## Sunday, December 24, 2006

The Mod Mod Infinite Rebounds

Ages ago, I'd written a post on Infinite Rebounds as part of a series discussing scoring formats in quizzes. That post has a couple of errors which came to notice in time, and is slightly obsolete now. This current post aims to correct those errors and record the current set of conventions that are followed in IR (at least in the BC circles). This post is also partly aimed at perhaps reaching a consensus on how we should run this method here - it is after all the most popular form of conducting a quiz in these parts - so as to avoid the usual raised eyebrows just before the finals of a quiz begin when the quizmaster announces his nth variation.

Infinite Rebounds a.k.a. Infinite Bounds (IR) is one of the algorithms to answer the timeless quizzing question of "who gets the next question?". One answer is in these commonly agreed conventions is IR:

(Some terms and abbreviations:
* Q{n-1} : the previous question which has been completed
* Q{n} : the next question which is to be asked
* S: the team that successfully answered Q{n-1}
* P: the team that started Q{n-1}
* Normal passing direction: The usual passing order that a quiz begins with i.e. A->B->...->F->A...
* Reverse: F->E->...->A->F...
* S-1(P-1): the team seated "before" S(P) in the Normal passing direction
* S+1(P+1): the team seated "after" S(P) in the Normal passing direction
)

Rule 1. Q{n} after Q{n-1} was correctly answered: The choice of the team to receive Q{n} depends on how Q{n-1} was answered. If Q{n-1} was answered correctly by a team S, Q{n} goes to S+1.

Analysis:
* IR attempts to keep things "fair" for all teams by keeping the number of attempts on questions as equal as possible. "Equality" not only over the entire length of a quiz, but also over short windows of time. In IR, this means that at any point, no team will have an unfairly greater number of attempts at questions than any other team.
* We know that the fairest possible system is for all teams to simultaneously attempt all questions. But this is not possible for all questions in a stage-final-format, and so every algorithm needs to address where a question starts and how it passes. Here 'fairness' means for teams to at least attempt almost, if not exactly, the same number of questions. For this, one needs to also convince oneself that theoretically there is no difference between attempting a question on the direct or on a pass.
* In earlier systems like Direct-and-Pass, some teams could enjoy several more attempts than others in a single round. Perhaps over the length of the quiz, the teams may have similar number of attempts, but in a local window, this will not be true.
* A Team is not penalised for sitting next to a "good" team that gobbles up all questions before they can get to it.
* IR always increments the next question marker, ensuring that no team can get an attempt again unless all teams after it have had one attempt since.
* We can state the fundamental principle of IR as: "keep the number of attempts equal". Earlier, we thought it would make sense to re-start next to the team that last got points. But on reflection, this is not quite correct.

Rule 2. Q{n} after Q{n-1} was not correctly answered:
(Before modification) If Q{n-1} went through unanswered, Q{n} will go to P+1 i.e. the team next to the one that started the previous question.
(Modification 1) If Q{n-1} went unanswered, Q{n} goes back to P i.e. the same team with which the previous question started.

Analysis:
* Earlier, it was felt that no team should start a question twice in a row. However, if we want to keep the number of attempts equal, then here the old rule violated the principle. This is because the team that started the last question is now pushed to the end of the new cycle. Unless the question is unanswered by the rest, this team loses an attempt while the others gain one.
* Applying the modification ensures that at any given point in the quiz, no team can have more than 1 additional attempts over any other team in the final. (Having "1 more" means that the other teams will attempt questions to square the count before this team can increment its count.) In the previous system, this could not be ensured. In fact, data shows that a difference of 2 or more is often seen with the older system.
* This modification is as simple to run, and more importantly, it does not penalise a team just because the quiz-setter asked a tough question that none of the teams could answer.
Rule 3. Next question after partial points:
(Before modification) Sometimes a question is made up of multiple sub-answers. Suppose such a question Q{n-1} is attempted by all teams, some of which who give partial correct answers. At the end, the quiz-master awards partial points to those teams with correct contributions. In this case, who gets Q{n}? Previously, the question would be awarded to the team next in passing order to the last team to get points.
(Modification 2) Here, since all teams have attempted the previous question, the question heads back again to P.
(There is no consensus on this point)

Analysis:
* Again, the previous system believed in re-starting next to the team that last got points. Instead, if we appeal to the spirit of the principle of equal attempts, Q{n} should go back to P, as everyone's had an attempt on the previous question.
Rule 4. Next question after round reversal: A question is asked and completed. The order of passing now reverses. Who gets the next question? If modification #1 is implemented and if the last question was unanswered, then Q{n} goes back to P. This is a trivial case. In all the other cases, there have been several variations here and no consensus has really emerged. The variations:

* Q{n} is given to S-1 (or P+1 in case no modification is applied) and the passing is in Reverse
* Q{n} is given to S+1 and the passing then Reverses
* Breaking with IR here, Q{n} starts at the other extreme i.e. Team F (assuming the previous half started with Team A)

Analysis:
* It seems to me that each of the above rules have their pros and cons, and hence their respective advocates. None of these seem to be able to give any firm guarantees, so I think this much dissimilarity in number of attempts can be tolerated. Perhaps the best way is to toss a coin and pick the starting team in each segment randomly. Random is better than Bad :-)

The rest of the 'philosophical' aspects can be read in the previous posts. I hope to bring out the emphasis on the "equality in number of attempts" to help guide all minor quibbles. Of course, there is no intention of imposing some kind of standard, but perhaps leading to a convention that say most open quizzes in Pune and even Bombay can adopt. My personal basket of conventions would be to include both the modifications on points 2 and 3. Hopefully, quizmasters will think clearly about the options available to them instead of making an arbitrary and dogmatic choice. Of course, there are those who say IR is dogma :-), but it's the best we've got at the moment, is it not?

Kindly comment on issues.

P.S.: I remarked that "theoretically, there is no difference between attempting a question on the direct or on a pass". This is not quite true. In fact, practically, there is a difference between all teams when they attempt. The earlier you attempt a question, you have less to work with from answers previously given, but you also benefit from lack of "Forward Bias" (example embedded in this post). But overall, does it matter? I think not.

P.P.S: Apologies for the use of short-forms etc. etc. - all in the name of brevity (esp. if you consider the size of this post!).

P.^3.S: Special mention of Arnold D'Souza (as promised) for independently re-discovering modification #1 in Pune circles :-)

Salil said...

While stating Rule 3, you are making an assumption that *all* teams have attempted that question. This may not be the case all the time.

Partial points can be awarded even if all the teams haven't attempted. Eg. questions started at A and ended at E, with partial points awarded to B & E. Here, F didn't get an attempt, so to keep no. of attempts equal, the next question should go to F.

Ramanand said...

Salil: I wrote it initially, then removed it, since it's a little similar to Rule 1 and I wanted to focus on partial points where everyone gets a chance like it happened in the last quiz. But yes, point to be noted.

Shamanth said...

Ref. Rule 4, the question immediately after round reversal:

The fairest way of doing this is to start from the other extreme, ie team F in a 6 team quiz.

Now, for instance, the last question before reversal is answered by team C. At this point, teams A, B and C have one attempt more than teams D, E and F. As the quiz progresses, any of A, B and C will have a minimum of 1 and a maximum of 2 attempts more than any of teams D, E and F. If the quiz finishes at A, all three of A, B and C will have two attempts more than all 3 of D, E and F.

However, if the last question before reversal is to C, and you reverse starting from team F, then the difference in number of attempts between teams D and E and the rest will be a minimum of 0 and a max of 1.

Though the above analysis is only for the case when round reverses at C, you'll see that it holds for other reversal points too. It simply is much fairer(=the difference in the number of attempts is minimal, 0 or 1 as against 1 or 2) to reverse rounds starting from the other extreme, ie team F or H as the case may be.

Arnold said...

Rule 3:
Salil's point is valid. Thought needs to go into this rule. There are other points that need to be considered too. For example, all teams are not getting a crack at equal points, since a team can never get points for parts that they answer correctly but have already been answered by some other team before it reached them.

Rule 4:
Starting with Team F after the Round Reversal is the best/fairest method.

Kunal said...

I think that the QM declaring what orecise variation he will be using during the quiz before he starts might be a good idea. Negotiating the rules in the middle of the quiz is not a very good isea, IMO.