Saturday, February 27, 2021

The Ring Announcer's Dilemma

The below feels like one of those math puzzles, though I don't actually know enough about math puzzles to know if it really is one. It is something I've genuinely noticed and wondered about when watching boxing.

In boxing, there are functionally six types of decisions: A unanimous decision, a split decision, a majority decision, as well as a unanimous draw, a split draw, and a majority draw.

  • In a unanimous decision, all three judges agree in scoring the bout in favor of one fighter.
  • In a split decision, two judges score the bout in favor of one fighter, while the third scores it favor of the other.
  • In a majority decision, two judges score the bout in favor of one fighter, while the third scores it draw.
  • In a unanimous draw, all three judges score the bout a draw.
  • In a split draw, one judge votes for fighter A, one for fighter B, and the third scores it a draw.
  • In a majority draw, one judge votes for one fighter, but the other two judges score it a draw.

When a ring announcer gets set to tell the audience the judges' decision, there are several pieces of information he needs to communicate. By the end of his announcement, the audience should know how each judge scored the fight, and for whom, and of course they need to know the actual result of the fight (who won, or that it was a draw). In general, however, he can announce the three judges' decision in any order he likes. Being a good performer, the announcer would like there to be as much suspense as possible. In practice that means he wants to the last piece of information he reveals to be the result.

The dilemma is as follows: how, if it all, can an announcer accomplish that goal in the case of a majority draw?

Start with a unanimous decision. A bad announcer might deliver the decision this way:

"Judge A had the bout 77-75 for Doe, Judge B had the bout 78-75 for Doe, and Judge C had the bout 78-74 for the winner, John Doe!"

Notice how once the crowd knows both A and B voted for Doe, they know the result even before it is announced. John Doe has at least a majority of the judges, so he won. In order to achieve the result of not tipping off the audience until the very end, a better announcement might go as follows:

"Judge A had the bout 77-75, Judge B had the bout 78-75, and Judge C had the bout 78-74, all for the winner, by unanimous decision, John Doe!"

Notice how by the end everyone knows how each judge voted, and for whom, but the last piece of information they got was the result. Until "John Doe" was said, the crowd didn't know the result of the match.

One can do this for most types of decision. A split decision can be announced like this:

Judge A had the bout 77-75 for Doe. Judge B had the bout 77-75 for Smith. And Judge C had the fight 77-75 for the winner, by split decision, John Doe!"

This works so long as the announcer is permitted to choose what order he delivers the judges' verdicts (i.e., he can make sure the one card for Smith is announced either first or second).

Here's a split draw:

Judge A had the bout 77-75 for Doe. Judge B had the bout 77-75 for Smith. And Judge C had the fight 76-76 -- this fight is a draw!

Here's a unanimous draw: 

All three judges scored the fight the same, 76-76 -- this bout is a draw!

And here's a majority decision:

Judge A scored the bout 76-76. Overruled by Judge B, who scored the bout 77-75, and Judge C, who scored the bout 78-74, for the winner, by majority decision, John Doe!

The majority decision is the toughest one so far -- the alert listener knows once the announcer says the word "overruled" that a majority decision is coming*, but still doesn't know who won.

But what of a majority draw? Consider the following:

Judge A scores the fight 77-75 for Doe. Overruled by judges B and C, who each score the bout even -- this fight is a majority draw!

Here one knows the result of the fight -- that it's a majority draw -- the moment the announcer says "overruled". Why? Well, there are only two ways that A's score for Doe could be overruled -- either B and C voted for Doe's opponent, or they voted for a draw. If it was the former, though, this would be a terrible way to announce it, as the audience would know who won as soon as just one of Judge B or C's card was announced for the other fighter. There'd be no suspense. Given that, we all know that the card was overruled by two judges voting for a draw even before we actually hear it.

What happens if you announce the cards in a different order?

Judge B and C score the fight 76-76, a draw, overruling Judge A, who scored the fight 77-75 Doe.

Nope -- that gives away the result before we ever hear Judge A's card. Similar problems emerge if you try to do something like going B (draw), A (Doe), C (draw) -- once you've revealed that B voted draw and A voted for Doe, then you know that if anyone won it has to be Doe (by majority decision), which means that if Doe did win you'd know as soon as the announcer gave a non-draw score even before they told you who the judge voted for -- and knowing that the announcer wouldn't do that, you know that C's score is going to be a draw and that the fight will be a majority draw. 

*Deep breath*

So ... is there a resolution to this? Is there a way for a ring announcer to announce a majority draw without sapping it of all the drama? I don't know. And I don't know if this "dilemma" reveals anything interesting. But I have noticed it, and haven't been able to solve it (if it can be solved).

* How does he know? Because Judge A's decision to score the fight a draw could only be "overruled" if the two other judges did not have it a draw and did vote for the same person to win. If one or both scored it a draw, then the fight would be either a majority or unanimous draw. And if they didn't score the fight even but voted for different fighters to win, then the result would be a split draw.


Erl said...

Is it obligatory or merely customary to say "overruled"?

Benjamin Lewis said...

Erl's questions makes sense. In a similar direction - why can't this be announced the same way the split decision is done, with the 2 majority cards announced individually from each other?

David Schraub said...

To Erl's question -- let's say customary. How does that change things?

BL: So the idea is going "Judge A has the fight even, Judge B has the fight 77-75 Doe, and Judge C has the fight even -- the fight's a draw"?

The problem is that after the first two announcements the drama is gone. Yes, it's possible that Judge C has the fight for Doe and Doe wins by majority decision -- but announcing a decision that way is a bad way to announce a majority decision because we know even before the last judge's card is announced that if anyone won, it's Doe (Smith has no chance at winning). It reveals more information than necessary, earlier than necessary (though in this case it's "who didn't win" rather than "who won"). Compare that to the "standard" way of announcing a majority decision, where one doesn't reveal until the last possible moment who has the majority cards. And because a majority decision consequently wouldn't be announced that way, we know that the only alternative is a draw.

Erl said...

If we have some flexibility around "overruled", it opens up my preferred strategy:

"Judge A has the fight even, Judge B has the fight 77-75, and Judge C has the fight even—THE FIGHT'S A DRAW, overruling Judge B's decision for Doe"

Treat the last clause as a sort of appendix—you can let them start clapping and just speak into the applause—and you've sustained suspense until the final result, with Doe, Roe, and draw outcomes all on the table.

The downside is, yeah, you have to sort of cram that extra data in there, and at a meta-level unless you call some majority decisions in this way people will realize that this level of ambiguity portends majority draws.

Unknown said...

I don't know the answer, but this is how I would approach the question.

In a classic game of 20 questions, you want to ask questions that maximize how much you learn on average with each question. This roughly the same as maximizing what you can exclude. In this case, a question that excludes exactly half of the possibilities gains you the most information on average. However, the order matters. Asking whether its green doesn't tell you much if you already know its a plant, for example.

A decision tree is a machine learning algorithm that finds a sequence of attributes that will eliminate the maximal number of possibilities as quickly as possible (see the Wikipedia page on information gain in decision trees).

Roughly speaking, you want to do the opposite of that (minimize instead of maximize information gain). Encode the announcements and outcome (very probably the real work involved in this problem), and apply the machine.