the (un)expected value

the mean and the nice

The term "expected value" has a very precise meaning in probability theory, but I think it often clashes with what we as everyday humans would really "expect" from certain probabilistic situations.

To avoid confusion, I will refer to the traditional definition of expected value as the mean (this is also standard terminology [wiki]) and contrast it with a new idea which I will here refer to as the nice.

Let's see why the mean is not always what's expected. Consider a game of chance which you can play for free. Here's how it works: you flip 20 coins. If any of them are heads, you win a million dollars. If they're all tails, you owe 10 trillion dollars (yes, you're really screwed if you lose). Would you play this game? I would, and I think that many people would happily do the same. The chances you'll lose are less than one in a million (since (1/2)^(20) < 1/million). Yet the mean value is negative, since the loss of -10 trillion outweighs the far-more-likely win of a million. Intuitively speaking, what value do you really expect? I think most people would agree that they are least surprised by the outcome of winning a million dollars.

Hence I am hoping to mathematically capture this notion of the "least surprising" outcome, which we will call the nice value. Here are some properties the nice value should have:

• The values close to the nice value should be far more likely than those far away from the nice value.
  
• The nice value should be an actually possible value in the probability distribution.
  
• It should be understood that some nice values are "expected" with much less confidence than others.
  

I mention the last point because the above game is relatively easy to predict. Consider rolling a six-sided die. What should be the nice value? I think in that case, intuitively speaking, there is no best value. We expect any of the 6 numbers to show up with equal probability. So there are many probability distributions in which we have low confidence about the outcome. The motivation for a nice value is targeted toward probability distributions which are concentrated around a particular value.

a gambling strategy

Part of my motivation for thinking about the nice value is the following old "guaranteed-to-work" gambling strategy. Suppose you're playing a double-or-nothing game with 50/50 odds. In other words, you bet a certain amount, flip a coin, and if it's heads you get twice your money, tails you lose all your money. (So the mean value is zero, since half the time you lose x dollars and the other half you gain as much.) Here is the strategy: first you bet x. If you win, you stop, and you've successfully won x dollars. If you lose, you bet 2x. If you win, your net gain is now 2x-x = x, and you stop. If you lose, you're down by 3x. So bet 4x. If you win, you're up by x, and you stop. Et cetera. You just keep doubling your bet until you win, and you're "guaranteed" to end up winning x dollars.

What's the catch? The problem is that you could run out of money before you ever win. Suppose you start with 6x dollars. Then there's a 25% chance that you'll lose twice in a row, in which case you won't be able to afford doubling your bet again. If you play this strategy long enough, you're essentially guaranteed to go broke.

However, what this strategy does achieve is a way to effectively turn this essentially unpredictable game into a (possibly longer) game with a single very likely outcome -- that you'll end up winning x dollars. Specifically, let's say you start with 63x dollars. Then you can afford to lose up to 5 times in a row and still ultimately win overall in the last game (with a bet of 32x on that last game). This means you'll net x dollars profit with probability 63/64 = 98.4375% chance. So I would call x the nice value of that game.

By the way, you may be wondering why people don't use this strategy all the time if it sounds so tempting? Well, in the long run, it's actually not a great strategy. I think it's a good one-time strategy, but not something you'd like to repeat. For example, suppose you go to a casino and use this strategy once a month. By the end of 4 years, it's most likely that you'll have lost at least once. Even if you only lose once, and win the other 47 times, you've still lost overall since a single loss means losing 6 times in a row, or a loss of 63x. This perspective shows that the mean value is the "right value" for long-term additive behavior, while the nice value is meant to express a one-time value which is the least surprising outcome.

toward a rigorous definition of the nice

Suppose we have a metric space X and a probability distribution P() on X. (I also assume we have a measure space on X including all spheres generated by the distance function of the metric.) Given any x in X, we can measure its niceness via the function f_x:R → [0,1] defined by

f_x(z) = P({y: dist(x,y) < z}).

Basically, f_x() is an increasing function which (if no distance in X is infinite) asymptotically approaches 1. The faster f_x() approaches 1, the nicer x is -- that is, the less surprising it would be that x is the outcome. The trick is that functions are not easily put into a linear ordering, so it is hard to say which value of x is the nicest. Beyond that, we can easily imagine situations in which many values of x have the same function f_x() -- for example, rolling a die. But when there is a value x in X which gives the greatest function f_x(), we can safely say that that value is the nicest.

Future work: try to define the nicest value in other cases!