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One million players participate in a game that has 10 levels. At first, all players are at level 1.

At the end of each turn, each player rolls a twelve-sided die, numbered 1 to 12. Player advance a level if he gets a higher number of its current level number. Otherwise, it remains at the same level.

Warning! At 10, "advance a level" means return to level 1!

After some time, the number of players in each level tends to stabilize.

What will be the average number of players at level 10? Be rounded off to the nearest integer.

Any help to that problem would be appreciated. it's not a homework problem.

JeanClaudeDaudin
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1 Answers1

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You need to set up a Markov Chain and find the transition probability matrix, based on the rules described. Then, find the stationary distribution. Then, average number at level 10 (or any other level) is found from this stationary distribution.

I'm not giving the details because they are widely available.

PA6OTA
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  • I have found the stationary distribution of 1/10 at every level, but I don't figure out how to find the average number at one level. Can you help on that point? – JeanClaudeDaudin Jun 16 '14 at 14:20
  • Well, it's just the number of players times the proportion you found (!?); the stationary distribution is "a fraction of time (in the limit) the Markov chain spends in that state". – PA6OTA Jun 16 '14 at 15:36
  • if 1/10 is right, the answer would be 100000 but this answer is wrong. – JeanClaudeDaudin Jun 16 '14 at 16:00
  • To think of it, it does seems strange that the probabilities are asymmetric, but your answer is $1/10$. It seems that lower numbers (0,1,2,..) will be less likely. What is the transition matrix you're using? – PA6OTA Jun 16 '14 at 16:08
  • my transition matrix is: (1 11 0 0 0 0 0 0 0 0) (0 2 10 0 0 0 0 0 0 0) (0 0 3 9 0 0 0 0 0 0) (0 0 0 4 8 0 0 0 0 0) (0 0 0 0 5 7 0 0 0 0) (0 0 0 0 0 0 6 6 0 0 0) (0 0 0 0 0 0 7 5 0 0) (0 0 0 0 0 0 0 8 4 0) (0 0 0 0 0 0 0 0 9 3) (2 0 0 0 0 0 0 0 0 10)*1/12 – JeanClaudeDaudin Jun 16 '14 at 16:34
  • The matrix is correct. This means the mistake is in computing your stationary distribution. Beware of transposition. – PA6OTA Jun 16 '14 at 16:58