New Solution to $100-Sklansky-Problem !

Two players (First and Late) are dealt a real number between 0 and 390 - just to get somewhat nice numbers ...


There have been some minor changes in First's strategy ...


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On Round 1 First should bet with 310.62 or better and his 47.63 worst hands and check/call with 149.85 or better.

Late should bet with 270 or better and his 72 worst hands and call with 130 or better.


Second round has to be separated in to 3 different cases ...


Case 1 ($2 in pot - Round 1 was checked around - First is marked with a hand in the range: 47.63 to 310.62 - Late is marked with a hand in the range: 72 to 270).

First should bet with 204 or better and bluff with 83.17 or worse and check/call with 138 or better.

Late should bet with 171 or better and bluff with 105 or worse and call with 138 or better.


Case 2 ($4 in pot - First betted Round 1 - First is marked with a hand in the range: 0 to 47.63 or 310.62 to 390 - Late is marked with a hand in the range: 130 to 390).

First should bet with 310.62 or better and bluff with 15.88 or worse and check/fold with anything else.

Late should call with 182 or better. If First checks Late should do some betting and bluffing - but thats not really important. He could 'bluff' with his 10 worst hands and 'value-bet' with 50-best - that should do it.


Case 3 ($4 in pot - First check/called Round 1 - First is marked with a hand in the range: 149.85 to 310.62 - Late is marked with a hand in the range: 0 to 72 or 270 to 390).

First should 'check to the raiser' and call with 182 or better.

Late should bet with 270 or better and bluff with 24 or worse.


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Sklansky also asked for the value of the game ...


EV for First as a function of x is:


EV-First(x)= 0 * x:0-105 * 0


2*x-210 * x:105-171 * 4356


4*x-552 * x:171-270 * 32670


6*x-1092 * x:270-390 * 106560


Total EV for First is: 0+4356+32670+106560=143586


EV for Late as a function of x is:


EV-Late(x)= 43.66 * x:0-105


2*x-166.34 * x:105-171


4*x-508.34 * x:171-270


6*x-1048.34 * x:270-390


We see that for any x the difference in EV is always 43.66. That is exactly the difference in how often they bluff times 2:


(105- 83,17)*2=43.66


Very interesting ! Go figure /images/wink.gif


Total EV for Late is: 143586+43.66*390=160616


Notice: 143586+160616=304202=2*390^2 * That looks NICE !!!


Translating this solution to the original zero-one-game we have to divide 160616 by 390^2 and subtract $1 (the ante):


EV-Late= 5.598948 cent


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Just my opinion. Any Q's ? Don't ask me how long it took me to come up with this /images/wink.gif Comments are welcomed.

Unfortunately I am leaving town for a while so I won't be able to look at this for several days. Others are of course welcome to disect it, especially Bill Chen.

just an idle question


who is the ppfkaj?


since ds set his problem on the 1st July he has posted only 4 times that i can see with a quick scan - once to say he didn't send a virus; once to answer a tongue-in-cheek jibe; and twice to respond directly to the ppfkaj's comments on the problem, even though numbers of others have posted answers of one quality or another, without receiving a single word from the master


we have mat here now - is the ppfkaj mrs s or someone similar, that ds would jump like this?

though no - i haven't tried to answer the problem - it's too difficult for me, i admit


but a lot of others have


i am just curious if the ppfkaj is some highly respected poker player, that he is worthy of two more replies than anyone else


but then i get a kick out of my autographs collection

Jack,


Interesting! I'm happy that your results and the other guy who seemed confident came up with a similar earn for player 'late'. Something that seemed odd to me was that if hand 260 should be the average winning hand, wouldn't it make sense for 'early' to bet more hands than >~310 given the money in the pot?

What does late call with if First comes out betting after checking and calling?

Late is marked with a hand in the range: 0 to 72 or 270 to 390.


Obviously Late can't call w/ his bluff-hands: 0 to 72.


I recommend Late calls w/ 270 to 390 ... but that's not really important.

One very important property of an optimal strategy is that no matter what the opponent does, your strategy must do better than some expected value v for you. This is true for both players.


Therefore, if one of the players, L for example, knows F's strategy, it should not be possible to improve the expected value of the game for L.


The solution jack put forward, is on the right track, but I don't think it can be the solution because both players can make improvements to their strategy if the other side's strategy did not change. The reason for this is in the way the strategy is formulated. It's based on a small number of fixed intervals of cards, not probability distributions over all cards. There should be a chance that with some card L bets, not that L always bet with some card y and doesn't bet with others.


For a concrete example, I think I have an improvment to L's second round calling strategy after F and L checked in the first round. I'm a little confused as to why jack used the interval (0,390) instead of (0,1). I'm going to try to make the discussion in terms of probabilities and percentages.


At the point when F bets, he has two intervals of cards he could have, either (47.63,83.17) or (204,310.62). This gives him a total range of holdings (83.17-47.63)+(310.62-204) = 142.16. We know F is bluffing with 25% of his possible card holdings at this point ((83.17-47.63)/142.16). We also know F is "betting for value" with the rest of his holdings (75% =(310.62-204)/142.16).


Similarly, since we checked the first round, our holdings are (72,270). How much of this range is helping us call the value bets, and how much is calling bluffs? The fancy way to do it is with integrals, but instead you can plot the F intervals on the X axis, and the L intervals on the Y axis of a graph. Then, identify boxes and triangles. Find the areas of these boxes and add them together. This is the total area of everything that can happen. Now draw the line y = x on the graph. Calculate the area for only the boxes & triangles above the line y = x. This is the region of space where y wins after calling. I'm sorry, but I can't do it all here. It's too complicated.


When I got done with that. I ended up at L winning 36.49% of the time using jack's posted calling strategy in round 2 after F and L checked in round 1.


If you instead say that L should only call with only the cards in (204,270) I calculate a win rate of 58.195%.


This does not including expected value calculations w.r.t. the fact that you're folding a lot more hands. But the range of cards (138,204) are hurting L more than they're helping him. These calls are 50% of the calls made by L and they're only good against F's bluffs (47.63,83.17) which are only 25% of F's bets.


L's win rate improved to 58.195% from 36.49%.


L's calling rate went down 50% meaning he'll be losing another 36.49%*.5 = 18.245% of the hands that he used to get by calling more bluffs.


Weighting these by the money involved for each outcome:

delta EV_L = $4*21.705%-$3*18.245% = + $0.32085


Despite the fact that I think I have come up with an example that shows a weakness in the solution (and others like it) I think the solution that jack presented is on the right track. I think what he probably ended up doing was walling off hte right proportions of probability for calling/betting/etc. for each side. If we use these probability masses to decorate a decision tree, then the next step is to assign a distribution over all the cards.


So, I still think Jack's answer is worth examining in that it may have the right proportions of probability mass at each decision point which definitely would contribute to the solution.


Before I get too carried away, someone please let me know about any mistakes, it's late now! /images/smile.gif