At the start of each month, Microsoft classifies Xbox Live users either as Gold, Silver or Bronze members. Gold members have a 70% chance of staying a Gold member, and can never become Bronze members by the end of the next month. If a user is a Silver member at the start of a given month, then they are 3 times as likely as becoming a Gold member compared with a Bronze member at the start of the next month. They also have a 40% chance of staying a Silver member.
Finally, 50% of Bronze members remain Bronze members, while 50% become Silver members at the start of the next month. For the 50% of Bronze members who would become Silver members at the start of the next month, Microsoft randomly offers 10% of these users a chance to become a Gold member at the start of the next months (hence it is possible for Bronze members to become Gold members in one month). To accept this offer, the user must pay a small fee, so not everybody accepts. Microsoft estimates that 90% of Bronze members accept this offer.
(a) Set up the probability transition matrix for this Markov chain, with a user’s membership considered at the beginning of each month.
(b) If a user is a Bronze member at the beginning of April, what is the probability that they will be a Gold member at the beginning of June?
(c) Determine the proportion of users who are Gold members in the long run
Finally, 50% of Bronze members remain Bronze members, while 50% become Silver members at the start of the next month. For the 50% of Bronze members who would become Silver members at the start of the next month, Microsoft randomly offers 10% of these users a chance to become a Gold member at the start of the next months (hence it is possible for Bronze members to become Gold members in one month). To accept this offer, the user must pay a small fee, so not everybody accepts. Microsoft estimates that 90% of Bronze members accept this offer.
(a) Set up the probability transition matrix for this Markov chain, with a user’s membership considered at the beginning of each month.
(b) If a user is a Bronze member at the beginning of April, what is the probability that they will be a Gold member at the beginning of June?
(c) Determine the proportion of users who are Gold members in the long run
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Set up a transition matrix. The rows are the "from": first row gold, second silver, third bronze. The columns are the "to", again in the order gold, silver, bronze.
The gold row is clearly: [0.7 0.3 0]
For the silver row, the middle entry is clearly 0.4. The 1st entrt (silver-to-gold) is 3 times the third entry, (silver to bronze), and they need to add to 0.6 so they are 0.45 and 0.15 respectively:
[0.45 0.4 0.15]
For the bronze row: bronze-to-bronze = 0.5. For the remaining 0.5, 0.5 * 0.1 * 0.9 go to gold, = 0.045. The remaining 0.5 - 0.045 go to silver So the bronze row is:
[0.045 0.455 0.5]
So the overall matrix is:
[0.7 0.3 0]
[0.45 0.4 0.15]
[0.045 0.455 0.5]
For bronze to gold in 2 months there are 3 possible paths:
1. bronze to silver then silver to gold, prob 0.455 * 0.45
2. bronze to gold, then stay gold: 0.045 * 0.7
3. stay bronze, then bronze to gold: 0.5 * 0.045
Add those 3 numbers to get the prob of bronze to gold in 2 months
c. You need to solve this steady state matrix:
[g s b][P] = [g s b]
... where [P] is the matrix I gave you in (a). Then g, s and b are the long term proportions of gold, silver and bronze respectively. That's simply a matter of solving 3 simultaneous equations.
The gold row is clearly: [0.7 0.3 0]
For the silver row, the middle entry is clearly 0.4. The 1st entrt (silver-to-gold) is 3 times the third entry, (silver to bronze), and they need to add to 0.6 so they are 0.45 and 0.15 respectively:
[0.45 0.4 0.15]
For the bronze row: bronze-to-bronze = 0.5. For the remaining 0.5, 0.5 * 0.1 * 0.9 go to gold, = 0.045. The remaining 0.5 - 0.045 go to silver So the bronze row is:
[0.045 0.455 0.5]
So the overall matrix is:
[0.7 0.3 0]
[0.45 0.4 0.15]
[0.045 0.455 0.5]
For bronze to gold in 2 months there are 3 possible paths:
1. bronze to silver then silver to gold, prob 0.455 * 0.45
2. bronze to gold, then stay gold: 0.045 * 0.7
3. stay bronze, then bronze to gold: 0.5 * 0.045
Add those 3 numbers to get the prob of bronze to gold in 2 months
c. You need to solve this steady state matrix:
[g s b][P] = [g s b]
... where [P] is the matrix I gave you in (a). Then g, s and b are the long term proportions of gold, silver and bronze respectively. That's simply a matter of solving 3 simultaneous equations.