Math. Appl. 9 (2020), 67–81 DOI: 10.13164/ma.2020.06 IN-GAME BETTING AND THE KELLY CRITERION ROBIN ANDERSEN, VEGARD HASSEL, LARS MAGNUS HVATTUM and MAGNUS STÅLHANE Abstract. When a bet with a positive expected return is available, the Kelly crite- rion can be used to determine the fraction of wealth to wager so as to maximize the expected logarithmic return on investment. Several variants of the Kelly cri- terion have been developed and used by investors and bettors to maximize their performance in inefficient markets. This paper addresses a situation that has not, hitherto, been discussed in academic literature: when multiple bets can be placed on the same object and the available odds, true probabilities, or both, vary over time. Such objects are frequently available in sports betting markets, for example, in the case of in-game betting on outcomes of soccer matches. We adapt the Kelly criterion to support decisions in such live betting scenarios, and provide numerical examples of how optimal bet sizes can sometimes be counter-intuitive. 1. Introduction The task of choosing the optimal wealth allocation amongst a set of candidate secu- rities in financial markets is usually referred to as portfolio optimization. This term was first discussed by Markowitz [21] in a paper considered to be the foundation of modern portfolio theory. The core of modern portfolio theory is the mean-variance model, which rests on the assumption that an optimal portfolio can be constructed in such a way that the financial return is maximized for a given risk level or vice- versa. The portfolio return is here defined as a linear combination of the returns on the individual investments and the risk coincides with the covariance matrix of these returns. Another approach to portfolio optimization was proposed by Kelly [15] based on a problem in information theory. The approach is to allocate fractions of wealth to each asset such that these fractions maximize the expected logarithmic growth rate of the investor’s wealth. In fact, this allocation also maximizes the expected utility for investors with a logarithmic utility function with respect to their wealth [4, 24]. This allocation is known as the Kelly criterion and does not necessarily lie on the efficient frontier of the mean-variance model [32]. The Kelly criterion has become a popular tool in betting [16] as well as a sup- plement for academic research on inefficiencies in betting markets. This paper is motivated by the evaluation of odds when betting on the final outcome of a soccer MSC (2010): primary 91B06, 91B16; secondary 91. Keywords: gambling, sport, probability, utility, live odds. The authors wish to thank an anonymous reviewer for providing comments that helped us to significantly improve an initial version of this manuscript. 67