Win rate with optimal strategy against limit raise bot

[mykey1961]

[ QUOTE ]
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And the results are in.

The numbers I produced are in small blind units.

But the final answer is EV = +1.848846 BB's per hand.

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There's really no need to spam the boards with all of the pre-flop holdings. People who want to verify your results are basically stuck redoing things anyway - it would have been more useful to describe your methodology or post source code.

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That's odd considering I both described my methodology, and posted source code earlier.

[nickabourisk]

Several things mykey.

1) I talked to Morgan Kan yesterday about his thesis and the claim that the always raise bot will make about 3 sb/g. He said that he ran some simulations with VexBot (which I believe is an exploitive bot) and it made over 3 sb/g, however he said that they didn't calculate the winrate exactly (and it would be better than whatever VexBot can achieve).

2) I was wondering if it is totally apparent why we might not want to get 2 or 3 bets in on different streets instead of just 1 or 4 (call or cap). Is this immediately obvious? Have you modified your code at all to test whether there is ever a situation where getting more than the min and less than the max bets in on SOME street has the highest EV?

I would think some hands that have a) high card value, b) straight draw possibilities, c) flush draw possibilities would at some point (on some boards) want to put in 2 or 3 bets on some street.

E.g. JsTs on 9s8s2c. I would guess that there is at least a possibility that you would want to get 2 or 3 bets in on a street here.

I apologize if it is completely apparent why it's ALWAYS either call or cap. If so, is there a way to prove it (aside from brute force calculations)?

[mykey1961]

[ QUOTE ]
Several things mykey.

1) I talked to Morgan Kan yesterday about his thesis and the claim that the always raise bot will make about 3 sb/g. He said that he ran some simulations with VexBot (which I believe is an exploitive bot) and it made over 3 sb/g, however he said that they didn't calculate the winrate exactly (and it would be better than whatever VexBot can achieve).

2) I was wondering if it is totally apparent why we might not want to get 2 or 3 bets in on different streets instead of just 1 or 4 (call or cap). Is this immediately obvious? Have you modified your code at all to test whether there is ever a situation where getting more than the min and less than the max bets in on SOME street has the highest EV?

I would think some hands that have a) high card value, b) straight draw possibilities, c) flush draw possibilities would at some point (on some boards) want to put in 2 or 3 bets on some street.

E.g. JsTs on 9s8s2c. I would guess that there is at least a possibility that you would want to get 2 or 3 bets in on a street here.

I apologize if it is completely apparent why it's ALWAYS either call or cap. If so, is there a way to prove it (aside from brute force calculations)?

[/ QUOTE ]

I didn't check against the possibility of 3 betting preflop, or 2 and/or 3 betting other streets.

As for something being obvious... I doubt anything should be considered obvious.

Also I did my calculations as the player being the small blind.

The position the player is in determines where they could stop the betting in each round.

For the small blind, the choices would be 2sb or 4sb preflop, 1sb, 3sb, or 4sb on the flop, and 1bb, 3bb or 4bb on the turn and river.

For the big blind, the choices would be 2sb or 4sb preflop, 1sb, 2sb, or 4sb on the flop, and 1bb, 2bb, or 4bb on the turn and river.

It wouldn't take very much recoding to find out if there ever were cases were the middle option was optimal.


As for Vexbot, it doesn't know the optimal strategy from the beginning so some of it's win rate would be less than optimal while it's learning.

[JDonk]

you can complete disregard this as it is in no way based on facts, but i would assume that folding preflop is only an option when your chances of improving against a random hand are slim depending on the flop texture as previously stated and that if you improve you can 2 bet and eventually complete when the bot 3 bets further giving you your good odds on good hands, and being able to fold when your hand stands slim chance of improving (ex. 10-9s on a K-2-3 flop all 3 the same suit or just 2 and 1 both of which arent your suits)

[pzhon]

Thanks for computing and posting the data. The values can be used to estimate the probability needed to make capping better than calling with a hand of a particular type such as Jxs or Qxo.

The percentages show the hot and cold equity against a random hand according to PokerStove. I used a linear fit to the difference between capping and calling as a function of the equity against a random hand for two hands of each "type." Sometimes this was an interpolation, and sometimes an extrapolation.

Mathematica code:
<ul type="square">interp[awin_, acall_, acap_, bwin_, bcall_, bcap_] := Solve[{m win + b == 0, m == ((bcap - bcall) - (acap - acall))/(bwin - awin), m awin + b == acap - acall}, win][/list]97s
49.118%
3.57828309 Call 9s7s
3.39590634 Cap4 9s7s

98s
50.801%
4.10159435 Call 9s8s
4.00392871 Cap4 9s8s

9xs threshold: 52.741%

T7s
50.639%
3.71125583 Call Ts7s
3.65590388 Cap4 Ts7s

T8s
52.334%
4.26844205 Call Ts8s
4.29225052 Cap4 Ts8s

Txs threshold: 51.824%

J6s
50.606%
3.37494360 Call Js6s
3.34855963 Cap4 Js6s

J7s
52.325%
3.91616227 Call Js7s
3.97595234 Cap4 Js7s

Jxs threshold: 51.132%

Q3s
51.019%
3.31406924 Call Qs3s
3.29880263 Cap4 Qs3s

Q4s
51.855%
3.52399630 Call Qs4s
3.55550247 Cap4 Qs4s

Qxs threshold: 51.292%


K2s
53.212%
3.74144948 Call Ks2s
3.84367895 Cap4 Ks2s

K3s
54.055%
3.96604578 Call Ks3s
4.10879528 Cap4 Ks3s

Kxs threshold: 51.085%

22
50.334%
2.31547685 Call 2h2s
2.31588600 Cap4 2h2s

33
53.693%
3.09300448 Call 3h3s
3.23221099 Cap4 3h3s

pair threshold: 50.324%

When the pair plays, you usually have an easy call. The times you fold and give up some chances to win the pot are on boards like KKQQ5 or JJJJ5, where you are tied with 53 and 42, but get bluffed out. That means you win the pot slightly less frequently than your hot and cold equity.

T8o
49.721%
3.49687136 Call 8hTs
3.41170156 Cap4 8hTs

T9o
51.532%
4.10140864 Call 9hTs
4.09621361 Cap4 9hTs

Txo threshold: 51.650%

J7o
49.682%
3.10112040 Call 7hJs
3.05732728 Cap4 7hJs

J8o
51.490%
3.70130303 Call 8hJs
3.72921507 Cap4 8hJs

Jxo threshold: 50.786%

Q5o
50.120%
2.89032620 Call 5hQs
2.86731632 Cap4 5hQs

Q6o
51.024%
3.16559710 Call 6hQs
3.19349036 Cap4 6hQs

Qxo threshold: 50.529%

K2o
50.509%
2.82466166 Call 2hKs
2.82056061 Cap4 2hKs

K3o
51.426%
3.06551953 Call 3hKs
3.10477006 Cap4 3hKs

Kxo threshold: 50.596%

I expect that in some sense, the true threshold decreases as you raise the higher card, giving the hand more showdown value unimproved. There are minor exceptions in the above estimates, but this agrees with the overall trend.

[nickabourisk]

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For the small blind, the choices would be 2sb or 4sb preflop

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In the case of the SB, you can get in 2, 3, or 4 sb. For 3 sb to go in, the action would go as follows:

SB raises 1.5sb to 2sb, BB reraises to 3sb, SB calls.

It would be interesting to investigate the scenarios with the middle betting option (getting either two or three bets in postflop depending on your position) and to determine how much the overall preflop strategies (if at all) vary depending on your position (SB or BB).

What programming language did you use to compute these and how long do the calculations take for each hand?

BTW, thanks for doing those calculations.

[mykey1961]

I thought about the preflop bets, over and over and still I missed 3sb... doh.

[nickabourisk]

No biggie. I thought about postflop bets and forgot that there is one bet size in the middle (as you pointed out) which is not available (depending on your position).

[mykey1961]

Programming language is Delphi (Pascal)

It takes about 12 mins per preflop hand so it's about 1.5 days of computing.

[mykey1961]

from the BB, you should be able to stop at 0, 1, 2, 3, and 4 bets postflop.

0 = check, fold
1 = check, call
2 = bet, call
3 = check, bet, call
4 = bet, bet, call