Strategy and Football: A Game Theoretic Model of a Football Match

Transcription

1 Strategy and Football: A Game Theoretic Model of a Football Match Joakim Ahlberg Supervisor: Harald Lang December 7, 2003 Abstract There can be unambiguous, yet non-trivial, theoretical insights about the behaviour of players in simple game theory. Sport constitute a perfect underlying subject for game theory. In this paper a football match is analysed with help of a game theoretic model. The idea of subgame perfect equilibrium is the method that has been used to gain insight into the strategy choices a team uses in different stages of the game. The optimal strategy in this model is depending on the state of the game, i.e. in which time the game currently is in and what the present score difference is. What emerge from the model are: When the score difference is zero: Both teams, under certain restriction, attacks. When a team is leading: It attacks if it is early in the match, but starts defending more and more as the times goes by. When a team is loosing: It enhances its attack along with the time. These results are also in unanimity with the sport itself. Through that they indicate that football teams behave consistently with rationality and equilibrium. However one must always have in mind that scoring in football is a stochastic process.

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3 Contents Preamble 5 2 Introduction 5 2. The Sports Industry The Game Theoretic approach Criticism The Name of the Game Rules of the game 8 4 The Model 8 4. The state of the game The strategy Probability of scoring Value functions Equilibriums A simplified game 5 5. Time (Half-time) Time 0 (Match-start) Equilibrium In the Game Conclusion Extended game 2 6. Equilibrium when the teams are tied Equilibrium when one team is leading Conclusion Discussion 27 3

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5 Preamble In this Thesis a football match is examined with the purpose to acquire knowledge about how a team should react in various situations during the game. The paper starts out with some introduction to the field. Then a model is presented which will be used throughout the examination. Like all models this of course is a crucial step for the forthcoming investigation. Here one has to balance the clarity (transparency) and the description of the core of the subject studied. With help from the model, there will be a simplified two-period game played. From this simplified game one can extract a lot of valuable information about the strategy choice a team would take. Three cases are of special interest here: When the teams are playing tie. 2/3 When a team is leading/loosing. When the game later is extended to a multi-period game, the informationtransparency eludes. The lack of empirical facts, which are needed for the extension of the model, are going to make the model a bit difficult to interpret, as will be seen below. But it has a theoretical value however. The readers of this text are assumed to have basic knowledge in game theory. 2 Introduction Sports is big business. The European Commission has estimated that trade in sports-related activities constitutes 3 per cent of world trade. Yet sports economics remains a surprisingly under-developed field. The game theoretic approach in sports-related research is especially rare. This is strange since game theory is the natural tool for sports-research. 2. The Sports Industry In most industries the output of competitors negatively affects the ability of a producer to sell its output and generate profits. But in sports contests each producer requires the input of its competitor in order to make a product at all. Because of this complementarity in production competitors will be in a position where their returns are increasing in the quality of their rivals. In team sports, the owners of teams have an incentive to construct tournament structures that may actually diminish the playing success of their own team in order to make a more attractive contest. More important, from an economic policy perspective, team owners have argued that 5

6 collusion among the owners is necessary to make a more attractive product for the supporters. In team sports, and especially in football, there are huge investments in human capital. A professional player is trained by the club several times each week in order to optimize teamwork and to extend the individual skill. The training cost and the players salaries are a huge part of the teams total cost. Clubs depend for their success on the ability of their players. If for some reasons a club wishes to, or is willing to part with a player it likes to be able to treat the player as an asset which the club can be compensated for when selling. These assets, or property rights, are the heart of the club. (The supporters can also be regarded as the heart of the club, they co-exist with the players.) The structure of the European leagues is not at all like the North- American football, or soccer as they call it. Clubs in the European leagues who end up at the bottom of a league are relegated to a minor league. As a result, clubs in the top of minor leagues can be promoted to a premier league due to their performance on the field. And in each country the top team(s) of the premier league are qualified, or semi-qualified, to the Champion League or the UEFA-cup the following season. Both arranged by the United European Football Association (U EF A). In addition players can get nominated for their respective national team. In North American soccer there is only one professional league N SL, i.e. the teams do not move between leagues, and the club has to buy their way in. 2.2 The Game Theoretic approach There can be unambiguous, yet non-trivial, theoretical insights about the behaviour of players in game theory. This contrasts from the commonly held view that many predictions in applied game theory are either obvious or inconclusive. Most games in the society are complex and possess multidimensional strategies and incomplete information. Sometimes the games are not even fully specified, when played. Professional sports, however, provides strategic competition in which the participants have devoted their lives to become experts at their games, and in which they are often very highly motivated as well. Moreover, in sports, coaches are fired and players lose market value when performance is not acceptable. There are concrete situations in sports where the game is simply and well defined and with a limited number of strategies for the players. This means that sports is an area which easily can be well specified and a place were game theoretic tools easily could be used to predict behaviour in the game. There have been a couple of authors who have recognized this advantage of sports and the use of Game Theory. Notable are Palomino et al. (999) [] who deal with football. They show that, although the behaviour of teams 6

7 is roughly consistent with rationality, there is still a substantial component of irrationality or, as they call it, passion, illustrated by the fact that teams perform better at home field than they do at away field. Also, there is a discussion paper from Brocas and Carillo (2002) [2] which also is about football. They analyse two big changes recently made in the rules of football; Three-point victory and the Golden Goal. They argue that with a rigorous application of simple game theory it is possible to obtain insights about the effects of game rules on the behaviour of participants. These conclusions could be used to shape simple and clear recommendations for the modification of rules. There are also a few more, for example Walker and Wooders (200) [3] who study the mini-max hypothesis in tennis games and Chiappori et al. (2000) [4] & Huerta (200) [5] who both examine penalty kicks in European Football leagues. 2.3 Criticism The boundary between pure and applied game theory is vague; some developments in the pure theory were motivated by issues that arouse in applications. That makes the testing of predictions of game theory that are basic though not fully intuitive extremely important. It is also reassuring to notice that the observed behaviour is roughly consistent with the predictions. Even though these kinds of papers are subject to criticism that the theoretical conclusion is rather straightforward, they can be seen as a test of the empirical relevance of game theoretic concept. In doing this, many author spend a substantial amount of effort in finding games with simple strategies. Then it is quite natural that knowledge of game theory is not required to find the optimal strategy. In other words, as Brocas and Carillo (2002) [2] point out: Criticizing these papers on the ground that predictions are straightforward is, in a sense, saying that the papers are too good at identifying simply games. 2.4 The Name of the Game The game tool that has been used in this thesis is the model of an extensive game with perfect information. An extensive game is an explicit description of the sequential structure of the decision problems encountered by the players in a strategic situation. The model allows the study of solutions in which each player can consider his plan of action not only at the beginning of the game but also at any point of time at which he has to make a decision. By contrast, the model of a strategic game restricts one to solutions in which each player chooses his plan of action once and for all; this plan can cover unlimited contingencies, but the model of a strategic game does not allow a player to reconsider his plan of action after some events in the game have unfolded. 7

8 A general model of an extensive game allows each player, when making his choices, to be imperfectly informed about what has happened in the past. But in this paper the investigation concerns a simpler model in which each player is perfectly informed about the players previous actions at each point in the game. The solution concept of Nash Equilibrium is unsatisfactory in this model, since it ignores the sequential structure of the decision problems. That is why the notion of subgame perfect equilibrium, in which a player is required to reassess his plans as play proceeds, is the solution concept that has been used in this model. 3 Rules of the game The basic rules of a football match are quite simple; eleven players on each side of the field attempt to get the ball in the net of the opposing side. If they succeed, they score a goal. The team with the highest number of goals at the end of the match wins, there can also be ties which imply they have scored equal number of goals (or no goal at all). Football is a low scoring sport. A single goal can change radically, and for a considerable amount of time, the strategic environment in which teams interact. 4 The Model One of the difficulties in making models is of course to get a realistic one. It should reflect the core of the thing studied or, at least, a harsh core of the subject. The starting point to this model is to assume, as usual, that the teams are rational in the sense that they, or, at least, their coach, are aware of the alternatives, forms expectations about any unknowns, have clear preferences and chooses their actions deliberately after some process of optimization. These are the standard arguments that ensure the validity of the game theoretic approach. 4. The state of the game The football match is modelled as a dynamic game between two teams, i {, 2}. Each team s objective is to maximize the expected payoff, i.e. the expected number of points collected in the game, which depend on the final score between the teams. The winner gets 3 points, the loser 0 and if the match ends as a tie, each team gets point. Time, the dynamic variable, will be discrete, with an instant labelled t {0,,..., T }. This is motivated by the fact that the scoring for the 8

9 teams is highly discrete, there is a time lag between a goal is scored until the match is to begin again. According to that, there is a maximum of goal that can be scored in an instant. The history of time t is the score difference up to that point. Thus, the game is finite due to the fact that there is a limit in time how many goals that actually can be scored. (The game has also a finite horizon as long as the time is discrete.) Then one can define the state of the game as the time elapsed and the current score difference, i.e. a state can be labelled as (t, n) for some t Z 0 and n Z. Hence, the match begins at state (0, 0) and ends at state (T, n) for some n and T being the final time. This is a game of simultaneous moves with perfect information and a strategy for each team is a plan that specifies the action chosen by the team for every history after which it is their turn to move. Each team will be able to choose (change) their strategy in every instant of time. The actions reflect the level of attack, or the intensity of the attack, selected by team i at time t and score difference n. It can be thought upon as the ability of the coach or the players positions in the field or, alternatively, the induced, e.g. by the coach, ability of the players, i.e. their mindset in the game. The strategy chosen is perfectly observable by the other team. 4.2 The strategy A strategy of a player is a function that assigns an action to each nonterminal history of the game. This does not correspond to a plan of action since it requires a player to specify his actions after histories that are impossible if he carries out his plan. One must remember that football, or most games, always contain a stochastically element. The coach must know what to do even if his plan does not work. One interpretation for the components of a player s strategy, corresponding to histories that are not impossible if the strategy is followed, is that they are the beliefs of the other player about what the first player will do in the event he does not follow his original plan. Since little is added by considering mixed strategies in games with perfect information they will be omitted. In every instant of time (at least as long as t T, since time T is the terminal history.), a teams strategy maps the state of the game into an attacking intensity, a i R i, where the strategy space, R i, is assumed to be a closed interval on the real line. (The boundaries of R i will be considered in section 5.) If the score difference is n and the time is t there is a mapping (for each team): s i : Z 0 Z R i s i (t, n) = a i, 9

10 where a i is the appropriate action for team i in state (t, n). The greater value of a i, the more offensive strategy. The more offensive strategy, the greater the chance to score a goal. But playing more intensive does not only affect the probability of scoring in any instant of the match, it also increases the risk of conceding a goal. This because the more offensive the team plays, the more it exposes itself and, consequently, the higher chance for the other team to score. 4.3 Probability of scoring Scoring is random in football, but the strategy choice influences the scoring probability. (There are in fact a lot of other factors that affect this probability; team skills and home-field advantage being the two most important.) The probability of scoring for e.g. team i depends not only on team i:s strategy, but at team j:s strategy as well, since it is more difficult to score against a defending team than an attacking one. (The team becomes vulnerable when it attacks since players move up in the field and therefore are withdrawn from the defence.) Define s = s s 2 S = S S 2, then the probability of scoring a goal can be defined as: p(s) = (p (s), p 2 (s)) = [0, ] [0, ]. Since football is a low scoring game, these probabilities are relative small. Palomino et al. (2000) [] have estimated them to be smaller than 5 per cent a minute. Since, as mentioned above, it is quite natural to assume that it is easier to score against an attacking team than a defending one the following is required. Assumption p i (s) is monotonic increasing, i.e. for i,j=,2 and for all s i, s j S : s j p i (s) 0. Assumption not only states that the probability of scoring increases if the attack enhances for e.g. team ( s p (s) 0), it also says that the risk to concede a goal increases as well ( s p 2 (s) 0), because of the exposing. The same is valid for team 2 too, i.e. when the coach chooses strategy he must try to find some equilibrium level of attack, a level which does not give the other team too great probability to score ( s 2 p (s) 0), whereas the own scoring probability increases sufficiently ( s 2 p 2 (s) 0). Note that, when a team is in state (t, n), i.e. the history n of time t, there is a probability for a history change if either one of the two teams 0

11 scores. From n to n +, or n, depending on which team that scores the goal. For simplicity, and transparency, there will be an assumption which will make the teams symmetric in their change in scoring probability: Assumption 2 s p (s) = s 2 p 2 (s), s = s 2 S. The above assumption states that the teams are homogeneous in the way that, given equal strategy, the change of strategy, for each team, brings about the same change in scoring probabilities. That is to say; the teams are not homogeneous, but the probability changes (of scoring) in their strategy are. They can, and certainly will have, different ability for scoring a goal. There will also be an assumption that concerns the marginal effect of scoring. Assumption 3 for i, j {, 2}. 2 p s 2 i (s) 0, 2 p i s 2 i (s) 0, 2 s j i s j p i (s) = 0 s, s 2 and According to this, the marginal probability to score, resp. concede, a goal is decreasing, resp. increasing, in the level of offensive play. Also, the marginal effect in the change of strategy for each team is independent of the other team s change of strategy. There are two interconnected reasons for maintaining these marginal assumptions. One is simplicity since they will assure a unique and easy to compute symmetric equilibrium which will give clear cut comparative statics about the effect of the strategy choices. The other (which is more important) is transparency; strategies can always have strange indirect effects in payoffs if one include some suitably chosen asymmetries in the teams and/or if the second and cross derivatives in scoring probabilities are sufficiently twisted. The probability functions are also assumed to be continuous. 4.4 Value functions The teams payoff depends on the final score at time T. If n > 0 team will receive three points and team 2 zero points, if n = 0 each team will get one point and if n < 0 team will receive zero points while team 2 gets three. (These points reflect the current rules in football.) But the value at time t, t < T, is the expected value, or the probability distribution, of the final score difference at the final time T. Consequently, the value at an arbitrary t must depend on both the time remaining, T t, and the score difference at that time. If the history at time t is defined as the number of goals scored until that time, i.e. the scoring difference up to time t. Then each such node defines

12 a subgame, starting at time t, with score difference n, denoted by Γ(t, n). To each such Γ(t, n) there is a value function denoted by v i (t, n) associated. This value function for team is (recursively) defined by: v (t, n) = max s [p (s)v (t, n + ) + p 2 (s)v (t, n ) + ( p (s) p 2 (s))v (t, n)], with v (T, n) = 3 if n > 0, v (T, n) = if n = 0 and v (T, n) = 0 if n < 0. This should be interpreted as; the value at Γ(t, n), for team, is a maximization problem over the expected values at time t +. There could only be three outcomes from one period to the next. The first is if team scores (n n + ), which happens with probability p (s). The second is if team 2 scores (n n ), this occur with probability p 2 (s) and finally if neither team scores (n n), where ( p (s) p 2 (s)) is the probability for that occurrence. The same can be stated for team 2, with an index change, giving: v 2 (t, n) = max s 2 [p 2 (s)v 2 (t, n ) + p (s)v 2 (t, n + ) + ( p (s) p 2 (s))v 2 (t, n)], with v 2 (T, n) = 3 if n < 0, v 2 (T, n) = if n = 0 and v 2 (T, n) = 0 if n > 0. Team 2:s value function differs from team :s only in n. Team 2 wants to minimize n when maximizing v 2. It decreases in the score difference, while v increases in n. Hence: for all t, v, resp v 2, is monotonic increasing, resp decreasing, in n. () This can be seen since v (T, ), resp v 2 (T, ), is increasing, resp decreasing, in n which makes v (T, ), resp v 2 (T, ), increasing, resp decreasing, in n, due to the fact of assumption. This property will then propagate recursively. The value functions are also assumed to be quasi-concave in p. 4.5 Equilibriums The standard solution concept when dealing with (extensive) games with perfect information is the method of subgame perfect equilibrium. It requires that the action prescribed by each player s strategy is optimal, given the other players strategies, after every history at any time, that is, in every state of the game. But since this is a game of simultaneous moves, the team s 2

13 strategy must actually be optimal without knowing the other teams strategy for the next period of time. Instead, the leading role of the optimization is here played by the state of the game (t, n). (The state of the game carries all the relevant information for the optimizer.) Equivalently one can define a subgame perfect equilibrium to be a strategy profile s in Γ which for every state (t, n), the strategy profile s (t, n) is a Nash Equilibrium of the subgame Γ(t, n). (Hence, every subgame perfect equilibrium also constitutes a Nash equilibrium but not the reverse though.) The algorithm used for calculating the set of subgame perfect equilibriums of a finite game is often referred to as backwards induction. It describes what appears to be a natural way for players to analyze a game with (relative) short horizon. The algorithm starts at the final state of the game, i.e. (T, n) for every (possible) n; from there one goes back one step in time, that is to T, and choose the optimal actions, for every possible history for the teams and replace the actions with a value function v(t, n), in which the payoff profile is that which results when the optimal actions are chosen. Then this procedure is repeated, all the way back to the initial state (0, 0). 3

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15 5 A simplified game The motivation for the discrete time approach is not only the non-playing time due to a goal, there is in fact a lot of non-playing time in football; such as corners, throw-ins and other breaks where the time still runs. Due to these built-in time-lags there is non-playing time where you can not score a goal, so the discrete approach is intuitively appealing. One of the problems with a discrete time though, is to decide how long an instant should be, i.e. what the value of T is. A football match is scheduled to be 90 minutes but in most cases there is a couple of minutes overtime due to injuries and changes of players. (Unlike other breaks in a match these two events are not regarded to be part of the play, so the referee clocks such time.) To get insight in, and understanding of, the model the starting point will be to set T = 2. That is to say, there will only be two instants in the game of approximately 45 minutes each. This, in turn, implies that there could only be two strategy changes during the whole match, not were realistic but transparent. Since there is half-time after 45 minutes in football, the two instant game is best understood in the way that the teams are only able to change their strategy before the game and at half-time. One can also look at the game as a two instant match without specifying the length of an instant, provided they are of equal length. This is actually more accurate, since it is more realistic and since what will come clear below. Formally, since there can only be one goal in an instant, the history set consists of: h = { 2,, 0,, 2}, or, equivalently, the set of states contains s = {(0, 0), (, ), (, 0), (, ), (2, 2), (2, ), (2, 0), (2, ), (2, 2)} The value functions are, with payoffs: v i (t, n) = max s i [p (s)v i (t, n + ) + p 2 (s)v i (t, n ) + ( p (s) p 2 (s))v i (t, n)], for i {, 2}. (2) v (2, n > 0) = v 2 (2, n < 0) = 3 v (2, n = 0) = v 2 (2, n = 0) = (3) v (2, n < 0) = v 2 (2, n > 0) = 0. The value function for each team is maximized over each teams strategy space. This strategy space is not yet fully defined. In section 4.2 all that was said, is that it was a closed set on the real line. Instead of defining it in advance, it will be implicitly defined by the value function. That is to 5

16 say, the upper and lower boundary is to be defined by the maximum and minimum requirement to score a goal. These occurrences, in the simplified game, take place in the second last period. The greatest need to score is when the team is behind by one goal in time, and the smallest need to score is when it is ahead by one goal in time. Hence, the boundaries for the teams are going to be: s = max [p (s)v (, ) + p 2 (s)v (, ) + ( p (s) p 2 (s))v (, )], s s 2 = max [p (s)v (, ) + p 2 (s)v (, ) + ( p (s) p 2 (s))v (, )], s s 2 = max [p (s)v 2 (, ) + p 2 (s)v 2 (, ) + ( p (s) p 2 (s))v 2 (, )] s 2 s = max [p (s)v 2 (, ) + p 2 (s)v 2 (, ) + ( p (s) p 2 (s))v 2 (, )]. s 2 This could also be stated through the following. 5. Time (Half-time) The maximization problem for team in time T (i.e. time ) is (due to eq (2) and eq (3) above): v (, ) = max [3 2p 2 (s)] s (4) v (, 0) = max [ + 2p (s) p 2 (s)] s (5) v (, ) = max [p (s)]. s (6) The solution to equation (4) demands minimizing p 2 (s), which is accomplished by minimize s, by assumption. Hence, the solution is achieved by choosing strategy s. Equation (6) claims that p (s) should be maximized, which is achieved by maximizing s, by assumption. Thus, in choosing strategy s this is reached. Finally, if equation (5) is differentiated to get the first order condition, the following emerges: s p (s) = 2 s p 2 (s). Denote the strategy which solves this F.O.C. by s. The same calculations on team 2:s value function renders similar boundaries, i.e. choose s 2 if team 2 is ahead with one goal and s 2 if it is behind. If the teams still are tied, i.e. if no goals have been scored, the first order condition gives: s 2 p 2 (s) = 2 s 2 p (s), which renders the solution s 2. If the solution for respective team, when the game is tied, is brought together with assumption 2 the outcome becomes: s p (s) = s 2 p 2 (s) s p 2 (s) = p 2 (s) = p (s) 2 s 2 s 2 p (s). (7) s 2 6

17 That is to say; the second assumption also makes the teams symmetric in change of scoring probabilities in the other team s strategy. This, like assumption 2, could of course be discussed. Are the teams really that symmetric? It is probably an exception that they are, but if there were not any restrictions on the probability function, the analyzing would go out of hand very quickly. Conclusion: if the match is tied at time both teams should choose strategy s i in accordance with: p i (s) = p i (s), for i, j {, 2}, (8) s i 2 s j i.e. the solution calculated above combined with equation (7). Otherwise the teams should choose one of the boundary strategies, depending on if it is in the lead of or if it is behind the other team. Due to assumption all these strategies are unique. 5.2 Time 0 (Match-start) The next step is to investigate what will happen in time 0, that is the beginning of the game, and what should be the starting strategy for the two teams. The only possible state is now (0, 0) since teams are tied when the match starts. Thus, the value functions to maximize are: v i (0, 0) = max s i [p (s)v i (, ) + p 2 (s)v i (, ) + ( p (s) p 2 (s))v i (, 0)], for i {, 2}, where the v :s are the solutions to the problem above. The optimal strategy ŝ, which solves the above optimization problem, provides the following first order conditions (only team :s condition is shown): p (s) (v s (, ) v(, 0)) = p 2(s) (v s (, 0) v(, )) / eq. (7) / = p (s) (v s (, 0) v(, )) 2 The equation is the restriction on the team s unique strategy in period 0. It can also be displayed as: p (s) = v (, 0) v (, ) p (s) s v (, ) v (, 0) (9) s 2 / eq. (4, 5, 6) / = + 2p (s, s 2 ) p 2(s, s 2 ) p (s, s 2 ) 2 2p 2 (s, s 2 ) 2p (s, s 2 ) + p 2(s, s 2 ) p (s) s 2 7

18 In order to compare the equilibrium result above, i.e. at time 0, with the equilibrium result at time, one has to know the teams relative probabilities to score. Three possible cases appears: s = s 2 p (s) > p 2 (s), p (s) = p 2 (s) or p (s) < p 2 (s). Or, stated in words, which team has the greatest probability to score for equal amount of attacking intensity. That is to say; which team is, on the average, the better one. This could be due to, e.g., more talent in the team or home-field advantage. The relevant figures must be taken from statistics for the teams in question and is not presented here. 5.3 Equilibrium In the Game Since the standing between the teams before the match is tied (0-0), the only result that can be compared between the two periods is just the tied result. As can be seen in equation 8 the corresponding equation (in time ) to the one above (in time 0) can be written: p (s) s p (s) s 2 = 2. (0) If equation (9) is less than /2 the attacking intensity will increase in the period to come, due to the monotonicity assumption. If it is equal to /2 the intensity will be unchanged and if it is less it will decrease. By letting equation (9) above be equal /2 (equation (0)) the following comes up: 3p (s, s 2) p (s, s 2 ) = 3 2 p 2(s, s 2) p 2 (s, s 2 ). () If the same calculations on team 2 are carried out, the below appears: 6p 2 (s, s 2) p 2 (s, s 2 ) = 2p (s, s 2) 4p (s, s 2 ) + 3. (2) If the left hand side of equation () and equation (2) is less than the right hand side the attacking intensity will increase and the opposite if not. If the equations are brought together, which they must in equilibrium, it becomes: p 2 (s, s 2) = 2 3 ( p (s, s 2 ) 3 p (s, s 2)) (3) This restriction is not so transparent. To make the investigation a bit easier, there will be an assumption about the team s relative probabilities. Team 2:s probability function is assumed to be an affine transformation of team :s probability function with the constant term zero. That is to say: p 2 (s(t, n)) = kp (s(t, n)), k R +. (4) 8

19 The n must change sign since the other teams objective is opposite in sign of n. If equation (3) and (4) are used together, the solution becomes: p (s, s 2) = p (s, s 2 ) k. (5) which becomes the boundary value between increase- or decrease of intensity. Since k always must be greater than zero and the probability function must lie between zero and one, there are two restrictions on the above equation, namely: 2 > 2p (s, s 2 ) + p (s, s 2) k > 4 6p (s,s 2 ) 9. (6) None of them should be any problem in real life since, as written before, the probabilities to score are very small, typical about or below Actually, the figure 0.05 is per minute, that is way it is better to look at the game as a two-period match within the hole match and not 45 minutes instants. The k restriction would then, if the probabilities are assumed to be about the small number mentioned, be about k > /3. This means that no team can be more than 3 times better than the other team, on the average, if the equation should be valid. This arouses partly because of the symmetric assumption (assumption 2). (The k can then, of course, never be greater than 3, because it would then mean that the other team is 3 times as bad.) 5.4 Conclusion What can be deduced is the following: If the attacking is to be greater in period, than in period 0, the restriction on team :s probability function is: p (s, s 2 ) < p (s,s 2 ) k. If the inequality are equal there will be no change in attacking intensity in the following period and if p (s, s 2 ) > p (s,s 2 ), the k attacking will decrease in the period that follows. The k in the equations tells that the restrictions will depend on how much better the other team is, not surprisingly. The greater k, the lower restriction boundary. This means that if a team is much better than the other one, it will have a low boundary value and therefore almost always will increase its attack, when tied, at half-time. While a comparative weaker team, lower k, maybe is happy to gain point and therefore chooses to defend in order to preserve the point they will have if the tied result will stand. They believe the 3 points for winning are to far away. Since p (s, s 2 ) always is less than p (s, s 2 ), due to the assumption, this restriction tells us, if k is small, that the probability function must be rather flat, if the attacking intensity is to decrease in the following period. This, in turn, would mean that the teams level of strategies are not to be so significant for the scoring probabilities. 9

20 It would, then, rather be the intrinsic quality, such as team skills (talent) or home field advantages, of the teams that count for the probability to score a goal. Since this paper is restricted to strategy that discussion ends here with a note that this is an interesting question and a possible idea for some more research. The same calculations on team 2 gives the same result, i.e. it will also enhance its attack if the match is still tied at half time but the restriction now looks like: p 2 (s, s 2 ) = p 2(s,s 2 ) k This simplified two-period game tells what actually is seen in football, that is to say; the attacking intensity enhances along with the match-time, as long as the game is tied. This is partly because of the three-point rule which was introduced for the purpose of giving more interesting matches. In former years there was only two points for a victory. This led, and would lead in this model too, to a more defensive play due to the fact that there was only one point difference between winning and play tied. This is also showed by Brocas and Carillo (2002) [2]. Most of the goals are in fact scored late in the match. According to Palomino et al. (2000) [] this increase over time is particularly significant when winning. They also state that winning teams switch to more defending strategies very early in the match, which could not be seen in this two-period game. 20

21 6 Extended game To get the game more realistic one has to increase the instants and thereby extend the strategy possibilities. But the problem is to decide the length of the instants or, equivalently (since the instants should be of equal length), how great T should be. Instead of deciding the value of T one can investigate what happens with the strategies when T gets larger. To begin with one has to notice that the history set now consists of: h = { T, (T ),..., T, T }. The extreme values of this set will of course never appear in reality but in the model they have a theoretical existence. The value functions are, as before: v i (t, n) = max s i [p (s)v i (t, n + ) + p 2 (s)v i (t, n ) + (7) and the payoffs are now: ( p (s) p 2 (s))v i (t, n)], for i {, 2}, (8) v (T, n > 0) = v 2 (T, n < 0) = 3 v (T, n = 0) = v 2 (T, n = 0) = v (T, n < 0) = v 2 (T, n > 0) = 0. By optimizing the above value functions one gets the generic solution for all t < T and n T (for greater n the match is already won): period t 2 : p i (s) s i p i (s) s j = v i (t, n) v i (t, n ) v i (t, n + ) vi, for i {, 2}. (9) (t, n) This equation is always positive due to the previous assumptions and it is also decreasing in n, the score difference. So within each period there is a decrease in the strategy with greater n. This means: the more a team is leading the lesser they will attack. This does not mean they will score less goals since the equations also tell: the more the other team is behind it chooses to increase its attack intensity (in order to catch up), which, in turn, means that they will expose itself. This vulnerability for attacks can then be taken advantage of by the first team. The problem is to decide what occur between the periods. There are three special cases: one of the teams can be in lead (with one or more goals), it can be tied or the team can be behind (with one or more goals) but the last case is the same as the first for the other team. (The cases are perfectly symmetric between the teams.) So there are just two cases to be examined. 2

22 In order to get something out of the results (the results are going to be rather difficult to interpret what so ever) the assumption of the relative probabilities, that is, equation (4) is assumed to be valid here too. Moreover, the teams are supposed to be homogeneous, that is, they will have the same ability to score, i.e. k = in equation (4). This, as been argued before, is probably more the exception than the rule. (This could be motivated by the fact that the matches are played between teams in the same league. But, of course, even in the same league, the strength between teams differs.) 6. Equilibrium when the teams are tied When the game is tied, n = 0. Equation 9 then becomes: period t 2 : p (s) s = v (t, 0) v (t, ) p (s) v (t, ) v (t, 0). (20) s 2 If the optimized value functions, i.e. equations 7, are substituted into the above equation it becomes: = v0 v + p 0 (v 2v0 + v ) p (v0 v ) + p (v v 2 ) v v0 + p0 (v 2v0 + v ) p (v v0 ) + p (v2 v ), where time, t, has been suppressed and the numeral raised in each term is the score-difference. The equation can also be written as: ( ) ( = v (t, 0) v (t, ) + p 0 v 2v 0+v p v 0 v + p v ) v 2 v 0 v ( ) ( ) v(t, ) v(t, 0) p 0 v 2v 0+v p + p v 2. (2) v v v0 v v0 Since this last equation is concerning time t and the right hand side of equation 20 is concerning time t the following can be said about the expression between the parenthesis (in equation (2)): If it is < The attack increases in the following period. = The attack stays unchanged in the following period. > The attack decreases in the following period. This since the teams are value-maximizers, i.e. they choose the strategy that provides the most value for the team. So if, for example, the value for the team is to decrease in the following period, the suitable strategy for avoiding this would be to decrease the intensity of the attack and thereby minimize the lost of value between the periods. If the expression between the parenthesis (in equation (2)) is equated to the below equation appears: ( (v p 0 = p 2 v )(v0 v ) (v v0 )(v v 2 ) ) (v 2v0 + v )(v v ) (22) 22

23 The three cases now apply to this equation, i.e. eq (22). But since the relation p 0 p always is valid due to assumption, that is to say if a team is leading it will decrease its attack (or at least, not increase it), the numerator is greater than the denominator. That restriction gives the following constraint on the value function: v 2 v v v0 v v 2 v 0. (23) v This constraint demonstrates that the value function must be a (quasi- )convex function in n. This could seem like a contradiction when assumption 3 tells that 2 p 0 and the value function is a function of the probability s 2 function. But if the value function is differentiated twice w.r.t. n one gets (where the first and second derivative of s(n) are assumed to exist): 2 v n 2 = max [ 2 v p s s p 2 s n + v 2 p s p s 2 n + v p 2 s p s n 2 ]. In this expression v p 0 and 2 v 0, because of the quasi-concave p 2 property of v(p). The same is true for p(s) due to assumptions and 3. Moreover, since the attacking intensity is non-increasing in n, s 0. So if 2 s n 2 is assumed to be 0 for all n, which is argued from the fact that there is no great strategy change for sufficient great n, i.e there is assumed to be so called diminishing return of marginal strategy in n. All this together gives the value function the desired properties. So, if the expression between the parenthesis in equation (22) is labelled α, the equation becomes p 0 = αp, which is the boundary value between the increase- and decrease of strategy for a team. If the equal sign holds the team will continue with unchanged strategy in the period to come, while, if p 0 > αp, the attack will decrease the following period and, finally, if p 0 < αp, the attack will increase the period to come. What actually will happen is then solely resting on each teams probability function, which comprise all the important information about the team in question. Since the value function for each team is recursively built up from the probability function they can be produced once all the probabilities are estimated. 23 n

24 6.2 Equilibrium when one team is leading When one of the teams, e.g. Team, is leading, n is greater than one. If n is assumed to be the counterpart for equation (20) and (2) are: = v v0 v 2 v p (s) s p (s) period t 2 : = v (t, ) v (t, 0) (24) v (t, 2) v (t, ) s 2 ( ) ( ) + p v 2 v p v v0 p 0 v 2v 0+v v v0 ( ) ( ) + p 2 v 3 v 2 p 2 v 2 v p + v. (25) p v 0 v 2 v The same concerning the increasing or decreasing of attack is valid as when the teams were tied. That is to say, since the right hand side of the equation (24) is concerning time t and equation (25) concerns time t the following can be said about the expression between the parenthesis: If it is < The attack increases in period t. = The attack stays unchanged in period t. > The attack decreases in period t. If the expression inside the parenthesis is equated to one, as before, it becomes: ( v p 2 v 0 ) ( v v p 0 v ) ( v v0 v + p 2 v0 = p 2 3 v 2 ) ( v v 2 + p 2 v 0 ) v v 2. v The three cases of attacking probabilities is valid to this equation as well. But from here on it is very hard to continue since in this expression there are so many unknown variables of p(s), the probability function. When the teams were tied the calculations rendered only two unknown variables of p(s), but here there are five. All that can be said is that if the left hand side of equation (26) is lesser than the right hand side the team will increase its attack. And the if the opposite is true, the team will decrease its attack. There is no need to check for other n, it will only render more and more unknown variables. From here, one need to get the probability function for a closer investigation and since that is not available the analysis stops here. 6.3 Conclusion In this extended, and more realistic, game it is difficult to see what strategy a team would adopt in different cases since the probability function for scoring is not accessible. But what can be deduced, is that there are going to be strategy changes in certain situations. This since, if a team is leading towards the end of the match, it will use its lowest possible strategy (which was seen in the simplified game) while, if this lead is taken in time t 2 and 24

25 t is sufficient low, the team will use a greater strategy than the lowest in period t if the left hand side of the above equation is greater than the right hand side. That is to say, a leading team will lower its strategy successively during the match as long as they are in the lead. What strategy the team will use is dictated from equation (25). Since this equilibrium is symmetric the opposite can be said if a team is loosing, i.e. if a team is behind by one goal at time t it will enhance its attack the period that follows if the probability function fulfils the corresponding requirement. 25

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27 7 Discussion Professional sports have become an important part in the society and of everyday life. Many individual decisions are affected by sport events. Optimizing the sport can therefore be of great interest but to do that one first has to find a method, a model, which could be used. Surprisingly, economists have almost completely neglected this possibility, even though sports seems to be one of the most natural application to game theory. This paper has presented a game theoretic model of a football match, characterized its equilibria both in a simplified game and in a more complex game. The model predicts the teams strategies in each moment of the match and by that, the probability to score depending on the prevailing time and score, i.e. on the prevailing state of the game. It also describes how strategy and probabilities are affected by the current state of the game. The simplified game, were all the interesting conclusions were made, is of course a too simplistic one since the teams are able to choose strategy only twice during the match. But it nevertheless succeeded to extract some key features that is noticed in football today. e.g. that the attacking intensity enhances along with the match-time, as long as the game is tied. This was exactly what the regulators wanted to achieve when they changed the points for winning, i.e. from two to three points. This, if anything, proves the necessity of a rigorous investigation of the game. The theoretical insights about the game achieved by the model is (maybe) not sensational but it could be of interest for certain groups. One thing that has not been mentioned earlier is the great assistance these types of papers are for the professional gamblers. If one is able to predict the probability for a goal at a certain state of the game, this information would have a great money-value. Since many bookmakers offer live betting on football matches this probability is of great interest; both for the bookmakers and for the players. The bookmaker market is also growing fast due to the Internet. During the last couple of years the Internet has been glutted with bookmakers but is now begin to reach a steady state in the number of bookies. From a positive viewpoint the conclusions that could be deduced from the model could then not only be of interest for the teams, they can also be of interest for the regulators and the bookmakers. So, given the availability of data on football, a natural next step would be to test the theory and the predictions, but that is out of scope of this thesis. 27

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29 References [] Palomino, F.,Rigotti, L. and Rustichini, A. (2000): Skill, Strategy and Passion: an Empirical Analysis of Soccer; Review of Economic Studies, vol 25, p [2] I Brocas and J.D. Carillo (2002): Do The Three-Point Victory and Golden Goal Rules Make Soccer More Exciting? CEPR Discussion Paper No [3] Walker, M. and Wooders, J. (200) Minimax Play at Wimbledon; American Economic Review [4] Chiappori, P.A. and Lewitt, T. and Groceclose, S. (2000) Testing mixed strategy equilibria when players are heteregeneous: the case of penalty kicks in football; American Economic Review [5] Palacios-Huerta, I. (200) Professionals play minimax; mimeo, Brown University [6] Osbourne, M, Rubinstein, A (994): A Course in Game Theory. Cambridge, Massachusetts: MIT Press. 29