On Fri, May 14, 2010 at 7:45 PM, Isaac Deutsch <[email protected]> wrote: > >>> In Tichu, there seem to be bids ( Tichu and Grand Tichu ) which also reveal >>> something about the bidder's hands. > > Yes, from these bids you are able to tell that a player's hand is good. I can > see it being helpful to give these players better cards in MC simulations. > There are of course other signs that may hint at which players has which > cards, which may be useful for biasing the simulation, too. > > >> Bidding actions are not different from any other actions when it comes >> to biasing distributions. At least the Math involved is the same. You >> need to consider each MC simulation with a weight that is the >> conditional probability that all the actions taken would have been >> taken, given this distribution (this is an application of Bayes' >> theorem). In order to make something like this work in practice, you >> sometimes need to find clever tricks to make sure you don't miss the >> distributions that have high weights. This is very important in poker, >> but I don't expect it to be a big deal in Tichu. >> >> So it looks like it would be very important to have a model that will >> quickly map a game situation to a probability distribution over the >> possible actions, which could then be used both as an MC policy and as >> a way to assign "likelihood" weights to the simulations. >> >> Álvaro. > > Can you elaborate how this model would work? So this would also tell which > distribution of cards was likely given the current situation, in addition to > telling which cards are likely to be played?
Yes, precisely. Perhaps this would be easier to explain with an example. Imagine we are playing Texas Hold'em, and we want to figure out the probability distribution over the possible hands that an opponent has. At the beginning of the round, he may have any 2 cards, and all combinations are equally likely (except that we have seen our own two cards, so we know he doesn't have those). Imagine now that it is his turn to play, he happens to raise. Now my estimate of what he is likely to have should change. For instance, how likely is he to have two aces? Before he raised, if we don't have an ace ourselves, we would believe the probability of him having two aces is 4/50*3/49, or about 0.005. After he raises, we have more information. Following Bayes' theorem: P(two_aces | raise) = P(two_aces) * P(raise | two_aces) / P(raise) We just said that P(two_aces) = 0.005. Imagine that our quick probabilistic model says that someone with two aces raises 80% of the time. And the probability of raising in general is 10%. Plug everything in the formula, and you'll get 0.04, which is the new probability for two aces. Notice that P(raise) = Sum_over_all_possible_hands ( P(raise | hand) ). So you don't need a separate model for that. You simply multiply the probability of each hand by the probability of the action given the hand, and when you have done it for each hand, you rescale everything so the sum continues to be 1. SInce in MC we are trying to compute the expected value of our utility function after each one of our possible actions, you should compute the average utility, weighted by the probability of each scenario (see the formal definition of "expected value" in Wikipedia). You could do this by biasing the dealing to give that opponent two aces 4% of the time, instead of the natural 0.5%. Since doing this will get complicated quickly (as other people take actions that further change the probabilities), it is easier to deal the cards normally, but then scale each scenario with its actual probability of being what happened (this is the method the formal definition of "expected value" suggests). Does that make it clear enough? _______________________________________________ Computer-go mailing list [email protected] http://dvandva.org/cgi-bin/mailman/listinfo/computer-go
