I think the difference in equities goes like this.

One person wants to run it once. The other 2 players agree to run it more than once if the first person DOES NOT win the first board.

Because the extra boards are CONDITIONAL on the first person losing, heads-up equities between the two players doing business matters a lot.

So if the first player has top set (and wants to run it only once), and the other two players (who want to make a deal) have Ace high NFD and pair+FD respectively...

The only way that the multiple boards will happen is when the first player loses. Probably because a flush came in on the first board. Remember that the pair+FD is basically drawing close to dead in terms of the 3-way ALLIN fest.

Now that the first board resulted in the first player losing (probably to the nut flush), there will be extra boards running out.

But in these extra boards that get run out, the pair + FD has very good headsup equity versus the Ace high NFD.

Therefore, the pair+FD player will receive a lot of extra equity with this deal. And all that extra equity will come from the Ace high NFD player.

So yeah, the Ace high NFD player is getting royally screwed by this deal.

Most of my experience is regulated rooms or true home games, but I've been to a few "home games" that were better described as underground or "semi-pro" in a few states in the West and Midwest. None in the Atlantic Seaboard/New England. Only casinos there.

A quick note that while I do appreciate Vernon's input, if anyone else has any strategy points to add regarding whether to RIT when one player will only RIO on a 3+ way all-in, feel free to chime in. I think we basically answered the question of how to gain equity in these cases... and don't believe the existence of side pots, or the existence of four players (instead of three) would change any part of the thought process.

Hi Garick- I've played at casinos across the country (1/3 and 2/5), and have never seen a "rock" at a regulated (casino) game. It could be I just didn't play at the places that have it. I played in Biloxi, Mississippi (where almost everyone Mississippi straddles), in Montreal (where they use "French Decks" of cards, which have R, D (Dame), and V (Valet) instead of K, Q, and J....), and in other places in which poker rules are slightly different- but have never seen a casino game with a rock. And "Kill" buttons I've seen in other poker games, but not the 1/3 and 2/5 that I play. I thought the rock was an underground game invention... but I guess it is at regulated games. I learned something new today.

Hi 009285832- still no. Equities definitely change when in a 3-way all in, one person runs in once, and the other two RIT. Feel free to read any of the other posts in this thread. I don't fault you, though- I didn't realize it either, over the first two years I played in the NYC underground games (all of them allow this), and even in the last 10-12 years in which I realized there was a change in equities by doing business, I couldn't determine how to quantify it. I've been missing out on a lot of +EV opportunities for almost 15 years, by not understanding... and I'm a certified actuary! If I didn't understand it, I don't think I'm exaggerating by saying that perhaps 90% of people that play in these underground NYC home games have no idea that equities change. Garick had to bring in his mathematician friend Vernon, to help explain what's going on here, with the changes in equities that occur.

Smoola- you didn't respond to the last message I addressed to you, but I think that if you saw this situation happen often, you might be more open to gaining equity by RIT when appropriate, if one player will only RIO. (I wasn't involved in either hand, but I saw it twice just last night, with different players involved in the two multiway all-ins, over the four hours I played poker. One all-in was pre-flop, and one was after the turn betting.) To you, this is an odd novelty that All Clear is trying to game... but to NYC underground players- we all see this happen many days, and players always choose whether or not to participate in RIT, if one RIO player doesn't win the first runout.

Have you read the examples above? It is clear that equity can change for the remaining players if the RIO V does not win the first run. What doesn't change is the RIO players equity, or the RIT players' collective equity. But their equity against each other can change wildly, especially for a player who was basically drawing dead against RIO guy.

For example, if you have a FD, but RIO guy has an overpair to your highest card and higher FD (like say, he has KdJd and you have 4d5d on a Qs2d6dJh board), and the flush doesn't hit on the first board, you went from drawing dead if you all did the same number of times to drawing live against the other V, basically no matter what he holds (other than another higher FD).

Let's say the other V has QcTc. When you run it once (or all of you run it any number of times more than once, but all the same number) you are drawing stone dead. 28.5% of the time, the higher FD will hit 2p+ or a higher flush than yours, and 71.5% of the time, the V with the pair of Qs will hold.

But now let's say that the river was a blank, and you get another river only against V2, something that will happen 71.5% of the time. You have 20.5% equity against V2 on the other river. .715x.205=14.7. So by agreeing to RIT with V2, but not V1, you just went from 0.00% to 14.7% equity.

This kind of thing is why most rooms don't allow such deals, I would assume. Because if you win it should be really obvious that could only have happened with an uneven number of runs.

The examples don't matter. You have the same equity as you had. Your equity doesn't change regardless of how many times you run things. 3 people in a main pot and 2 people in a side pot. You have the same equity. It doesn't change.

You're confusing outcome with equity in the example you give.

Most rooms don't want anything that slows down hands as they make money on a rake. The running it more than once idea slows things down and by the time you're doing that, they've capped their rake anyways. Running it twice isn't favorable to the better player as you're more likely to be ahead. It just leads to idiots calling or making dumb semi-bluffs as, if you run it twice, they're not losing money 40% of the time anyways.

That's not how math works. If an example falsifies your theory, the theory is wrong

You are correct if and only if everyone runs it the same number of times. If, in OP's odd ruleset, people are running it different numbers of times, it gets much more complicated.

Hi Garick- if every NYC underground game allows a certain rule, while I don't mind if you disagree, I think most would hardly label it as "odd". I suspect most people reading this thread have only played poker in casinos or friendly (non-rake) home games, but for those that play underground games, the house doesn't care how many times any player does a runout, if those players agree to RIT if the RIO player doesn't win the first runout.

There was a Poker After Dark episode around a decade ago, in which three players were all-in, and the commentator said something like "and Phil Ivey will only let them run it once..." - at the time, I remember wondering why the other two players couldn't just do business if Ivey lost the first runout. I only understood years later, that this isn't allowed. For underground players, in contrast, the "odd" ruleset is that everyone must agree, to run business.

And even though this "odd" rule-set is so common (it's allowed at every underground game I've played at in NYC- around 15 different ones, give or take), it's very possible that almost no one knows the proper strategy... those who read this thread have a huge advantage when this situation comes up. And as mentioned earlier, I'd suspect this rule-set would be allowed in underground games outside NYC, as well.