#41
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Re: Why Position Matters Using Game Theory
I finally spotted my error which is why I was producing different results .
Fact #1 : Player 1 gains EV from his fold equity Fact #2 : Player 1 gains EV when both hands get checked Fact # 3 : If the pot gets contested by both players then it's neutral EV using game theory strategy . Solution: Let a be your optimal betting 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 If player 1 checks [0,a] then player 2 should bet with [(5a-2)/3 ,1] . Note that this comes from a - 2/3*(1-a) = (5a-2)/3 . Now we will compute player 1's EV under the assumption that EV(fold) =0 . We may subtract $1 at the end . The algebra is brutal so stay with me . EV= 3*(1-x)/2*2x + 2*[(1-a)*2/3 +(5a-2)/3*1/2] EV= 3*(1-x)*x + (a+2)/3 write everything in terms of a . EV= (-4a^2 +3a +4)/3 EV' = 1/3*(-8a+3) Set EV'=0 and we get that a=3/8 . So player one bets [3/8,1] checks with [0,3/8]. Player two calls with [7/12,1] If player one checks then player two always checks behind . EV=1.52083333333 If we subtract $1 we get EV =0.5208333333 |
#42
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Re: Why Position Matters Using Game Theory
I forgot to mention one other thing .
(5a-2)/3 >=0 implies that a>=2/5 . Since the EV function is convex , we have to evaluate the maximum at the endpoints . EV(2/5) = 1.52 EV(1) = 1 so a=2/5 . Final answer [img]/images/graemlins/smile.gif[/img] |
#43
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Re: Why Position Matters Using Game Theory
[ QUOTE ]
So player one bets [3/8,1] checks with [0,3/8]. Player two calls with [7/12,1] If player one checks then player two always checks behind . EV=1.52083333333 If we subtract $1 we get EV =0.5208333333 [/ QUOTE ] What does player 1 do if player 2 decides not to check behind? |
#44
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Re: Why Position Matters Using Game Theory
The goal as I see it:
Pick a strategy for P1, calculate the maximal opponent to P1. Calculate EV_P1_vs_P1MO Pick a strategy for P2, calculate the maximal opponent to P2. Calculate EV_P2_vs_P2MO if EV_P1_vs_P1MO = EV_P2_vs_P2MO then you have a set of optimal strategies. |
#45
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Re: Why Position Matters Using Game Theory
I calculated a maximal opponent for your P2 strategy
with Range1[0] do begin Start := 0; Stop := 1/6; Action := P1_Bet; end; with Range1[1] do begin Start := 1/6; Stop := 19/24; Action := P1_Check_Call; end; with Range1[2] do begin Start := 19/24; Stop := 1; Action := P1_Bet; end; with Range2[0] do begin Start := 0.00; Stop := 7/12; Action := P2_Fold_Check; end; with Range2[1] do begin Start := 7/12; Stop := 1; Action := P2_Call_Check; end; [0.0000,0.1667] vs [0.0000,0.1667] 31 0.0278 1.00000 0.02778 [0.0000,0.1667] vs [0.1667,0.5833] 31 0.0694 1.00000 0.06944 [0.0000,0.1667] vs [0.5833,0.7917] 33 0.0347 -3.00000 -0.10417 [0.0000,0.1667] vs [0.7917,1.0000] 33 0.0347 -3.00000 -0.10417 [0.1667,0.5833] vs [0.0000,0.1667] 21 0.0694 1.00000 0.06944 [0.1667,0.5833] vs [0.1667,0.5833] 21 0.1736 0.00000 0.00000 [0.1667,0.5833] vs [0.5833,0.7917] 23 0.0868 -1.00000 -0.08681 [0.1667,0.5833] vs [0.7917,1.0000] 23 0.0868 -1.00000 -0.08681 [0.5833,0.7917] vs [0.0000,0.1667] 21 0.0347 1.00000 0.03472 [0.5833,0.7917] vs [0.1667,0.5833] 21 0.0868 1.00000 0.08681 [0.5833,0.7917] vs [0.5833,0.7917] 23 0.0434 0.00000 0.00000 [0.5833,0.7917] vs [0.7917,1.0000] 23 0.0434 -1.00000 -0.04340 [0.7917,1.0000] vs [0.0000,0.1667] 31 0.0347 1.00000 0.03472 [0.7917,1.0000] vs [0.1667,0.5833] 31 0.0868 1.00000 0.08681 [0.7917,1.0000] vs [0.5833,0.7917] 33 0.0434 3.00000 0.13021 [0.7917,1.0000] vs [0.7917,1.0000] 33 0.0434 0.00000 0.00000 EV_P2_vs_P2MO = 0.114583 The EV is measured in terms of P1, so positive EV for P1 means P2 is losing money. |
#46
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Re: Why Position Matters Using Game Theory
I think I may be able to do better than this . If you come up with something else , let me know .
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#47
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Re: Why Position Matters Using Game Theory
The only part i'm having difficult with is that if player A checks [0,a] , then what numbers does player B bet with ?
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#48
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Re: Why Position Matters Using Game Theory
with Range1[0] do begin Start := 0.0; Stop := 3/8; Action := P1_Check_Call; end;
with Range1[1] do begin Start := 3/8; Stop := 1; Action := P1_Bet; end; vs with Range2[0] do begin Start := 0.00; Stop := 3/16; Action := P2_Fold_Check; end; with Range2[1] do begin Start := 3/16; Stop := 7/12; Action := P2_Fold_Bet; end; with Range2[2] do begin Start := 7/12; Stop := 1; Action := P2_Call_Bet; end; gives [0.0000,0.1875] vs [0.0000,0.1875] 21 0.0352 0.00000 0.00000 [0.0000,0.1875] vs [0.1875,0.3750] 22 0.0352 -3.00000 -0.10547 [0.0000,0.1875] vs [0.3750,0.5833] 22 0.0391 -3.00000 -0.11719 [0.0000,0.1875] vs [0.5833,1.0000] 24 0.0781 -3.00000 -0.23438 [0.1875,0.3750] vs [0.0000,0.1875] 21 0.0352 1.00000 0.03516 [0.1875,0.3750] vs [0.1875,0.3750] 22 0.0352 0.00000 0.00000 [0.1875,0.3750] vs [0.3750,0.5833] 22 0.0391 -3.00000 -0.11719 [0.1875,0.3750] vs [0.5833,1.0000] 24 0.0781 -3.00000 -0.23438 [0.3750,0.5833] vs [0.0000,0.1875] 31 0.0391 1.00000 0.03906 [0.3750,0.5833] vs [0.1875,0.3750] 32 0.0391 1.00000 0.03906 [0.3750,0.5833] vs [0.3750,0.5833] 32 0.0434 1.00000 0.04340 [0.3750,0.5833] vs [0.5833,1.0000] 34 0.0868 -3.00000 -0.26042 [0.5833,1.0000] vs [0.0000,0.1875] 31 0.0781 1.00000 0.07813 [0.5833,1.0000] vs [0.1875,0.3750] 32 0.0781 1.00000 0.07813 [0.5833,1.0000] vs [0.3750,0.5833] 32 0.0868 1.00000 0.08681 [0.5833,1.0000] vs [0.5833,1.0000] 34 0.1736 0.00000 0.00000 EV_P1_vs_P1MO = -0.669271 and with Range1[0] do begin Start := 0.0; Stop := 3/8; Action := P1_Check_Fold; end; with Range1[1] do begin Start := 3/8; Stop := 1; Action := P1_Bet; end; vs with Range2[0] do begin Start := 3/16; Stop := 7/12; Action := P2_Fold_Bet; end; with Range2[1] do begin Start := 7/12; Stop := 1; Action := P2_Call_Bet; end; gives [0.0000,0.3750] vs [0.0000,0.3750] 12 0.1406 -1.00000 -0.14063 [0.0000,0.3750] vs [0.3750,0.5833] 12 0.0781 -1.00000 -0.07813 [0.0000,0.3750] vs [0.5833,1.0000] 14 0.1563 -1.00000 -0.15625 [0.3750,0.5833] vs [0.0000,0.3750] 32 0.0781 1.00000 0.07813 [0.3750,0.5833] vs [0.3750,0.5833] 32 0.0434 1.00000 0.04340 [0.3750,0.5833] vs [0.5833,1.0000] 34 0.0868 -3.00000 -0.26042 [0.5833,1.0000] vs [0.0000,0.3750] 32 0.1563 1.00000 0.15625 [0.5833,1.0000] vs [0.3750,0.5833] 32 0.0868 1.00000 0.08681 [0.5833,1.0000] vs [0.5833,1.0000] 34 0.1736 0.00000 0.00000 EV_P1_vs_P1MO = -0.270833 |
#49
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Re: Why Position Matters Using Game Theory
For some reason, we assume each range has only 1 strategy.
I have a feeling that optimal would have some ranges that have mixed strategies. |
#50
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Re: Why Position Matters Using Game Theory
As reference
Marv's P1 00/01 -> Bet 01/10 -> Check,Fold 03/10 -> Check,Bet 04/10 -> Check,Fold 06/10 -> Check,Bet 08/10 -> Bet vs 00/01 -> Fold,Bet 01/10 -> (Call/Fold),Bet 01/05 -> (Call/Fold),Check 07/20 -> (Call/Fold),Bet 01/02 -> (Call/Fold),Check 13/20 -> (Call/Fold),Bet 04/05 -> Call,Bet EV_P1_vs_P1MO = -0.10000 |
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