Ultimate X Puzzle

Before you read this question look at a pay table for ultimate x and examine how the multipliers differ for three five and ten handed games. This seems a bit weird at first but igt gives better multipliers for the hands above full houses when you play more hands. What many people don't know is that this isn't designed to reward people for playing more hands, it's designed to compensate people for the equity they would lose if they played more hands with the same multiplier table. That's right, the return of ultimate x, holding the multiplier table constant, decreases as the number of hands increases. The puzzle is this: why does the return of the game depend on the number of hands? This isn't true in triple play versus ten play or almost any other game. I recently figured it out and I think the answer is interesting. I will give people a chance to think about it and then post my answer if there is any interest in this topic.

Because the more hands you play the more variance you're going to have between the multipliers of the individual hands. Which means you're playing more hands incorrectly.

Quote: vetsen

Because the more hands you play the more variance you're going to have between the multipliers of the individual hands. Which means you're playing more hands incorrectly.

This sounds like a reasonable answer to me. I knew that the return reduced with increased number of lines with consistent multipliers from the academic paper by Koehler, but he didn't bother to explain exactly why this was the case.

Quote: vetsen

Because the more hands you play the more variance you're going to have between the multipliers of the individual hands. Which means you're playing more hands incorrectly.

That's definitely getting closer, but this discussion is about the return of the game under optimal play, not about mistakes. The answer does definitely have to do with how you change your play as the number of hands increases.

The statement about variance is very tricky, but thinking about multiplier variance is the way to go. Try thinking about how multiplier variance effects strategy choices.

When I say "playing more hands incorrectly" I don't mean that the player is holding cards that he shouldn't be holding.

What I mean is with 10-play you're going to have times where say 5 of the hands have a 2x multiplier and the other 5 hands have a 12x multiplier. For various starting hands, the optimal strategy is different for a 2x multiplier than for a 10x multiplier. Which means the player has no choice but to play one of those groups of 5 hands incorrectly. Contrast this to single line where the player is able to every hand optimally.

Quote: vetsen

When I say "playing more hands incorrectly" I don't mean that the player is holding cards that he shouldn't be holding.

What I mean is with 10-play you're going to have times where say 5 of the hands have a 2x multiplier and the other 5 hands have a 12x multiplier. For various starting hands, the optimal strategy is different for a 2x multiplier than for a 10x multiplier. Which means the player has no choice but to play one of those groups of 5 hands incorrectly. Contrast this to single line where the player is able to every hand optimally.

That's the correct logic. Imagine you are dealt two pair. You hit a full house approximately one out of ten times. In the course of your play, imagine you are dealt two pair once in a ten handed game and you play twice as long in a five handed game and are dealt two pair twice. The most common outcome in the ten handed game is to hit one full house for an average multiplier of 3.9. Now in the five handed game the most common outcomes are getting one full house or getting zero. Imagine over two series of dealt two pairs you get one full house the first time and zero the second time. The average multiplier in one case is 4.8 and the other is 3. Now since optimal strategy depends on the average multiplier, you can play different strategies in those two five handed games. You have had the same total coin in and the same distribution of hands, but you have more strategy flexibility in the five handed game.

The variance statement you made is the opposite of where I would suggest is the correct way to think about it. The distribution of the average multiplier conditional on any dealt hand is smoother in the ten handed game. In this case the extra standard deviation on the five handed game is what gives you the advantage. A wider distribution of multipliers is your friend.

This is very similiar to a classic question in economics. Imagine you are going to the grocery store to buy a weeks worth of food and the manager asks you if you want to increase half the prices in the store by 50% and decrease the other half by 50%. Do you accept the deal? Of course you do, because this is another case where variance is your friend. When you have a choice, you can substitute away from more expensive goods and towards the cheaper ones. In other words, when you can substitute, variance in prices is a good thing.

Excellent topic! Without considering this idea, I had assumed it was merely a way of "rewarding" those who bet more by playing more hands. Averaging the multiplier over more hands gives the optimal strategist control in fewer situations. So what affect do the "better" set of multipliers on 10-Play give you when using a single Basic Strategy designed to fit Triple-Play, Five-Play, and Ten-Play?

Quote: camapl

Excellent topic! Without considering this idea, I had assumed it was merely a way of "rewarding" those who bet more by playing more hands. Averaging the multiplier over more hands gives the optimal strategist control in fewer situations. So what affect do the "better" set of multipliers on 10-Play give you when using a single Basic Strategy designed to fit Triple-Play, Five-Play, and Ten-Play?

If you use a strategy that does not depend on the current multipliers when playing ultimate x, then your return would be the same for three five and ten handed games if the multiplier table was the same. This means that the effect of the additional multipliers on your return is even bigger as you increase the number of hands. There is no offsetting downside to balance it out.

Quote: camapl

Excellent topic! Without considering this idea, I had assumed it was merely a way of "rewarding" those who bet more by playing more hands. Averaging the multiplier over more hands gives the optimal strategist control in fewer situations. So what affect do the "better" set of multipliers on 10-Play give you when using a single Basic Strategy designed to fit Triple-Play, Five-Play, and Ten-Play?

Well according to JB, it costs you about 0.05% for 9/6 DDB ten-play. Still a good game at 99.82% if you can afford to play $25/hand at a few Vegas casinos, including SouthPoint.

https://wizardofodds.com/games/video-poker/strategy/ultimate-x/double-double-bonus/