This post examines modeling ideas related to the timing of an apology in a two-person scenario that results in a maximally effective 'sorry'. We optimize timing here not to maximize own benefits (user optimal), but on the basis of mutual respect, to express regret and maximally repair the damage in a timely manner that most helps the subject (recipient optimal). We start with the findings in Frank Partnoy's book "Wait: the Art and Science of Delay". It's one of the many useful books in the last couple of years that analyze human decision making. We introduce a mental decision support model for a timely apology that is derived from decision analytical methods employed in an industrial setting.
Objectives and Constraints
Justice delayed may be justice denied, but an apology that is optimally delayed may not be such a bad thing. The 'Wait' book recognizes the existence of a suitable time to apologize, and notes that the fastest apology in not necessarily the most effective. Given that we may have to apologize more than once, in general we have to determine an optimal trajectory of timed apologies. Thus, our goals are to:
i) apologize at least once,
ii) in a timely manner, and
iii) within a finite time horizon, such that
iv) a measure of the recipient's benefits is maximized
"... Saying you are sorry is always better than not apologizing at all. But as with the first study, the students felt better about a delayed apology: “Improvement in the late apology condition was significantly greater than improvement in the early apology condition.” In fact, a statistically significant improvement in the students’ reactions occurred only in the late apology condition, when there was a chance for them to discuss what had happened and why. Overall, these studies suggest that the relationship between apologies and timing follows a “bell curve” distribution: effectiveness is low at first, then rises, peaks, and ultimately declines."
(It seems that these ideas are related to the complementary 'problem' of delivering the most time-effective 'Thank You')
We can see that the timing-effectiveness curve described in the book extract above is related to the subject's level of distress/angst (which we represent as 'entropy') that follows a similar trajectory of rise, cruise, and a gradual demise. Depending on the person, the 'cruise' and 'demise' portions can last long and result in a very fat-tailed distribution. But before we get into 'when', a quick comment from the book on the what/why/how questions:
".... effective apologies typically contain four parts:
1. Acknowledge that you did it.
2. Explain what happened.
3. Express remorse.
4. Repair the damage, as much as you can."
Searching for the Optimal Timing
"The art of the apology centers on the management of delay. For most of us, the lesson is that the next time we do something wrong to a close friend or family member, or say something at work we wish we could take back, we should try to imagine how the victim might react to an apology tomorrow instead of today, or in a few hours instead of right now. If delay will give a friend or relative or coworker a chance to react, to voice a response and prepare themselves to hear our regret, the apology will mean more later than right away."
In other words, the timing has to take into account where the subject is located in their entropic life cycle: is the person likely to be getting angrier by the hour now (positive entropic gradient), or has reached the peak and is calming down (negative entropic gradient). To formulate a model based on these observations, we borrow ideas from a classical inventory management problem analyzed in retail operations research: Markdown Optimization (MDO).
An Optimization Model
MDO is employed to manage an inventory of short-life cycle (SLC) products that are manufactured pre-season, with the (sunk) costs paid up-front. Thus MDO typically focuses on total revenue earned in-season. Analogously, we already messed up in the beginning incurring an irreversible cost, and thereafter it costs relatively little to issue a sincere apology. Retailers employ a cadence of optimally delayed price cuts to smartly boost the end-of-season demand rate so as to maximize revenue over the remaining life of the product. Like MDO, we eventually have to solve an entropic inventory depletion problem: optimally alter the entropic gradient via one or more carefully timed apologies, which will (ideally) reduce the inventory level to zero within a finite time period.
Disclaimer: The postulated model is not assumed to be the most suitable or even a "correct" one for this problem, but merely a useful starting point. Some brief comments on the modeling elements, next.
a. Life-cycle of the entropy
SLC products (like designer fashion apparel) often have little to no historic data early in the season, and retailers may borrow results for a comparable historical "like-item" to produce an initial prediction and then continually update their sales projection based on in-season demand. Here, we play the role of a 'like-item' and place ourselves in the recipient's shoes to better appreciate the degree of distress caused and the impact it will have on the recipient over time. The entropy level is an uncertain quantity that must be learned, but its 'mean value' is assumed to representable using an approximately concave function like the one shown in the figure below. Note that unlike the MDO case where inventory is always non-increasing, entropic inventory initially increases before gradually decreasing.
Elasticity ~ % change in entropy / % change in regret and effort, as perceived by the subject
A simple model like the inverse square law that abounds in nature (elasticity = -2) may be a good starting point. Ill-timed and empty-sounding apologies may have zero elasticity and do little to reduce entropic inventory. A careless apology can result in an entropic spike ("adding insult to injury"). On the other hand, an apology that is 'deep and sincere' and well-timed can be expected to have a calming effect.
Optimally timing a single apology requires impeccable timing. On the other hand, randomly distributed, and incessant apologies may not be helpful either. A premature apology (e.g. around an increasing entropic gradient) that kicks the can down the road is a greedy approach that may be counter-productive. Thus, optimally timing multiple apologies can require a degree of coordination between decisions. Today's apologize-or-delay decision will impact the timings of future decisions, so we have to holistically manage the impact on the entropic life-cycle.
Often, despite our best efforts, the damage can never be fully repaired. Note that our objective function was setup to be indifferent to personal benefits. To paraphrase a profound Indian saying: "You have the right to optimize, but not to the fruits of your actions". Regardless of the outcome, a sincere and optimally timed apology is good Karma.