In a game, players take actions according to the rewards they expect to gain from their behavior. Game theory is the science of optimal decision-making of independent and competing players in a strategic setting. For those wishing to find out more about game theory and relative historical grounds, reference should be made to Owen (1968) and Gambarelli and Owen (2004). Although game theory is deeply rooted in economic issues, there are several applications to sports strategy, including e.g. play calling in American Football (Jordan, Melouk, and Perry 2009) pitting in NASCAR racing (Deck, Deck, and Zhen 2014), or first serves in tennis (Walker and Wooders 2001).

Game theory also received a fair share of attention in soccer [see for instance the entertaining book by Palacios-Huerta (2014)]. Hiller (2015) applied the approach of Hernández-Lamoneda and Sánchez-Sánchez (2010) which is based on cooperative game theory to soccer, creating a player performance metric which captures the contribution of the player to the success of his team. Auer and Hiller (2015) computed this measure for the German Bundesliga teams in the season 2012–2013, and found that it correlates positively with classic player performance measures such as e.g. overall player scores published by prominent sports magazines.

A penalty kick can also be studied from a game theory perspective, when seen as a two-person zero-sum game (the goalkeeper loses whilst the penalty taker wins, or vice versa). Using data from penalty kicking in the Bundesliga, Dohmen and Sonnabend (2018) provide evidence that experienced players in high-stakes environments behave according to game theoretic predictions. They verify empirically that the main requirements for the existence of a Nash equilibrium in mixed strategies are met (i.e. the expected payoffs are equal across strategies and players choose their actions randomly). A game theoretic analysis of penalty kicks was also performed by Chiappori, Levitt, and Groseclose (2002), Coloma (2007), and Palacios-Huerta (2003).

Game theory has also been applied to rule design in soccer. For instance, Brams and Ismail (2018) show that the current rule for penalty shootouts, where a coin toss determines which team kicks first on all five penalty kicks, gives a substantial advantage to the first-kicking team, both in theory and in practice. They present the so-called catch-up rule to mitigate this bias.

When we focus on the use of game theory for strategical/tactical decisions to be taken by soccer coaches of two opposing teams, including the effect of interaction between their decisions, the literature includes just a few contributions.

Hirotsu and Wright (2006) study the interaction between both teams’ tactical changes before and during the game, which boil down to a change of formation imposed by the manager. They see the selection of a formation as one of the main tactics employed by managers to make their team play offensively or defensively. The authors model a soccer match as two-player zero-sum game, where one player’s loss is the other’s gain, in order to quantify the effect of decisions to change the team formation on the probability of winning the match. They demonstrated that such tactical decisions can indeed influence the probability of winning the game, using real data from the Japanese professional soccer league, and illustrating it on a number of match cases. The results show that the difference between the combinations of the best and worst tactics leads to a significant difference in terms of the probability of winning.

Hirotsu et al. (2009) extend the model introduced by Hirotsu and Wright (2006) to non-zero sum games, this time focusing on the expected number of points gained in a match. A non-zero sum game is natural given the 3-1-0 scoring rule, which distributes three points for a match with a winner (all points go to the winner), whereas a draw only yields two points shared among the teams. They demonstrate the effect of the point score system on tactical changes, as well as the effect of cooperation between managers which could make both teams benefit in terms of the expected number of league points gained in a match. Note that neither Hirotsu and Wright (2006), nor Hirotsu et al. (2009) deal with the case where the game is solved using mixed strategies, in which teams select their tactics according to probabilities.

The point system, and in particular the switch from the 2-1-0 to the 3-1-0 system, has been the topic of several studies, theoretical as well as empirical. Using a game theoretic approach, Brocas and Carrillo (2004) find that the 3-1-0 rule results in more defensive play in the first half and more attacking play in the last part of a match, as opposed to the 2-1-0 rule where the teams do not change their strategy. Haugen (2008) proves that, under some reasonable assumptions, in a match between two equal teams, no team will choose to alter its strategy in a more defensive direction after the switch to the 3-1-0 system. Moreover, certain parameter value (team differences) will result in more offensive play.

While the theoretical work on the scoring rule points more or less in the same direction, empirical studies offer more diverse results. Fernandez-Cantelli and Meeden (2002) performed an empirical study involving soccer leagues in 10 countries. They find little evidence to support that the switch to the 3-1-0 system led to more goals or fewer ties; they report a noteworthy decrease in ties only for Italy and Turkey. Dilger and Geyer (2009) do find evidence for a decrease in matches ending in a tie after the rule change for the German Bundesliga. This difference is significant, even when compared with matches from the German Cup (being a knock-out tournament, the German Cup was not influenced by the switch to the 3-point system). Then again, Aylott and Aylott (2007) show that the average number of goals per match (with the exception of the German Bundesliga), as well as the number of draws increased, since the introduction of the 3-1-0 rule. Dewenter and Emami Namini (2013) argue that when the reward for a win is increased, the opportunity costs of playing extremely offensively rise, and consequently, the home team chooses a more defensive playing style. Hence, an increase in the rewards for a win leads to the counterintuitive result of more defensive play if the home bias is sufficiently strong and if the levels of offensiveness of the teams are strategic substitutes. The authors find evidence for their theoretical hypotheses in the German Bundesliga: the 3-1-0 rule leads to a reduction in both the number of goals and the number of victories for home teams. The away teams, in contrast, choose a more offensive strategy (the total number of goals as well as the total number of victories have not been affected significantly).

In this section, we develop a game-theoretical model that, given estimates on the win probabilities for each combination of strategies adopted by the opponents, results in an strategic advice for each opponent. We assume that each team has two discrete strategic choices: an offensive (O) or defensive (D) approach.

We will now describe the input parameters, labelling the two teams A and B. We indicate α_DD and β_DD, respectively as the probabilities of a win for team A and team B if both adopt a defensive strategy (α_DD + β_DD ≤ 1); the probability of a draw will, therefore, correspond to 1 −α_DD − β_DD. In the same way, we define α_OO and β_OO (both teams with an offensive approach), α_DO and β_DO (A in defense and B in attack), α_OD and β_OD (A in attack and B in defense).

We define the expected payoff a_ij (b_ij) for team A (B) where team A follows strategy i and team B adopts strategy j (with i, j ∈ {O, D}). The payoffs are calculated as follows, based on the hypothesis that a win awards p points and a draw yields q points:

$$\begin{array}{ccccc}{a}_{ij}\hfill & =p{\alpha }_{ij}+q\left(1-{\alpha }_{ij}-{\beta }_{ij}\right)=q+\left(p-q\right){\alpha }_{ij}-q{\beta }_{ij}\hfill & & & \\ b_{ij}\hfill & =p{\beta }_{ij}+q\left(1-{\alpha }_{ij}-{\beta }_{ij}\right)=q+\left(p-q\right){\beta }_{ij}-q{\alpha }_{ij}\hfill & & & \end{array}$$

We are facing a normal-form variable-sum game. The competitive solution by John Nash (1950) requires the use of an algorithm of Linear Programming on the matrices [α_ij] and [β_ij] based on John von Neumann’s Minimax Theorem (1928). However, the specificity of the problem considered here allows closed solutions which we will present in this section for the reader’s convenience. Firstly, it is necessary to verify whether any team has a dominant strategy (i.e. a strategy that results in a higher expected payoff than the other strategy, regardless of the opponent’s strategy), in which case the solution can easily be found.

If there is no domination, the solution consists of in probability distributions ($q_{\text{D}}^{\text{A}}$, $q_{\text{O}}^{\text{A}}$) on the strategies of A and ($q_{\text{D}}^{\text{B}}$, $q_{\text{O}}^{\text{B}}$) on the strategies of B. Such assignments of probabilities to the offensive and defensive strategy correspond to mixed strategies, and can also be interpreted as different levels of adoption of both strategies.

Let’s define h = (a_OD − a_OO)/(a_DO + a_OD − a_DD − a_OO). The probability distribution for the strategy of team A is then as follows:

$$\begin{array}{ccccc}{q}_{\text{D}}^{\text{A}}\hfill & =h\hspace{1em}\text{if \hspace{1em}0}\le h\le \text{1, and}\hfill & & & \\ q_{\text{D}}^{\text{A}}\hfill & =\{\begin{array}{c}\text{0 if min}\hspace{1em}(a_{\text{DD}},a_{\text{DO}})<\text{min}\hspace{1em}(a_{\text{OD}},a_{\text{OO}}),\hfill \\ \text{1 if min}\hspace{1em}(a_{\text{DD}},a_{\text{DO}})>\text{min}\hspace{1em}(a_{\text{OD}},a_{\text{OO}})\hspace{1em}\text{elsewhere,}\hfill \\ \text{any value between 0 and 1, otherwise.}\hfill \end{array}\hfill & & & \end{array}$$

In all cases, $q_{\text{O}}^{\text{A}}=1-q_{\text{D}}^{\text{A}}$.

Similarly, for team B, with k = (b_OD − b_OO)/(b_DO + b_OD − b_DD − b_OO), the distribution is given by:

$$\begin{array}{ccccc}{q}_{\text{D}}^{\text{B}}\hfill & =k\hspace{1em}\text{if}\hspace{1em}0\le k\le 1,\text{and}\hfill & & & \\ q_{\text{D}}^{\text{B}}\hfill & =\{\begin{array}{c}\text{0 if min}\hspace{1em}(b_{\text{DD}},b_{\text{OD}})<\text{min}\hspace{1em}(b_{\text{OD}},b_{\text{OO}})\hfill \\ \text{1 if min}\hspace{1em}(b_{\text{DD}},b_{\text{OD}})>\text{min}\hspace{1em}(b_{\text{OD}},b_{\text{OO}})\hspace{1em}\text{elsewhere,}\hfill \\ \text{any value between 0 and 1, otherwise.}\hfill \end{array}\hfill & & & \end{array}$$

In all cases, $q_{\text{O}}^{\text{B}}=1-q_{\text{D}}^{\text{B}}$.

The expected payoffs of the two teams are:

$$\begin{array}{ccccc}{p}^{\text{A}}\hfill & =q_{\text{D}}^{\text{A}}\mathit{\hspace{1em}}{q}_{\text{D}}^{\text{B}}\mathit{\hspace{1em}}{a}_{\text{DD}}+q_{\text{D}}^{\text{A}}\mathit{\hspace{1em}}{q}_{\text{O}}^{\text{B}}\mathit{\hspace{1em}}{a}_{\text{DO}}+q_{\text{O}}^{\text{A}}\mathit{\hspace{1em}}{q}_{\text{D}}^{\text{B}}\mathit{\hspace{1em}}{a}_{\text{OD}}+q_{\text{O}}^{\text{A}}\mathit{\hspace{1em}}{q}_{\text{O}}^{\text{B}}{a}_{\text{OO}}\hfill & & & \\ p^{\text{B}}\hfill & =q_{\text{D}}^{\text{A}}\mathit{\hspace{1em}}{q}_{\text{D}}^{\text{B}}\mathit{\hspace{1em}}{b}_{\text{DD}}+q_{\text{D}}^{\text{A}}\mathit{\hspace{1em}}{q}_{\text{O}}^{\text{B}}\mathit{\hspace{1em}}{b}_{\text{DO}}+q_{\text{O}}^{\text{A}}\mathit{\hspace{1em}}{q}_{\text{D}}^{\text{B}}\mathit{\hspace{1em}}{b}_{\text{OD}}+q_{\text{O}}^{\text{A}}\mathit{\hspace{1em}}{q}_{\text{O}}^{\text{B}}{b}_{\text{OO}}\hfill & & & \end{array}$$

Consider for example a setting where the expected payoffs for team A and B are as presented in Table 1.

This means that for team A, h = (1.5 − 1.1)/(1.8 + 1.5 − 1.0 − 1.1) = 1/3. Hence $q_{\text{D}}^{\text{A}}=1/3$ and $q_{\text{O}}^{\text{A}}=2/3$, or in other words, in the long run, team A should attack in 2/3 of such cases and defend in 1/3 of such cases. For team B, k = (0.7 − 1.2)/(1.0 + 0.7 − 2.5 − 1.2) = 1/4, such that team B should attack with a probability of 75% ($q_{\text{O}}^{\text{B}}=3/4$) and choose a defensive approach with a probability of 25% ($q_{\text{D}}^{\text{B}}=1/4$).

An important remark concerns the incompleteness of the information on which the table of win probability estimates (i.e. α_ij and β_ij) are based. In fact, every coach has at his disposal inside information regarding the psychophysical condition of his players, but may not have the same information about the opposing team. Consequently, his win probability estimates could differ from that of the opponent’s coach. Except in the cases when the two tables of probabilities lead to different results in terms of dominance, such differences do not influence the strategy chosen by the coach. In fact, only in such situations can the table of the opponent’s coach cause problems. The possibility to repeat the calculation given by the algorithm that we will present in the next section, helps to overcome this problem.

In this section, we present an algorithm for the automatic calculation of strategy advice (i.e. the solution to the model presented in section 3), with a test of the robustness of this advice with respect to uncertainty involved in estimating the win probabilities for each combination of strategies.

The input of the algorithm consists of the four pairs of probabilities of a win of team A and team B: α_ij and β_ij, with i, j ∈ {O, D}. While reasonably accurate predictions for the outcome (win, draw, loss) of a match can be derived from the odds published by bookmakers, this is much less the case for α_ij and β_ij, as they require insight in the impact of the strategies on the match outcome. As this input is chosen on the basis of a subjective evaluation of the available data, it is characterized by uncertainty. Obviously, wrong initial considerations can lead to poor strategic advice, and forecasting is strongly based on a priori achievable assessment. In order to give an indication of the robustness of the resulting strategy advise, we vary the starting conditions in a suitable neighborhood (necessarily a discrete set of points). We use the mean of the results in the neighborhood as a tractable way to summarize the strategies in the neighborhood. By comparing this mean with the initial result for increasing an increase size of the neighborhood (corresponding to an increase degree of uncertainty), we can see how much the resulting strategy can change due to increasing variations of the initial estimate of the win probabilities.

The neighborhood is generated as follows. We specify a value r indicating how far we deviate from the original estimates for α_ij and β_ij, For each pair of values α_ij and β_ij, for some choice of i, j ∈ {O, D}, we are interested in 9 discrete points, centered around the values α_ij and β_ij, given in Table 2. As there are 4 pairs of (α_ij, β_ij) given the values that i and j can take, this results in a neighborhood with 9^4 (=6561) points. The algorithm computes the solution for each of these points. A solution consists of the probability distribution of the strategies ($q_{\text{D}}^{\text{A}}$, $q_{\text{O}}^{\text{A}}$, $q_{\text{D}}^{\text{B}}$, $q_{\text{O}}^{\text{B}}$) as well as the resulting payoffs (p^A, p^B) for both teams, and is computed in the way described in section 3. The algorithm aggregates the information obtained in the neighborhood by outputting the mean of the solution parameters over all points in that neighborhood.

A representation of the algorithm in pseudo code is given in Figure 1. The algorithm uses the data structure storage to store all solutions obtained in a neighborhood determined by the value r, where r takes on values from 0.01 to radius, which establishes the maximum radius of the neighborhood to be searched. This allows us to evaluate how the solution varies as the win probabilities α_ij and β_ij move further away from our original estimate. Note that in case radius = 0, only the solution for the given values of α_ij and β_ij is calculated.

Figure 1:

Algorithm pseudo code.

Note that this robustness analysis is not intended to account for future match events match (e.g. a goal scored or a red card). Indeed, when events occur during the match, the analysis should be redone based on updated values of α_ij and β_ij. Nevertheless, these new win probabilities will also be subject to uncertainty. Hence, redoing the robustness analysis after each such event is advised. A complete implementation of the algorithm in Python v3 is provided in the Supplementary material.

Arena Pernambuco, Recife, June 20th, 2014: it is 11 o’clock in the morning. In two hours, the Chilean referee Enrique Osses will blow his whistle for the kick-off of the match between Italy and Costa Rica for the second round of Group D in the World Cup in Brazil 2014. Coaches Cesare Prandelli (Italy) and Jorge Luis Pinto (Costa Rica) must put aside all reservations and reveal their formations and initial tactics to their players. Both teams won on the first day of the tournament and now the time has come to choose whether to play offensively in an attempt to qualify immediately for the next round, or draw up a formation with more defensive tactics, thus reducing the risk of a defeat.

5.6 The impact of the point system

Our approach can also be used to assess the impact of various point system on the willingness of teams to attack. We illustrate this again with the World Cup game between Italy and Costa Rica. Figure 2 presents the willingness of Italy (left) and Costa Rica (right) to opt for attacking play at the start of the game. We compare the current rule (3 points for a win, 1 point for a draw), with the previous rule, which awarded 2 points for a win, and a rule that awards 4 points for a win. We also included a rule that only rewards a win (0 points for a draw or a loss). For the case with r = 0.00, we see that the FIFA rule change awarding 3 points for a win instead of 2 has had a huge impact on the strategy of Italy: they move from 0% attacking play under the old rule to 100% attacking play under the new rule. For Costa Rica, none of the rules can persuade them to abandon their defensive stance. If we include more uncertainty on our parameter estimates (i.e. larger values for r), we see that the difference between both rules is less pronounced for Italy, and that it remains negligible for Costa Rica.

Figure 2:

%O for Italy (left) and Costa Rica (right) for various point systems (win – draw – loss), for r ranging from 0.00 to 0.10 on the x-axis.

Figure 2 also show that in this case, rules that further favor a win compared to a draw will indeed result in more attacking play. However, even in the most extreme case, where only a win counts, there will be situations where a team’s best strategy is to defend.

The method presented here provides insight in the choice of tactics to be adopted at the start of the match, and can be used as a tool to support coaches. As the input of our algorithm is based on a subjective evaluation of the available data (e.g. bookmaker’s odds), it is subject to uncertainty. To some extent, our algorithm can deal with this uncertainty, by computing the optimal strategies for a wide range of probabilities around the original estimates. On top of that, during the match, the situation on the pitch (e.g. a goal scored) may require appropriate changes to be made. At this point, our method should be rerun based on updated win probabilities, which take into account the most recent match events. As our computer program runs very quickly, our method can be used during an ongoing game to give real-time strategic advice.

Although the model has its limitations (as is the case for any theoretical model), we believe that game theory provides a highly suitable framework for analyzing this type of strategic considerations between two opponents in sports. Furthermore, while in this text the focus is on soccer, our approach can be applied to any sport with a win(-draw)-loss structure and a scoring rule, where two opponents faces a choice between two (or more) strategies.

Our method also allows to obtain some insight in the effect of various point systems on the strategic considerations for a particular match. Obviously, we are not making any claims in favor or against the switch to the 3-1-0 rule based on a single case study. However, our results show that even in settings where anything other than a win does not yield any points, for some teams a defensive stance is still the best way to achieve a win. In this sense, no point system will be able to guarantee offensive play.

Finally, while our case study hints the possibilities of our game-theoretical approach, a large-scale empirical study of the success of coaches following this approach is left for future research. This is mostly due to the lacking of a straightforward tool to quantify a team’s offensive or defensive play, which would in our opinion require a thorough analysis of the formation, the players, and their behavior on the pitch. Given a way to quantify offensive/defensive play, one could evaluate the success of coaches that follow our strategic advice, and obtain a better consideration of the merits of our approach.