The Stationary Distributions of a Class of Markov Chains

Transcription

1 Appled Mathematcs Publshed Onlne May 0 ( The Statonary Dstrbutons of a Class of Marov Chans Chrs Cannngs School of Mathematcs and Statstcs Unversty of Sheffeld Sheffeld UK Emal: ccannngs@shefacu Receved February 8 0; revsed March 8 0; accepted Aprl 5 0 Copyrght 0 Chrs Cannngs Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense whch permts unrestrcted use dstrbuton and reproducton n any medum provded the orgnal wor s properly cted ABSTRACT The objectve of ths paper s to fnd the statonary dstrbuton of a certan class of Marov chans arsng n a bologcal populaton nvolved n a specfc type of evolutonary conflct nown as Parer s model In a populaton of such players the result of repeated nfrequent attempted nvasons usng strateges from 0 m s a Marov chan The statonary dstrbutons of ths class of chans for m4 are derved n terms of prevously nown nteger sequences The asymptotc dstrbuton (for m ) s derved Keywords: Parer s Model; Marov Chans; Integer Sequences Introducton In classcal model n conflct theory [] Parer s model [] two ndvduals compete for a reward V by selectng tmes from some set Here we suppose that the avalable tmes are nteger values n 0 m If a player chooses and hs opponent y then the payoff to the player E y s gven by V f y E yv f y f y The scenaro envsaged s as follows An ndvdual choosng tme dsplays for that length of tme ncurrng a cost If eceeds hs opponents choce y then he collects the reward In the event of a te the reward s shared In a populaton n whch ndvduals are restrcted to play ether u or v where u v then the payoff matr P s smply V u u P V v V v Thus f V vu the frst row strctly domnates the second (that s p p and p p ) and f V vu the second strctly domnates the frst We consder a populaton of ndvduals playng Parer s model We suppose that the populaton evolves as follows Suppose at some tme there s a populaton all of whom are playng a sngle strategy u (e the populaton s monomorphc) A new strategy v arses by some random process If u domnates v then the strategy v wll be elmnated under any reasonable dynamc On the other hand f v domnates u t wll rapdly ncrease n frequency and dsplace u We wll suppose that the ntroducton of new strateges s nfrequent compared wth the tme taen for ths replacement process For a more detaled dscusson of ths model see [] The Class of Marov Chans We nvestgate here the followng class of Marov chans [4] motvated by the above scenaro We suppose the avalable strategy set s M 0 m and the reward V The use of V rather than V = ensures that n every par of strateges u and v where u v one s domnant The case V allows partcularly neat forms for the dstrbutons whereas other values of V requre more comple less elegant analyss and wll be presented elsewhere New strateges arse from the set M If the current strategy s and a new strategy j arses ths latter wll nvade ff j 0 If we suppose that the strateges arse wth equal probabltes m then we have a Marov chan wth transton matr A m aj gven by Copyrght 0 ScRes

2 770 C CANNINGS a j m f j m f j 0 f j 0 f j jaj f j Clearly ths chan s rreducble We nvestgate the statonary dstrbuton of ths class of Marov processess for m4 (the cases m and m are trval) We derve a ratonal epresson for these statonary dstrbutons worng throughout prmarly n ntegers For ths reason we gve the epresson for the matr A m mam below A m m m m m The Statonary Dstrbuton Now the domnant egenvalue s m and we derve a recurrence relaton for the correspondng left egenvector um the statonary dstrbuton where we set the rght-most element equal to It s straghtforward to demonstrate that the fnal three elements of the egenvector um are m m and Observe that A m Im c m A m rc where throughout I s the c s a element column vector and row vector We then have dentty matr r s a element um0 A mmum0 rm Also and so A Now consder We have so A m Im c c m m m rm rm um 00 A m mum rm 00 m um 0 um rm v 00 m m mum 0 r m m m m m m um 0 um m m va u 00 r 0 r mv v s the requred egenvector m We now have a recurrence relaton for the u m whch s u m mum 0 um rm 00 Ths s vald for 4 usng u 0 and u Suppose we wrte y m for the sum of the elements of u m so that u m ym s the statonary dstrbuton of the Marov chan We have mmedately that m y m m y m y m y wth y Ths s sequence A00040 [5] specfed as an where our ynan where the sequence s ntated wth a 0 0 and a We can etract ndvdual elements of the statonary dstrbutons Suppose that u m s the th element Copyrght 0 ScRes

3 C CANNINGS 77 of u m Then we have that as one can see easly by consderng the lntng case so the above epresson tends to zero as um mum um and so y y as for 0 m wth ntal values u 0 and Comment Much weaer condtons are necessary than u 40 u and u 4 4 and for those stated above for y y as we have u and u The sequence We can apply the lemma mmedately to the elements for u m0 s A05807 and u m s A05879 n of the statonary dstrbuton epressed n the nteger [5] form The ratos for 0 and elements for m 8 are Table gves some values of um llustratng the speed wth whch convergence taes place 4 The Asymptotc Egenvector We have no epresson for the asymptotc value but for m = 00 the rato s appromately The Havng derved recurrence relatons for the elements of ratos for and elements for m 8 are 05 the egenvectors we now consder the lmt as m and for m = We begn wth a smple Lemma 00 appromately Lemma In the absence of a smple way of evaluatng the lmtng ratos dscussed above analytcally we adopt a df- Suppose we have a recurrence relaton of the form y y y where the s not dependent ferent method to derve the asymptotc statonary dstrbuton agan epressed n ntegers Suppose ths s gven on the y and 0 Suppose we have two sequences z and z satsfyng the recurrence relatonshp by 0 and defne j but ntated by dfferent values e by and We have z0 z that z0 z respectvely Then z z c as where Thus 0 and c s a constant whch depends on the ntal values Proof In a smlar way we can obtan and Snce and so on It s clear that the sgns alternate For ease we ntroduce the followng notaton; we Z zz zz wrte c d when 0 c d z zzzzz so that the sequences c and d for consst of postve ntegers Smlarly we wrte g h zz zzz when g0 h so the sequences We have g and h consst of postve ntegers Thus we have zz z z Z z z z z 4 6 and so on whle 0 zz zz 4 4 and so on The theorem below gves recurrence relatons for c Z Z d g and h n terms of A05879 and A05807 [5] zz zz Theorem For we have c e e where Now the denomnator ncreases at a rate greater than en n en e n wth e0 e ([5] A05879) Table Egenvectors for m = ()8 m Total Copyrght 0 ScRes

4 77 C CANNINGS and d f f where fn n fn f n wth f0 0 and f ([5] A05807) Further g e and h f Proof We have that en 4789 and fn 0 68 and we have already shown that whch satsfy the formula gven n the statement of the theorem We prove that f the formula holds for for K then t holds for for K and thence for for K Thence by nducton Note frst that c c c c c c c c c Suppose then that the formula for holds for K Then snce we have and substtutng for the epressons gven n the statement of the theorem we have c c ( c c d dd d c c c d d d c d Now clearly we also have have c d c c d d so we Table gves the frst 5 elements for the egenvector for m 00 Some dea of the speed of convergence can be ganed by observng that these values agree wth the elements of the egenvector for m 5 ecept n the fnal decmal places 5 Concluson We have derved the statonary dstrbuton of the frequences of the avalable strateges n a populaton n whch mutatons occur nfrequently for Pare s model when the reward s + and for nteger valued strateges These relate to certan nown nteger sequences Ths wor provdes a base for further nvestgatons for other values of the reward and more comple nvason processes 6 Dscusson Parer s model whch s also nown as the Scotch Aucton s often used n the conflct theory lterature as an eample of a smple model n whch there s no ESS (evolutonarly stable strategy) The mplcaton of ths s that there s no populaton assembly whch s resstant to nvason Of course f such a contests actually occurs t s mportant to as what wll happen n the populaton Ths s the queston whch s addressed n [] and whch generates the class of matrces consdered here The statonary dstrbuton then corresponds to the frequency wth whch one would observe a populaton to be playng a specfc strategy ecept f one happened on a populaton n transton The class of cases dscusses above arses from Parer s model when we consder a fed reward value V and when the value of m the range of possble strateges s allowed to vary It would be of nterest to eamne Table Frst 5 egenvector elements for m = Copyrght 0 ScRes

5 C CANNINGS 77 other possble values of V as m vares For eample for V 4 the Marov matrces have s for j 0 s for j and dagonal elements to mae the row sums m It s hoped to treat these models n a subsequent paper We observe from the numercal values that the most frequent strategy value played s that the dstrbuton s un-modal and that the strateges 0 are played over 90% of the tme; asymptotcally appromately whch agrees to fve decmal places to the value for m 6 whle the mean value s asymptotcally appromately 07 whch agrees to fve decmal places to the value for m 8 These latter fgures confrm the rapdty of the convergence REFERENCES [] M Broom and C Cannngs Evolutonary Game Theory In: Encyclopeda of Lfe Scences John Wley & Sons Ltd Chchester 00 [] G A Parer Seual Selecton and Seual Conflct In: M S Blum and N A Blum Eds Seual Selecton and Reproductve Competton Academc Press New Yor 979 pp -66 [] C Cannngs Populatons Playng Parer s Model In: Preparaton [4] J Norrs Marov Chans Cambrdge Unversty Press Cambrdge 998 [5] The On-Lne Encyclopeda of Integer Sequences Copyrght 0 ScRes