Tuesday, January 19, 2016

You should be random, so carry dice!

Why should you act in a random, unpredictable, or arbitrary way? It seems no rational adult would want to do that. But as we'll see, there are situations that arise in our lives — surprisingly often — where acting randomly is exactly the right thing to do.

What is random?

What does random mean? By this we mean unpredictable. A random variable is some quantity that cannot be predicted in advance. A good example is the result of the roll of a six-sided die.

If randomness is useful at all (which is yet to be seen) then dice may be good enough for some purposes. But for others you might want something that is truly random: something that is impossible to predict in principle. Many modern computers can produce random numbers. For the purposes of this essay we will pretend that dice are truly random.

Acting randomly

Would it make sense to act in a random manner? You'd be howling at the moon, bouncing off the walls, jumping through windows, and generally making a nuisance of yourself. We'd probably have to keep knives far away from you. We'd probably have to keep people far away from you! No, I think we agree that this is not the kind of random behavior any of us would want.

Perhaps it would make sense to act randomly sometimes. Perhaps there are specific, well-defined situations in your daily life where you would benefit from making some decision that cannot be predicted. Perhaps it makes sense to keep dice in your pocket.

Philosophers have proven that this is impossible. Suppose you have a decision to make, and you want to make it rationally. If there is a rational reason to choose one option over the other, then that is, of course, what you should do. There would be no need to act randomly in that case. On the other hand if there is no rational reason to choose one option over the other, throwing dice would just be a waste of your time. You might as well just pick the first option, whichever that is. In either case, the dice would remain in your pocket. And since you will never use them, you might as well travel light and just leave them at home. Or at the game store. So philosophers have "proven" that we should never want to act randomly.

But philosophers are wrong.

Board Games

The most obvious time that you would want to act randomly is when you are playing a board game like Monopoly or Backgammon. The rules of the game require it. You simply cannot play these games without rolling dice and basing your actions on the outcome. Admittedly, this is a trivial example. Are there serious situations where rolling dice would help us in our daily lives? Yes, there are.

Shopping for toothpaste

Imagine that you are shopping for toothpaste. You go to the corner store and head down the toothpaste aisle. There are hundreds of brands to choose from. Every one of them is different from every other in some way. Every advertised feature seems relevant. Extra flouride? Whitening? Tartar control? Baking soda? Do you care what sweetener is used, or even if a sweetener is used at all? There are dozens of flavors. How will you make a rational selection? Surely, unless you inspect each and every choice, it seems impossible that you will select the optimal choice.

This is a difficult problem. If you only have fifteen minutes to make your selection, you might not be able to make the optimial choice. You could spend your fifteen minutes gathering what information you can, and return at a later time (or several times) to complete the job. In the meantime you'll just have to settle for having bad breath and not caring for your teeth. No, that doesn't sound like a very attractive option. Perhaps you should just get the same brand that you got last time - even though you aren't really very happy with it. Difficult.

This is so serious and common a problem that it has been extensively studied. Overchoice, also referred to as "choice overload", describes a cognitive process in which philosophers have a difficult time making a decision when faced with many options. From Wikipedia:

The phenomenon of overchoice occurs when many equivalent choices are available. Making a decision becomes overwhelming due to the many potential outcomes and risks that may result from making the wrong choice. Having too many [approximately] equally good options is mentally draining because each option must be weighed against alternatives to select the best one. As the number of choices increases, people tend to feel more pressure, confusion, and potentially dissatisfaction with their choice. Although larger choice sets can be initially appealing, smaller choice sets lead to increased satisfaction and reduced regret. Another component of overchoice is the perception of time. Extensive choice sets can seem even more difficult with a limited time constraint.
Fortunately, those of us who carry dice have an easy solution to this problem. Here is what you do. First, roll a die. If the result is any number one through five, just buy your favorite brand from among all those that you have previously tried. If the die shows a six, use your dice again to select a small random sampling from among all of the available choices. Two or three choices will do. Then look at those and if any of them seem plausible alternatives — if one might be your new favorite if only you had the chance to try it — then buy the best from among that small selection. If none of them seem likely, just buy your old favorite.

This strategy doesn't guarantee that you will get the optimal choice. But it does give you a very good chance of selecting a nearly optimal choice. Over time you will be more and more satisfied with the toothpaste that you end up using because you will be trying new ones from time to time. As the toothpaste industry produces new innovations, you will find yourself enjoying them. Most importantly, it reduces the amount of time you spend selecting toothpaste, leaving you more time for important choices, such as Ice Cream.

Now you know why philosophers have bad breath. It is because they do not carry dice.

The narrow bridge problem

Consider a two lane road that has one lane running in each direction. There is a place where the road narrows to a single lane to go over a bridge. Traffic on this road is not very heavy, and visibility is good, so there is no traffic signal. The obvious thing is to enter the bridge only when there is no traffic heading in the opposite direction. But what if two cars arrive at opposite ends of the bridge at about the same time? Each can wait for the other, but then they will both wait forever. Each can try going, and then back off if the other does the same. But then they will be going and backing off again and again, forever.

Philosphers have specified that the drivers are to exit their vehicles, meet in the middle of the bridge, and play chess until one of them has won a game. The winning driver is entitled to cross the bridge first. Philosophers find this a just solution, as it gives an advantage based only on the driver's ability to apply logic. Unfortunately, philosophers are perfect chess players. When they play each other, they always reach a stalemate.

The local engineers have devised a different solution. When you reach the bridge, you stop and roll a die. You then wait a number of minutes indicated by the number on the die. Then you enter the bridge. If you encounter another car on the bridge, you back out to the beginning and try again. This is the purpose for the fuzzy dice hanging from the rear-view mirror of many cars.

A more familiar example occurs when cars reach an intersection in which there is a stop sign for each of the four roads entering the intersection. It is customary — in fact a requirement under the law — that if you arrive at the intersection around the same time as another car on the cross street, the car to the right goes first. This works fine except when cars arrive on all four roads. In that case obeying the rule, even though mandated by law, would lead to deadlock. That is when the procedure used on the bridge works very well.

Now you know why philosophers do not drive. It is because they do not carry dice.

The Scientific Method

There are applications in science where the intentional use of randomness is necessary to the proper conduct of an experiment. Chief among them is the double-blind trial. This is an especially stringent way of conducting an experiment that eliminates subjective bias both on the part of the scientist and on the part of the subject of the experiment. Double-blind studies are frequently used to test drugs for their effectiveness. It is the gold standard for scientific rigor.

In order to conduct a double-blind study, a population of test subjects is divided into two groups: those who will receive the drug to be tested, and those who will receive a placebo in its place. The subjects are assigned to one or the other group randomly. Neither the scientific investigator nor the test subjects know who is in which group. It is only after the outcome has been evaluated for all test subjects does it become known to the investigator which was which. Consequently, there is no opportunity for any bias to arise due to such knowledge.
Now you know why philosophers do not conduct scientific experiments. It is because they do not carry dice.

Game theory

Even if you playing a "deterministic" game like Chess, there may be value in making some choices randomly. As when selecting toothpaste, you usually cannot select the optimal move because you are operating under a time constraint. A technique called Monto Carlo Game Tree Search, which uses randomness in selecting moves to consider, has resulted in a revolution in the quality of computer play for Go, a board game that is widely played in Korea, Japan, and China.

Consider the game rock-paper-scissors. What if you want an optimal strategy, or procedure, for playing the game? What we mean by "optimal" is that there is no other strategy that has a winning edge over it, in the long run. There is a field of mathematics called game theory that studies games, and the game rock-paper-scissors is well understood. One strategy is to choose "rock", "paper", or "scissors" randomly, and it has been proven, formally, that this is an optimal strategy. It has also been proven that any optimal strategy for rock-paper-scissors must, necessarily, use randomness. In fact many games require randomness to be played optimally.

Game theory has wide applications in the real world, including war strategy, political science, economics, and negotiation. Now you know why philosophers are not rich. It is because they do not carry dice.


How can the philosophers have been so wrong? After all, didn't they prove that basing a decision on the roll of the dice is irrational? Yes, but they made a hidden assumption: that the decision is a one-time decision made in isolation. If you only have one decision to make in your whole life, the philosophers may have been right. But most decisions we make are part of a series of decisions, and the logic that the philosophers used does not work in that case. When we have a series of decisions to make, we need a strategy for making them, and it has been shown that randomness is a necessary part of the strategy for many problems.

While it seems that basing your behavior on the roll of the dice might be irrational, there are situations that arise every day where it is the most rational thing to do. Those people who refuse to do so are systematically weeding themselves out of the human gene pool. Don't be one of them. Carry dice.

"View source" and search for "Dining Philosophers" to find out why philosophers are hungry.
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Stefan W. said...

Americans don't have dice, too.
That's why they have guns.

Chandler Klebs said...

Neal! This is hilarious! I love the part about philosophers being perfect chess players. I've quite a chess player myself and that's what often happens when I play with my friend Jerry.

I can totally see the advantage of behaving in a way that your opponent cannot predict, but of course that is yet another reason for doing so. I absolutely hate games with dice because I can't use my logic very well.

Edward Herzog said...

Nice read, Gafter. I love games with dice and spinners and anything that adds a little unpredictability.
Embrace the Random!!
Belief that you can really optimize a thousand+ micro-decisions every day would be an exhausting conceit.

Neal Gafter said...

Of course, Sheldon knew this before I did... https://www.youtube.com/watch?v=5yk8ixiKUEw