Win rate with optimal strategy against limit raise bot

[nickabourisk]

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

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Okay, I thought I remembered every possibility when I was thinking it through last night. Then when you posted about the middle bets, I checked the SB case and didn't think too much about the BB case and forgot that all of these were possible.

[pzhon]

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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).

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This is a good question. I don't think it is obvious, but here is a rigorous argument that you don't need to consider such strategies:

<ul type="square">Suppose you have a strategy S that puts in strictly between the minimum call and maximum on a street. If you win the pot less than 50% of the time by following S, you would be better off putting in the minimum, and then following the same actions that you did in strategy S. If you win the pot more than 50% of the time, you would be better off capping, and then following the same actions that you did in strategy S. In case the probability that you win the pot is exactly 50%, you can change to either the minimum or the maximum without harming your expected value. Since this is a finite problem, any strategy can be improved (or not made worse) by eliminating any actions between calling and capping. [img]/images/graemlins/spade.gif[/img][/list]To be clear, the probability of winning the pot means the probability of winning while following S, not the hot and cold equity.

[rufus]

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This is a good question. I don't think it is obvious, but here is a rigorous argument that you don't need to consider such strategies:

<ul type="square">Suppose you have a strategy S that puts in strictly between the minimum call and maximum on a street. If you win the pot less than 50% of the time by following S, you would be better off putting in the minimum, and then following the same actions that you did in strategy S. If you win the pot more than 50% of the time, you would be better off capping, and then following the same actions that you did in strategy S. In case the probability that you win the pot is exactly 50%, you can change to either the minimum or the maximum without harming your expected value. Since this is a finite problem, any strategy can be improved (or not made worse) by eliminating any actions between calling and capping. [img]/images/graemlins/spade.gif[/img][/list]
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This doesn't consider the issue that the chance to see the pot, and thus the chance to win, is affected by the size of the pot. It also fails to account for the cost of increased calling rates. (Although I'm not sure that either can actually occur in a way that eliminates can or min as a possibility.)

[RobNottsUk]

May be it's possible to construct a board, where you want to 2 or 3 bet that round, so RaiseBot is making a mistake, but if a number of 'bust' cards come, then you have the option to fold. If you'ld jammed the pot earlier, pushing a marginal edge, then you'ld now have to call off (most likely) more chips due to high pot odds.

The NLHE strategy, against a RaiseBot that bet 2/3 pot say, would be to play it very deep stacked and keep it cheap until, you have a big edge on the river.

With capped betting the strategy is actually more interesting, the fundamental need in FL to push more marginal edges, rather than trap a weaker opponent.

[marv]

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This is a good question. I don't think it is obvious, but here is a rigorous argument that you don't need to consider such strategies:

<ul type="square">Suppose you have a strategy S that puts in strictly between the minimum call and maximum on a street. If you win the pot less than 50% of the time by following S, you would be better off putting in the minimum, and then following the same actions that you did in strategy S. If you win the pot more than 50% of the time, you would be better off capping, and then following the same actions that you did in strategy S. In case the probability that you win the pot is exactly 50%, you can change to either the minimum or the maximum without harming your expected value. Since this is a finite problem, any strategy can be improved (or not made worse) by eliminating any actions between calling and capping. [img]/images/graemlins/spade.gif[/img][/list]
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This doesn't consider the issue that the chance to see the pot, and thus the chance to win, is affected by the size of the pot. It also fails to account for the cost of increased calling rates. (Although I'm not sure that either can actually occur in a way that eliminates can or min as a possibility.)

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pzhon is right.

The extra bets can't effect the hole cards or the upcoming board cards, nor do they effect oppo's strategy (he always raises). We can choose not to let the extra money in the pot effect our strategy and still change S's actions to be fold/call/cap on each street without decreasing our EV.

(Image that the extra bets are placed to one side - which could be a negative amount if S puts in 3 bets and we're considering the effect of putting in just 1 - and we continue to use S with the original betting sequence. Then at then end of the hand if we win we get the extra money. If we lose the hand we lose the extra money. Cleary if S wins &gt;50% we want the extra money to be as much as possible (i.e. cap) and is P(S wins) &lt; 50% we want the extra money to be as small as possible.)

Marv

[rufus]

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pzhon is right.

The extra bets can't effect the hole cards or the upcoming board cards, nor do they effect oppo's strategy (he always raises). We can choose not to let the extra money in the pot effect our strategy and still change S's actions to be fold/call/cap on each street without decreasing our EV.

(Image that the extra bets are placed to one side - which could be a negative amount if S puts in 3 bets and we're considering the effect of putting in just 1 - and we continue to use S with the original betting sequence. Then at then end of the hand if we win we get the extra money. If we lose the hand we lose the extra money. Cleary if S wins &gt;50% we want the extra money to be as much as possible (i.e. cap) and is P(S wins) &lt; 50% we want the extra money to be as small as possible.)

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I may be a bit slow here, but you seem to be assuming that the strategy is independent of the pot in a certain sense. What prevents the scenario of a large pot where the expected win rate after one raise is 1% but the expected win rate after 2-4 raises is 49%? It's easy to construct blatantly non-optimal strategies where a scenario of that nature can occur, and I don't think it happens for optimal strategies, but it represents a problem with that argument.

[marv]

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pzhon is right.

The extra bets can't effect the hole cards or the upcoming board cards, nor do they effect oppo's strategy (he always raises). We can choose not to let the extra money in the pot effect our strategy and still change S's actions to be fold/call/cap on each street without decreasing our EV.

(Image that the extra bets are placed to one side - which could be a negative amount if S puts in 3 bets and we're considering the effect of putting in just 1 - and we continue to use S with the original betting sequence. Then at then end of the hand if we win we get the extra money. If we lose the hand we lose the extra money. Cleary if S wins &gt;50% we want the extra money to be as much as possible (i.e. cap) and is P(S wins) &lt; 50% we want the extra money to be as small as possible.)

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I may be a bit slow here, but you seem to be assuming that the strategy is independent of the pot in a certain sense. What prevents the scenario of a large pot where the expected win rate after one raise is 1% but the expected win rate after 2-4 raises is 49%? It's easy to construct blatantly non-optimal strategies where a scenario of that nature can occur, and I don't think it happens for optimal strategies, but it represents a problem with that argument.

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Because our EV using the modified S from this point in the hand is:

Old EV + p(win pot)*extra money

since we 'follow S' pretending that we hadn't put in the exra bets. The modified S may not be optimal, we just want to replace S with a strategy at least as good which only ever folds/calls/caps, so that we can justify only looking for optimal strategies with that property.

Marv

[pzhon]

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What prevents the scenario of a large pot where the expected win rate after one raise is 1% but the expected win rate after 2-4 raises is 49%? It's easy to construct blatantly non-optimal strategies where a scenario of that nature can occur, and I don't think it happens for optimal strategies, but it represents a problem with that argument.

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Because our EV using the modified S from this point in the hand is:

Old EV + p(win pot)*extra money

since we 'follow S' pretending that we hadn't put in the exra bets.

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Exactly.

I think the problem is that the definition of a "strategy" is ambiguous. The notion that I used was a function from (hand,board) combinations to the number of bets to put in. The actions don't have to depend on the pot size directly, since the pot size can be determined from actions on earlier streets.

Adjusting the number of bets on one (hand/board) combination from an intermediate value to a call or a cap without changing any other actions does not change the probability of winning the pot (against this mindless opponent whose actions do not depend on the size of the pot). It changes your EV as described above, so you don't need to consider intermediate values.

[rufus]

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I think the problem is that the definition of a "strategy" is ambiguous. The notion that I used was a function from (hand,board) combinations to the number of bets to put in. The actions don't have to depend on the pot size directly, since the pot size can be determined from actions on earlier streets.

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Thanks, now it makes sense.

[RobNottsUk]

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The extra bets can't effect the hole cards or the upcoming board cards, nor do they effect oppo's strategy (he always raises). We can choose not to let the extra money in the pot effect our strategy and still change S's actions to be fold/call/cap on each street without decreasing our EV.


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Hang on you make money in Poker when your opponent makes mistakes.

You may make more money on a hand, by keeping pot small, and then letting RaiseBot fire away later in the hand, when you have a bigger advantage, than by pushing a razor thin edge eg) 22, early in hand, and then getting forced into being a CallBot.

Not having the river betting capped would be the ideal situation, as you just cover RaiseBots stack and only call until you're sure you are winning on the River. Effectively it'd be NLHE on last betting round, with you just deciding if you should buy the card, on every previous street.