[CaptLego]

A hand I got into with SpaceMonkey in the QToC has me thinking... a little...

Basically, there was some money in the pot, I bet all-in, and he called.
What I'm thinking about is a different way to analyze the decision to call an all-in bet heads-up. (This could be generalized, but for now I'm trying to keep it simple.) I haven't carefully worked through the math yet to see if this makes any difference, but here's the direction of the thinking:

The normal way to make this decision (call the all-in bet vs. fold) is based on pot odds. You look at the ratio of the potsize to the betsize, and compare that to your probability of winning the hand.

I'm thinking that the probability of winning the hand is important, but instead of pot odds, the important ratio is the ratio of the potsize to your remaining chips. And rather than just the probability of winning the hand, you want to base your decision on the probability of winning the tournament.

So for example:
Let's call the first player "SM", with a chipstack of 7000.
The second player, "CL" has a chipstack of 5000.

After some rounds of betting, there's 6000 in the pot when CL goes all-in for his last 2000. Should SM call?

Option A:
fold. Then CL would have 8000 chips, and SM would have 4000. So CL would have a 2:1 chiplead. What's the prob ability of SM winning the tournament? In this case, let's say it's even money (YMMV), or 50%. (SM is better than CL)

Option B:
call.

Now if SM wins the hand, he wins the tourney. What's the probability of this? It depends on # of outs, etc. Let's say, just for the sake of argument, that he's getting exactly the correct pot odds for a call. The pot is 8000, and it takes 2000 to call, so the pot is laying 4:1. So SM needs a 20% chance to win to match those pot odds. Assume he has the 20% chance. So he has a 20% chance to win the tourney right here.

Then there's a (1 - .2) = 80% chance SM will lose that hand. In that case, he's got 2000 chips facing a stack of 10,000. CL would have a 5:1 chiplead. So if 1:2 is 50% to win the tourney, then 1:5 would be -- what, maybe 32% to win the tourney? (.32=.5^(log(5+1)/log(2+1))

So the probability of SM winning the tourney if he calls is:
.20 + .80*.32 = 46%

So in this case, an even-money bet based on pot odds actually reduces SM's probability of winning the tournament.

Suppose SM started with 10,000 chips. Should he more inclined, or less inclined to make that call? He's getting the same pot odds.
In this case,
Option A: fold, would leave him with 7000 chips vs. 8000. So he'd be a 62% favorite to win the tourney.
Option B: call If he loses, he'd be 5000 vs 10000 chips, or 50% to win the tourney, for a total call expectation of (.2 + .8*.5) = 60%

This is a closer decision, but it's still 62% for fold vs 60% for call, even though the pot odds are precisely correct.

On the other hand, suppose SM started with only 6000 chips.
Then:
Fold: he has 3000:8000 chips, or 44% chance to win the tourney.
Call: (lose means 1000:10,000 chips or 21% chance) so the probability of winning the tourney by calling is (.2 +.8*.21) = 37% , or 7% lower than folding.


Playing around with the numbers in this formula:

Against an evenly-match opponent, it makes no difference if you call this bet or not. (Makes sense, since it's an even-money bet.)

Playing against a weaker opponent:
This is a bad bet. But the bigger your chiplead is to start, the less the penalty for making this call.

Playing against a stronger opponent:
This is a good call. Further, the smaller your starting stack is, the greater gain you get in %chance to win the tournament. For example, with a starting chipstack of 6000 against an opponent over whom you'd need a 2:1 chipstack advantage for a 50% chance of a tourney win, you'd be doubling your chances to win the tourney by calling this bet.

[scottoc]

Using your example, getting 3000 of your 7000 chips in the pot is pretty much commiting yourself to it. Unless the blinds were stupid high your pretty much only going to get yourself in this position w/ some kind of hand your willing to go all the way with or w/ a semi-bluff like a flush or open-ended stright draw. And if you were making a semi-bluff you'd be setting the pot odds so that you'd be forced to call if your opponent went all-in w/ his remaining chips.

[gmunny]

I didn't go through all your numbers, but shouldn't you take into account the payouts for first and second place and work them into your calculations as to which is better from an EV standpoint? Regardless, I would be inclined to agree with your line of thinking. If the chip count is close and the blinds are relatively small, why take coinflips that can damage your stack when you think you can outplay your opponent? If the blinds are large, though, you may as well call. I remember a thread on 2+2 from one of the online sng pros, gigabet, and he said he would fold even if pot odds were in his favor if he knew he can out play his opponents and he wanted to be the one betting and raising, not calling. Of course if you have a big chip lead, it would be worth it to call to bust the player as you can afford to gamble a little.
G$

[CaptLego]

Originally Posted by gmunny

I didn't go through all your numbers, but shouldn't you take into account the payouts for first and second place and work them into your calculations as to which is better from an EV standpoint? Regardless, I would be inclined to agree with your line of thinking. If the chip count is close and the blinds are relatively small, why take coinflips that can damage your stack when you think you can outplay your opponent? If the blinds are large, though, you may as well call. I remember a thread on 2+2 from one of the online sng pros, gigabet, and he said he would fold even if pot odds were in his favor if he knew he can out play his opponents and he wanted to be the one betting and raising, not calling. Of course if you have a big chip lead, it would be worth it to call to bust the player as you can afford to gamble a little.
G$

I think you've summed it up pretty well.
As for EV, as long as 1st place pays more than 2nd place, then anything that increases your chances of winning is +EV. I don't think the ratio of 1st to 2nd $ would change the decision point (but I haven't really thought about it much.)

As for the blinds -- I did think about that a bit. In my calculation, there's an evaluation of the relative strength of the players. The number I use is the ratio of chipstacks that makes you 50% to win. The correct number to use would be a function of the skill level of the players and the size of the blinds. For example, if the blinds are bigger than the stacks this number should be 1:1 no matter what the skill level of the players.

The easiest example I came up with to understand my analysis is the case of a weak player against a strong player. For the weak player, coin flip gambles are a good bet, since that's better than they do on average. So they increase their chances of winning by making even-money bets.
Similarly, a good player is giving up their skill advantage if they make even money bets against a weaker player -- so their probability of winning the tournament would decrease.

[SpeakEasy]

Interesting exercise. I like the concept of trying to factor in the probability of winning the tournament, as well as the hand, based on one decision in a big pot when its heads-up.

Other factors to consider:

1. Difficulty in projecting outs.

When its heads-up, there is a much greater probability that your opponent is bluffing, or at least does not have a solid hand. In any calculation, its safe to assume that your opponent is bluffing at least 10% of the time (as Harrington recommends). During heads-up play, the probability is likely much higher than 10% on any given hand. This would need to be added to the calculations.

In addition, its more difficult to distinguish a “bluff” from a regular bet. Say you are on a draw with K8 on a board of 976. Your opponent might go all in with QT, thinking you are on a draw with low cards like 54. Is this a bluff? Not really, since his bet is premised upon the fact that he thinks he has the best hand at the moment with queen high. The point is, this makes counting your outs more difficult, AND you may already have the winning hand with king high.

2. Absolute cash value of remaining tournament chips at a short table.

At the start of our QTOC, we each get T1500 for $20, so T1500 = $20.

First place was $90 and second place was $54. Each of the final 2 players is guaranteed $54, and they are now playing for $36 as the difference between first and second. There are T13,500 chips on the table, so when its heads-up, T13,500 = $36.

Consider the value when three players are left: each player is guaranteed $36, and the three players are playing for the remaining $72. So, T13,500 = $72 in the aggregate. Further, when there are 3 players left and if a player is down to 100 chips and is faced with a decision to call all-in or fold, but one of the other players is all-in and might bust out on this hand, then that T100 = $18, as the difference between third and second place money, and leaves the door open to another potential $36.

Tournament chips can have wildly different values late in the game, depending on the circumstances. This analysis is important to keep in mind when doing these exercises involving all-in plays late in a tournament. A decision you make with 3 or 4 players left can be vastly more important, in terms of cash in your pocket, than when you are heads-up, and does not necessarily relate to a comparison of your remaining tournament chips to the pot odds on a particular hand.