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
The player seeks to choose
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
Thus, the optimal choice of
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 We have technically misspecified the payoff associated with hitting the jackpot, since the jackpot doesn't scale with the bet size
(and therefore would not be equal to ). 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 The upper bound on
is necessary to ensure that .Alternatively, we could define , in which case the upper limit of would just be the player's endowment , which might feel more natural. - 3 The player's optimization problem would become
where rounds 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 The bet size
is constrained to be between . This implies for all . It is clear that the first two terms of must always be nonnegative. To see that the third term must also be nonnegative, let and take the derivative with respect to :