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	SSC	SCS	SCC	CSS	CSC	CCS	CCC
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.

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.

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.

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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:

http://en.wikipedia.org/wiki/Modern_portfolio_theory

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.