Apologies bundle several functions together in one speech act – a set of assertions, an expression of a feeling, perhaps a request, and at least one promise. When someone selects only part of the list they are seen as evading a full apology. One would think it might be useful on some occasions to say, “I honestly don’t think I was to blame, but now that I know how you feel, I regret doing it and promise never to repeat it.” This might reassure the offended party, yet there is no special word that does it, and often it might be taken as a refusal to “really” apologize.
The game below suggests a possible reason why. The element of promising that there will be no repetition is of prime importance, so the model treats that as the only element of apologizing. The promise turns out to be unconvincing. Some offenders are more conscientious about keeping a promise and others less so, and the latter have a greater willingness to make one, since they know it is less of a restriction on their future options. Like Akerlof’s “market of lemons (1970), the selection effect means that in equilibrium promises would not be believed, and effectively no one, scrupulous or nonchalant, would make a promise. Adding the expression of regret makes the promise more believable. It requires the apologizer to show an emotion, which may be hard to fake, so the equilibrium with promising appears.
In the game, Player 1 decides whether to apologize, which here means whether to promise Player 2 that he will not do action X again. If Player 1 made no promise he would receive payoff d > 0 from doing X, but if 1 did it after promising not to, he would bear a cost of c, for a net payoff d - c. Whether the cost c represents his conscience or worries about reputation or reprisals from 2 is not important, only that it measures his reluctance to break a promise. Player 1’s motive to make a promise is that he would receive value bp from Player 2’s belief p that he will refrain from X. Perhaps 1 wants 2 to take some action that requires 2’s trust, and the parameter b measures his value per unit of probability of being believed.
At the time of his decision Player 1 knows the values b and c but not d, and Player 2 is uncertain about all three values. Making a promise is, in a sense, betting that the value of d will not be too great. The three values are the realizations of three random variables B, C and D, which have mutually independent uniform distributions on [0,1].