In a game of Hearts played by 4 good or expert playing agents, what percentage of rounds will end with a player shooting the moon?

Because nobody knows the game-theoretical-optimal strategy in Hearts, I am looking for an approximation based on empirical data. For example, based on logs of games between expert players from an online server, or from records of championship events.

I'm particularly curious about the difference between J♦︎ games and non-J♦︎ games. (In the the J♦︎ variant, the jack of diamonds is worth -10 points).

NOTE: This is not a duplicate of this question. That question doesn't qualify in what context the game is being played. This question does, and as such, it will have an objective answer, but only if someone has access to a decent data set of games.

• @JoeW No, 'shooting the moon' involves getting all of the hearts and the queen of spades (all the cards that would normally give you points), and instead of you getting the points everyone else does. – diego Jul 8 '16 at 0:30
• Possible duplicate of What are the chances of shooting the moon in Hearts? – Alex Robinson May 1 '17 at 8:30
• This is not quite a duplicate of that question - that one asks what the probability is from a theoretical standpoint, and this from an empirical one. – Benjamin Cosman May 1 '17 at 15:56
• I got in touch with Einar. He says their site only allows playing against computer opponents. So his data is not useful for this question. – dshin Aug 9 at 17:00
• He told me his bots are simple and never try to shoot the moon. If they do so it is always by accident. Also he says that the bots are not good at detecting and stopping others from shooting the moon. – dshin Aug 9 at 21:25

Undefined.

You have made the question a contradiction: There are no "perfectly playing agents", and thus there can be no empirical data on their results.

Reason: Hearts, like Contract Bridge, is a stochastic game of incomplete information: There is incomplete data throughout most of the play of each hand, meaning statistical likelihoods need to be estimated by each player for several rounds. There is no theoretic framework for such, and no evidence that even AI can do so in a foreseeable future.

Even in Contract Bridge, where additional information about the hidden hands is available from the bidding and one hand is face down on the table, there is o such thing as a perfectly playing agent, even considering just the play of the cards. The closest that is available is termed Double Dummy Analysis, which presents an assessment of makeable contracts when both opponent hands are assumed visible for the play.

So, until such time as a perfectly playing agent has at least been constructed for Contract Bridge, a game with one hand on the table for all players to see, there can be no such thing in the much more challenging conditions of a Hearts game, with all opponents always hidden.

In card games such as Hearts and Contract Bridge, one of the most powerful analysis techniques is that of:

• drawing correct inferences from what other players have not done;

• then from an assessment of their habits and presumed skill making an assessment of holdings they don't have; and

• from this finally creating an assessment of what they might have by removing the eliminated possibilities.

This is an extremely challenging skill to master, even in Contract Bridge where at least one partnership, and often both, have participated in the auction and described their hands in considerable detail to both partnerships - as there are no secret bidding or play agreements in Bridge.

If this cannot even be approached perfectly in Bridge, imagine how much harder it will be in Hearts, where only the three cards you received on the Pass is available information about the other hands when the play starts.

• John Nash proved the existence of a perfect strategy in hearts in 1950. You can read the proof here: en.m.wikipedia.org/wiki/Nash_equilibrium – dshin Aug 10 at 2:15
• Also, in 2019, being a stochastic game of imperfect information is no longer a reason the assume a game is intractable for AI. Poker shares these same characteristics and much progress has been made recently in that area leading to superhuman agents. – dshin Aug 10 at 2:22
• Also, I did not ask for empirical data between games played by perfect agents. Of course no such data can exist, as such agents do not exist, and probably never will. I’m asking for an educated guess based on the assumption that current world class players might shoot the moon at similar frequencies as perfect agents (which again are known to be theoretically possible due to John Nash’s 1950 proof). – dshin Aug 10 at 2:25
• @JoeW Imperfect information is a rigorously defined mathematically concept. Poker is an imperfect information game according to this definition. – dshin Aug 10 at 2:28
• @dshin but you have a a massive amount of information in poker compared to hearts. – Joe W Aug 10 at 2:34