I'm trying to define which hotel offers the best value.
Let's say we have two hotels - A and B.
For A, you pay $10 a night and the rating for the hotel is 9.8.
For B, you pay $8 a night and the rating for the hotel is 9.6.
So for A, you pay $2 more and get one with a higher rating of 0.2. But is it the better value?
How would you solve this?
Thanks!
This depends on the person you are asking.
Suppose an individual's utility function only depends on hotel rating scores and money (which is used to purchase other things). Then it is completely consistent for two individuals to prefer A=(9.8,m-10) to B=(9.6,m-8) (and vice versa). Thus, there is no value metric for all individuals. However, if you focus on one individual, you could ask how much cheaper/costlier you would need to make each hotel such that the individual would be exactly indifferent to A. This idea is captured by the concepts of equivalent and compensating variation. However, for this you would need to know a lot about each consumer.
A simpler measure of value (but less meaningful to each individual) would be to measure the proportion $p(A \succsim B)$ of individuals which would prefer A over B. To each individual, this number would of course be meaningless. Notice also that the numbers obtained need not be consistent across hotels. Indeed, $p(A\succsim B)> .5$ and $p(B \succsim C)>.5$ does not imply $p(A \succsim C)>.5$. This follows from the cyclicality of pairwise majority voting. Arrow's impossibility theorem tells us that under some reasonable assumptions, there exists no aggregation procedure of ordinal preferences. Thus, unless you are willing to make stronger assumptions on preferences (than most economists are willing to make), it will be impossible to measure the value of hotels.
