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A More Mathematically Rigorous Analysis of the Optimal Bet Size

Zedekiah G. Higgs

In another post I discuss the optimal bet size for winning a progressive jackpot. I conclude that a player with a finite budget who is interested in maximizing their chances of winning the jackpot should play the largest bet size possible. While I lay out an intuitive argument using a simple example, the analysis lacks mathematical rigor. In this post I provide a more mathematically rigorous analysis of the problem.

Mathematical Model

A player with endowment \(w\) plays a lottery in which they place a bet \(b\) and receive a payoff of \(x_{i}b\), where \(x_{i} \in X\) for \(i = 1, 2, ..., J\) and \(X \subset \mathbb{R}_{+}\). The payoffs are determined by a probability measure \(\mu \in \Delta(X)\), where \(\mu(x_{i}) = p_{i}\) for each \(i = 1, 2, ..., J-1\). That is, each \(x_{i} \in X\) for \(i = 1, 2, ..., J-1\) has a fixed probability of occurring. However, the final element of \(X\), \(x_{J}\), occurs with probability \(\mu(x_{J}) = p_{J}b\), where \(p_{J} \in (0,1)\). We refer to to \(x_{J}\) as the jackpot.1

The player seeks to choose \(b\) to maximize their chances of realizing state \(x_{J}\) in at least one game of the lottery, where the number of games they are able to play is determined by \(\frac{w}{b}\). The probability of realizing state \(x_{J}\) in at least one game is given by \[ \begin{align} P[E_{J} \geq 1] &= 1 - (1 - \mu(x_{J}))^{w/b}\\ &= 1 - (1 - p_J b)^{w/b} \end{align} \] where \(E_{J}\) denotes the event where \(x_{J}\) is realized and \(0 \leq b \leq \min\{\frac{1}{p_J}, w\}\).2

Note that we are not forcing the number of games played to be an integer (e.g., the player can play 1.345 games). This is obviously an abstraction from the real-world setting, but forcing the number of games played to be an integer would complicate the math without contributing anything to our understanding of the problem.3

Since we want to find the optimal bet size \(b\) that maximizes \(P[E_J \geq 1]\), we take the derivative with respect to \(b\). This gives us \[ \begin{align} P'[E_J \geq 1] &= \frac{\partial}{\partial b} \left(1 - (1-p_J b)^{w/b}\right) \\ &= - \frac{\partial}{\partial b} \left((1-p_J b)^{w/b}\right) \\ &= - \frac{\partial}{\partial b} \left(e^{(w/b) ln(1 - p_J b)}\right) \\ &= (1 - p_J b)^{w/b} \cdot \left(\frac{w}{b^2}\right) \cdot \left[ ln(1 - p_J b) + \frac{p_J b}{1 - p_J b} \right] \end{align} \] \(P'[E_J \geq 1] \geq 0\) over the entire domain of \(b\).4 Therefore, \(P[E_J \geq 1]\) is monotonically increasing in the bet size \(b\), and the optimal solution occurs at the boundary where \(b\) is largest.

Thus, the optimal choice of \(b\) will be the lesser of either \(b = \frac{1}{p_J}\) (where the probability of hitting the jackpot, \(\mu(x_J)\), is equal to 1) or \(b = w\) (where the player bets their entire endowment in a single game). If there is an exogenously set maximum bet amount, \(0 < \hat{b} < \min\{\frac{1}{p_J}, w\}\), the player's optimal strategy will be to bet the maximum, setting \(b = \hat{b}\). This is what we would expect to occur in the real world, where the probability of hitting the jackpot is typically very small (and \(\frac{1}{p_J}\) is therefore very large) and the maximum allowable bet is typically much smaller than a player's endowment.

Of course, in the real world the number of games played would also need to take an integer value, which deviates slightly from the model presented. The final conclusion, however, will still hold: if a player wishes to maximize the probability of winning a progressive jackpot, they are better off maximizing their bet amount, all else equal.

  1. 1 We have technically misspecified the payoff associated with hitting the jackpot, since the jackpot doesn't scale with the bet size \(b\) (and therefore would not be equal to \(x_J b\)). Since we are only interested in calculating the probability of hitting the jackpot, this detail is unimportant. However, if we were interested in, say, deriving the expected value of the lottery or the variance of the payoffs associated with different bet sizes, we would then need to correct this issue. But I'm not going to do those things here, so I'm also not going to complicate the notation by properly specifying the payoffs from hitting the jackpot. (So much for being rigorous, eh?)
  2. 2 The upper bound on \(b\) is necessary to ensure that \(\mu(x_J) \leq 1\).Alternatively, we could define \(p_J \in (0, \frac{1}{w})\), in which case the upper limit of \(b\) would just be the player's endowment \(w\), which might feel more natural.
  3. 3 The player's optimization problem would become \[ \begin{align} P[E_{J} \geq 1] &= 1 - (1 - \mu(x_{J}))^{\text{floor}\{w/b\}}\\ &= 1 - (1 - p_J b)^{\text{floor}\{w/b\}}, \end{align} \] where \(\text{floor}\{w/b\}\) rounds \(w/b\) down to the nearest integer. The optimal solution will be the same. However, if there is an exogenously set maximum bet value (such as a bet limit imposed by the Lottery), we would conclude that the player should maximize their bet size while also maximizing the portion of their endowment used (i.e., they want to avoid "leaving money on the table").
  4. 4 The bet size \(b\) is constrained to be between \(0 \leq b \leq \min\{\frac{1}{p_J}, w\}\). This implies \(0 \leq p_J b \leq 1\) for all \(b\). It is clear that the first two terms of \(P'[E_J \geq 1]\) must always be nonnegative. To see that the third term must also be nonnegative, let \(m = p_J b\) and take the derivative with respect to \(m\): \[ \begin{align} \frac{d}{dm} \left( ln(1-m) + \frac{m}{1-m} \right) &= \frac{m}{(1-m)^2} > 0 \text{ for all } m \in (0,1) \end{align} \]