Gambling as investment a generalized Kelly criterion for
Gambling as investment: a generalized Kelly criterion for optimal allocation of wealth among risky assets 2 nd Asia Pacific Conference on Gambling David C J Mc. Donald Ming-Chien Sung Johnnie E V Johnson Centre for Risk Research Southampton Management School University of Southampton United Kingdom & Commercial Gaming Research Dec 2013
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 2
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 3
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) 4
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) 5
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. ” 6
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) 7
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. 8
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) 9
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 10
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 11
General concavity conditions • A single betting event consists of n mutually exclusive outcomes i, each with probability pi 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 12
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 13
General concavity conditions • So the Lagrangian for the constrained maximization problem is • Proposition. Suppose that all wealth factor functions gi and all active constraint functions Gj are linear. Then Λ is concave. • Rough proof: The Ui 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. 14
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) 15
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! 16
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 17
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 18
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 algorithmic solutions 19
Thank you • Any questions? • d. mcdonald@soton. ac. uk 20
- Slides: 20