There is a subtle difference between "being more likely to have a larger value" and "having a larger expected value" from the point of view of transitivity.
Say you can assign arbitrary integer numbers to faces of a dice.
Scenario 1: Assume dice A beats dice B if it has a larger expected value. An obviously transitive relation, i.e. if B beats C, A beats C as well.
Scenario 2: Now assume A beats B if it is more likely to roll a higher value. This is to say if 2 players roll these dices simultaneously and the one with the higher value gains one point, then dice A would be the winner in the long run.
Now the question is; if A beats B and B beats C, will A beat C necessarily?
5 comments:
اون سوال آخرت استفهام انکاری بود دیگه، نه؟ چون مثال نقض که راحت میشه زد
Yes. But do share your counter-example :) or a method for generating one
Suppose there are 3 different 3-faceted dice, having these numbers on them: A={3, 5, 6}, B={2, 4, 9}, C={1, 7, 8}. It's obvious that in the second scenario, A beats B, B beats C, and C beats A!
The first scenario is like the election in Iran, and the second one is like the election in US! it doesn't matter how much larger is the value of the dice! the only thing that matters is that most of the time, it's value is larger that the other's.
Right, the US election doesn't have that "monotonicity" property, but it's proven that if you insist that your election mechanism be monotonic, you'll lose some other desired properties.
Post a Comment