Being Optimally Sorry: When to Apologize?

A Delayed Apology
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

'Wait' notes:
"... 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
'Wait' notes:
"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.

b. Elasticity of the entropy with respect to an apology

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.

c. Timings
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.

Gender Shaping - II

This tab examined the issue of 'gender shaping' last year and we continue the discussion here. This time we analyze simple probability models related to this issue. Imagine a population in a geographical area where parents adopt a policy of 'stop having children after the first boy'. Surprisingly (or maybe not), this practice in itself cannot really 'shape' or affect the stability of the population, as neatly explained by Prof. Thomas C. Schelling in his book 'Micromotives and Macrobehavior':  no “stopping rules,” like stopping after the first boy, can affect the ultimate proportions. At the first round, half the babies will be boys. At the second round, only half the families have children, but they will be half boys. The half with only girls will proceed to the third round and again, by the 50–50 hypothesis, half will have boys and half girls. If at each round half are boys and half girls the total—no matter where it stops—will be half boys and half girls. (A corollary is that we know, without adding, how many children will be born. In the end, every family will have one boy; girls will equal boys; and, the average will be two children per family.)

Dr. Schelling also mentions: "It has occasionally been proposed that this motivation might explain a slight excess of boys over girls in some populations. Where female infanticide is practiced it is bound to have that result."


Thus when one sees F-M ratios like 89:100 in some pockets of Northern India, it's a scary indicator that a sizable percentage of baby girls have been murdered (the Gov of India has had in place a strict ban on sex-determination tests for many years now). Female infanticide is a relatively recent phenomenon in certain sections of society within India's 7000+ year culture where women were typically accorded an equal (perhaps higher) status compared to men. Russel Ackoff has discussed a related issue in his classic book many decades ago.

Although the boy-driven stopping rule does not affect the stability of the population and the resultant average family looks pretty normal, the internal distribution is asymmetric (another example of the flaw of averages?). For example, a boy will either be the only kid or the youngest kid in the family. In the latter case, the parents are 'focused' on producing a boy and then tending to his needs and thus more likely to ignore the needs of their girl babies, and as the family gets bigger, this situation, on a per-capita basis is likely to get worse. These conclusions are largely confirmed in a recent NBER econometric/statistical study that uses data-driven analytical models to answer the question "Are boys and girls treated differently". Girls brought up to adulthood in such a biased environment may well help perpetuate this vicious cycle in certain parts of India. The U.S. does not appear to suffer from the problem of gender-shaping, although the pro-abortion groups have required some deft arguments to enunciate their stance on the selective gender-based abortion question posed by anti-abortionists. On the other hand, there may be some issues to be overcome with respect to investments in girl children as far as their career choices, as very briefly touched upon in a prior post.

Book review: Choke

This post reviews Sian Beilock's recent book:  
"Choke: What the Secrets of the Brain Reveal About Getting It Right When You Have To"  from an O.R. perspective, as well as from the p.o.v of Dharmic philosophy that has some deep connections to some of the mind-training techniques mentioned in the book.

Scholastic/Intellectual test situations
In addition to aspiring sport stars and business leaders who certainly want to avoid 'choking' at any cost, this book can be particularly useful to students who plan to take competitive scholastic tests. Beilock characterizes 'choking' as suboptimal performance, which implies the existence of a clearly superior level that becomes feasible via practical mind-training techniques. When it comes to tests like SAT and GRE, 'worry' can negatively affect the parts of the brain that are most involved ("working memory" in the prefrontal cortex) in the Q&A process and this can directly result in choking due to a suboptimal allocation of brain power. In fact when constrained by 'worry', lab experiments show that smarter students who routinely ace practice tests are likely to drop to a more seriously suboptimal level relative to 'average' students who face a similar worry. In particular, the book presents evidence that shows that the effect of negative gender stereotyping (e.g. "girls can't do math") has been devastating in the US, inducing a lot of female exam takers to 'choke' in such situations. Subsequently, many of these candidates reconsider their original choice of a STEM career. The author presents a systematic rebuttal of Larry Summers' controversial gender-related remarks in Harvard a few years ago that looks compelling.

An important technique that is proven to minimize the chances of choking during intense time-constrained testing situations is the ancient Indian method of Yoga and meditation that is freely available to anybody (Vipasana in particular, is recommended by the author. Even three months of adopting such methods are known to have beneficial effects). Now if one were to, over an extended period of time, move along this positive Yogic meditation gradient to maximize its benefits, one can practically experience higher states of consciousness and self-realization. This is a central truth-claim of the Dharmic thought system (DTS) of India. The 2011 book 'Being Different' by Rajiv Malhotra is a scholarly and well-researched book that expounds on DTS and is particularly useful for western minds that seek to understand what Yoga and Sanskrit (the language of Yoga) truly mean.

Sports  
Finely honed motor-skills and fluid movements are critical to achieving optimal performance. Here, one can think of the task of winning a contest as constantly solving two nested decision optimization problems. The meta-problem is to manage tactics and overall strategy, while the inner problem is how best to 'operationalize' the chosen objectives in real-time. A conclusion in this book is that one must certainly think about the meta ("what") problem using working memory. On the other hand, it is better not to (like Yogi Berra said) intellectually analyze the "how" part where a player makes real-time play decisions and executes a sequence of precise movements since these have been optimized (objectified?) over years of careful practice and then 'outsourced' to the brain's 'procedural memory'. It's like trying to analyze your legs as you descend a staircase in a hurry.

When it comes to crunch free-throws in basketball, it appears that an important 'choke' statistic is the conditional probability that a player will make the shot given that his/her team is one point behind. Apparently, this conditional probability differs by about 7% on average from its unconditional counterpart. As far as crunch-time soccer penalty kicks, well-established European league stars are more likely to choke and have a success rate of 65%, which is much less than future stars, whose conversion rate was above 90%. In baseball, home teams that are a game away from winning a series, win that game only about 38% of the time. Clearly, heightened expectations from supporters increases the chances of choking.

 

A useful point to remember that will help minimize the chances of suboptimal performance in any situation is present in a Sanksrit Mantra that was uttered after what can be viewed as the world's first ever choke, when Arjuna, the hero of the Mahabharatha, on the eve of battle, is consumed by self-doubt and initially decides against fighting the good fight and plans on simply walking away, before Krishna who was selected by Arjuna to be his charioteer in this battle, reminds him of his Dharma (a Sanskrit untranslatable, roughly means 'fundamental duty') and says, among other things:  

 

Karmanyeva adhikaraste ma phaleshu kadachana
Ma karmaphalahetur bhurma te sangostvakarmani.


"Your attention must be directed toward the action alone, never with its fruits. Let not the fruits of action be your motive, neither should you be inclined toward inaction".

As we can see, even heroes can choke, but the truly great ones have a reliable 'corner man' like support system that helps them find a way to turn it around.

 

[update: fixed format]