{VERSION 4 0 "IBM INTEL NT" "4.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 
2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 
{CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 
4 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier
" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }
{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 
2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot
" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }
3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 
"Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 
19 1 }}
{SECT 0 {PARA 18 "" 0 "" {TEXT -1 31 "Recurrent and Transient Sstates
" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 25 "set up transition matrix " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "#load packages\nrestart;\nwi
th(linalg):\nwith(plots):\n\n# set up parameters\ncount := 100;\nN := \+
4;\n" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm
 and trace have been redefined and unprotected\n" }}{PARA 7 "" 1 "" 
{TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&countG\"$+\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"NG\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
232 "#set up transition matrix\n# states 1 and 2 are recurrent, states
 3 and 4 are transient \nP := matrix(N,N,[ 0.5,0.4,0.1,0.1, \n        \+
          0.5,0.6,0.1,0.1,\n                  0.0,0.0,0.3,0.2,\n      \+
            0.0,0.0,0.5,0.6]);\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"PG-%'matrixG6#7&7&$\"\"&!\"\"$\"\"%F,$\"\"\"F,F/7&F*$\"\"'F,F/F/7&
$\"\"!F6F5$\"\"$F,$\"\"#F,7&F5F5F*F2" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 94 "#set up initial probabilities\nU0 := vector(N,1);\nS \+
:= sum(U0[k],k=1..N);\nU0 := evalm(U0/S):; \n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#U0G-%'vectorG6#7&\"\"\"F)F)F)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"SG\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
83 "#check\nTV := matrix(1,N,1):\ncheckP := evalm(TV&*P-TV); \ncheckU0
 := evalm(TV&*U0-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'checkPG-%'m
atrixG6#7#7&$\"\"!F+F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(check
U0G-%'vectorG6#7#\"\"!" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 34 "calcu
late probabilites and display" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 131 "#compute probabilities\nU := U0;\nUt := matrix(N,1,U0):\nfor j \+
from 1 to count do\n   U := evalm(P&*U);\n   Ut := augment(Ut,U);\n   \+
od:\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"UG%#U0G" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 61 "#matrixplot\nmatrixplot(Ut,axes=boxed,ori
entation = [-60,70]);" }}{PARA 13 "" 1 "" {GLPLOT3D 622 558 558 
{PLOTDATA 3 "6&-%%GRIDG6%;\"\"\"\"\"%;F'\"$,\"7&7aq#F'F($\"++++]F!#5$
\"++++vIF0$\"+++]ZLF0$\"+++vmNF0$\"++]FUPF0$\"++vq#)QF0$\"+]Z0&*RF0$\"
+vE$\\3%F0$\"+G]$o:%F0$\"+7pN9UF0$\"+=WPgUF0$\"+C%)=(H%F0$\"+G'RmK%F0$
\"+!f+-N%F0$\"+i$\\!pVF0$\"+y$GTQ%F0$\"+#f\">'R%F0$\"+i@%eS%F0$\"+>Ec8
WF0$\"+&)*Q(>WF0$\"+x+oCWF0$\"+]HjGWF0$\"+\\_zJWF0$\"+)3DVV%F0$\"+f*[j
V%F0$\"+d!ozV%F0$\"+MLERWF0$\"+b&*HSWF0$\"+M&G6W%F0$\"+;<zTWF0$\"+iAKU
WF0$\"+*pYFW%F0$\"+\\i3VWF0$\"+()yNVWF0$\"+*>vNW%F0$\"+[!\\PW%F0$\"+F
\"))QW%F0$\"+#R**RW%F0$\"+.%)3WWF0$\"+7'fTW%F0$\"+zl@WWF0$\"+`@EWWF0$
\"+6')HWWF0$\"+yxKWWF0$\"+76NWWF0$\"+z(pVW%F0$\"+8ZQWWF0$\"+gmRWWF0$\"
+=iSWWF0$\"+kQTWWF0$\"+\")*>WW%F0$\"+u[UWWF0$\"+)yGWW%F0$\"+?>VWWF0$\"
+EWVWWF0$\"+IkVWWF0$\"+L!QWW%F0$\"+:$RWW%F0$\"+U.WWWF0$\"+j6WWWF0$\"+?
=WWWF0$\"+XBWWWF0$\"+lFWWWF0$\"++JWWWF0$\"+qLWWWF0$\"+&eVWW%F0$\"+cPWW
WF0$\"+%*QWWWF0$\"+/SWWWF0$\"+#4WWW%F0$\"+jTWWWF0$\"+?UWWWF0$\"+lUWWWF
0$\"+,VWWWF0$\"+IVWWWF0$\"+`VWWWF0$\"+qVWWWF0$\"+&QWWW%F0$\"+'RWWW%F0$
\"+1WWWWF0$\"+9WWWWF0$\"+?WWWWF0$\"+DWWWWF0$\"+GWWWWF0$\"+JWWWWF0$\"+M
WWWWF0$\"+PWWWWF0$\"+QWWWWF0$\"+RWWWWF0$\"+SWWWWF0$\"+TWWWWF0F]wF]wF[w
F[wF[wF[wF[wF[wF[w7aqF-$\"++++]KF0$\"++++DPF0$\"+++]#4%F0$\"+++D&Q%F0$
\"++]K>YF0$\"++Dd1[F0$\"+]#pj&\\F0$\"+Dl?w]F0$\"+Mj2s^F0$\"+y@x[_F0$\"
+a)G,J&F0$\"+%>9#f`F0$\"+mC[)R&F0$\"+%3(*)HaF0$\"+y(G]X&F0$\"+MT8vaF0$
\"+=%=7\\&F0$\"+Ye3/bF0$\"+*yzV^&F0$\"+U\\hAbF0$\"+lI?HbF0$\"+jNZMbF0$
\"+i**oQbF0$\"+!3j?a&F0$\"+v:wWbF0$\"+s.#pa&F0$\"+3uk[bF0$\"+P!H+b&F0$
\"+TV8^bF0$\"+&e=?b&F0$\"+zfs_bF0$\"+%*=H`bF0$\"+FYu`bF0$\"+7o5abF0$\"
+hlRabF0$\"+h$GYb&F0$\"++Q\"[b&F0$\"+_@'\\b&F0$\"+L33bbF0$\"+zd<bbF0$
\"+M<DbbF0$\"+*\\7`b&F0$\"+56ObbF0$\"+++SbbF0$\"+76VbbF0$\"+,gXbbF0$\"
+9fZbbF0$\"+U=\\bbF0$\"+'e/bb&F0$\"+#y9bb&F0$\"+PH_bbF0$\"+h%Hbb&F0$\"
+\"oMbb&F0$\"+d)Qbb&F0$\"+(>Ubb&F0$\"+p[abbF0$\"+1qabbF0$\"+<([bb&F0$
\"+&3]bb&F0$\"+!=^bb&F0$\"+c?bbbF0$\"+dFbbbF0$\"+;LbbbF0$\"+kPbbbF0$\"
+ATbbbF0$\"+4WbbbF0$\"+PYbbbF0$\"+@[bbbF0$\"+p\\bbbF0$\"+&3bbb&F0$\"+!
=bbb&F0$\"+c_bbbF0$\"+<`bbbF0$\"+k`bbbF0$\"+-abbbF0$\"+LabbbF0$\"+dabb
bF0$\"+wabbbF0$\"+#\\bbb&F0$\"+/bbbbF0$\"+9bbbbF0$\"+AbbbbF0$\"+HbbbbF
0$\"+LbbbbF0$\"+QbbbbF0$\"+UbbbbF0$\"+XbbbbF0$\"+ZbbbbF0$\"+[bbbbF0$\"
+]bbbbF0$\"+^bbbbF0$\"+_bbbbF0FdblFdblFdblFdblFdblFdblFdblFdbl7aqF-$\"
++++]7F0$\"++++]#*!#6$\"++++DtF]cl$\"+++]_eF]cl$\"+++D\"o%F]cl$\"++]#
\\u$F]cl$\"++D$f*HF]cl$\"+]_u'R#F]cl$\"+DhR<>F]cl$\"+#*o\"R`\"F]cl$\"+
8N8F7F]cl$\"+0\"oq\")*!#7$\"+%[aO&yFbdl$\"+(eBHG'Fbdl$\"+q)Qj-&Fbdl$\"
+(4r5-%Fbdl$\"+xo&o@$Fbdl$\"+,b[tDFbdl$\"+,%)ye?Fbdl$\"+?2.Z;Fbdl$\"+w
Xi<8Fbdl$\"+h'*4a5Fbdl$\"+*G(zK%)!#8$\"+JyBYnFiel$\"+j-*pR&Fiel$\"+6Af
<VFiel$\"+pP2aMFiel$\"+;!fKw#Fiel$\"+8sg5AFiel$\"+qd[o<Fiel$\"+;')y99F
iel$\"+$*3$=8\"Fiel$\"+Vrka!*!#9$\"+:xrVsF^gl$\"+sT(\\z&F^gl$\"+Q$zfj%
F^gl$\"+pMy3PF^gl$\"+wn-nHF^gl$\"+@9itBF^gl$\"+Or*))*=F^gl$\"+4x6>:F^g
l$\"+nTH:7F^gl$\"+PLNA(*!#:$\"+pE)yx(Fchl$\"+NhIAiFchl$\"+1\\%y(\\Fchl
$\"+EfF#)RFchl$\"+T2#e=$Fchl$\"+#fc'[DFchl$\"+u_#*Q?Fchl$\"+>-9J;Fchl$
\"+v@\"\\I\"Fchl$\"+S(HR/\"Fchl$\"+?zV^$)!#;$\"+O.:\"o'Fjil$\"+p-#\\M&
Fjil$\"+:i$fF%Fjil$\"+s*[2U$Fjil$\"+y\"*fOFFjil$\"+U$z#*=#Fjil$\"+uMU^
<Fjil$\"+z(Q6S\"Fjil$\"+B5\"47\"Fjil$\"+&=)Gn*)!#<$\"+[0$Q<(F_[m$\"+QW
1RdF_[m$\"+\\:D\"f%F_[m$\"+R7+tOF_[m$\"+\"*4SQHF_[m$\"+$z?2N#F_[m$\"+N
md!)=F_[m$\"+38Y/:F_[m$\"+Y!pN?\"F_[m$\"+qBbG'*!#=$\"+(*=%Gq(Fd\\m$\"+
<NFihFd\\m$\"+8)=)H\\Fd\\m$\"+]]&Q%RFd\\m$\"+SS3bJFd\\m$\"+Ks1CDFd\\m$
\"+'y`#>?Fd\\m$\"+HIS:;Fd\\m$\"+BCK#H\"Fd\\m$\"+Qz&Q.\"Fd\\m$\"+1N'3F)
!#>$\"+13p;mF[^m$\"+WEN$H&F[^m$\"+:@oMUF[^m$\"+#pXxQ$F[^m$\"+al>5FF[^m
$\"+Vs:o@F[^m$\"+&zDXt\"F[^m$\"+O1i(Q\"F[^m$\"+4l456F[^m$\"+r?x!)))!#?
$\"+dwh/rF`_m$\"+FTp$o&F`_m$\"+-`&pa%F`_m$\"+VUcPOF`_m$\"+%R^+\"HF`_m7
aqF-F.$\"++++vAF0$\"+++]F=F0$\"+++vi9F0$\"++]Fq6F0$\"++]Fi$*F]cl$\"++v
#)*[(F]cl$\"+]F'=*fF]cl$\"+v-\\$z%F]cl$\"+GAzMQF]cl$\"+$yLy1$F]cl$\"+E
qEaCF]cl$\"+@OTj>F]cl$\"+(*3tq:F]cl$\"+=Zec7F]cl$\"+uxE05F]cl$\"+#>U@/
)Fbdl$\"+`PrLkFbdl$\"+-5(p9&Fbdl$\"+,od<TFbdl$\"+T91%H$Fbdl$\"+`\"\\_j
#Fbdl$\"+A$*>3@Fbdl$\"+d%flo\"Fbdl$\"+mvC\\8Fbdl$\"+`!)Rz5Fbdl$\"+CW=N
')Fiel$\"+Qv93pFiel$\"+J!=l_&Fiel$\"+DW@@WFiel$\"+S:(p`$Fiel$\"+KsdHGF
iel$\"+'yhOE#Fiel$\"+H%H4\"=Fiel$\"+VNu[9Fiel$\"+M[**e6Fiel$\"+t'e>F*F
^gl$\"+Qpc<uF^gl$\"+^N0MfF^gl$\"+TGCZZF^gl$\"+tUz(z$F^gl$\"+=aBQIF^gl$
\"+M$)eICF^gl$\"+n1ZW>F^gl$\"+Lldb:F^gl$\"+F7YW7F^gl$\"+:)*ob**Fchl$\"
+_=bkzFchl$\"+\"[T;P'Fchl$\"+&=8t4&Fchl$\"+[0&y2%Fchl$\"+Q/GiKFchl$\"+
]V#)4EFchl$\"+![fy3#Fchl$\"+%e(Gq;Fchl$\"+n+BO8Fchl$\"+aS)*o5Fchl$\"+K
C(=b)Fjil$\"+Xz\\ToFjil$\"+c$)>taFjil$\"+&oe&yVFjil$\"+[p%G]$Fjil$\"+e
vF-GFjil$\"+Y?#=C#Fjil$\"+PwX$z\"Fjil$\"+4hwM9Fjil$\"+()G\"y9\"Fjil$\"
+'4.D=*F_[m$\"+yC+YtF_[m$\"+$)>!o(eF_[m$\"+'eT9q%F_[m$\"+qK:hPF_[m$\"+
;E#*3IF_[m$\"+$4QrS#F_[m$\"+u/rD>F_[m$\"+z$o0a\"F_[m$\"+.ZXK7F_[m$\"+C
wjf)*Fd\\m$\"+*45x)yFd\\m$\"+z!o,J'Fd\\m$\"+jW8[]Fd\\m$\"+rv]QSFd\\m$
\"+eg!3B$Fd\\m$\"+Y[k%e#Fd\\m$\"+xern?Fd\\m$\"+,F<a;Fd\\m$\"+h\"QLK\"F
d\\m$\"+H0ne5Fd\\m$\"+KUOp%)F[^m$\"+&Q\"\\vnF[^m$\"+3JR?aF[^m$\"+([9jL
%F[^m$\"+!f^!pMF[^m$\"+s7CvFF[^m$\"+=I>?AF[^m$\"+:W:w<F[^m$\"+KN#4U\"F
[^m$\"+E)Qn8\"F[^m$\"+21\"R4*F`_m$\"+'[G^F(F`_m-%*AXESSTYLEG6#%$BOXG-%
+AXESLABELSG6%%$rowG%'columnG%\"AG-%+PROJECTIONG6%$!#g\"\"!$\"#qFa]nF'
" 1 2 0 1 10 0 2 1 1 2 2 1.000000 70.000000 -60.000000 1 0 "Curve 1" }
}}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 
30 "count payoff for transitions  " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 351 "R :=  matrix(4,4,[0,0,200,200,\n                  0,
0,200,200,\n                  0,0,1,1,\n                  0,0,1,1]);\n
# $200 payoff when trasiting to recurrent state \n# $1 payoff in betwe
en transient states\n# no payoff from a recurrent state\n\nRP := matri
x(4,4):for j from 1 to 4 do\nfor k from 1 to 4 do\n    RP[j,k] := R[j,
k]*P[j,k];\n    od:\n    od:   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "print(RP); " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"RG-%'matrixG6#7&7&\"\"!F*\"$+#F+F)7&F*F*\"\"\"F-F,
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&$\"\"!F)F($\"$+#!
\"\"F*F'7&F(F($\"\"$F,$\"\"#F,7&F(F($\"\"&F,$\"\"'F," }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 147 "tnr := 0;\nU := U0;\ncount2 := 100;\nfor
 j from 1 to count2 do\n    cnr := evalm(TV&*(RP)&*U);\n    tnr := tnr
 + cnr[1];\n    U := evalm(P&*U); \n    od: " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 4 "tnr;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$tnrG\"\"!" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"UG%#U0G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'count2G\"$+\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"
+'*****>5!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 306 "RPtr := s
ubmatrix(RP,1..4,3..4); #throw out zero columns\nQ := submatrix(P,3..4
,3..4);     #transition matrix among transient states \nE2 := diag(1,1
);                  \nW0 := vector(2,[U0[3],U0[4]]);   #original proba
bility on the transient states \nevalm(TV&*RPtr&*inverse(E2-Q)&*W0); #
compute the expected " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%RPtrG-%'matrixG6#7&7$$\"$+#!\"\"F*F)7$$\"
\"$F,$\"\"#F,7$$\"\"&F,$\"\"'F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"QG-%'matrixG6#7$7$$\"\"$!\"\"$\"\"#F,7$$\"\"&F,$\"\"'F," }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#E2G-%'matrixG6#7$7$\"\"\"\"\"!7$F+F*" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#W0G-%'vectorG6#7$#\"\"\"\"\"%F)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7#$\"++++?5!\"(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "4 3 0 0" 305 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }