Head Up Theory Question

[PokrLikeItsProse]

Lets add a few more questions.

Same problem, except you and your opponent both have a finite amount of money, $100. Would that change your strategy?

In hold em, you hold A2 of spades. The board comes 345 of spades with two other meaningless cards. The only hand that can beat you is 67 of spades. Given unlimited money between you and your opponent and unlimited raising allowed, how many times should you reraise before you just call against an expert player in a limit game? How about in a pot limit game?

[MicroBob]

[ QUOTE ]
strategy of calling and folding after getting reraised the 8th time at a ratio of what the pot odds are offering (21:1 I think). This wouldn't allow your opponent any opportunity to exploit your strategy.

[/ QUOTE ]


mixing calling with folding after 8 re-raises is still lousy.

folding is especially terrible.
calling is only mostly terrible.


you're going to fold getting 1:21 odds on the LITERALLY 1-in-a-million chance that he has you beat?



i still very much look forward to any ideas David has on this.
The fact that he says he hasn't figured out the answer and is just watching us concerns me though.
What if there is no answer?!?!
How will I go on?!?!

[M.B.E.]

Our decision whether to keep raising, at any given point, is based on these three values:

p:= the probability your opponent is not holding card 1,000,000, given the action so far;

q:= the probability your opponent will reraise if he is not holding card 1,000,000 and you now reraise, given the action so far; and

r:= the probability your opponent will fold if he is not holding card 1,000,000 and you now reraise, given the action so far.

I think we want to continue raising as long as

p(q-r+3) > 2.

If we want the game-theoretically optimal result, then we should be able to derive p and q using Bayes' Theorem recursively; r will quickly approach zero so we can probably ignore it safely.

[MtDon]

I posted that late last night, and somewhat off the top of my head. After a little more thought, I'd like to withdraw my comment that you need to consider whether the expert misread his hand. The chance that he has a hand that could be misread as 1,000,000, mulytiplied by the probability that he misread it, is too low to matter in deciding whether to raise or not.

The problem with the question is the statement that you have unlimited money, which is an immposibility. The total money in the USA (M3) is about 9,616.3 billion dollars according to this site: http://www.federalreserve.gov/releas...st/h6hist1.txt

I didn't find how much money there is in that world, but certainly not more than the US M3 times the number of countries in the world, which is about 200 (just under that many). So if both players have half the money in the world, they'd have about 961,630 billion dollars, say, 1,000,000 billion dollars. In scientific/engineering notation: 1.0E15.

Playing pot limit, a raise will exceed this on the 37 betting round, according to my calculations. At a speed of one raise per second, after the first few raises (when you might think or pretend to think awhile), the world's money supply could easily be exhausted before you are a minute into the hand.

Betting
Round Bet/Raise Pot after bet, before call or raise
1 2 4
2 .......... 6 ....... 10
3 ......... 16 ....... 26
4 ......... 42 ....... 68
5 ........ 110 ...... 178
6 ........ 288 ...... 466
7 ........ 754 .... 1,220
8 ...... 1,974 .... 3,194
9 ...... 5,168 .... 8,362
10 .... 13,530 ... 21,892

20 2.047E+07 3,312E+08

30 3.096E+12 5.009E+12

35 3.808E+14 6.161E+14
36 9.969E+14 1.613E+15
37 2.610E+15 4.223E+15

40 4.683E+16 7.578E+16


-- Don

[9cao]

I don't think I am anywhere close to the correct answer. However, I am quite certain that your logic is off. It is not a question of what the odds of him having 1,000,000 are (1 in a million) it is what are the odds of him reraising you 9 times without 1,000,000 (optimal should be 1/22, given the 8th reraise, or something like that).

[Shoe]

How many times do you think an expert would re-raise with hands like 999998, 999997, 999996.... 995000, 990000? I would think each of those hands would be worth well more than 9 or 10 rounds of betting as well.

[MicroBob]

precisely.

if someone played this for 100 or 200 hands or so with all the 576,893 losing to 724,358 and so on then many would most certainly be inclined to put in more than 8 bets with 999,987.


If I have 999,001 and knew that there was a 1:21 chance that you would be folding after my 8th re-raise then I shouldn't be calling at that point.
i should be re-raising because of the combined chances that I'm still best OR that you would fold the better hand.


Otherwise, I admit to my complete idiocy on this situation and suspect David is reading this and shaking his head wondering how the hell any of us can possibly win at the tables using logic as ridiculous and flawed as ours in our feeble attempts to figure out the answer.

(although it looks like MBE may very well be onto something I suspect).

[FishAndChips]

Well this certainly is a tricky problem. While I'm not quite sure what the correct answer is, I think a few deductions can be made:

First, if either player is going to fold, it would be right off on the first or second bet. Any additional betting would end with a CALL and not a fold. Thus, I assume that after there has been a bet and a raise, there will be AT LEAST a call to end the action. (This satisfies everyone who says the pot odds are to good to fold.) Therefore, the fold equity for either player is tantamount to zero after a bet and a raise etc. (You will just call at the moment that you no longer think its likely that you are in the lead when called or raised by your opponent-- and your "expert" opponent will do the same)

Second, the money is hypothetically unlimited, but we are still looking to make the play which has the greatest EV.

Third, you call based on pot odds, and raise based on pot equity. Your opponent is an "expert" and knowing that there is no fold equity after a few bets, everytime he raises it's for what he perceives as value. Likewise, you only raise when you believe you have have a pot equity edge when called by your opponent. Claiming you raise based on pot odds is wrong when there is no way your opponent will fold a better hand. The size of the pot, after the first few bets in this case, only really matters when you think you may no longer be the favorite when called and are considering the chances your beat to the pot odds you're getting to call.

Finally, your hand is strong there is no denying this, but your opponent won't keep raising without a strong hand himself, and eventually he would just call at the point where he feels he is no longer the favorite when his raise is called or reraised. This is why I believe the "halfing" method is correct, because without the 1,000,000, your opponent would just call at some point before you would be able to get in 20 bets and raises. [img]/images/graemlins/smile.gif[/img]

[MicroBob]

this is a very good post.


and since the opponent also assumes that we are never going to fold either then I do now see some validity for the halving method.


Still...for some reason just calling after 8 re-raises FEELS incorrect here because I suspect the chances that he would re-raise 10 or more times with anything from 999,990 to 999,998 is still high enough to justify keeping the heat on.



After 8 bets are in there do I feel there is a 50/50 chance that my opponent would re-raise with me 4 more times with just 999,997 or 999,998?
If the answer is yes then I need to keep re-raising because there is a 2x greater chance that he would have one of those numbers than he would have the nuts.

Even if I think there is only a 35% chance that he would take it to 12 bets with EITHER 999,998 OR 999,997 that still gives value to it because the chances he has ONE of those hands are greater than the chances that he has 1-million.


My continued re-raising is based on the fact that I believe HE is very possibly not so conservative as to only call after 8 bets with 999,997 or 999,998.


Throw in the 10-20% chance that he would keep going with hands down to 999,995 or even 999,990 and I think you need to keep it going well past 8 bets.

[pheasant tail (no 18)]

[ QUOTE ]
That looks like log2 rather than log4.

Not that I'm saying log4 is the correct answer.

[/ QUOTE ]

Not that there is anything wrong w/ using log4, of course. But I included metagame considerations in my decision to use log2. Considering the fact that I'll still have well over $999,000 in my bankroll after hand (should I lose), I should have plenty of opportunity to capitalize on it in our fine game.

I'd hate to have my opponent know that I will only put in the 10th raise w/ the nuts. I'd get no action [img]/images/graemlins/grin.gif[/img]