 2007 Royal Statistical Society 0035–9254/07/56607 Appl. Statist. (2007) 56, Part 5, pp. 607–623 Algorithms for optimal allocation of bets on many simultaneous events Chris Whitrow Imperial College London, UK [Received September 2006. Revised June 2007] Summary. The problem of optimizing a number of simultaneous bets is considered, using pri- marily log-utility. Stochastic gradient-based algorithms for solving this problem are developed and compared with the simplex method. The solutions may be regarded as a generalization of ‘Kelly staking’to the case of many simultaneous bets. Properties of the solutions are examined in two example cases using real odds from sports bookmakers. The algorithms that are developed also have wide applicability beyond sports betting and may be extended to general portfolio optimization problems, with any reasonable utility function. Keywords: Gambling; Kelly staking; Log-utility; Portfolio optimization; Sports betting; Stochastic gradient ascent 1. Introduction Two algorithms are presented to solve the problem of maximizing long-term return from a series of bets on events, which may occur in simultaneous groups. The solution to this problem has widespread applications in investment strategy and gambling games. In particular, this paper considers the problem that is faced by those betting on fixed odds sports events. An investor may identify many bets on a given day, all of which offer value in the odds (i.e. positive expectation). For example, over 40 matches are played, mostly at the same time, on a typical Saturday of the English football league season. It is possible that many of these will offer the chance to make a profitable bet. For example, the investor may have a statistical model which gives good estimates of the probabilities of each result {p i }. For models relating to football, see for example Dixon and Coles (1997), Crowder et al. (2002) or Goddard (2005). The result probabilities from such models can be compared with the best odds that are offered on each result, represented as the return on a unit stake {r i }, which is referred to as the ‘decimal’ odds by bookmakers. The set of ‘value’ bets, V , is {i : r i p i > 1}. These are the events for which the odds are more generous than they should be, giving the gambler a positive expectation. In what follows, we must assume that we have a good model in the sense that it gives unbiased estimates for the event probabilities {p i }. This may be difficult in practice but, with sufficient data and care, it should be possible to reduce the bias to a sufficiently small level. This still does not guarantee that the model will be sufficiently powerful to identify many good bets, given that the odds are never ‘fair’. In the author’s own experience, however, it is possible to develop good models for football betting and to use these in conjunction with the algorithms that are presented in this paper to produce substantial profits in the course of a season. Of course, sports betting is not the only context in which these methods are of interest. Address for correspondence: Chris Whitrow, Institute for Mathematical Sciences, Imperial College London, 53 Princes Gate, South Kensington, London, SW7 2PG, UK. E-mail: c.whitrow@imperial.ac.uk