It is generally accepted that the most effective form of staking is based upon Kelly staking. The underlying assumption is that the bettor seeks to maximise his bankroll in the long run. While this is a perfectly valid objective, it is possible to have other valid objectives also. For example, an alternative objective is simply to win the most possible money at any given time. A more complex objective might be to maximise the amount of money won, while constraining the risk to be limited to a given maximum value. In Kelly's original paper he recognises this and discusses an alternative hypothetical bettor whose wife only allows him to gamble a set amount of money per week. In this latter case, the optimal strategy is simply to wager all the money on the outcome with the highest expectation.
Economists would describe these two cases, using the concept of utility functions. A utility function is a function which seeks to put a numerical value upon the utility or benefit that an individual gets from a particular course of action. There are an infinite number of possible utility functions. The correct/appropriate utility function for use in any particular case will depend upon the individual. In Kelly's original paper, the utility function used is the natural logarithmic function.
The use of the logarithmic function as utility function dates all the way back to Daniel Bernoulli early in the early 18th century. In Bernoulli's own words: "The determination of the value of an item must not be based on the price, but rather on the utility it yields....... there is no doubt that a gain of 1000 ducats is more significant to the pauper than to the rich man, though both gain the same amount."
Bernoulli's formulation of the logarithmic utility function was motivated by an effort to solve the so-called St. Petersburg's paradox. This paradox proposes a game of chance, which you pay a fixed fee to enter. A fair coin is tossed repeatedly until the tail's first appears in the game. The pot starts at one dollar and is doubled every time a head appears. You win whatever is in the pot after the game ends. The question is how much you should be willing to pay to enter this game. The conventional method of solving this problem would be to look at all the possible outcomes and how much cost is associated with each. With probability 1/2, you win one dollar; with probability 1/4 you win two dollars; with probability 1/8, you win four dollars, etc. The expected value is calculated by summing all of these. Thus 1/2 multiplied by one plus 1 such four report by the two plus 1/8 multiplied by four. This equals 1/2 plus 1/2 plus 1/2 etc. i.e. it sums to infinity - infinite expectation!
The naive approach would suggest that this game should be played irrespective of the entry fee. Bernoulli's insight was that there was a diminishing marginal utility of money, i.e. 1000 ducats to the rich man was worth less than to the pauper. Bernoulli wasn't the only mathematician around this time to investigate the paradox. Several years earlier, a Swiss mathematician, Gabriel Cramer, wrote a letter to Bernoulli's cousin Nicolas, outlining how the problem could be solved by using a square root utility function.
So, which is right? In fact, neither! Those utility functions could be exploited by changing the rules of the game with a different payout structure. The subject of appropriate utility functions is still a hot topic in behavioural finance today. Nobel prizewinners lineup on opposite sides of the ring and square off intellectually about the merits and demerits of the logarithmic utility function. In the language of the punters’ paradise regular, some of these heavyweight support the Kelly criterion, while some don't. The best advice we can give here is simply to decide upon the utility function that describes your preferences. That may be Kelly staking, fractional Kelly staking, or some other format - maybe even staking all your money on the outcome of the highest expectation, just like the hen-pecked husband in Kelly's original paper!
Whatever you choose, best of luck!
BBC
