Gambling as investment: a generalized Kelly criterion for optimal allocation of wealth among risky assets David C J McDonald Ming-Chien Sung Johnnie E V Johnson Centre for Risk Research Southampton Management School University of Southampton United Kingdom 2 nd Asia Pacific Conference on Gambling & Commercial Gaming Research Dec 2013

2 Gambling as investment Most gambling has a negative expected return, i.e., the gambler loses wealth in the long term Suppose instead you found a positive expected return gamble: –You are a skilled sports or horserace bettor –Card counting in blackjack –A biased roulette wheel –Even a faulty slot machine

3 Gambling as investment If the gamble is available for the long term (so you can bet on it repeatedly), then that gamble is an investment But how much of your wealth do you allocate to that investment? This is like a classic portfolio allocation problem in finance

4 The wrong criteria The answer (of how much to bet) depends on what question is being asked: what is the criterion? An extreme answer is to bet all your wealth This is equivalent to maximizing your expected return from the bet But the first time you lose the bet you will go bust! At the other extreme, if you minimize the probability of going bust, your expected return is minimized (wealth is increased too slowly)

5 The Kelly criterion The answer was given by John Kelly Jr (1956): bet the fixed proportion of your wealth that maximizes the log of expected return Bet = edge / odds This is the principle of: ‘Bet your beliefs’ (moderated by your return)

6 Kelly investing Warren Buffett thinks like a Kelly investor, making concentrated bets on a small number of assets, rather than diversification: “If you are a professional and have confidence, then I would advocate lots of concentration.”

7 The advantages of Kelly Maximizes long term growth rate Used by Ed Thorp in his blackjack card-counting system Advocated by professional sports and horserace gamblers (e.g., Benter, 1994)

8 Background to the problem There are many different Kelly problems, depending on the type of market and economic condition - not all have been solved! Bookmaker (Kelly’s original 1956 formula) Pari-mutuel (Isaacs, 1953; Levin, 1994) Exchange (backing/laying and commission) Restricted bet size Simultaneous games (Insley et al., 2004; Grant, 2008) etc.

9 An encompassing framework Our goal is to solve all possible Kelly problems efficiently To do this, we look at whether the problem is concave – if it is, it has a unique solution We establish concavity to (a) know that the optimized Kelly fractions are unique, (b) know that it is possible to find the optimal solution using numerical methods Literature: sufficient conditions for concavity in some cases (Algoet and Cover, 1988; Kallberg and Ziemba, 1994)

10 Some notation It costs r to make a bet that returns 1 if an event occurs and returns 0 otherwise The probability of the event occurring is p > r Fraction of wealth to bet given by x Return (‘odds’) given by R = 1/r

11 The general Kelly problem The general problem is over a range of n bets where: for at least one bet overall (the ‘sub-fair’ case) Order the bets by ‘merit order’: Then

12 General concavity conditions A single betting event consists of n mutually exclusive outcomes i, each with probability p i of occurring. Up to m bets, given by the vector are placed on any single outcome or combination of outcomes occurring or not occurring, with constraints on bet sizes given by

13 General concavity conditions If outcome i occurs, the factor by which the bettor’s wealth changes is given by the function and the utility function of the bettor is a concave function where Hence the utility of wealth after the event is given by

14 General concavity conditions So the Lagrangian for the constrained maximization problem is Proposition. Suppose that all wealth factor functions g i and all active constraint functions G j are linear. Then Λ is concave. Rough proof: The U i are concave functions of the linear g, so are also concave, and their concavity is preserved under positive scaling (by p) and summation. The G are linear.

15 What does this mean? Most Kelly problems you can think of can be shown to satisfy the general concavity conditions So, they can be solved by a computer (using numerical methods)

16 General concavity conditions The general conditions apply in all these cases: Bookmaker Exchange backing/laying (without commission) Restricted bet size Simultaneous games … and hopefully many other cases you could think of!

17 General concavity conditions They do NOT apply in: Pari-mutuel – bet size changes returns However, this problem has been solved (Isaacs, 1953; Levin, 1994) Exchange with commission – bet selection and size changes returns This one can also be solved separately

18 Market efficiency Kelly betting solutions are mostly theoretical, since they require that the gambler has an advantage, but in general, this is not the case However, a good way to show that a market is inefficient (e.g., a horserace betting market), is to construct a Kelly strategy So these conditions allow us to test market efficiency in most markets

19 Conclusion Kelly betting (log-optimal betting) is in many senses the ‘optimal’ allocation of wealth among risky assets (bets), provided you have some advantage There are many different market settings with different types and combinations of bet, payoff structures, etc. Many of these problems can be shown to be concave (using our general conditions), i.e., there is a unique global optimum that can be found by numerical methods Those that aren’t concave can be fiendish, but there are algorithmic solutions

20 Thank you Any questions?