![]() So no, the fact that there is private knowledge in hearts means that even with a perfect memory you cannot always guarantee optimal play. If you had perfect knowledge however the right play would be obvious, but without perfect knowledge the right play depends on knowledge you don't have. Given a fairly nominal distribution of clubs, the odds of someone being out of clubs on the second round (so they can sluff a heart or the Queen) AND someone else not playing a higher club are not insignificant but low and usually worth the risk. If you had only three clubs in your hand, the Ace, a fairly high card and a low rank card, a typical play in the first round is to play the Ace and take the trick and then to lead the higher card to burn it before playing the low ranking card or switching suits. Since the scores are tied at zero, a good definition of optimal play is to get the lowest score in the round. ![]() I'll prove the point with an obvious example: the first play of the first round. Even given a stochastic play model (where you're looking for the play most likely to give you the best results) it is possible for two different distributions of the hidden cards to suggest different plays, which means the hypothesized algorithm must make a random selection. This private knowledge means there is no deterministically optimal play you don't have enough information to know what move is always best. But that's only 16 cards out of 52 and you don't know how the remaining cards are distributed. You know about the cards in your hand and those you've passed (if applicable), and you can potentially infer something from the cards you've received. On the first turn of the game, you have limited knowledge of how the cards have been distributed. Having a perfect memory of cards that have been played and in what order is insufficient to guarantee optimal play, and we can say this without even having to define optimal play. The answer to this question is a categorical no. No, because in hearts there is private knowledge But it could be very hard to find without some pretty slick game-theoretical analysis and/or a lot of computational power. If you want to write a computer program to play near-optimal Hearts, I would suggest using random samples and Markov-type simulations to estimate optimal moves. So much more sophisticated techniques must be used. So using Zermelo's algorithm to "solve" hearts (though theoretically possible) is - in practice - not possible. Even a rough calculation shows that there are far more nodes in the game tree than there are atoms in the known universe. Then each of those have yet more branhces, and so on. Each of those nodes has a huge number of branches (the number of branches at each of those nodes is the number of ways four players can choose three cards from 13). That's just how many nodes there are at the second level of the game tree. ![]() There are 52!/(13!^4) hands that could be dealt (each of the 4 players gets 13 cards). Like most nontrivial games, Hearts has a HUMONGOUS game tree. I was actually just googling this question myself, trying to see if any journal articles have been written about it (I was thinking about doing a game-theoretic analysis of Hearts for a thesis paper). Now that that's out of the way, I want to commend the questioner on a great question. An optimal strategy is one that maximizes your expected payoff. ![]() Here's another one: an optimal strategy means you win every time. ![]() In fact, Game Theory has a very rich body of results about games of imperfect information. I just want to say, as a mathematician who has studied Game Theory for several years, that you DO NOT need perfect information to have an optimal strategy. ![]()
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