How does FTOP apply in multiway pots ?

[Cezar]

As I was playing PLO a few days ago, I found myself in a 4-way pre-flop all-in fest with this beast :7 [img]/images/graemlins/spade.gif[/img] 8 [img]/images/graemlins/spade.gif[/img]9 [img]/images/graemlins/diamond.gif[/img]J [img]/images/graemlins/diamond.gif[/img]

In post portem, I put this hand through Cardplayer's Omaha calculator. Against two high pairs (and one donk with card salad) my 35% was actually the best pot equity of the 4 hands.

This got me thinking. The Fundamental Theorem of Poker would seem to imply that there always is a an objective, mathematically correct way to play your hand (save for big bet poker, where you may be correct to bet, but you still get to choose exactly how much).

How would this apply in multiway pots, where correctness of your decision depend on the people to act behind you ? A hand like ragged aces will always be over 50% favorite heads up, except for better aces. But fail to isolate, and you quickly end up with less than 33% against two fat drawing hands.
Or the reverse situation, where you have less than 50% heads up, but lose very little equity from multiple hands entering the pot behind you?
And just to add further to the puzzle, it is possible to concieve a rock-paper-scissors scenario, each hand being favorite over one other.

Of course, in real life this decision would be based on what you think your opponents are likely to do. But if we ignore psychological considerations, is there still an objective mathematical decision ?

Cezar.

[AaronBrown]

You are correct that there is no obvious mathematical solution in this case. John Nash won a Nobel prize (despite being insane) for coming up with the most common solution in this case, it's called the Nash Equilibrium. Although mathematical economists love Nash Equilibria, it's not a useful idea for poker (or much else outside of ivory tower economics).

To simplify your situation, you can construct hands where if you player A does something different than he would do if he knew all the cards, and player B does something different, you are worse off than if both players had played perfectly.

Outside of pure theory, this does happen. If you read Frank Wallace's extraordinary "Advanced Concepts of Poker," you see him using other players at the table. If you're one of his victims, you collect money from other players, only to deliver it to Frank. You can induce errors in other players, say getting them to put too much money in the pot when you have a better hand, only to find that Frank has a better hand than either of you. Or you can get them to fold the best hand, so Frank can win with the second-best hand.

I think this kind of analysis is the interesting part of poker, rather than the two-player/one-hand analysis that you read so much about.

[ohnonotthat]

I think this is what you meant.

2-2 beats A-K/off.

A-K/off beats JT/suited (or off)

J-T/suited (or off) beats 2-2.

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An good example for multiway play would be the virtual uselessness of hands such as A-Q/off when facing a couple of small pocket pairs and a couple of low suited connectors.

A-Q rarely survives an encounter with 3-3, 4-4, 8-7s and J-Ts, and when it does its wins tend to be small.

Another good example from a multiway pot would be the rapid decline in the worth of A-A as more and more ocket pairs enter the pot. It's not just that each additional PP costs the Aces valuable "hot-cold" EV, it's also the punishment the Aces take when they get beaten in this scenario. When they get taken off a set it's almost always going to be expensive; however, when they beat the collective field of 2-2, 4-4, etc., they seldom earn much since these baby pairs rarely go past the flop . In other words, 2-2 can sneak up on, and molest, A-A; the reverse is not true.

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In holdem there are very few "absolute" best hands; this is part of what makes big money pre-flop decisions so important (and fascinating).

If a [slightly] loose cannon makes a big raise I am thrilled to call with any pair if I know he has A-K (well, 2-2 and 3-3 are marginal but they are profitable if his A-K is not suited) but I can get into trouble if he has the "lessor" hand of J-T.

This revelation tends to make the stud or draw player say to himself "no, that can't be right" - and is one of many reasons why competent (even some good) stud players dump huge amounts of money at holdem before they figure it out.

- Some never figure it out. (Most of those who never do can be found at PartyPoker's 3-6 tables [img]/images/graemlins/confused.gif[/img]).

Do a search for "Morten's Theorum", either here or on Google for a detailed description of how the FToP applies (or fails to apply) in multi-way pots.

[damaniac]

Your ideas make some sense but I believe are vastly overstated. AQ is hardly worthless in the first scenario, you have a small equity edge against the field, about equal with JTs at about 23%. Your point about postflop would be true if your opponents all play well. Most of the time, they don't, and if you are in a game where everyone is pretty level-headed, you might want to find a better one. Fact is, plenty of people will call down with 33 UI, or bottom pair or a gutshot draw. In the example none of them will be incorrect to do so (other than 33), but you're at least getting value on each round by betting, it's not like people just up and fold unless they have the nuts or a nut draw.

Same thing with AA. Your comment is generally true of PP's, but other bad top pair hands or the like are usually going to come along too, earning you nice pots.

Finally, this

[ QUOTE ]


If a [slightly] loose cannon makes a big raise I am thrilled to call with any pair if I know he has A-K (well, 2-2 and 3-3 are marginal but they are profitable if his A-K is not suited) but I can get into trouble if he has the "lessor" hand of J-T.

[/ QUOTE ]

just confused me. Yes, I'd call a raise if I knew my opponent had AK...how often do you "know" they have exactly AK? Ever?

The real problem is that you don't know what the other players have. The drawing hands could actually be in bad shape against a better draw, or they may not convince anybody to call behind them. If all the cards are face up there should be a correct way to play, and you just solve it out using standard game theory to find the (most reasonable if there are more than 1)Nash Equilibrium. Incomplete information throws a big wrench into things and (I think) usually favors the hands that play well heads up because the multiway hands have a coordination problem and are often at risk of being in bad shape against another draw. There might be some situations where the draws can signal to each other (via their betting) what the situation is so that they can cultivate a situation profitable to both of them at the expense of the other hand, though I'm still not entirely sure how you deal with the problem of being up against a better draw (say mostly the same straight draws but their flush draws dominate yours). Take this with a grain of salt because I'm mostly a HE player and don't fully understand the omaha situation.

[ohnonotthat]

I meant to put the word "know" in quotes - obviously you never know.

I stand by the rest; my response was intended to address the OP's question about the FToP in multi-way pot (specifically his rock-paper-scissors question ), not to debate strategy, and I think I did this well. The "best" hand in multiway pots often suffers when an "inferior" hand is added to the mix in spite of the fact that heads-up the former is a clear favorite over the latter.

Re. A-Q: Of course it holds it's own hot/cold; the best hand almost always does. That said, of the five hands in this example - two low PPs, two suited connectors and A-Q - I would prefer any of the first four to the final. If we add one more PP and/or another SC this point becomes even stronger.

By the way, if I were to defer to your judgement in regard to this matchup (I do not but let's pretend), would you care to have a go at answering the original question ? [img]/images/graemlins/confused.gif[/img]

- Or did I answer that one to your satisfaction ? [img]/images/graemlins/smirk.gif[/img]

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O.K. That does it; I am adding some bran to my diet.

- I am WAY to cranky of late. [img]/images/graemlins/mad.gif[/img]

[Cezar]

Thanks for all the responses, especially AaronBrown, that book definitely goes on my to-read list.

Ohnonotthat is definitely onto something with his stud players comment, Hold'em is probably my weakest game.

MathEconomist opened a whole can of worms [img]/images/graemlins/smirk.gif[/img]
I don't think that incomplete information fundamentaly benefits the hands that do better heads up. In Omaha doublesuited rundowns would have to put a lot of faith into their flush draws. Even if a good read puts me squarely on aces, they have no way of knowing how coordinated they are, so your opinion seems to hold well for Omaha.

Hold'em on the other hand would pose a really hard problem. A small pair may technically be a better hand than two high cards, but he really needs to hit his set to know where he stands. To play this correctly would require a good read. Otherwise he needs to hit a silly calldown mode, or be outplayed by unpaired overcards, both -EV.

Cezar