Response by Doron Zeilberger
Posted: Feb. 24, 2020.
The paper A Multi-Computational Exploration of Some Games of Pure Chance by Thotsaporn Aek Thanatipanonda and myself has recently been accepted to the Journal of Symbolic Computation by member of Editorial Board Manuel Kauers. Howver, one of the reports showed that the referee misunderstood our paper, and this is our response.
Dear Anonymous referee
It seems that you completely missed the "big picture", and also the "little picture". Unfortunately, you are anonymous. I asked Manuel Kauers to ask you to email me, identifying yourself, so that I can directly communicate with you. He kindly did it. (In that regard, see my Opinion 3).
I also offered you the option to respond to this response, via Manuel, staying anonymous. Manuel just informed me that you decided not to respond.
I decided to make my response public, so other people may learn from your mistakes. Manuel Kauers kindly agreed to email you a link to this webpage.
Your original text is in red and my response is in black.
In this article, the authors consider "games of pure chance" and give algorithms to compute quantities related to the probability distribution of the number of rounds required to win in the single-player
So far correct
and two-player situation
This is a much harder problem, and we give algorithms for the probability of the first player winning, that, has not, as far as we know, been considered before, and is one of the novelties of our paper, that you apparently missed.
Typical quantities considered are the expectation, variance and higher moments. In Chapter 1, a single game, such as "Snakes and Ladders" is analyzed. Later chapters can be phrased as random walk problems. In Chapter 2, a family of walks indexed by the integers, where the jumps are positive, is analyzed. In Chapter 3, a variant of the games in Chapter 2, with one positive and one negative jump step, is analyzed. Finally, in Chapter 4, arbitrary positive and negative jump steps are allowed.
I agree with this summary.
Mathematically, this amounts to studying Markov chains with an absorbing state and computing the time needed to get absorbed. This is a classical subject and is a topic of standard textbooks. I'm surprised that the authors don't cite standard textbooks on Markov chains. For example, see https://en.wikipedia.org/wiki/Absorbing_Markov_chain.
So much is obviously true. This subject (NUMERICAL Markov chains) is so standard, that no reference is needed.
It is clear from standard theory that such probabilities are rational functions of the parameters, which, unless I am mistaken, is one of the key points of Chapter 1.
Even in this simple case, that is mostly "warm up", it goes beyond standard treatment. We give algorithms, fully implemented in Maple, to actually compute the probability generating functions for the duration.
In particular, the example of Snakes and Ladders is mentioned in Section 3.2 and has been studied in this article: S. C. Althoen, L. King and K. Schilling (1993), "How Long Is a Game of Snakes and Ladders?", The Mathematical Gazette, Vol. 77, No. 478: 71–76.
They only do expectation, using a NUMERICAL Markov chain, with a one-player game. To do the two-player game, with their approach one would need a Markov chain with 1002=10000 states, and hence a matrix of that dimension, to do the three-player version one would need 1003=1000000 states etc. The novelty of our beautiful approach is that we only need 100 states, using symbolic-computation rather than numerical computation. Our Theorem 2 is novel, and and is surprising: the probability of the first player winning is a rational number. This follows from the theory of C-finite sequences. It is clear that the learned referee missed that point (see below)
In Chapters 2, 3 and 4, the calculations are done using the bivariate generating function of the probabilities. Some of the results follow from standard machinery of Grobner bases.
Groebner bases is an important implementation tool, but the results themselves do not need them.
If I understand correctly, the real innovation is in Chapter 4, where this generating function is shown to be algebraic.
You did not understand correctly, all the chapters are innovative.
However, that seems to use results from the thesis of Bryan Ek. Therefore, the main value of this article is in combining many ideas and packaging it in a potentially useful way.
Bryan Ek did it in the enumerative context, we extend it to the probabilistic context. Also this was only one of the points. We also used the defining algebraic equation to deduce expected length and higher moments, using sophisticated symbolic computation for the duration, and using Wilf-Zeilberger algorithmic Proof theory to deduce
"probability of the first player winning"
is entirely novel, and was missed by the learned referee.
Here are some other minor issues.
- p 4, l 6: degree _two_.
This "correction" shows that the referee completely misunderstood our paper, the degree is obviously one, as stated in the paper
- p 7, l -5: I don't understand the calculation leading to that unnumbered equation. Some explanation is needed here.
This, once again, shows that the referee is unqualified. The deduction is trivial, and spelling it out would have offended the reader. The sum of the ak is 1, hence their square is 1, and the rest follows by symmetry (have you heard of the commutative rule for multiplying real numbers? ab=ab, hence ak1ak2=ak2ak1)
- p 8: The (sub)section titled "A short user's manual for the Maple package SnakesAndLadders.txt" fits in better as a part of "Supplementary material" rather than the main text.
There is lots of stuff in the "supplementary material", but it is nice to have this here.
- p 10: Define F_n(t) carefully, perhaps with the coin-toss example.
It is defined carefully. We said "prob. generating function according to duration"
- p 10, l -9: I don't see how the claim "..., and some polynomial Q_1(x) whose roots are all larger than 1 in absolute value." holds.
We added a short sentence in the revised version.
- p 17, Prop. 5 and later: Replace "numer" by "number"
FINALLY you made a good point. Corrected.
One final general comment: our paper is also remarkable in that, everything is fully implemented (in Maple)
THE IMPLEMENTATION IS THE MESSAGE