# A game of two halves? Football's more a linear Markov process, Brian

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The conventional wisdom has it that football is very much a game of two halves. Actually, it's not. It is a four-state, log-linear Markov process.

On the eve of the World Cup, scientists at Lancaster University have used software based on formulae devised by a Russian mathematician to help to predict the best time during the game to make a substitute, the optimum time to switch from defence to attack, and even to judge the probability of scoring in the time left.

Although the mathematician, Andrei Andreyevich Markov, died in 1922, his equations enable researchers to estimate the optimal time to change tactics, either by making a substitution or by some other change of plan.

"It makes it possible, too, to estimate the probable score,'' said Mike Wright, who has worked on the research with his colleague, Nobuyoshi Hirotsu. In the calculations, honed on an analysis of data from Premiership games, the team developed a series of equations to deal with variables and probabilities.

The variables they took into account in the formulae include any goals that are already scored, the amount of time each team is in possession, the number of successful passes and those that go to the opposition, as well as tackles and unsuccessful tackles, crosses to own players and to the opposition, goal assists, dribbles and goalkeeper passes.

The formulae also include data on how the team performed in the past, and when individual players had been put on or taken off. All this is fed into a computer. At least in theory, out comes the kind of answers that Premiership managers are paid millions to make.

"It is at an early stage, and there are always going to be limitations because you have to make assumptions and you would need to account for players getting tired, or not playing as well as usual,'' said Mr Wright, whose research appears in the Journal of the Operational Research Society. "It could one day be used at the pitchside, more as a guide than a precise rule. But we are not at that stage yet. We need to include more analysis, more on players and formations and so on. Whether or not it does eventually get used will depend on football's culture.

"If you look at American football or baseball, they have people analysing things to death before, during and after a game. Some of the top sides have their own full-time analysts. When a baseball manager is thinking of substituting a pitcher, as well as considering how the player is playing, he will also chat to the analyst sitting next to him.''

Managers who can't wait for the finished thing could try working out the Markov equations on the back of a match programme, but as there are dozens of formulae involved, it is not advised. It could well take longer than 90 minutes to work out.