Expected Number of Rolls Until All Rental Properties Have Been Landed On (Hence Purchased)

• A player who lands on an unowned rental property may purchase it from the bank at the printed price. Otherwise, the property is auctioned by the bank to the highest bidder. Since there is no minimum bid, we may assume that some player purchases the property.
  
  
• An exact computation would require enlarging the size of the state space by a factor of 2²⁸, since there are this many possible subsets of rental properties visited. So we will content ourselves with an approximation and some bounds.
  
  
• The expected number of visits to state i during the first n rolls is given by the ith component of the vector Σ a · P^k, where a is the initial distribution vector and the sum is taken over 0 ≤ k ≤ n.
  
  
• Now to get an approximation to the expected number of rolls, find the smallest value of n such that the components corresponding to rental properties are ≥ 1.
  
  
• Table
  
  
• A trivial lower bound is 28.
  
  
• To get an upper bound, we first compute the matrix T whose (i,j) entry is the expected number of rolls to get from state i to state j. The jth column of T is obtained by first computing (I - Q_j)^-11 (where I is the identity matrix, Q_j is the matrix obtained by removing from P the jth row and column, and 1 is a column vector of ones) and then inserting a zero in the jth position.
  
  
• Now consider any tour (starting from Go) visiting all rental properties. Use T as the distance matrix and sum the total distance along the tour to obtain an upper bound on the expected number of rolls until all rental properties have been landed on. A minimum tour gives the best upper bound obtainable by this approach.
  
  
• For the LJ model, a minimum tour (with 1145.15 expected rolls) is:
  
  Go, ConnecticutAve, StJamesPlace, IndianaAve, MarvinGardens, ReadingRR, ElectricCo, NewYorkAve, AtlanticAve, ParkPlace, VirginiaAve, KentuckyAve, WaterWorks, ShortLineRR, BalticAve, StCharlesPlace, TennesseeAve, BandORR, NorthCarolinaAve, Boardwalk, OrientalAve, StatesAve, VentnorAve, PennsylvaniaAve, MediterraneanAve, VermontAve, PennsylvaniaRR, IllinoisAve, PacificAve, Go,
  
  where the number of doubles thrown is zero for each property.
  
  
• For the RIJ model, a minimum tour (with 1203.88 expected rolls) is:
  
  Go, ConnecticutAve, StJamesPlace, IndianaAve, MarvinGardens, ParkPlace, ReadingRR, ElectricCo, NewYorkAve, AtlanticAve, VirginiaAve, KentuckyAve, WaterWorks, ShortLineRR, BalticAve, StCharlesPlace, TennesseeAve, BandORR, NorthCarolinaAve, Boardwalk, OrientalAve, StatesAve, VentnorAve, PennsylvaniaAve, MediterraneanAve, VermontAve, PennsylvaniaRR, IllinoisAve, PacificAve, Go,
  
  where the number of doubles thrown is zero for each property.
  
  
• Note that these two minimum tours differ only in the position of ParkPlace.
  
  
• In practice, many fewer rolls (about 100) are needed. Here, the assumption of a single player does not apply. The expected number of rolls decreases as the number of players increases.

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