Replicator Dynamics and Mathematical Description of Multi-Agent Interaction in Complex Systems

Transcription

1 Journal of Nonlnear Mathematcal Physcs Volume 11, Number 1 (2004), Artcle Replcator Dynamcs and Mathematcal Descrpton of Mult-Aent Interacton n Complex Systems Vasyl V GAFIYCHUK and Anatoly K PRYKARPATSKY Insttute for Appled Problems of Mechancs and Mathematcs Natonal Academy of Scences, Naukova St, 3 b, Lvv 79601, Ukrane E-mal: vaaf@yahoo.com Department of Appled Mathematcs at the AGH Unversty of Scence and Technoloy Cracow, Poland, and Insttute for Appled Problems of Mechancs and Mathematcs Natonal Academy of Scences, Naukova St, 3 b, Lvv 79601, Ukrane E-mal: prykanat@cyberal.com, pryk.anat@ua.fm Receved May 27, 2003; Accepted Auust 24, 2003 Abstract We consder the eneral propertes of the replcator dynamcal system from the standpont of ts evoluton and stablty. Vector feld analyss as well as spectral propertes of such system has been studed. A Lyaponuv functon for the nvestaton of the evoluton of the system has been proposed. The eneralzaton of replcator dynamcs to the case of mult-aent systems s ntroduced. We propose a new mathematcal model to descrbe the mult-aent nteracton n complex system. 1 Introducton Replcator equatons were ntroduced by Fsher to capture Darwn s noton of the survval of the fttest [1]. Replcator dynamcs s one of the most mportant dynamc models arsn n boloy and ecoloy [2, 3], evolutonary ame theory [4] and economcs [5, 6], traffc smulaton systems and dstrbuted computn [7] etc. It s derved from the stratees that fare better than averae and thrvn s on the cost of others at the expense of others (see e.. [4]). Ths leads to the fundamental problem of how complex mult-aent systems wdely met n nature can adapt to chanes n the envronment when there s no centralzed control n the system. For a complex lvn system such a problem has been consdered n [8]. By the term complex mult-aent systems ben consdered we mean one that comes nto ben, provdes for tself and develops, pursun ts own oals [8]. In the same tme replcator dynamcs arses f the aents have to deal wth conflctn oals and the behavor of such systems s qute dfferent from the adaptaton problem consdered n [8]. In ths artcle we develop a new dynamc model to descrbe the mult-aent nteracton n complex systems of nteractn aents sharn common but lmted resources. Copyrht c 2004 by V V Gafychuk and A K Prykarpatsky

2 114 V V Gafychuk and A K Prykarpatsky Based on ths model we consder a populaton composed of n Z + dstnct competn varetes wth assocated ftnesses f (v), = 1,n, where v [0, 1] n s a vector of relatve frequences of the varetes (v 1,v 2,..., v n ). The evoluton of relatve frequences s descrbed by the follown equatons: where dv /dt = v (f (v) f(v) ), = 1,n, (1.1) f(v) = v f (v). (1.2) =1 The essence of (1.1) and (1.2) s smple: varetes wth an above-averae ftness expand, those wth a below-averae ftness contract. Snce v [0, 1] has to be nonneatve for all tme the system (1.1),(1.2) s defned on the nonneatve orthant R n + = {v R n : v 0}. (1.3) The replcator equaton descrbes the relatve share dynamcs and thus holds on the unt a(n 1) dmensonal space smplex { } S n = v R n + : v =1. (1.4) =1 We wrte Fsher s model n the follown form dv /dt = v a j v j j=1 a jk v j v k, (1.5) j,k=1 where a j R, v [0, 1],,j= 1,n, n Z +. One can check that the system (1.5) can be wrtten n the matrx commutatve form: dp/dt =[[D; P ],P]. (1.6) Here by defnton P = {(v v j ) 1 2 :, j = 1,n }, (1.7) { } D = 1 2 da jk v k : j = 1,n. (1.8) k=1a It can be checked easly that the matrx P EndR n s a projector, that s P 2 = P for t R. Ths s very mportant for our further studyn the structure of the vector feld (1.6) on the correspondn projector matrx manfold P [9, 10].

3 Replcator dynamcs of mult-aent nteracton n complex systems Vector feld analyss In order to study the structure of the flow (1.6) on the projector matrx manfold P P we consder a functonal Ψ : P R, where by defnton the usual varaton δψ(p ):=Sp(D(P )δp), (2.1) wth D : P EndR n,sp: EndR n R ben the standard matrx trace. Takn nto account the natural metrcs on P, we consder the projecton of the usual radent vector feld Ψ to the tanent space T (P) under the follown condtons: ϕ(x; P ):=Sp(P 2 P, X) =0, Sp( ϕ, ϕ Ψ) P =0, (2.2) holdn on P for all X EndR n. The frst condton s evdently equvalent to P 2 P =0, that s P P. Thereby we can formulate the lemma. Lemma 1. The functonal radent ϕ Ψ(P ), P P, at condton (2.2) has the follown commutator representaton: ϕ Ψ(P )=[[D(P ),P],P]. (2.3) Proof. Consder the projecton of the usual radent Ψ(P ) to the tanent space T (P) on the manfold P havn assumed that P EndR n : ϕ Ψ(P )= Ψ(P ) ϕ (Λ,P), (2.4) where Λ EndR n s some unknown matrx. Takn nto account the condtons (2.2) we fnd that ϕ Ψ(P ) = D Λ P (D Λ) (D Λ)P + PD+ DP (2.5) = PD+ DP +2PΛP, (2.6) where we have made use of the condtons ϕ Ψ(P )=D Λ+PΛ+ΛP and P (D Λ) + (D Λ)P +2PΛP = D Λ. Now one can see from (2.6) and the second condton n (2.4) that P ΛP = PDP (2.7)

4 116 V V Gafychuk and A K Prykarpatsky for all P P, vn rse to the fnal result ϕ Ψ(P )=PD+ DP 2PDP, (2.8) concdn exactly wth the commutator (2.3). It should be noted that the manfold P s also a symplectc Grassmann manfold ([9, 10]), the canoncal symplectc structure of whch s ven by the expresson: ω (2) (P ):=Sp(PdP dp P ), (2.9) where dω (2) (P ) = 0 for all P P, and the dfferental form (2.9) s non-deenerate [9, 11] on the tanent space T (P). We assume that ξ : P Rs an arbtrary smooth functon on P. Then the Hamltonan vector feld X ξ : P T (P) onp enerated by ths functon relatve to the symplectc structure (2.9) s ven as follows: X ξ =[[D ξ,p],p], (2.10) where D ξ EndR n s some matrx. The vector feld X ξ : P T (P) enerates on P flow the dp/dt = X ξ (P ) (2.11) whch s defned lobally for all t R. Ths flow by constructon s evdently compatble wth the condton P 2 = P. Ths means n partcular that X ξ + PX ξ + X ξ P =0. (2.12) Thus we state that the dynamcal system (1.6) ben consdered on the Grassmann manfold P s Hamltonan whch makes t possble to formulate the follown statement. Statement 1. A radent vector feld of the form (2.10) on the Grassmann manfold P s Hamltonan wth respect to the canoncal symplectc structure (2.9) and a Hamltonan functon ξ : P R, satsfyn ξ(p )=[D ξ,p]+zp + PZ Z, D ξ = D, (2.13) vn rse to the equalty ϕ Ψ(P )=X ξ (P ) for all P P some stll undetermned matrx. wth Z : P EndR n ben

5 Replcator dynamcs of mult-aent nteracton n complex systems 117 For the expresson (2.13) to be the radent of some functon ξ : P R t must satsfy the Volterra condton [9, 11] ( ξ) =( ξ). The latter nvolves lnear constrant equatons on the mappn Z : P EndR n whch are always solvable. Namely, the follown relatonshp on the mappn Z : P EndR n 2[., D]+D [P,.]+Z (P.)+Z (.P ) Z (.) =[D., P ]+Z (.)P +PZ (.) Z (.), (2.14) where the sn means the standard Frechet dervatve, holds now for any P EndR n. The lnear matrx equaton (2.14) s n realty a system of n 2 Z + equatons on n 2 Z + unknown scalar functons. Concernn the matrx D : P EndR n (1.8) the follown useful representaton holds: D = da{sp(a j P )δ jk : j, k = 1,n}, (2.15) where daonal matrces A j = da{a jk δ ks : k, s = 1,n} for all j = 1,n. In eneral, t s well known [9, 11] that all reductons of Le-Posson matrx flows the Grassmann (projector) manfolds are Hamltonan subject to the cannoncal symplectc structure (2.9), ben exactly the case under our consderaton. In the case the explct form of the Hamltonan functon ξ : P R s obtaned by means of some tedous and a bt cumbersome calculatons whch we shall delver n our next report. Consder now the (n-1)-dmensonal Remannan space M n 1 = { v R + : = 1,n, } v =1 =1 wth the metrc ds 2 (v) :=d 2 Ψ P (v) = where j (v)dv dv j P,,j=1 j (v) = 2 Ψ(v) v v j,,j= 1,n, v =1. =1 Subject to the metrcs on M n 1 we can calculate the radent ϕ Ψ of the functon Ψ : P Rand set on Mϕ n 1 the radent vector feld

6 118 V V Gafychuk and A K Prykarpatsky dv/dt = ψ Ψ(v), (2.16) where v M n 1 ψ, or n v = 1 s satsfed. Havn calculated (2.16) we can formulate the follown statement. =1 Statement 2. The radent vector felds ϕ Ψ on P or n another words vector felds and Ψ on M n 1 are equvalent dv/dt = Ψ(v) (2.17) and dp (v)/dt =[[D(v),P(v)],P(v)] (2.18) enerates the same flow on M n 1. As a result of the Hamltonan property of the vector feld ϕ Ψ on the Grassmann manfold P we et a fnal statement. s Haml- Statement 3. The radent vector feld Ψ (2.16) on the metrcspace M n 1 tonan subject to the nondeenerate symplectc structure ω (2) (v) :=ω (2) (P ) M n 1 (2.19) for all v M 2m wth the Hamltonan functon ξ ψ : M 2m R, whereξ ψ := ξ M n 1,ξ: P R s the Hamltonan functon of the vector feld X ξ (2.10) on P. Otherwse, f n Z + s arbtrary, our two flows (2.17) and (2.18) are on P only Possonan. 3Spectral propertes Consder the eenvalue problem for a matrx P P, dependn on the evoluton parameter t R: P (t)f = λf, (3.1) where f R n s an eenfuncton, λ R s a real eenvalue P = P,.e. matrx P P s symmetrc. It s seen from expresson (2.17) that specp (t) ={0, 1} for all t R. Moreover, takn nto account the nvarance of SpP = 1, we can conclude that only one eenvalue of the matrx P (t), t R, s equal to 1, all others ben zero. So we can formulate the next lemma. Lemma 2. The mae ImP R n of the matrx P (t) P for all t R s k-dmensonal k = rankp, and the kernel KerP R n s (n-k)-dmensonal, where k Z + s constant, not dependn on t R.

7 Replcator dynamcs of mult-aent nteracton n complex systems 119 As a consequence of the lemma we establsh that at k = 1 there exsts a unque vector f 0 R n /(KerP)forwhch Pf 0 = f 0, f 0 R n /(KerP). (3.2) Due to the statement above for the projector P : R n R n we can wrte down the follown expanson n the drect sum of mutually orthoonal subspaces: R n = KerP ImP. Take now f 0 R n satsfyn condton (3.2). Then n accordance wth (2.17) the next lemma holds. Lemma 3. The vector f 0 R n satsfes the follown lnear evoluton equaton: df 0 /dt =[D(v),P(v)] f 0 + C 0 (t)f 0, (3.3) where C 0 : R R s a certan functon dependn on the choce of the vector f 0 Im P. At some value of the vector f 0 ImP we can evdently ensure the condton that C 0 0 for all t R n. Moreover one easly observes that for the matrx P (t) P one has [10] the representaton P (t) =f 0 f 0, f 0,f 0 = 1, vn rse to the system (1.5) f f 0 := { vj R + : j = 1,n } R n +. 4 Lyapunov Functon Consder the radent vector felds ϕ ΨonP and ΨonM n 1. It s easy to state that the functon Ψ : P R ven by (2.1) and equal on M n 1 subject to the follown expresson n 1 Ψ(v) = a j v v j c v, (4.1) 4,j=1 =1 whch has to be consdered under the condton c R +, a Lyapunov functon for the vector felds ϕ ΨonP and ΨonM n 1 dψ dt dψ dt v =1, ben at the same tme =1. Indeed: = Ψ, dv dt T (M n 1 ) (4.2) = Ψ, Ψ T (M n 1 ) 0, = Sp(DdP)/dt = Sp(D dp dt ) (4.3) = Sp(D, [[D, P],P]) = Sp([P, D], [P, D]) = Sp([P, D], [P, D] ) 0 for all t R, where.,. T (M n 1 ) s the scalar product on T (M n 1 reducton of the scalar product.,. on R n upon T (M n 1 ) under the constrant ) obtaned va the v =1. =1

8 120 V V Gafychuk and A K Prykarpatsky 5 Mult-aent replcator system In the above we have consdered the case when rankp(t) =1,t R. It s naturally to study now the case when n rankp(t) =k>1ben evdently constant for all t R too. Ths means therefore that there exsts some orthonormal vectors f α ImP, α = 1,k, such that P (t) = f α (t) f α (t) (5.1) α=1 for all t R. Put now f α := =1 v (α) { f () α } := θ (α) v (α) R + : θ (α) := sn f α (), = 1,n ImP, =1forallα = 1,k. The correspondn radent flow on P then takes a modfed commutator form wth the Lyapunov functon varaton δψ = Sp(D α δp (α) ), α=1 δψ :P R as follows D α := da{sp(a (α) j P (α) ):j = 1,n} (5.2) where by defnton, P (α) = f α (t) f α (t), P (α) P (β) = P (α) δ αβ,α,β= 1,k, The resultn flow on the projector manfold P s ven as P (α) = P. α=1 dp/dt = ϕ Ψ(P ), (5.3) where ϕ : D(P) T (P) s calculated takn nto account the set of natural constrants: ϕ α,β (X; P ) : = Sp((P α P β P α δ α,β )X α,β )=0, (5.4) Sp( ϕ α,β, ϕ Ψ) : = Sp[X α,β ( ϕ Ψ α P (β) δ αβ P β )] = 0 for all X α,β EndR n and every α, β = 1,k.Assume now that k = rankp(t) for all t R. Then the radent flow brns about the follown form: ϕ Ψ α = D α + (P β Λ βα +Λ αβ P β ) Λ αα, (5.5) β=1 where Λ βα EndR n,α,β= 1,k, are the correspondn matrx Laranan multplers. To fnd them take nto account that vector feld (5.5) must satsfy the determnn constrant ϕ Ψ α P β + P α ϕ Ψ β = δ α,β P β (5.6) for all α, β = 1,k. Substtutn (5.5) and (5.6) nto the second equaton of (5.4) one fnds easly that ϕ Ψ α = P α D α + D α P α + (P β Λ βα P α + P α Λ αβ P β ). (5.7) β=1

9 Replcator dynamcs of mult-aent nteracton n complex systems 121 Based now once more on the relatonshp (5.4) one can obtan that the expresson P β Λ βα P α + P α Λ αβ P β = (P β D α P α + P α D α P β ) (5.8) holds for all α, β = 1,k. As a result of (5.7) and (5.8) one fnally obtans the follown radent vector feld on the space of projectors P: dp α /dt =[[D α,p α ],P α ] β=1,β α (P α D α P β + P β D α P α ), (5.9) or n the coordnate form dv (α) /dt = v (α) j=1 a (α) j v(α) j a (α) jk v(α) j v (α) k j,k=1 θ (α) θ (α) j θ (β) θ (β) j K (α,β) j,j a(α) js v(α) s, β α=1 j,s=1 (5.10) where the two-speces correlaton tensor K α,β j,ls =2 v (α) v (α) j v (β) l v s (β),,j,l,s= 1,n, α,β = 1,k, must satsfy the next nvarant due to (5.4) sn condtons s=1 θ (α) θ s (α) θ (β) j θ s (β) K α,β s,js =0 (5.11) for all α β = 1,k and, j = 1,n. One can also obtan the flow (5.9) as a Hamltonan flow wth respect to the follown canoncal symplectc structure on the manofold P : ω (2) (P )= Sp(P α dp α dp α P α ) (5.12) α=1 wth the element P = k P α P and some Hamltonan functon H D(P )whch α=1 should be found makn use of the expresson ϕψω (2) = dh. (5.13) Strahtforward calculatons of (5.13) ve rse to the same expresson (5.9). The system of equatons (5.10) s a natural eneralzaton of the replcator dynamcs for descrpton of a model wth mult-aent nteracton. They can descrbe, for example, economc communtes tryn to adapt to chann envronment. In ths approach aents update ther behavor n order to et maxmum payoff under ven matrces of stratees {a (α) k } n response to the nformaton receved from other aents. The structure of the system (5.4) s qute dfferent from the system of equatons (1.5). Frst term on the rht hand sde descrbes the ndvdual evoluton of each economc aent α n accordn to ts own ndependent replcator dynamcs and the second term descrbes averae payoffs of all others k aents except α.

10 122 V V Gafychuk and A K Prykarpatsky The obtaned mathematcal model has also a clear physcal nterpretaton. In framework of ths approach there are k speces n the system wth v (α) ben the frequency of the pursun stratey <nfor speces α. In ths case each type of aents (speces) may exhbt a number of possble stratees n. The state of the evolutonary system s then ven by the vector v =(v1 1,...,v1 n,v (α) 1,...,v n (α),..., v1 k,...,vk n) T ben the soluton of the k n equatons (5.10). Mult-aent approach s also an approprate concept for understandn behavor of lvn systems consstn from molecules and sells and characterzn ther stron specalzaton, ecolocal and human socetes as well as economc communtes especally n the realm of aent-medated electronc commerce. Some of applcatons of the mult-aent system approach have been also dscussed n detal concernn another mult-aent models [12, 13], ben of nterest too concernn eventual mbeddn them nto the pcture suested n ths work. References [1] Fsher RA, The Genetcal theory of natural selecton, Clarendon Press, Oxford, [2] Camazne S, Deneubour J-L, Franks N R, Sneyd J, Theraulaz J G, Bonabeau E, Self- Oranzaton n Bolocal Systems, Prnceton Unversty Press, New Jersey, [3] Mchod RE, Darwnan Dynamcs, Prnceton Unversty Press, New Jersey, [4] Hofbauer J and Smund K, Evolutonary Games and Populaton Dynamcs, Cambrde Unversty Press, London, [5] Youn H P, Indvdual stratey and Socal Structure, Prnceton Unversty Press, New Jersey, [6] Frank S A, Foundatons of Socal Evoluton, Prnceton Unversty Press, New Jersey, [7] Nael K, Dstrbuted ntellence n lare scale traffc smulatons on parallel computers. ntellence.ps.z [8] Lubashevsky I A and Gafychuk V V, Cooperatve mechansm of self-reulaton n herarchcal lvn systems, SIAM J. Appl. Math. 60 (2000), [9] Prykarpatsky A K and Mykytuk I V, Alebrac Interablty of Dynamcal Systems on Manfolds, Kluwer Academc Publsher, The Netherlands, [10] Gafychuk V V, Prykarpatsky A K and Prytula M M, Symplectc analyss of the dynamcal system assocated wth a stochastc replcator Fsher type model, Proceedns of the Lvv Natonal Unversty, Appled Mathematcs and Informatcs seres 6 (2003), [11] Prykarpatsky A K, Zarodznsk J A and Blackmore D L, Lax-type flows on Grassmann manfolds and dual momentum mappns, Reports Math. Physcs 40 (1997), [12] Borkar V S, Jan S and Ranarajan G, Collectve Behavour and Dversty n Economc Communtes: Some Inshts from an Evolutonary Game. [13] Borkar V S, Jan S and Ranarajan G, Dynamcs of Indvdual Specalzaton and Global Dversfcaton n Communtes.