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Markov-opoly

Markov-opoly. Markov chains: an Applied Approach By Daniel Huang and Mo Dwyer. What is a Markov Chain, Anyway?. It is a sequence of states where every future state is independent of the preceding ones, except for the n-1 state. The Process. Create a “transition matrix.”

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Markov-opoly

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  1. Markov-opoly Markov chains: an Applied Approach By Daniel Huang and Mo Dwyer

  2. What is a Markov Chain, Anyway? • It is a sequence of states where every future state is independent of the preceding ones, except for the n-1 state.

  3. The Process • Create a “transition matrix.” • This defines the probability of something being in any given location in the state you are interested in. • Raise the transition matrix to the nth power.

  4. Confused? Have an example… The state: a car rental agency has three locations in LA: • Downtown location (A) • East end location (B) • West end location (C).

  5. The agency's statistician has determined the following: Of the calls to the Downtown location, 30% are delivered in Downtown area, 30% are delivered in the East end, and 40% are delivered in the West end Of the calls to the East end location, 40% are delivered in Downtown area, 40% are delivered in the East end, and 20% are delivered in the West end Of the calls to the West end location, 50% are delivered in Downtown area, 30% are delivered in the East end, and 20% are delivered in the West end.

  6. T= T7= T2= The Transition Matrix: Notice how it starts to converge!

  7. Tn= You now have the probability distribution of the drivers at time n!

  8. So where will you be next? • Take the transition matrix and multiply by the current state. • That is: XO+1=TXO

  9. So where will you be next? • Take the transition matrix and multiply by the current state. • That is: XO+1=TXO • After time n approaching infinity, the resulting vector is not dependent on Xo P=TnX

  10. Our Project: Creating a Markov chain to predict monopoly moves.

  11. For Monopoly • First, find the probability of rolling a certain number, and thus landing on a certain square. • Add the prob of Chance or Community Chest sending you somewhere.

  12. Thank goodness for MATLAB • You end up with a 40x40 matrix because there are 40 squares to land on.

  13. Et c. …

  14. Alternatively: • You can also use eigenvectors to solve steady-states!

  15. Alternatively: • You can also use eigenvectors to solve steady-states! • Take λ=1 • Tp=p • p is the probability vector of dimension mx1. Its elements add to equal 1.

  16. THE END! Any questions?

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