Posted on July 24, 2018

We continue our examples but now with a game of chance with uneven pay-off. This part will mirror Part 2 and it will be important to understand so we can finally conclude our soccer betting example that we will develop in Part 4.

Similar to Part 2, let our random variable $X$ be the number of heads obtained after $n = $ 5 tosses. The expected value of heads obtained is 2.5.

Now we want to calculate the expected value of our winnings with an uneven payoff. This is also a random variable $W$ that is a function of $X$, and is defined as the following

$\begin{align*} W = 2 \cdot X - 1 \cdot (5-X) \end{align*}$

$W$ is equal to 2 times the number of Heads we land minus 1 unit times the number of Tails that land. Remember we established earlier that we will win 2 units if the coin lands Heads. Therefore, whatever amount of Heads land, our winnings will be double that amount. However, if the coin lands Tails (or Not Heads), we will lose 1 unit. If our total number of tosses is 5, and $X$ refers to the number of Heads that land, then the difference between our total number of tosses and whatever number of Heads is the number of Tails that landed, in short, $5-X$.

Computing the expected value of W is straightfoward,

$\begin{align*} E[W] & = E[2 \cdot X - 1 \cdot (5-X)] \\ & = E[2 \cdot X - 5 \cdot + X] \\ & = E[3X] - 5 \\ & = 3 \cdot E[X] - 5 \\ & = 2.5 \end{align*}$

Let make sense of this. Since our coin is fair, with every toss we have 50% probabilities of landing Heads or Tails. Therefore, the expected value of our winnings in a single coin toss would be equal to $0.5 \cdot 2 - 0.5 \cdot 1 = 0.5$ (the midpoint between 2 and -1). After 5 successive coin tosses, our expected value increases to $2.5$.

Again, what does Kelly have to say about this? We slightly re-word the example above by adding an initial capital, call it $X_0$, of 100 units. Our new game is different than the above because now we are risking our wealth for a possible profit or loss, whereas before we simply won or loss after every coin toss. Lets define our uneven pay-off as 1.5 units for every 1 unit bet on Heads to land. Contrary, if Tails land, we only lose 1 unit. We refer to this as European Odds of $1.50$ in sports betting.

From Part 1, we already worked out a game of chance with **uneven pay-off**. We stand to profit $0.5$ unit if we have odds of $1.5$, such that for every Heads that lands, our wealth will increase by $0.5$ unit times whatever the fraction of our wealth Kelly's Criterion proposes. Therefore, we define $b = 0.5$ and our wealth will increase by $1+bf$ or decrease by $1-f$ (please refer to Part 1 if you are unsure where these formulas came from). Our wealth after 5 tosses would like the following,

$\begin{align*} X_5 = X_0(1+bf)^H (1-f)^T \quad\quad \text{H for Heads, T for Tails} \end{align*}$

where $f^* = \frac{pb - q}{b} = \frac{0.5 \cdot 0.5 - 0.5}{0.5} = -0.5$, and is the fraction of our wealth that maximizes the expected value in increase of wealth defined as $\frac{H}{5}\log (1+bf) + \frac{T}{5} \log(1-f)$.

Lets compute the expected value of increase of our wealth maximized by $f^*$

$\begin{align*} & E[\frac{H}{5}\log (1+bf^*) + \frac{T}{5} \log(1-f^*)] \\ & = 0.5 \cdot \log (1 + 0.5 \cdot (-0.5)) + 0.5 \cdot \log (1 + 0.5) \\ & = 0.5 \cdot \log (1 - 0.25) + 0.5 \cdot \log (1 + 0.5) \\ & = 0.02558 \end{align*}$

So let backtrack. Kelly's Criterion suggests betting a negative fraction of our wealth when our probabilities are 50%-50%.In other words, Kelly is suggesting that we **hedge** in the opposite direction, and it makes complete sense. At 50%-50% probabilities, we stand to profit $0.5$ units for every $1$ unit we risk. In other words, we stand to lose more than what we would profit!

Another way to look at this is to simply commute the quantities of the equation

$\begin{align*} & 0.5 \cdot \log (1 - 0.25) + 0.5 \cdot \log (1 + 0.5) \quad \text{left quantity is Heads, right quantity is Tails}\\ & = 0.5 \cdot \log (1 + 0.5) + 0.5 \cdot \log (1 - 0.25) \quad \text{we commute quantities} \end{align*}$

and interpret accordingly. We bet 50% of our wealth on Tails landing (hedge) on every coin toss in order to maximize our increase of wealth.

Now lets suppose we have analyzed the historic performance of this coin that pays out 1.5 units for every unit we bet. We have determined with great confidence that the probability of winning (obtaining Heads) is 80% after 5 tosses. Now $f^* = \frac{0.8*0.5 - 0.2}{0.5} = 0.4$. That is, in order to maximize our expected value of increase of our wealth in each trial we must bet $40$% of our wealth. The expected value of increase of our wealth after each trial, maximized by $f^*$, would be

$\begin{align*} & E[\frac{H}{5}\log (1+bf^*) + \frac{T}{5} \log(1-f^*)] \\ & = 0.8 \cdot \log (1 + 0.5 \cdot 0.4) + 0.2 \cdot \log (1 - 0.4) \\ & = 0.01898 \\ & \approx 2.0\% \end{align*}$

So after 5 successive bets on Heads the **log** average of our wealth is expected to be $0.01898\cdot 5$ times bigger than our initial capital. That is, after 5 tosses betting 40% of our wealth on Heads our final wealth is expected to be on average $e^{0.01897524691 * 5} = 1.09952276377$ times bigger than our initial wealth, that is $X_5 = 100 * 1.04469367879 \approx 109.95$

Lets remember that our increase of wealth after each trial was equal to $\log (\frac{X_n}{X_0})^\frac{1}{n}$. This was derived in Part 1

This concludes Part 3. It is not unheard of, generally in sports betting, that if Kelly's Criterion suggests a negative value the bet should be immediately discarded. However, with the rise of betting exchanges that allow us to back or lay outcomes, a negative value takes on a new light as a hedge. As we have all heard, financial investments heavily rely on hedging as it provides a way to manage risk.