Choices: Crunching numbers

The Commonwealth has attracted many adventurers who were drawn to tropical paradise. It inspires thoughts about the risks and rewards of following our rainbows in life. In fact, many types of seemingly simple choices can present some interesting situations. In last week’s column (see SaipanBlog.com for a handy link if you missed it), I posed the following question:

Consider a game of chance that a player gets to play only once, and then he has to walk away from the table for good. Here is the game, in which only one of two choices can be selected:

Choice 1 is a coin toss, in which the player gets $1 million if it’s heads, but no money at all if it’s tails.

Choice 2 is even simpler: The player gets a briefcase containing $300,000.

Which choice would a “rational” person select? And which would you take?

Sure, we have to open a can of math to analyze this question, but it’s not hard, so here we go…

* * *

First of all, let’s place a value on the choices.

Let’s start with Choice 2. The value of this proposition is simple; it’s $300,000.

Choice 1, on the other hand, is a bit murkier. It carries an “expected value” that is the probability of heads multiplied by the outcome of getting heads, plus the probability of getting tails multiplied by the outcome of getting tails (if you don’t follow that, just take my word for it). So the value of Choice 1 is, statistically, $500,000.

So here we are: Choice 1 is more valuable than Choice 2. Period.

So wouldn’t any rational player select Choice 1? Here’s where the fun starts.

Keep in mind that the player gets only one trial in the game (just as we get only one trial in life, eh?). If you were going to toss the coin a lot of times, the law of averages would assert itself, and, for Choice 1, that $500,000 expected value would be pretty solid in the long run.

But, since the player can only play the game once, the odds are fully 50 percent that he will walk away empty handed with Choice 1, when he could have otherwise taken that sweet $300,000 in Choice 2. In other words, though Choice 1 is more valuable mathematically, it also carries a 50 percent probability of giving the player a serious dose of regret.

Ah, regret! It’s no mere emotional pang, it is a valid player in economics as well. So our simple probability math gets a bit more complicated when we back up to a more basic economic step and ask this: What is our goal here? It is to maximize our gains, or, by contrast, to minimize our regret?

If we go with the latter case, minimizing regret, we’ll take Choice 2 and are paying, in theory, $200,000 for that luxury, since that’s how much of increased “expected value” we are forsaking from Choice 1. This is known as “risk aversity,” where we pay a “premium” in order to lock in to a solid outcome. Most people are risk averse and are willing to pay to avoid risk. That’s what insurance is, for example.

So, what is rational? If you’re looking for a clean economic answer, here it is: The rational choice, here, is the one that you make for yourself. Nobody else can make a rational choice for the player in this example; not me, not Big Brother, not even Hillary, because the optimal tradeoff between risk and reward is a personal thing.

Personally, by the way, I’d take Choice 2, but if you would opt for Choice 1, your choice is no less rational than mine.

This is no trivial matter. Risk and reward are the very underpinnings of capitalism, and this is why only free markets can create substantial wealth: They allow risks, rewards, effort, and capital to seek their most rational allocations, which cannot be centrally determined.

Yes, I readily admit that most people prefer socialism to freedom of choice: They need an authority, a surrogate parent, to make their choices for them, and they also enjoy seeing their more capable neighbors getting smothered by the gray and drab gloom of bureaucracy. That’s human nature, and why freedom and free markets are such very, very rare things.

Still, for the few outliers who enjoy making free choices, it’s an interesting exercise to look into how risk aversity, rewards, regrets, expected values, and probabilities can all be seen in my simple mathematical riddle of the coin toss game. If you rode a rainbow to tropical paradise, thus avoiding the tyranny and safety of a cubicle, then you know what I mean.