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show up in intermediate steps of the algorithms for computing an optimal policy However, in most practical applications the Markov chain {Xn } is unichain for each stationary policy Policy-improvement idea A stationary policy R is said to be average cost optimal if gi (R ) gi (R) for each stationary policy R, uniformly in the initial state i It is stated without proof that an average cost optimal stationary policy R always exists Moreover, policy R is not only optimal among the class of stationary policies but it is also optimal among the class of all conceivable policies In most applications it is computationally not feasible to nd an average cost optimal policy by computing the average cost for each stationary policy separately For example, if the number of states is N and there are two actions in each state, then the number of possible stationary policies is 2N and this number grows quickly beyond any practical bound However, several algorithms can be given that lead in an effective way to an average cost optimal policy Policy iteration and value iteration are the most widely used algorithms to compute an average cost optimal policy The rst method works on the policy space and generates a sequence of improved policies, whereas the second method approximates the minimal average cost through a sequence of value functions In both methods a key role is played by the so-called relative values The relative values are the basis for a powerful improvement step The improvement step is motivated through a heuristic discussion of the relative values of a given policy R In the next section a rigorous treatment will be presented for the relative values Let us x any stationary policy R It is assumed that the Markov chain {Xn } associated with policy R has no two disjoint closed sets Then the average cost gi (R) = g(R), independently of the initial state i I The starting point is the obvious relation limn Vn (i, R)/n = g(R) for all i, where Vn (i, R) denotes the total expected costs over the rst n decision epochs when the initial state is i and policy R is used This relation motivates the heuristic assumption that bias values vi (R), i I , exist such that, for each i I , Vn (i, R) ng(R) + i (R) for n large (628).

Note that i (R) j (R) Vn (i, R) Vn (j, R) for n large Thus i (R) j (R) measures the difference in total expected costs when starting in state i rather than in state j , given that policy R is followed This explains the name of relative values for the i (R) We have the recursion equation Vn (i, R) = ci (Ri ) +

ISUP[B-party] ISUP[C-party]

De ne the m m diagonal matrices = diag( 1 , , m )

(1) (m) and Ak = diag(ak , , ak ),

Let the m m matrix Dk for k = 0, 1, be de ned by D0 = Q and Dk = Ak ,

k = 1, 2, (141) (142)

.

k = 1, 2,

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assist in the company s management The report concluded that the company s secret, such as it was, boiled down to 30 years of aggressive yet conservative management in other words, bedrock values coupled with bold ventures, the very same equation Walgreens has relied on for over a century No sooner had the analysts from Davis and Company departed than the reporters from Fortune magazine arrived to produce the lead story for the September 1934 issue on the chain35 Titled, 500 Corner Drugstores, the piece extolled Walgreens merchandising prowess A druggist s technique nowadays is to put before you everything that you can conceivably need in your home, at a price calculated to attract you Walgreens decision to sell much more than drugs, which began when Charles first put together his display of pots and pans years earlier, had caught on Across the fountain [of the average Walgreens store] each year flow 455 gallons of chocolate [and] 45 gallons of vanilla syrup, the author said Tables for food service alone, the magazine reported, accounted for half the floor space in many stores Although the pharmacy department brought in a paltry 3 percent of profits, Charles remained committed to staffing a fullservice pharmacy in every store The Fortune staffers were smart enough to look beyond the soda fountains that hooked readers and customers alike Walgreens success, they said, was due in part to some very smart financial decisions that customers never see The once powerful Liggett chain and the United Drug network were in trouble, Fortune explained, because they had locked themselves into long-term leases in the 1920s that had become oppressive in the 1930s Walgreens executives, in contrast, signed much more favorable leases and only leased what they needed without taking on extra space they would then have to sublease to others, putting the onus on the chain Fortune gave particular credit to Robert Knight, whom it called the financial brain of the outfit, who had analyzed, trimmed, and coordinated Walgreens financial methods to weather the storm that broke in 1929 In.

Using (Dk )ij to denote the (i, j )th element of the matrix Dk , de ne the generating function Dij (z) by

Dij (z) =

(Dk )ij zk ,

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