## Monday, December 15, 2014

### A Tiny Bit of Game Theory

I’m fascinated by game theory and I’ve lately been thinking about retirement finances through that lens.

You may be familiar with three products of game theory, whether you realize it or not. The first is the strategy of Mutually-Assured Destruction, with the appropriate acronym MAD, that was developed from game theory in the 1960's as a response to the threat of nuclear war. The second is "Nash equilibrium", suggested in the book and movie, "A Beautiful Mind". (John Nash won a Nobel Prize for his work on game theory.) The other is called "Prisoner's Dilemma", a game that pits two "perps" against one another to obtain a confession that seems to part of every TV crime drama ever created.

Game theory is the study of strategic decision-making or, according to expert Roger Myerson, "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers.”

I keep his book, Game Theory: Analysis of Conflict, on my desk. On days when I want to humble myself, I try to understand the math. But, there is a lot to learn from game theory even if you wouldn’t touch linear algebra with a ten-foot pole.

Game theory can model strategic decisions in a number of ways, but the simplest is by using a matrix of Player A’s strategies versus those of Player B’s. The cells of the matrix contain the payoffs for each player when each chooses a particular strategy. A pair of numbers describes the payoffs. The first of the pair (boldface) is Player A’s payoff and the second is Player B’s.

Here’s an example. In this “game”, if Player A chooses Strategy 1 and Player B chooses Strategy 2, then Player A will receive a payoff of 3 “points” and Player B will receive 0 points. Each player will look at the potential payoffs for each strategy available to her, guess what Player B might do, and choose a strategy accordingly. The outcome of the game will be determined by the contents of the cell at the intersection of the two strategies.

(In case you're interested, the game above is “Prisoner’s Dilemma”, where each player’s Strategy 2 is to confess and rat out his partner in crime. Strategy 1 is to keep silent. The payoffs are the number of years in prison, so I suppose they should be negative numbers.)

In financial planning, we rarely are interested in a game between two individuals, but a game, instead, of an individual against a system of markets with random outcomes. Game theory refers to these as “stochastic games against nature”, a phrase you may never need to hear again. On the other hand, when someone asks what you're doing about retirement, you could impress them by answering, "I'm playing a stochastic game against nature."

In these games, there will be only one payoff in each cell, since “nature” doesn’t need payoffs.

Here’s an example. Let’s say that nature has two possible strategies in a game: it can rain or not rain where you are. You, in turn have two strategies. You can carry an umbrella, or leave it at home.

If you leave your umbrella at home, there are two possible outcomes. It may rain, in which case you will get wet, or it may not rain, and you will have a good outcome. You stay dry and don’t have to lug an umbrella around for no reason.

If you choose the umbrella strategy instead of leaving it at home, you also have two possible outcomes. If it rains, you stay dry. If it doesn’t rain, you will have to carry an umbrella around all day, looking stupid and encumbered for no good reason.

A matrix to describe this game might look like this:

The correct strategy choice in this game, of course, depends on the weather forecast’s probability of rain and its accuracy. It is called a stochastic game because the outcome depends on chance. It is called a “game against nature”, not because we’re talking about rain, but because we are playing against a complex system and not against an individual. The stock market, for example, would also be included in this definition of nature.

If these were the payoffs (I made them up), at what probability of rain would you switch strategies?

I think we can gain some insight into some retirement financial decisions if we look at them from a game theory perspective. In particular, I think game theory can make us focus on all possible results of our financial decisions and not just the most likely outcomes. In my next few blogs, I’ll provide some examples from retirement finance and we’ll find out if you agree, starting with Game Theory and Social Security Benefits.

1. I think I can see where this is going. I bought a plain-vanilla annuity 4 years ago at a payout rate I liked, but of course if I die tomorrow, that money will be gone, none of it available to my kids. If interest rates move up sharply, I will have stupidly short-changed myself.

On the other hand, in any case, I still have the payout that I bargained for, and if I live to 105 I'll still get it (albeit much devalued by inflation).

Have I "won" or have I "lost"? If I put the correct % of my available funds into this flooring strategy, in my mind I will have won in any case, but perhaps others would feel differently, especially if it mattered to them not to look "dumb", or if they interpret winning differently from me.

2. That is SOOOO not where this is going.

I have no knowledge of your financial situation but, if anything, game theory would probably suggest that you buy the annuity. Many game theory decision rules are based on avoiding the worst case loss (MAD and Prisoner's Dilemma, for example), which an annuity would do.

Game theory lets you assign your own payoffs, so strategies work differently for different people.

"Looking dumb" refers to walking around the Sahara in a rain coat. It has nothing to do with financial decisions.

Stick with me a while longer.

3. I guess I wasn't clear: I DO think I've won the game that I happen to be playing, which is to have a known, steady income stream, even if it isn't optimal. However, others may be playing a completely different one: make as much money as possible, even if there's a significant possibility of failure.

1. You were quite clear about how you feel about your decision and I agree with you. You were unclear about where I'm going with this discussion.

4. Good, I will check in later to find out. Thanks for interesting posts.

5. Excellent! I promise to let you know where this is headed the instant that I know.