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#31
05-05-2007, 04:04 AM
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Re: If You Are Right-
[ QUOTE ]
Regarding solving the game originally, there are systematic algebraic ways to do it, but I solved this game by inspection, basically just fiddling with the strategies until I got a Nash equilibrium. [/ QUOTE ] "Fiddling with the strategies" is still a valid method: see Fictitious Play. Juk [img]/images/graemlins/smile.gif[/img] 
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#32
05-05-2007, 04:08 AM
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Re: If You Are Right-
Cool. What would be interesting, to get a true measure of the value of position in this game (it's still clearly valuable, just apparently not enough to make the game +EV for the BB), would be to compute the EV if the BB had to act first instead of the SB. The difference is the value of position.
It's my hunch that if you crunch numbers in more detail, the player in position's advantage WHEN THE POT IS RAISED BY EITHER PLAYER continues to increase (approaching some maximum value that's obviously capped at or below winning $3 back from the pot), but the amount of times the pot can actually be raised is inversely proportional to the raise size (a 10-fold increase in raise size will result in a roughly 10-fold decrease in raise frequency for meaningful raise sizes). That would mathematically account for the distribution with a peak advantage (well, minimum disadvantage) for the BB at 7BB.
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#33
05-05-2007, 01:55 PM
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Re: Important, Simple, Hard, Game Theory Problem
[ QUOTE ]
If both player's play the above strategies Player 1 has an expected win of 1/16. [/ QUOTE ] I checked your result, and I'm not sure it's correct. Checking is much easier than solving, I just computed the expected value for each player, then checked if a slight change to any of the parameters increased the expected value for the player controlling that parameter. To make progress, I suggest we start with the calculated edge. Based on your strategy, I think the results should be as shown in the table below. The first number is the frequency out of 384 hands that the result should occur, the second number just states the frequency as a percentage, the third number is the small blind's profit (negative for a loss) and the fourth number is the profit to the big blind. The average profit for SB is 35/32, meaning a gain of 3/32 (not 1/16 as you said) after posting the small blind. The average profit for BB is 61/32, meaning a loss of 3/32 after posting. Do you agree with these outcomes? If so, we can move on to slight variations. If not, perhaps I've made a mistake in entering your strategy or computing the results. 112 29% 0 3 SB folds 64 17% 3 0 SB Raises, BB Folds 60 16% 3 0 SB Calls, BB Calls, SB wins 36 9% 5 -2 SB Raises, BB Calls, SB wins 33 9% -3 6 SB Calls, BB Raises, SB Calls, BB wins 28 7% -3 6 SB Raises, BB Calls, BB wins 24 6% -1 4 SB Calls, BB Raises, SB Folds 15 4% 5 -2 SB Calls, BB Raises, SB Calls, SB wins 12 3% 0 3 SB Calls BB Calls, BB wins
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#34
05-05-2007, 04:05 PM
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Re: Important, Simple, Hard, Game Theory Problem
[ QUOTE ]
[ QUOTE ] If both player's play the above strategies Player 1 has an expected win of 1/16. [/ QUOTE ] I checked your result, and I'm not sure it's correct. Checking is much easier than solving, I just computed the expected value for each player, then checked if a slight change to any of the parameters increased the expected value for the player controlling that parameter. To make progress, I suggest we start with the calculated edge. Based on your strategy, I think the results should be as shown in the table below. The first number is the frequency out of 384 hands that the result should occur, the second number just states the frequency as a percentage, the third number is the small blind's profit (negative for a loss) and the fourth number is the profit to the big blind. The average profit for SB is 35/32, meaning a gain of 3/32 (not 1/16 as you said) after posting the small blind. The average profit for BB is 61/32, meaning a loss of 3/32 after posting. Do you agree with these outcomes? If so, we can move on to slight variations. If not, perhaps I've made a mistake in entering your strategy or computing the results. 112 29% 0 3 SB folds 64 17% 3 0 SB Raises, BB Folds 60 16% 3 0 SB Calls, BB Calls, SB wins 36 9% 5 -2 SB Raises, BB Calls, SB wins 33 9% -3 6 SB Calls, BB Raises, SB Calls, BB wins 28 7% -3 6 SB Raises, BB Calls, BB wins 24 6% -1 4 SB Calls, BB Raises, SB Folds 15 4% 5 -2 SB Calls, BB Raises, SB Calls, SB wins 12 3% 0 3 SB Calls BB Calls, BB wins [/ QUOTE ]Should be -1 4, that's the reason for the discrepancy.
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#35
05-05-2007, 07:16 PM
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Re: Important, Simple, Hard, Game Theory Problem
[ QUOTE ]
<font class="small">Code:</font><hr /><pre> 112 29% 0 3 SB folds 64 17% 3 0 SB Raises, BB Folds 60 16% 3 0 SB Calls, BB Calls, SB wins 36 9% 5 -2 SB Raises, BB Calls, SB wins 33 9% -3 6 SB Calls, BB Raises, SB Calls, BB wins 28 7% -3 6 SB Raises, BB Calls, BB wins 24 6% -1 4 SB Calls, BB Raises, SB Folds 15 4% 5 -2 SB Calls, BB Raises, SB Calls, SB wins 12 3% 0 3 SB Calls BB Calls, BB wins</pre><hr /> [/ QUOTE ] It's difficult to check your work. You didn't provide action vectors for SB and BB. [ QUOTE ] Player 1's strategy: [0,6/24] raise [6/24,12/24] call and call a raise [12/24,15/24] call and fold to a raise [15/24,17/24] raise [17/24,1] fold [/ QUOTE ]
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#36
05-05-2007, 07:20 PM
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Re: Important, Simple, Hard, Game Theory Problem
Also I'd offer the following argument that my proposed strategy pair is an equilibrium:
The BB's strategy is a best response to the SB's because: Following an SB raise, the best response is to call with any hand that's good 25% of the time against the SB's raising range (since he's getting 3:1). This means he has to call with hands < 6/24, has to fold hands > 15/24, and can call or fold with hands in between. BB's calling threshold is 12/24 so that checks out. Following a SB call, SB has a hand in [6/24,15/24], and will call a raise with 12/24 or better. The best response to this is to value raise with 9/24 or better (the old rule that if you're closing the action, you should value raise with any hand that will be good 50% of the time that you're called. SB will call with [6/24,12/24], so 9/24 is the threshold hand for being good 50% of the time when called). The BB will break even on bluffs with his hopeless hands (his hands > 15/24 which cannot win at showdown) since he's getting 2:1 on a bluff and the SB will fold 1/3 of the time. BB can therefore raise or fold with his hopeless hands. And hands that have some showdown value but can't be raised for value have to be checked down. My strategy has BB raising with [0,9/24] and [21/24,1], and checking with everything else, so that checks out. The SB's strategy is a best response to the BB's because: Notice that with the SB's hands < 9/24, raising and call-calling have the same expectation (with a hand x, SB will lose $4 with probability x, win $4 with probability .5-x, and win $2 with probability .5, whether he raises or call-calls). And call-calling is better than call-folding since all these hands are good > 25% of the time against BB's raising range. And open folding is pretty clearly wrong. So to be a best response SB must raise or call-call with hands < 9/24. My strategy for the SB does this, so that checks out. With hands [9/24,12/24], call-calling is better that raising because with a hand x, both options have the SB winning with probability x and losing with probability 1-x, but raising always loses $4 when it loses and call-calling sometimes only loses $2 when it loses, and when they win, call-calling wins $4 more often than raising does (and both, whenever they don't win $4, win $2). Call-calling is equal in expectation to call-folding since these hands are good exactly 25% of the time against BB's raising range. And call-folding is better than folding since call-folding will result in SB winning the pot more than 25% of the time (SB must win the pot against BB's [12/24,21/24]), and SB is getting 3:1 on a call-fold. So the SB's best response with [9/24,12/24] is to call-call or call-fold. I have SB call-calling with [9/24,12/24] so that checks out. With hands [12/24,15/24], raising is a pure bluff and is breakeven since the SB is getting 1:1 on a bluff and the BB calls exactly 50% of the time. Raising is therefore equal in expectation to folding. Call-folding is still better than folding since call-folding with these hands will still win at least 25% of the time (SB must win the pot against BB's [15/24,21/24]), and SB is getting 3:1 on a call-fold. And call-folding is equal in expectation to call-calling since these hands are good exactly 25% of the time against BB's raising range. So the SB's best response with [12/24,15/24] is to call-call or call-fold. I have SB call-folding with these hands so that checks out. With hands [15/24,1], raising is still equal in expectation to folding. Call-folding is now worse than folding since call-folding won't win the pot 25% of the time anymore. And call-calling is equal to or worse than call-folding since over this interval an SB hand is good <= 25% of the time against a BB raise. So the SB's best response with [15/24,1] is to raise or fold. My strategy has SB raising with [15/24,17/24] and folding [17/24,1], so that checks out.
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#37
05-05-2007, 11:18 PM
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Re: Important, Simple, Hard, Game Theory Problem
[ QUOTE ]
Also I'd offer the following argument that my proposed strategy pair is an equilibrium: The BB's strategy is a best response to the SB's because: Following an SB raise, the best response is to call with any hand that's good 25% of the time against the SB's raising range (since he's getting 3:1). This means he has to call with hands < 6/24, has to fold hands > 15/24, and can call or fold with hands in between. BB's calling threshold is 12/24 so that checks out. [/ QUOTE ] Your own link to Tom Ferguson says BB must call halfway between SB's value bet and bluff points. Else SB can be exploited. Whether you're using low or high values as best hand. The call point is not halfway. It's closer to the weaker hands. SB will bet more value bets than bluffs.
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#38
05-07-2007, 12:37 AM
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Re: If You Are Right-
[ QUOTE ]
[ QUOTE ] [ QUOTE ] How large does the size of the raise have to be (as opposed to the two dollars it is now), with 1 and 2 blinds, for the game to be even? [/ QUOTE ]I don't see a way to do this which doesn't involve a ton of algebra (solving for optimal strategies in terms of r, calculating the outcome when the optimal solutions are played against each other as a function of r, then setting that function equal to 0 and solving for r), which I'm too lazy to do. Anyone have any clever ideas? [/ QUOTE ] Well how bout solving for a few guesses, perhaps five dollars, to see approximately where the edge switches. You do agree it would switch don't you? [/ QUOTE ] Why would it switch?
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#39
05-07-2007, 11:35 PM
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Re: Important, Simple, Hard, Game Theory Problem
To solve. Break the game into parts.
Part A. SB may fold or raise. Fold, game over. Raise BB may fold or call. SB wins 1. SB's action vector [0,.5] bet [.5,5/6] fold [5/6,1] bluff BB's action vector [0,2/3] call [2/3,1] fold =========== Part B. SB may fold, complete, or raise. Game over if SB folds or completes. BB may fold or call raise. SB wins 1.25 SB's action vector [0,.25] bet [.25,.75] complete [.75,5/6] fold [5/6,1] bluff ============ Complete game. SB wins 1.0933 Still tweaking the results.
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Linear Mode
