta ^k_ix^k_i + \theta ^k_jx^k_j)^2}(w^k_i-\ell ^k_i)-1=0, \end{aligned}$$
yielding
$$\begin{aligned} x^k_i = \frac{\theta ^k_i\theta ^k_j(w^k_i-\ell ^k_i)^2(w^k_j-\ell ^k_j)}{[\theta ^k_i(w^k_i-\ell ^k_i)+\theta ^k_j(w^k_j-\ell ^k_j)]^2} \end{aligned}$$
(4)
for \(i,j \in \{A,B\}\) with \(i \not = j\). The resulting equilibrium winning probabilities equal
$$\begin{aligned} p^k_i = \frac{\theta ^k_i(w^k_i-\ell ^k_i)}{\theta ^k_i(w^k_i-\ell ^k_i)+\theta ^k_j(w^k_j-\ell ^k_j)}. \end{aligned}$$
(5)
Inserting Eqs. (4) and (5) into Eq. (3) yields the expected equilibrium payoffs:
$$\begin{aligned} E^k_i = \ell ^k_i + \frac{(\theta ^k_i)^2(w^k_i-\ell ^k_i)^3}{[\theta ^k_i(w^k_i-\ell ^k_i)+\theta ^k_j(w^k_j-\ell ^k_j)]^2}. \end{aligned}$$
(6)
The possibility of self-restriction
In practice, most tournaments offer the possibility that players restrict their current resources to increase the probability of having unrestricted resources available in future matches. In team sports, such self-restriction might involve giving particular important players a break, so they can rest for a future contest. As a new element, we now introduce this self-restriction opportunity into the above tournament model. To keep the analysis tractable, we make the following assumptions.
Players differ in their basic abilities.Footnote 28 Let \(\theta _i > 0\) denote the basic ability of player \(i \in \{1,2,3\}\). Before the tournament starts, players 1 and 2 make a binary choice to restrict themselves (\(\text {R}\)) or not (\(\text {NR}\)) in their first match (match 1). If player \(i \in \{1,2\}\) chooses \(\text {R}\), his ability in the first match will be restricted to \(\theta _i^F = r \theta _i\) with some \(r \in (0,1)\), but he will have unrestricted ability \(\theta _i^k=\theta _i\) in the second match (\(k \in \{D,E\}\) if \(i=1\), and \(k \in \{A,B,C,C'\}\) if \(i=2\)). In contrast, if player \(i \in \{1,2\}\) chooses \(\text {NR}\), his ability in the first match will be unrestricted, \(\theta _i^F = \theta _i\), but he faces some positive probability \(\pi \in (0,1)\) of having to compete with restricted ability in the second match. To capture the related uncertainty in reduced form, we assume that the player’s (expected) ability in the second match equals \(\theta _i^k = q\theta _i\), with \(q=1-\pi (1-r) \in (r,1)\). For simplicity, we assume that player 3 has unrestricted ability in both matchesFootnote 29 which is normalised such that \(\theta _3^k=\theta _3=1\) for all \(k \in \{A,\ldots ,E\}\).Footnote 30
This round-robin tournament with three players and an opportunity of self-restriction (for players 1 and 2) constitutes a sequential game \(\Gamma\) with four stages: In Stage 0, players 1 and 2 decide simultaneously whether to self-restrict or not in match 1; their choices are observed by all players. In Stage 1, players 1 and 2 decide simultaneously about their effort in match 1, and the outcome of match 1 is observed by all players. In Stage 2, players 1 and 3 decide simultaneously about their effort in match 2, and the outcome of match 2 is observed by all players. Finally, in Stage 3, players 2 and 3 decide simultaneously about their effort in match 3.
Results
For each feasible combination of exogenous parameters \((\theta _1,\theta _2,r,q)\), the game \(\Gamma\) can be solved by backward induction for its subgame perfect equilibrium (SPE) through repeated use of Eqs. (4)–(6). Appendix 2 illustrates this procedure for the example of \((\theta _1,\theta _2,r,q)=(1,1,1/2,3/4)\).
We are particularly interested in identifying the conditions in which players 1 or 2 choose self-restriction in the first match as part of an equilibrium strategy. Because the comparative statics are analytically not tractable, we study the effect of variations of the exogenous parameters \((\theta _1,\theta _2,r,q)\) on equilibrium behaviour numerically. We use Microsoft Excel to compute the SPE of \(\Gamma\) on a grid with more than 200,000 grid points, increasing \(\theta _i\) from 0.1 to 10 in steps of \(1\%\) (and from 0.001 to 1000 in steps of \(10\%\), respectively) for \(i \in \{1,2\}\), together with r and q from 0.5 to 0.9 in steps of 0.1, subject to \(r<q\).Footnote 31 The calculations demonstrate that, depending on the parameters, all kinds of equilibrium behaviours by players 1 and 2 may arise in the first stage.
Proposition 1
For each of the following combinations of first-stage behaviour by players 1 and 2, there are parameters \((\theta _1,\theta _2,r,q)\), such that the respective combination is part of the players’ strategies in the SPE of game \(\Gamma\):
-
(a)
No player chooses self-restriction \(\text {R}\).
-
(b)
Only the weaker player chooses self-restriction \(\text {R}\).
-
(c)
Both players choose self-restriction \(\text {R}\).
-
(d)
Only the stronger player chooses self-restriction \(\text {R}\).
-
(e)
Both players choose self-restriction \(\text {R}\) with positive probability (equilibrium in mixed strategies).
Figure 4 illustrates Proposition 1. The players’ behaviour only depends on their relative basic abilities, such that the diagrams are symmetric around the \({45}^{\circ }\)-line on which \(\theta _1=\theta _2\). We thus can focus on cases in which player 1 is at least as able as player 2, with grid points on and below the \({45}^{\circ }\)-line. Moreover, the left panel of Fig. 4 encompasses a range of parameters in which the maximum difference in relative abilities of players 1 and 2 is 1:1,000,000, whereas, the right panel zooms in on the practically more relevant range in which this maximum difference is 1:100.
Around the \({45}^{\circ }\)-line, players 1 and 2 have similar basic abilities, and they never self-restrict. Self-restriction by one or both players occurs only if the difference between their abilities \(\theta _1\) and \(\theta _2\) is sufficiently large.Footnote 32 Therefore,
Hypothesis 1
The larger the difference in basic abilities
\(\theta _1\)
and
\(\theta _2\)
in the current match, the more likely a player chooses self-restriction.
The underlying intuition is that players cannot afford to restrict themselves in their first match if the match will be close. Only if they are sufficiently sure they will win or lose, due to a large gap in abilities, will players conserve their strength for their second match.
The scale on both axes of the diagrams in Fig. 4 also is exponential, such that the ratio of the basic abilities of players 1 and 2 is constant along any parallel to the \({45}^{\circ }\)-line. Moving from the lower left to the upper right on any such parallel, the ability of player \(i \in \{1,2\}\) increases relative to the ability of player 3. Because the region below the \({45}^{\circ }\)-line in which player \(i \in \{1,2\}\) self-restricts is convex in the relevant range (see the right panel of Fig. 4), there is some ratio \(\alpha _i\) of the basic abilities of players 1 and 2 for which the corresponding parallel \(\theta _1=\alpha _i\theta _2\) is tangential to the region in which player \(i \in \{1,2\}\) self-restricts. In this sense, the relative ability \(\delta _i=\theta _i/\theta _3\) that characterises the tangent point is that for which player i’s probability of self-restriction is maximal (see Fig. 5). These observations suggest:
Hypothesis 2
The greater the difference in (weighted) basic abilities
\(\theta _i\)
and
\(\delta _i\theta _3\)
in the future match, the less likely player
\(i \in \{1,2\}\)
chooses self-restriction.
Here, the intuition is that it will not pay off for players in the first match to conserve their strength for their second match if they expect the first match to be closer than the second match or, more formally, if the (weighted) heterogeneity in the second match is too pronounced.Footnote 33
Finally, if we vary the (expected) levels of restriction r and q, the number of grid points \((\theta _1,\theta _2)\) for which the players choose self-restriction \(\text {R}\) in the SPE of game \(\Gamma\) increases in r and decreases in q. Figure 6 illustrates this finding. Again, the intuition is straightforward: An increase in r means that the restriction in the players’ first match is less severe, and therefore, choosing \(\text {R}\) is less expensive. An increase in q instead means that the possible restriction in the players’ second match is less severe or less likely, so the incentive to conserve their full strength by choosing \(\text {R}\) is less pronounced. Moreover, the variations of r and q in Fig. 6 make clear that our hypotheses are stable across diverse specifications of the model.
Appendix 2: Backward Induction of Game \(\Gamma\)
This appendix illustrates how to solve game \(\Gamma\) (see Fig. 3) by backward induction, considering the parameters \((\theta _1,\theta _2,r,q)=(1,1,1/2,3/4)\).
Case \((\text {R},\text {R})\)
Suppose that both players 1 and 2 have chosen \(\text {R}\) in stage 1 and thus restrict themselves in the first match.
4th stage: Player 2 vs player 3
In the third match, player 2 is unrestricted, and thus \(\theta _2^k=\theta _3^k=1\) in all nodes \(k \in \{A,B,C,C'\}\).
In node A, player 2 wins the first match, and player 3 wins the second match. Thus \(w_2^A=w_3^A=1\), \(\ell _2^A=\ell _3^A=1/2\), which yields
$$\begin{aligned} x_2^A = x_3^A = \frac{(1-\frac{1}{2})^{2} \cdot (1-\frac{1}{2})}{[(1-\frac{1}{2})+(1-\frac{1}{2})]^2} = \frac{1}{8}, \end{aligned}$$
\(p_2^A = p_3^A = \frac{1}{2}\), and \(E_2^A = E_3^A = \frac{1}{2} + \frac{1}{8} = \frac{5}{8}\) by Eqs. (4)–(6).
In node B, player 2 wins the first match, and player 1 wins the second match. Thus \(w_2^B=1\), \(w_3^B=1/2\), \(\ell _2^B=1/2\) and \(\ell _3^B=0\), which yields \(x_2^B=x_3^B=1/8\), \(p_2^B=p_3^B=1/2\), \(E_2^B = 5/8\), and \(E_3^B = 1/8\).
In node C, player 1 wins the first match, and player 3 wins the second match. Thus \(w_2^C=1/2\), \(w_3^C=1\), \(\ell _2^C=0\), and \(\ell _3^C=1/2\), which yields \(x_2^C=x_3^C=1/8\), \(p_2^C=p_3^C=1/2\), \(E_2^C = 1/8\), and \(E_3^C = 5/8\).
In node \(C'\), player 1 wins the first and the second matches. Thus \(w_2^{C'}=w_3^{C'}=1/2\), and \(\ell _2^{C'}=\ell _3^{C'}=0\), which yields \(x_2^{C'}=x_3^{C'}=1/8\), \(p_2^{C'}=p_3^{C'}=1/2\), and \(E_2^{C'}=E_3^{C'} = 1/8\).
3rd stage: Player 1 vs player 3
In the second match, player 1 is unrestricted, and thus \(\theta _1^k=\theta _3^k=1\) in both nodes \(k \in \{D,E\}\).
In node D, player 2 wins the first match. Thus, \(w_1^D=\frac{1}{2}p_2^B+\frac{1}{2}p_3^B=\frac{1}{2}\), \(w_3^D=E_3^A=\frac{5}{8}\), \(\ell _1^D=0\), and \(\ell _3^D=E_3^B=\frac{1}{8}\), which yields \(x_1^D=x_3^D=1/8\), \(p_1^D=p_3^D=1/2\), \(E_1^D = 1/8\), and \(E_3^D = 1/4\).
In node E, player 1 wins the first match. Thus, \(w_1^E=1\), \(w_3^E=E_3^C = \frac{5}{8}\), \(\ell _1^E = \frac{1}{2}p_2^C+\frac{1}{2}p_3^C=\frac{1}{2}\), and \(\ell _3^E=E_3^{C'} = \frac{1}{8}\), which yields \(x_1^E=x_3^E=1/8\), \(p_1^E=p_3^E=1/2\), \(E_1^E = 5/8\), and \(E_3^E = 1/4\).
2nd stage: Player 1 vs player 2
In the first match, players 1 and 2 are restricted, and thus \(\theta _1^F=\theta _3^F=\frac{1}{2}\) in node F. Moreover \(w_1^F=E_1^E = \frac{5}{8}\), \(w_2^F=p_1^DE_2^B+p_3^DE_2^A = \frac{5}{8}\), \(\ell _1^F=E_1^D = \frac{1}{8}\), and \(\ell _2^F=p_1^{E}E_2^{C'}+p_3^EE_2^C = \frac{1}{8}\), which yields \(x_1^F=x_2^F=1/8\), \(p_1^F=p_2^F=1/2\), \(E_1^F = 1/4\), and \(E_2^F = 1/4\). Moreover, player 3’s expected payoff equals \(E_3^F = p_1^FE_3^E+p_2^FE_3^D = 1/4\).
Cases \((\text {NR},\text {R})\), \((\text {R},\text {NR})\), and \((\text {NR},\text {NR})\)
Using analogous procedures, we can calculate the players’ expected payoffs in cases in which only one or none of them self-restricts in the first match.
1st stage: Decision on self-restriction
The results for the expected payoffs of players 1 and 2 are in Table 6 (rounded to two decimal places). For the parameters under consideration, \(\text {NR}\) is a dominant strategy for both players, and thus, \((\text {NR},\text {NR})\) is the only Nash equilibrium in the first stage of game \(\Gamma\).
Table 6 Game matrix of stage 1 Appendix 3: Additional tables
See Tables 7, 8, 9, 10, 11 and 12.
Table 7 Self-restriction and the starting line-up decision—OLS regressions Table 8 Descriptive statistics—starting11 (betting odds) Table 9 Self-restriction and the starting line-up decision—logit regressions (betting odds) Table 10 Descriptive statistics–yellow card (betting odds) Table 11 The effect of heterogeneity on yellow card suspensions—OLS regressions Table 12 The effect of heterogeneity on yellow card suspensions–logit regression (betting odds) Table 13 Self-restriction and the substitution of yellowplayers–logit regressions