Calculating Odds Of Coming In Places Other Than First

I'm new to these forums, so if this has already been posted, feel free to flame and/or delete at will. But, I saw some people using approximations and stuff, and i'm 95% sure this is actually correct, so enjoy.

I think it's easiest with an example, so here's a theoretical tourny with easy numbers:

Player 1: t100
Player 2: t200
Hero: t700
Total: t1000

Your odds of first are obviously 700/1000 = 70%

Your odds of second are slightly more complicated. You get them by figuring each other player's odds of winning multiplied by the "tournament" you play out with the remaining field, then summing. Thus:

Odds of Player 1 winning and you coming in second =
(Player 1 chips) / (Total Chips) * (Hero Chips)/ (Total Chips - Player 1 chips) = 10% * 78% = 7.8%

Odds of Player 2 winning and you coming in second =
(Player 2 chips) / (Total Chips) * (Hero Chips)/ (Total Chips - Player 2 chips) = 20% * 88% = 17.6%

Total Odds of Second Place = 17.6% + 7.8% = 25.4%

Now, it follows that the odds of coming in 3rd are 4.6%. You can check for yourself that the formula is consistent with the odds of player 1 and player 2 placing in second and third. Note that for more than 3 players, the odds of coming in third can be found by the very long calculation of

SUM over X( SUM over Y (Player X) * (Player Y) * (Hero)/((Total Chips)*(Total Chips-Player X) * (Total Chips - Player X - Player Y)) ) ).


I noticed people in this forum using approximations that resulted in the following:

Player 1: 500
Player 2: 500
Hero: 9000

Hero's odds of first: 90%
Hero's odds of second: 5%
Hero's odds of third: 5%

This is clearly wrong because hero, with his monstrous (and exaggerated) chip advantage clearly has a much larger chance of coming in second than last. The real values are

1st: 90%
2nd: ~9.4%
3rd: ~.6%

Does anyone know what SnG Power Tools uses?

[citanul]

[ QUOTE ]
Player 1: t100
Player 2: t200
Hero: t700
Total: t1000

Your odds of first are obviously 700/1000 = 70%

[/ QUOTE ]

I stopped there, as that's when you reached the point of saying something I felt was incredibly wrong. Hope I didn't miss much in the rest.

citanul

[USCSigma1097]

He used to be Citanul...

He's now called the flamethrower.

That is all.

[ QUOTE ]
[ QUOTE ]
Player 1: t100
Player 2: t200
Hero: t700
Total: t1000

Your odds of first are obviously 700/1000 = 70%

[/ QUOTE ]

I stopped there, as that's when you reached the point of saying something I felt was incredibly wrong. Hope I didn't miss much in the rest.

citanul

[/ QUOTE ]

I'd like it if you checked the rest, but ok I guess.

[citanul]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
Player 1: t100
Player 2: t200
Hero: t700
Total: t1000

Your odds of first are obviously 700/1000 = 70%

[/ QUOTE ]

I stopped there, as that's when you reached the point of saying something I felt was incredibly wrong. Hope I didn't miss much in the rest.

citanul

[/ QUOTE ]

I'd like it if you checked the rest, but ok I guess.

[/ QUOTE ]

But, but, you appear from my skim to go on to make a bunch of conclusions that "follow" or are based on this original assumption of the "obvious." So I have little interest in following the logic.

FWIW, SNGPT uses ICM, which appears to be sort of what you're trying to get at.

citanul

[AKQJ10]

I see the problem that the OP is conflating odds with probability, and misusing the former term instead of the latter. Beyond that, though, I don't see the logical fallacy. I thought Sklansky's TPFAP went to great lengths to demonstrate that probability of finishing first was proportional to chip count, provided of course that all remaining players play equally well.

Can you help me understand the OP's error in logic?

[FlyWf]

ICM calculator shows chance of first at 70% in the OP's example.

Oh, I didn't realize there was any debate about the % chance to win = chips/total chips. I tested that with a comp program, and it's in a lot of books, so it's right. I'm pretty sure I'm right here, so I'm not a big fan of being criticized unless this was old hat to you guys.

[citanul]

a) players don't play equally well (not that icm actually takes this into account)

b) i haven't looked in the last few days, but i think that it says 1st is proportional to chips, but not linearly

c) i think that analysis like this that doesn't consider blinds levels is inherently flawed, even though adding blinds levels into the equation is difficult.

d) i'm sure there's more, but meh.

citanul

[ QUOTE ]
a) players don't play equally well (not that icm actually takes this into account)

b) i haven't looked in the last few days, but i think that it says 1st is proportional to chips, but not linearly

c) i think that analysis like this that doesn't consider blinds levels is inherently flawed, even though adding blinds levels into the equation is difficult.

d) i'm sure there's more, but meh.

citanul

[/ QUOTE ]

You're correct in that the model assumes random amounts of chips moving from one player to another. Skill, as well as things like position and blind structure, and big bullying chip stacks aren't being taken into acount, but they never are in these models and shouldn't be IMHO.

I've seen people using less accurate models, so here's a better one.