Sunday, July 4, 2010

Regret Minimization

I did a mini research paper for Game Theory on the topic of Regret, it's pretty interesting when considered alongside Nash Equilibrium, and financial markets. While Equilibrium proves the existence of a single mixed strategy that guarantees a minimum payoff, this payoff could still be negative, and this would only serve as the maximum payoff if everybody else is achieving equilibrium. In real life games, players often deviate quite a bit from equilibrium.

The whole buy & hold thing, if done with ETFs to avoid bankruptcy risk, is an equilibrium strategy where the player is guaranteed a return that roughly equals to the actual rate of inflation. This would be an optimal strategy, if everybody else is doing the same, including hedge funds and investment banks. Obviously, that is not so, resulting in the buy & hold folks considerable opportunity loss or "Regret". A change of strategy to minimize "Regret" then is needed to exploit the inefficiencies from other players deviating from equilibrium.

Here's a basic example of regret minimization from Wikipedia,

Minimax example

Suppose an investor has to choose between investing in stocks, bonds or the money market, and the total return depends on what happens to interest rates. The following table shows some possible returns:

Return Interest rates rise Static rates Interest rates fall Worst return
Stocks −4 4 12 −4
Bonds −2 3 8 −2
Money market 3 2 1 1
Best return 3 4 12

The crude minimax choice based on returns would be to invest in the money market, ensuring a return of at least 1. However, if interest rates fell then the regret associated with this choice would be large. This would be −11, which is the difference between the 1 received and the 12 which could have been received if the outturn had been known in advance. A mixed portfolio of about 11.1% in stocks and 88.9% in the money market would have ensured a return of at least 2.22; but, if interest rates fell, there would be a regret of about −9.78.

The regret table for this example, constructed by subtracting best returns from actual returns, is as follows:

Regret Interest rates rise Static rates Interest rates fall Worst regret
Stocks −7 0 0 −7
Bonds −5 −1 −4 −5
Money market 0 −2 −11 −11

Therefore, using a minimax choice based on regret, the best course would be to invest in bonds, ensuring a regret of no worse than −5. A mixed investment portfolio would do even better: 61.1% invested in stocks, and 38.9% in the money market would produce a regret no worse than about −4.28.