Is There an Optimal Bet Size for Winning a Progressive Jackpot?
Zedekiah G. Higgs
Some of the Online Instant games offered by the Virginia Lottery have progressive jackpots. Unlike regular jackpot prizes, progressive jackpots continuously grow--with a percentage of all bets placed contributing to the size of the jackpot--until the jackpot is won (at which point the jackpot returns to zero and begins growing again).
As I have previously discussed, the prizes in Online Instant games are typically calculated as multiples of the amount bet. For example, if a player bets \(b\) on a game, the potential prizes they might win will range from 0.1x\(b\) to 1000x\(b\). Because of this, changing the bet size simply scales the prize amounts and has no effect on the probabilities of winning.
Unlike standard prizes, rolling jackpots are the same for all players, regardless of how much they choose to bet. Since the size of the prize no longer scales with the size of the bet, the Lottery instead scales the probability of receiving the prize. Thus, a player who bets twice as much will have 2x the chance of winning the jackpot, and a player who bets 10x as much will be 10x more likely to win the jackpot, etc.
This design naturally leads one to ask, "Is there an optimal bet size for maximizing your chances of winning a progressive jackpot?"
To make this question interesting, we must assume a finite budget (which I think is a fair assumption). If we had an infinite budget, the size of our bets shouldn't really matter since we could continue playing for an infinite number of games, guaranteeing that we would eventually win the jackpot. However, with a finite budget we are forced to make a tradeoff: increasing our bet size increases our odds of winning the jackpot on any given game, but it also means that we can play fewer games (giving us fewer chances to win the jackpot). Decreasing our bet size has the opposite effect--lowering the chances we win the jackpot in any given game but giving us a greater number of chances to do so.
So which strategy is optimal if we want to maximize our chances of winning the jackpot? Should we make the smallest bets possible and maximize the number of games we get to play? Or should we make the largest bets possible, giving ourselves the fewest number of games but maximizing the odds of winning the jackpot in any given game? Or does the size of our bets have no effect on our probability of winning the jackpot?
The answer turns out to be that we should make the largest bets possible, but it's interesting to see why this is the case. The following example provides some insight into why larger bets are optimal.
An Example Setting
Suppose you are walking down the street one day and see me sitting in a lawn chair under a tent in my yard. I have a cardboard sign that says "Lottery Game: Progressive Jackpot". You're always looking for opportunities to gamble, especially illegally, so you stop to see what the game is. You only have $1 on you but you're feeling lucky and you're looking to grow your money.
The game's simple. There are two different price points you can play at: $0.50 or $1. Both price points give you a chance at winning the rolling jackpot, which happens to be up to $10. If you play at the $0.50 price point, I'll flip a fair coin and if the coin lands on heads you'll win the jackpot. At this price point you can afford to play the game twice. And the expected payoff from playing the game is clearly positive, so you're ready to put your money down.
But wait. If you play at the $1 price point, I'll use my other coin that has heads on both sides. That is, at the $1 price point you will have a 100% chance of winning the jackpot. By doubling your bet size, you double your probability of winning the jackpot.
So what should you do? Bet the smallest amount and maximize the number of games you get to play, or bet the largest amount and give yourself the highest probability of winning the jackpot in any given game?
Hopefully it's obvious that you should take the second gamble, betting the larger amount of $1 and giving yourself the greatest chance of winning the jackpot in any given game (which happens to be 100% in this example). Betting $1 guarantees you will win the jackpot. On the other hand, betting $0.50 leaves open the possibility that you fail to win the jackpot. (It's entirely possible that the coin lands on tails twice in a row.)
This result is somewhat intuitive, especially since the expected payoffs associated with the two strategies is equal. If the expected payoffs are equal, then how can one of them provide a better chance of winning the jackpot? Well, comparing the expected payoffs is a little misleading, especially in the case of a rolling jackpot.
Why comparing expected payoffs is misleading
First, let's see why the expected payoffs are the same for both price points. At the price point of $1, you play 1 game and the only possible outcome is that you win the jackpot, so the expected payoff is $10. At the price point of $0.50, you get to play 2 games, each providing a 50% chance of winning the jackpot. The expected payoff is therefore given by \[ P(T,T) \cdot 0 + P(T,H) \cdot 10 + P(H,T) \cdot 10 + P(H,H) \cdot 20 \] \[= (1/4)*0 + (1/4)*10 + (1/4)*10 + (1/4)*20\] \[= 0 + 2.5 + 2.5 + 5\] \[=10\]
There is one outcome where you win nothing (tails, tails) and three outcomes where you win the jackpot: (tails, heads), (heads, tails), and (heads, heads). In total, only 3 out of 4 possible outcomes end up with you winning the jackpot. The only reason the expected payoffs are the same for the two price points is because the $0.50 price point provides an opportunity to win the jackpot twice (heads, heads). But you "pay" for this possibility by also introducing an outcome where you win nothing (tails, tails). Since we are only interesting in maximizing the probability of winning the jackpot, the $1 price point is preferable because it provides a greater chance of winning the jackpot.
And things get even worse for the $0.50 price point. Because the rolling jackpot resets once somebody wins it, winning the jackpot twice won't actually pay $20. Instead, it will still only pay $10. So the expected payoff associated with the $0.50 price point is actually only $7.50. Even more reason to prefer the larger gamble.
Conclusion
If a player has a fixed budget and wishes to maximize their chances of winning a rolling jackpot where the odds of winning scale proportionally with the size of the bet placed, their optimal strategy is to play with the largest bet size possible. There are two reasons for this. First, smaller bet sizes increase the number of games played, which increases the odds of hitting the jackpot multiple times but reduces the odds of hitting the jackpot at least once. Since the objective of the player is to win the jackpot at least once (and not to win the jackpot as many times as possible), larger bet sizes are preferable. Second, because rolling jackpots reset after a player wins them, hitting the jackpot multiple times is no better than winning it a single time. Thus, smaller bet sizes provide lower expected payoffs since they increase the odds of winning multiple times (which provides no added benefit) at the cost of lowering the overall probability of winning the jackpot. Combined, these two effects make it clear that players who wish to maximize their odds of hitting a progressive jackpot should maximize their bet sizes.
Of course, these effects are miniscule in practice because the odds of hitting the jackpot are tiny. At the minimum bet size of $0.50, the progressive jackpot game Jackpot Spectacular provides a 1 in 80 million chance of winning the jackpot. At the maximum bet size of $50, the odds of winning the jackpot increase to 1 in 800,000. Therefore, in practice there is no meaningful difference between strategies, and players should base their decision of what bet size to use on other factors, such as how much money they wish to lose. While larger bet sizes theoretically provide a greater chance of winning the jackpot, players should recognize that they will never win the jackpot, that any beliefs that they might win the jackpot are delusional, and that they are better off investing their money somewhere other than the lottery.