What if my preferences for making an extra few dollars outweigh the risk of losing the same amount? I will stop on why one's preferences might be asymmetric in a little while in this section, and there is scientific evidence that this asymmetry is ingrained in our minds for evolutionary reasons, but you are right, I have to optimize the expected value of the asymmetric function of the parameterized utility now, as follows:

Why would an asymmetric function surface in the analysis? One example is repeated bets or re-investments, also known as the Kelly Criterion problem. Although originally, the Kelly Criterion was developed for a specific case of binary outcome as in a gambling machine and the optimization of the fraction of money to bet in each round (A New Interpretation of Information Rate, Bell System Technical Journal 35 (4): 917–926, 1956), a more generic formulation as an re-investment problem involves a probabilistic distribution of possible returns.

The return over multiple bets is a product of individual return rates on each of the bets—the return rate is the ratio between the bankroll after the bet to the original bankroll before each individual bet, as follows:

This does not help us much to optimize the total return as we don't know how to optimize the product of i.i.d. random variables. However, we can convert the product to a sum using log transformation and apply the central limit theorem (CLT) to approximate the sum of i.i.d. variables (provided that the distribution of r _i is subect to CLT conditions, for example, has a finite mean and variance), as follows:

Thus, the cumulative result of making N bets would look like the result of making N bets with expected return of , and not

As I mentioned before, the problem is most often applied for the case of binary bidding, although it can be easily generalized, in which case there is an additional parameter: x, the amount of money to bid in each round. Let's say I make a profit of W with probability p or completely lose my bet otherwise with the probability (1-p). Optimizing the expected return with respect to the following additional parameter:

The last equation is the Kelly Criterion ratio and gives you the optimal amount to bet.

The reason that one might bet less than the total amount is that even if the average return is positive, there is still a possibility to lose the whole bankroll, particularly, in highly skewed situations. For example, even if the probability of making 10 x on your bet is 0.105 (W = 10, the expected return is 5%), the combinatorial analysis show that even after 60 bets, there is roughly a 50% chance that the overall return will be negative, and there is an 11% chance, in particular, of losing (57 - 10 x 3) = 27 times your bet or more:

akozlov@Alexanders-MacBook-Pro$ scala
Welcome to Scala version 2.11.7 (Java HotSpot(TM) 64-Bit Server VM, Java 1.8.0_40).
Type in expressions to have them evaluated.
Type :help for more information.27 

scala> def logFactorial(n: Int) = { (1 to n).map(Math.log(_)).sum }
logFactorial: (n: Int)Double

scala> def cmnp(m: Int, n: Int, p: Double) = {
     |   Math.exp(logFactorial(n) -
     |   logFactorial(m) +
     |   m*Math.log(p) -
     |   logFactorial(n-m) +
     |   (n-m)*Math.log(1-p))
     | }
cmnp: (m: Int, n: Int, p: Double)Double

scala> val p = 0.105
p: Double = 0.105

scala> val n = 60
n: Int = 60

scala> var cumulative = 0.0
cumulative: Double = 0.0

scala> for(i <- 0 to 14) {
     |   val prob = cmnp(i,n,p)
     |   cumulative += prob
     |   println(f"We expect $i wins with $prob%.6f probability $cumulative%.3f cumulative (n = $n, p = $p).")
     | }
We expect 0 wins with 0.001286 probability 0.001 cumulative (n = 60, p = 0.105).
We expect 1 wins with 0.009055 probability 0.010 cumulative (n = 60, p = 0.105).
We expect 2 wins with 0.031339 probability 0.042 cumulative (n = 60, p = 0.105).
We expect 3 wins with 0.071082 probability 0.113 cumulative (n = 60, p = 0.105).
We expect 4 wins with 0.118834 probability 0.232 cumulative (n = 60, p = 0.105).
We expect 5 wins with 0.156144 probability 0.388 cumulative (n = 60, p = 0.105).
We expect 6 wins with 0.167921 probability 0.556 cumulative (n = 60, p = 0.105).
We expect 7 wins with 0.151973 probability 0.708 cumulative (n = 60, p = 0.105).
We expect 8 wins with 0.118119 probability 0.826 cumulative (n = 60, p = 0.105).
We expect 9 wins with 0.080065 probability 0.906 cumulative (n = 60, p = 0.105).
We expect 10 wins with 0.047905 probability 0.954 cumulative (n = 60, p = 0.105).
We expect 11 wins with 0.025546 probability 0.979 cumulative (n = 60, p = 0.105).
We expect 12 wins with 0.012238 probability 0.992 cumulative (n = 60, p = 0.105).
We expect 13 wins with 0.005301 probability 0.997 cumulative (n = 60, p = 0.105).
We expect 14 wins with 0.002088 probability 0.999 cumulative (n = 60, p = 0.105).

Note that to recover the 27 x amount, one would need to play only additional rounds on average with these favourable odds, but one must have something to bet to start with. The Kelly Criterion provides that the optimal is to bet only 1.55% of our bankroll. Note that if I bet the whole bankroll, I would lose all my money with 89.5% certainty in the first round (the probability of a win is only 0.105). If I bet only a fraction of the bankroll, the chances of staying in the game are infinitely better, but the overall returns are smaller. The plot of expected log of return is shown in Figure 02-3 as a function of the portions of the bankroll to bet, x, and possible distribution of outcomes in 60 bets that I just computed. In 24% of the games we'll do worse than the lower curve, in 39% worse than the next curve, in about half—44%—a gambler we'll do the same or better than the black curve in the middle, and in 30% of cases better than the top one. The optimal Kelly Criterion value for x is 0.0155, which will eventually optimize the overall return over infinitely many rounds:

Figure 02-3. The expected log of return as a function of the bet amount and possible outcomes in 60 rounds (see equation (2.2))

The Kelly Criterion has been criticized for being both too aggressive (gamblers tend to overestimate their winning potential/ratio and underestimate the probability of a ruin), as well as for being too conservative (the value at risk should be the total available capital, not just the bankroll), but it demonstrates one of the examples where we need to compensate our intuitive understanding of the "benefit" with some additional transformations.

From the financial point of view, the Kelly Criterion is a much better description of risk than the standard definition as volatility or variance of the returns. For a generic parametrized payoff distribution, y(z), with a probability distribution function, f(z), the equation (2.3) can be reformulated as follows. after the substitution r(x) = 1 + x y(z), where x is still the amount to bet:

It can also be written in the following manner in the discrete case:

Here, the denominator emphasizes the contributions from the regions with negative payoffs. Specifically, the possibility of losing all your bankroll is exactly where the denominator is zero.

As I mentioned before, interestingly, risk aversion is engrained in our intuitions and there seems to be a natural risk-aversion system of preferences encoded in both humans and primates (A Monkey Economy as Irrational as Ours by Laurie Santos, TED talk, 2010). Now enough about monkeys and risk, let's get into another rather controversial subject—the exploration-exploitation trade-off, where one might not even know the payoff trade-offs initially.