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Adding note to not neglect other answers
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DMGregory
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Note to readers: just because this answer sorts to the top at the moment does not mean it's the best. This is still a young question, so check out the other answers and vote on them too, or consider posting your own analysis.

Note to readers: just because this answer sorts to the top at the moment does not mean it's the best. This is still a young question, so check out the other answers and vote on them too, or consider posting your own analysis.

Clearer phrasing
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DMGregory
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The game can get more interesting once you've narrowed the range above your minimum sale price or below your maximum buying price, as you try to estimate how much farther you can push. But since the penalty for a failed deal is always the same for both players, you have a strongeryour incentive to avoid a deal that's even slightly more favourable to your opponent than to yourself is stronger than you doyour incentive to avoid no deal at all. That means you should reject any price moredeal that gives you less than about a quarter of your number away from yoursmaximum theoretical profit, resulting in successful negotiations that move the game along only about one time in eight.

You should look for ways to break this symmetry and punish a player who forces no-deal by being too greedy or drags out the negotiation well past their threshold of profit, so there's an incentive to take a deal over nothing, and an opportunity to bluff/play chicken with that penalty / opponent's rewwrdreward. Again, other answers have great suggestions for specific mechanisms to do this, so I'll defer to them.

Despite all of the above, you probably find that the games you've played with your friends - at least the runs you've enjoyed most - deviate significantly from this behaviour. It's likely you've invented unspoken "house rules" about what kinds of bids are "sporting" play. If someone started negotiating in the manner described above, they'dthe group would likely protest that this person isn't playing "right". Those tacit norms can give hints into how to shape play in a way that's more fun.

The game can get more interesting once you've narrowed the range above your minimum sale price or below your maximum buying price, as you try to estimate how much farther you can push. But since the penalty for a failed deal is always the same for both players, you have a stronger incentive to avoid a deal that's even slightly more favourable to your opponent than to yourself than you do to avoid no deal at all. That means you should reject any price more than about a quarter of your number away from yours, resulting in successful negotiations that move the game along only about one time in eight.

You should look for ways to break this symmetry and punish a player who forces no-deal by being too greedy or drags out the negotiation well past their threshold of profit, so there's an incentive to take a deal over nothing, and an opportunity to bluff/play chicken with that penalty / opponent's rewwrd. Again, other answers have great suggestions for specific mechanisms to do this, so I'll defer to them.

Despite all of the above, you probably find that the games you've played with your friends - at least the runs you've enjoyed most - deviate significantly from this behaviour. It's likely you've invented unspoken "house rules" about what kinds of bids are "sporting" play. If someone started negotiating in the manner described above, they'd likely protest that person isn't playing "right". Those tacit norms can give hints into how to shape play in a way that's more fun.

The game can get more interesting once you've narrowed the range above your minimum sale price or below your maximum buying price, as you try to estimate how much farther you can push. But since the penalty for a failed deal is always the same for both players, your incentive to avoid a deal that's even slightly more favourable to your opponent than to yourself is stronger than your incentive to avoid no deal at all. That means you should reject any deal that gives you less than about a quarter of your maximum theoretical profit, resulting in successful negotiations that move the game along only about one time in eight.

You should look for ways to break this symmetry and punish a player who forces no-deal by being too greedy or drags out the negotiation well past their threshold of profit, so there's an incentive to take a deal over nothing, and an opportunity to bluff/play chicken with that penalty / opponent's reward. Again, other answers have great suggestions for specific mechanisms to do this, so I'll defer to them.

Despite all of the above, you probably find that the games you've played with your friends - at least the runs you've enjoyed most - deviate significantly from this behaviour. It's likely you've invented unspoken "house rules" about what kinds of bids are "sporting" play. If someone started negotiating in the manner described above, the group would likely protest that this person isn't playing "right". Those tacit norms can give hints into how to shape play in a way that's more fun.

Elaboration
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DMGregory
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You should look for ways to break this symmetry and punish a player who forces no-deal by being too greedy or drags out the negotiation well past their threshold of profit, so there's an incentive to take a deal over nothing, and an opportunity to bluff/play chicken with that penalty / opponent's rewwrd. Again, other answers have great suggestions for specific mechanisms to do this, so I'll defer to them.

 

However, in playing with your friendsDespite all of the above, you probably find that the games you've played with your friends - at least the runs you've enjoyed most - deviate significantly from this behaviour. It's likely you've invented unspoken "house rules" about what kinds of bids are "sporting" play. If someone started negotiating in the manner described above, they'd likely protest that person isn't playing "right". Those tacit norms can give hints into how to shape play in a way that's more fun.

You should look for ways to break this symmetry and punish a player who forces no-deal by being too greedy, so there's an incentive to take a deal over nothing, and an opportunity to bluff/play chicken with that penalty.

However, in playing with your friends, you probably find that the games - at least the runs you've enjoyed most - deviate significantly from this behaviour. It's likely you've invented unspoken "house rules" about what kinds of bids are "sporting" play. If someone started negotiating in the manner described above, they'd likely protest that person isn't playing "right". Those tacit norms can give hints into how to shape play in a way that's more fun.

You should look for ways to break this symmetry and punish a player who forces no-deal by being too greedy or drags out the negotiation well past their threshold of profit, so there's an incentive to take a deal over nothing, and an opportunity to bluff/play chicken with that penalty / opponent's rewwrd. Again, other answers have great suggestions for specific mechanisms to do this, so I'll defer to them.

 

Despite all of the above, you probably find that the games you've played with your friends - at least the runs you've enjoyed most - deviate significantly from this behaviour. It's likely you've invented unspoken "house rules" about what kinds of bids are "sporting" play. If someone started negotiating in the manner described above, they'd likely protest that person isn't playing "right". Those tacit norms can give hints into how to shape play in a way that's more fun.

Directing to other excellent answer
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DMGregory
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Precision
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DMGregory
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Precision on number-choosing incentives, chance of acceptable deal
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DMGregory
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DMGregory
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