Ordering preferences are a way of prioritizing alternative choices and attaching expected utility to outcomes. We measure the magnitude of differences between alternate choices so we can determine how much choice A is preferred to choice B. I prefer choice A two times greater than choice B and choice B three times greater than choice C. This tells us the ratio of difference.

Ordered preferences tells how how agents make their choices. There are different ways to rank and measure preferences.

Ordinate scales are the weakest form of ordered preferences. I like chocolate ice cream more than vanilla and hate lemons. It’s useful in a vague sense. This tells us that an agent prefers A>B>C. We can assume that A>C but we do not know the magnitude of the difference. There is very little information.

We can compare it with different sets if we preserve the relative positions of the ordinal set. Monotone transformations convert one set into a different set that preserves their relative positions. Function y=10x or y=x^3 both work.

So 10A>10B>10C. We still don’t know the magnitude of difference.

Why does x cubed work but x squared does not?
Here’s an example: 10, 3, -5
if we use x^2, then we get 100, 9, 25 so the relative positions are not preserved. x^3 gives 1000, 27, -125, so the relative positions are preserved.

Ratio Scales provide the most information about the magnitude of difference. This is the most precise information and is most often used in the ‘harder’ sciences where precision is more valuable. Ratio scales are used in measurements, energy, time, money, etc. Similarity transformation preserves the ratios. The Function is y=ax. This one is the most obvious – the ratio of difference is twice as much, three times as much, etc.

Interval scales are more commonly used in game theory. This in when the differences of magnitude may be known but not the ratio of the differences. Interval scales use linear transformation. The function is y=ax+b (where a>0, otherwise you’d have y=b which is stupid).

In example A=10, B=3, C= -5.
We can measure the differences this way:
(10-3)/(3-(-5)) = 7/8. This gives us the ratio of the differences. The linear transformation (y=ax+b) preserves these positions. So if a=2 and b=1, then we get the new set.
(21-7)/(7-(-9))= 14/16 (or 7/8).

Interval scales and ratio scales allow us to measure the differences between choices A, B, and C. This is how you measure the utility of each decision relative to the alternatives.

Probability can change the order of preference.
The utility of A is 10, B is 3, and C is -5.
What if there is only a 10% chance of the agent getting choice A, but a 100% chance of getting choice B and C?
The expected utility changes to B (3), A (1), C (-5) and will change the order of preferences to B>A>C

Backwards Induction
In complex games or real life situations, backwards induction is really the only tool to empirically evaluate the decisions made. At the start of a complex game – like war, crime, public policies, the utilities and probabilities of the consequences are unknown to the agents themselves.

We can look at the end results of a game and ask, how did the agents reach this point? We trace this backwards from the endpoint to the starting point of the game.

Very often, agents use ordinate scale to rank their preferences. Agents may not know what the magnitude of difference between the own preferences until later when they discover the utilities through trial and error.

Historically, we can trace their decisions and describe them through game theory, including their rational and emotional errors. They may have inaccurately guessed at the probabilty of a course of action and made a poor decision in hindsight.

Game Theory is a descriptive tool, not a predictive one. It does not really change our decisions, since we’re dealing with incomplete information and uncertainty. The idea that we can model human behavior deep into the future is nothing more than fortune telling.