So a losing poker player has a prop bet with 2 pros. Player has $2,500 in front of them in a 1/3/15/30 NLHE game and they have a bet that if the losing player gets their stack up to $5,000 by the end of their session, they will both pay him $200 each, and if he doesn't, he will pay the $200 each. Also, the idea is that the player will not rebuy if they get stacked.

Unrelated, losing player and I agree to flip for $1,000 (run a dry plo hand for $1,000). Then the pros want to amend the bet so that any money won through flips and 0 EV prop bets don't count towards their bet. My initial reaction is that is absurd. The EV is clearly 0 so the net impact to their prop bet is 0.

The 2 pros make the contention that because the player is a losing player, the losing player will lose more than 50% of the time, so allowing him to take bets where he must win 50% of the time improves his odds.

In some sense, I see where they are coming from. Imagine if the bet was that the losing player will get their stack up to $5,000 and they only were going to take 50/50 flips. In this case, if the player is down 3 flips cumulatively, they lose, but if they are up 3 flips cumulatively, they win. There really is no edge to be had by the pros.

Imagine that the player is allowed to do 1 flip for every 10 holdem hands. In some sense the flips do seem like they have 0 EV and really don't have any impact on the prop bet. But when you start to say 5 flips for every 5 holdem hands, 10 flips for every 5 holdem hands, 100 flips for every holdem hand... all flips and no holdem hands, at some point it seems like all edge to the pros has been lost and they are really just participating in a flip.

I think I am leaning towards agreeing with the pros here. It does not seem like it is in their interest to allow the flips. But I do have some cognitive dissonance about this. On the micro level, bankroll managent aside, I feel like we should be risk neutral in poker, always trying to make the highest EV play, even if some times that EV is only slightly positive and the amount we risk can be substantial, such as when we overbet shove the river as a bluff. Similarly, I like gambling when given even odds, and I like to show people that I am willing to gamble with others and not always their to play with them because of my edge. As long as my bankroll is comfortable, I don't see a problem with 0 EV flips. In fact, if anything, I force myself to do these flips to face my fear head on, because if I am not willing to take a 50/50 risk, then maybe I won't be willing to make other aggressive plays because of the emotional risk of losing. I am more afraid of being afraid of losing than being afraid of losing, because I feel like if I am afraid of losing, I am more apt to make mistakes from being under-aggressive.

So every ounce of me is like, suck it up guys, let the guy take his flips, they are 0 EV so they have no impact. But I also can't deny that it does seem like the flips do diminish the edge the pros have. Feels a little like a paradox.

Am I missing something here?

I would side with the pros here. The 0EV flops indeed have no bearing on the losing playerÂ’s stack in the long run, but this is not a long run prop. It is a short run prop, so variance plays a pretty major role in the result. Allowing flips will increase the variance of the losing playerÂ’s win rate. Increasing the variance will make him more likely to actually post a large enough short term win to win the prop.

A good way to look at it is that the loser is playing a casino game with a house edge - that is the same thing as having a negative expected value, the very definition of losing player. If you owed a loan shark $5000, you had only $2500, and playing blackjack was your only chance of escaping the situation without broken bones, what is your optimal strategy? It would be (assuming house limits allow) to bet the entire 2500 on a single hand. This is true regardless of house edge.

For example (and to make the math easier), compare this strategy with the strategy of betting half your stack and betting the same if you win. Suppose that you have only a 40% chance to win. Betting it all gives you (obviously) a 40% chance of continued good health. If you bet half, you lose 60% of the time on hand 1. Of your 40% wins, you win 40% on hand 2, a total of 16%. The other 24% of the time we get back to the starting point, so we can ignore these and calculate our probability of doubling up as follows. In 100 plays we lose 60 and win 16, with 24 “do-overs”. Of the cases that actually result in a win or loss we win 16 out of 76 which is a bit over 21% - a significant decrease from the 40% we get from just betting it all.

The underlying reason is that a larger number of smaller bets has a smaller variance than a smaller number of larger bets. We need variance to win a losing bet, so increasing variance is a good thing in such situations.

I would side with the pros here. The 0EV flops indeed have no bearing on the losing playerÂ’s stack in the long run, but this is not a long run prop. It is a short run prop, so variance plays a pretty major role in the result. Allowing flips will increase the variance of the losing playerÂ’s win rate. Increasing the variance will make him more likely to actually post a large enough short term win to win the prop.

A good way to look at it is that the loser is playing a casino game with a house edge - that is the same thing as having a negative expected value, the very definition of losing player. If you owed a loan shark $5000, you had only $2500, and playing blackjack was your only chance of escaping the situation without broken bones, what is your optimal strategy? It would be (assuming house limits allow) to bet the entire 2500 on a single hand. This is true regardless of house edge.

For example (and to make the math easier), compare this strategy with the strategy of betting half your stack and betting the same if you win. Suppose that you have only a 40% chance to win. Betting it all gives you (obviously) a 40% chance of continued good health. If you bet half, you lose 60% of the time on hand 1. Of your 40% wins, you win 40% on hand 2, a total of 16%. The other 24% of the time we get back to the starting point, so we can ignore these and calculate our probability of doubling up as follows. In 100 plays we lose 60 and win 16, with 24 “do-overs”. Of the cases that actually result in a win or loss we win 16 out of 76 which is a bit over 21% - a significant decrease from the 40% we get from just betting it all.

The underlying reason is that a larger number of smaller bets has a smaller variance than a smaller number of larger bets. We need variance to win a losing bet, so increasing variance is a good thing in such situations.

You’re definitely killing their edge. What if you just flipped for the full 5K?

This is a weird question, but here's one way to think of it. Suppose we extended the coin flip scenario so that all hands played would be coin flips and rebuys were allowed. Then it's obvious that their edge is eliminated, because given a sufficiently large bankroll, the losing player could just continue to play until variance took him to $5k, even though he would expect to make no money in the long run. I just don't know if this logic tracks when you take only one flip at 0ev instead of all of them.

The pros got owned if the fish figured out a way to get a 50% shot at +2900 or -2500. That's on them for not specifying a rule against that. It is against the spirit of the bet.
There's no paradox, because "they are 0 EV so they have no impact" isn't true. They're 0 EV for the size of his stack, but + EV for the probability of making it to $5000 (if it's sub-50% by conventional play)

By the way, this is already a kind of a scummy bet for the pros to make. You don't even have a fair 50% shot at making it to exactly twice your buyin, because occasionally you go significantly over the threshold, for example if you go allin at $4000 stack depth. That still puts your prop bet at exactly 50% risk again, despite the fact you were in a good looking position. So the fish needs and deserves every advantage here.

Your random flips are not part of the nle game and therefore not part of the bet. What’s the difference if he just went to roulette and put it all on black. Pros are correct