Dynamc Analyss of Sharng of Agents wth Heterogeneous Kazuyo Sato Akra Namatame Dept. of Computer Scence Natonal Defense Academy Yokosuka 39-8686 JAPAN E-mal {g40045 nama} @nda.ac.jp Abstract In ths paper we consder knowledge sharng as basc consttutes of socal nteracton. The problem of knowledge sharng of self-nterested agents wth heterogeneous knowledge s formulated as knowledge tradng games. Agents consder to trade a set of heterogeneous knowledge on whch they have dfferent value judgments. The knowledge transactons are formulated as symmetrc coordnaton games. We can aggregate ther dosyncratc value judgments wth threshold. We nvestgate how knowledge types held by agents may nfluence knowledge sharng.. Introducton Many socal nteractons can be treated as transacton [] [7]. We focus on knowledge sharng as the consttute part of socal nteractons and formulate knowledge tradng game as a basc methodology. For nstance let consder a group of agents G = { A: N} wth a set of heterogeneous knowledge to be transacted. Agents desre to exchange ther knowledge on whch they may have dfferent value judgments. They exchange ther prvate knowledge wth other agents of nterests and they beneft by exchangng ther knowledge f ther utlty wll be ncreased. They do transacton on the bass of ther own utlty through acqurng the new knowledge [3]. In Fg we show a conceptual framework for the creaton of new knowledge. If they succeed to share knowledge at the hgh level t creates the postve feedback to the creaton of new knowledge at the ndvdual level. Such agents are ratonal n the sense that they only do what they want to do and what they thnk s n ther own best nterests. Wth the knowledge transacton among self-nterested agents they mutually exchange ther prvate knowledge such a way that ther utltes can be mproved. Each agent needs to reason about the value of knowledge held by the other agent before the transacton. The factors such as the value (worth) of the knowledge possessed by each agent the utlty through acqurng the tem and the transacton cost also provdes effect on the mutual an agreement for knowledge transacton. The goal of our research s to formalze an economc model of knowledge creaton by focusng the quanttatve aspects of the value of knowledge. And the central ssue n ths paper s the relatonshp between the value of knowledge whch agents have and the propertes of knowledge tradng. As agents trade and receve knowledge they are able to ntegrate t wth ther exstng stock and create new knowledge. But ths vew can be benefcal to those agents who are at least partly capable of understandng and ntegratng t [5]. We nvestgate the knowledge tradng between ndvduals and how the varous types of knowledge nfluences knowledge tradng.
Creaton of New Sharng Common Prvate (Ω ) Prvate Prvate (Ω ) (Ω ) Prvate (Ω n ) Accumulaton Transacton Common (K) Fg. The process of Creaton through Transacton. Sharng through Tradng In ths secton we formulate knowledge transacton as knowledge tradng games. Agent A and agent B have the followng two strateges: S : Trade a pece of knowledge : Does not trade (.) We assume that the utlty functon for agent = A Bwth the prvate knowledge Ω and the common knowledge K n Fg s gven as the sem-lnear functon as follows. U ( Ω K) = Ω + v ( K) = A B (.) Utlt y of Agent (Valu e of prvat e knowledge) (Valu e of commo n knowledge) Knowledg X e transacton Y Utlt y of Agent A afte r transacton Prvate knowledge Ω A Agen t A Common knowledge K Prvate knowledge Ω B Agen t B Common knowledge K U A (S S ) = Ω A X + v A ( X Y ) Fg The llustraton of the knowledge transacton between two agents We consder a tradng stuaton n whch agent A trades wth hs knowledge X and agent B trades wth hs knowledge Y. The utlty functon defned over the common knowledge v ( K) = A B can be classfed as the followng three types: Defnton: For a par of knowledge X and Y ( X Y) ()If v( X Y) = v( X) + v( Y) then the value functon v ( X ()If v( X Y) v( X) + v( Y) then the value functon v ( X (3)If v ( X Y) v ( X) + v ( Y) then the value functon v X ( )s lnear. )s convex. )s concave. Factors such as the value (worth) of knowledge possessed by each agent the loss for dsclosng the knowledge to the other should be consdered. Each agent has the dfferent value judgment. The assocated payoffs of both agents when they choose the strategy S or are gven as shown n Table
Agent B Agent A S U A U A 3 S 3 U B U B U A U B U B 4 U A 4 Table The payoff matrx Each payoff n Table s gven as follows: U ( S S ) = Ω X + v ( X Y) U A A A A 3 U ( S S ) = Ω + v ( Y) U A A A A U ( S S ) = Ω Y+ v ( X Y) U B B B B 3 U ( S S ) = Ω + v ( X) U B B B B U ( S S ) = Ω X + v ( X) U A A A A 4 U ( S S ) = Ω U A A A U ( S S ) = Ω Y+ v ( Y) U B B B B 4 U ( S S ) = Ω U. B B B (.3) (.4) The above assocated payoffs are nterpreted as follows: Once they decde to transact ther prvate knowledge t s dsclosed to the other agent and t becomes common knowledge. When both agents decde to trade ther prvate knowledge the payoffs of both agents are defned as ther values of common knowledge mnus ther values of prvate knowledge. If agent A trades hs prvate knowledge X and agent B does not trade hs prvate knowledge X becomes common knowledge and he may lose some value from ths change. On the other hand f agent A does not trade and agent B trades hs prvate knowledge Y he receve some payoff snce the prvate knowledge Y becomes common knowledge. Ths s dstngushed dfference of knowledge tradng from physcal commodty. Wth the trade of knowledge agents do not lose the value of hs traded tem. Furthermore they may receve some value even f they do not trade and ther partner trades. 3 4 Subtractng U from U and U from U n the payoff matrx of Table we defne the followng payoff parameters: α β α β U U = X + v ( X Y) v ( Y) 3 A A A A A U U = X v ( X) 4 A A A A U U = Y+ v ( X Y) v ( X) 3 B B B B B U U = Y v ( Y) 4 B B B B (.5) If agent A does not trade and agent B trades he receves the postve payoff by acqurng new knowledge Y. If both agents do not trade they receve nothng. The parameter α represents the mert of tradng. On the other hand the parameter β represents the rsk of tradng. By aggregatng those payoffs we defne the followng parameters whch represent the values of ntegratng two ndependent knowledge X and Y. α + β = v( X Y) v( X) v( Y) = A B (.6) If the value functon v ( K ) = A B defned over ther tradng knowledge X and Y are convex then we have α + β >0 = A B. And we assume the parameter β = A B s not negatve. We ntroduces the followng parameters defned as thresholds.
θ = β /( α + β ) { X v ( X)}/{ v ( X Y) v ( X) v ( Y)} A A A A A A A A θ = β /( α + β ) { Y v ( Y)}/{ v ( X Y) v ( X) v ( Y)} B B B B B B B B (.7) By usng these parameters the payoff matrx n Table can be transformed the payoff matrx n Table. Agent B S Agent A S (Trade) (Not to trade) θ B 0 θ A 0 0 0 θ B θ A Table The Payoff Matrx of Agent If the probablty of the other agent to trade s gven by p the expected utlty of agent when he chooses the strategy S or s gven as follows: U ( S ) = p( θ ) U ( S ) = ( p) θ. (.8) The optmal tradng rule s obtaned as the functons of ther threshold θ = A B as follows: () If p θ then trade ( S ). () If p<θ then do not trade ( ). (.9) Snce each agent has dfferent threshold θ reflectng hs dosyncratc value judgment over heterogeneous knowledge the optmal transacton rule of each agent n (.9) dffers n general. 3. Characterzng Heterogeneous by Threshold In ths secton we characterze knowledge by threshold. The threshold defned n (.7) s assocated wth these peces of knowledge whch reflects ther value judgment at knowledge tradng. Here we assume that each agent reasons the value of other agents n term of that of hs own knowledge and the value of knowledge of hs partner can be approxmated n terms of the value of hs own knowledge. For nstance an agent wth knowledge of the value X reasons the value of the other agent of knowledge Y as Y = α X ( α > 0 ). If 0< α < that agent transacts wth the agent of havng low value knowledge and f α > that agent transacts wth the agent of havng hgh value knowledge. As example we specfy the value functon of an agent as follows: Convex functon : v ( X ) = kx ln( X ) (3.) If the value functon of agent A s convex as gven n (3.) we can approxmate t as follows: v( X αx) v( X) v( αx) = kx(ln( + α) + αln( + / α) (3.) X v( X) = X( kln X) (3.3) Therefore f value functon of agent A s convex hs threshold n (.7) can be approxmated by / k ln X θ( X α) (3.4) ln( + α) + αln( + / α) The threshold of agent A s the functon of both the value of hs own knowledge X and the relatve value of hs tradng partner gven α. Fg 3 shows the relaton between the value of knowledge and threshold. In ths fgure we fnd that f the value of hs own knowledge X ncreases then her threshold decreases. Ths mples that the agent of convex value functon s wllng to trade hs knowledge f he has knowledge of hgh qualty.
Furthermore n Fg 3 f agent estmates that tradng partner have more valuable knowledge threshold decreases and agent wllng to trade hs knowledge. The agent of convex value functon decdes whether he trades or not from the value of own knowledge and tradng partner s. Fg 3 The relaton between the value of knowledge and threshold We consder the knowledge transacton between agent A wth the set of knowledge Ω A = { X : N} and agent B wth the set of knowledge Ω B = { Y : N}. Each agent makes hs decson on each pece of hs knowledge whether he trades t or does not trade as the order lsted n the sets K A and K B. As examples we llustrate several threshold dstrbutons defned over the sets of heterogeneous knowledge whch are approxmated as the contnuous functons. Type s the agent wth hgh value of knowledge. Ths agent has a lot of knowledge of low threshold and he or she wllng to dsclose hs knowledge. Type s the agent wth knowledge of ntermedate value. He or she has only knowledge of mean value. Type 3 s the agent wth knowledge of low value. Ths agent has a lot of knowledge of hgh threshold and he or she does not trade. (a) Type (b) Type (d) Type 3 Fg 4 dstrbuton n thresholds 4. The Dynamc Process of Sharng of Agents wth Heterogeneous In ths secton we characterze repeated knowledge transacton between heterogeneous agents. Because each agent has varous types and level of knowledge. That means both agents have approxmately dfferent value of knowledge. We show the property of knowledge transacton changes by the tradng partner and own knowledge value. We denote the proporton of knowledge whch has the same threshold θ by n ()/ θ N = A B. We approxmate the dscrete functons n ()/ θ N = A B by the contnuous functon f () θ A B = whch are defned as the densty functon of threshold. Then the proportons of knowledge whch threshold s less than θ are gven by
F (θ) = f (λ)dλ = A B (4.) λ θ whch are defned as the accumulatve dstrbutons of threshold of agent θ = A B. We denote the proporton of the successful tradng by the t-th transacton by xt () for agent A by yt () for agent B. Snce the optmal transacton rule was gven n (.9) agent A wll transact hs knowledge whch threshold satsfes yt () θ. Smlarty agent B wll transact hs knowledge wth the threshold satsfyng A xt () θ. The proporton of knowledge by the next tme perod t +are gven by B FA ( y( t))for agent A and by FB ( x( t)) for agent B. Then the proportons of knowledge to be traded are descrbed by the followng dynamcs: xt ( + ) = FA ( yt ( )) yt ( + ) = FB ( xt ( )) (4.) The dynamcs are at the fxed pont * * x = FA ( y ) * * y = FB ( x ) (4.3) We consder the knowledge transacton of heterogeneous agents who are characterzed threshold dstrbuton n terms of value of knowledge. (case )Agent wth knowledge of ntermedate value and agent wth knowledge of hgh value Frst we consder the knowledge transacton between agent wth knowledge of ntermedate value and agent wth knowledge of hgh value. Fg 5(c) denotes the portrat of the dynamc process of knowledge transactons. The x-axs represents of the proporton of tradng for agent A (xt ()) and y-axs represents of the proporton of tradng for agent B (yt ()). The dynamcs have two stable equlbrums E 0 and E 3. At the lowest equlbrum E 0 where ( xy ) = ( 00 ) both agents do not trade any knowledge. On the other hand at the hghest equlbrum E 3 where ( xy ) = ( ) both agents trade all ther knowledge. If the ntal estmaton ( x( 0) y( 0)) s n the area of the regon (I) the dynamcs converge to E 0. On the other hand f t s n the regon (IV) the dynamcs converge to E 3. In ths case the proporton of convergng to E 3 s so hgh because agent trade wth the partner who has hgh valuable knowledge. The partner actvely trades hs knowledge agent also wllng to trade. (a) agent A ( of ntermedate value) (b) agent B ( of hgh value) (c) The portrat of the dynamc process Fg 5 The portrat of the dynamc knowledge transacton process
(case )Agent wth knowledge of ntermedate value and agent wth knowledge of low value Next we consder the knowledge transacton between agent wth ntermedate value of knowledge and agent wth low value of knowledge. In comparng a prevous example we fnd that the regon (I) s larger and the regon (IV) s so smaller. Because agent B wth knowledge of low value does not trade postvely. If agent A trade so larger proporton of hs knowledge that means only f agent A also trades knowledge of low value agent B have same level of knowledge and he also trade hs knowledge. In ths case we fnd that t s dffcult to trade wth each other. (a) agent A ( of ntermedate value) (b) agent B ( of low value) (c) The portrat of the dynamc process Fg 6 The portrat of the dynamc knowledge transacton process (case 3)Agent wth knowledge of hgh value and agent wth knowledge of low value Fnally we consder the knowledge transacton between the agent wth knowledge of hgh value and the agent wth knowledge of low value. In ths case these agents have knowledge of completely dfferent level. We fnd that t s dffcult for them to ntegrate ther knowledge. They repeat the mss-coordnaton because the exstence of small asymmetry n the ntal stage. (a) agent A ( of hgh value) (b) agent B ( of low value) (c) The portrat of the dynamc process Fg 7 The portrat of the dynamc knowledge transacton process
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