AN ANALYTICAL METHOD TO CALCULATE THE ERGODIC AND DIFFERENCE MATRICES OF THE DISCOUNTED MARKOV DECISION PROCESSES

det (I - βP) = (1 - 0, 5β) (1 - β²) = (1 - β)(1 + β) (1 - 0,5β)

Hence

^ν∞ (^β) = ^{(1 - β} ∙ ^P)  ^{1 q} =C-j----X [∙] ⁺ 1 I [ [∙ ∙ ∙] ⁺ ^j----+~^ξ+ [∙ ∙ ∙]

y 1 - β 1 + β 1 - 0, 5β

Now we can calculate total finite expected rewards for given values β, β₁ = 0, 5
and β₂ = 0, 99. For β₁ = 0, 5 we obtain

and next

ν₁,∞ (0, 5) = 2 ∙ 5, 5 + 0, 666 ∙ 2, 5 + 1, 333 ∙ 0 = 11 + 1, 666 + 0 = 12, 666,
ν2,∞ (0, 5) = 2 ∙ 5, 5 - 0, 666 + 0 = 9, 334,

11

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