European Journal of Operational Research 205 (2010) 661–669

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European Journal of Operational Research

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Decision Support

Optimizing strategic behaviour in a dynamic setting in professional team sports

Stephen Dobson a,*, John Goddard b

a Division of Economics, Nottingham Trent University, Burton Street, Nottingham NG1 4BU, United Kingdomb Bangor Business School, Bangor University, Gwynedd LL57 2DG, United Kingdom

a r t i c l e i n f o

Article history:Received 10 February 2009Accepted 18 January 2010Available online 22 January 2010

Keywords:EconomicsGame theorySport (Football)

0377-2217/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.ejor.2010.01.024

* Corresponding author. Tel.: +44 115 848 2109.E-mail addresses: stephen.dobson2@ntu.ac.uk (S.

ac.uk (J. Goddard).1 The 90-min duration is notional, because there a

play for various reasons: play is interrupted when the bof the pitch, when a foul is committed or a player is pewhen a goal is scored, and when a player requires treaAlthough a few minutes of stoppage time are usually adperiod of play, the time added is invariably less thanstoppages.

a b s t r a c t

This article develops a dynamic game-theoretic model of optimizing strategic behaviour by football (soc-cer) teams. Teams choose between defensive and attacking formations and between a non-violent and aviolent playing style, and can vary these choices continuously throughout each match. Starting from theend of the match and working backwards, the teams’ optimal strategies conditional on the current stateof the match are determined by solving a series of two-player non-cooperative subgames. Numerical sim-ulations are used to explore the sensitivity of strategic behaviour to variations in the structural param-eters. The analysis demonstrates that the strategic behaviour of football teams can be rationalized inaccordance with game-theoretic principles of optimizing strategic behaviour by agents when payoffsare uncertain and interdependent.

� 2010 Elsevier B.V. All rights reserved.

2 The payoff structure, however, is other than zero-sum, for two reasons. First, if thescores at the end of the match are unequal, three league points are awarded to the

1. Introduction

It is widely accepted among researchers in the fields of opera-tions research, management science and economics that profes-sional sport offers a fruitful arena for formulating hypothesesconcerning strategic behaviour and risk taking (Chiappori et al.,2002; Palacios-Huerta, 2003; Annis, 2006; Hirotsu and Wright,2006; Swartz et al., 2006; de Dios Tena and Forrest, 2007; Bekkerand Lotz, 2009; Scarf et al., 2009; Wright, 2009; Frick et al.,2010). In many sports, only two teams or players are involved ineach contest. The contest takes place within a specific and clearlydefined time-frame. Strategies and payoffs are relatively simpleto identify. Outcomes matter because large sums of money are atstake for the winners and losers. Large data sets are readily avail-able and new data are being continually generated.

The sport of football (soccer) is dynamic and strategic, and soprovides an ideal setting for analysing optimizing strategic behav-iour. In every match, the two teams are pitched into direct opposi-tion for a period of play of 90 min’ notional duration.1 Each teamstarts the match with one goalkeeper and 10 outfield players. Teammanagers or coaches are at liberty to deploy their outfield players in

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Dobson), j.goddard@bangor.

re frequent discontinuities inall travels outside the confinesnalized under the offside law,tment on the pitch for injury.ded at the end of each 45-minthe time lost through routine

any formation of their choosing, and to adjust their team’s formationand style of play at any stage of the match. The immediate objectivesof the two teams throughout the match are symmetric: each teamattempts to score goals and prevent its opponent from scoring.2

Accordingly there is a high level of interdependence: both teams’strategies have implications for both teams’ probabilities of scoringand conceding goals.

The dynamic game-theoretic model of optimizing strategicbehaviour that is developed in this paper has two important ante-cedents in the literature. First, in an unpublished working paper,Palomino et al. (2000) (henceforth, PRR) introduce a dynamicgame-theoretic model, in which two opposing football teamschoose continuously between a defensive and an attacking forma-tion. PRR show that teams tend to play more defensively whenthey are leading than they play when they are either level or trail-ing, especially during the later stages of matches. Second, a similarmodel has been used by Banerjee et al. (2007) (henceforth, BSW) toexamine the impact on team strategies in the National Hockey

winning team and none to the losing team. If the scores are level, one point isawarded to each team. In our data set, 27.4% of matches were drawn having finishedlevel after 90 min’ play (including stoppage time). In English league football, matchdurations are never extended beyond 90 min (plus stoppage time), and penaltyshootouts are never used to settle drawn matches. Second, a player who commitsserious foul play may be dismissed while the match is in progress. No replacement ispermitted, and play resumes with the dismissed player’s team at a numericaldisadvantage in players. A dismissal also results in the player’s suspension from up tothree future matches; in this case, replacements are permitted. However, asuspension still imposes a cost upon the team concerned, which is obliged to selectfrom a smaller or weaker pool of players. The corresponding gain accrues to theteam’s future opponents and not its current opponent.

662 S. Dobson, J. Goddard / European Journal of Operational Research 205 (2010) 661–669

League of a recent change in the league points scoring systemaffecting matches that were tied at the end of regular time.

The PRR and BSW models contain several simplifying assump-tions that are either controversial or counterfactual, or which limitthe generality of their analysis. PRR rely upon an unprovenassumption that if one team adopts an attacking strategy and theother adopts a defensive strategy, the goal scoring rate of theattacking team increases by more than that of the defending team.BSW relax this assumption, by introducing a parameter allowingfor comparative advantage in either attack or defense. Both PRRand BSW consider the special case in which the attacking anddefensive strengths of the two teams are the same; and most ofPRR’s results are based on a counterfactual assumption that thepayoff structure at the end of the match is zero-sum. PRR are un-able to account for what appears to be a strong empirical regular-ity, that after controlling for team quality and duration effects, thescoring rates of both teams when the scores are level tend to belower than those of teams that are trailing, and not significantlydifferent from those of teams that are leading.

Both PRR and BSW restrict the teams’ strategic choices to twooptions: attack and defense. Neither considers a second strategicdimension: the choice between a violent and non-violent style ofplay. Teams that play violently commit foul play and other actsof violence or aggression in an attempt to disrupt or sabotage theopposing team.3 By doing so, however, they incur an increased riskof being subject to disciplinary sanction, which may involve the dis-missal of a player from the field of play for the remainder of thematch. By allowing the teams to exercise choices concerning the le-vel of violence or foul play, the model that is developed in this paperincorporates the link between strategic behaviour and disciplinarysanction.4

In the model that is developed below, it is assumed that theavailable strategic choices are discrete: teams choose between‘defensive’ and ‘attacking’ formations, and between ‘non-violent’and ‘violent’ styles of play. These strategic choices influence theprobabilities of scoring and conceding goals at the current stageof the match, and the probabilities that players are dismissed.Using numerical simulations based on a dynamic game-theoreticmodel incorporating a broad spectrum of structural assumptions,it is shown that the optimal strategic choices at each stage of thematch are dependent on the current difference in scores and onthe amount of time that has elapsed since the start of the match.The numerical simulations provide a simple and flexible apparatus

3 In English football, anecdotal evidence that violence provokes retaliation in kindis abundant. In the 1970s Chelsea and Leeds had a long-running rivalry, starting witha bad-tempered FA Cup final replay in 1970. At its centre were Norman Hunter andPeter Osgood. ‘There was a tremendous rivalry between Ossie and myself,’ saidHunter. ‘You started getting into him a bit and to be fair he’d come back at you. Hewasn’t afraid at all.’ In 2001 Roy Keane (Manchester United) carried out his ownpunishment on Alf Inge Haaland (Manchester City), having been taunted by himearlier. Keane said of his infamous tackle in his autobiography, ‘I’d waited longenough. I hit him hard ... And don’t ever stand over me again sneering about fakeinjuries.’ In 2001 Graeme Le Saux (Chelsea) had to be physically restrained fromattacking Mauricio Taricco (Tottenham) after the final whistle. Le Saux alleged he hadbeen punched in the face. In March 2002 Le Saux was sent off for a bad tackle onTaricco. Three days later the two teams played again, and Taricco was sent off for afoul on Le Saux.

4 In previous empirical studies of the incidence of foul play and disciplinarysanction in football, Garicano and Palacios-Huerta (2005) find that following a changein the league points scoring system in Spain’s La Liga from 2–1–0 to 3–1–0,introduced in 1995, indicators of both attacking intensity and foul play increasedoverall. However, after taking the lead, teams tended to play more defensively in anattempt to prevent their opponents from equalizing. Dawson et al. (2007) find thatthe incidence of disciplinary sanction is inversely related to team quality (measuredby ex ante win probabilities for the two teams) and is higher for away teams than forhome teams. However, the tendency for away teams to be penalized more frequentlythan home teams also appears to reflect a refereeing bias favouring the home team.

for exploring the effects on strategic behaviour and outcomes ofchanges in any of the structural assumptions.5

The remainder of the paper is structured as follows: Section 2motivates the development of a theoretical model by identifyinga number of key patterns in the data on goal scoring and disciplin-ary sanction in English professional football. Section 3 develops thetheoretical model. Section 4 presents numerical simulations whichidentify the theoretical model’s predictions for strategic behaviourover a range of structural assumptions. Finally, Section 5 concludes.

2. Player dismissals and goal scoring in English football

In order to illustrate several of the key strategic issues that areexamined in this paper, data were compiled for all 12,216 matchesplayed in the English Premier League (the Premiership) and thethree divisions of the English Football League (currently knownas the Championship, League One and League Two) during thesix football seasons from 2001/02 to 2006/07 (inclusive).6 Table 1reports the average rates of player dismissal per minute and theaverage rates at which goals were scored by the home and awayteams per minute, at various durations within matches.7 Table 2 re-ports the same averages conditional on the current difference be-tween the scores of the two teams.

Key features of Tables 1 and 2 are as follows. First, there is a pro-nounced upward trend in the rates of player dismissal over theduration of matches. For example, the rates for 85 < t 6 89 (dura-tions between 85 and 89 min) are around 18 times and nine timesthose for 0 < t 6 10 for home teams and for away teams, respec-tively. Second, there is also a clear but less heavily pronounced up-ward trend in the goal scoring rates. The goal scoring rates for thehome and away teams for 85 < t 6 89 are around 1.5 times thosefor 0 < t 6 10.

Third, except in the case where either team is leading by morethan two goals (for which the data are relatively sparse), the prob-ability of having a player dismissed is lowest when the scores arelevel; is of intermediate value for teams leading by one or twogoals (s = 1, 2 for the home team, s = �1, �2 for the away team);and is highest for teams trailing by one or two goals (s = �1, �2for the home team, s = 1, 2 for the away team). Fourth and finally,the probability of either team scoring is lowest when the scores arelevel, and is somewhat increased when the scores are unequal. Inthe latter case, and provided the current scores difference is notmore than two goals, the scoring probabilities of both teams in-crease in similar proportions (no matter which team is leadingand which is trailing). However, a team that is leading by three

5 In the model and throughout this paper, it is assumed that the objectives of theteam manager (or coach) on the one hand, and the players on the other hand, areidentical. Therefore the model does not incorporate agency problems arising frompossible divergence of interests between manager and players. The consensus amongsports economists is that agency costs are less significant in professional sports clubsthan in many other organizations. In football, as in other professional sports,individual effort and performance are transparent and easy to monitor. Win bonusesand other forms of performance-related remuneration are employed widely toprovide the required incentives for players to maximize their efforts.

6 The data source is www.4thegame.com. Match reports posted on this websiterecord the match durations (in minutes) at which player dismissals occurred andgoals were scored. The data set was compiled by transcribing these details for eachmatch individually by hand.

7 In Table 1 the data for the 45th and the 90th minutes are displayed separatelyfrom those for other durations. These cells include all player dismissals and goalsscored during the 45th and 90th minutes, and during stoppage time (playedimmediately after the 45th and 90th minutes have elapsed). The data source does notrecord the amount of stoppage time played, which is variable but typically of between2 and 5 min’ duration at the end of each 45-min period. Accordingly the rates ofplayer dismissal and goal scoring recorded in these cells are several times themagnitudes of those in adjacent cells.

Table 1Rates of player dismissal and goal scoring conditional on current duration.

Match duration, t (min) Players dismissed per minute Goals scored per minute

Home team Away team p-Value Home team Away team p-Value

0 < t 6 10 .00007 .00028 .000 .0121 .0089 .00010 < t 6 20 .00039 .00043 .619 .0136 .0101 .00020 < t 6 30 .00041 .00061 .031 .0146 .0103 .00030 < t 6 40 .00049 .00083 .001 .0145 .0111 .00040 < t 6 44 .00043 .00111 .000 .0149 .0105 .00044 < t 6 45 .00360 .00434 .361 .0558 .0380 .00045 < t 6 55 .00064 .00088 .033 .0157 .0122 .00055 < t 6 65 .00102 .00151 .001 .0162 .0124 .00065 < t 6 75 .00110 .00195 .000 .0166 .0126 .00075 < t 6 85 .00130 .00215 .000 .0172 .0130 .00085 < t 6 89 .00129 .00250 .000 .0174 .0127 .00089 < t 6 90 .00950 .01441 .000 .0842 .0625 .000

Note: Based on data for all 12,216 matches played in the English professional leagues (Premiership, Championship, League One and League Two) during seasons 2001/02 to2006/07 (inclusive).p-Values are for two-tail tests for the significance of the difference between the home team and away team rates of player dismissal and goal scoring, conditional on thematch durations listed in the left-hand columns. In these tests, the arrival rates of player dismissals and goals are assumed to be Poisson processes.

Table 2Rates of player dismissal and goal scoring conditional on current difference in scores.

Difference in scores, s(home team goals – away team goals)

Players dismissed per minute Goals scored per minute

Home team Away team p-Value Home team Away team p-Value

s 6 �3 .00139 .00052 .033 .0177 .0163 .419s = �2 .00214 .00135 .007 .0178 .0136 .000s = �1 .00129 .00132 .811 .0172 .0122 .000s = 0 .00060 .00096 .000 .0154 .0115 .000s = 1 .00078 .00192 .000 .0166 .0130 .000s = 2 .00083 .00207 .000 .0177 .0134 .000s P 3 .00066 .00214 .000 .0205 .0130 .000

Note: Based on data for all 12,216 matches played in the English professional leagues (Premiership, Championship, League One and League Two) during seasons 2001/02 to2006/07 (inclusive).p-Values are for two-tail tests for the significance of the difference between the home team and away team rates of player dismissal and goal scoring, conditional on thematch durations listed in the left-hand columns. In these tests, the arrival rates of player dismissals and goals are assumed to be Poisson processes.

8 Garicano and Palacios-Huerta (2005) assume that an increase in foul play by oneteam results in a reduction in its opponent’s probability of scoring. However, thisassumption seems difficult to reconcile with Tables 1 and 2, which suggest that ateam that is leading by one goal, whose main priority is to prevent its opponent fromscoring, tends to play less violently (and not more so) than a team that is trailing byone goal.

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goals or more appears to experience a large increase in its probabil-ity of scoring further goals.

The tendency for scoring rates to increase over the duration ofthe match (see Table 1) is well known to commentators, punditsand spectators, and is commonly attributed to a tendency for play-ers to become fatigued and commit defensive errors, which lead togoals being scored. Errors due to fatigue could also produce morefouls in last-ditch attempts to prevent goals, which could in turnaccount for the upwardly-trended rates of player dismissal.

Since goals in football are relatively infrequent events (incomparison with scores in other team sports such as rugby orbasketball), observations in Table 2 for level scores tend to beweighted towards the earlier match durations, while observa-tions for a non-zero difference in scores tend to be weightedtowards later durations. Therefore player fatigue could also ac-count for some of the variation shown in Table 2. However,the data in Table 2 could also reflect a form of team qualityselection effect. Observations of scoring rates when the awayteam is already leading (for example) are drawn predominantlyfrom matches in which the away team dominates the hometeam in terms of quality. Therefore it is expected that the awayteam’s probability of scoring further goals is higher than average.This team quality selection effect might explain why teams scoremore frequently when leading than when scores are level, but itdoes not explain why teams that are trailing also tend to scoremore frequently.

Distinct from the player fatigue and team quality selection ef-fects, an alternative explanation for the variations in scoring rates

shown in Tables 1 and 2 relates to optimizing strategic behaviouron the part of the teams throughout the duration of the match.In the dynamic game-theoretic model that is developed below, ateam that switches from a defensive to an attacking formation in-creases both its own and its opponent’s probabilities of scoring. Byleaving itself open to a counter-attack and the possibility of havingto commit a last-ditch foul in order to prevent an opponent fromscoring, it also increases its own probability of having a player dis-missed. A team that switches from a non-violent to a violent styleof play disrupts the pattern of play, in a manner that increases bothteams’ scoring probabilities.8

The dismissal of a player imposes two separate costs upon hisown team. First, the team must complete the match at a numericaldisadvantage in players (unless an opposing player is also dis-missed). Table 3 indicates that there is a large reduction in thescoring probability for a team that is playing at a numerical disad-vantage, and a still larger increase in the probability of conceding agoal. This portion of the cost of a player dismissal is highest for dis-missals that occur at the early stages of matches, because the teamthat loses a player is exposed to these adverse probabilities forlonger. Second, a dismissal results in the suspension of the player

Table 3Rates of goal scoring conditional on numerical disparity in players.

Goals scored per minute

Home team Away team p-Value

Away team with disadvantage in player numbers .0285 .0090 .000Neither team with disadvantage in player numbers .0160 .0121 .000Home team with disadvantage in player numbers .0119 .0229 .000

Note: Based on data for all 12,216 matches played in the English professional leagues (Premiership, Championship, League One and League Two) during seasons 2001/02 to2006/07 (inclusive).p-Values are for two-tail tests for the significance of the difference between the home team and away team rates of player dismissal and goal scoring, conditional on thematch durations listed in the left-hand columns. In these tests, the arrival rates of player dismissals and goals are assumed to be Poisson processes.

664 S. Dobson, J. Goddard / European Journal of Operational Research 205 (2010) 661–669

concerned from one or more future matches, reducing his team’sprospects of obtaining league points because the team manageror coach has to select from a weaker pool of players for thosematches. This portion of the cost is independent of the stage ofthe (current) match at which the dismissal occurs. Therefore theoverall cost of a dismissal in the current match, expressed in termsof expected league points foregone, is higher for dismissals duringthe early stages of matches. A tendency for dismissal rates to in-crease over the duration of matches (see Table 1) seems consistentwith optimizing strategic behaviour in the light of this cost struc-ture. The following section develops a game-theoretic model thatseeks to capture the dynamic aspects of the interplay betweeninterdependence, risk and outcomes.

3. A dynamic game-theoretic model of strategic behaviour andrisk taking

In the theoretical model, it is assumed that each match consistsof a number of discrete unit time intervals, such that durationswithin the match can be represented by t = 0, . . . ,T where T is thecomplete match duration. In the numerical simulations that followin the next section, each time interval represents 1 min, and T = 90.For any time interval within the match denoted (t, t + 1), each teammanager or coach can select either a non-violent or a violent styleof play, and either a defensive or an attacking team formation. Thehome and away teams’ strategic choices are represented by i and j,respectively: i = 1 or j = 1 denotes (non-violent, defend); i = 2 orj = 2 denotes (violent, defend); i = 3 or j = 3 denotes (non-violent,attack); and i = 4 or j = 4 denotes (violent, attack).

The payoff for each team at the end of the match is dependenton the number of league points gained and the cost of future playersuspensions arising from any dismissals incurred during the cur-rent match. To allow for the possibility of risk-averse behaviouron the part of the teams, the payoffs from the league points gainedare determined through a utility function, which may be either lin-ear in points gained (risk-neutrality) or concave (risk-aversion).The available league points are 3 for a win, 1 for a draw (tie),and 0 for a loss. Without any loss of generality, the utility functionis specified as follows: Uð0Þ ¼ 0; Uð1Þ ¼ 1; Uð3Þ ¼ 1þ 2k, wherek ¼ 1 represents risk-neutrality and 0 < k < 1 represents risk-aver-sion. For simplicity the utility cost of a dismissal (arising from fu-ture player suspensions) is a fixed utility deduction per playerdismissed, denoted x.

The remaining notation that is used to specify the theoreticalmodel is as follows.

s difference between scores of home team and away team atduration t

dh number of dismissals incurred by home team prior toduration t

da number of dismissals incurred by away team prior to dura-tion t

�htðs;dh;daÞ home team value function, denoting the final payoff tothe home team at end of the match (t = T), and the expec-tation of this payoff at earlier durations (t = 0, . . . ,T � 1)

�atðs;dh; daÞ corresponding away team value functioni home team’s strategic choice: i = 1 (non-violent, defend);

i = 2 (violent, defend); i = 3 (non-violent, attack); i = 4 (vio-lent, attack)

j away team’s strategic choice, defined in the same way as ipi;j;x;t probability that home team scores a goal during (t, t + 1),

conditional on x ¼ dh � da and on i and jqi;j;x;t corresponding probability of a goal scored by the away

teamui;t probability that home team has a player dismissed during

time interval (t, t + 1), conditional on iv j;t corresponding probability of an away team player dis-

missalhi;j

t ðs;dh;daÞ expectation at duration t of the home team’s final pay-off as a function of s, dh and da, conditional on the teams’choices of i and j over (t, t + 1)

ai;jt ðs;dh; daÞ expectation at t of the away team’s final payoff

At the end of the match, �hTðs; dh; daÞ ¼ 1þ 2k�xdh for s > 0(and for any daÞ; 1�xdh for s = 0; and �xdh for s < 0. Similarly,�aTðs; dh; daÞ ¼ �xda for s > 0 (and for any dhÞ; 1�xda for s = 0;and 1þ 2k�xda for s < 0. For any time interval within the matchdenoted (t, t + 1) for t = 0, . . . ,T � 1, it is assumed that each teammanager or coach selects either a non-violent or a violent styleof play, and either a defensive or an attacking team formation.The home and away teams’ strategic choices are represented byi and j, respectively. In the model, the sole criterion for selectionof team formation and style of play is the maximization of theteam’s expected payoff, as defined above. Therefore the analysisexcludes several other factors that might, in practice, influencethe team manager’s decision, or the implementation by the play-ers of the manager’s instructions. Such factors could include, forexample: pursuit of grudges arising from previous encounterswith opposing teams or individual players; stronger incentivesfor players to maximize their efforts when playing at home thanwhen playing away; pressures emanating from the crowd’s reac-tions to incidents during the match; or attempts by individualplayers to manipulate match outcomes due to bribery or illicitgambling.

Since the teams interact with each other continuously over thecourse of the match, the situation is modelled as a repeated non-cooperative game with a finite time horizon. It is assumed thatthe teams do not cooperate at any point, because cooperationwould unravel from the end of the match backwards. For example,both teams are aware that the subgame starting at T � 1 is the fi-nal subgame. Both may defect, because defection creates no long-term loss to offset the short-term gain. Similarly, both teams areaware that the subgame starting at T� 2 will be repeated atT � 1, but both are aware that their opponent will defect atT � 1. Therefore neither team has any incentive to cooperate at

S. Dobson, J. Goddard / European Journal of Operational Research 205 (2010) 661–669 665

T � 2. Once again, defection imposes no long-term loss to offsetthe short-term gain.9

For simplicity, it is assumed that only one goal may be scoredand one player dismissed within each time interval. For each timeinterval (t, t + 1) for t = 0, . . . ,T � 1, the following expressions gov-ern the relations between the expectations at t of the teams’ finalpayoffs conditional on the values of i and j selected over the inter-val (t, t + 1), denoted hi;j

t ðs; dh; daÞ and ai;jt ðs; dh; daÞ, and the expecta-

tions at t + 1 of the teams’ final payoffs unconditional on i and j,denoted �htþ1ðs; dh; daÞ and �atþ1ðs; dh; daÞ:

hi;jt ðs; dh; daÞ ¼ C0XK; ai;j

t ðs;dh;daÞ ¼ C0UK;

where C0 ¼ ðqi;j;x;t 1� pi;j;x;t � qi;j;x;t pi;j;x;tÞ;K0 ¼ ðv j;t 1� ui;t � v j;t ui;tÞ

X¼�htþ1ðs�1;dh;daþ1Þ �htþ1ðs�1;dh;daÞ �htþ1ðs�1;dhþ1;daÞ

�htþ1ðs;dh;daþ1Þ �htþ1ðs;dh;daÞ �htþ1ðs;dhþ1;daÞ�htþ1ðsþ1;dh;daþ1Þ �htþ1ðsþ1;dh;daÞ �htþ1ðsþ1;dhþ1;daÞ

0B@

1CA;

ð1Þ

U¼�atþ1ðs�1;dh;daþ1Þ �atþ1ðs�1;dh;daÞ �atþ1ðs�1;dhþ1;daÞ

�atþ1ðs;dh;daþ1Þ �atþ1ðs;dh;daÞ �atþ1ðs;dhþ1;daÞ�atþ1ðsþ1;dh;daþ1Þ �atþ1ðsþ1;dh;daÞ �atþ1ðsþ1;dhþ1;daÞ

0B@

1CA:

ð2Þ

According to these expressions, each team’s expected payoff att, conditional on the values of s, dh and da at t and the teams’ strat-egies (i and j) over the interval (t, t + 1), is the sum of the expectedpayoffs at t + 1 for all feasible values of s, dh and da at t + 1, multi-plied by the transition probabilities from the values of s, dh and da

at t to the feasible values at t + 1. The transition probabilities de-pend on the teams’ strategies (i and j) over the interval (t, t + 1).From these expressions, the solutions for the teams’ optimal strat-egies over each time interval are obtained through a process ofbackwards induction. Starting at duration t = T � 1, the teams’ opti-mal strategies (choices of i and j) over the interval (T � 1,T) can bedetermined, since �hTð Þ and �aTð Þ are pre-determined. This estab-lishes �hT�1ðÞ and �aT�1ð Þ. The optimal choices of i and j over(T � 2,T � 1) are then similarly determined. By iterating back-wards, the optimal choices of i and j at any stage of the matchare determined.

4. Numerical simulations

This section illustrates the behavioural properties of the theo-retical model by means of numerical simulation. Some of the mod-el’s parameters can be calibrated accurately from inspection of thedata that are summarized in Tables 1–3. Some parameters, how-ever, cannot be calibrated in this manner. In the case of the latter,simulations of the model are run over ranges of plausible parame-ter values.

9 Therefore the possibility that defection might be deterred through the use ofpunishment strategies is not considered. With infinite horizons, each team mightdeter defection by threatening punishment that would reduce the opponent’s payoffover the entire duration. A pair of punishment strategies might constitute a Nashequilibrium, so that cooperation is achieved in every subgame. Where cooperation isa dominant strategy equilibrium, however, there is no need to invoke punishmentstrategies. With finite horizons, punishment strategies are relevant only if there isbounded rationality and the teams ignore the fact that the duration is finite. In Section4 of this paper, cooperation is shown to break down mainly during the closing stagesof matches; cooperation is typically a dominant strategy equilibrium in the earlystages. Therefore punishment strategies are generally irrelevant at the stage when abounded rationality assumption might be plausible, and implausible at the stagewhen cooperation breaks down and punishment would otherwise (in an infinitegame) be relevant.

The rates of goal scoring within each time interval for the homeand away teams are observable at an aggregate level, but the ratesconditional on the teams’ strategic choices i and j are unobservable,because i and j are themselves unobservable. Let Ii and Jj denoteindicator variables for strategic choices i (home team) and j (awayteam), respectively. The parameterization of the scoring rates perminute is as follows:

Home team:

pi;j;0;t ¼ 0:011þ 0:0015hþ 0:00004t þ 0:005/½2ðI2 þ I4Þ þ J2 þ J4�þ 0:005w½ðI3 þ I4Þuþ ðJ3 þ J4Þð2�uÞ�;

pi;j;x;t ¼ ð1� 0:25xtÞpi;j;0;t for x > 0;

pi;j;x;t ¼ ð1þ 0:75xtÞpi;j;0;t for x < 0:

Away team:

qi;j;0;t ¼ 0:008� 0:0015hþ 0:00004t þ 0:005/½I2 þ I4 þ 2ðJ2 þ J4Þ�þ 0:005w½ðI3 þ I4Þuþ ðJ3 þ J4Þð2�uÞ�;

qi;j;x;t ¼ ð1� 0:75xtÞqi;j;0;t for x > 0;

qi;j;x;t ¼ ð1þ 0:25xtÞqi;j;0;t for x < 0:

Naturally the scoring rates of both teams depend on relativeteam quality, represented by the parameter h. Positive values of hrepresent matches in which the home team is stronger than theaway team, h ¼ 0 represents a match between two evenly balancedteams, and negative values of h represent matches in which theaway team is stronger than the home team. When h ¼ 0, the hometeam’s scoring rate is higher than that of the away team due tohome-field advantage. h ¼ �1 represents a match in which thehome and away teams have identical scoring rates, because theaway team’s team quality advantage is just sufficient to offsetthe home-field advantage. In the numerical simulations h ¼ 0 isthe baseline value, and variations in the range h ¼ �2; �1; 0;1; 2 are examined.

It is apparent from the raw data that the scoring rates increasewith a near-linear trend over the duration of the match, and it ap-pears unlikely that the existence of this trend is explained in its en-tirety by changes in strategic behaviour during the match. In thesimulations the scoring rates are assumed to follow a linear trend.For simplicity, the trend component is the same for home and wayteams, and it does not vary with either relative team quality or theteams’ strategic choices.

The assumed impact of the teams’ strategic choices on theirscoring rates is as follows. First, if either or both teams play vio-lently (i,j = 2 or 4), the effect on the scoring probabilities is deter-mined by the parameter /. If one team is violent and the other isnon-violent, the violent team’s scoring rate increases by 0:01/and the non-violent team’s by 0:005/. If both teams are violent,both scoring rates increase by 0:015/. In the numerical simulations/ ¼ 1 is the baseline value, and variations in the range/ ¼ 0:5; 1; 1:5 are examined.

Second, by attacking (i,j = 3 or 4), a team increases its own andits opponent’s scoring rates by amounts that are proportional tothe parameter w. In the case where both teams attack, both teams’scoring rates increase by the same amount. In the case where oneteam defends and the other attacks, the teams’ scoring rates in-crease by smaller amounts. The relativities depend upon whichteam (if either) has a comparative advantage when one team at-tacks and the other defends. The parameter u controls for compar-ative advantage: u ¼ 1 implies there is no comparative advantage;u > 1 implies the home team has a comparative advantage when itattacks and the away team defends; and u < 1 implies the hometeam has a comparative advantage when it defends and the awayteam attacks. In the numerical simulations the baseline values

Table 4Analysis of two-person non-cooperative subgame in the 75th minute.

Note: s denotes (home team score minus away team score) at the end of the 74th minute.It is assumed no players have been dismissed previously during the match ðdh ¼ da ¼ 0Þ.Home team strategies are i = 1 (defend, non-violent), i = 2 (defend, violent), i = 3 (attack, non-violent), i = 4 (attack, violent).Away team strategies are j = 1, 2, 3, 4, defined as for home team.Table 4 shows expected payoffs for the home and away teams, denoted hi;j

74ðs;dh; daÞ and ai;j74ðs;dh; daÞ in the text, respectively.

Payoffs for dominant strategies are shown in bold font. The solution to each two-person non-cooperative subgame is shaded grey.

666 S. Dobson, J. Goddard / European Journal of Operational Research 205 (2010) 661–669

are w ¼ 1 and u ¼ 1, and variations in the ranges w ¼ 0:5; 1; 1:5and u ¼ 1:33; 1; 0:67 are examined.

The impact on the scoring rates if either team is currently play-ing with a numerical disadvantage in players is observable in theraw data. For simplicity it is assumed that the team with a numer-ical disadvantage experiences a reduction in its own scoring rateand an increase in its opponent’s scoring rate, both of which areproportionate to the size of the numerical disadvantage. In prac-tice, it is unusual for the numerical disparity in players to exceedone player, and almost unknown for the disparity to exceed two.

In common with the scoring rates, the rates of player dismissalwithin each time interval are observable at an aggregate level. Theassumed expressions are as follows:

Home team :

ui;t ¼ ½0:0004þ 0:00001t þ 0:0002ðI3 þ I4Þ�½1þ bðI2 þ I4Þ�;Away team :

v j;t ¼ ½0:0004þ 0:000012t þ 0:0002ðJ3 þ J4Þ�½1þ bðJ2 þ J4Þ�:

The player dismissal rates appear to be trended over the dura-tion of the match, and the trend appears to be more pronouncedfor away teams. For simplicity any relationship between relativeteam quality and the dismissal rates is ignored, but a home-fieldeffect is incorporated through the coefficient on the trend. As be-fore, the impact of strategic behaviour on the dismissal rates isunobservable. In the simulations, the parameter b controls theamounts by which the dismissal rate increases when a teamswitches from non-violence (i,j = 1 or 3) to violence (i,j = 2 or 4).In the numerical simulations b ¼ 2 is the baseline value, and vari-ations in the range b ¼ 1; 2; 3 are examined. It is assumed that aswitch from defense (i,j = 1 or 2) to attack (i,j = 3 or 4) producesan increase in the team’s own dismissal rate.10

The utility cost of future player suspensions arising from dis-missals is unobservable. In the numerical simulations x ¼ 0:5 isthe baseline value, and variations in the range x ¼ 0:25; 0:5; 1are examined. The risk-aversion parameter k is also unobservable.At the baseline value of k ¼ 0:5, the difference in utility between awin and a draw is the same as the difference between a draw and aloss, even though the league points difference between a win and a

10 Teams that attack are vulnerable to counter-attack, and run an increased risk ofbeing obliged to commit a last-ditch foul in order to prevent an opposing player fromscoring. The punishment for a player who denies an opponent a goal scoringopportunity in this manner is dismissal.

draw is twice the difference between a draw and a loss.11 In thenumerical simulations variations in the range k ¼ 0:25; 0:5;0:75; 1 are examined. At the maximum value of k ¼ 1, the relation-

ship between league points and utility values is linear.Table 4 presents a full analysis of the two-player non-coopera-

tive subgame at one arbitrarily selected stage of a match, the 75thminute, with the variable parameters set to their baseline values asdetailed above. The analysis is presented conditional on the differ-ence in scores at the end of the 74th minute, for s = �1 (away teamleading by one goal), 0 (scores level) and 1 (home team leading byone goal). It is assumed that no player from either team has beendismissed prior to t = 74.

The left-hand panel of Table 4 shows the expected payoffs tothe home and away teams at t = 74 for s = �1, conditional on theteams’ strategic choices for the 75th minute, and assuming thatoptimal choices will be made in each subsequent minute. Thehome team needs to score at least one goal to collect any leaguepoints from the match. Its expected payoffs from i = 4 (violent, at-tack) are higher than those from i = 1, 2, 3, because i = 4 maximizesits probability of scoring. Therefore i = 4 is the home team’s domi-nant strategy. In contrast, the away team’s interests are best servedby keeping the scoring rates as low as possible, and j = 1 (non-vio-lent, defend) is the away team’s dominant strategy. Accordinglythe outcome in the 75th minute is (i = 4, j = 1).

The central panel of Table 4 shows the expected payoffs fors = 0. In this case, i = 2, j = 2 (violent, defend) is the dominant strat-egy for both teams. In comparison with this non-cooperative solu-tion, both teams would be made better off by cooperating toachieve either i = 1, j = 1 (non-violent, defend) or i = 3, j = 3 (non-violent, attack). This subgame has a prisoner’s dilemma structure.Finally, the right-hand panel of Table 4 shows the expected payoffsfor s = 1. In this case, the priorities of the home and away teams arethe reverse of those in the case s = �1, and the dominant strategiesare i = 1 (non-violent, defend) and j = 4 (violent, attack).

Table 5 summarizes the outcomes of the two-person non-coop-erative subgames at three different stages of the match: the 25th,50th and 75th minutes, allowing for variations in each of theparameters h; /; w; u; b; x and k in turn, while the other param-eters are set to their baseline values. The main findings from thisnumerical simulation exercise can be summarized as follows:

11 Until the 1981/82 season the league points scoring scheme in the English leaguewas 2–1–0 (win–draw–lose). Subsequently, the scheme was changed to 3–1–0, in anattempt to encourage attacking play.

Table 5Outcomes of two-person non-cooperative subgame in the 25th, 50th and 75th minutes over various parameter values.

25th minute 50th minute 75th minute

s! �3 �2 �1 0 1 2 3 �3 �2 �1 0 1 2 3 �3 �2 �1 0 1 2 3h / w u b x k

1. Variation in h (relative team quality)�2 1.0 1.0 1.0 2 0.5 0.5 a/– va/– va/– v/– –/va –/va –/a a/– va/– va/– –/– –/va –/va –/a a/– a/– va/– v/– –/va –/a –/a�1 1.0 1.0 1.0 2 0.5 0.5 a/– va/– va/– –/– –/va –/va –/a a/– va/– va/– –/– –/va –/a –/a a/– a/– va/– v/v –/va –/a –/a0 1.0 1.0 1.0 2 0.5 0.5 a/– va/– va/– –/– –/va –/va –/a a/– va/– va/– –/– –/va –/a –/a a/– a/– va/– v/v –/va –/a –/a1 1.0 1.0 1.0 2 0.5 0.5 a/– va/– va/– –/v –/va –/va –/a a/– va/– va/– –/a –/va –/a –/a a/– a/– va/– v/v –/va –/a –/–2 1.0 1.0 1.0 2 0.5 0.5 a/– va/– v/– –/a –/va –/a –/a a/– va/– va/– –/a –/va –/a –/a a/– a/– va/– –/va –/va –/a –/–

2. Variation in / (effect of violent play on scoring probabilities)0 0.5 1.0 1.0 2 0.5 0.5 a/– a/– a/– –/a –/a –/a –/a a/– a/– a/– –/a –/a –/a –/a a/– a/– va/– –/– –/va –/a –/–0 1.0 1.0 1.0 2 0.5 0.5 a/– va/– va/– –/– –/va –/va –/a a/– va/– va/– –/– –/va –/a –/a a/– a/– va/– v/v –/va –/a –/a0 1.5 1.0 1.0 2 0.5 0.5 va/– va/– va/– v/v –/va –/va –/a a/– va/– va/– v/v –/va –/va –/a a/– va/– va/– v/v –/va –/va –/a

3. Variation in w (effect of attacking play on scoring probabilities)0 1.0 0.5 1.0 2 0.5 0.5 a/– va/– v/– –/v –/va –/va –/a a/– va/– va/– –/– –/va –/a –/a –/– a/– va/– v/v –/va –/a –/–0 1.0 1.0 1.0 2 0.5 0.5 a/– va/– va/– –/– –/va –/va –/a a/– va/– va/– –/– –/va –/a –/a a/– a/– va/– v/v –/va –/a –/a0 1.0 1.5 1.0 2 0.5 0.5 a/– va/– va/– –/– –/va –/va –/a a/– va/– va/– –/a –/va –/a –/a a/– a/– va/– v/a –/va –/a –/a

4. Variation in u (attack/defense comparative advantage)0 1.0 1.0 1.33 2 0.5 0.5 a/– va/– va/– –/– –/va –/va –/a a/– va/– va/– –/– –/va –/a –/a a/– a/– va/– v/v –/va –/a –/–0 1.0 1.0 1.0 2 0.5 0.5 a/– va/– va/– –/– –/va –/va –/a a/– va/– va/– –/– –/va –/a –/a a/– a/– va/– v/v –/va –/a –/a0 1.0 1.0 0.67 2 0.5 0.5 a/– va/– va/– –/a –/va –/va –/a a/– va/– va/– –/a –/va –/a –/a –/– a/– va/– v/v –/va –/a –/a

5. Variation in b (effect of violent play on probability of dismissal)0 1.0 1.0 1.0 1 0.5 0.5 va/– va/– va/– v/v –/va –/va –/va va/– va/– va/– v/v –/va –/va –/a a/– va/– va/– v/v –/va –/va –/a0 1.0 1.0 1.0 2 0.5 0.5 a/– va/– va/– –/– –/va –/va –/a a/– va/– va/– –/– –/va –/a –/a a/– a/– va/– v/v –/va –/a –/a0 1.0 1.0 1.0 3 0.5 0.5 a/– a/– a/– –/a –/a –/a –/a a/– a/– a/– –/– –/a –/a –/a a/– a/– va/– –/– –/va –/a –/–

6. Variation in x (utility cost of dismissals, incurred in future matches)0 1.0 1.0 1.0 2 0.25 0.5 va/– va/– va/– v/v –/va –/va –/va a/– va/– va/– v/v –/va –/va –/a a/– va/– va/– v/v –/va –/va –/a0 1.0 1.0 1.0 2 0.5 0.5 a/– va/– va/– –/– –/va –/va –/a a/– va/– va/– –/– –/va –/a –/a a/– a/– va/– v/v –/va –/a –/a0 1.0 1.0 1.0 2 1.0 0.5 a/– a/– a/– –/– –/a –/a –/a a/– a/– va/– –/– –/a –/a –/a –/– a/– va/– –/– –/va –/a –/–

7. Variation in k (risk-aversion or risk-neutrality)0 1.0 1.0 1.0 2 0.5 0.25 a/– va/– a/– –/– –/va –/ a –/a a/– va/– va/– –/– –/va –/a –/a a/– a/– va/– –/– –/va –/a –/a0 1.0 1.0 1.0 2 0.5 0.5 a/– va/– va/– –/– –/va –/va –/a a/– va/– va/– –/– –/va –/a –/a a/– a/– va/– v/v –/va –/a –/a0 1.0 1.0 1.0 2 0.5 0.75 a/– va/– va/– –/v –/va –/va –/a a/– va/– va/– v/a –/va –/va –/a a/– a/– va/– va/va –/va –/a –/a0 1.0 1.0 1.0 2 0.5 1.0 va/– va/– va/– v/v –/va –/va –/a a/– va/– va/– v/va –/va –/va –/a a/– a/– va/– va/va –/va –/a –/a

Note: s denotes (home team score minus away team score) at the end of the 24th, 49th or 74th minute.It is assumed no players have been dismissed previously during the match ðdh ¼ da ¼ 0Þ.Subgame outcomes are reported in the following format: ‘home team strategic choice’/‘away team strategic choice’. Strategic choices are reported as follows:– denotes (non-violent, defend); v denotes (violent, defend); a denotes (non-violent, attack); va denotes (violent, attack).

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1. In general teams that are trailing tend to attack, and teams thatare trailing tend to play violently for as long as there is a realis-tic chance of salvaging the match. Teams that are trailing bythree goals tend not to play violently. The stage at which a teamthat is trailing by two goals ceases to play violently, implicitlyconceding that the match is beyond salvation, depends uponrelative team quality and upon home-field advantage: strongerteams when trailing play violently for longer than weakerteams, and home teams when trailing play violently for longerthan away teams. Towards the closing stages of matches inwhich scores are level, when the impact of a dismissal on thematch outcome is relatively small, there is a tendency for bothteams to defect from non-violence to violence. This feature ofthe theoretical model seems consistent with the empirical ten-dency for the number of dismissals to increase significantly inthe later stages of matches (see Table 1).

2. The propensity for teams that are either trailing or level to playviolently varies positively with the effect of violent play on thescoring probabilities. The propensity for teams that are eithertrailing or level to attack varies positively with the effect ofattacking play on the scoring probabilities. In a few situations,the existence of a comparative advantage in attack or defenseinfluences the choice of strategy, but in most situations the cur-rent state of the match seems to be the decisive factor. A teamthat is trailing is very likely to choose to attack, even if by sodoing it increases its opponent’s scoring probability by morethan it increases its own scoring probability.

3. The propensity for teams that are either trailing or level to playviolently varies negatively with the effect of violent play on theprobability of a dismissal, and negatively with the utility cost ofa dismissal in terms of league points foregone in futurematches.

4. The propensity for teams that are either trailing or level to playviolently and/or to attack is increasing in k, the risk-aversionparameter. For k ¼ 0:75 or 1 (mild risk-aversion or risk-neutral-ity), teams are likely to accept additional risk when the scoresare level, because scoring and proceeding to win the matchyields twice as many league points as are lost by concedingand proceeding to lose. However, for k ¼ 0:25 or 0.5 (severerisk-aversion), both teams tend to opt for caution when thescores are level.

5. Conclusion

This paper develops a dynamic game-theoretic model of opti-mizing strategic behaviour for football teams. At any stage of thematch, teams choose between defensive and attacking formationsand between a non-violent and a violent style of play, so as to max-imize their expected payoffs at the end of the match. Their strate-gic choices determine the probabilities of scoring and concedinggoals at the current stage the match, and the probabilities of hav-ing a player dismissed.

In the theoretical model, optimal strategic behaviour at anystage of the match is found to be dependent on the current differ-ence in scores, and on the amount of playing time that has elapsed.Teams that are trailing tend to attack, and teams that are trailingtend to play violently for as long as there is a realistic chance of sal-vaging the match. Stronger teams when trailing play violently forlonger than weaker teams, and home teams when trailing play vio-lently for longer than away teams. In the latter stages of matcheswhen the scores are level, the subgames have a prisoner’s dilemmastructure, and there is a tendency for the teams to defect from non-violence to violence. This feature of the model is consistent with anobserved tendency for the number of dismissals to increase mark-edly during the closing stages of matches. The paper’s main results

suggest that the strategic behaviour of football teams can be ratio-nalized in accordance with game-theoretic principles of optimizingstrategic decision-making by agents when payoffs are uncertainand interdependent.

For football players, coaches and managers, the game-theoreticmodel presented in this paper identifies explicitly the interplay be-tween, on the one hand, fluctuations in the state of the match (timeelapsed, current score, players dismissed), and the on the otherhand, the optimal choice of team formation and style of play fora team that seeks to maximize its expected (probability-weighted)payoff from the match. By clarifying the trade-offs between, forexample, the disruptive effect on the opposing team of a switchto a more violent style of play, and the costs to the team if, as a con-sequence, one of its own players is dismissed, the model provides arational basis for strategic decision-making in football. The modelalso highlights the advantages of a fluid strategic or tactical ap-proach, that adapts flexibly to changes in the state of the matchthat are both continuous (the elapse of time), and discrete (changesin the score, or player dismissals).

For football’s governing bodies and referees, the model providesa novel explanation for an observed tendency for player disciplineto deteriorate or sometimes break down altogether during the lat-ter stages of matches. Rather than follow media pundits in accus-ing footballers who succumb to their violent inclinations of‘behaving irrationally’ or ‘losing their heads’, the analysis suggeststhat violence during the closing stages of matches may be a conse-quence of a phenomenon that is second-nature to game-theorists:a tendency, in a non-cooperative game with a prisoner’s dilemma-type payoff structure, for behaviour that is individually rational toproduce outcomes that are collectively sub-optimal. The frame-work developed in this paper might be used subsequently to ex-plore the effectiveness of changes to the structure of rewards andpenalties in football, such as tougher refereeing or league pointsdeductions for dismissals, in mitigating what might be considereda perfectly rational descent towards violent conduct during theclosing stages of many football matches.

Acknowledgements

John Goddard gratefully acknowledges the support of the MarieCurie Transfer of Knowledge Fellowship of the European Union’sSixth Framework Program at the Department of Economics at theUniversity of Crete (contract number MTKD-CT-014288), duringSpring 2008.

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