#11
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Re: Why Position Matters Using Game Theory
I thought I mentioned that .
There are two antes so you should bet the pot which is 2 antes . So your opponent is getting 2:1 odds to call . |
#12
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Re: Why Position Matters Using Game Theory
Oh. Duh. You're right. I missed the "pot" part of "bet the pot" in one sentence, and everywhere else you just say "bet". [img]/images/graemlins/smile.gif[/img]
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#13
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Re: Why Position Matters Using Game Theory
why dont u become a math teacher ?
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#14
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Re: Why Position Matters Using Game Theory
[ QUOTE ]
Here is another interesting problem that shows why position matters . There are two players who pick random numbers from the closed interval [0,1] . Each player posts an ante and has to make a decision whether or not to play for the pot or fold . The first player to act has three choices to make . He may either bet the pot, check-call , or check and fold . Player two can only call a bet , check or fold . There is no raising in this game . a) What numbers should player one bet with ? b) What numbers should player one check and call with ? c) What numbers should player one check and fold with ? *Bonus Question* What is the EV of this game for both players ? [/ QUOTE ] I'm not 100% sure but here goes: Player 1 should bet with 3/7 thru 1 Player 1 should never check and call. Player 1 should check and fold with 0 thru 3/7. If Player 1 checks: Player 2 should check with 0 thru 1/7 Player 2 should bet with 1/7 thru 1 If Player 1 bets: Player 2 should fold with 0 thru 13/21 Player 2 should call witn 13/21 thru 1 EV is -4/21 for Player 1 |
#15
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Re: Why Position Matters Using Game Theory
Here is one possible solution . I will follow with the same set up as I did for my Game Theory Resolution problem .
Solution: Let a be your optimal pushing range ; a>=0 Let x be your opponent's optimal calling range . 1/3<=x<=1 We can write a in terms of x . Notice that (1-x)/(x-a) = 2 x=(2a+1)/3 The probability that your opponent wins given that he calls is 2/3 which is verified since 1/3 + 2/3*1/2 = 2/3 So the probability that you win given that he calls is 1/3 . Lets compute your optimal EV when you bet . EV = 3*(1-x)/2*[2x + 4*1/3*(1-x) -2*2/3*(1-x)] EV= 3x*(1-x) after simplifying . Take the derivative of the EV function EV' = -6x+3 which means that x>=1/2 if we set the derivative =0 . Therefore a=1/4 . So we should only check and call or possibly check and fold if our number is less than 1/4 and that we will always bluff with hands [1/4,1/2]. Note that there are variations to one's bluffing interval . If we check , then our opponent knows that our number is less than 1/4 . I haven't worked out the check and call or check and fold scenario but I'll come up with an answer shortly . |
#16
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Re: Why Position Matters Using Game Theory
So player one should check and fold 100% of the time with numbers less than 1/4 .
Clearly , player two's EV is positive just for being in position ! |
#17
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Re: Why Position Matters Using Game Theory
Analysis of (a = 1/4)
P1 bets (1/4..1) P2 fold (0..1/2), call (1/2..1) P1 checks (0..1/4) P2 checks (0..1/12) P2 bets (1/12..1) P1 folds (0..7/18) P1 calls (7/18..1) but if P1 had (7/18..1) he would have bet. Calculating EV from P1's perspective: P1 (0..1/12), P2 (0.,1/12) : Check Check Tie 1/144 * ($0) = 0/144 P1 (0..1/12), P2 (1/12..1) : Check Bet Fold P2 Win 11/144 * (-$1) = -11/144 P1 (1/12..1/4), P2 (0..1/12) : Check Check P1 Win 2/144 * ($1) = 2/144 P1 (1/12..1/4), P2 (1/12..1) : Check Bet Fold P2 Win 22/144 * (-$1) = -22/144 P1 (1/4..1/2), P2(0..1/2) : Bet Fold P1 Win 18/144 * ($1) = 18/144 P1 (1/4..1/2), P2(1/2..1) : Bet Call P2 Win 18/144 * (-$3) = -54/144 P1 (1/2..1), P2 (0..1/2) : Bet Fold P1 Win 36/144 * ($1) = 36/144 P1 (1/2..1), P2 (1/2..1) : Bet Call Ties 36/144 * ($0) = 0/144 P1's EV = -31/144 That's worse for P1 than (a = 3/7) |
#18
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Re: Why Position Matters Using Game Theory
Mykey , you have my EV wrong .
If player one shoves with 1/4 + , then his EV= 3*0.5*0.5 - 1 = -0.25 Just plug the numbers into my EV equation . EV using your strategy is : EV = 4/7*2*13/31 ~ 0.7074 . If we subtract the $ fee then we get ~ -0.2925. Again , my strategy is optimal . |
#19
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Re: Why Position Matters Using Game Theory
Your doing it again.
You plug the numbers into a formula, get a number and assume that's the right answer to the question. Your formula doesn't consider the times player 1 checks, then player 2 bets. |
#20
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Re: Why Position Matters Using Game Theory
[ QUOTE ]
Mykey , you have my EV wrong . If player one shoves with 1/4 + , then his EV= 3*0.5*0.5 - 1 = -0.25 Just plug the numbers into my EV equation . EV using your strategy is : EV = 4/7*2*13/31 ~ 0.7074 . If we subtract the $ fee then we get ~ -0.2925. Again , my strategy is optimal . [/ QUOTE ] Hmmm. This is a jam/call/fold problem with ante=1 and stacksize=3 My program doesn't handle the continuous [0,1] game but can do the game with a finite number of cards (like your previous problem). The optimal EV for various numbers of cards in the deck looks like: ncards EV 10 -0.0888889 50 -0.0847619 100 -0.0841414 150 -0.0838628 200 -0.0837353 250 -0.0836627 300 -0.0836046 350 -0.0835667 So I'm pretty sure your SB strategy isn't optimal. Marv |
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