I continually play Wii game Tennis with three players P1, P2, P3 (they are built

Question

I continually play Wii game Tennis with three players P1, P2, P3 (they are built-in and controlled by the Wii) whose skill levels are 1, 2, 3 respectively. At each game, I play with one of these three players and my winning probabilities against them are 0.8, 0.5, and 0.2 respectively. After each game, Wii software selects my opponent for the next game as follows: If my current opponent has level i and I lose the game, my next opponent will equally likely to have levels j = i; If my current opponent has level i and I win the game, my next opponent will equally likely to have levels j = i. For instance, if I win the game and current opponent has level 2, then my next opponent will equally likely to have level 2 or 3; if I lose a game and current opponent has level 1, then my next opponent is level 1, which is the lowest level.

A. Construct an appropriate Markov chain, specify its state space, and find its transition probability matrix.

B. Is the Markov Chain irreducible? Are the states transient or recurrent?

C. Explain how to find the long-run fraction of times that my opponent has level 2. (You only need to list the equations, no need to solve. You should specify which of the unknown numbers in the equation is the answer.)

Explanation / Answer

OK, I'm not sure that I understand the rules. Here is how I take them. If I win against level three, I always play level three again. If I win against level two, I play level two half the time and level three half the time. If I lose against level two, I play level two half the time and level one half the time. (A) Here are the States and transistion probabilities: Playing level one: If I play level one, the chance I play level one again next time is the chance that I lose plus half the chance that I win, or 0.2 +0.8*(1/2) = 0.6 If I play level one, the chance that I play level two next time is half the chance that I win this time, or 0.8*(1/2) = 0.4 If I play level one, the chance that I play level three next time is zero. Playing level two: If I play level two, the chance I play level one next time is one half the chance that I lose, or 0.5*(1/2) If I play level two, the chance I play level two next time is one half the chance that I lose, plus one half the chance that I win, or 0.5*(1/2) + 0.5*(1/2) If I play level two, the chance I play level three next time is one half the chance that I win, or 0.5*(1/2) Playing level three: If I play level three, the chance that I play level one next time is zero. If I play level three, the chance that I play level two next time is one half the chance that I lose or 0.8*(1/2) If I play level three, the chance that I play level three next time is one half the chance that I lose plus the chance that I win, or .8*(1/2) + 0.2 Now we can add up the transistion probabilities and put them into a matrix. P1 P2 P3 P1 0.6 0.4 0 P2 0.25 0.5 0.25 P3 0 0.4 0.6 On the left are the initial states, across the top (if the formatting stays how it is now) are the next states, and the values in the matrix are the transition probabilities. Note that the values in each row add to one. This means that there is a 100% chance that I will have a game once I finish this game. So if I start in P2 there is a 25% chance I will play P3 next game. I put this into MS Excel and raised the matrix to higher powers until the matrix stopped changing (you could make your own or I could send you the one I made if you'd like) matrix squared 0.46 0.44 0.1 0.275 0.45 0.275 0.1 0.44 0.46 matrix to 64th power 0.277777778 0.444444444 0.277777778 0.277777778 0.444444444 0.277777778 0.277777778 0.444444444 0.277777778 Here we see that it doesn't matter which level player you started playing against, after many games, the probability that you are playing a level one player next is 28%, the probability you are playing a level two player next is 44%, and the probability you are playing a level three player next is 28%. (B) Since we can (not right away, but eventually) go from playing level one to playing level three, or from three to one, the Markov Chain is irreducible. There are no forbidden transitions that stay out of reach after many games. If we start playing a level one player, we will (eventually) go back to playing a level one player. The same can be said for each possibility, so the states are recurrent, not transient. (C) To find the long-run fraction of times you would keep multiplying the matrix until the values stoppped changing and then look at the probability to be in starte P2 (it is 44.44% no matter where you started). Since the matrix is made from the probability equations, those are the equations that you solve, and you would keep multiplying them out. Just to have somewhere to begin, assuming that there was a 1/3 chance of starting with each level, the first iteration of this this would then look like: Prob of playing P2 = (1/3).8*(1/2) + (1/3)[.5*(1/2) + .5*(1/2)] +(1/3).8*(1/2) and then you would keep this going to more and more games. It is interesting to note that the probability is already (.4+.5+.4)/3 = 43% after this first iteration, so all of the other iterations just get us that last percent accuracy.

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