Thanks to Bruce Foutch who found the video above. Transitivity is familiar with ordinary numbers. If B > A and C > B and D > C, then D > A. But only if the numbers A, B, C and D behave themselves. They don’t always, as the video shows.
What’s nice about this demonstration is the probability and not expected value ordering. Hence the “10 gazillion” joke. “Expected” is not exactly a misnomer, but it does have two meanings. The plain English definition tells you an expected value is a value you’re probably going to see sometime or another. The probability definition doesn’t match that, or matches only sometimes.
Expected value is purely a mathematical formalism. You multiply the—conditional: all probability is conditional—probability of a possible outcome by the value of that possible outcome, and then sum them up. For an ordinary die, this is 1/6 x 1 + 1/6 x 2 + etc. which equals 3.5, a number nobody will ever see on a die, hence you cannot plain-English “expect” it.
It’s good homework to calculate the probability expected value for the dice in the video. It’s better homework to calculate the probabilities B > A and C > B and D > C, and D > A.
It’s not that expected values don’t have uses, but that they are sometimes put to the wrong use. The intransitive dice example illustrates this. If you’re in a game rolling against another playing and what counts is winning then you’ll want the probability ordering. If you’re in a game and what counts is some score based on the face of the dice, then you might want to use the expected value ordering, especially if you’re going to have a chance of winning 10 gazillion dollars. If you use the expected value ordering and what counts is winning, you will in general lose if you pick one die and your opponent is allowed to pick any of the remaining three.
Homework three: can you find a single change to the last die such that it’s now more likely to beat the first die?
There are some technical instances using “estimators” for parameters inside probability models which produce intransitivity and which I won’t discuss. As regular readers know I advocate eschewing parameter estimates altogether and moving to a strictly predictive approach in probability models (see other other posts in this class category for why).
Intransitivity shows up a lot when decisions must be made. Take the game rock-paper-scissors. What counts is winning. You can think of it in this sense: each “face” of this “three-sided die” has the same value. Rock beats scissors which beats paper which beats rock. There is no single best object in the trio.
Homework four: what is the probability of one R-P-S die beating another R-P-S die? Given that, why is it that some people are champions of this game?
R-P-S dice in effect are everywhere, and of course can have more than three sides. Voting provides prime cases. Even simple votes, like where to go to lunch. If you and your workmates are presented choices as comparisons, then you could end up with a suboptimal choice.
It can even lead to indecision. Suppose it’s you alone and you rated restaurants with “weights” the probability of the dice in the video (the weights aren’t necessary; it’s the ordering that counts). Which do you choose? You’d pick B over A, C over B, and D over C. But you’d also pick A over D. So you have to pick A. But then you’d have to pick B, because B is better than A. And so on.
People “break free” of these vicious circles by adding additional decision elements, which have the effect of changing the preference ordering (adding negative elements is possible, too). “Oh, just forget it. C is closest. Let’s go.” Tastiness and price, which might have been the drivers of the ordering before, are jettisoned in favor of distance, which for true distances provides a transitive ordering.
That maneuver is important. Without a change in premises, indecision results. Since a decision was made, the premises must have changed, too.
Voting is too large a topic to handle in one small post, so we’ll come back to it. It’s far from a simple subject. It’s also can be a depressing one, as we’ll see.