Winner-take-all theory

[M.B.E.]

In a NLHE winner-take-all one-table satellite, is your equity determined solely by your chips as a fraction of the total in play, or does it also depend on the number of players remaining?

For example, if there are five players left and then one of your opponents eliminates another one so you are down to four, does that please you?

Another example: if you have 6000 in your stack, would you rather be the chip leader threehanded (6000-4000-3500), or be trailing headsup (6000-7500). Or does it make no difference?

I don't know the answer to this question and I'm interested in arguments on both sides.

[eastbay]

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In a NLHE winner-take-all one-table satellite, is your equity determined solely by your chips as a fraction of the total in play, or does it also depend on the number of players remaining?


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Good question and one I've wondered about myself. I don't know.

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For example, if there are five players left and then one of your opponents eliminates another one so you are down to four, does that please you?


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Intuitively, I think if anything it displeases me. You still need to win those chips somehow, and I think it is easier to win them from two stacks my size than from one stack twice my size. The only reason I can conjure up to support that is that it is easier to get a comparable stack to fold, since I can threaten his tournament, whereas the double stack feels less pressure since he's still alive even if he loses an all-in with me.

eastbay

[pzhon]

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In a NLHE winner-take-all one-table satellite, is your equity determined solely by your chips as a fraction of the total in play, or does it also depend on the number of players remaining?

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It also depends on your position, particularly if the blinds are large.

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Another example: if you have 6000 in your stack, would you rather be the chip leader threehanded (6000-4000-3500), or be trailing headsup (6000-7500). Or does it make no difference?

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I think you are better off with your opponents' chips consolidated. In general, it is an advantage to have a short stack with several opponents. It is a disadvantage to have a large stack.

A short stack can imagine that the larger stacks are all the same size as the short stack. From this perspective, the large stacks play badly, including sometimes folding while "all-in." This helps the short stack.

The sizes of bets above the size of the short stack can convey information about whether the short stack should call or not. For example, if there is T300 in the pot, and a short stack has T300 left, a bet of T300 and a raise to T1200 in front of the small stack may suggest more strength than a bet and call of T300.

[Sheriff Fatman]

If I remember correctly, the equity is directly related to stack sizes only when Heads Up. With more than 2 players the maths gets more complicated.

There's a section about this in Mason's book, Gambling Theory and Other Topics. The theory very relevant when dealmaking at a final table as it can be used to determine whether a particular deal offered is beneficial for you or not.

I'm sure the theory applies equally to structured payouts and winner take all events.

Sheriff

[La Brujita]

I have been mulling this over a bit over the last few weeks and I just can't get a comfortable answer. The following are things I have been thinking about and I don't know which way each or all cut:

1. The concept that you can't reach into your pocket to buy more chips has to have some implications that make this different from a ring game. I think tournament theory has to apply to some extent, particularly the gap concept as it applies to tournaments.

2. Isn't it still true in this format that if you have more chips than your opponents let's say 6000 3000 3000 each individual chip for you has less value (because for any bet you are risking less of your stack)? This is a point I am not clear of for the following reason: for any given size bet you have the potential to increase your original stack size by x percent. What I am talking about of course is implied odds but I guess my point is in the 6000 3000 3000 example if one big stack and short stack are involved and there is a 300 bet, ss can double his stack moving from 25% of all chips to 50% while bs can increase his stack 50% moving from from 50% to 75%. In some sense although the ss had to risk a larger portion of his stack isn't he deriving a bigger benefit? Should the third stack be indifferent to the result assuming equally skilled opponents?

3. This point is purely from my experience but I would much rather have 3000 chips toa single opponent's 9000 rather than 3000, 3000, 6000. Perhaps it depends on blind size and perhaps it is because I am a better heads up player rather than three handed.

This is just me thinking out loud and I have no real answer.

Regards

[thomastem]

Believe it or not being happy that someone got eliminated by somebody other than yourself is a simple one.

Did the stronger of the 2 players get eliminated or build their stack? I just cheer for the weaker player.

[eastbay]

I think to have any kind of tractable discussion here you have to assume all players are of equal skill.

eastbay

[Wayne]

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For example, if there are five players left and then one of your opponents eliminates another one so you are down to four, does that please you?


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It would please me. I play a decent HU game, better than most SnG players.

Ignoring the skill levels of HU/shorthanded/full table, the theory says your chance of winning is equal to your percentage of chips. So your EV doesn't change in the examples you give.

[M.B.E.]

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Ignoring the skill levels of HU/shorthanded/full table, the theory says your chance of winning is equal to your percentage of chips. So your EV doesn't change in the examples you give.

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That is the conventional wisdom, but is it correct? I am not persuaded by the "purely logical proof" David offers in TPFAP (pp. 105-06) for more than two players.

Pzhon and La Brujita suggest some good arguments for why your probability of winning might not be simply the fraction of the chips owned by you. Their arguments would need to be refuted before we can accept the conventional wisdom as correct.

[M.B.E.]

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I think to have any kind of tractable discussion here you have to assume all players are of equal skill.

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It's useful to start with the scenario where all players are equally skilled and are playing similar strategies. And of course Thomastem is generally correct that if skills are unequal, our equity increases when a stronger-than-average player is eliminated (even when our stack size stays the same).

But there are some non-trivial examples with unequal skill levels that may be interesting to think about. Suppose we are the weakest player at the table, and our opponents are equally skilled with each other. Now one opponent knocks out another opponent. Has our equity increased or decreased (or remained constant)?