From: AAAI Technical Report FS-01-03. Compilation copyright © 2001, AAAI (www.aaai.org). All rights reserved.
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yt BIZ{ $BtS GS sS $tS YS }B{RRS BPS AZ$jI$tS *VNyVS stS t%$Bt8t S PBS I$R $AZ IS s $P${$sjS stIS j$tZ$R ${sjjkS {B8} t S $t jj$t S st R S GY$jS YS$RStBS{BtRtRZRSBtSbYs S-s{ jkS$RSstS$t jj$t S st S YS {jsjkS $RS sS {BS 8st$tS Ys S YS st S {B88Zt$ kS sRS Z}BtS bY${YS s S sS 8$t$8Z8S $t{jZIRS YSPBjjBb$tUSw$DSstSst S$RSstSsZ BtB8BZRS 8BIZj\RkR 8S Ys S $RS -}{ IS BS AS stS -} S s S }PB8$tS sS R}{$P${S sR, S w$$DS st RS sS R$ Zs IS $ S YkSB}s S$tSsS Ikts8${S t%$Bt8t SBPSbY${YS YkS 8ZR S AS sbs S stIS w$$$DSst RSsS-}{ IS BS AS{s}sAjSBPS}PB8$tS{B88BtRtRSsRBt$tSstIS BS -Y$A$ S Pj-$AjS }BAj8S RBj%$tS AYs%$B S ! YS {Yss{ $R ${RS $t{jZIS jst$tS 8BA$j$ kS stIS {B88Zt${s $BtS wRS sIRYsbS wdDS stIS Ftt$tRS stISGBBjI$ISwdMDD SS S ytS BZS %$bS $t jj$t S st RS 8ZR S sjRBS Ys%S sS { s$tS j%jS BPS j$tZ$R ${S {B8} t{kS stIS 8ZR S AS sAjS BS }PB8S {B88BtRtRS sRBt$tS $tS sS Y$YjkS Ikts8${SstISZt{ s$tSt%$Bt8t SqYSsS{jsjkS R%sjS {YsjjtRS $tS PB8sj$9$tS Y$RS k}S BPS {B88BtRtRS sRBt$tS bYS %s$BZRS 8}BsjS stIS 8BIsjS sR}{ RS 8ZR S AS s,tS $t BS {BtR$Is $BtS $tS sS Y$YjkSIkts8${SstISZt{ s$tSt%$Bt8t S!ZSjBtT 8SBA`{ $%S$RS8B $%s ISAkS YSPBjjBb$tSR{ts$BS $t%Bj%$tSsS{ s$tSAZkS US S
Sww%kSj$,jkS_]S}${RSb$jjS,}SB$tSIBbtSPBSsSbY$jD ∧ w S{stSbs$ SPBSstB YSPbS8Bt YRS BSAZkSsS_] DD
S SSSSSS⊃ w¬Sw SR Z8AjRSBtSsS%kSBBISIsj DS⊃ SSSSSSSSSSSSSSSS SRYBZjIS}BAsAjkSbs$ SsSPbS8Bt YRS BSAZkSsS_] D S GSb$jjStB SAS{Bt{tISb$ YS Y$RSsR}{ SBPS YSbB,S Yd S ytR sIS BZS PB{ZRS $tS Y$RS }s}S b$jjS AS BtS YS 8BIj$tS BPS stS st :RS 8t sjS R s S ws $ ZIDS $tS st S tB $s $BtR S VRS sZ BtB8BZRS st RS 8$Y S }B t $sjjkSAStsIS$tSsStZ8ASBPSI$sjBZS k}RS RZ{YSsRSw{B8} $ $%DStB $s $BtS$tJZ$kS}RZsR$BtS BS Ij$As $BtS BS ts8S sS PbS wRS Gsj BtS stIS sAAS dxDS YS ts ZS sRS bjjS sRS YS BjS BPS 8t sjS R s RS 8$Y S I$PPS RZAR st $sjjkS $tS I$PPt S k}RSBPSI$sjBZR SG$ YSR}{ S BS YSst R:S8t sjS R s RS I$PPt S k}RS BPS I$sjBZRS {stS sZsAjkS AS ABsIjkS {jsRR$P$IS $t BS bBS k}RUS {B8} $ $%S stIS {BB}s $%S I$sjBZR S ytS sS {BB}s $%S R $tS wRZ{YS sRS $tS sS Ij$As $BtS I$sjBZDS st RS k}${sjjkS {BB}s S BS s{YS sS {BtRtRZRS BtS RB8S {BZRS BPS s{ $Bt SytS Y$RSPs8bB,S YS$RStBS$t$ $sjSP$-ISBsjS stIS YS BA`{ S BPS YS w{BB}s $%DS tB $s $BtS $RS BS s{YS RZ{YS sS {B88BtS Bsj S ytS sS {B8} $ $%S R $tS wRZ{YS sRS tB $s $BtS A btS AZk$tS stIS Rjj$tS st RDS YBb%S st RS sS sRRZ8IS BS Ys%S {B8} $tS $t R R S ytS YS PB8S {sRS YPBS YS sS tBS Y$IItS BsjRS bY$jS $tS YS js S st RS 8ZR S$tIISYs%SsSY$IItSstIs SS
]BB}s $%S%R S]B8} $ $%S*$sjBZRS ytS sS {B8} $ $%S t%$Bt8t S w S tB $s $BtS A btSAZk$tSstISRjj$tSst RDS}s ${$}st RSYs%S sS P$-IS BsjS stIS sS bjjTIP$tIS Z $j$ kS PZt{ $BtS Ys S }s ${$}st RS ZRS BS 8sRZS Y$S }BRRS BbsIRS s{Y$%$tS Y$S Bsj S ytS tB $s $tS PBS sS { s$tS RBZ{S PBS -s8}jS st RS k}${sjjkS {B8}Z S sS PZt{ $BtS Ys S 8sRZRS YS Z $j$ kS BPS BtS BS 8BS s $AZ RS w S {BR D S qYRS s $AZ RS sS k}${sjjkS d S 7S w7sAsS stIS ]B$%sZS d S 7sAsS stIS ]B$%sZS 1((dDS PBS BZS {Bt{Zt S bB,S BtS sRBt$tS stIS jstZsS ZtIR stI$t S
sRRZ8IS BS s,S BtS %sjZRS $tS sS }s $sjjkS BIIS IB8s$t S ytS RZ{YS sS Ps8bB,S }s ${$}st RS Ys%S sS P$-IS BsjS wRskS sS }${S stDS stIS }BRRS wBPS YS tB $s $BtDS $RS 8sRZIS AkS YBbS PsS wBS YBbS {jBRDS YS}s ${$}st RSsSPB8S Y$SR}{ $%SBsjR SqYZRS }BRRS stIS 8$ts $BtS {$ $sS {stS AS sR$jkS PB8Zjs IS$tSRZ{YSsS8BIj S S ytSsSIj$As $BtSw SRS<$ {Y{B{,S Ssj Sw1((dDDS bY${YS$RSsSPB8SBPS{BB}s $%StB $s $BtS}s ${$}st S st RS Ys%S tBS P$-IS $t$ $sjS {B88$ 8t S BS stkS }B t $sjS {BZRS BPS s{ $BtS stIS YS BZ {B8S BPS YS Ij$As $BtS $RS tsjjkS sS {BtRtRZRS BtS stS B} $8sjS {BZRS BPS s{ $Bt S qY$RS t{RRs$jkS 8stRS Ys S YS BZ {B8S BPS sS tB $s $BtS $tS sS {BB}s $%S t%$Bt8t S $RS $tS }$t{$}jS RZATB} $8sjS PB8S YS }R}{ $%S BPS stkS $tI$%$IZsjS st S VRS sZIS $tS w7sAsS 1((dsDS $%tS YS sARt{S BPS $tI$%$IZsjS BsjRS stIS YSsARt{SBPSstkS$t$ $sjSBsjS$ S$RSJZ$ SI$PP${Zj S BSPB8Zjs S}BRRSstIS 8$ts $BtS{$ $sS$tSRZ{YS sS8BIj SytSB YSbBIRSR$t{S$tI$%$IZsjS}s ${$}st RS Ys%StBSP$-ISBsjSss$tR SbY${YS YkS{stS8sRZSsS { s$tS }B}BRsjS $ S $RS tB S {jsS YBbS }BRRS {stS AS 8sRZS stIS {BtRJZt jkS $ S $RS tB S sjjS Ys S {jsS YBbS 8$ts $BtS {stS AS Zsst I S GS sZS YBb%S Ys S%tS$tSsS{BB}s $%SR $tSRZ{YSsRSsS Ij$As $BtS I$sjBZS }s ${$}st RS 8ZR S Ys%S RB8S I$PP$tS BsjRS bY${YS 8ZR S AS Pj{ IS $tS Y$S $tI$%$IZsjS 8t sjS R s R S GYs S 8s,RS sS {BB}s $%S R $tS I$PPt S PB8S Ys S BPS sS {B8} $ $%S R $tS $RS Ys S $tS YS PB8S }s ${$}st RS 8ZR S RYsS s S jsR S BtS sR}{ S BPS Y$S BsjS bY$jS Y$RS {BtR s$t S {sttB S AS-}{ IS$tS YSjs S{sR S S ytS Y$RS }s}S bS YZRS sZS Ys S sIjRRS BPS YS I$sjBZS k}S st S I$sjBZRS sS sS PZt{ $BtS BPS YS }s ${$}st :RS 8t sjS R s RS w$ S YS }s ${$}st :RS BsjRSstISs $ ZIRD SBB%SbSsZS Ys SsSR$tjS 8t sjS R s S 8BIjS PBS AB YS YS {B8} $ $%S stIS YS {BB}s $%S 8BIjRS {stS AS I%jB}I S VRRZ8$tS Ys S sS8t sjSR s SwPj{ $tSstSst :RSs $ ZIDS$RS$%tS sRS stS tT Z}jS 〈sdS S S st〉S bYS sdS S S stS sS s $AZ RS PB8S sS }s $sjjkS BIIS IB8s$tS sS {BB}s $%S 8BIjS $RS BtS bYS sjjS YS }s ${$}st RS sS BtS s S jsR S BtS BPS YRS s $AZ R S ytS YS {B8} $ $%S R $tS tBS RZ{YS {BtR s$t S {stS AS sRRZ8I SS
AS sZIS AjBbS }jskRS stS $8}B st S BjS $tS YS tB $s $BtS }B{RRS YsRS {$%IS j$ jS s t $Bt S 7}{$P${sjjkS bS sZS Ys S stS st :RS 8t sjS R s S IP$tRS stS B%sjjS {Bt - S BPS tB $s $BtS bY${YS {BtRJZt jkS IP$tRS stS st :RS R s kS stIS Zj $8s jkS I 8$tRS YBbS sS tB $s $BtS }B{IR S ytS YS{Zt S8BIjSstSst :RS8t sjSR s S$RS s,tS BS AS stS st :RS j%jS BPS {B88$ 8t S stS st :RS }$BS -}$t{S stIS YS IS BS bY${YS $8S stIS }${S sS $8}B st S BS YS AZkS stI\BS Rjj S y S 8ZR S AS tB IS YS Ys S $tS -}jB$tS YS PP{ S stS st :RS 8t sjSR s SYsRSBtS YSB%sjjStB $s $BtS}B{RRSbS bStB S}$8s$jkS{Bt{tISb$ YSYBbS{B8}YtR$%S BZS 8t sjS R s S 8BIjS $RS AZ S $tS AZ$jI$tS YS Ps8bB,S BPS sS tB $s $BtS 8BIjS bYS }$BS -}$t{RS stIS YS {Zt S 8t sjS R s S BPS stS st S }jskS sS $8}B st S Bj S 3$tsjjkS stIS sj YBZYS RB8S PB8S BPS jst$tS PB8S -}$t{S YsRS AtS }%$BZRjkS RZR IS w S GBtS S sj S 1(((DS YS $t s{ $BtS A btSstSst :RS8t sjSR s SstISstSst :RS}$BS -}$t{S$tSPB8Zjs $tStbStB $s $BtSR s $RS$RS tB%jS$tSBZS8BIj S
sR${S]B8}Btt RSBPS YSBIjS qYS %$ ZsjS 8s, }js{S bS Ys%S I%jB}IS $RS stS t%$Bt8t S Ys S s S stkS }B$t S $tS $8S YsRS sS j$R S BPS AZkRS stIS RjjR S ZkRS stIS RjjRS $tS YS t%$Bt8t S sS sjRBS sRRZ8IS BS Ys%S s{{RRS BS bBS ,tBbjISRBZ{RUSw$DSstSBt BjBkSPBSIB8s$tTR}{$P${S }BIZ{ S $tPB8s $Bt S stIS w$$DS sS R S BPS tsjS {B88BtRtRSZjR SVt RSsSsRRZ8IS BSjstSPB8S -}$t{S ZR$tS {sRTAsRIS sRBt$tS w] iDS {Yt$JZRUS tB $s $BtRS sjBtS b$ YS }BIZ{ S $tPB8s $BtS 8s, S {BtI$ $BtRS stIS YS BZ {B8S BPS YS tB $s $BtS sS R BIS sRS -}$t{RS PBS ZRS $tS PZ ZStB $s $BtRd SVSY$YTj%jS%$bSBPS YS8BIjS$RS RYBbtS$tSP$ZSdSAjBb
buyers sellers
negotiation1
buyerClone1 buyer
buyerClone2
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seller1Clone
seller1
seller2Clone
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negotiation 2
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d S !PS {BZRS BtS {stS $8s$tS B YS s $AZ RS Ys S {BZjIS sjRBSASsIIIS BS YS8t sjSR s SS
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S Vj YBZYS YS bBS TtB8S PZt{ $BtRS {B88BtjkS ZRIS PBS {Bt`Zt{ $BtS w8$t$8Z8S stIS }BIZ{ DS R8S JZsjjkS }jsZR$AjS YS bS Ys%S wsI8$ IjkDS sA$ s$jkS {YBRtS BSZRS}BIZ{ S$tS YS{Zt S8BIj S S *P$t$ $BtS 1 1S _ZAj${S _${S istUS VjjS st RS $tS YS 8s, }js{S sS sRRZ8IS BS Ys%S s{{RRS BS sS }ZAj${jkS s%s$jsAjS }${S stS Ys S {stS AS BA s$tIS PB8S sS }BIZ{ TR}{$P${S Bt BjBkUS __iw}BIDS )S >}8$t}8s-H S VS 8sRZS BPS R$8$js$ kS A btS bBS }ZAj${S }${S stRS (S ≤S i7wstdst1DS ≤S dS $RS {B8}Z ISsRSPBjjBbRUS S Si7 ([8$td 8s- d ] [8$t1 8s- 1 ]) S 8$td − 8$t 1 8s- d − 8s- 1 ∧ d − S= d − 8$td + 8$t1 8s- d + 8s- 1 S S *P$t$ $BtS 1 nS VS WB $s $BtS $RS sS }B{RRS Ys S {stS AS $tS stkS BPS YS PBjjBb$tS R s RUS w$DS DONE+ S w$$DS DONE- SBSw$$$DSDONE0S{BR}BtI$tSR}{ $%jkS BSsS tB $s $BtS Ys S {B8}j IS RZ{{RRPZjjkS wIsjS bsRS 8sIDS sS tB $s $BtS Ys S {B8}j IS ZtRZ{{RRPZjjkS wtBSIsjS$RS}BRR$AjDSstIS BSsStB $s $BtS Ys S$RSR $jjS $tS}BRR SVStB $s $BtS{BIS$RSstSwBIIDSj$R SNS BPS BPPRS stIS {BZt S BPPRS stIS stS BZ {B8US NS )S 〈{〈!PP]BZt !PP〉}!Z {B8〉 S bYS !Z {B8S ∈S {DONE+DONE-} S ytS sII$ $BtS BS YS BZ {B8S stS $8}B st SPs ZSBPSsStB $s $BtS$RS YS $8S$ S BB,S bY${YS {BZjIS AS {ZIjkS s}}B-$8s IS sRS jt YwNDS $ SsRS YSjt YSBPS YStB $s $BtS{BI S S qYS R}{$P${S BPPRS stIS {BZt TBPPRS sS sjRBS $8}B st S Ps ZRS BPS sS tB $s $BtS {BI S 7$t{S YS jt YSBPS YStB $s $BtSYsRSsjsIkSAtS s,tS$t BS s{{BZt S BtS BtjkStIRS BS{BtR$IS YSs%sSBPPS stIS{BZt TBPPSBPSsStB $s $Bt SqYs S$R
d
s%!PPw< N BZ {B8 >D =
jt YwND
∑ BPP ∈N
BPP {BZt !PP
S Ss%]BZt !PPw< N BZ {B8 >D S d = ∑ {BZt !PP S jt YwND ∈ S S qs,tS YS s%sS BPPS stIS {BZt S BPPS BPS sS tB $s $BtS YS $8S sS tB $s $BtS s,RS sRS bjjS sRS YS tB $s $BtS BZ {B8S sS R$8$js$ kS 8sRZS A btS bBStB $s $BtRS(S≤SW7w〈NdS!Z {B8d〉〈SN1S !Z {B81〉DS≤SdS{stStBbSASIP$tISsRSPBjjBbRUS S SW7 ( Nd BZ {B8d N1 BZ {B81 ) 9s0(( 7$8 (Nd N1 ) − ε ) $P SSBZ {B8d ≠ BZ {B81 S S= B Yb$R S 7$8 (Nd N1 ) BPP {BZt !PP
N
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SSSSSSSSS∧ (d − s%]BZt !PP wNd D − s%]BZt !PP wN1 D )
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S VS R$8$jsS R s kS YsRS AtS RZR IS $tS wZIsjS stIS zsRR$j%sS 1(((DS sj YBZYS BtjkS YS }${S bsRS{BtR$IISsS Ps{ BS$tSPB8Zjs $tSsSA$II$tSR s k S
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