[0,1] game #6

[Aisthesis]

Ok, after reviewing methods, it's back to a game with obvious parallels to heads-up play in actual poker situations:

SB of $1 and BB of $2. A posts the BB and B posts the SB, with restrictive rules: B can limp, fold, or raise to $4. A can call a raise or fold to a raise but can't re-raise. But if B limps, A can also raise to $4, and B can either call or fold.

Hopefully someone will check my math on what I get as optimal strategies for both players:

B: limps on [3/8,3/4]
bluff-raises on [7/24,3/8] (better than [0,1/12] but still 1/12 of his hands]
value-raises [3/4,1]
calls a raise on [1/2,3/4]

A: calls a raise on [1/2,1]
bluff-raises [0,1/8]
value-raises [5/8,1]

I haven't figured up the EV yet. Hopefully in the meantime, someone can confirm that these figures are right or come up with better ones.

[well]

Same results, EV for player B here is 1/16.

[Aisthesis]

Great! Although I'm getting an EV for B of 1/8. I'll double-check my math there. When you added up EV, you did set the value of BB at $2 and SB at $1, right? I only ask since your result there is exactly 1/2 of the result I get...

Basically, I get 1/8 as the total value when B bluffs, value-raises or folds, and then that B breaks even on all hands where he limps.

[Aisthesis]

Game #7 is the same as game #6 except the raises for both players are larger by rule: With blinds of $1 (SB) and $2 (BB) with player B in SB again, the only raise allowed here is to $6. Limps, checks, and folds still allowed as before.

I'll try and solve this game later today, but I feel like it's a good one for seeing (by comparison to game #6) how the quantity of the raise will influence hand selection in a simplified simulation of heads-up play.

[Aisthesis]

The numbers look pretty ugly on the solution I got for this one. So, I really don't know whether it's because of clerical errors somewhere (I've checked it over several times, but there are lots of strange fractions to add) or because the blind structure relative to the raising quantities simply makes the fractions inherently rather complicated. Anyhow, here's what I got:

B: bluff-raises [0,3/32] (or that fraction moved to the top of his fold hands)
limps [19/48,13/16]
calls a raise [29/48,13/16]
value-raises [13/16,1]

A: calls a raise [5/8,1]
bluff-raises [0,7/48]
value-raises [17/24,1]

I get EVb as 17/192.

Hopefully someone can check these results, as there was a lot of arithmetic involving a lot of fractions on this one.

One general remark concerning the location of the bluff-raises (following up Jerrod's remarks there in the thread on David's problems #3 and #4):

When the bluff-raises are coming from hands that would be folded anyway (EV=0), then it seems to me that it's always "better" to choose the top end of those hands that would otherwise be folded--just as a preventive against possible mistakes by your opponent.

But you definitely want to take the worst hands of all if they are coming out of a group of hands that you can check. The worst hands are always the ones where checking has the least overall value (in case of opponents' errors).

[well]

[ QUOTE ]
Anyhow, here's what I got:

B: bluff-raises [0,3/32] (or that fraction moved to the top of his fold hands)
limps [19/48,13/16]
calls a raise [29/48,13/16]
value-raises [13/16,1]

A: calls a raise [5/8,1]
bluff-raises [0,7/48]
value-raises [17/24,1]

I get EVb as 17/192.

Hopefully someone can check these results, as there was a lot of arithmetic involving a lot of fractions on this one.

[/ QUOTE ]

Hmmm, I got this:

A:

[0,1/8] fold/raise
[1/8,5/8] fold/check
[5/8,3/4] call/check
[3/4,1] call/raise

B:

[0,5/16] fold
[5/16,3/8] raise
[3/8,5/8] limp/fold
[5/8,7/8] limp/call
[7/8,1] raise

And the EV for B is 1/32.

But hey, at least we got the 0, the 1 and the 5/8 in common!

[Aisthesis]

Hmmmm... I'll redo mine. Your numbers look a lot cleaner in any case. The only thing that seems suspicious to me is that B is raising so rarely (only [7/8,1]) and that this doesn't correspond to the top half of A's calling hands...

Anyhow, that makes me pretty sure that my numbers are off, but it seems possible that we're both wrong here. My "raising lemma" should mean that B definitely raises the top half of A's call hands, I would assume...

Anyhow, please don't go to any trouble checking until I've redone it with the indifference equations. I'll post again when I'm done...

[Aisthesis]

Yes, your figures are right. Actually, it was my "raising lemma" that was throwing me off: Its conditions are not fulfilled here, as I discovered when explicitly writing out the indifference equations.

As a result, A's raising option drives the value of limping at z (B's raise threshold) up, because B wins a lot there when A bluffs and some on A's value-raises. In fact, the only variables that even play a role in the indifference equation for z are the values for A's bluff- and value-raises. Oddly enough, A's call threshold plays no role at all in this case.

So, I guess the moral of the story is to be careful assuming that the proper raise value is the top 1/2 of the call hands. That's only the case if there is no further action to come in case of a limp.

After that correction, all the rest of the equations fell into place.

[well]

Rather than checking each case again, I thought I'd solve it for the case of a fixed raise to $n.

So A puts in $2, B puts in $1.
B has option to fold, call or raise to $n.
If called, A has option to check or raise to $n.
If raised, A has option to check or fold.

Strategies

Player A
[0,a] fold/raise
[a,b] fold/check
[b,c] call/check
[c,1] call/raise

Player B
[0,w] fold
[w,x] raise
[x,y] limp/fold
[y,z] limp/call
[z,1] raise

These numbers are in ascending order:

[ 0, a, w, b, x, y, z, c, 1]

EV for player B:

-n*w*b-2*a*y+4*a*x+n*y+n*z*b+2*z*c+2*y*c-n*z*c-n*b-2*z*b+n*w-n*x-2*x^2+2*x*b+n*x*b-2*w*b-2*a*z-a*n*y+a*n*z-n*y*c+n*c^2-2*c^2-w+2*b-2*y+2*x

partial deratives:

d/da: -2*y+4*x-2*z-n*y+n*z
d/db: -n*w+n*z-n-2*z+2*x+n*x-2*w+2
d/dc: 2*z+2*y-n*z-n*y+2*n*c-4*c
d/dw: -n*b+n-2*b-1
d/dx: 4*a-n-4*x+2*b+n*b+2
d/dy: -2*a+n+2*c-a*n-n*c-2
d/dz: n*b+2*c-n*c-2*b-2*a+a*n

solving these to zero gives

a = 3/2*(n-2)/n/(n+2)
b = (n-1)/(n+2)
c = 1/2*(2*n-3)/n
w = 1/4*(n^3+10*n^2-20*n+24)/n/(4+4*n+n^2)
x = 1/4*(8*n-12+n^2)/n/(n+2)
y = (n-1)/(n+2)
z = (n^2+2*n-6)/n/(n+2)

In this case, player B has an EV of

1/8*(n^3-14*n^2+76*n-72)/n/(4+4*n+n^2)

Here are some instances, for [n,a,b,c,w,x,y,z,EV]


4 1/8 1/2 5/8 7/24 3/8 1/2 3/4 1/16
6 1/8 5/8 3/4 5/16 3/8 5/8 7/8 1/32
8 9/80 7/10 13/16 127/400 29/80 7/10 37/40 19/800
10 1/10 3/4 17/20 19/60 7/20 3/4 19/20 1/40
12 5/56 11/14 7/8 123/392 19/56 11/14 27/28 23/784
14 9/112 13/16 25/28 139/448 37/112 13/16 109/112 31/896
16 7/96 5/6 29/32 265/864 31/96 5/6 47/48 23/576
18 1/15 17/20 11/12 91/300 19/60 17/20 59/60 9/200
20 27/440 19/22 37/40 1453/4840 137/440 19/22 217/220 481/9680
22 5/88 7/8 41/44 157/528 27/88 7/8 87/88 19/352
24 11/208 23/26 15/16 797/2704 63/208 23/26 103/104 313/5408
26 9/182 25/28 49/52 745/2548 109/364 25/28 361/364 313/5096
28 13/280 9/10 53/56 1219/4200 83/280 9/10 139/140 181/2800
30 7/160 29/32 19/20 369/1280 47/160 29/32 159/160 173/2560
32 45/1088 31/34 61/64 787/2747 317/1088 31/34 541/544 193/2747

In decimals:


4 0.1250 0.5000 0.6250 0.2917 0.3750 0.5000 0.7500 0.0625
6 0.1250 0.6250 0.7500 0.3125 0.3750 0.6250 0.8750 0.0313
8 0.1125 0.7000 0.8125 0.3175 0.3625 0.7000 0.9250 0.0238
10 0.1000 0.7500 0.8500 0.3167 0.3500 0.7500 0.9500 0.0250
12 0.0893 0.7857 0.8750 0.3138 0.3393 0.7857 0.9643 0.0293
14 0.0804 0.8125 0.8929 0.3103 0.3304 0.8125 0.9732 0.0346
16 0.0729 0.8333 0.9063 0.3067 0.3229 0.8333 0.9792 0.0399
18 0.0667 0.8500 0.9167 0.3033 0.3167 0.8500 0.9833 0.0450
20 0.0614 0.8636 0.9250 0.3002 0.3114 0.8636 0.9864 0.0497
22 0.0568 0.8750 0.9318 0.2973 0.3068 0.8750 0.9886 0.0540
24 0.0529 0.8846 0.9375 0.2947 0.3029 0.8846 0.9904 0.0579
26 0.0495 0.8929 0.9423 0.2924 0.2995 0.8929 0.9918 0.0614
28 0.0464 0.9000 0.9464 0.2902 0.2964 0.9000 0.9929 0.0646
30 0.0437 0.9063 0.9500 0.2883 0.2938 0.9063 0.9938 0.0676
32 0.0414 0.9118 0.9531 0.2865 0.2914 0.9118 0.9945 0.0703

You might want to paste it to notepad or something for readability.

Next Time.

[Aisthesis]

If y = b, then b, x, y are not in ascending order anyway... unless we always have b = x = y, which isn't usually the case according to your formulae.