Abstract
In a previous article, I showed how to derive the Kelly criterion, which is a method for determining bet amounts in gambling.
This is a method of determining a portfolio of bets to maximize a logarithmic utility function, and is a special case of a mathematical and economic problem. In this article, we extend the Kelly criterion to a more general utility function (CRRA-type utility function) and explain how to determine a betting portfolio based on a gambler’s risk aversion.
Expected Utility Function
Let assume that the measure of how desirable an individual is regarding the wealth or income realized as a result of making an uncertain decision such as gambling or investing (called utility in economics) can be expressed as the function \(\small u(W)\). This function is called the utility function, and the function:
\[ \small U(W) = E[u(W)], \]
which calculates the expected value for uncertainty, is called the expected utility function. When
\[ \small u(C)=E[u(W)] \]
holds true for the wealth \(\small W\) after gambling and the wealth \(\small C\) if one holds the cash without gambling, then \(\small C\) is called certainty equivalent. The certainty equivalent should be smaller than the expected value \(\small E[W]\) of wealth \(\small W\) after gambling, so
\[ \small u(C) = E[u(W)] < u(E[W]) \]
would hold. If this equation is equal, then the gambler is risk-neutral; conversely, if \(\small E[u(W)] > u(E[W])\), then the gambler is risk-seeking.
In this sense, the shape of the utility function with respect to wealth can be thought of as expressing an individual’s preference for risk. A utility function that satisfies the condition for a risk-averse utility function is concave, i.e. \(\small u”(W)<0\). The smaller the second derivative of the utility function with respect to wealth, the more risk-averse the utility function is. There are two measures to measure the degree of this risk aversion:
\[ \begin{align*} &\small A(W) = -\frac{u”(W)}{u'(W)} \\ &\small R(W) = -\frac{u”(W)}{u'(W)}W. \end{align*} \]
\(\small A(W)\) is called absolute risk aversion, and \(\small R(W)\) is called relative risk aversion. Since absolute values of utility functions do not necessarily have meaning, they are evaluated by standardizing them as the rate of increase in utility in response to an increase in wealth. Absolute risk aversion is risk aversion where the amount of risk is the same regardless of wealth. To illustrate, if your wealth is 10 thousand dollars or 1 million dollars, you evaluate the risk of gaining or losing 1 thousand dollars in the same way. Relative risk aversion is a way of assessing risk as a ratio to the level of wealth, and it can be imagined that the Kelly criterion considers risk aversion in this way. Generally speaking, relative risk aversion is more likely to fit human perception, so this is the measure used in most cases.
A utility function in which the degree of relative risk aversion remains constant regardless of the level of wealth is called a Constant Relative Risk Aversion (CRRA) utility function, that is,
\[ \small R(W) = -\frac{u”(W)}{u'(W)}W = \gamma = \text{const.}. \]
The specific shape of the utility function can be found by solving differential equation:
\[ \small u”(W) = -\frac{\gamma}{W}u'(W). \]
This equation is called the Euler-Cauchy differential equation, and it is known that the solution can be given by power functions or logarithmic functions. In practice,
\[ \small u(W) = \left\{ \begin{array}{ll}\frac{W^{1-\gamma}-1}{1-\gamma},& \quad \gamma \neq 1 \\ \log(W), & \quad \gamma=1 \end{array}\right. \]
is often used as a CRRA-type utility function. When \(\small \gamma=0\), \(\small u(W)=W-1\), which would be a risk-neutral utility function. Also, when \(\small \gamma=1\), the utility function is logarithmic, which corresponds to the Kelly criterion. If we actually calculate the derivative, we can confirm that the assumption of constant relative risk aversion is satisfied. Under this utility function assumption, let us find the optimal betting strategy in the same way as with the Kelly criterion.
Optimal Betting Strategies for CRRA-type Utility Functions
As with the general Kelly formula, let us consider the case where there is a house edge. In other words, we assume that one of the \(\small n\) choices will win, and the amount bet on each choice will be denoted as \(\small A_1,\cdots,A_n\). In this case, the odds of each option are given by
\[ \small \alpha_i = \frac{(1-\theta)\sum_{i=1}^n A_i}{A_i}. \]
Here, \(\small \theta\) is the house’s edge ratio. Note that in this case, the sum of the winning probabilities calculated backwards from the odds will not be 1, but will be:
\[ \small \sum_{i=1}^n q_i = \sum_{i=1}^n \frac{1}{\alpha_i} = \frac{1}{1-\theta} > 1. \]
Let the gambler’s estimated probability of winning be denoted as \(\small p_1,\cdots,p_n\) and the proportion of the funds he or she will keep be denoted as \(\small b\). In this case, the gambler’s optimization problem to maximize the CRRA-type utility function can be expressed as:
\[ \begin{align*} \small \max_{b, f_1, \cdots, f_n} \; & \small E[u(W)] = \sum_{i=1}^n p_i u \left(b+f_i \alpha_i \right) \\ \small \text{s.t.} \; & \small b + \sum_{i=1}^n f_i = 1, \; b \geq 0, \; f_i \geq 0 \; \forall i. \end{align*} \]
Here,
\[ \small u(W) = \left\{ \begin{array}{ll}\frac{W^{1-\gamma}-1}{1-\gamma},& \quad \gamma \neq 1 \\ \log(W), & \quad \gamma=1 \end{array}\right.. \]
Let us solve the above optimization problem in the same way as the Kelly criterion. Introducing the slack variable \(\small k,\lambda_1,\cdots,\lambda_n \), defining the Lagrangian as:
\[ \small L(b, f_1,\cdots, f_n, k, \lambda_1,\cdots,\lambda_n) = \qquad \qquad \qquad \qquad \qquad \\ \small \qquad \qquad \sum_{i=1}^n p_i u \left(b+ f_i \alpha_i \right) + k \left( 1-b- \sum_{i=1}^n f_i \right) + \sum_{i=1}^n \lambda_i f_i \]
and finding the Karush-Kuhn-Tucker (KKT) condition, we get
\[ \begin{align} & \small \frac{\partial L}{\partial f_i} = \frac{p_i \alpha_i}{(b+f_i \alpha_i)^\gamma}-k + \lambda_i = 0, \quad \forall \; i \\ & \small \frac{\partial L}{\partial b} = \sum_{i=1}^n \frac{p_i}{(b+f_i \alpha_i)^\gamma} – k = 0\\ & \small \frac{\partial L}{\partial k} = 1 -b -\sum_{i=1}^n f_i = 0 \\ & \small \lambda_if_i = 0, \quad \forall \; i. \end{align} \]
Note that if \(\small f_i > 0 \), then \( \small \lambda_i = 0 \), and if \(\small f_i = 0 \), then \( \small \lambda_i > 0 \). By rearranging the first equation of the KKT condition,
\[ \small f_i = \max\left\{\frac{p_i^{\frac{1}{\gamma}} q_i^{\frac{\gamma-1}{\gamma}}}{k^{\frac{1}{\gamma}}}-bq_i, 0 \right\} \]
holds. For the second equation, if we calculate the subscripts for \( \small f_i >0 \) and \( \small f_i =0 \) separately,
\[ \small \sum_{f_i>0}\frac{p_i}{(b+f_i \alpha_i)^\gamma}+\frac{1}{b^\gamma}\sum_{f_i=0} p_i = k \]
holds. Therefore,
\[ \small \sum_{f_i>0}\frac{p_i}{(b+f_i \alpha_i)^\gamma}+\frac{1}{b^\gamma}\left(1-\sum_{f_i>0} p_i\right) = k \]
can be obtained. If we sum the first equation of the KKT condition over \(\small f_i>0\), we get
\[ \small \sum_{f_i>0}\frac{p_i}{(b+f_i \alpha_i)^\gamma} = k\sum_{f_i>0}\frac{1}{\alpha_i}, \]
so by substituting and rearranging the equation we get
\[ \small b = \frac{1}{k^{\frac{1}{\gamma}}}\left(\frac{1-\sum_{f_i>0} p_i}{1-\sum_{f_i > 0} q_i}\right)^\frac{1}{\gamma}. \]
By calculating the sum of \(\small f_i\) for the subscript of \( \small f_i >0 \), we get
\[ \small \sum_{f_i>0}f_i = 1-b = \sum_{f_i>0}\frac{p_i^{\frac{1}{\gamma}} q_i^{\frac{\gamma-1}{\gamma}}}{k^{\frac{1}{\gamma}}}-b\sum_{f_i>0}q_i. \]
By substituting the equation for \(\small b\) and rearranging for \(\small k\), we can get
\[ \small k^{\frac{1}{\gamma}}=\sum_{f_i > 0}p_i^{\frac{1}{\gamma}} q_i^{\frac{\gamma-1}{\gamma}}+\left(\frac{1-\sum_{f_i>0} p_i}{1-\sum_{f_i > 0} q_i}\right)^\frac{1}{\gamma}\left(1-\sum_{f_i > 0} q_i\right). \]
Therefore, the ratio of funds to be kept on hand, \(\small b\), can be calculated as:
\[ \small b = \frac{\left(\frac{1-\sum_{f_i>0} p_i}{1-\sum_{f_i > 0} q_i}\right)^\frac{1}{\gamma}}{\sum_{f_i > 0}p_i^{\frac{1}{\gamma}} q_i^{\frac{\gamma-1}{\gamma}}+\left(\frac{1-\sum_{f_i>0} p_i}{1-\sum_{f_i > 0} q_i}\right)^\frac{1}{\gamma}\left(1-\sum_{f_i > 0} q_i\right)}.\]
Without loss of generality, we reorder the subscripts to favor the gambler. That is, let assume \( \small p_1 /q_1 > p_2 /q_2 > \cdots > p_n / q_n \). In this case, by calculating \( \small t^{\ast} \) that satisfies
\[ \small t^{\ast} = \arg \min_t \frac{\left(\frac{1-\sum_{i=1}^t p_i}{1-\sum_{i=1}^t q_i}\right)^\frac{1}{\gamma}}{\sum_{i=1}^tp_i^{\frac{1}{\gamma}} q_i^{\frac{\gamma-1}{\gamma}}+\left(\frac{1-\sum_{i=1}^t p_i}{1-\sum_{i=1}^t q_i}\right)^\frac{1}{\gamma}\left(1-\sum_{i=1}^t q_i\right)} \quad \text{s.t.} \;\; \sum_{i=1}^t q_i < 1 \]
(Here, \( \small \arg \min_t f(t) \) denotes the value of \( \small t \) that minimizes \( \small f(t) \).), the optimal betting ratio for the gambler can be calculated as:
\[ \begin{align*} & \small f_i = \max\left\{\frac{p_i^{\frac{1}{\gamma}} q_i^{\frac{\gamma-1}{\gamma}}}{k^{\frac{1}{\gamma}}}-bq_i, 0 \right\} \\ & \small b=\frac{1}{k^\frac{1}{\gamma}}\left(\frac{1-\sum_{i=1}^{t^\ast} p_i}{1-\sum_{i=1}^{t^\ast} q_i}\right)^\frac{1}{\gamma} \\ &\small k^{\frac{1}{\gamma}}=\sum_{i=1}^{t^\ast} p_i^{\frac{1}{\gamma}} q_i^{\frac{\gamma-1}{\gamma}}+\left(\frac{1-\sum_{i=1}^{t^\ast} p_i}{1-\sum_{i=1}^{t^\ast} q_i}\right)^\frac{1}{\gamma}\left(1-\sum_{i=1}^{t^\ast} q_i\right). \end{align*} \]
The above is the optimal betting strategy for a gambler with a CRRA-type utility function. It can be confirmed that when \(\small \gamma = 1\), \(\small k=1\) holds, and the solution coincides with Kelly’s formula.
Generally, the Kelly criterion has a high betting ratio, and it is said that many gamblers find the fluctuation of their funds painful. However, by using this formula, you can choose the appropriate leverage by determining \(\small \gamma\) according to your risk appetite. In economics, the value of relative risk aversion \(\small \gamma\) is said to be around 2 to 5, so the Kelly criterion may appear to be a more risk-loving standard compared to the average person. Furthermore, it is important to estimate the probability \(\small p_i \) that each choice will occur, and I will consider this together at some point.
Reference
[1] Ethier, Stewart N. The Doctrine of Chances: Probabilistic Aspects of Gambling. Springer-Verlag, 2010.
[2] Huang, Chi-Fu and Robert H. Litzenberger, Foundations for Financial Economics, North Holland, 1988.
[3] Kelly, John L. A New Interpretation of Information Rate. Bell System Technical Journal, 1956.
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