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The Mighty Maximin and worst-case analysis

Last modified: Sunday, 18-Apr-2010 00:47:30 MST

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This is one of my more recent campaigns. Indeed, it is a mini-campaign in its own right. Still, this particular project has an affinity with my Solution-Method-Free-Modeling campaign in that, both projects are concerned with the role of mathematical modeling in decision making. I might also add that this campaign inspired my Voodoo Decision-Making campaign.

The direct trigger for this project was the realization that there is something suspect in the interpretation that naive, and not so naive, users of Info-Gap Decision Theory — a purportedly new theory for decision making under severe uncertainty — give this theory.

In any case, the objective of this campaign is to clarify conceptual and technical issues associated with the modeling aspects of the classical Maximin paradigm, especially in the context of decision-making in the face of severe uncertainty.

Table of contents

Overview

However incomprehensible this may appear to seasoned analysts/scholars working in the area of decision-making under uncertainty, there are those who profess to be experts in the field yet they are not conversant with Worst-Case Analysis and Wald's Maximin Principle. This can range from ignorance about the existence of these two central modeling tools, to a shaky grasp of their role and place in decision theory, to a total lack of understanding of the mathematical modeling issues that come into play in the application of these important modeling tools.

So here are some thoughts on this topic.

The basic idea underlying the Maximin paradigm can be summarized as follows:

Maximin Rule

Rank alternatives by their worst possible outcomes: adopt the alternative the worst outcome of which is at least as good as the worst outcome of the other alternatives.

This paradigm was conceived by the mathematician Abraham Wald (1902-1950). It is in fact an adaptation of John von Neumann's (1903-1957) Maximin paradigm that he developed in the context of game theory. Wald hit on the idea of adapting this paradigm to the framework of decision-making under uncertainty thus casting Uncertainty (call it Nature) as the player pit against the decision maker. The idea here is that, to play it safe, the decision maker assumes that Nature constitutes an antagonistic adversary.

So, if the decision maker maximizes her "utility", Nature will attampt to minimze the "utility", whereas if the decision maker minimizes the "utility", Nature will attempt to maximize the "utility". The first case is captured by the term Maximin and the second case by the term Minimax.

The Maximin and Minimax models are essentially equivalent (subject to multiplying the "utility" by -1) and the choice between them is a matter of convenience.

Note that this dark view of uncertainty is not meant to be taken as a philosophical statement about the nature of reality. Rather, the point of the Maximin/Minimax model is that Nature is used as a device for expressing the decision maker's attitude towards the risks associated with uncertainty. So, like my dear wife, in this framework the decision maker takes an extremely pessimistic stance by assuming that the "worst-case-scenario" pertaining to her decision will be realized.

Not surprisingly, therefore, Maximin/Minimax has become almost synonymous with robust decision-making not only in classical decision theory but in other areas as well. For instance, here is the abstract of the entry Robust Control by Noah Williams in the New Palgrave Dictionary of Economics, Second Edition, 2008:

Robust control is an approach for confronting model uncertainty in decision making, aiming at finding decision rules which perform well across a range of alternative models. This typically leads to a minimax approach, where the robust decision rule minimizes the worst-case outcome from the possible set. This article discusses the rationale for robust decisions, the background literature in control theory, and different approaches which have been used in economics, including the most prominent approach due to Hansen and Sargent.

The following quote is from the book Robust Statistics by Huber (1981, pp. 16-17):

But as we defined robustness to mean insensitivity with regard to small deviations from assumptions, any quantitative measure of robustness must somehow be concerned with the maximum degradation of performance possible for an e-deviation from the assumptions. The optimally robust procedure minimizes this degradation and hence will be a minimax procedure of some kind.

Keeping in mind that "minimax" signifies the reverse of Maximin, in this discussion, we are concerned for the most part, with the Maximin option.

Math formulation

The two most prevalent (equivalent) formal mathematical formulations of the Maximin paradigm are: the classic formulation and the mathematical programming (MP) formulation. Here they are side by side:

where the double-lined R denotes the real line.

Note that an instance of a Maximin model is specified by a triplet (D,S,f) whose elements can be interpreted as follows:

In short, these formulations describe a game played by two players: one player (the decision maker) tries to maximize the objective function by controlling d in D, while the other player (Nature) tries to minimize the objective function by controlling s in S(d). Observe that when Nature decides on her best s she knows the value of d selected by the decision maker. In plain language, in this game the decision maker plays first whereupon Nature responds to his choice of a d in D.

In cases where robustness is sought with respect to constraints, rather than "utility", the Maximin model would be formulated as follows:

where — without loss of generality — the constraint requires the value of c(d,s) to be an element of some given set C. In other words, here the decision maker's goal is to maximize the utility f(d) over all d in D subject to the condition that c(d,s) is in C for all s in S(d). For her part, Nature will attempt to violate this constraint — if possible — by a proper choice of s in S(d).

And if robustness is sought with repsect to the objective function and the constraint, the Maximin model would be formulated as follows:

The Minimax formulations are similar except that the min and max operations are interchanged. In this case the decision maker seeks to minimize the objective function and therefore antagonistic Nature attempts to minimize it.

For our purposes here, the above discussion is sufficient. We need point out, however, that Wald's Maximin model plays a key role in classical decision theory, robust optimization, economics, statistics, and control theory.

To some this paradigm is second nature, to others it is a puzzle.

Remark:

The fact that a "max" and a "min" appear in a formulation of a decision-making model does not automatically render the model a Maximin model. For example, consider the following decision-making model:

In general, this is not a Maximin model because we do not have here a situation where Nature (deciding on s) "antagonizes" the decision maker (deciding on d). Or, in other words, the idea here is not to identify the "best worst-case".

This is so, because here not all s in S(d) are required to satisfy the constraint K ≥ c(d,s). So, by selecting s to minimize c(d,s) over s in S(d), Nature does not select the worst s in S(d) with regard to the constraint K ≥ c(d,s), which means that some s in S(d) are "allowed" to violate the constraint K ≥ c(d,s).

In a word, all that is required here is that "at least one" element of S(d) satisfy the constraint K ≥ c(d,s).

This, as a matter of fact is a Maximax model, observing that

where

Note that in this framework Nature is not antagonistic towards the decision maker. To the contrary, here Nature cooperates with her.

In contrast, the model

is a Maximin model even though no "min" occurs in the formulation. Note that here all s in S(d) must satisfy the constraint K ≥ c(d,s). Thus,

where

Variations on the same theme

Now, the pessimistic stance to uncertainty prescribed by Wald's Maximin paradigm proves far too conservative for many applications. Not surprisingly, therefore, a number of attempts have been made to modify the paradigm's grim approach to uncertainty with the view to mitigate its excessive conservatism.

The reader is advised that modeling the Maximin can involve a significant effort. Indeed, stating the paradigm in terms of a particular situation can often require of the modeler/analyst considerable insight, imagination and invention (see my paper The Mighty Maximin!).

Decision Tables

In cases where the decision and state spaces are finite sets, the Maximin model can be depicted as a Decision Table. The convention is that the rows of the table represent decisions, the columns represent states and the entries of the table represent the rewards r(d,s), say in AU$.

For example, consider the following simple decision table:

  s1     s2     s3     s4  
d1 7569
d2 4865
d3 9178

We now append an additional column to the table to list the security levels of the decisions. The security level of decision dj is the smallest reward in the j-th row of the table. This means that if we select decision dj, then no matter what state is realized, the actual reward will be no smaller than the security level.

So here is our decision table with the associated security levels (AU$):

  s1     s2     s3     s4     SL(j)  
d1 75695
d2 48654
d3 91781

What is left then is to find the largest security level. We indicate this in the usual/unusual way:

  s1     s2     s3     s4     SL(j)  
d1 75695♥
d2 48654
d3 91781

In short, the Maximin rule selects decision d1. The optimal (maximal) security level is AU$5.

Bayes Rule vs Maximin Rule

Recently, I was greatly surprised to learn that apparently, there is a perception out there, that a Maximin decision rule can be simulated by a degenerate Bayes Rule that selects (with certainty) one of the states.

It is important therefore to set the record straight on this matter by showing that this is not so.

Recall that in the framework of Bayes Rule, some probability distribution function is assumed to operate on the state space and the performance of a decision, say d, is measured by the expected value of the reward r(d,s) determined by this distribution. Thus let E[d] denote the expected value of r(d,s) generated by decision d and the assumed probability distribution function of s.

In keeping with Bayes rule, decisions are ranked by their E[d] values — the larger the better. So the optimal decision according to this rule is one that maximizes E[d] over all the feasible values of d.

For example, consider the following (familiar) decision table:

  s1     s2     s3     s4  
d1 7561
d2 4865
d3 9178

To illustrate Bayes Rule in action we have to postulate a probability distribution over the four states. Suppose that we consider the vector p=(0.1, 0.2,0.4,0.3) for this purpose. The convention is to list these probabilities above the states:

p(i)  0.1     0.2     0.4     0.3  
  s1     s2     s3     s4  
d1 7561
d2 4865
d3 9178

We now append an additional column to the table where we list the value of E[dj], j=1,2,3.

p(i)  0.1     0.2     0.4     0.3  
  s1     s2     s3     s4   E[dj]
d1 75614.4
d2 48655.8
d3 91786.3

Thus, the best decision according to Bayes Rule is d3. It yields an expected reward of 6.3.

p(i)  0.1     0.2     0.4     0.3  
  s1     s2     s3     s4   E[dj]
d1 75614.4
d2 48655.8
d3 91786.3♥

The point to note then is that here, all the decisions are evaluated in accordance with the same assumed probability distribution (over the state space). But in the framework of the Maximin model, each individual decision is evaluated on the basis of the worst case (state) pertaining to this decision alone. This means that, in general, there is no guarantee that a single probability distribution over the state space will be able to simulate the Maximin Rule.

If you remain unconvinced, consider the following simple case: there are two states, say s1 and s2, and three decisions, call them d1 , d2, and d3. Suppose that the reward table is as follows:

  s1     s2  
d1 75
d2 48
d3 91

Appending the security levels column we have:

  s1     s2   SL(j)
d1 755♥
d2 484
d3 911

We conclude then that the optimal decision prescribed by Maximin is d1, yielding the maximum security level of AU$5.

In short, in this case the Maximin decision rule yields the following result:

Optimal decision: d1
Guaranteed reward: AU$5

Note that there is no degenerate distribution on S = {s1,s2} that can simulate this result under Bayes Rule. If Nature selects s1 with probability 1, then clearly decision d3 is better than decision d1. If Nature selects s2 with probability 1, then clearly d2 is better than d1.

The two decision tables are as follows:

p(i)  1     0
  s1     s2   E[dj]
d1 757
d2 484
d3 919♥
  
p(i)  0     1
  s1     s2   E[dj]
d1 755
d2 488♥
d3 911

In short, in this case there is no degenerate distribution on the state space that can force Bayes Rule to select d1.

The conclusion is therefore that the Maximin Rule is not an instance of Bayes Rule. This, however, does not mean that the two rules are rivals. Rather, what this means is that, they complement one another.

So the moral of the story is that both are essential. Indeed, don't leave home without either of them. It is best to have both. Even if you travel to Australia to take up the best job in the world!

That said, we must not neglect to mention in this context another pillar of classical decision theory:

Laplace's Principle of Insufficient Reason

This Principle — attributed to the French mathematician and astronomer Pierre-Simon, marquis de Laplace (1749 - 1827) — argues, by symmetry, that if there are n > 1 mutually exclusive and collectively exhaustive possibilities, then each possibility should be assigned the same probability (equal to 1/n).

To put it in more general terms, this Principle, which is also known as the Indifference Principle — so named by the British economist John Maynard Keynes, 1st Baron Keynes (1883 - 1946) — argues that under severe uncertainty all the states are equally likely. This means that often it is possible to postulate a uniform probability distribution function over the state space.

Of course, there are situations where this is impossible: for instance, if the state space is the real line, then we cannot formulate a uniform probability distribution function on the state space.

From the standpoint of Bayesian decision theory, the state-of-affairs captured by this principle can be viewed as the simplest non-informative prior.

The idea captured by this Principle is so fundamental that it is appealed to widely, often invoked implicitly, one might say automatically. For example, since the possible two outcomes of a (fair) coin toss are mutually exclusive, exhaustive, and interchangeable, we assign each of these outcomes a probability of 1/2.

Still, it is important to make sure that the Principle is applied correctly for otherwise one faces the prospect of embarrassing nonsensical results. The following example illustrates a typical misuse of the Principle (see WIKIPEDIA):

  • Suppose that we know that a cube inside a closed safe has a side length between 3 and 5 cm, but the true value of this length is subject to severe uncertainty.
  • We can safely conclude that the surface area of the cube is between 54 and 150 cm2.
  • Similarly, we can safely conclude that the volume of the cube is between 27 and 125 cm3.
  • Applying the Principle to each of these three intervals, we may (erroneously) conclude that the following three assertions are true:
    • The side of the cube is uniformly distributed on the interval [3,5].
    • The surface area of the cube is uniformly distributed on the interval [54,150].
    • The volume of the cube is uniformly distributed on [27,150].

To see clearly why this collective conclusion is wrong, let X denote the random variable representing the length of the side of the cube. Then, the surface area of the cube is represented by the random variable Y=6X2, and the volume of the cube is represented by the random variable Z=X3.

Since clearly these three random variables are not stochastically independent — in fact they are "highly" dependent on each other, so much so that the value of one uniquely determines the values of the other two. And since the relation between them is not linear, it follows that if one of these random variables is uniformly distributed, then clearly the other two are definitely not uniformly distributed.

In short, if we want to use the Principle to deal with the uncertainty associated with X, Y and Z, then we can apply it to one of these three variables and then use standard tools for the transformation of random variables to determine the distributions of the other two variables. The Principle does not tell us to which one of these three variables it should be applied.

The moral of the story narrated in this example is that one must make sure that the Principle is not being applied simultaneously to a number of random variables unless these variables are stochastically independent, or the dependence between them is linear.

The Info-Gap saga

And to illustrate the kind of trouble one might fall into if one is not at home with the modeling aspects of the Maximin paradigm, consider the following:

Assignment 1 :
Formulate a Maximin model for the following optimization problem:

This is the Info-Gap robustness model considered in Davidovitch and Ben-Haim (2008). The authors claim that this is not a Maximin/Minimax model. So your assignment is to prove them wrong.

And while you are at it, consider also the following:

Assignment 2 :
Explain in detail why it is absurd to consider the Maximin model
as a Maximin/Minimax representation of the optimization problem given in Assignment 1.

The Minimax model given in Assignment 2 is the model developed by Davidovitch and Ben-Haim (2008) as a "proof" that Info-Gap's robustness model is not a Maximin/Minimax model.

Solutions to these assignments and a critique of Davidovitch and Ben-Haim's (2008) Maximin/Minimax modeling effort are available in my paper

Anatomy of a Misguided Maximin formulation of Info-Gap's Robustness Model

In brief, the fatal error in Davidovitch and Ben-Haim's (2008) analysis is the uncalled for assumption that in Minimax/Maximin modeling the parameter "alpha" of the Info-Gap model must be treated as a fixed number. There are other serious Maximin/Minimax modeling errors in Davidovitch and Ben-Haim (2008).

In any case, the picture is this:

Similar erroneous Maximin/Minimax formulations of Info-Gap's robustness model can be found in the following publications of the Norges Bank. For your convenience I include the abstracts as well. The full papers are available on line:

And talking about Banks, how about the De Nederlandsche Bank?

" ... Info-gap robust-satisficing is motivated by the same perception of uncertainty which motivates the min-max class of strategies: lack of reliable probability distributions and the potential for severe and extreme events. We will see that the robust-satisficing decision will sometimes coincide with a min-max decision. On the other hand we will identify some fundamental distinctions between the min-max and the robust-satisficing strategies and we will see that they do not always lead to the same decision.

First of all, if a worst case or maximal uncertainty is unknown, then the min-max strategy cannot be implemented. That is, the min-max approach requires a specific piece of knowledge about the real world: "What is the greatest possible error of the analyst's model?". This is an ontological question: relating to the state of the real world. In contrast, the robust-satisficing strategy does not require knowledge of the greatest possible error of the analyst's model. The robust-satisficing strategy centers on the vulnerability of the analyst's knowledge by asking: "How wrong can the analyst be, and the decision still yields acceptable outcomes?" The answer to this question reveals nothing about how wrong the analyst in fact is. The answer to this question is the info-gap robustness function, while the true maximal error may or may not exceed the info-gap robust satisficing. This is an epistemic question, relating to the analyst's knowledge, positing nothing about how good that knowledge actually is. The epistemic question relates to the analyst's knowledge, while the ontological question relates to the relation between that knowledge and the state of the world. In summary, knowledge of a worst case is necessary for the min-max approach, but not necessary for the robust-satisficing approach.

The second consideration is that the min-max approaches depend on what tends to be the least reliable part of our knowledge about the uncertainty. Under Knightian uncertainty we do not know the probability distribution of the uncertain entities. We may be unsure what are typical occurrences, and the systematics of extreme events are even less clear. Nonetheless the min-max decision hinges on ameliorating what is supposed to be a worst case. This supposition may be substantially wrong, so the min-max strategy may be mis-directed.

A third point of comparison is that min-max aims to ameliorate a worst case, without worrying about whether an adequate or required outcome is achieved. This strategy is motivated by severe uncertainty which suggests that catastrophic outcomes are possible, in conjunction with a precautionary attitude which stresses preventing disaster. The robust-satisficing strategy acknowledges unbounded uncertainty, but also incorporates the outcome requirements of the analyst. The choice between the two strategies — min-max and robust-satisficing — hinges on the priorities and preferences of the analyst.

The fourth distinction between the min-max and robust-satisficing approaches is that they need not lead to the same decision, even starting with the same information. ..."

Confidence in Monetary Policy
Yakov Ben-Haim and Maria Demertzis
DNB Working Paper 192, 2008.
pp. 17-18

What a mess!

My point is then that the authors demonstrate a gross misapprehension of the modeling aspects of the Minimax/Maximin paradigm. As a result, they attribute to the Minimax paradigm "undesirable" properties that are in fact the properties of the misguided ad hoc instances of Wald's model that they constructed for this comparison.

Had they set out a proper (equivalent) Maximin formulation of their Info-Gap's robust-satisficing model, their analyses would have vanished into thin air.

Moreover, what is the point in reinventing the wheel and a faulty one at that?

This is another illustration of Info-Gap proponents discoursing at length on purported "similarities and differences" between Info-Gap's robustness model and the Maximin model while the clear proof that Info-Gap's robustness model is a simple instance of Wald's Maximin model is staring them in the face.

What a waste of time!

More details on the on-going Info-Gap/Maximin saga can be found in

WIKIPEDIA article on Info-Gap Decision Theory

The interesting material is in the associated WIKIPEDIA Discussion page.

If you wish to join the campaign or just be on my mailing list for this project, send me note.

I am now working on a short book entitled "Worst-Case Analysis for Decision Making Under Severe Uncertainty". I plan to post it here when it is ready. But do not hold your breadth, I have plenty of other more urgent tasks to complete.

However, if you are eager to read this material now, I suggest the following bits and pieces that eventually will be incorporated in the book.

Warning:

This is work in progress. I plan to update the files regularly. Make sure that you have the latest update.

Modern Alchemy, Freudian Slips, Quick-Fixes and Suchlike

If you are taking it for granted that the quest for a magic formula capable of transforming severe lack of knowledge / information into substantial knowledge was abandoned with the Enlightenment, I have news for you!

Apparently, against all scientific odds, Info-Gap scholars were successful in imputing likelihood to results generated by a non-probabilistic model that is completely devoid of any notion of likelihood!

Recall that Info-Gap decision theory prides itself on being non-probabilistic and likelihood-free. Yet, Info-gap scholars -- the Father of Info-Gap included -- now claim that Info-Gap's robustness model is capable of identifying decisions that are most likely to satisfy a given performance requirement.

Consider for instance the following quote from ACERA Endorsed Core Material (emphasis is mine):

Information-gap (henceforth termed 'info-gap') theory was invented to assist decision-making when there are substantial knowledge gaps and when probabilistic models of uncertainty are unreliable (Ben-Haim 2006). In general terms, info-gap theory seeks decisions that are most likely to achieve a minimally acceptable (satisfactory) outcome in the face of uncertainty, termed robust satisficing. It provides a platform for comprehensive sensitivity analysis relevant to a decision.

Burgman, Wintle, Thompson, Moilanen, Runge, and Ben-Haim (2008, p. 8).
Reconciling uncertain costs and benefits in Bayes nets for invasive species management
ACERA Endorsed Core Material: Final Report, Project 0601 - 0611.
(PDF file, Downloaded on March 21, 2009)

This is a major scientific breakthrough.

For, until now we have been warned repeatedly by Info-Gap scholars that no likelihood must be attributed to results generated by Info-Gap decision models. Indeed, we have been advised that this would be deceptive and even dangerous (emphasis is mine):

However, unlike in a probabilistic analysis, r has no connotation of likelihood. We have no rigorous basis for evaluating how likely failure may be; we simply lack the information, and to make a judgment would be deceptive and could be deceptive and dangerous. There may definitely be a likelihood of failure associated with any given radial tolerance. However, the available information does not allow one to assess this likelihood with any reasonable accuracy.

Ben-Haim (1994, p. 152)
Convex models of uncertainty: applications and implications
Erkenntnis, 4, 139-156.

This point is also made crystal clear in the second edition of the Info-Gap book (emphasis is mine):

In info-gap set models of uncertainty we concentrate on cluster-thinking rather than on recurrence or likelihood. Given a particular quantum of information, we ask: what is the cloud of possibilities consistent with this information? How does this cloud shrink, expand and shift as our information changes? What is the gap between what is known and what could be known. We have no recurrence information, and we can make no heuristic or lexical judgments of likelihood.

Ben-Haim (2006, p. 18)
Info-Gap Decision Theory: Decisions Under Severe uncertainty
Academic Press.

So the question is: have Info-gap scholars managed to accomplish a major feat in the area of decision-making under severe uncertainty?

Of course the answer is that this new claim (Burgman et al's 2008) is not due to a breakthrough in decision-making under severe uncertainty, but rather to a serious misrepresentation of Info-gap’s robustness model, culminating in a thoroughly incorrect representation of the results.

My view on this episode -- based as it is on numerous discussions with Info-Gap scholars over the past five years -- is that this new claim is simply -- but not surprisingly -- ... a Freudian slip.

The point is that -- see my FAQs about Info-Gap -- without imputing some sort of "likelihood" to Info-Gap's decision model, Info-Gap decision theory is, and cannot escape being, a voodoo decision theory.

So, all that this Freudian slip manages to do is to extend the already existing error -- an alternative that some Info-Gap scholars seem to prefer to an admission to a mistake.

It is interesting to note, though, that some Info-Gap scholars have taken note of my criticism of Info-Gap's robustness analysis to the effect that they now introduce an assumption that explicitly imputes "likelihood" to Info-Gap's uncertainty model. For instance, consider this (emphasis is mine):

An assumption remains that values of u become increasingly unlikely as they diverge from û.

Hall, J. and Harvey, H. (2009, p. 2)
Decision making under severe uncertainty for flood risk management: a case study of info-gap robustness analysis.
Eighth International Conference on Hydroinformatics
(January 12-16, 2009, Concepcion, Chile)
(PDF file)

Although this attempt at a quick-fix fails to fix the problem (see FAQ # 78), it does attest to a recognition that without such an assumption, conducting an analysis of the kind prescribed by Info-Gap's robustness model is utterly senseless.

One can only wonder then: how long will it take other Info-Gap scholars such as Burgman et al (2008) to reach this unavoidable conclusion?

Only time will tell (March 21, 2009).

The Black Swan

 

Only time will tell what impact (if any) Nassim Taleb's recent popular and controversial book The Black Swan: The Impact of the Highly Improbable will have on the field of decision-making under severe uncertainty.

I, for one, hope that the issues raised in this book and in its predecessor, Fooled by Randomness: The Hidden Role of Chance in the Markets and in Life, will be instrumental in helping decision-makers to identify voodoo decision theories -- such as Info-Gap decision theory -- that promise robust decisions under severe uncertainty.

I fear though -- in view of my experience of the past 40 years - that the danger is that the huge success of the Black Swan will inspire a new wave of voodoo decision theories, purportedly capable of ... "domesticating" black swans and preempting the discovery of ... purple swans!

We shall have to wait and see.

For those who have "been in hiding" I should note that Taleb has become quite a celebrity. According to the Prudent Investor Newsletters (Tuesday, June 3, 2008):

  • Mr. Taleb charges about $60,000 per speaking engagement and does about 30 presentations a year to "to bankers, economists, traders, even to Nasa, the US Fire Administration and the Department of Homeland Security" according to Timesonline’s Bryan Appleyard.

  • He recently got $4million as advance payment for his next much awaited book.

  • Earned $35-$40 MILLION on a huge Black Swan event-on the biggest stockmarket crash in modern history-Black Monday, October 19,1987.

So, if you haven’t heard him in person you can easily find on the WWW numerous videos of his interviews.

Here is a link to a very short (2:45 min) clip, recorded by Taleb himself, apparently at Heathrow Airport, of 10 tips on how to deal with Black Swans, and life in general.

  1. Scepticism is effortful and costly. It is better to be sceptical about matters of large consequences, and be imperfect, foolish and human in the small and the aesthetic.

  2. Go to parties. You can't even start to know what you may find on the envelope of serendipity. If you suffer from agoraphobia, send colleagues.

  3. It's not a good idea to take a forecast from someone wearing a tie. If possible, tease people who take themselves and their knowledge too seriously.

  4. Wear your best for your execution and stand dignified. Your last recourse against randomness is how you act -- if you can't control outcomes, you can control the elegance of your behaviour. You will always have the last word.

  5. Don't disturb complicated systems that have been around for a very long time. We don't understand their logic. Don't pollute the planet. Leave it the way we found it, regardless of scientific 'evidence'.

  6. Learn to fail with pride -- and do so fast and cleanly. Maximise trial and error -- by mastering the error part.

  7. Avoid losers. If you hear someone use the words 'impossible', 'never', 'too difficult' too often, drop him or her from your social network. Never take 'no' for an answer (conversely, take most 'yeses' as 'most probably').

  8. Don't read newspapers for the news (just for the gossip and, of course, profiles of authors). The best filter to know if the news matters is if you hear it in cafes, restaurants ... or (again) parties.

  9. Hard work will get you a professorship or a BMW. You need both work and luck for a Booker, a Nobel or a private jet.

  10. Answer e-mails from junior people before more senior ones. Junior people have further to go and tend to remember who slighted them.

It is interesting to juxtapose Prof. Taleb’s thesis in The Black Swan that severe uncertainty makes (reliable) prediction in the Socio/economic/political spheres impossible, with the polar position taken by his colleague, Prof. Bruce Bueno de Mesquita, who actually specializes in predicting the future.

New Nostradamuses

One need hardly point out that not only professionals who make "decision under uncertainty" their metier, but also the proverbial "man in the street", take it for granted that the ability to accurately predict future events is one of the most onerous challenges facing humankind — especially persons in authority, persons responsible for the management of business or economic organizations etc.

But apparently no longer.

For, according to Good Magazine, predicting future events — at least in the area of international conflicts — is now possible thanks to the efforts of the New Nostradamus: Prof. Bruce Bueno de Mesquita, a political science professor at New York University and Senior Fellow at the Hoover Institution.

The claim is that this distinguished political scientist can actually predict the future — more specifically the outcome of any international conflcit!

And this he does not with the aid of the age old Crystal Ball, but through the use of a scientific method that, apparently, is grounded in a branch of applied mathematics called Game Theory.

According to GoodReads.com,

" ... Bruce Bueno de Mesquita is a political scientist, professor at New York University, and senior fellow at the Hoover Institution. He specializes in international relations, foreign policy, and nation building. He is also one of the authors of the selectorate theory.

He has founded a company, Mesquita & Roundell, that specializes in making political and foreign-policy forecasts using a computer model based on game theory and rational choice theory. He is also the director of New York University's Alexander Hamilton Center for Political Economy.

He was featured as the primary subject in the documentary on the History Channel in December 2008. The show, titled Next Nostradamus, details how the scientist is using computer algorithms to predict future world events ..."

Here is an interview with Prof. Bueno de Mesquita (with Riz Khan - The art and science of prediction - 09 Jan 08):


And here is a 20-minute lecture on the ... future of Iran (TED, February 2009):

Apparently, all you need to accomplish this is a computer, expert-knowledge on Iran, and game theory!

Some of the predictions attributed to Prof. Bueno de Mesquita are:

  1. The second Palestinian Intifada and the death of the Mideast peace process, two years before this came to pass.

  2. The succession of the Russian leader Leonid Brezhnev by Yuri Andropov, who at the time was not even considered a contender.

  3. The voting out of office of Daniel Ortega and the Sandanistas in Nicaragua, two years before this happened.

  4. The harsh crack down on dissidents by China's hardliners four months before the Tiananmen Square incident.

  5. France's hairs-breadth passage of the European Union's Maastricht Treaty.

  6. The exact implementation of the 1998 Good Friday Agreement between Britain and the IRA.

  7. China's reclaiming of Hong Kong and the exact manner the handover would take place, 12 years before it happened.

Impressive, isn't it!

As might be expected, these and similar claims by Prof. Bueno de Mesquita have sparked a vigorous debate not only in the professional journals but also on the WWW. Interested readers can consult this material to see for themselves, whether Bueno de Mesquita's claims attest to a major scientific breakthrough or ... voodoo mathematics.

Also, in addition to consulting this material you may want to have a look at a short video clip by Matt Brawn (right) which, he compiled in response to a short note entitled This man can actually predict the future!.

Of particular interest is, of course, the "success" rate of the Prof. Bueno de Mesquita's predictions: over 90% — yes over ninty percent!

Here is Trevor Black's common sense reaction to this claim:

I am a little skeptical about anyone who claims to have a 90% success rate. I just don't buy it. Especially when they say that they can explain away a lot of the other 10%.

If you come to me and tell me you have a model that gets it right 60% or 70% of the time, I may listen. Skeptically, but I will listen. 90% and I start to smell something.

All I wish to add here is that Prof. Bueno de Mesquita (left) makes his predictions under conditions of "severe uncertainty" which of course render them hugely vulnerable to what Prof. Naseem Taleb (right) dubs the Black Swan phenomenon.

Hence, the very proposition that such predictions can be made at all, let alone be reliable, is diametrically opposed to Nassim Taleb's categorical rejection of any such position. For his thesis is that Black Swans are totally outside the purview of mathematical treatment, especially by models that are based on expected utility theory and rational choice theory.

Interesting, though, this is precisely the stuff that Prof. Bueno de Mesquita's method is made of: expected utility theory and rational choice theory!

Even more interesting is the fact that Nassim Taleb (right) and Bueno de Mesquita (left) are staff members of the same academic institution, namely New York University. So, all that's left to say is: Go figure!

As indicated above, the debate over Bueno de Mesquita's theories is not new. It has been ongoing, in the relevant academic literature, at least since the publication of his book The War Trap (1981).

Note, therefore, that Bueno de Mesquita's work has attracted a considerable amount of criticism. For an idea of the kind of criticism sparked by his work, take a look at the quotes I provide from articles that are critical of Bueno de Mesquita theories.

Of course, there are other New Nostradamuses around.

According to the Associated Press, the latest (Mar 4, 4:39 AM EST) news from Russia about the future of the USA is that

" ... President Barack Obama will order martial law this year, the U.S. will split into six rump-states before 2011, and Russia and China will become the backbones of a new world order ..."

Apparently this prediction was made by Igor Panarin (right), Dean of the Russian Foreign Ministry diplomatic academy and a regular on Russia's state-controlled TV channels (see full AP news report).

Regarding the future of Russia,

"You don't sound too hopeful".
"Hopeful? Please, I am Russian. I live in a land of mad hopes, long queues, lies and humiliations. They say about Russia we never had a happy present, only a cruel past and a quite amazing future ..."
Malcolm Bradbury
To the Hermitage (2000, p. 347)

We should therefore be reminded of J K Galbraith's (1908-2006) poignant observation:

There are two classes of forecasters: those who don't know and those who don't know they don't know.

And in the same vein,

The future is just what we invent in the present to put an order over the past.

Malcolm Bradbury
Doctor Criminale (1992, p. 328)

So, we shall have to wait and see.

And how about this more recent piece by Heath Gilmore and Brian Robins in the Sydney Morning Herald (March 27, 2009):

"... COUPLES wondering if the love will last could find out if theirs is a match made in heaven by subjecting themselves to a mathematical test.

A professor at Oxford University and his team have perfected a model whereby they can calculate whether the relationship will succeed.

In a study of 700 couples, Professor James Murray, a maths expert, predicted the divorce rate with 94 per cent accuracy.

His calculations were based on 15-minute conversations between couples who were asked to sit opposite each other in a room on their own and talk about a contentious issue, such as money, sex or relations with their in-laws.

Professor Murray and his colleagues recorded the conversations and awarded each husband and wife positive or negative points depending on what was said. ..."

Such interviews should perhaps be made mandatory for all couples registering their marriage.

More details on the mathematics of marriage can be found in The Mathematics of Marriage: Dynamic Nonlinear Models by J.M. Gottman, J.D. Murray, C. Swanson, R. Tyson, and K.R. Swanson (MIT Press, Cambridge, MA, 2002.)


On a more positive note, though, here is an online Oracle from Melbourne (Australia: the land of the real Black Swan!).

You may wish to consult this friendly 24/7 facility about important "Yes/No" questions that you no doubt have about the future.

Enter your "Yes/No" question:
    

 
 

More on this and related topics can be found in the pages of the Worst-Case Analysis / Maximin Campaign, Severe Uncertainty, and the Info-Gap Campaign.

Your comments, please!

I am most interested in feedback on the contents of this web page. I promise to respond to comments within ... a week or so!

This feature is experimental, so I am not sure how robust it is. I checked a number of platforms and they seem to work fine. But, if it does not work well for you, .... let me know!

Recent Articles, Working Papers, Notes

Also, see my complete list of articles
  • Sniedovich, M. (2010) A bird's view of Info-Gap decision theory, Journal of Risk Finance, 11(3), 268-283.

  • Sniedovich, M. (2010) Dynamic programming: introductory concepts, in Wiley Encyclopedia of Operations Research and Management Science (EORMS), Wiley.

  • Caserta, M., Voss, S., Sniedovich, M. (2010) Applying the corridor method to a blocks relocation problem, OR Spectrum, in press.

  • Sniedovich M. (2009) Modeling of robustness against severe uncertainty, pp. 33- 42, Proceedings of the 10th International Symposium on Operational Research, SOR'09, Nova Gorica, Slovenia, September 23-25, 2009.

  • Sniedovich M. (2009) A Critique of Info-Gap Robustness Model. In: Martorell et al. (eds), Safety, Reliability and Risk Analysis: Theory, Methods and Applications, pp. 2071-2079, Taylor and Francis Group, London.
    • .
  • Sniedovich M. (2009) A Classical Decision Theoretic Perspective on Worst-Case Analysis, Working Paper No. MS-03-09, Department of Mathematics and Statistics, The University of Melbourne, (PDF File)

  • Caserta, M., Voss, S., Sniedovich, M. (2008) The corridor method - A general solution concept with application to the blocks relocation problem. In: A. Bruzzone, F. Longo, Y. Merkuriev, G. Mirabelli and M.A. Piera (eds.), 11th International Workshop on Harbour, Maritime and Multimodal Logistics Modeling and Simulation, DIPTEM, Genova, 89-94.

  • Sniedovich, M. (2008) FAQS about Info-Gap Decision Theory, Working Paper No. MS-12-08, Department of Mathematics and Statistics, The University of Melbourne, (PDF File)

  • Sniedovich, M. (2008) A Call for the Reassessment of the Use and Promotion of Info-Gap Decision Theory in Australia (PDF File)

  • Sniedovich, M. (2008) Info-Gap decision theory and the small applied world of environmental decision-making, Working Paper No. MS-11-08
    This is a response to comments made by Mark Burgman on my criticism of Info-Gap (PDF file )

  • Sniedovich, M. (2008) A call for the reassessment of Info-Gap decision theory, Decision Point, 24, 10.

  • Sniedovich, M. (2008) From Shakespeare to Wald: modeling wors-case analysis in the face of severe uncertainty, Decision Point, 22, 8-9.

  • Sniedovich, M. (2008) Wald's Maximin model: a treasure in disguise!, Journal of Risk Finance, 9(3), 287-291.

  • Sniedovich, M. (2008) Anatomy of a Misguided Maximin formulation of Info-Gap's Robustness Model (PDF File)
    In this paper I explain, again, the misconceptions that Info-Gap proponents seem to have regarding the relationship between Info-Gap's robustness model and Wald's Maximin model.

  • Sniedovich. M. (2008) The Mighty Maximin! (PDF File)
    This paper is dedicated to the modeling aspects of Maximin and robust optimization.

  • Sniedovich, M. (2007) The art and science of modeling decision-making under severe uncertainty, Decision Making in Manufacturing and Services, 1-2, 111-136. (PDF File) .

  • Sniedovich, M. (2007) Crystal-Clear Answers to Two FAQs about Info-Gap (PDF File)
    In this paper I examine the two fundamental flaws in Info-Gap decision theory, and the flawed attempts to shrug off my criticism of Info-Gap decision theory.

  • My reply (PDF File) to Ben-Haim's response to one of my papers. (April 22, 2007)

    This is an exciting development!

    • Ben-Haim's response confirms my assessment of Info-Gap. It is clear that Info-Gap is fundamentally flawed and therefore unsuitable for decision-making under severe uncertainty.

    • Ben-Haim is not familiar with the fundamental concept point estimate. He does not realize that a function can be a point estimate of another function.

      So when you read my papers make sure that you do not misinterpret the notion point estimate. The phrase "A is a point estimate of B" simply means that A is an element of the same topological space that B belongs to. Thus, if B is say a probability density function and A is a point estimate of B, then A is a probability density function belonging to the same (assumed) set (family) of probability density functions.

      Ben-Haim mistakenly assumes that a point estimate is a point in a Euclidean space and therefore a point estimate cannot be say a function. This is incredible!


  • A formal proof that Info-Gap is Wald's Maximin Principle in disguise. (December 31, 2006)
    This is a very short article entitled Eureka! Info-Gap is Worst Case (maximin) in Disguise! (PDF File)
    It shows that Info-Gap is not a new theory but rather a simple instance of Wald's famous Maximin Principle dating back to 1945, which in turn goes back to von Neumann's work on Maximin problems in the context of Game Theory (1928).

  • A proof that Info-Gap's uncertainty model is fundamentally flawed. (December 31, 2006)
    This is a very short article entitled The Fundamental Flaw in Info-Gap's Uncertainty Model (PDF File) .
    It shows that because Info-Gap deploys a single point estimate under severe uncertainty, there is no reason to believe that the solutions it generates are likely to be robust.

  • A math-free explanation of the flaw in Info-Gap. ( December 31, 2006)
    This is a very short article entitled The GAP in Info-Gap (PDF File) .
    It is a math-free version of the paper above. Read it if you are allergic to math.

  • A long essay entitled What's Wrong with Info-Gap? An Operations Research Perspective (PDF File) (December 31, 2006).
    This is a paper that I presented at the ASOR Recent Advances in Operations Research (PDF File) mini-conference (December 1, 2006, Melbourne, Australia).

Recent Lectures, Seminars, Presentations

If your organization is promoting Info-Gap, I suggest that you invite me for a seminar at your place. I promise to deliver a lively, informative, entertaining and convincing presentation explaining why it is not a good idea to use — let alone promote — Info-Gap as a decision-making tool.

Here is a list of relevant lectures/seminars on this topic that I gave in the last two years.

Disclaimer: This page, its contents and style, are the responsibility of the author (Moshe Sniedovich) and do not represent the views, policies or opinions of the organizations he is associated/affiliated with.


Last modified: Sunday, 18-Apr-2010 00:47:30 MST