Markov chains and equilibrium distribution

guklu

New Member
#1
Two yachts, sailed by "Yacht 1" and "Yacht 2" respectively are sailing around a course. If the teams are even at the beginning of a lap then during that lap they embark on a duel and one team gains a 1 boat length advantage by the end of the lap. Otherwise the leading yacht always sails according to the current wind, while the other yacht sails separately, hoping for a wind-shift. This means that on each lap of the course the leading yacht gains one boat length over the other yacht if there is no wind-shift, while the team behind gains one boat length of the wind shifts on that lap. Suppose that Yacht 2 wins any given duel with probability p1 ∈ (0, 1) (independent of all previous duels and wind-shifts) and that with probability p2 ∈ (0, 1) there is a wind-shift on any given lap (independent of all previous wind-shifts and duels).
There is a pre-race duel, and the winner of the pre-race duel starts with a 1 boat length lead.
Let Xn be the number of boat lengths that Yacht 2 leads by the end of n laps.

For what values of p2 would this Markov chain Xn have an equilibrium distribution, and show that the probability that the race is tied at the end of 3 laps is increasing in p2.

Can anyone explain to me how to draw the transition diagram for Xn so that I can find the equilibrium distribution? Or should this require a different approach here? I've found the probability that Yacht 2 is leading the race at the end of the first lap, which is p1(1-p2), but I have no idea how to proceed from this. Any help would be appreciated.