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A More Realistic Analysis of the Optimal Bet Size, with Simulation

Zedekiah G. Higgs

In my previous analyses of the optimal bet size, I considered a player with a fixed budget who wishes to maximize the odds of winning the jackpot within a predetermined number of games. The total number of games the player plays is given by \(w/b\), where \(w\) is the player's budget and \(b\) is the bet size. For example, a player with a budget of $50 who is playing $5 games will get to play a total of 50/5 = 5 games. The odds of winning the jackpot in any given game scale with the bet size, so that placing a bet that is twice as large double the player's odds of winning the jackpot. In this way, a player seeking to maximize their odds of winning the jackpot faces a tradeoff: larger bets increase the odds of winning the jackpot in a given play, but they also reduce the total number of games played; smaller bets decrease the odds of winning the jackpot, but allow the player to play more games.

As I showed in my previous two posts on this topic, if the player is solely interested in maximizing their chances of winning the jackpot, their optimal strategy is to use the largest bet size allowable. Using smaller bet sizes provides the player with a greater number of plays, but it lowers the probability of winning the jackpot at least once across all of their games played. While this analysis is correct for the problem as it was specified in those previous posts, there are two important ways in which it does not realistically capture the problem faced by a real-life player of the Virginia Lottery.

First, I did not consider any payoffs other than the jackpot, making the total number of games played a fixed function of their starting budget and the bet size. In reality, players receive many other payoffs when playing, and these payoffs can be used to increase the total number of games played (and therefore the total number of opportunities for winning the jackpot). If a player has a budget of $50 that they are willing to spend to try and win the jackpot, they should continue to play games until they have fully expended their $50.

Second, even if the optimal strategy is to play the largest bet size possible, if a player finds themself with a budget that is insufficient to make the largest bet (but is still able to cover the minimum bet size), they should play a smaller bet size. This is another way of saying that they should make full use of their budget. For example, consider a player with a budget of $50 who places a bet of size $50 in their first play. If they end up winning $45, they may not have enough to play another game at $50, but they can move down to the next largest bet size and continue to play. As they lose money and their budget shrinks, they should continue to move down to lower bet sizes until they are unable to meet the minimum bet.

Incorporating these considerations into the problem make it much more difficult to solve. The average number of games players get to play at each bet size will depend on the payoff distribution of the game being played. As I have shown in previous work, this is very difficult to predict. Using the game Jackpot Spectacular as the example, on average players can expect a return of .86 on their bets. Of course, there is a lot of variance in the distribution of outcomes. Furthermore, the problem is also complicated by the fact that their earnings are non-ergodic.

Due to the complexity of the problem, no analytical solution can be derived that utilizes the actual payoff probabilities present. Therefore, to understand how the probability of winning the jackpot varies with bet size, I build a simulation model of the problem and simulate outcomes for several different bet sizes.

Simulation Model

The simulation model is built to replicate the game Jackpot Spectacular. The possible payoffs, and the probabilities associated with each of the payoffs, are identical to those provided in the real game. I simulate 8 different bet sizes: $0.50, $1, $2, $5, $10, $20, $30, $50. This follows the bet sizes allowed in the actual game. In all simulations, players begin with a budget of $50.

The total number of games played by each simulated player is not fixed. Each time they play a game, the bet size is deducted from their budget and a payoff is randomly drawn. The payoff they receive (if any) is then added to their budget. If a player's budget decreases to the point that they are no longer able to purchase their desired bet size, they then move down in bet sizes. For example, if a player who begins by playing bets of $50 ends up with only $35 in their budget, they will then move down to the $30 bet size and continue playing. As their budget shrinks, they will continue to move to smaller bet sizes until they are unable to meet the required amount for the minimum bet size (i.e., their budget drops below $0.50). However, if their budget ever increases, they will continue to move up in bet sizes until they return to their desired bet size (i.e., their starting bet size). In the example of the player who had to move to playing $30 bets, if they were to win a large payoff that increased their budget back above $50, they would return to playing $50 games.

By following the above process, all players will continue playing games until they have fully expended their budgets, giving them the maximum number of chances to win the jackpot.

To make the results easier to interpret and reduce the number of simulations needed, I increase the odds of winning the jackpot. In the actual Jackpot Spectacular game, the odds of winning the jackpot on a $1 bet are 1 in 40 million. However, in my simulations, the odds of winning the jackpot on a $1 bet are 1 in 100,000. Just as in the actual game, the odds of winning the jackpot scale with the bet size. At the maximum bet size of $50, the odds of winning the jackpot are 1 in 2,000. At the minimum bet size of $0.50, the odds are 1 in 200,000.

Simulation Results

The results of the simulations make clear that the optimal strategy is to play the smallest bet size possible. The probability of winning the jackpot increases significantly as the bet size is decreased. Using the minimum bet size of $0.50 leads to a more than 20% increase in the number of jackpots won compared to the maximum bet size of $50.

The superiority of smaller bet sizes is driven by the number of games played. While non-jackpot payoffs are scaled by the bet size and therefore always return 86% of the amount bet in expectation, using smaller bets provides more opportunities to receive those payoffs, reducing the variance in outcomes. A single bet at $50 that fails to pay anything (which happens about 75% of the time) ends the player's chances at winning the jackpot. In constrast, a player using $0.50 bets has much more room to absorb such losses. As previously mentioned, the outcomes of the lottery are non-ergodic, so smaller bets are advantageous because they reduce the probability of being knocked out early.

The advantage of smaller bets is relected in the average number of games played at each bet size. Players using only $0.50 bets play on average 708 games. That's more than 7 times what their starting balance covers ($50 x $0.50 per game = 100 games). In contrast, players using the maximum bet of $50 on average only get to play 6.4 games.

The simulated outcomes for each bet size are listed below.

  1. Bet Size = $0.50: Avg. games played = 708.02; Number of Jackpots Won = 3,503
  2. Bet Size = $1: Avg. games played = 353.77; Number of Jackpots Won = 3,563
  3. Bet Size = $2: Avg. games played = 176.44; Number of Jackpots Won = 3,526
  4. Bet Size = $5: Avg. games played = 71.14; Number of Jackpots Won = 3,463
  5. Bet Size = $10: Avg. games played = 35.44; Number of Jackpots Won = 3,405
  6. Bet Size = $20: Avg. games played = 17.98; Number of Jackpots Won = 3,333
  7. Bet Size = $30: Avg. games played = 11.92; Number of Jackpots Won = 3,206
  8. Bet Size = $50: Avg. games played = 6.43; Number of Jackpots Won = 2,977

Summary

The optimal strategy for maximizing your odds of winning a progressive jackpot is to play the minimum bet size possible. This is because, due to the non-ergodicity of the problem, smaller bet sizes provide you with a proportionally greater number of total plays. This proportional increase in the total number of plays more than offsets the theoretical reduction in the probability of winning the jackpot discussed in my previous posts. In the actual online lottery games, where the odds of winning the jackpot are significantly more rare than what I have modeled in my simulations, the advantage of smaller bets is likely even larger than what I have found. This is because the theoretical disadvantage of smaller bets that I discuss in my previous work becomes increasingly smaller as the probability of winning the jackpot decreases. As a result, the theoretical disadvantage is not meaningful in this setting, and one should instead be interested in increasing the number of games played, since the number of games played directly affects your chances of winning the jackpot.

Furthermore, using smaller bet sizes also reduces the overall variance of outcomes, reducing the chances that a player loses a large amount. Given that smaller bet sizes also provide the greatest chance of winning the jackpot, I would argue that using the minimum bet size is an optimal strategy more generally, not just for players seeking to maximize their odds of winning the jackpot.