The Mathematics of Social Ethics: Reflexive Theory and 'The Torturer's Dilemma'

I recently read a fascinating opinion piece by Jonathan David Farley, a mathematician working on developing mathematical models of terrorism -- San Francisco Chronicle: The torturer's dilemma: the math on fire with fire. The conclusion of the op-ed was that, if the United States adopted torture as an acceptable tactic in the war on terrorism, based on mathematical and logical criteria (as opposed to purely ethical or moral considerations), America will become the kind of strife-ridden society -- e.g., Saudi Arabia, the Soviet Union, etc. -- that the U.S. has traditionally been at odds with. Prof. Farley relies on a somewhat obscure branch of applied mathematics (or, perhaps it's better to call it mathematical psychology) called "reflexive theory" to make his points.

Applying reflexive theory to social ethics (or morality) leads to the conclusion that a society that is willing to compromise on issues of good and evil is a society where "individuals would often seek the path of confrontation with each other." This conclusion is somewhat counterintuitive because many people tend to assume that the willingness to compromise on principle is an almost necessary condition to cooperation, and the lack of compromise is what leads to conflict.

Unfortunately -- in an otherwise excellent article -- Prof. Farely doesn't directly address the last point. Let me try to explain how reflexive theory could work in spite of the objection just raised by using an example Prof. Farley uses in his piece. Prof. Farley uses the example of two roads to illustrate the conclusions of reflexive theory. In the first road, drivers are more highly esteemed by society if they are willing to yield to other drivers on the road. On the second road, drivers "lose face when they yield." Not surprisingly, Prof. Farley argues that it "is clear that traffic will move faster on the first road than on the second."

On the surface, the example of the two roads may seem more consistent with the intuitive objection to reflexive theory rather than bolstering the theory. However, deeper thought reveals that the two roads example is consistent with reflexive theory. In the case of the first road, the drivers are holding fast to the principle that courtesy and politeness on the road is a societal 'good' that should not be compromised. This unwillingness to compromise on a moral/ethical principle (paradoxically to those with more cynical intuitions) leads to the emergence of cooperative behavior. In the case of the second road, the willingness to compromise on a moral/ethical principle for self-serving purposes leads to conflict and, thus, gridlock.

All of this leads to Prof. Farley's stark conclusion -- again, based on mathematical logic:

What does this mean? If Americans begin to accept the use of torture, American society might turn into a society of individuals in conflict.

....

It can be argued that repressive states like Saudi Arabia, which bred most of the Sept. 11 hijackers, are on the second road. If the United States moved to accept torture, it could veer toward the second road, too -- the road of the Soviet Union.

And we know where that road ends. The Soviet Union no longer exists.

In other words, America would be heading down a dark and perilous road -- both morally and mathematically -- if the U.S. accepts torture as a legitimate tactic in the war on terror.

By the way, reflexive theory was developed by Vladimir Lefebvre, a mathematical psychologist formerly of the Soviet Union but now at University of California, Irvine. One of the advantages that reflexive theory has over its better known alternative, game theory (at least the traditional approaches to game theory), is that reflexive theory, from its inception, attempted to take into account behavioral and moral considerations into its calculus. Traditional approaches to game theory (unmodified by behavioral economics) made little or no attempt to take those important considerations into account.

VIX and the Recent Market Downturn

When I started this blog, I vowed to myself that I would try to avoid writing about the daily fluctuations of the markets. Why? There are a variety of good reasons for trying not to be too focused on day-to-day market movements (trying to avoid the unsatisfactory mumbo-jumbo that try to 'explain' the ups and downs of the markets on any particular day that one sees on TV or in the newspapers, trying to stick to the higher purposes of this blog, etc.). I believe that I have done a pretty good job of trying to be current but not too current with the Econophysics Blog.

However, I felt I needed to make an exception in this case. Recently, there have been some down movements in world equity markets. By now, this downturn has even spread to the commodity markets (something that goes against most people's intuitions about correlations between the different markets). There is a general unease in the air.

This reminded me about what I had been reading and seeing about a month or so ago. Back in those 'good old days' (which really wasn't so long ago if you think about it), there was a lot of talk about how low the VIX was (around 11 or so ... it abruptly jumped higher since then). VIX is the acronym for the Chicago Board Options Exchange's Volatility Index. The VIX is essentially the measure of the 'implied volatility' (as define by the Black-Scholes options pricing formula) of the options on the S&P 500 index. The VIX is widely considered to be (for very valid reasons) the best gauge of the market's consensus guess on the future direction of volatility (hence 'risk'); based on rational expectations reasoning, this would make VIX the best measure of future volatility (at least the best measure that is widely accepted and available).

It should be obvious by now that the VIX is a flawed measure of future risk since the VIX has failed to anticipate the current wide-scale downturn in the financial markets (across geographic and asset-class boundaries). Only a few short weeks ago, the VIX was relatively low and was 'predicting' tranquil markets. Now serious people are beginning to think we might be in crises mode (although I'm not sure if I'd go that far at this point). Of course, the VIX has abruptly jumped to higher levels since then, but this hardly justifies putting much faith in it as an accurate predictor of future risk ... it's like closing the barn doors after the horses have ran away.

What's been happening lately should be considered as more evidence of the wild and jumpy nature of financial risk.

-- Addendum --

Since writing the above posting, I found one of the articles I had read talking about the recently low levels of the VIX (from the May 11, 2006 -- almost exactly a month ago -- issue of The Economist): The fear gauge (this article actually did an excellent job of predicting what has transpired over the last month or so). According to that article:

For instance, on only eight days this year has the S&P 500 index moved by more than 1%, compared with 12 times a month in the wake of the dotcom bust, and 11 times a month during the great sell-off in the 1970s ... Meanwhile, the Chicago Board Options Exchange's Volatility Index (VIX), which measures the share movements implied in stock index options, is at record lows. It predicts that the S&P 500 will move by less than 1%, up or down, over the next month.

To put it mildly, over the last month, the markets have moved down by more than 1% since May 11th of this year!

The article goes onto deliver what has turned out to be some perceptive Jeremiads:

To stockmarket bears, however, low volatility is a warning sign: too much stability may, paradoxically, be destabilising. Ed Easterling of Crestmont Holdings, a Dallas investment firm, calls it “the calm before the storm”. He worries that speculators may have become overly complacent. “The current state of volatility is an indicator of potentially sharp stockmarket decline,” he says.

Episodes of extremely low volatility rarely last long, says Mr Easterling, and are usually followed by periods of exceptionally jumpy prices. There is, indeed, evidence that an increase in volatility often means a sell-off in markets.

As it turns out, the contrarians -- rather than VIX -- turned out to be right.

Predictive Markets for the World Cup?

I just read the latest Buttonwood column on The Economist magazine's website -- Trading World Cup volatility (June 6, 2006). The article discusses various electronic exchanges that mimic stock and derivatives exchanges. The key difference is that the 'shares' are of the 32 national teams playing in this year's World Cup in Germany.

The pricing of these 'shares' on almost all of the various electronic markets are based on the probabilities of a particular team winning. In theory, one could use the information gleaned from these sites in order to make predictions on how the various soccer/football teams will do during the World Cup. At the very least, one can study what type of information affects prices/subjective probabilities and where the consensus of opinion seem to be headed toward.

The World Cup market is one example of a predictive market. Predictive (or prediction or information) markets have been set up by economists (and others) in order to use market mechanisms to determine the most likely outcomes of political contests, sporting events, and other non-economic events. For the most part, these markets have done a pretty good job (at least better than the alternatives) at making predictions. A good overview of prediction/predictive markets can be found in the following working paper on Justin Wolfer's website: http://bpp.wharton.upenn.edu/jwolfers/Papers/PredictionMarkets(Palgrave).pdf