jump to navigation

Varoufakis on game theoretic analysis of asymmetric expectations among the powerful & the powerless March 22, 2014

Posted by larry (Hobbes) in economics, Game Theory, Psychology, social justice.
add a comment

This is for those who have some experience, knowledge or interest in game theoretic analyses of social interaction. Varoufakis is a political economist who is intimately familiar with game theory, having written a book about it with Hargreaves Heap some years ago. Nevertheless, economically, this research falls within the domain of what has become known as microeconomics, as distinct from macroeconomics. But this research focuses on social group expectations, not the processes of the economic system as a whole.

There are links in the paper to more detailed and mathematical discussions of the issues. I don’t think that the PowerPoint slides are self-explanatory, as they assume more understanding of the mathematics underlying game theory than is perhaps possessed by the general population. But the conclusions Varoufakis reaches require no math to understand. They are:

Varoufakis-ConclusionsEvolutionaryGameTheoryExperiments-Spr2014

There are a number of books and articles dealing with applications of game theory to biological situations, one famous one by the late John Maynard Smith, Evolution and the Theory of Games, mentioned by Varoufakis. I think I should point out that Richard Lewontin, the population geneticist, has written that he thought that the appropriation of game theory to evolutionary contexts altered the character of the theory to such an extent that it ought to be called something other than “evolutionary game theory”. History and social convention have made the decision for him. The appellation has stuck.

To be fair to Varoufakis, he doesn’t think that these results are extraordinarily original, though they would seem to be unknown to mainstream economists, whose theoretical paradigm Varoufakis is concerned to attack. And he feels that one of the best ways of doing this is from within, as it were, a tactic utilized successfully by logicians for centuries to attack positions they don’t (didn’t) like. (Have a look at the way Socrates, as set out by Plato, operates.)

Here is the link to a non-technical discussion of this topic, including links mentioned.

http://yanisvaroufakis.eu/2014/03/21/how-do-the-powerful-get-the-idea-that-they-deserve-more-lessons-from-a-laboratory/

Reprise

Getting back to my simple example of being able to breathe, we know that the presence of oxygen is a necessary though not sufficient for breathing. We also know that the presence of CO2 is a necessary but not sufficient condition for breathing.

So, here we have two necessary conditions for breathing. For B for breathing, O for oxygen, and C for CO2, we have (if B, then O) and (if B, then C). Since we know that they must occur together for a person being able to breathe, we have (if B, then C & O).

Realizing that the system under discussion is more complex than this discussion, can we nevertheless go on to contend that the presence of oxygen and CO2 are individually necessary and jointly sufficient for breathing? That is, that B iff C & O? How should we then interpret this equivalence? As a kind of “law of breathing”, a kind of scientific regularity, or as a definition of the conditions of being able to breathe?

This question may seem to be a triviality, but it arises in the discipline of macroeconomics all the time. When is a statement to be interpreted as a substantive assertion that has a truth-value or as a definition of terms, which has no truth-value but is only useful or not? Too many economic discussions are not at all clear about this issue. And it can make a difference to how you treat what they are saying.

Necessary & Sufficient conditions: A Medical Example March 22, 2014

Posted by larry (Hobbes) in Logic, Medicine, Science.
add a comment

A sufficient condition A for B is one where A being the case is sufficient for bringing about B.

A necessary condition B for A is one where if B is not the case, then A won’t be either.

Logically it looks like this. A is sufficient for B: if A, then B.

B is necessary for A: if not-B, then not-A.

Ex. of a necessary condition: oxygen (B) is necessary for being able to breathe (A). Therefore, if not-B, then not-A.

It is easier to come up with necessary conditions than it is sufficient conditions. For instance, what is sufficient for being able to breathe?

A set of necessary and sufficient conditions for A is often considered to be equivalent to, or for, A.

A slightly more realistic and complicated way of expressing this set of relationships is this. A is sufficient for D and B is necessary for D. This translates to (if A, then D) and (if D, then B). The contrapositive of each yields (if not-D, then not A) and (if not-B, then not-D). It is relatively clear, I think, that A and B each have a distinct relationship to D (I am ignoring the issue of transitivity illustrated in the example.). The potential complexity of this relationship is borne out is the following medical example from research into the dementias.

Here is a quote from a medical investigation of causes of Alzheimer’s and other dementias (from Daily Kos).

“Researchers have found that a protein active during fetal brain development, called REST, switches back on later in life to protect aging neurons from various stresses, including the toxic effects of abnormal proteins. But in those with Alzheimer’s and mild cognitive impairment, the protein — RE1-Silencing Transcription factor — is absent from key brain regions.”

“Our work raises the possibility that the abnormal protein aggregates associated with Alzheimer’s and other neurodegenerative diseases may not be sufficient to cause dementia; you may also need a failure of the brain’s stress response system,” said Bruce Yankner, Harvard Medical School professor of genetics and leader of the study, in a release.”

While the situation is more complicated than the simple example I gave initially, the logic is the same.

From the quote, we have: protein aggregate A; failure brain stress response system B; absence of RE1 (=R); dementia D.

Second paragraph of the above quote may be saying that A & B is sufficient for D.

But from the first paragraph, we also have absence of R (RE1) as a necessary condition for D. I.e., if D, then not-R (absence of RE1).

So, A&B is sufficient for D, hence (if A&B, then D). But, not-R is necessary for D. Or, equivalently, (if R, then not-D). I.e., R is sufficient for not-D.

Similarly as in the simple example: A, B, and R are related in a complex way to D, a relationship that is not entirely spelled out in the quote.

We are, therefore, left with an important question: what is status of A & B with respect to R? Is R a component of either A or B? The quote doesn’t link all these factors together. While this may be obvious from the quote, the logical situation underlying this hiatus may not be clear. Hopefully, it now is and also clearer what additional relationships need to be explored in order to lead to better understanding of the dementias, particularly Alzheimer’s, and thereby better control of their onset and progression if not complete defeat.

Bayes’ theorem: a comment on a comment March 10, 2014

Posted by larry (Hobbes) in Bayes' theorem, Logic, Philosophy, Statistics.
add a comment

Assume the standard axioms of set theory, say, the Zermelo-Fraenkel axioms.

Then provide a definition of conditional probability:

1. BayesEq1, which yields the identity

1a. BayesEq2, via simple algebraic cross multiplication.

Because set intersection is commutative, you can have this:

1b. BayesEq3, which equals

2. BayesEq4.

What we have here is a complex, contextual definition relating a term, P, from probability theory with a newly introduced stroke operator, |, read as “given”, so the locution becomes, for instance, the probability, P, of A given B. Effectively, the definition is a contextual definition of the stroke operator, |, “given”.

Although set intersection (equivalent in this context to conjunction) is commutative, conditional probability isn’t, which is due to the asymmetric character of the stroke operator, |. This means that, in general, P(A|B) ≠ P(B|A). If we consider the example of Data vs. Hypothesis, we can see that in general, for A = Hypothesis and B = Data, that P(Hypothesis|Data) ≠ P(Data|Hypothesis).

Now, from the definition of “conditional probability” and the standard axioms of set theory which have already been implicitly used, we obtained Bayes’ theorem trivially, mathematically speaking, via a couple of simple substitutions.

BayesEq5.

Or the Bayes-Laplace theorem, since Laplace discovered the rule independently. However, according to Stigler’s rule of eponymy in mathematics, theorems are invariably attributed to the wrong person (Stigler, “Who Discovered Bayes’ Theorem?”. In Stigler, Statistics on the Table, 1999).

Now, since we have seen that Bayes’ theorem follows from the axioms of set theory plus the definition of “conditional probability”, the following comments from a recent tutorial text on Bayes’ theorem can only be interpreted as being odd. The following quote is from James V. Stones’ Bayes’ Rule: A Tutorial Introduction to Bayesian Analysis (3rd printing, Jan. 2014).

If we had to establish the rules for calculating with probabilities, we would insist that the result of such calculations must tally with our everyday experience of the physical world, just as surely as we would insist that 1+1 = 2. Indeed, if we insist that probabilities must be combined with each other in accordance with certain common sense principles then Cox (1946) showed that this leads to a unique set of rules, a set which includes Bayes’ rule, which also appears as part of Kolmogorov’s (1933) (arguably, more rigorous) theory of probability (Stone: pp. 2-3).

Bayes’ theorem does not form part of Kolmogorov’s set of axioms. Strictly speaking, Bayes’ rule must be viewed as a logical consequence of the axioms of set theory, the Kolmogorov axioms of probability, and the definition of “conditional probability”.

Whether Kolmogorov’s axioms for probability tally with our experience of the real world is another question. The axioms are sometimes used as indications of non-rational thought processes in certain psychological experiments, such as the Linda experiment by Tversky and Kahneman.  (For an alternative interpretation of this experiment that brings into question the assumption that people either do or should reason according to a simple application of the Kolmogorov axioms, cf. Luc Bovens & Stephan Hartmann, Bayesian Epistemology, 2003: 85-88).

A matter of interpretation

In the discussion above, the particular set theory and the Kolmogorov axioms mentioned and used were interpreted  via the first-order extensional predicate calculus. This means that both theories can be viewed as not involving intensional contexts such as beliefs. The probability axioms in particular were understood by Kolomogorov and others using them as relating to objective frequencies and applicable to the real world, not to beliefs we might have about the world. For instance, an unbiased coin and die, in the ideal case admittedly, are considered to have a .5 and 1/6 (or .1666) probability for the side of the coin and a side of a six-sided die, respectively, on each flip or throw of the object in question. In these two particular cases, it is only via behavior observed over a long period of time that can produce data that will show whether in fact our assumption that the coin and the die are unbiased is true or not.

Why does this matter. Simply because Bayes’ theorem has been interpreted in two distinct ways – as a descriptively objective statement about the character of the world and as a subjective statement about a users’ beliefs about the state of the world. The derivation above derives from two theories that are considered to be non-subjective in character. One can then reasonably ask: where does the subjective interpretation of Bayes’ theorem come from? Two answers suggest themselves, though these are not the only ones. One is that Bayes’ theorem is arrived at via a different derivation than the one I considered, relying, say, on a different notion of probability than that of Kolmogorov’s. The other is that Bayesian subjectivity is introduced by means of the stroke (or ‘given’) operator, |.

Personally, I see nothing subjective about statements concerning the probability of obtaining a H or a T on the flip of a coin as being .5 or that of obtaining one particular side of a 6-sided die being .166. These probabilities are about the objects themselves, and not about our beliefs concerning them. Of course, this leaves open the possibility of alternative interpretations of probabilities in other contexts, say the probability of guilt or non-guilt in a jury trial. Whether the notions of probability involving coins or dice are the same as those involving situations such as jury trials is a matter for further debate.

%d bloggers like this: