Kelly's fractional staking updated for betting exchanges

In 1956, Kelly developed a staking system to calculate the optimal fraction of wealth to bet on each of a series of favourable bets. In line with historical betting markets, Kelly considered only back bets on events with a single favourable outcome. The recent advent of betting exchanges makes a much wider range of bets available. We extend Kelly's algorithm to betting on a single market, with both back and lay bets available, and we demonstrate the superiority of doing so with betting data collected during the 2009/2010 English Premiership season. We further show that a more general problem with multiple dependent markets can be described as linearly constrained convex optimization. As numerical solutions to these problems are nearly as advanced as those for linear systems, we argue that it is possible to consider this more general case to be solved (at least numerically). We demonstrate how it is possible to formulate such a problem in the case of football bets and we demonstrate the superior growth rate of such a strategy. Two further issues which Kelly did not have to consider are commission and the variability of market size. We modify the objective function which we have been maximizing to take account of commission and further improve the growth rate of wealth in our test data. We begin a discussion on market size, adding constraints to take account of it in our numerical solution and observe the effect. We note the level of wealth at which further growth is impeded by the market size.