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In the first hand of a four handed Hearts game, someone is trying to shoot the moon and the only way It can be stopped is me taking the Q♠. Should I take the queen or let them shoot the moon? In other words, is 0-26-26-26 worse for me than 0-0-13-13?

To make the question precise, lets assume:

  1. All three opponents are strong bots with equal strength.
  2. I want to maximize the chance to win the game.
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  • That is really going to depend on the person
    – Joe W
    Feb 24 at 1:57
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    At the very least to provide a quantitative answer here, you'll need to provide a metric for how you rank coming first, second, third or fourth; my intuition is that the answer is different if the only thing you care about is coming first vs a metric where coming last is ranked lower than coming second. Feb 24 at 8:37

2 Answers 2

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Take the Queen

I did a Monte Carlo simulation of how often you would win a game of Hearts, assuming each point card had an equal and independent probability of being taken by each player after the first hand.

In the "Shoot the Moon" scenario, the viewpoint player won ~11% of the games.
In the "Take the Queen" scenario, the viewpoint player won ~14% of the games.

Intuitively, this makes sense - in either case, you need to deliver 26 points above the average to specific players, but in scenario 1 you need to deliver them all to one player, while in the other you need to deliver them to two players. If we're assuming a random distribution, the second is more probable. Also, you have slightly more hands in which to deliver the points, although I don't think that really affects my perfect random simulation much.

Of course, my random distribution is not a great simulation of actual Hearts gameplay. But I think it provides a useful baseline.

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    One factor no-one has yet mentioned: if the score is 0-26-26-26, the three 26s will tend to gang up on the low score until things are brought back into equilibrium. I wonder if the "ganging up" tendency is as strong or focused if the scores are 0-0-13-13.
    – David K.
    Mar 4 at 4:58
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The answer is "it depends." In order to perform a game theoretical analysis of this scenario, you have to know the relative strengths of the players, which for most casual scenarios is going to massively influence the game. Some questions you have to be able to answer for an analysis are:

  • Are you a better player than the person shooting the moon?
  • Are you a better player than anyone else at the table?
  • How much better or worse are you than the other players?
  • What are the politics of your play group?
  • If you are better than everyone else at the table, do they know it or not?

I know someone who won 80% of the hearts games we played as a group, because he'd always stay in second place, until the last hand, and set the person who was in the lead in some way to win the game. He was that good.

Also, game theory in general doesn't handle political situations well at all, which is often what hearts ends up being near the end.

TLDR: Basically, it's what JoeW says. There's too much context missing from your question to give a mathematical answer.

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  • Thanks for your answers. But assume that the four players are of exactly equal strength, or better yet, each player is a computer that plays every hand perfectly according to the ongoing information it has access to. Then which is better, 0-0-0-26 or 0-0-13-13?
    – David K.
    Feb 24 at 5:38
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    This isn't a great answer as there is a long list of opinion based answers as to what to do that have nothing to do with the skill level of players or politics of the group as it could easily not be an established group.
    – Joe W
    Feb 25 at 16:14

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