Archive for December, 2008

Risk and Ignorance, Keynes

Tuesday, December 30th, 2008

Financial Markets, Money and the Real World By Paul Davidson, page 46, section 3.4:

The two fundamentally different concepts of uncertainty in economics are the classical theory concept where an uncertain future is actuarially certain and the Keynes concept where the future is unknown and unknowable.

Chapter 3: Uncertainty and reality in economic models

Bronchitis Urgent Care in New York

Saturday, December 13th, 2008

I just had my first encounter with the US medical system outside of university. The place was part of the Beth Israel Medical Group, called Docs Physicians, and I went as a walk-in to the branch at 55 E 34th. They opened at 9am on saturday morning, so I arrived a bit early and had a coffee and bagel next door before I went in. There were two patients before me, and the wait was short (30 min) even though there was only one attending physician. I told the doctor that the last time I had suffered a prolonged cough from a cold, I had been prescribed an inhaler, and may have accelerated the diagnosis a bit that way. Docter was Keith Uleis, he ended up prescribing an antibiotic (Azithromycin), an inhaler (Albuterol) and a steroid (Prednisone) for inflammation control, after putting me on a nebulizer for 15 min to gauge the possible asthmatic effects. The charge for the consultation was $225, and filling my prescription next door at Pasteur Pharmacy cost $10 (steroid) + $40 (antibiotic) + $50 (inhaler) = $100.

I can’t believe it took me this long to figure out that I needed to go to urgent care – am seriously wondering about the purpose of having a primary care physician, which I spent part of yesterday trying to pick out. The differences: one, familiarity with medical history;  two,  a more relationship-based, as opposed to transaction-based, incentive system; three, most importantly, scheduling flexibility.

Giving doctors uniform access to standardized medical records gets rid of the first point. As for the second, I dislike decision by anecdotal evidence, which is what the relationship model encourages. What I want is for my accessible doctor pool to be more effectively commoditized, by virture of some standard success record. Urgent care centers may provide that by acruing reputation as a collective and not on a doctor basis.

Free will, Rationality and Intelligence

Thursday, December 11th, 2008

Free will, rationality and intelligence are inseparable in definition.

In FREE WILL-EVEN FOR ROBOTS, McCarthy gives a definition of free will which boils down to the ability to say “I can, but I won’t”. To be able to say “I can”, the system has to have built a representation of the world, complete with counterfactuals representing the way the individual parts of the world link and react to each other. To be able to say “I won’t”, the system needs to have a preference, that is, in its actions it is using its understanding of its cans and cannots (the counterfactuals) to drive the world towards a state more preferable to itself.

Seen this way, free will is a design pattern / framework. It is any representation-building goal-seeking system. Rationality, then, is a statement about the quality of the representations – if a free will makes choices that most effectively seek its goals, it is considered rational. Note that you need to know both the actions and the goals to determine rationality. Whenever I see papers on irrational behavior, I look carefully to see what goals they assume.

The study of human rationality poses problems because we often don’t know what people want, and it isn’t certain that you get the right answer by asking them. However, even when the goals sought are not known, there is progress that can be made towards assessing the presence of rationality. For example, if I assume that a person is walking with the goal of going from point A to point B, I don’t have to know what those points are to observe that any path with a U-turn is suboptimal. (I think this underlies the unwillingness to make U-turns even when they are optimal going forward, because they provide everyone around you with an undeniable proof of suboptimality.) I would be wrong to conclude this for a sight-seeing tourist though, or an oil tanker that gets diverted because it receives news that the price of crude is now higher elsewhere.

What of intelligence then? Well, counterfactuals are built by processing data from the senses / memory. I consider all quality difference attributable to the processing, and not the data, to be intelligence. This is often described in terms of speed – by locking a person up in a room, the time needed to reach the final conclusion cannot be due to new data, and must therefore be due to the processing, i.e. how fast or slow the person is. This is for cases where a final conclusion exists – where given enough time all people arrive at the same answer. In cases where the answers are persistently different, it is more difficult to examine intelligence by itself – I believe this is why the slow/fast terminology persists.

Risk and Ignorance, the way I understand it

Wednesday, December 10th, 2008

There are limits to what we know, and a lot of the time we are trying to figure out ways in which to make outcomes insensitive to our ignorance. Risk avoidance in the daily sense, to me, is ignorance avoidance, the seeking of profitable action crafted to avoid areas of ignorance. My ignorance being special to me, by paying attention to it I make maximal use of my Richardian comparative advantage with respect to others.

I don’t understand what others do or do not know very well, so I use the shortcut of estimating comparative advantage by virtue of my absolute level of certainty in the subject matter. Sometimes, however, I come across subjects which nobody really knows anything about. If others also maximize their comparative advantage by sticking to opportunities of high absolute certainty, then opportunities of low absolute certainty and high relative certainty would be one type of under-exploited opportunity I know to expect to find.

This, I feel, is the risk-tolerance of the entrepreneur. It is quite distinct from financial risk as we treated it two posts ago, which in fact is a rather strange animal because it is simultaneously uncertainty about the outcome and certainty about probabilities, an unlikely state in the real world.

I am guessing that the mind machinery for dealing with possibilities and future counterfactuals is more like Monte Carlo than anything else. We think of detailed alternate future paths and use those to weigh the value of an option – sounds like the algorithm a particular go-playing AI uses. What makes for a good Monte Carlo sampling strategy? How does one become a better Monte Carlo sampler? Answers to those would make for big advances in thinking.

The twice-iterated prisoner’s dilemna

Wednesday, December 10th, 2008

Familiar with prisoner’s dilemna? It’s a game, played between two people, with two responses available to each side, making for a total of 4 possible outcomes. The context is that of two accomplices caught by the police, and the choice is to either stay silent or confess. Outcomes are as follows (from the wikipedia page):

B stays silent B confesses
A stays silent Each serves 6 months A serves 10 years
B goes free
A confesses B serves 10 years
A goes free
Each serves 5 years

What should one do as A? In this case it is easy to decide, because no matter what B does, A ends up either a better result if A confesses. Thus A will confess. Symmetrically, B will also confess, and we predict the lower-right outcome.

Now consider what happens if we know we are going to play the game twice. Naively, you may think that B now has 4 possible strategies, (silent-silent, confess-silent, silent-confess, confess-confess). In reality, B has 8 possible strategies, because on the second round he would have seen A’s first round move, and thus the second round response can be conditioned on A’s first round move, which means a total of 4 ways of responding in the second round. That, combined with the first round’s 2 ways, makes for a total 8 possible strategies.

Let’s consider one of those 8 strategies, where B will first remain silent, and then imitate A’s first move for the second round. In this case, if A follows the same strategy, they will end up with silent plays the whole way, and get away with serving one year each (2 x 6 months). What if A decides to confess and then confess? In that case, the result is (confess,silent) for the first round, and (confess,confess) for the second round. A then serves 5 years time, while B serves 15 years time. Note that even though A does better than B, it does considerably worse than the 1 year in the other scenario.

I can encode all 8 possible 2-game strategies using 3 bits: (round 1 response, round 2 response if opponent remained silent, round 2 response if opponent confessed) and this would be the resulting payoff table:

SSS 1.0,1.0 1.0,1.0 10.5,0.5 10.5,0.5 10.5,0.5 10.5,0.5 20.0,0.0 20.0,0.0
SSC 1.0,1.0 1.0,1.0 10.5,0.5 10.5,0.5 10.0,10.0 10.0,10.0 15.0,5.0 15.0,5.0
SCS 0.5,10.5 0.5,10.5 5.5,5.5 5.5,5.5 10.5,0.5 10.5,0.5 20.0,0.0 20.0,0.0
SCC 0.5,10.5 0.5,10.5 5.5,5.5 5.5,5.5 10.0,10.0 10.0,10.0 15.0,5.0 15.0,5.0
CSS 0.5,10.5 10.0,10.0 0.5,10.5 10.0,10.0 5.5,5.5 15.0,5.0 5.5,5.5 15.0,5.0
CSC 0.5,10.5 10.0,10.0 0.5,10.5 10.0,10.0 5.0,15.0 10,10 5.0,15.0 10,10
CCS 0.0,20.0 5.0,15.0 0.0,20.0 5.0,15.0 5.5,5.5 15.0,5.0 5.5,5.5 15.0,5.0
CCC 0.0,20.0 5.0,15.0 0.0,20.0 5.0,15.0 5.0,15.0 10,10 5.0,15.0 10,10

Looking at the table, we realize that “always confess”, CCC, is still the only Nash equilibrium for the twice-iterated case. For more iterations, however, the “tit-for-tat” startegy eventually becomes another Nash equilibrium. For more reading, go here.

The Risk Premium

Wednesday, December 10th, 2008

Investments involve predictions about the future, and those predictions can be uncertain. The formal representation of that uncertainty is accomplished through the use of probability theory, such that every uncertain number X is now represented as the probability distribution function P(X).

Given two investments which give uncertain returns X and Y respectively, how does one choose between them? Under decision theory, a utility function U is used to compare <U(X)> and <U(Y)>, and the investment with the higher utility is chosen. For the sake of simplicity, modern portfolio theory models U using a quadratic function, where U(X) = X – r*X^2, with a greater r meaning more risk adversity.

Given a quadratic utility function, it is possible to evaluate the expected utility of all linear combinations of X and Y by just knowing <X>, <Y>, <XY>, <X^2>, <Y^2>. This result generalizes to any number of investments. Using this information, for any given return, one can find the linear combination which minimizes the variance, forming the optimal portfolio for that return. The set of all optimal portfolios forms the efficient frontier in the diagram below.

Now examine the effect of adding a risk-free asset. Taking a linear combination of the risk-free asset and any given portfolio, you can achieve any new portfolio with the same Sharpe ratio (ratio of the difference between return and risk-free rate to the standard deviation) as the given portfolio. The capital market line represents the best portfolios that can be formed this way – they consist of combinations of the risk-free asset and the market portfolio, which is the efficient frontier portfolio with the largest Sharpe ratio.

Markowitz Frontier

Under the CAPM pricing model, it is assumed that stocks function as components for building market portfolios, and stockholders are only compensated for risk which cannot be diversified away by the other components. Approximating the market portfolio with the SP500, we approximate this non-diversify-able risk using the covariance between the stock and the SP500. That covariance is scaled by the variance of the market portfolio and called beta.

This is the standard classroom treatment of the concept of risk, and I am uncomfortable with it in ways that I will describe in the next post. For additional reading, you have wikipedia:

Time Value of Money

Tuesday, December 9th, 2008

Given the choice between money today and money tomorrow, one would prefer the money today. This is true because money today can be hoarded until tomorrow, or alternately used for other purposes before tomorrow – “money tomorrow” is just one out of many options available to someone with “money today” and hence “money today” must be of greater than or equal value to “money tomorrow”. Similarly, a greater sum of money is always preferable to a lesser sum, because I have the option of discarding a part of the greater sum to get the smaller sum. It is then plausible that there be a greater sum of money tomorrow (s2,t2) which is of equal value to a smaller sum today (s1,t1).

What is the relationship between (s2,t2) and (s1,t1)? So far we have found that (s2-s1)(t2-t1) >= 0. Only by assuming that investment opportunities are of much shorter duration and of much smaller size than the quantities under consideration, and also available equally throughout time (i.e.  having multiple instantiations at (t1+dt,t2+dt) for all dt), do we get the conventional discounting rule, where (s2/s1) = r^(t2-t1) for some r > 0.

To go from the inequality to the equality requires work. This specific equality only emerges because of the additional assumptions made. Given access to a pool (defined by our assumptions of time-invariance and scale-invariance) of investments with discount rate r, we can now arbitrage any cash flow to its present value. The net present value (NPV) is this concept, and when positive it represents situations in which choosing the investment over choosing the pool results in a cash gain.

So far, I have only dealt with the opportunity cost of money, as represented by participation in the pool of investments. I will discuss the risk of the investment itself next.

Two Envelope Problem

Monday, December 8th, 2008

You are given two envelopes to choose from, and are told that one has twice the amount of money as in the other. You pick one and open it, finding $20. The host asks you if you would like to switch. It is apparent from the random nature of the initial choice that switching would not change the payoff.

The only possible amounts in the other envelope are $10 and $40. I became tempted to assign 2/3 probability to $10 and 1/3 probability to $40, so that the expected value came out to $20. This is wrong, however, because the host could announce that he would pay a square of whatever amount of money I ended up with ($100 for $10, $400 for $20, $1600 for $40) and still switching would not matter, yet those probabilities (2/3, 1/3) would prescribe switching.

This is apparently a state of ignorance in which the possibilities are finite and known, and yet probabilities cannot be assigned to them. This disturbs me.

HT: Lee Hsien-Yang’s New York talk last Friday


Monday, December 8th, 2008

When I went from undergrad to grad school, the introduction of a monthly stipend into my life gave me a natural unit in which to measure my monthly consumption. I never did itemized budgeting, but net positive savings was definitely a goal. This goal was easily reached, mostly due to my realization that scrimping on rent got me very far. As of a month ago, I started working, and for the first time ever, have substantial savings.

This introduces a degree of freedom, and with that, one more thing to optimize. There were a few decisions necessitated by the living environment, like eating out and using the wash-n-fold service instead of the laundromat. Other than that, however, it seems like the other decisions are going to be made by my choice of company more than anything else. Tonkatsu curry is the most delicious dish I can think of, and I really don’t think anything will change that. I do follow along when people go to fancy restaurants though.

Don’t listen to Noise

Tuesday, December 2nd, 2008

I’ve stopped reading the news recently because I think it is terribly time-consuming given how little knowledge it conveys. The credibility of news depends a lot on social proof, I find – just because everyone around you has the same sources and believes the same things about far away worlds doesn’t mean those beliefs are more dependable.

News informs belief choice, and is most useful when surprising. It can be unsurprising because it is predictable, or because it is noise. In either case, uninteresting because the candidate beliefs are not differentiated by the evidence.

My particular candidate beliefs at this moment mean that news does not interest me.