# Eight to Late

Sensemaking and Analytics for Organizations

## Complex decision making as an infinite game

A decision is the act of choosing between two or more options.

There are two kinds of decisions, computable and non-computable [1]. In the former, options are well-defined and finite in number, and there are unambiguous facts (data) available based on which options can be rated. In the latter, options are neither clear nor enumerable and facts, if available at all, are ambiguous.

Computable decisions are simple, non-computable decisions are complex. We’ll refer to the two decision types by these names in the remainder of this article.

An example of a simple decision is buying a product (TV, car or whatever) based on well-defined criteria (price, features etc.). An example of a complex decision is formulating a business strategy.

It should be clear that simple decisions involve smaller temporal and monetary stakes – i.e. the cost of getting things wrong is limited and the effects of a bad decision wear off in (a relatively short) time. Neither is true for complex decisions: the cost of a poor choice can be significant, and its negative effects tend to persist over time.

A key feature of complex decisions is that they (usually) affect multiple parties. That is, they are socially complex. This has implications regarding how such decisions should be approached. More on this later.

Conventional decision theory is based on the notion of maximizing benefit or utility. For simple decisions it is assumed that utility of each option can be computed; for complex decisions it is assumed they can be estimated, or at least ranked. The latter assumption is questionable because each party affected by a complex decision will have its own notion of utility, at least at the outset. Moreover, since neither options nor facts are unambiguous at the start, it makes little sense to attempt to estimate utility upfront.

The above being the case, it is clear that complex decisions cannot be made on the basis of maximizing utility alone.  Something else is needed.

–x–

James Carse’s classic book, Finite and Infinite Games, begins with the following lines:

There are at least two kinds of games. One could be called finite, the other infinite. A finite game is played for the purpose of winning, an infinite for the purpose of continuing the play.

A finite game ends when a player or team wins. However, “just as it is essential for a finite game to have a definitive ending, it must also have a precise beginning. Therefore, we can speak of finite games as having temporal boundaries.”

The parallel between simple decisions and finite games should be evident. Although less obvious, it is useful to think of a complex decision as an infinite game.

When making a complex decision – such as a business strategy – decision-makers will often focus on maximising potential benefits (aka utility). However, as often as not, the outcome of the decision will fall far short of the expected benefits and may, in some cases, even lead to ruin. This being so, it is perhaps more fruitful to focus on staying in the game (keep playing) rather than winning (maximising utility).

The aim of a complex decision should be to stay in the game rather than win.

How does one ensure that one stays in the game? Heinz von Foerster’s ethical imperative offers an answer”

Always act to increase your choices.

That is, one should decide in such a way that increases one’s options in the future thereby increasing chances of staying in the game. One can frame this discussion in terms of adaptability:  the greater the number of options the greater the ability to adapt to unexpected changes in the environment.

How can one “act to increase one’s choices”?

One way to do this is to leverage social complexity: get different parties to articulate their preferred options. Some of these options are likely to contradict each other. Nevertheless, there are ways to handle such a diversity of potentially contradictory views in an inclusive manner (for an example, see this paper; for more, check out this book). Such an approach also ensures that the problem and solution spaces are explored more exhaustively than if only a limited number of viewpoints are considered.

The point is this: there are always more options available than apparent. Indeed, the number of unexplored options at any stage is potentially infinite. The job of the infinite player (decision-maker) is to act so as surface them gradually, and thus stay in the game.

–x–

Traditionally, decision-making is seen as a logical undertaking based on facts or data. In contrast, when viewed as an infinite game, complex decision-making becomes a matter of ethics rather than logic.

Why ethics?

The answer lies in von Foerster’s dictum to increase one’s choices.  By doing so, one increases the chances that fewer stakeholders’ interests are overlooked in the decision-making process.

As Wittgenstein famously said, “It is clear ethics cannot be articulated.” All those tedious classes and books on business ethics miss the point entirely. Ethical matters are necessarily oblique:  the decision-maker who decides in a way that increases (future) choices, will be acting ethically without drawing attention to it, or even being consciously aware of it.

–x–

Any honest discussion of complex decision-making in organisations must address the issue of power.

Carse asserts that players (i.e. decision-makers in the context of this article) become powerful by acquiring titles (e.g. CEO, Manager etc.). However, such titles can only be acquired by winning a finite game– i.e. by being successful in competitions for roles. Power therefore relates to finite rather than infinite games.

As he notes in his book:

Power is a concept that belongs only in finite play. To speak meaningfully of a person’s power is to speak of what that person has already achieved, the titles they have already won.

Be that as it may, one cannot overlook the reality that those in powerful positions can (and often do) subvert the decision-making process by obstructing open and honest discussion of contentious issues. Sometimes they do so by their mere presence in the room.

How does a complex decision-maker deal with the issue of power?

Carse offers the following answer:

How do infinite players contend with power? Since the outcome of infinite play is endlessly open, there is no way of looking back to make an assessment of the power or weakness of earlier play. Infinite players look forward, not to a victory but toward ongoing play. A finite player plays to be powerful; the infinite player plays with strength. Power is concerned (and a consequence of) what has happened, strength with what has yet to happen. Power will be always restricted to a relatively small number of people. Anyone can be strong.

What strength means is context-dependent, but the following may help clarify its relationship to power:

Late last year I attended an end-of-year event at the university I teach at. There I bumped into a student I had mentored some time ago. We got talking about his workplace (a large government agency).

At one point he asked, “We really need to radically change the way we think about and work with data, but I’m not a manager and have no authority to initiate changes that need to be made.”

“Why don’t you demonstrate what you are capable of? Since you are familiar your data, it should be easy enough to frame and tackle a small yet meaningful data science problem.” I replied.

“What if my manager doesn’t like my taking the initiative?”

“It is easier to beg forgiveness than seek permission.”

“He might feel threatened and make life difficult for me.”

“If management doesn’t like you’re doing, it’s their loss. What’s the worst that could happen? You could lose your job. With what you are learning at university you should have no trouble moving on to another role. Indeed, by doing so, you will diversify your experience and increase your future options.”

–x–

To summarise:  when deciding on complex matters, act in a way that maximises possibility rather than utility. Such an approach is inherently ethical and enhances one’s chances of staying in the game.

Complex decision making is an infinite game.

[1] There are many other terms for this classification:  tame and wicked (Horst Rittel), programmed and non-programmed (Herbert Simon), complicated and complex (David Snowden). Paul Culmsee and I have, perhaps confusingly, used the terms uncertain and ambiguous to refer to these in our books.  There are minor contextual differences between how these different authors interpret these terms, but for the most part they are synonymous with computable/non-computable.

Written by K

January 21, 2020 at 4:09 am

## 3 or 7, truth or trust

with one comment

“It is clear that ethics cannot be articulated.” – Ludwig Wittgenstein

Over the last few years I’ve been teaching and refining a series of lecture-workshops on Decision Making Under Uncertainty. Audiences include data scientists and mid-level managers working in corporates and public service agencies. The course is based on the distinction between uncertainties in which the variables are known and can be quantified versus those in which the variables are not known upfront and/or are hard to quantify.

Before going any further, it is worth explaining the distinction via a couple of examples:

An example of the first type of uncertainty is project estimation. A project has an associated time and cost, and although we don’t know what their values are upfront, we can estimate them if we have the right data.  The point to note is this: because such problems can be quantified, the human brain tends to deal with them in a logical manner.

In contrast, business strategy is an example of the second kind of uncertainty. Here we do not know what the key variables are upfront. Indeed we cannot, because different stakeholders will perceive different aspects of a strategy to be paramount depending on their interests – consider, for example, the perspective of a CFO versus that of a CMO. Because of these differences, one cannot make progress on such problems until agreement has been reached on what is important to the group as a whole.  The point to note here is that since such problems involve contentious issues, our reactions to them  tend to be emotional rather than logical.

The difference between the two types of uncertainty is best conveyed experientially, so I have a few in-class activities aimed at doing just that. One of them is an exercise I call “3 or 7“, in which I give students a sheet with the following printed on it:

Circle either the number 3 or 7 below depending on whether you want 3 marks or 7 marks added to your Assignment 2 final mark. Yes, this offer is for real, but there a catch: if more than 10% of the class select 7, no one gets anything.

Write your student ID on the paper so that Kailash can award you the marks. Needless to say, your choice will remain confidential, no one (but Kailash) will know what you have selected.

3              7

Prior to handing out the sheet, I tell them that they:

• should sit far enough apart so that they can’t see what their neighbours choose,
• are not allowed to communicate their choices to others until the entire class has turned their sheets.

Before reading any further you may want to think about what typically happens.

–x–

Many readers would have recognized this exercise as a version of the Prisoner’s Dilemma and, indeed, many students in my classes recognize this too.   Even so, there are always enough of “win at the cost of others” types in the room who ensure that I don’t have to award any extra marks. I’ve run the exercise about 10 times, often with groups comprised of highly collaborative individuals who work well together. Despite that,15-20% of the class ends up opting for 7.

It never fails to surprise me that, even in relatively close-knit groups, there are invariably a number of individuals who, if given a chance to gain at the expense of their colleagues, will not hesitate to do so providing their anonymity is ensured.

–x–

Conventional management thinking deems that any organisational activity involving several people has to be closely supervised. Underlying this view is the assumption that individuals involved in the activity will, if left unsupervised, make decisions based on self-interest rather than the common good (as happens in the prisoner’s dilemma game). This assumption finds justification in rational choice theory, which predicts that individuals will act in ways that maximise their personal benefit without any regard to the common good. This view is exemplified in 3 or 7 and, at a societal level, in the so-called Tragedy of the Commons, where individuals who have access to a common resource over-exploit it,  thus depleting the resource entirely.

Fortunately, such a scenario need not come to pass: the work of Elinor Ostrom, one of the 2009 Nobel prize winners for Economics, shows that, given the right conditions, groups can work towards the common good even if it means forgoing personal gains.

Classical economics assumes that individuals’ actions are driven by rational self-interest – i.e. the well-known “what’s in it for me” factor. Clearly, the group will achieve much better results as a whole if it were to exploit the resource in a cooperative way. There are several real-world examples where such cooperative behaviour has been successful in achieving outcomes for the common good (this paper touches on some). However, according to classical economic theory, such cooperative behaviour is simply not possible.

So the question is: what’s wrong with rational choice theory?  A couple of things, at least:

Firstly, implicit in rational choice theory is the assumption that individuals can figure out the best choice in any given situation.  This is obviously incorrect. As Ostrom has stated in one of her papers:

Because individuals are boundedly rational, they do not calculate a complete set of strategies for every situation they face. Few situations in life generate information about all potential actions that one can take, all outcomes that can be obtained, and all strategies that others can take.

Instead, they use heuristics (experienced-based methods), norms (value-based techniques) and rules (mutually agreed regulations) to arrive at “good enough” decisions.  Note that Ostrom makes a distinction between norms and rules, the former being implicit (unstated) rules, which are determined by the cultural attitudes and values)

Secondly, rational choice theory assumes that humans behave as self-centred, short-term maximisers. Such theories work in competitive situations such as the stock-market but not in situations in which collective action is called for, such as the prisoners dilemma.

Ostrom’s work essentially addresses the limitations of rational choice theory by outlining how individuals can work together to overcome self-interest.

–x–

In a paper entitled, A Behavioral Approach to the Rational Choice Theory of Collective Action, published in 1998, Ostrom states that:

…much of our current public policy analysis is based on an assumption that rational individuals are helplessly trapped in social dilemmas from which they cannot extract themselves without inducement or sanctions applied from the outside. Many policies based on this assumption have been subject to major failure and have exacerbated the very problems they were intended to ameliorate. Policies based on the assumptions that individuals can learn how to devise well-tailored rules and cooperate conditionally when they participate in the design of institutions affecting them are more successful in the field…[Note:  see this book by Baland and Platteau, for example]

Since rational choice theory aims to maximise individual gain,  it does not work in situations that demand collective action – and Ostrom presents some very general evidence to back this claim.  More interesting than the refutation of rational choice theory, though, is Ostrom’s discussion of the ways in which individuals “trapped” in social dilemmas end up making the right choices. In particular she singles out two empirically grounded ways in which individuals work towards outcomes that are much better than those offered by rational choice theory. These are:

Communication: In the rational view, communication makes no difference to the outcome.  That is, even if individuals make promises and commitments to each other (through communication), they will invariably break these for the sake of personal gain …or so the theory goes. In real life, however, it has been found that opportunities for communication significantly raise the cooperation rate in collective efforts (see this paper abstract or this one, for example). Moreover, research shows that face-to-face is far superior to any other form of communication, and that the main benefit achieved through communication is exchanging mutual commitment (“I promise to do this if you’ll promise to do that”) and increasing trust between individuals. It is interesting that the main role of communication is to enhance or reinforce the relationship between individuals rather than to transfer information.  This is in line with the interactional theory of communication.

Innovative Governance:  Communication by itself may not be enough; there must be consequences for those who break promises and commitments. Accordingly, cooperation can be encouraged by implementing mutually accepted rules for individual conduct, and imposing sanctions on those who violate them. This effectively amounts to designing and implementing novel governance structures for the activity. Note that this must be done by the group; rules thrust upon the group by an external authority are unlikely to work.

Of course, these factors do not come into play in artificially constrained and time-bound scenarios like 3 or 7.  In such situations, there is no opportunity or time to communicate or set up governance structures. What is clear, even from the simple 3 or 7 exercise,  is that these are required even for groups that appear to be close-knit.

Ostrom also identifies three core relationships that promote cooperation. These are:

Reciprocity: this refers to a family of strategies that are based on the expectation that people will respond to each other in kind – i.e. that they will do unto others as others do unto them.  In group situations, reciprocity can be a very effective means to promote and sustain cooperative behaviour.

Reputation: This refers to the general view of others towards a person. As such, reputation is a part of how others perceive a person, so it forms a part of the identity of the person in question. In situations demanding collective action, people might make judgements on a person’s reliability and trustworthiness based on his or her reputation.’

Trust: Trust refers to expectations regarding others’ responses in situations where one has to act before others. And if you think about it, everything else in Ostrom’s framework is ultimately aimed at engendering or – if that doesn’t work – enforcing trust.

–x—

In an article on ethics and second-order cybernetics, Heinz von Foerster tells the following story:

I have a dear friend who grew up in Marrakech. The house of his family stood on the street that divide the Jewish and the Arabic quarter. As a boy he played with all the others, listened to what they thought and said, and learned of their fundamentally different views. When I asked him once, “Who was right?” he said, “They are both right.”

“But this cannot be,” I argued from an Aristotelian platform, “Only one of them can have the truth!”

“The problem is not truth,” he answered, “The problem is trust.”

For me, that last line summarises the lesson implicit in the admittedly artificial scenario of 3 or 7. In our search for facts and decision-making frameworks we forget the simple truth that in many real-life dilemmas they matter less than we think. Facts and  frameworks cannot help us decide on ambiguous matters in which the outcome depends on what other people do.  In such cases the problem is not truth; the problem is trust.  From your own experience it should be evident it is impossible convince others of your trustworthiness by assertion, the only way to do so is by behaving in a trustworthy way. That is, by behaving ethically rather than talking about it, a point that is squarely missed by so-called business ethics classes.

Yes,  it is clear that ethics cannot be articulated.

Notes:

1. Portions of this article are lightly edited sections from a 2009 article that I wrote on Ostrom’s work and its relevance to project management.
2.  Finally, an unrelated but important matter for which I seek your support for a common good: I’m taking on the 7 Bridges Walk to help those affected by cancer. Please donate via my 7 Bridges fundraising page if you can . Every dollar counts; all funds raised will help Cancer Council work towards the vision of a cancer free future.

Written by K

September 18, 2019 at 8:28 pm

## Peirce, Holmes and a gold chain – an essay on abductive reasoning

It has long been an axiom of mine that the little things are infinitely the most important.” – Sir Arthur Conan Doyle (A Case of Identity)

The scientific method is a systematic approach to acquiring and establishing knowledge about how the world works. A scientific investigation typically starts with the formulation of a hypothesis – an educated, evidence-based guess about the mechanism behind the phenomenon being studied – and proceeds by testing how well the hypothesis holds up against experiments designed to disconfirm it.

Although many philosophers have waxed eloquent about the scientific method, very few of them have talked about the process of hypothesis generation. Indeed, most scientists will recognise a good hypothesis when they stumble upon one, but they usually will not be able to say how they came upon it. Hypothesis generation is essentially a creative act that requires a deep familiarity with the phenomenon in question and a spark of intuition. The latter is absolutely essential, a point captured eloquently in the following lines attributed to Einstein:

“…[Man] makes this cosmos and its construction the pivot of his emotional life in order to find in this way the peace and serenity which he cannot find in the narrow whirlpool of personal experience.  The supreme task is to arrive at those universal elementary laws from which the cosmos can be built up by pure deduction. There is no logical path to these laws; only intuition, resting on sympathetic understanding of experience can reach them…”  – quoted from Zen and The Art of Motorcycle Maintenance by Robert Pirsig.

The American philosopher, Charles Peirce, recognised that hypothesis generation involves a special kind of reasoning, one that enables the investigator to zero in on a small set of relevant facts out of an infinity of possibilities.

Charles Sanders Peirce

As Pierce wrote in one of his papers:

“A given object presents an extraordinary combination of characters of which we should like to have an explanation. That there is any explanation of them is a pure assumption; and if there be, it is [a single] fact which explains them; while there are, perhaps, a million other possible ways of explaining them, if they were not all, unfortunately, false.

A man is found in the streets of New York stabbed in the back. The chief of police might open a directory and put his finger on any name and guess that that is the name of the murderer. How much would such a guess be worth? But the number of names in the directory does not approach the multitude of possible laws of attraction which could have accounted for Kepler’s law of planetary motion and, in advance of verification by predications of perturbations etc., would have accounted for them to perfection. Newton, you will say, assumed that the law would be a simple one. But what was that but piling guess on guess? Surely vastly more phenomena in nature are complex than simple…” – quoted from this paper by Thomas Sebeok.

Peirce coined the term abduction (as opposed to induction or deduction) to refer to the creative act of hypothesis generation.  In the present day, the term is used to the process of justifying hypotheses rather than generating them (see this article for more on the distinction). In the remainder of this piece I will use the term in its Peircean sense.

–x–

Contrary to what is commonly stated, Arthur Conan Doyle’s fictional detective employed abductive rather than deductive methods in his cases.  Consider the following lines, taken from an early paragraph of Sherlock Holmes’ most celebrated exploit, The Adventure of the Speckled Band. We join Holmes in conversation with a lady who has come to him for assistance:

…You have come in by train this morning, I see.”

“You know me, then?”

“No, but I observe the second half of a return ticket in the palm of your left glove. You must have started early, and yet you had a good drive in a dog-cart, along heavy roads, before you reached the station.”

The lady gave a violent start and stared in bewilderment at my companion.

“There is no mystery, my dear madam,” said he, smiling. “The left arm of your jacket is spattered with mud in no less than seven places. The marks are perfectly fresh. There is no vehicle save a dog-cart which throws up mud in that way, and then only when you sit on the left-hand side of the driver.”

“Whatever your reasons may be, you are perfectly correct,” said she. “I started from home before six, reached Leatherhead at twenty past, and came in by the first train to Waterloo…

Notice what Holmes does: he forms hypotheses about what the lady did, based on a selective observation of facts. Nothing is said about why he picks those particular facts – the ticket stub and the freshness / location of mud spatters on the lady’s jacket.  Indeed, as Holmes says in another story, “You know my method. It is founded upon the observation of trifles.”

Abductive reasoning is essentially about recognising which trifles are important.

–x–

I have a gold chain that my mum gave me many years ago.  I’ve worn it for so long that now I barely notice it. The only time I’m consciously aware that I’m wearing the chain is when I finger it around my neck, a (nervous?) habit I have developed over time.

As you might imagine, I’m quite attached to my gold chain. So, when I discovered it was missing a few days ago, my first reaction was near panic,  I felt like a part of me had gone missing.

Since I hardly ever take the chain off, I could not think of any plausible explanation for how I might have lost it. Indeed, the only time I have had to consistently take the chain off is when going in for an X-ray or, on occasion, through airport security.

Where could it have gone?

After mulling over it for a while, the only plausible explanation I could think of is that I had taken it off at airport security when returning from an overseas trip a week earlier, and had somehow forgotten to collect it on the other side.  Realising that it would be near impossible to recover it, I told myself to get used to the idea that it was probably gone for good.

That Sunday, I went for a swim. After doing my laps, I went to the side of the pool for my customary shower. Now, anyone who has taken off a rash vest after a swim will know that it can be a struggle. I suspect this is because water trapped between skin and fabric forms a thin adhesive layer (a manifestation of surface tension perhaps?).  Anyway, I wrestled the garment over my head and it eventually came free with a snap, generating a spray of droplets that gleamed in reflected light.

Later in the day, I was at the movies. For some reason, when coming out of the cinema, I remembered the rash vest and the flash of droplets.  Hmm, I thought, “a gleam of gold….”

…A near forgotten memory: I vaguely recalled a flash of gold while taking of my rash vest in the pool shower some days ago. Was it after my previous swim or the week earlier, I couldn’t be sure. But I distinctly remembered it had bothered me enough to check the floor of the cubicle cursorily. Finding nothing, I had completely forgotten about it and moved on.

Could  it have come off there?

As I thought about it some more, possibility turned to plausibility: I was convinced it was what had happened. Although unlikely I would find it there now,  it was worth a try on the hope that someone had found the chain and turned it in as lost property.

I stopped over at the pool on my way back from the movies and asked at reception.

“A gold chain? Hmm, I think you may be in luck,” he said, “I was doing an inventory of lost property last week and came across a chain. I was wondering why no one had come in to claim something so valuable.”

“You’re kidding,” I said, incredulous. “You mean you have a gold chain?”

“Yeah, and I’m pretty sure it will still be there unless someone else has claimed it,” he replied. “I’ll have a look in the safe. Can you describe it for me?”

I described it down to the brief inscription on the clasp.

“Wait here,” he said, “I’ll be a sec”

It took longer than that but he soon emerged, chain in hand.

I could not believe my eyes; I had given up on ever recovering it. “Thanks, so much” I said fervently, “you won’t believe how much it means to me to have found this.”

“No worries mate,” he said, smiling broadly. “Happy to have helped.”

–x–

Endnote: in case you haven’t read it yet, I recommend you take ten minutes to read Sherlock Holmes’ finest adventure and abductive masterpiece, The Adventure of the Speckled Band.

Written by K

December 4, 2018 at 6:05 am

## A gentle introduction to Monte Carlo simulation for project managers

This article covers the why, what and how of Monte Carlo simulation using a canonical example from project management –  estimating the duration of a small project. Before starting, however, I’d like say a few words about the tool I’m going to use.

In keeping with the format of the tutorials on this blog, I’ve assumed very little prior knowledge about probability, let alone Monte Carlo simulation. Consequently, the article is verbose and the tone somewhat didactic.

### Introduction

Estimation is key part of a project manager’s role. The most frequent (and consequential) estimates they are asked deliver relate to time and cost.  Often these are calculated and presented as point estimates: i.e. single numbers – as in, this task will take 3 days. Or, a little better, as two-point ranges – as in, this task will take between 2 and 5 days.  Better still, many use a PERT-like approach wherein estimates are based on 3 points: best, most likely and worst case scenarios – as in, this task will take between 2 and 5 days, but it’s most likely that we’ll finish on day 3.  We’ll use three-point estimates as a starting point for Monte Carlo simulation, but first, some relevant background.

It is a truism, well borne out by experience, that it is easier to estimate small, simple tasks than large, complex ones. Indeed, this is why one of the early to-dos in a project is the construction of a work breakdown structure. However, a problem arises when one combines the estimates for individual elements into an overall estimate for a project or a phase thereof. It is that a straightforward addition of individual estimates or bounds will almost always lead to a grossly incorrect estimation of overall time or cost. The reason for this is simple: estimates are necessarily based on probabilities and probabilities do not combine additively. Monte Carlo simulation provides a principled and intuitive way to obtain probabilistic estimates at the level of an entire project based on estimates of the individual tasks that comprise it.

### The problem

The best way to explain Monte Carlo is through a simple worked example. So, let’s consider a 4 task project shown in Figure 1. In the project, the second task is dependent on the first, and third and fourth are dependent on the second but not on each other. The upshot of this is that the first two tasks have to be performed sequentially and the last two can be done at the same time, but can only be started after the second task is completed.

To summarise: the first two tasks must be done in series and the last two can be done in parallel.

Figure 1; A project with 4 tasks.

Figure 1 also shows the three point estimates for each task – that is the minimum, maximum and most likely completion times. For completeness I’ve listed them below:

• Task 1 – Min: 2 days; Most Likely: 4 days; Max: 8 days
• Task 2 – Min: 3 days; Most Likely: 5 days; Max: 10 days
• Task 3 – Min: 3 days; Most Likely: 6 days; Max: 9 days
• Task 4 – Min: 2 days; Most Likely: 4 days; Max: 7 days

OK, so that’s the situation as it is given to us. The first step to  developing  an estimate is to formulate the problem in a way that it can be tackled using Monte Carlo simulation. This bring us to the important topic of the shape of uncertainty aka probability distributions.

### The shape of uncertainty

Consider the data for Task 1. You have been told that it most often finishes on day 4.  However, if things go well, it could take as little as 2 days; but if things go badly it could take as long as 8 days.  Therefore, your range of possible finish times (outcomes) is between 2 to 8 days.

Clearly, each of these outcomes is not equally likely.  The most likely outcome is that you will finish the task in 4 days (from what your team member has told you). Moreover, the likelihood of finishing in less than 2 days or more than 8 days is zero. If we plot the likelihood of completion against completion time, it would look something like Figure 2.

Figure 2: Likelihood of finishing on day 2, day 4 and day 8.

Figure 2 begs a couple of questions:

1. What are the relative likelihoods of completion for all intermediate times – i.e. those between 2 to 4 days and 4 to 8 days?
2. How can one quantify the likelihood of intermediate times? In other words, how can one get a numerical value of the likelihood for all times between 2 to 8 days?  Note that we know from the earlier discussion that this must be zero for any time less than 2 or greater than 8 days.

The two questions are actually related. As we shall soon see, once we know the relative likelihood of completion at all times (compared to the maximum), we can work out its numerical value.

Since we don’t know anything about intermediate times (I’m assuming there is no other historical data available), the simplest thing to do is to assume that the likelihood increases linearly (as a straight line) from 2 to 4 days and decreases in the same way from 4 to 8 days as shown in Figure 3. This gives us the well-known triangular distribution.

Jargon Buster: The term distribution is simply a fancy word for a plot of likelihood vs. time.

Figure 3: Triangular distribution fitted to points in Figure 1

Of course, this isn’t the only possibility; there are an infinite number of others. Figure 4 is another (admittedly weird) example.

Figure 4: Another distribution that fits the points in Figure 2.

Further, it is quite possible that the upper limit (8 days) is not a hard one. It may be that in exceptional cases the task could take much longer (for example, if your team member calls in sick for two weeks) or even not be completed at all (for example, if she then leaves for that mythical greener pasture).  Catering for the latter possibility, the shape of the likelihood might resemble Figure 5.

Figure 5: A distribution that allows for a very long (potentially) infinite completion time

The main takeaway from the above is that uncertainties should be expressed as shapes rather than numbers, a notion popularised by Sam Savage in his book, The Flaw of Averages.

[Aside:  you may have noticed that all the distributions shown above are skewed to the right – that  is they have a long tail. This is a general feature of distributions that describe time (or cost) of project tasks. It would take me too far afield to discuss why this is so, but if you’re interested you may want to check out my post on the inherent uncertainty of project task estimates.

### From likelihood to probability

Thus far, I have used the word “likelihood” without bothering to define it.  It’s time to make the notion more precise.  I’ll begin by asking the question: what common sense properties do we expect a quantitative measure of likelihood to have?

Consider the following:

1. If an event is impossible, its likelihood should be zero.
2. The sum of likelihoods of all possible events should equal complete certainty. That is, it should be a constant. As this constant can be anything, let us define it to be 1.

In terms of the example above, if we denote time by $t$ and the likelihood by $P(t)$  then:

$P(t) = 0$ for $t< 2$ and  $t> 8$

And

$\sum_{t}P(t) = 1$ where $2\leq t< 8$

Where $\sum_{t}$ denotes the sum of all non-zero likelihoods – i.e. those that lie between 2 and 8 days. In simple terms this is the area enclosed by the likelihood curves and the x axis in figures 2 to 5.  (Technical Note:  Since $t$ is a continuous variable, this should be denoted by an integral rather than a simple sum, but this is a technicality that need not concern us here)

$P(t)$ is , in fact, what mathematicians call probability– which explains why I have used the symbol $P$ rather than $L$. Now that I’ve explained what it  is, I’ll use the word “probability” instead of ” likelihood” in the remainder of this article.

With these assumptions in hand, we can now obtain numerical values for the probability of completion for all times between 2 and 8 days. This can be figured out by noting that the area under the probability curve (the triangle in figure 3 and the weird shape in figure 4) must equal 1, and we’ll do this next.  Indeed, for the problem at hand, we’ll assume that all four task durations can be fitted to triangular distributions. This is primarily to keep things  simple. However, I should emphasise that you can use any shape so long as you can express it mathematically, and I’ll say more about this towards the end of this article.

### The triangular distribution

Let’s look at the estimate for Task 1. We have three numbers corresponding to a minimummost likely and maximum time.  To keep the discussion general, we’ll call these $t_{min}$, $t_{ml}$ and $t_{max}$ respectively, (we’ll get back to our estimator’s specific numbers later).

Now, what about the probabilities associated with each of these times?

Since $t_{min}$ and $t_{max}$ correspond to the minimum and maximum times,  the probability associated with these is zero. Why?  Because if it wasn’t zero, then there would be a non-zero probability of completion for a time less than $t_{min}$ or greater than $t_{max}$ – which isn’t possible [Note: this is a consequence of the assumption that the probability varies continuously –  so if it takes on non-zero value, $p_{0}$,  at $t_{min}$ then it must take on a value slightly less than $p_{0}$ – but greater than 0 –  at $t$ slightly smaller than $t_{min}$ ] .   As far as  the most likely time,  $t_{ml}$,  is concerned:  by definition, the probability attains its highest value at time $t_{ml}$.    So, assuming the probability can be described by a triangular function, the distribution must have the form shown in Figure 6 below.

Figure 6: Triangular distribution redux.

For the simulation, we need to know the equation describing the above distribution.  Although Wikipedia will tell us the answer in a mouse-click, it is instructive to figure it out for ourselves. First, note that the area under the triangle must be equal to  1 because the task must finish at some time between $t_{min}$ and $t_{max}$.   As a consequence we have:

$\frac{1}{2}\times{base}\times{altitude}=\frac{1}{2}\times{(t_{max}-t_{min})}\times{p(t_{ml})}=1\ldots\ldots{(1)}$

where $p(t_{ml})$ is the probability corresponding to time $t_{ml}$.  With a bit of rearranging we get,

$p(t_{ml})=\frac{2}{(t_{max}-t_{min})}\ldots\ldots(2)$

To derive the probability for any time $t$ lying between $t_{min}$ and $t_{ml}$, we note that:

$\frac{(t-t_{min})}{p(t)}=\frac{(t_{ml}-t_{min})}{p(t_{ml})}\ldots\ldots(3)$

This is a consequence of the fact that the ratios on either side of equation (3)  are  equal to the slope of the line joining the points $(t_{min},0)$ and $(t_{ml}, p(t_{ml}))$.

Figure 7

Substituting (2) in (3) and simplifying a bit, we obtain:

$p(t)=\frac{2(t-t_{min})}{(t_{ml}-t_{min})(t_{max}-t_{min})}\dots\ldots(4)$ for $t_{min}\leq t \leq t_{ml}$

In a similar fashion one can show that the probability for times lying between $t_{ml}$ and $t_{max}$ is given by:

$p(t)=\frac{2(t_{max}-t)}{(t_{max}-t_{ml})(t_{max}-t_{min})}\dots\ldots(5)$ for $t_{ml}\leq t \leq t_{max}$

Equations 4 and 5 together describe the probability distribution function (or PDF)  for all times between $t_{min}$ and $t_{max}$.

As it turns out, in Monte Carlo simulations, we don’t directly work with the probability distribution function. Instead we work with the cumulative distribution function (or CDF) which is the probability, $P$,  that the task is completed by time $t$. To reiterate, the PDF, $p(t)$, is the probability of the task finishing at time $t$ whereas the CDF, $P(t)$, is the probability of the task completing by time $t$. The CDF, $P(t)$,  is essentially a sum of all probabilities between $t_{min}$ and $t$. For $t < t_{min}$ this is the area under the triangle with apexes at   ($t_{min}$, 0), (t, 0) and (t, p(t)).  Using the formula for the area of a triangle (1/2 base times height) and equation (4) we get:

$P(t)=\frac{(t-t_{min})^2}{(t_{ml}-t_{min})(t_{max}-t_{min})}\ldots\ldots(6)$ for $t_{min}\leq t \leq t_{ml}$

Noting that for $t \geq t_{ml}$, the area under the curve equals the total area minus the area enclosed by the triangle with base between t and $t_{max}$, we have:

$P(t)=1- \frac{(t_{max}-t)^2}{(t_{max}-t_{ml})(t_{max}-t_{min})}\ldots\ldots(7)$ for $t_{ml}\leq t \leq t_{max}$

As expected,  $P(t)$  starts out with a value 0 at $t_{min}$ and then increases monotonically, attaining a value of 1 at $t_{max}$.

To end this section let’s plug in the numbers quoted by our estimator at the start of this section: $t_{min}=2$, $t_{ml}=4$ and $t_{max}=8$.  The resulting PDF and CDF are shown in figures 8 and 9.

Figure 8: PDF for triangular distribution (tmin=2, tml=4, tmax=8)

Figure 9 – CDF for triangular distribution (tmin=2, tml=4, tmax=8)

### Monte Carlo in a minute

Now with all that conceptual work done, we can get to the main topic of this post:  Monte Carlo estimation. The basic idea behind Monte Carlo is to simulate the entire project (all 4 tasks in this case) a large number N (say 10,000) times and thus obtain N overall completion times.  In each of the N trials, we simulate each of the tasks in the project and add them up appropriately to give us an overall project completion time for the trial.  The resulting N overall completion times will all be different, ranging from the sum of the minimum completion times to the sum of the maximum completion times.  In other words, we will obtain the PDF and CDF for the overall completion time, which will enable us to answer questions such as:

• How likely is it that the project will be completed within 17 days?
• What’s the estimated time for which I can be 90% certain that the project will be completed? For brevity, I’ll call this the 90% completion time in the rest of this piece.

“OK, that sounds great”, you say, “but how exactly do we simulate a single task”?

Good question, and I was just about to get to that…

### Simulating a single task using the CDF

As we saw earlier, the CDF for the triangular has a S shape and ranges from 0 to 1 in value. It turns out that the S shape is characteristic of all CDFs, regardless of the details underlying PDF. Why? Because, the cumulative probability must lie between 0 and 1 (remember, probabilities can never exceed 1, nor can they be negative).

OK, so to simulate a task, we:

• generate a random number between 0 and 1, this corresponds to the probability that the task will finish at time t.
• find the time, t, that this corresponds to this value of probability. This is the completion time for the task for this trial.

Incidentally, this method is called inverse transform sampling.

An example might help clarify how inverse transform sampling works.  Assume that the random number generated is 0.4905. From the CDF for the first task, we see that this value of probability corresponds to a completion time of 4.503 days, which is the completion for this trial (see Figure 10). Simple!

Figure 10: Illustrating inverse transform sampling

In this case we found the time directly from the computed CDF. That’s not too convenient when you’re simulating the project 10,000 times. Instead, we need a programmable math expression that gives us the time corresponding to the probability directly. This can be obtained by solving equations (6) and (7) for $t$. Some straightforward algebra, yields the following two expressions for $t$:

$t = t_{min} + \sqrt{P(t)(t_{ml} - t_{min})(t_{max} - t_{min})} \ldots\ldots(8)$ for $t_{min}\leq t \leq t_{ml}$

And

$t = t_{max} - \sqrt{[1-P(t)](t_{max} - t_{ml})(t_{max} - t_{min})} \ldots\ldots(9)$ for $t_{ml}\leq t \leq t_{max}$

These can be easily combined in a single Excel formula using an IF function, and I’ll show you exactly how in a minute. Yes, we can now finally get down to the Excel simulation proper and you may want to download the workbook if you haven’t done so already.

### The simulation

Open up the workbook and focus on the first three columns of the first sheet to begin with. These simulate the first task in Figure 1, which also happens to be the task we have used to illustrate the construction of the triangular distribution as well as the mechanics of Monte Carlo.

Rows 2 to 4 in columns A and B list the min, most likely and max completion times while the same rows in column C list the probabilities associated with each of the times. For $t_{min}$ the probability is 0 and for $t_{max}$ it is 1.  The probability at $t_{ml}$ can be calculated using equation (6) which, for $t=t_{max}$, reduces to

$P(t_{ml}) =\frac{(t_{ml}-t_{min})}{t_{max}-t_{min}}\ldots\ldots(10)$

Rows 6 through 10005 in column A are simulated probabilities of completion for Task 1. These are obtained via the Excel RAND() function, which generates uniformly distributed random numbers lying between 0 and 1.  This gives us a list of probabilities corresponding to 10,000 independent simulations of Task 1.

The 10,000 probabilities need to be translated into completion times for the task. This is done using equations (8) or (9) depending on whether the simulated probability is less or greater than $P(t_{ml})$, which is in cell C3 (and given by Equation (10) above). The conditional statement can be coded in an Excel formula using the IF() function.

Tasks 2-4 are coded in exactly the same way, with distribution parameters in rows 2 through 4 and simulation details in rows 6 through 10005 in the columns listed below:

• Task 2 – probabilities in column D; times in column F
• Task 3 – probabilities in column H; times in column I
• Task 4 – probabilities in column K; times in column L

That’s basically it for the simulation of individual tasks. Now let’s see how to combine them.

For tasks in series (Tasks 1 and 2), we simply sum the completion times for each task to get the overall completion times for the two tasks.  This is what’s shown in rows 6 through 10005 of column G.

For tasks in parallel (Tasks 3 and 4), the overall completion time is the maximum of the completion times for the two tasks. This is computed and stored in rows 6 through 10005 of column N.

Finally, the overall project completion time for each simulation is then simply the sum of columns G and N (shown in column O)

Sheets 2 and 3 are plots of the probability and cumulative probability distributions for overall project completion times. I’ll cover these in the next section.

### Discussion – probabilities and estimates

The figure on Sheet 2 of the Excel workbook (reproduced in Figure 11 below) is the probability distribution function (PDF) of completion times. The x-axis shows the elapsed time in days and the y-axis the number of Monte Carlo trials that have a completion time that lie in the relevant time bin (of width 0.5 days). As an example, for the simulation shown in the Figure 11, there were 882 trials (out of 10,000) that had a completion time that lie between 16.25 and 16.75 days. Your numbers will vary, of course, but you should have a maximum in the 16 to 17 day range and a trial number that is reasonably close to the one I got.

Figure 11: Probability distribution of completion times (N=10,000)

I’ll say a bit more about Figure 11 in the next section. For now, let’s move on to Sheet 3 of workbook which shows the cumulative probability of completion by a particular day (Figure 12 below).  The figure shows the cumulative probability function (CDF), which is the sum of all completion times from the earliest possible completion day to the particular day.

Figure 12: Probability of completion by a particular day (N=10,000)

To reiterate a point made earlier,  the reason we work with the CDF  rather than the PDF is that we are interested in knowing the probability of completion by a particular date (e.g. it is 90% likely that we will finish by April 20th) rather than the probability of completion on a particular date (e.g. there’s a 10% chance we’ll finish on April 17th). We can now answer the two questions we posed earlier. As a reminder, they are:

• How likely is it that the project will be completed within 17 days?
• What’s the 90% likely completion time?

Both questions are easily answered by using the cumulative distribution chart on Sheet 3 (or Fig 12).  Reading the relevant numbers from the chart, I see that:

• There’s a 60% chance that the project will be completed in 17 days.
• The 90% likely completion time is 19.5 days.

How does the latter compare to the sum of the 90% likely completion times for the individual tasks? The 90% likely completion time for a given task can be calculated by solving Equation 9 for $t$, with appropriate values for the parameters $t_{min}$, $t_{max}$ and $t_{ml}$ plugged in, and $P(t)$ set to 0.9. This gives the following values for the 90% likely completion times:

• Task 1 – 6.5 days
• Task 2 – 8.1 days
• Task 3 – 7.7 days
• Task 4 – 5.8 days

Summing up the first three tasks (remember, Tasks 3 and 4 are in parallel) we get a total of 22.3 days, which is clearly an overestimation. Now, with the benefit of having gone through the simulation, it is easy to see that the sum of 90% likely completion times for individual tasks does not equal the 90% likely completion time for the sum of the relevant individual tasks – the first three tasks in this particular case. Why? Essentially because a Monte Carlo run in which the first three tasks tasks take as long as their (individual) 90% likely completion times is highly unlikely. Exercise:  use the worksheet to estimate how likely this is.

There’s much more that can be learnt from the CDF. For example, it also tells us that the greatest uncertainty in the estimate is in the 5 day period from ~14 to 19 days because that’s the region in which the probability changes most rapidly as a function of elapsed time. Of course, the exact numbers are dependent on the assumed form of the distribution. I’ll say more about this in the final section.

To close this section, I’d like to reprise a point I mentioned earlier: that uncertainty is a shape, not a number. Monte Carlo simulations make the uncertainty in estimates explicit and can help you frame your estimates in the language of probability…and using a tool like Excel can help you explain these to non-technical people like your manager.

### Closing remarks

We’ve covered a fair bit of ground: starting from general observations about how long a task takes, saw how to construct simple probability distributions and then combine these using Monte Carlo simulation.  Before I close, there are a few general points I should mention for completeness…and as warning.

First up, it should be clear that the estimates one obtains from a simulation depend critically on the form and parameters of the distribution used. The parameters are essentially an empirical matter; they should be determined using historical data. The form of the function, is another matter altogether: as pointed out in an earlier section, one cannot determine the shape of a function from a finite number of data points. Instead, one has to focus on the properties that are important. For example, is there a small but finite chance that a task can take an unreasonably long time? If so, you may want to use a lognormal distribution…but remember, you will need to find a sensible way to estimate the distribution parameters from your historical data.

Second, you may have noted from the probability distribution curve (Figure 11)  that despite the skewed distributions of the individual tasks, the distribution of the overall completion time is somewhat symmetric with a minimum of ~9 days, most likely time of ~16 days and maximum of 24 days.  It turns out that this is a general property of distributions that are generated by adding a large number of independent probabilistic variables. As the number of variables increases, the overall distribution will tend to the ubiquitous Normal distribution.

The assumption of independence merits a closer look.  In the case it hand,  it implies that the completion times for each task are independent of each other. As most project managers will know from experience, this is rarely the case: in real life,  a task that is delayed will usually have knock-on effects on subsequent tasks. One can easily incorporate such dependencies in a Monte Carlo simulation. A formal way to do this is to introduce a non-zero correlation coefficient between tasks as I have done here. A simpler and more realistic approach is to introduce conditional inter-task dependencies As an example, one could have an inter-task delay that kicks in only if the predecessor task takes more than 80%  of its maximum time.

Thirdly, you may have wondered why I used 10,000 trials: why not 100, or 1000 or 20,000. This has to do with the tricky issue of convergence. In a nutshell, the estimates we obtain should not depend on the number of trials used.  Why? Because if they did, they’d be meaningless.

Operationally, convergence means that any predicted quantity based on aggregates should not vary with number of trials.  So, if our Monte Carlo simulation has converged, our prediction of 19.5 days for the 90% likely completion time should not change substantially if I increase the number of trials from ten to twenty thousand. I did this and obtained almost the same value of 19.5 days. The average and median completion times (shown in cell Q3 and Q4 of Sheet 1) also remained much the same (16.8 days). If you wish to repeat the calculation, be sure to change the formulas on all three sheets appropriately. I was lazy and hardcoded the number of trials. Sorry!

Finally, I should mention that simulations can be usefully performed at a higher level than individual tasks. In their highly-readable book,  Waltzing With Bears: Managing Risk on Software Projects, Tom De Marco and Timothy Lister show how Monte Carlo methods can be used for variables such as  velocity, time, cost etc.  at the project level as opposed to the task level. I believe it is better to perform simulations at the lowest possible level, the main reason being that it is easier, and less error-prone, to estimate individual tasks than entire projects. Nevertheless, high level simulations can be very useful if one has reliable data to base these on.

There are a few more things I could say about the usefulness of the generated distribution functions and Monte Carlo in general, but they are best relegated to a future article. This one is much too long already and I think I’ve tested your patience enough. Thanks so much for reading, I really do appreciate it and hope that you found it useful.

Acknowledgement: My thanks to Peter Holberton for pointing out a few typographical and coding errors in an earlier version of this article. These have now been fixed. I’d be grateful if readers could bring any errors they find to my attention.

Written by K

March 27, 2018 at 4:11 pm

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## The improbability of success

Anyone who has tidied up after a toddler intuitively understands that making a mess is far easier than creating order. The fundamental reason for this is that the number of messy states in the universe (or a toddler’s room) far outnumbers the ordered ones.  As this point might not be obvious, I’ll demonstrate it via a simple thought experiment involving marbles:

Throw three marbles onto a flat surface.  When the marbles come to rest, you are most likely to end up with a random configuration  as in Figure 1.

Figure 1: A random configuration of 3 marbles

Indeed, you’d be extremely surprised if the three ended up being collinear as in Figure 2.   Note that Figure 2 is just one example of many collinear possibilities, but the point I’m making is that if the marbles are thrown randomly, they are more likely to end up in a random state than a lined-up one.

Figure 2: an unlikely (ordered) configuration

This raises a couple of questions:

Question: On what basis can one claim that the collinear configuration is tidier or more ordered than the non-collinear one?

Naive answer:  It looks more ordered. Yes, tidiness is in the eye of the beholder so it is necessarily subjective. However, I’ll wager that if one took a poll, an overwhelming number of people would say that the configuration in Figure 2 is more ordered than the one in Figure 1.

More sophisticated answer : The “state” of collinear marbles can be described using 2 parameters, the slope and intercept of the straight line that three marbles lie on (in any coordinate system) whereas the description of the nonlinear state requires 3 parameters. The first state is tidier because it requires fewer parameters.  Another way to think about is that the line can be described by two marbles; the third one is redundant as far as the description of the state is concerned.

Question: Why is a tidier configuration less likely than a messy one?

Answer:  May be you see this intuitively and need no proof, but here’s one just in case. Imagine rolling the three marbles one after the other. The first two, regardless of where they end up, will necessarily lie along a line (two points lie on the straight line joining them). Now, I think it is easy to see that if we throw the third marble randomly, it is highly unlikely end up on that line. Indeed, for the third marble to end up exactly on the same straight line requires a coincidence of near cosmic proportions.

I know, I know, this is not a proof, but I trust it makes the point.

Now, although it is near impossible to get to a collinear end state via random throws, it is possible to approximate it by changing the way we throw the marbles. Here’s how:

1. Throw the marbles consecutively rather than in one go.
2. When throwing the third marble, adjust its initial speed and direction in a way that takes into account the positions of the two marbles that are already on the surface. Remember these two already define a straight line.

The third throw is no longer random because it is designed to maximise the chance that the last marble will get as close as possible to the straight line defined by the first two. Done right, you’ll end up with something closer to the configuration in Figure 3 rather than the one in Figure 2.

Figure 3: an “approximately ordered” state

Now you’re probably wondering what this has to do with success. I’ll make the connection via an example that will be familiar to many readers of this blog: an organisation’s strategy. However, as I will reiterate later, the arguments I present are very general and can be applied to just about any initiative or situation.

Typically, a strategy sets out goals for an organisation and a plan to achieve them in a specified timeframe. The goals define a number of desirable outcomes, or states which, by design, are constrained to belong to a (very) small subset of all possible states the organisation can end up in.  In direct analogy with the simple model discussed above it is clear that, left to its own devices, the organisation is more likely to end up in one of the much overwhelmingly larger number of “failed states” than one of the successful ones.  Notwithstanding the popular quote about there being many roads to success, in reality there are a great many more roads to failure.

Of course, that’s precisely why organisations are never “left to their own devices.” Indeed, a strategic plan specifies actions that are intended to make a successful state more likely than an unsuccessful one. However, no plan can guarantee success; it can, at best, make it more likely. As in the marble game, success is ultimately a matter of chance, even when we take actions to make it more likely.

If we accept this, the key question becomes: how can one design a strategy that improves the odds of success?  The marble analogy suggests a way to do this is to:

1. Define success in terms of an end state that is a natural extension of your current state.
2. Devise a plan to (approximately) achieve that end state. Such a plan will necessarily build on the current state rather than change it wholesale. Successful change is an evolutionary process rather than a revolutionary one.

My contention is that these points are often ignored by management strategists. More often than not, they will define an end state based on a textbook idealisation, consulting model or (horror!) best practice. The marble analogy shows why copying others is unlikely to succeed.

Figure 4 shows a variant of the marble game in which we have two sets of marbles (or organisations!), one blue, as before, and the other red.

Figure 4: Two distinct configurations of marbles (or organisations)

Now, it is considerably harder to align an additional marble with both sets of marbles than the blue one alone. Here’s why…

To align with both sets, the new marble has to end up close to the point that lies at the intersection of the blue and red lines in Figure 5. In contrast, to align with the blue set alone, all that’s needed is for it to get close to any point on the blue line.

QED!

Figure 5: Why copying others is not a good idea (see text for explanation)

Finally, on a broader note, it should be clear that the arguments made above go beyond organisational strategies. They apply to pretty much any planned action, whether at work or in one’s personal life.

So, to sum up: when developing an organisational (or personal) strategy, the first step is to understand where you are and then identify the minimal actions you need to take in order to get to an “improved” state that is consistent with  your current one. Yes, this is akin to the incremental and evolutionary approach that Agilistas and Leaners have been banging on about for years. However, their prescriptions focus on specific areas: software development and process improvement.  My point is that the basic principles are way broader because they are a direct consequence of a fundamental fact regarding the relative likelihood of order and disorder in a toddler’s room, an organisation, or even the universe at large.

Written by K

April 4, 2017 at 9:16 pm

## Uncertainty, ambiguity and the art of decision making

A common myth about decision making in organisations is that it is, by and large, a rational process.   The term rational refers to decision-making methods that are based on the following broad steps:

1. Identify available options.
2. Develop criteria for rating options.
3. Rate options according to criteria developed.
4. Select the top-ranked option.

Although this appears to be a logical way to proceed it is often difficult to put into practice, primarily because of uncertainty about matters relating to the decision.

Uncertainty can manifest itself in a variety of ways: one could be uncertain about facts, the available options, decision criteria or even one’s own preferences for options.

In this post, I discuss the role of uncertainty in decision making and, more importantly, how one can make well-informed decisions in such situations.

### A bit about uncertainty

It is ironic that the term uncertainty is itself vague when used in the context of decision making. There are at least five distinct senses in which it is used:

1. Uncertainty about decision options.
2. Uncertainty about one’s preferences for options.
3. Uncertainty about what criteria are relevant to evaluating the options.
4. Uncertainty about what data is needed (data relevance).
5. Uncertainty about the data itself (data accuracy).

Each of these is qualitatively different: uncertainty about data accuracy (item 5 above) is very different from uncertainty regarding decision options (item 1). The former can potentially be dealt with using statistics whereas the latter entails learning more about the decision problem and its context, ideally from different perspectives. Put another way, the item 5 is essentially a technical matter whereas item 1 is a deeper issue that may have social, political and – as we shall see – even behavioural dimensions. It is therefore reasonable to expect that the two situations call for vastly different approaches.

### Quantifiable uncertainty

A common problem in project management is the estimation of task durations. In this case, what’s requested is a “best guess” time (in hours or days) it will take to complete a task. Many project schedules represent task durations by point estimates, i.e.  by single numbers. The Gantt Chart shown in Figure 1 is a common example. In it, each task duration is represented by its expected duration. This is misleading because the single number conveys a sense of certainty that is unwarranted.  It is far more accurate, not to mention safer, to quote a range of possible durations.

Figure 1: Gantt Chart (courtesy Wikimedia)

In general, quantifiable uncertainties, such as those conveyed in estimates, should always be quoted as ranges – something along the following lines: task A may take anywhere between 2 and 8 days, with a most likely completion time of 4 days (Figure 2).

Figure 2: Task completion likelihood (3 point estimates)

In this example, aside from stating that the task will finish sometime between 2 and 4 days, the estimator implicitly asserts that the likelihood of finishing before 2 days or after 8 days is zero.  Moreover, she also implies that some completion times are more likely than others. Although it may be difficult to quantify the likelihood exactly, one can begin by making simple (linear!) approximations as shown in Figure 3.

Figure 3: Simple probability distribution based on the estimates in Fig 2

The key takeaway from the above is that quantifiable uncertainties are shapes rather than single numbers.  See this post and this one for details for how far this kind of reasoning can take you. That said, one should always be aware of the assumptions underlying the approximations. Failure to do so can be hazardous to the credibility of estimators!

Although I haven’t explicitly said so, estimation as described above has a subjective element. Among other things, the quality of an estimate depends on the judgement and experience of the estimator. As such, it is prone to being affected by errors of judgement and cognitive biases.  However, provided one keeps those caveats in mind, the probability-based approach described above is suited to situations in which uncertainties are quantifiable, at least in principle. That said, let’s move on to more complex situations in which uncertainties defy quantification.

### Introducing ambiguity

The economist Frank Knight was possibly the first person to draw the distinction between quantifiable and unquantifiable uncertainties.  To make things really confusing, he called the former risk and the latter uncertainty. In his doctoral thesis, published in 1921, wrote:

…it will appear that a measurable uncertainty, or “risk” proper, as we shall call the term, is so far different from an unmeasurable one that it is not in effect an uncertainty at all. We shall accordingly restrict the term “uncertainty” to cases of the non-quantitative type (p.20)

Terminology has moved on since Knight’s time, the term uncertainty means lots of different things, depending on context. In this piece, we’ll use the term uncertainty to refer to quantifiable uncertainty (as in the task estimate of the previous section) and use ambiguity to refer to nonquantifiable uncertainty. In essence, then, we’ll use the term uncertainty for situations where we know what we’re measuring (i.e. the facts) but are uncertain about its numerical or categorical values whereas we’ll use the word ambiguity to refer to situations in which we are uncertain about what the facts  are or which facts are relevant.

As a test of understanding, you may want to classify each of the five points made in the second section of this post as either uncertain or ambiguous (Answers below)

Answer: 1 through 4 are ambiguous and 5 is uncertain.

### How ambiguity manifests itself in decision problems

The distinction between uncertainty and ambiguity points to a problem with quantitative decision-making techniques such as cost-benefit analysis, multicriteria decision making methods or analytic hierarchy process. All these methods assume that decision makers are aware of all the available options, their preferences for them, the relevant evaluation criteria and the data needed. This is almost never the case for consequential decisions. To see why, let’s take a closer look at the different ways in which ambiguity can play out in the rational decision making process mentioned at the start of this article.

1. The first step in the process is to identify available options. In the real world, however, options often cannot be enumerated or articulated fully. Furthermore, as options are articulated and explored, new options and sub-options tend to emerge. This is particularly true if the options depend on how future events unfold.
2. The second step is to develop criteria for rating options. As anyone who has been involved in deciding on a contentious issue will confirm, it is extremely difficult to agree on a set of decision criteria for issues that affect different stakeholders in different ways.  Building a new road might improve commute times for one set of stakeholders but result in increased traffic in a residential area for others. The two criteria will be seen very differently by the two groups. In this case, it is very difficult for the two groups to agree on the relative importance of the criteria or even their legitimacy. Indeed, what constitutes a legitimate criterion is a matter of opinion.
3. The third step is to rate options. The problem here is that real-world options often cannot be quantified or rated in a meaningful way. Many of life’s dilemmas fall into this category. For example, a decision to accept or decline a job offer is rarely made on the basis of material gain alone. Moreover, even where ratings are possible, they can be highly subjective. For example, when considering a job offer, one candidate may give more importance to financial matters whereas another might consider lifestyle-related matters (flexi-hours, commuting distance etc.) to be paramount. Another complication here is that there may not be enough information to settle the matter conclusively. As an example, investment decisions are often made on the basis of quantitative information that is based on questionable assumptions.

A key consequence of the above is that such ambiguous decision problems are socially complex – i.e. different stakeholders could have wildly different perspectives on the problem itself.   One could say the ambiguity experienced by an individual is compounded by the group.

Before going on I should point out that acute versions of such ambiguous decision problems go by many different names in the management literature. For example:

All these terms are more or less synonymous:  the root cause of the difficulty in every case is ambiguity (or unquantifiable uncertainty), which prevents a clear formulation of the problem.

Social complexity is hard enough to tackle as it is, but there’s another issue that makes things even harder: ambiguity invariably triggers negative emotions such as fear and anxiety in individuals who make up the group.  Studies in neuroscience have shown that in contrast to uncertainty, which evokes logical responses in people, ambiguity tends to stir up negative emotions while simultaneously suppressing the ability to think logically.  One can see this playing out in a group that is debating a contentious decision: stakeholders tend to get worked up over issues that touch on their values and identities, and this seems to limit their ability to look at the situation objectively.

### Tackling ambiguity

Summarising the discussion thus far: rational decision making approaches are based on the assumption that stakeholders have a shared understanding of the decision problem as well as the facts and assumptions around it. These conditions are clearly violated in the case of ambiguous decision problems. Therefore, when confronted with a decision problem that has even a hint of ambiguity, the first order of the day is to help the group reach a shared understanding of the problem.  This is essentially an exercise in sensemaking, the art of collaborative problem formulation. However, this is far from straightforward because ambiguity tends to evoke negative emotions and attendant defensive behaviours.

The upshot of all this is that any approach to tackle ambiguity must begin by taking the concerns of individual stakeholders seriously.  Unless this is done, it will be impossible for the group to coalesce around a consensus decision. Indeed, ambiguity-laden decisions in organisations invariably fail when they overlook concerns of specific stakeholder groups.  The high failure rate of organisational change initiatives (60-70% according to this Deloitte report) is largely attributable to this point

There are a number of techniques that one can use to gather and synthesise diverse stakeholder viewpoints and thus reach a shared understanding of a complex or ambiguous problem. These techniques are often referred to as problem structuring methods (PSMs). I won’t go into these in detail here; for an example check out Paul Culmsee’s articles on dialogue mapping and Barry Johnson’s introduction to polarity management. There are many more techniques in the PSM stable. All of them are intended to help a group reconcile different viewpoints and thus reach a common basis from which one can proceed to the next step (i.e., make a decision on what should be done). In other words, these techniques help reduce ambiguity.

But there’s more to it than a bunch of techniques.  The main challenge is to create a holding environment that enables such techniques to work. I am sure readers have been involved in a meeting or situation where the outcome seems predetermined by management or has been undermined by self- interest. When stakeholders sense this, no amount of problem structuring is going to help.  In such situations one needs to first create the conditions for open dialogue to occur. This is precisely what a holding environment provides.

Creating such a holding environment is difficult in today’s corporate world, but not impossible. Note that this is not an idealist’s call for an organisational utopia. Rather, it involves the application of a practical set of tools that address the diverse, emotion-laden reactions that people often have when confronted with ambiguity.   It would take me too far afield to discuss PSMs and holding environments any further here. To find out more, check out my papers on holding environments and dialogue mapping in enterprise IT projects, and (for a lot more) the Heretic’s Guides that I co-wrote with Paul Culmsee.

The point is simply this: in an ambiguous situation, a good decision – whatever it might be – is most likely to be reached by a consultative process that synthesises diverse viewpoints rather than by an individual or a clique.  However, genuine participation (the hallmark of a holding environment) in such a process will occur only after participants’ fears have been addressed.

### Wrapping up

Standard approaches to decision making exhort managers and executives to begin with facts, and if none are available, to gather them diligently prior to making a decision. However, most real-life decisions are fraught with uncertainty so it may be best to begin with what one doesn’t know, and figure out how to make the possible decision under those “constraints of ignorance.” In this post I’ve attempted to outline what such an approach would entail. The key point is to figure out the kind uncertainty one is dealing with and choosing an approach that works for it. I’d argue that most decision making debacles stem from a failure to appreciate this point.

Of course, there’s a lot more to this approach than I can cover in the span of a post, but that’s a story for another time.

Note: This post is written as an introduction to the Data and Decision Making subject that is part of the core curriculum of the Master of Data Science and Innovation program at UTS. I’m co-teaching the subject in Autumn 2018 with Rory Angus and Alex Scriven.

Written by K

March 9, 2017 at 10:04 am

## The dark side of data science

Data scientists are sometimes blind to the possibility that the predictions of their algorithms can have unforeseen negative effects on people. Ethical or social implications are easy to overlook when one finds interesting new patterns in data, especially if they promise significant financial gains. The Centrelink debt recovery debacle, recently reported in the Australian media, is a case in point.

Here is the story in brief:

Centrelink is an Australian Government organisation responsible for administering welfare services and payments to those in need. A major challenge such organisations face is ensuring that their clients are paid no less and no more than what is due to them. This is difficult because it involves crosschecking client income details across multiple systems owned by different government departments, a process that necessarily involves many assumptions. In July 2016, Centrelink unveiled an automated compliance system that compares income self-reported by clients to information held by the taxation office.

The problem is that the algorithm is flawed: it makes strong (and incorrect!) assumptions regarding the distribution of income across a financial year and, as a consequence, unfairly penalizes a number of legitimate benefit recipients.  It is very likely that the designers and implementers of the algorithm did not fully understand the implications of their assumptions. Worse, from the errors made by the system, it appears they may not have adequately tested it either.  But this did not stop them (or, quite possibly, their managers) from unleashing their algorithm on an unsuspecting public, causing widespread stress and distress.  More on this a bit later.

Algorithms like the one described above are the subject of Cathy O’Neil’s aptly titled book, Weapons of Math Destruction.  In the remainder of this article I discuss the main themes of the book.  Just to be clear, this post is more riff than review. However, for those seeking an opinion, here’s my one-line version: I think the book should be read not only by data science practitioners, but also by those who use or are affected by their algorithms (which means pretty much everyone!).

### Abstractions and assumptions

‘O Neil begins with the observation that data algorithms are mathematical models of reality, and are necessarily incomplete because several simplifying assumptions are invariably baked into them. This point is important and often overlooked so it is worth illustrating via an example.

When assessing a person’s suitability for a loan, a bank will want to know whether the person is a good risk. It is impossible to model creditworthiness completely because we do not know all the relevant variables and those that are known may be hard to measure. To make up for their ignorance, data scientists typically use proxy variables, i.e. variables that are believed to be correlated with the variable of interest and are also easily measurable. In the case of creditworthiness, proxy variables might be things like gender, age, employment status, residential postcode etc.  Unfortunately many of these can be misleading, discriminatory or worse, both.

The Centrelink algorithm provides a good example of such a “double-whammy” proxy. The key variable it uses is the difference between the client’s annual income reported by the taxation office and self-reported annual income stated by the client. A large difference is taken to be an indicative of an incorrect payment and hence an outstanding debt. This simplistic assumption overlooks the fact that most affected people are not in steady jobs and therefore do not earn regular incomes over the course of a financial year (see this article by Michael Griffin, for a detailed example).  Worse, this crude proxy places an unfair burden on vulnerable individuals for whom casual and part time work is a fact of life.

Worse still, for those wrongly targeted with a recovery notice, getting the errors sorted out is not a straightforward process. This is typical of a WMD. As ‘O Neil states in her book, “The human victims of WMDs…are held to a far higher standard of evidence than the algorithms themselves.”  Perhaps this is because the algorithms are often opaque. But that’s a poor excuse.  This is the only technical field where practitioners are held to a lower standard of accountability than those affected by their products.

‘O Neil’s sums it up rather nicely when she calls algorithms like the Centrelink one  weapons of math destruction (WMD).

### Self-fulfilling prophecies and feedback loops

A characteristic of WMD is that their predictions often become self-fulfilling prophecies. For example a person denied a loan by a faulty risk model is more likely to be denied again when he or she applies elsewhere, simply because it is on their record that they have been refused credit before. This kind of destructive feedback loop is typical of a WMD.

An example that ‘O Neil dwells on at length is a popular predictive policing program. Designed for efficiency rather than nuanced judgment, such algorithms measure what can easily be measured and act by it, ignoring the subtle contextual factors that inform the actions of experienced officers on the beat. Worse, they can lead to actions that can exacerbate the problem. For example, targeting young people of a certain demographic for stop and frisk actions can alienate them to a point where they might well turn to crime out of anger and exasperation.

As Goldratt famously said, “Tell me how you measure me and I’ll tell you how I’ll behave.”

This is not news: savvy managers have known about the dangers of managing by metrics for years. The problem is now exacerbated manyfold by our ability to implement and act on such metrics on an industrial scale, a trend that leads to a dangerous devaluation of human judgement in areas where it is most needed.

A related problem – briefly mentioned earlier – is that some of the important variables are known but hard to quantify in algorithmic terms. For example, it is known that community-oriented policing, where officers on the beat develop relationships with people in the community, leads to greater trust. The degree of trust is hard to quantify, but it is known that communities that have strong relationships with their police departments tend to have lower crime rates than similar communities that do not.  Such important but hard-to-quantify factors are typically missed by predictive policing programs.

### Blackballed!

Ironically, although WMDs can cause destructive feedback loops, they are often not subjected to feedback themselves. O’Neil gives the example of algorithms that gauge the suitability of potential hires.  These programs often use proxy variables such as IQ test results, personality tests etc. to predict employability.  Candidates who are rejected often do not realise that they have been screened out by an algorithm. Further, it often happens that candidates who are thus rejected go on to successful careers elsewhere. However, this post-rejection information is never fed back to the algorithm because it impossible to do so.

In such cases, the only way to avoid being blackballed is to understand the rules set by the algorithm and play according to them. As ‘O Neil so poignantly puts it, “our lives increasingly depend on our ability to make our case to machines.” However, this can be difficult because it assumes that a) people know they are being assessed by an algorithm and 2) they have knowledge of how the algorithm works. In most hiring scenarios neither of these hold.

Just to be clear, not all data science models ignore feedback. For example, sabermetric algorithms used to assess player performance in Major League Baseball are continually revised based on latest player stats, thereby taking into account changes in performance.

### Driven by data

In recent years, many workplaces have gradually seen the introduction to data-driven efficiency initiatives. Automated rostering, based on scheduling algorithms is an example. These algorithms are based on operations research techniques that were developed for scheduling complex manufacturing processes. Although appropriate for driving efficiency in manufacturing, these techniques are inappropriate for optimising shift work because of the effect they have on people. As O’ Neil states:

Scheduling software can be seen as an extension of just-in-time economy. But instead of lawn mower blades or cell phone screens showing up right on cue, it’s people, usually people who badly need money. And because they need money so desperately, the companies can bend their lives to the dictates of a mathematical model.

She correctly observes that an, “oversupply of low wage labour is the problem.” Employers know they can get away with treating people like machine parts because they have a large captive workforce.  What makes this seriously scary is that vested interests can make it difficult to outlaw such exploitative practices. As ‘O Neil mentions:

Following [a] New York Times report on Starbucks’ scheduling practices, Democrats in Congress promptly drew up bills to rein in scheduling software. But facing a Republican majority fiercely opposed to government regulations, the chances that their bill would become law were nil. The legislation died.

Commercial interests invariably trump social and ethical issues, so it is highly unlikely that industry or government will take steps to curb the worst excesses of such algorithms without significant pressure from the general public. A first step towards this is to educate ourselves on how these algorithms work and the downstream social effects of their predictions.

### Messing with your mind

There is an even more insidious way that algorithms mess with us. Hot on the heels of the recent US presidential election, there were suggestions that fake news items on Facebook may have influenced the results.  Mark Zuckerberg denied this, but as this Casey Newton noted in this trenchant tweet, the denial leaves Facebook in “the awkward position of having to explain why they think they drive purchase decisions but not voting decisions.”

Be that as it may, the fact is Facebook’s own researchers have been conducting experiments to fine tune a tool they call the “voter megaphone”. Here’s what ‘O Neil says about it:

The idea was to encourage people to spread the word that they had voted. This seemed reasonable enough. By sprinkling people’s news feeds with “I voted” updates, Facebook was encouraging Americans – more that sixty-one million of them – to carry out their civic duty….by posting about people’s voting behaviour, the site was stoking peer pressure to vote. Studies have shown that the quiet satisfaction of carrying out a civic duty is less likely to move people than the possible judgement of friends and neighbours…The Facebook started out with a constructive and seemingly innocent goal to encourage people to vote. And it succeeded…researchers estimated that their campaign had increased turnout by 340,000 people. That’s a big enough crowd to swing entire states, and even national elections.

And if that’s not scary enough, try this:

For three months leading up to the election between President Obama and Mitt Romney, a researcher at the company….altered the news feed algorithm for about two million people, all of them politically engaged. The people got a higher proportion of hard news, as opposed to the usual cat videos, graduation announcements, or photos from Disney world….[the researcher] wanted to see  if getting more [political] news from friends changed people’s political behaviour. Following the election [he] sent out surveys. The self-reported results that voter participation in this group inched up from 64 to 67 percent.

This might not sound like much, but considering the thin margins of recent presidential elections, it could be enough to change a result.

But it’s even more insidious.  In a paper published in 2014, Facebook researchers showed that users’ moods can be influenced by the emotional content of their newsfeeds. Here’s a snippet from the abstract of the paper:

In an experiment with people who use Facebook, we test whether emotional contagion occurs outside of in-person interaction between individuals by reducing the amount of emotional content in the News Feed. When positive expressions were reduced, people produced fewer positive posts and more negative posts; when negative expressions were reduced, the opposite pattern occurred. These results indicate that emotions expressed by others on Facebook influence our own emotions, constituting experimental evidence for massive-scale contagion via social networks.

As you might imagine, there was a media uproar following which  the lead researcher issued a clarification and  Facebook officials duly expressed regret (but, as far as I know, not an apology).  To be sure, advertisers have been exploiting this kind of “mind control” for years, but a public social media platform should (expect to) be held to a higher standard of ethics. Facebook has since reviewed its internal research practices, but the recent fake news affair shows that the story is to be continued.

### Disarming weapons of math destruction

The Centrelink debt debacle, Facebook mood contagion experiments and the other case studies mentioned in the book illusrate the myriad ways in which Big Data algorithms have a pernicious effect on our day-to-day lives. Quite often people remain unaware of their influence, wondering why a loan was denied or a job application didn’t go their way. Just as often, they are aware of what is happening, but are powerless to change it – shift scheduling algorithms being a case in point.

This is not how it was meant to be. Technology was supposed to make life better for all, not just the few who wield it.

So what can be done? Here are some suggestions:

• To begin with, education is the key. We must work to demystify data science, create a general awareness of data science algorithms and how they work. O’ Neil’s book is an excellent first step in this direction (although it is very thin on details of how the algorithms work)
• Develop a code of ethics for data science practitioners. It is heartening to see that IEEE has recently come up with a discussion paper on ethical considerations for artificial intelligence and autonomous systems and ACM has proposed a set of principles for algorithmic transparency and accountability.  However, I should also tag this suggestion with the warning that codes of ethics are not very effective as they can be easily violated. One has to – somehow – embed ethics in the DNA of data scientists. I believe, one way to do this is through practice-oriented education in which data scientists-in-training grapple with ethical issues through data challenges and hackathons. It is as Wittgenstein famously said, “it is clear that ethics cannot be articulated.” Ethics must be practiced.
• Put in place a system of reliable algorithmic audits within data science departments, particularly those that do work with significant social impact.
• Increase transparency a) by publishing information on how algorithms predict what they predict and b) by making it possible for those affected by the algorithm to access the data used to classify them as well as their classification, how it will be used and by whom.
• Encourage the development of algorithms that detect bias in other algorithms and correct it.
• Inspire aspiring data scientists to build models for the good.

It is only right that the last word in this long riff should go to ‘O Neil whose work inspired it. Towards the end of her book she writes:

Big Data processes codify the past. They do not invent the future. Doing that requires moral imagination, and that’s something that only humans can provide. We have to explicitly embed better values into our algorithms, creating Big Data models that follow our ethical lead. Sometimes that will mean putting fairness ahead of profit.

Excellent words for data scientists to live by.

Written by K

January 17, 2017 at 8:38 pm