Case A: Sometimes the optimal play is to make a threat. In Ultimatum Game, suppose Player One will offer $8 if Player Two threatens to turn down offers less than $8, and $2 otherwise. Then making that threat is optimal for Player Two.
Any “standard theory” that says to never make a threat is not optimal.
Case B: Sometimes the optimal play is to respond to a threat. Going first in Ultimatum Game, suppose Player Two threatens to turn down offers of less than $8. If Player One ignores the threat and offers $5 then she gets a zero payout. A better option is a probabilistic response. Player One offers $8 sometimes, but rarely, so that Player Two would have been better off not making the threat, but Player One is able to salvage some expected value.
Any “standard theory” that says to never give in to a threat is not optimal.
Case C: Going first in Ultimatum Game, suppose that Player Two says: “I will sometimes reject offers less than $8, with this formula”, and the formula is a typical probabilistic rejection such that Player One can maximize her naive expected value by offering $8. Player One responds to this probabilistic threat with her own probabilistic strategy, where she offers $8 sometimes, but rarely.
I call both of these threats. Threats can be good.
Case D: Player One is in an environment where most players accept any offer that’s at least $2. Therefore she is planning to offer $2, like most players offer. Then she learns that Player Two has threatened probabilistic rejection of offers less than $5. The correct play for Player One now depends on other aspects of the situation, beyond the scope of this comment, optimal decision theory is an unsolved problem.
Case A: Sometimes the optimal play is to make a threat. In Ultimatum Game, suppose Player One will offer $8 if Player Two threatens to turn down offers less than $8, and $2 otherwise. Then making that threat is optimal for Player Two. Any “standard theory” that says to never make a threat is not optimal.
Case B: Sometimes the optimal play is to respond to a threat. Going first in Ultimatum Game, suppose Player Two threatens to turn down offers of less than $8. If Player One ignores the threat and offers $5 then she gets a zero payout. A better option is a probabilistic response. Player One offers $8 sometimes, but rarely, so that Player Two would have been better off not making the threat, but Player One is able to salvage some expected value. Any “standard theory” that says to never give in to a threat is not optimal.
Case C: Going first in Ultimatum Game, suppose that Player Two says: “I will sometimes reject offers less than $8, with this formula”, and the formula is a typical probabilistic rejection such that Player One can maximize her naive expected value by offering $8. Player One responds to this probabilistic threat with her own probabilistic strategy, where she offers $8 sometimes, but rarely. I call both of these threats. Threats can be good.
Case D: Player One is in an environment where most players accept any offer that’s at least $2. Therefore she is planning to offer $2, like most players offer. Then she learns that Player Two has threatened probabilistic rejection of offers less than $5. The correct play for Player One now depends on other aspects of the situation, beyond the scope of this comment, optimal decision theory is an unsolved problem.
See also How to give in to threats without incentivizing them for more discussion on this.