Why Position Matters Using Game Theory

[jay_shark]

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

[jay_shark]

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]

[mykey1961]

[ 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?

[mykey1961]

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.

[mykey1961]

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.

[jay_shark]

I think I may be able to do better than this . If you come up with something else , let me know .

[jay_shark]

The only part i'm having difficult with is that if player A checks [0,a] , then what numbers does player B bet with ?

[mykey1961]

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

[mykey1961]

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.

[mykey1961]

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