At 6/23/19, 5:25 PM, "Keith F. Lynch" <kfl@KeithLynch.net> wrote,
If all 8 players play each other, and any result is possible, including ones that violate consistency and transitivity, that conveys 28 bits. With consistency and transitivity, most patterns of 28 bits are impossible. A complete ordering conveys log2(8!) bits, about 15.3 bits. Either way, with just 16 games we can't get more than 16 bits.
It seems to me prior probabilities are necessary to talk about information. An idea like that matchups *tend* to be *somewhat* consistent and transitive is the beginning of a prior. If I think that the total probability of one or more violation of transitivity or consistency is one in a binary million (2**20), then that probability gets divided between all the violating outcomes, so there would be more than 20 bits of information in any one of them. On the other hand, the more unlevel the playing field, so to speak, the less *expected* surprise. From my impression of statistics, what you do is start with a general model of how things work, which has hidden parameters, and then given an actual outcome, try to derive the hidden parameters most likely to give that outcome. (I actually don't understand why the peak of the curve is given such importance.) So if each team has a "skill level", and the odds of A wins/B wins/tie are just a function of the two skills, then once the tournament is over you can back-figure the most likely vector of 8 skills, and from that pick the most skillful team! Easy peasy. But, even having no prejudices about the teams to start with (not to mention ideas about how they interact), the amount of information the outcome represents depends on both the model (of behind-the-scenes Truth, not the tournament plan) and the actual outcome. Or, you can calculate the expected number of bits and turn off the TV satisfied. I understand the Google pagerank algorithm is equivalent to applying one (linear?) model to this problem. Of course, the information everyone wants to know is, who's the best team? Which for 8 teams is 3 bits without any prejudices. Then the question is, what are the odds a given method of matching teams gives the right answer (given some model of "best teams")? Because the less likely you're right, the fewer bits you actually delivered. On the other hand, if the best you can deliver is three bits, there are diminishing... returns to even... talking about... this. --Steve