> &(%7 bjbjUU 7|7|l*,,,,,,$m P!P\q\\\:*\*\\**`iQ!.**0*L*\Referee's report on "RISKy Business: An In-Depth Look at the Game of RISK" by S. Blatt
Overview: This paper presents an updated mathematical model for the board game RISK that is more consistent with the actual rules of the game than other papers. It answers two probability questions related to the game without assuming any independence on dice rolls. It is easy to read and illustrates its results with nice examples. However, some interesting mathematics has been left out that I believe should appear in the paper. If the author makes some minor changes outlined below, and adds the requested mathematics, I believe the paper should be accepted.
Mathematical content: For the most part, the author does a good job explaining her results. However, there are a few times when results appear out of nowhere.
In Section 3 (Markov Chains and State-Space Models), the author should describe Markov chains and absorbing Markov chains with greater detain. Most readers are not going to be familiar with this construct, but they can easily be described. Start by indicating that the transition probabilities can be formulated as a transition matrix (do not mention them as one-step, this will only confuse readers new to Markov chains). Then describe what transient and aborbing states are. At that point, describe the general form of the transition matrix for an absorbing chain. Make sure to note that each matrix R, Q gives the probability of going from one specific transient state to a specific absorbing/transient state. Finally, the author defines the matrix S later in the paper. This needs to be defined in this section, and the result of the Theorem on page 4 needs to include S. Also, in the statement of the theorem, the author needs to either define the spectral radius of a matrix, or indicate that the result holds (specifically) for Markov chains.
In Section 4, there are only a few corrections and additions I feel would make the paper better. The last paragraph on page 4 should start "For each value of Y(1) and Y(2)", since the probability of Y(1) >Y(2) also depends on Y(2) . The same issue arises on page 7 when three die are considered. Finally, the general formulas given in Table 1 on page 9 for any s-sided die need some more explanation. Perhaps some of these can be described in the paper (such as Cases I, II, and/or III), while others included in an appendix for interested readers.
Other corrections: There are only a few minor corrections that need to be made. On page 1, line -4 from bottom, should at least 1 army remain in the currently occupied territory, while others go to the attacking territory? On Page 3, first line of the first full paragraph, should it read "an e" 2"? Also on page 3, in the last sentence of the second full paragraph, the probabilities are given in "the next section." On Page 6, the last paragraph should start "Since the total probability". Finally, on page 12, the last sentence should end with a period, not a question mark.
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