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A134815
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Leader-Follower four person 4 X 4 matrix zero sum game Markov in a 16 by 16 matrix: Follower matrix: MA={{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {1, 0, 0, 1}}: game value =1/3 Leader Matrix: MB={{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {1, 0, 0, 5}}: game value=-1 Characteristic polynomial: 1 + 8 x^3 - 4 x^4 + 18 x^6 - 24 x^7 + 6 x^8 + 16 x^9 - 36 x^10 + 24 x^11 + x^12 - 16 x^13 + 18 x^14 - 8 x^15 + x^16.
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0
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4, 9, 35, 162, 790, 3923, 19602, 98108, 491242, 2460009, 12319475, 61695247, 308967153, 1547295326, 7748795485, 38805671816, 194337325053, 973233918963, 4873918388052, 24408397608972, 122236325365629, 612154860741196
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OFFSET
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1,1
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COMMENTS
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Ratio is: 5.00796 Roots: three copies of the theta1 4b4 Pisot and new pisot NSolve[ -1 - 5 x^3 + x^4 == 0, x] {{x -> -0.564325}, {x -> 0.278181 - 0.525792I}, {x -> 0.278181 + 0.525792I]}, {x -> 5.00796}} Total game value: Det[M]/(Sum[Sum[If[i == j, M[[i, j]], 0], {i, 1, 16}], {j, 1, 16}] - Sum[Sum[If[i ==j, 0, M[[i, j]]], {i, 1, 16}], {j, 1, 16}])=-1/8 It seems possible that this kind of game cam be generalized to: Follower:Game_Value[MA]=1/(n-1) Leader:Game_Value[MB]=-1 Where the leader gets an (n+1) point payoff.
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LINKS
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FORMULA
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M = {{0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 5}} v[1] = {1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1}; v[n]=M.v[n-1]; a(n) = Sum[v[n][[i]],{i,1,16}]
G.f.: -x*(3*x^7+2*x^6+x^5+15*x^4+3*x^3-x^2+15*x-4)/((x^4+x-1)*(x^4+5*x-1)). [Colin Barker, Nov 01 2012]
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MATHEMATICA
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M = {{0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 5}}; v[1] = {1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1}; v[n_] := v[n] = M.v[n - 1]; a = Table[Apply[Plus, v[n]], {n, 1, 50}] Det[M - x*IdentityMatrix[16]] Factor[%] aaa = Table[x /. NSolve[Det[M - x*IdentityMatrix[16]] == 0, x][[n]], {n, 1, 16}] a1 = Table[N[a[[n]]/a[[n - 1]]], {n, 7, 50}]
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CROSSREFS
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KEYWORD
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nonn,uned,easy,changed
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AUTHOR
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STATUS
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approved
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