"When a sinner implores one for pardon, one must grant pardon wholeheartedly and soulfully." —Maimonides, Hilchot T’shuvah 2:10s
"An apology is a gesture through which an individual splits himself into two parts, the part that is guilty of an offense and the part that dissociates itself from the delict and affirms a belief in the offended rule." —Goffman, 1971, 113 
Forgiveness is popular, lucrative, and de rigueur. Two forces behind this trend are Desmond Tutu and Martin Seligman. Tutu (1999) famously promoted forgiveness in South Africa’s search for Truth and Reconciliation at a time of crisis and transition. At the same time, Seligman kicked off the positive psychology movement, in the wake of which forgiveness became one of the signposts for better mental health and interpersonal relations (but see McNulty & Fincham, 2012, for downsides). Now, forgiveness has its own handbook (Worthington, 2005), and Psychology Today has its own rubric dedicated to this topic.
Forgiveness has tremendous intuitive appeal. Holding a grudge is psychologically taxing (Krueger, 2014), so forgiveness promises not only to liberate a contrite transgressor but also the wounded party. Ideally, forgiveness sets the stage for renewed interaction that will benefit all. So what is all the fuss about?
It turns out that asking for forgiveness and granting it are psychologically complex tasks. Asking for forgiveness is the more difficult task cognitively and behaviorally, whereas granting forgiveness is primarily an emotional challenge (Ricciardi et al., 2013). Not surprisingly, most research attention has gone to the question of how one might ask for forgiveness effectively (Lewicki, Pollin, & Lount, 2016; Scher & Darley, 1997). The basic finding is that successful apologies satisfy several criteria, from a sincere expression of regret and the acceptance of responsibility to credible commitments to make restitution.
What about the apology’s receiver? Once the seeker of forgiveness has done the work to get it right, what is there left to do? If the seeker makes an incomplete apology or sends signals casting doubt on its sincerity, it is the receiver’s challenge to come to a judgment as to whether the apology, as it stands, is enough. In other words, the seeker engages in constructive work, whereas the receiver engages in deconstructive work.
Assuming that each element of a good apology comes at a cost to the seeker, be it in terms of reputation, self-image, or material investment, the seeker might be motivated to offer a minimally effective apology (May 2013). Offering more would be a waste or a loss. The seeker must, therefore, predict what sort of apology will be sufficient. In contrast, the receiver is motivated to signal that only a costly apology will be accepted.
Both players—in the game-theoretic sense—have an interest in not seeing the game of forgiveness fail (Krueger, 2015). By these lights, seeking and granting forgiveness becomes a kind of strategic bargaining. Therefore, the analytical tools available from research on bargaining do apply.
However, apology and forgiveness are rarely viewed through this strategic lens, or if they are, the papers are not cited. The dominant view is a moral, or moralistic, one. This perspective, at least in its deontological form, has little use for strategic calculation. Instead, this morality is about obligation, that is, obedience.
In my limited research on this matter, I have found the Judaic tradition to be most elaborate. The Talmud contains views on forgiveness, and Maimonides provided a summary that has become accepted as authoritative. Here I quote, in italics, a Rabbi’s response to my query about apology and forgiveness.
 One is obliged to appease and implore the offended party until they offer forgiveness.
 If the offended party refuses, the offender should bring a committee of three friends to witness the apology.
 If the offended party still refuses to forgive, the offender is instructed to bring a second, and if necessary, a third group of three friends to witness the apology.
 At this point, if the offended party still refuses to forgive, Maimonides makes it clear that the apologist has done all that they can. “If the hurt one remains obstinate, the apologist may leave and pass on, for the sin then rests upon the one who refuses forgiveness.” (Mishneh Torah, Hilchot T’shuvah 2:9)
The key to this moral perspective is that both the seeker and the grantor of forgiveness have an obligation to each other (see epigraph as to the latter). In other words, Judaism, as well as other religious traditions, seek to maximize positive interpersonal outcomes by regulating the relevant behavior through social norms, and thereby relieve individuals from difficult strategic cognitive work. But here lies the Achilles heel of the normative approach: A recognition of the sociological or normative source of apology-making and apology-accepting undercuts the theory-of-mind effort to determine whether the other party really means it.
The question that brought on this bit of research and the query to the Rabbi actually referred to a commonly ignored marginalium of the apology-and-forgiveness world. Researchers, bloggers, and self-help authors tend to assume that an apology, if expressed, will be heard. But this may not be so.
Suppose A did or said something to offend B, and A embarks on a journey of apology. Now suppose that A cannot knock on B’s door or mailbox, but instead needs first to signal the intent to apologize. Now B has the option to decline to receive an apology, thereby scuttling the process so meticulously mapped by Maimonides and latter-day psychologists. I take it to be implicit in the moral (and the psychological) view that the offended party ought not to reject to even hear the apologetic argument.
If such refusal occurs nonetheless, the apology seeker is left in an ill-defined state. The seeker may not conclude to be off the hook as this would seem to be too convenient. Nor can it be said, however, that the seeker’s effort amounts to nothing. The difficulty is that when the potential grantor of forgiveness exits the scene, who is to make the judgment? It would have to be the surrounding social community. Apology and forgiveness transcend the I and thou. They are truly social processes.
Note : Goffman’s implication is that the unrepentant remains whole. Not better, but whole.
Goffman, E. (1971). Relations in public. New York: Harper & Row.
Krueger, J. I. (2014). Grudge: Holding one. Psychology Today Online. https://www.psychologytoday.com/us/blog/one-among-many/201404/grudge-holding-one
Krueger, J. I. (2015). Game of forgiveness. Psychology Today Online. https://www.psychologytoday.com/us/blog/one-among-many/201507/game-forgiveness
Lewicki, R. J., Pollin, B., & Lount, R. B. (2016). An exploration of the structure of effective apologies. Negotiaiton and Conflict Management Research, 9, 177–196.
May, C. (2013, July). The advantages of not saying you are sorry. Scientific American. https://www.scientificamerican.com/article/advantages-of-not-saying-you-are-sorry/
McNulty & Fincham (2012). Beyond positive psychology? Toward a contextual view of psychological processes and well-being. American Psychologist, 67, 101–110.
Ricciardi, E., Rota, G., Sanil, L., Gentili, C., Gaglianese, A., Guazzelli, M., & Pietrini, P. (2013). How the brain heals emotional wounds: The functional neuroanatomy of forgiveness. Frontiers in Human Neuroscience, 7. doi:10.3389/fnhum.2013.00839
Scher, S. J., & Darley, J. M. (1997). How effective are the things people say to apologize? Effects of the realizationof the apology speech act. Journal of Psycholinguistic Research, 26, 127–140.
Tutu, D. M. (1999). No future without forgiveness. New York: Doubleday.
Worthington, E. L., Jr. (2005). Handbook of forgiveness. New York: Routledge.