When is a multiple bet better than a single ?

The paper addresses an apparent paradox observed in betting on football scores, drawing on 20-year data from the English 2nd tier division (Championship). While accumulator bets have a lower Net Expected Value than single bets, ‘cross double’ bets on the scores, placed over successive playing rounds, produce distinctly better results and indeed a positive return overall. It is argued that this effect rests on the essential stability of the score frequencies across playing seasons and on the bookmakers’ failure, in setting the odds, to allow for occasional and temporary deviations from long run average frequencies. A betting strategy based on overdue scores occurring with compensating frequencies, and possibly clustered together, can then produce positive returns. Neglect of overdue scores can be expressed formally as a bias augmenting the probability of these and turning the odds in the bettor’s favour. It is shown that, while normally the bettor’s disadvantage is compounded in multiple bets, a compounded advantage results once the odds become better than fair. The paper also discusses certain quasi binomial characteristics of the betting involved and explores possible ways of hedging such bets ‘in running’.