Tournament Deals--fundamental questions
wc...@my-deja.com
some consensus on fundamental assumptions.
First, is the random walk model a good model for tournaments?
Secondly, does the Malmuth model *really* underestimate equity for tall
stacks?
First, random walk.
This is equivalent to lots of simple models. For instance, everyone
antes, everyone gets one card and the high card takes the pot. We
assume the ante is small in relation to the stack sizes. Or similarly,
we have a bunch or pairwise coin flips.
Most convincingly, this is also equivalent to the model where the bet
size is small compared to the stack sizes and we are playing 7-stud
amongst equally skilled players playing the same strategy.
The thing about all of these models is that after a number of games, the
number of chips each player has approaches the same gaussian whoose mean
is the number of chips each player started with. The only thing is that
a player is eliminated when his chips reach zero (and then the iteration
is restarted again with one fewer player).
In fact, this is equivalent to each player being assigned a variable
x_i(0) at the beginning corresponding to his # of chips. Each x_i takes
a random walk. The number of chips player i has at time t is
x_i(t) - (average(x_i(t)) - average(x_i(0))).
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Ok, Malmuth model. I'm talking about the one that assumes that chips of
the player who busts out is evenly divided amongst the remaining
players.
(BTW, this is equivalent to the model where the probability of a player
busting out next is *inversely proportional* to the number of chips he
has.)
Let's take a specific example, A=80, B=10, C=10. The overriding
assumption is that A has a 0.80 chance of winning, right? Then the two
questions are equivalent:
(1) Do you think A should have on average more or less than 85 chips if
B or C busts out third?
(2) Do you think A should have more or less than a 1/17 chance of 3rd?
See, the probability of A not finishing third (16/17 in the Malmuth
model) times A's expected chip count when B or C busts (85 in the MM)
has to be 80.
If you think A should have *more* than 85 chips when B or C busts, then
he has less than 16/17 chance of surviving, and thus *more* than 1/17 of
busting third, and thus the Malmuth model overestimates A's equity.
This does make some intutive sense, I mean when a short stack busts you
expect the tall stack to take most of his chips right?
Conversely, if you think A should have *less* than 85 chips when B or C
busts out, than the Malmuth model underestimates A's equity as Barbara
Yoon claims. This actually agrees with random walk.
If we take the random walk model, surprisingly A has less than 85 chips
when B or C busts. Intuitively, one might expect C's chips to be
randomly distributed, but the fact that C did *not* bust from a low chip
count (conditional probability here) skews his result more toward the
positive end than A.
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James Hoffmann
to the conclusion that there are too many variables that you cannot take
into consideration to model exactly. I think the best indicator is whether
or not the person who made the deal feels that they got at least a fair
shake. As an example:
Scenario 1:
Players left:
Yours truly with 40% of the chips and the button.
Tom McEvoy with 30% of the chips in the small blind.
T.J. Cloutier with 30% of the chips in the big blind.
Someone says:
Let's chop 1/3 each. I say "Yahoo" and am happy to take it.
Scenario 2:
Players left:
Yours truly with 30% of the chips and the big blind.
Mel from Reseda with 35% of the chips in the small blind.
Smedley from Lancaster with 35% of the chips in the button.
Someone says:
Let's chop 1/3 each. I say "Deal 'em" and figure I still probably have the
best of it.
I guess my point is that there are other considerations besides chip
position that go along way to my assessment of whether or not it is a good
deal for me. In scenario 1, even though I have more chips and a the best
position for that hand, I would be more than happy to give a small
percentage based on the relative strength and experience of my opponents.
In sceneario 2, despite my poor position I would not deal. Given the
algorithms, I would be giving up too much in the first and not taking the
"good deal" in the second. I believe that each situation needs to be
evaluated independently. Many things need to be considered such as the
relative position of the blinds, large vs. small stacks, position on who I
believe are the better players and/or the more aggressive players, my
relative strength of a player, size of the blinds, who's on tilt, my
psychological outlook and other factors may well be worth a couple of
hundred dollars one way or another. While I believe that many of the models
give a decent approximation, if I think I got a fair and equitable deal (or
better yet a deal that favored me based on the above considerations) I would
be happy, whether or any specific algorithm indicated I was justified in my
feelings.
James
wc...@my-deja.com wrote in message <7nh7cs$367$1...@nnrp1.deja.com>...
Barbara Yoon
> Ok, Malmuth model. ...the one that assumes that chips of the player
> who busts out is evenly divided amongst the remaining players.
> next is *inversely proportional* to the number of chips he has.)
> chance of winning... ...two questions are equivalent:
> (1) Do you think A should have on average more or less than 85 chips
> if B or C busts out third?
> (2) Do you think A should have more or less than a 1/17 chance of 3rd?
> has less than 16/17 chance of surviving, and thus *more* than 1/17 of busting
> third, and thus the Malmuth model overestimates A's equity. This does make
> most of his chips right? Conversely, if you think A should have *less* than 85
> chips when B or C busts out, than the Malmuth model underestimates A's equity
> model, surprisingly A has less than 85 chips when B or C busts. Intuitively, one
> might expect C's chips to be randomly distributed, but the fact that C did *not*
OK, William, my answers to your two questions are definitely "LESS and LESS."
And as you point out, the "more and more" answers can appear to be "right"
superficially, but expecting A (starting 80 chips) to have more than 85 chips after
B or C busts (starting 10 chips each) overlooks that the process of a player
busting is NOT just a matter of that player directly losing his chips in some
proportions to the other two players, but also in the meantime, those other two
players are winning and losing chips from and to each other too, and also think
of how B or C could bust after first winning his way up to 98 chips, and then
dramatically losing it all! OK, so now, what are YOUR answers?!
Barbara Yoon
> ...tournament deals...too many variables that you cannot take into consideration
> to model exactly. ...best indicator is whether or not the person who made the
> deal feels that they got at least a fair shake. ... ...other considerations besides
> chip position that go along way to my assessment of whether or not it is a good
> blinds, large vs. small stacks, position on who I believe are the better players
> and/or the more aggressive players, my relative strength of a player, size of
> the blinds, who's on tilt, my psychological outlook and other factors may well
> give a decent approximation, if I think I got a fair and equitable deal I would be
> happy, whether any specific algorithm indicated I was justified in my feelings.
Well stated, James...and your points are all valid ones, though real buggers
to mathematically model! All we are trying to get with all the analysis and
modeling are some reliable guidelines -- like in your own examples, if such
guidelines indicated that your stack was "worth $X" in prize money, then you
should definitely accept that amount if you are about to be the big blind, with
Mike Caro on the button, and Amarillo Slim on your left...but then if Mike's
been sipping something out of a paper bag, and slurring incredible stories
about some imaginary "talking bird," and the guy you thought was Slim is
actually just a tourist wearing a cowboy hat, in town for a Chico State alumni
reunion, then as you say, "Deal 'em!"
Barbara Yoon
>>> Ok, Malmuth model. ...the one that assumes that chips of the player
>>> who busts out is evenly divided amongst the remaining players.
By the way, William, this "Malmuth model" that you describe is one of the
two given in his book, this one being the one that the book amusingly calls
"more correct," when in truth, it's actually more WRONG! The other model
in the book works by assuming that whenever a player wins, it is simply by
'declaring victory,' and leaving the other two players to go heads-up with
their original chips, for second place. This second Malmuth model can be
shown to be flawed against the big stacks, and the first model (that you
describe) to be flawed in that same way even worse...
Robert
wrote:
I prefer to call the "Malmuth model" which William refers to as the
"Weitzman method" (or model). Mark Weitzman developed and
described this model in Mason Malmuth's "Gambling Theory and
Other Topics". He also described the second method which was
suggested to him by Mason Malmuth. I refer to this as the
"Malmuth method".
The "Malmuth method" is not as silly as you suggest. It can be
derived from two very reasonable assumptions:
(1) The probability of a player finishing first is equal to the
ratio of his chips to the total number of chips.
(2) The probability of player A finishing ahead of player
B is equal to the ratio of player A's chips to the sum of
A's and B's chips.
If the Random Walk tournament model is accepted as accurate,
then the Weitzman and Malmuth methods can be shown to
favor the small stacks at the expense of the large stacks.
Otherwise, I think it's an open question.
Robert Zimmerle
Barbara Yoon
>>> Ok, Malmuth model. ...the one that assumes that chips of the player
>>> who busts out is evenly divided amongst the remaining players.
B.Y.:
>> By the way, William, this "Malmuth model" that you describe is one of the
>> two given in his book, this one being the one that the book amusingly calls
>> "more correct," when in truth, it's actually more WRONG! The other model
>> in the book works by assuming that whenever a player wins, it is simply by
>> 'declaring victory,' and leaving the other two players to go heads-up with
>> their original chips, for second place. This second Malmuth model can be
>> shown to be flawed against the big stacks, and the first model (that you
>> describe) to be flawed in that same way even worse...
Robert Zimmerle:
> I prefer to call the "Malmuth model" which William refers to as the "Weitzman
> method" (or model). Mark Weitzman developed and described this model
> in Mason Malmuth's "Gambling Theory and Other Topics". He also described
> the second method which was suggested to him by Mason Malmuth. I refer to
> this as the "Malmuth method". The "Malmuth method" is not as silly as you
> suggest. It can be derived from two very reasonable assumptions:
>
> (1) The probability of a player finishing first is equal to the ratio of his chips
> to the total number of chips.
> (2) The probability of player A finishing ahead of player B is equal to the
> ratio of player A's chips to the sum of A's and B's chips.
>
> If the Random Walk tournament model is accepted as accurate, then the
> Weitzman and Malmuth methods can be shown to favor the small stacks
> at the expense of the large stacks. Otherwise, I think it's an open question.
Yes...the first, allegedly "more correct" (but actually more wrong!) model
(described by William Chen above) given in Malmuth's book could be called,
"Weitzman," and the second model (described by me above) given in the
book could be called, "Malmuth." But as you point out, both of these models
are flawed in such ways as "to favor the small stacks at the expense of the
large stacks" -- of your "two very reasonable assumptions" above, the second,
while superficially appealing, under closer scrutiny, does not quite pass muster.
However, the characterization of "silly" was not by me, but by another poster
in the "Idiot's Guide to Tournament Deals" thread on July 21, in response to
Mason Malmuth's July 20 post, excerpted below, in which Malmuth bafflingly
describes something entirely different from either method in his book as
"my [his] method" -- and I would agree that what he said there truly is "SILLY":
Mason Malmuth:
>>>>> Years ago I recommended to Card Player Magazine that they drop
>>>>> [W. Lawrence Hill's] writing because it was so misleading at times.
>>>>> [*** Evidently, the publisher didn't take you very seriously in that. ***]
>>>>> ...in my method of settling up it is assumed that the person who busts
>>>>> out loses his chips to the other players in proportion to their stack size.
Barbara Yoon
>>> Mason Malmuth's [book]...second method... ...can be derived from
>>> two [superficially] very reasonable assumptions:
>>> (1) The probability of a player finishing first is equal to the ratio of
>>> his chips to the total number of chips.
>>> (2) The probability of player A finishing ahead of player B is equal
>>> to the ratio of player A's chips to the sum of A's and B's chips.
>>> can be shown to favor the small stacks at the expense of the large stacks.
Yes, Robert, the "second model" does work on the basis that you describe here
(you've evidently done some hard digging into it!), and also equivalently on the
basis that I expressed earlier in this thread -- "works by assuming that whenever
a player wins [in accordance with your 'assumption (1)' above], it is simply by
'declaring victory,' and leaving the other two players to go heads-up with their
original chips, for second place." Check it out, and see if you agree that our
respective descriptions of how this model works indeed work out to the same
thing. The flaw in this model manifests itself in effect as "the LARGER a player's
stack gets, the WORSE he plays" -- equally true for all players, but a relative
advantage for a player with a small stack...see?!