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Re: Math is ...apparently not that important?
Hi FullyCompletely,
First thanks a lot for responding in this thread. You've really sparked my curiosity in the topic and I've been having fun trying out CFR for some toy games (with a 4GB laptop I don't expect much else =P). I have 2 questions, one theoretical and one technical.
(1) I had a thought about Nash equilibrium strategies and the limits it has on exploiting opponents, and wanted to confirm if I was thinking about this correctly.
Let's say we are playing against a Nash-eq strat and have a decision to make. Theoretically, each action we take has an EV associated with it. For example:
Fold: -2.0
Call: 3.6645
Raise: 3.854
We would have to raise in this situation or else we would lose out on EV.
Similarly, in this decision above, the Nash-eq strat itself must be 100% raise, because it can never take an EV hit.
That means that the only time when the Nash-eq strat can give a mixed strategy (like 80% call 20% raise) is if the EVs of all non-0% actions are the same, right? Like:
Fold: -2.0
Call: 2.645
Raise: 2.645
Basically, any decision where the Nash-eq strategy has a mixed strategy, the opponent could just pick any one of those actions and not take a loss.
It seems like a lot of the toughest decisions in poker are ones where multiple actions seem ok and you need to vary between them to hide your range (for example, betting out vs check-with-intention-raising on the flop). Playing against a Nash-eq strategy would actually take the meaning out of a lot of those tough spots because either one is good.
Have you noticed what % of the decisions in your strategies are mixed as opposed to 100%?
(2) Hope this one is not too technical. I was recently trying to implement imperfect recall (only way I know to move forward with 4GB), and had trouble thinking about how to calculate the pi_i/-i once histories are merged.
One of the key CFR calculations requires calculating pi_-i for each information set. In a perfect recall game tree, it is easy to calculate all of the pi's, since each history can only have one parent. However, once histories get merged, it seems like a player's contribution to pi gets fuzzied.
For example, let's assume that player A makes a decision on the flop. We keep this information until the river, where we want to 'forget' that decision, and merge the 2 history trees that spawned from it.
How can we then compute the pi_i of both player A and B for the merged history, given that they would have done different things based on A's action that we are 'forgetting'?
I have thought about just weighting both player's pi's based on A's original probability distribution on the decision we are forgetting, but that is just a shot in the dark.
Thanks again!
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Two Plus Two
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