When somebody offers you a 7:5 split, instead of the 6:6 split that would be fair, you should accept their offer with slightly less than 6⁄7 probability. Their expected value from offering you 7:5, in this case, is 7 * slightly less than 6⁄7, or slightly less than 6. This ensures they can’t do any better by offering you an unfair split; but neither do you try to destroy all their expected value in retaliation. It could be an honest mistake, especially if the real situation is any more complicated than the original Ultimatum Game.
If they offer you 8:4, accept with probability slightly-more-less than 6⁄8, so they do even worse in their own expectation by offering you 8:4 than 7:5.
Having thought about this strategy a bit, one thing I remain uncertain about is how to generalize it to a situation where the other party can try offering you a better deal if you reject the initial offer (or one where you get to make counteroffers, for that matter, which might or might not have the same problem).
If you evaluate subsequent offers independently, then it would seem to be in their interest to open with an offer of 1:11. With slightly less than 6⁄11 probability, you accept and they profit; with slightly greater than 6⁄11 probability, you say “no deal” and they say “fine, how about 2:10″, and so on until a deal is reached, which will never be better for you than fair, and may be worse.
So obviously you won’t do that.
In this scenario, it’s a pretty transparent exploit, and you can just consider yourself committed to not even consider subsequent worse-than-fair offers after you’ve rejected the first one. But in real life, there may be situations where there’s a perfectly legitimate reason for your counterparty to make you a better offer once you’ve rejected their initial one, and committing to reject those seems like it’ll result in missed opportunities.
you can just consider yourself committed to not even consider subsequent worse-than-fair offers after you’ve rejected the first one
Note that if you do that, I can exploit you by offering 9:1 split, getting it accepted 50% of the time, and getting a 5:5 split another 50% of the time, which leaves me with 7 in expectation (and you with 3 in expectation).
If you interact with the kind of entity that offers you a better deal if you reject their first offer, and they make an offer before you explain what you’ll do, you just always reject their first offer, explain your strategy to them, and say that it’s in their interest to make a fair offer, and if you probabilistically reject that one, they won’t be able to make more offers.
Or, a thing you can do in reality might be, e.g., deterministically rejecting all of their initial offers, making counteroffers that are better-than-fair for you, and bargaining this way until they reach a point where they don’t go lower because this is their best offer, and then probabilistically accepting or rejecting that best offer.
If you interact with the kind of entity that offers you a better deal if you reject their first offer, and they make an offer before you explain what you’ll do, you just always reject their first offer, explain your strategy to them, and say that it’s in their interest to make a fair offer, and if you probabilistically reject that one, they won’t be able to make more offers.
I’m not convinced that this remains optimal when you’re bargaining with an entity that might offer you a better deal if you reject their first offer for legitimate reasons, rather than in an attempt to exploit your bargaining strategy.
Suppose a deal falls through, your counterparty tries their BATNA, it goes worse than they thought, so they come back and make a better offer than they did the first time. Or maybe they, or you, learn something new regarding the value of the goods being negotiated for. Shouldn’t it be possible to take this into account and reopen negotiations? If not, a lot of mutually beneficial trades become impossible.
But I don’t see a specific strategy that allows for this without being exploitable, since you can’t necessarily tell whether a new offer is prompted by new information or whether it’s just a bargaining tactic.
Having thought about this strategy a bit, one thing I remain uncertain about is how to generalize it to a situation where the other party can try offering you a better deal if you reject the initial offer (or one where you get to make counteroffers, for that matter, which might or might not have the same problem).
If you evaluate subsequent offers independently, then it would seem to be in their interest to open with an offer of 1:11. With slightly less than 6⁄11 probability, you accept and they profit; with slightly greater than 6⁄11 probability, you say “no deal” and they say “fine, how about 2:10″, and so on until a deal is reached, which will never be better for you than fair, and may be worse.
So obviously you won’t do that.
In this scenario, it’s a pretty transparent exploit, and you can just consider yourself committed to not even consider subsequent worse-than-fair offers after you’ve rejected the first one. But in real life, there may be situations where there’s a perfectly legitimate reason for your counterparty to make you a better offer once you’ve rejected their initial one, and committing to reject those seems like it’ll result in missed opportunities.
Note that if you do that, I can exploit you by offering 9:1 split, getting it accepted 50% of the time, and getting a 5:5 split another 50% of the time, which leaves me with 7 in expectation (and you with 3 in expectation).
If you interact with the kind of entity that offers you a better deal if you reject their first offer, and they make an offer before you explain what you’ll do, you just always reject their first offer, explain your strategy to them, and say that it’s in their interest to make a fair offer, and if you probabilistically reject that one, they won’t be able to make more offers.
Or, a thing you can do in reality might be, e.g., deterministically rejecting all of their initial offers, making counteroffers that are better-than-fair for you, and bargaining this way until they reach a point where they don’t go lower because this is their best offer, and then probabilistically accepting or rejecting that best offer.
I’m not convinced that this remains optimal when you’re bargaining with an entity that might offer you a better deal if you reject their first offer for legitimate reasons, rather than in an attempt to exploit your bargaining strategy.
Suppose a deal falls through, your counterparty tries their BATNA, it goes worse than they thought, so they come back and make a better offer than they did the first time. Or maybe they, or you, learn something new regarding the value of the goods being negotiated for. Shouldn’t it be possible to take this into account and reopen negotiations? If not, a lot of mutually beneficial trades become impossible.
But I don’t see a specific strategy that allows for this without being exploitable, since you can’t necessarily tell whether a new offer is prompted by new information or whether it’s just a bargaining tactic.