Are Operations Researchers Eulerian Decision Makers?

It is interesting to read Venkatesh Rao's book 'Tempo' (and the blog) from an Operations Research (OR) perspective and explore its ideas on human decision making. To better comprehend the themes in the book, I've been trying to map the examples in the book to familiar OR-type problems. For example, timing the purchase of consumer products at Amazon.com over a period of time while also managing to satisfy the free-shipping threshold to the extent possible feels like playing a game of tetris where we try to optimally delay the inevitable ("entropic optimization").

A recent tempo blog post compares Lagrangian versus Eulerian decision makers (let's call them LDMs and EDMs). These two approaches appear to have their origin as choices of frames of reference for studying fluid flow. An LDM tracks a single 'parcel' of flow though its entire journey while the EDM examines many parcels that flow through a particular location. Let's apply this to an airline OR (crew scheduling) scenario:

an LDM follows a particular crew operating a sequence of flights through a network before returning to her/his base, while the EDM looks at many crews arriving and departing through a particular airport. LDMs track flow-paths, while EDMS zoom in on activities within a specific node in the network. Another example from airline network revenue optimization: an LDM may focus on a passenger flowing and the accumulated revenue through an itinerary consisting of multiple flights, whereas the EDM zooms in on the variable price and fixed cost associated with a seat on a specific flight that is a part of many passenger itineraries.

Intuitively, it feels as if the LDMs focus on the clearly visible primal decision variables, whereas the EDM focuses on the latent, underlying dual problem. Column generation as an iterative optimization approach works well when the EDMs compute and transmit useful 'dual' signals from specific locations to LDMs (e.g., the rate of pilots arrive and depart at their node) who in turn, collect and apply this updated information to stitch together new and improved flow-paths across the network (i.e., columns), which in turn ensures more balanced and smoother flows across the nodes (i.e., rows). For some problem classes, if we solve the dual problem, our primal decisions are also at hand, and if the dual is unsolvable, the primal is likely to be problematic as well. Conjecture: Lagrangian and Eulerian decision makers build mental models that represent (informal) primal and dual formulations of the same problem. Depending on the context, one or the other approach may be more convenient or effective. 

The ability to extract information from noisy dual signals is useful in solving complex or large-scale industrial optimization problems. For example, in the airline revenue optimization example, it is known to be more convenient and reliable for airline revenue/scheduling planners to take decisions based on the shadow prices of flights, rather than rely on the optimized prices of the millions of possible itineraries. Dual methods are insightful and can help solve a decision problem where a frontal assault fails. To the best of my knowledge, the default out-of-box method invoked by the leading mathematical optimization software packages today for solving linear programs is their dual algorithm.

In general, Eulerian decision making or as this post sees it, 'dual-driven decision making' may be the more attractive first-choice approach for human decision making. The tempo blog notes:
"... When flow gets turbulent, the fluid mixes a lot. To properly follow a “parcel”, you have to let it expand as flow lines diverge and churn. This means there is more fluid in your parcel than you started with, more “noise.” Eventually you are trying to analyze world hunger — the entire body of fluid.
But the Eulerian static parcel stays the same size. It just bleeds causal structure and gets more entropic. The action gets a lot more random and choppy, but still tractable in size. It is also easier to shrink what you’re paying attention to when things get complex — it’s called focusing – than it is to reduce the ambition of a model you’re tracking (generally called pruning)..."

Are Operations Researchers Eulerian Decision Makers?

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.

The Scamster versus the Statesman (or why the Indian National Congress fancies its chances in 2014 and 2019)

A Scam Response Model
The loss due to government-blessed scams and failed populist schemes in India run in the order of Billions and Trillions of Indian Rupees in a country where the impoverished still form a sizable percentage and earning under or around the government designated poverty level of ₹20+ a day (50₹ ~ 1$). To many Indians, such twelve-zero numbers seem fantastic. Thus a person caught stealing thousands in a small town may receive a sound thrashing because people more clearly recognize the utility of amounts that actually show up in their historical experience. However, as the size of the theft increases beyond imaginable sums, the magnitude of the public response does not appear to proportionally increase. The law of diminishing returns seems to kick in, resulting in a concave response function. We postulate that:
Change in Public Response (ΔR) ∝ Change in # of Zeros (ΔZ) in Scam amount S , i.e


R = k log S, where k is a constant to be estimated using historical data.

The assumption of a logarithmic response model turns to be quite useful in the Indian context. For example, if political party A pulls off a 12-zero scam, and party B a 6-zero scam (using a log-10 base):
response(A) = 12k, and response(B) = 6k.

The response is only a factor of 2 more for the scam that was a million times bigger. This makes it easier for party-A to equate itself to party-B in the eyes of the public. It can have its cake and perhaps eat it too. Most Indians would not find it hard to map A & B to their real-life representatives.

Statesman versus Scamster - Round 1
A subjugated, under-informed electorate may exhibit a strong concave response that is characterized by relatively high sensitivity to small thefts at one end, and a disregard for mega-scams at the other end.  We postulate that such a population's mental decision making model tends to be (1) myopic, (2) local optimality seeking, and (3) pessimistic (MYLOP). It accepts its terrible local living conditions provided the situation doesn't get perceptibly worse, and votes in polls based on this mental model. The immediate cause-and-effect tied to a local ₹100 theft represents a clear and present risk to current survival and is likely to elicit a swift and violent response, just like a populist cash-equivalent freebie elicits an equally enthusiastic response. However, a gigantic national scam that all but ensures that its future generations will see no improvement is shrugged off. MYLOP behavior is akin to a frog that allows itself to boiled alive because it does not register the gradual increase in water temperature until it is too late.

On the flip side, a government that is largely scam-free and claims to work on long-term growth, but is perceived to have marginally worsened the status quo can get booted out of power in the next election. Asking such a population to endure temporary 'hard choices' in order to be rewarded with medium-to-long term improvement ("no free lunch", "it will get worse before it gets better") implies a non-convex decision model - a hard sell since it clashes with the MYLOP attitude of the population. Scamsters + freebies trumps the Statesman here. Let's dig a little deeper into scams.

How to Hide a Scam?
Answer: A new scam that is an order of magnitude bigger. This can be explained via a mathematical model based on an interesting technical report by John Cook, M. D. Anderson Cancer Center (and author of the Endeavor blog):

The soft maximum of two variables is the function
g(x; y) = log(exp(x) + exp(y))

Given scams S1 and S2, the cumulative response from the public per our postulated model is log(S1 + S2). For example, the Commonwealth Games of India (CWG) plus a few other scams put together was a 9-zero scam, and this was followed by the 2G bandwidth sell-off, which was a thousand times bigger, i.e., a 12-zero scam. The cumulative response is:
log(10^12 + 10^9) = log(10.001 ^ 12) = 12.0005211273k,
a value that is barely more than the response to just the 2G scam itself (12k). In other words, the cumulative response to scams is approximately its soft maximum. The 2G scam simply eclipses the CWG scam in the mind of the public and the media. Even if there were 10 CWG-level scams, they'd be cumulatively eclipsed by a new 2G-level scam for all practical purposes and drop off the radar.

Scamster versus Scamster
Suppose a generally honest political party-B gets greedy seeing the muted public response and its forgiveness of prior scams. If it imitates party-A and pulls off a CWG-level scam, the public's anger would not be a 1000 times smaller. It would only be (12-9)/12 = 25% smaller. Having a clean prior record is not very helpful. Furthermore, if Party-B campaigned on a plank of 'transparent party that is different', it will be perceived as being not very different from party-A (75% alike) despite being 99.9% less corrupt compared to party-A, measured in terms of money siphoned off.

A MYLOP population that is forced to choose between two scamsters (even if their scam sizes are different) will pick the one that offers better freebies. If both provide equal freebies, the MYLOPs are likely to alternate between the two choices, every electoral cycle (e.g state of Tamil Nadu or Uttar Pradesh).


 
Statesman v/s Scamster - Round 2
Getting out a terrible locally-optimal situation may require a statesman to bring out and maintain a series of organic (sustainable and permanent) and highly visible improvements to first earn the trust of its MYLOP people over a sufficiently long period before unveiling any risky long-term development models. This means a lot of very hard infrastructure work: providing reliable 24x7 electricity, good roads, clean water, primary schools for girls, security for women, etc., to fundamentally alter the decision model of the population from MYLOP to a more optimistic and forward-looking model where people begin to see the possibilities within their grasp. An electorate that is moving along such a positive gradient is more likely to vote to continue along this path of incremental improvement without settling for "dont rock the old boat" and "here and now" populist freebies provided by the scamster.  So the statesman can win (e.g. State of Gujarat), but it takes a lot of heavy lifting to be done upfront.

The two seemingly contradictory outcomes from the two state elections that were announced yesterday (Gujarat and Himachal Pradesh), as well as many other Indian elections in the recent past can be plausibly explained using these simple models.

Update 1: 01/21/2013
Interesting quote from 'Steve Jobs' top 5 mistakes" that directly relates to the above idea of hiding scams using bigger scams:
"... The lesson that I take from these defunct products is that people will soon forget that you were wrong on a lot of smaller bets, so long as you nail big bets in a major way (in Jobs's case, the iPod, iPhone, iPad, etc) .."

Entropic Decision Optimization using Amazon's Add-on Item Constraint

When we shop at Amazon, we often end up with about 23-24$ worth of stuff in our cart, short of the 25$ threshold that gives us free shipping. As far as I could remember, we could always add small items (like a box of paper-clips) to satisfy this knapsack constraint. In other words, we could reasonably approximate this discrete threshold satisfaction problem as a continuous knapsack constraint, which is amenable to "greedy" approaches.

A Tougher Knapsack Problem
Amazon has now designated many items whose values are roughly $5 or less, to be "add on" items, and is marketed as follows:
"The new Add-on program allows Amazon to offer thousands of items at a low price point that would be cost-prohibitive to ship on their own... Add-on Items ship with orders that include $25 or more of items shipped by Amazon, and you can get them delivered to your doorstep with free shipping ....Previously it was only possible for us to offer many of the items like these in large multi-pack sizes... "

Speaking purely from a personal point of view, it feels like this "benefit" actually hurts customers by placing additional restrictions (on the other hand, there may well be an incremental benefit to a section of customers) since such low-priced items cannot be used to satisfy our knapsack constraint, and we may end up purchasing a bit more than the 25-26$ we achieved before, or pay for shipping. This new 'add on item' constraint requires us to solve a discrete knapsack problem to best satisfy the free-shipping threshold. Although NP-Hard in theory, it is well-known that we can find optimal or near-optimal solutions to practical integer knapsack problems in quick time using specialized algorithms.

Entropic Decision Optimization (EDO)
This add-on constraint can be used to illustrate the EDO approach that we discussed just a couple of days ago. Consider the following heuristic algorithm:

Start with a super-set of items (wish list) we intend to purchase over the next few weeks/months. This list is constantly updated over time. Any time we want to make a purchase, assign a priority score to each item that is positively correlated with the time-sensitivity of its purchase (an item that we 'must' purchase right away goes directly into the cart). Then solve the following integer knapsack problem:

Maximize (sum of scores of chosen items)
subject to:
1) sum($ value of chosen items) ≤ 30
2) Bounds on the quantity of purchase of each item.

We picked 30$ as the RHS assuming that we can easily find a 30$ or higher-valued basket using our old "continuous" approximation. If we use a dynamic programming approach, we also get as a by-product, the solutions for a range of RHS values less than $30. Pick the solution corresponding to the smallest capacity at or exceeding $25. If we have very few choices, we can simply enumerate the possibilities.

We chose as a measure of "entropy" in our super set, the total time-sensitivity of super set, and selecting relatively time-critical items reduces entropy. Our EDO algorithm will usually do a good job of limiting the entropy in our wish list, but over time we may end up with quite a few "hot" items that never seem to fit the knapsack well and kept getting rejected, indicating that our list has reached a state of high entropy. We are forced to make suboptimal choices and stray far away from the optimal $25 target. It does feel like we are playing Tetris.

Managing Black Swans using Operations Research

LPHC Events
The importance of modeling Low-Probability High Consequence (LPHC) events, popularized in recent times by Nassim Taleb as "Black Swans" has been recognized by Operations Researchers for quite some time. As this recent article (via Reddit) implies, it is both demystifying and useful to talk of consequential rare events in terms of a range of probabilities rather than a potentially misleading binary classification of B/W Swans. For example, LPHC modeling is critical to identifying min-risk Hazmat routes, a popular research area within OR. In addition to standard risk measures such as the expected probability of an accident occurring on a given route, or the total expected cost of a chosen route, planners also explicitly limit the conditional expected cost associated with a catastrophic event along a prescribed route if in fact such an event were to occur along any link in the route. Practical OR models aim for robustness by never focusing on just one objective given that extremal solutions tend to faithfully follow the no-free-lunch rule. OR Planners typically  track a variety of risk objectives and seek alternative optimal solutions that adequately address the multiple modes of failure that can be triggered over time.

Narrative-Rational, Time-Dependent World
Until recently, I thought that coin tosses were 50-50 events. In an unbiased, narrative-free world they are, but as Venkatesh Rao argues in his book 'Tempo' (and this understanding is subject to my limited ability to grasp and interpret the dense writing style employed in this book) that there is no absolutely narrative-independent useful model of rational decision-making: "... there is no meaningful way to talk about specific decisions outside of a narrative frame and a concrete context, any more than it is possible to talk about physics without reference to a specific, physical coordinate frame ..."
Rao agrees with Taleb's 'Black Swan' book claim that introducing a narrative injects bias, but is disinclined to adopt the approach of severing the narrative from the decision making process. Instead, Rao chooses to embrace narrative and his book examines elegant and sound decision-making within a given enactment over the narrative clock. On the other hand, Venkatesh Rao feels that the narrative-independent "Calculative rational decision-making finesses situatedness by working only with highly controlled clocks. When you start this way, seductive general theories of decision-making take center stage, and local conditions are forgotten, minimized or dismissed as “details.”. Perhaps, by zooming in on the narrative of a guy who makes decisions based on the outcome of 'vigorously tossing and catch a coin', researchers were able to recognize these subtle 'local conditions' that allowed them to predict the bias in real-life coin tosses.

Risk and Analytics
How do we deal with risk associated with decisions prescribed by today's predictive analytical models embedded within decision support systems? This is an important issue since our underlying predictive analytical models are calibrated using noisy data generated from past decisions. As long as our recommendations lie within the historical range, its consequences can be expected to be reasonably bounded. However, once we create new history (e.g. pricing a product at a value never ever done before), our predictive model is forced to extrapolate. Per 'Tempo', we now exit a closed world and enter an open one, where "The open world is a world that includes what Donald Rumsfeld called “unknown unknowns” and Nicholas Nassim Taleb calls “black swans” (rare, highly consequential events)." In such instances, history is of limited use as a guide, and statistically driven analytics can fail. Rao says that such calculative rational [analytical models]: "... can be viewed as a way of systematically leveraging or amplifying intuition. But intuition amplified through calculative-rational models, while necessary, is not sufficient for true risk management.".

Welcome to Entropic Decision Optimization
One option to survive the open world is to have in place a theoretical model that anticipates the consequences of such unforeseen events.  'Tempo' suggests that "Informal human mental models can comprehend open worlds because they can contain things that haven’t been understood: a high entropy state." Although Rao rightly suggests that analytical methods for such weakly understood "high entropy" situations are in its infancy, having even a simple and reasonable model helps. For example, in a recently patented pricing optimization method, we attempt to reduce a user-specified measure of 'entropy' (in the form of business disruption) associated with recommendations, which has worked reasonably well within a retail narrative so far. However, in the narrative-rational world of 'Tempo', the laws of thermodynamics dictate that all good things must come to an end regardless of whether we encounter a black swan or not. A simple "Frankenstein design principle" that we discussed a couple of years ago suggests that the design of particularly complex systems should explicitly assume a failure at some point in time. 'Tempo' chooses the example of 'Tetris' (rather than Blackjack/Poker, that analytics folks more commonly use) to illustrate that we can maximally delay the inevitable by chalking up elegant but temporary wins via better entropic decision making. A time-knapsack, if you will.