vnm Stable Sets for Totally Balanced Games

581 Center for Mathematical Economics Working Papers April 18, 2018 vnm Stable Sets for Totally Balanced Games Joachim Rosenmüller Center for Mathematical Economics (IMW) Bielefeld University Universitätsstraße 25 D-33615 Bielefeld Germany e-mail: ISSN: 0931-6558

! "# $ # & "' ( ) *+**+***+*,+, -./+ -0/+ -1/+ -2/+ -3/! + " r 4 5 # #! # q < r # ' 4 5! " "# #" #' # # 6+ ( " # "# 7 + $ $ $ + "+ 8" # ( " #9 :" "#5! ;# # + + " ( < 4 5 #9+ # 7 " #9 " + # $ " $ 8 4 5 #9 + $ "# + + + $ #+ ; 4 5= + " " ##" # #9 & ( $ 4 5 ( # # ( > 9 8 #+ $ $ 1 + ( # & $ + :; #5!+? $ " # " " $

!"#$&'()*+,!*c() 99! & # 8 "# # # 8 # # * ( 8 # ; # 9 "# "# -./01234-.5678-9:/! ; 9 # -./+ -0/+ -1/+ -2/ -3/ < # $ $ 4= ; "5 + >?@ " # A 7B3 2C /1D-E/ F GH2BIJ112K /BL -.7CGM7CN-9O/ -99/ ) + $! PQRS $ # + + (I,F0,v) $ I T +?" UVQWSXY+ F0 σ 8 Z! # I [\QV]^]\_Y!+ v [\QV]^]\_QV `a_[^]\_! #" v : F0 T + $ $ b" # λ & 4 "#5+ + v 8 #! # λ ρ (ρ R) v(s) := min{λ ρ (S) ρ R} (S F 0 ), *d"efc"g 9:! v = r Rλ ρ. C ρ λ ρ (ρ R) & # v #' v(i) = 1 + # + # Q R, Q R 9E! 1 = v(i) = λ ρ (I) < λ σ (I) (ρ Q, σ R\Q). # "" v UX\ha[^]\_ `Q[^\XY [\RR\h]^]SY "# v #9 "#! #' # λ ρ (ρ Q) [\XS "#+ λ ρ (ρ R \ Q! " 7 >?@+ $ $ ijklm^qnvs ms^ MGB o25i/bb4pgkq2bhc2kb -9E/! < # $ 8 rstuvwvxu y z y z { }~ (I,F0,v) ~ ƒ~ ] RUa^Q^]\_ ƒ~ ˆ Š~ ξ Œ ξ(i) = v(i) Œ ~ ~ Ž ƒ Ž ~ Ž ~ J(v) Œ ~ [ \XS Œ ~ ~ Ž ƒ Ž C(v) := {ξ J(v) ξ v} ƒ Ž ξ h\r]_q^sy ƒ Ž η Š S F 0 ξ S S[^]iS ŽŠ S, ~ 9! λ(s) > 0 ξ(s) v(s)

9.! ξ(t) > η(t) (T F 0, T S,λ(T) > 0) ŒŽ Š ~ Œ ~ ~Š Ž Ž Ž S ƒž ~ ~Š ~Š S Š ƒ ŠŽ ~ Ž ξ ~Š η. ~ Š ~ ξdoms η Ž ~ Žƒ Ž Š Ž ~ ξdomη Œ ~ ~ ~Š ξdoms η ŒŽ Š ~ Ž Š # Ž Ž S F0 rstuvwvxu y z z }~ v ~ ƒ~ ~ S J(v) ~ ijklm^qnv S ms^ "!*f)f+ 90! Œ ~Š~ Ž Š ξ,µ S Œ Œ ξdomµ ŒŽ Š ~ 4 5 Ž Š ~ ~Š η J(v)\S Œ ~ Š~ ~ ξ S Œ Œ ξdomη ~ 4 ( 5 & " $ a_]`\xr # [\_^]_a\ay R\hSV < $ # #+ + $ # # λ ρ $ + + $ 8 ( # t T := {1,...,t} $ UQX^]^]\_ I D = D T = {D τ } τ T # < #! I + + I = {τ T} Dτ # # ; b" #+ λ(d τ ) = 1 (τ T) t # t # 4 # 5! # # I ( + 1 λ(i) = τ T λ(d τ ) = τ T 1 t = t t > 1 ; + $ t t! # # $ + + T ρ $ T 91! C ρ = τ T ρ D τ (ρ R). < $ σ " F = F T " # 4 95! D # I F C ρ F (ρ R)

1 λ 1 λ 2 λ r D 1 D t D τ D t C 1 C 2 C r 1 t <" 99> # ()* f+ "! # # # 8" $ # λ ρ # c ρ = {c ρ τ } τ T T t + d+d,"!*! 92! λ ρ = c ρ τ D τ. τ T 6 = S S λ ρ # $ F + x T t " + ϑ x # $ F 93! ϑ x = τ T x τ D τ ϑ x # $ x UXSl]RUa^Q^]\_+ + $ 99O! #,+"),+c 999! + ϑ x dλ = τ Tλ(D τ )x τ = 1 t x τ = t τ T x τ = 1, #' v # # # λ ρ (ρ Q) C(v) λ ρ (I) = τ Tλ ρ (D τ )c ρ τ = 1 t c ρ τ = t τ T τ T c ρ τ = 1 = v(i), τ T <" 99 # # $ λ r λ 1 λ 2

λ ρ # # # # " 9 + #+ & " 4 5+ # Š { + + -./ + $ 4 5+ 4 #5+ 4 "#5+ 4 5+ < ( #+ UXSl[\QV]^]\_ " a = (aτ ) τ T T t +. 7 $ " b" # *+ λ # 8 99:! λ( ) = {λ( D τ )} τ T. + + # T F + f" * 99E! a = (a τ ) τ T := {λ(t D τ )} τ T = λ(t) T $ 99! $ e = (1,...,1)! 7 $ λ(t) = e λ(t) = ea. a τ = λ(t D τ ) = 1 λ(t D τ ) (τ T). t λ(d τ ) + T # $ + F λ(t F)( ) = t τ T a τ D τ( ), a # #' $ F ; + a T t + ε > 0 # εaτ λ(d τ ) (τ T) $ b< #! T εa " λ(t εa D τ ) = εa τ (τ T) + $ " 99E!+ f+ ** 99.! λ(t εa ) = εa, λ(t εa ) = εea. λ(t εa F)( ) = t τ T εa τ D τ( ). 7" + $ # λ ρ (ρ R) c ρ = {c ρ τ } τ T 8 λ ρ " 92! " " λ ρ # $ 990! λ ρ ( ) = {λ ρ ( D τ )} τ T = {c ρ τ λ( Dτ )} τ T

991! f" f++, λ ρ (T εa ) = {c ρ τλ(t εa D τ )} τ T = {c ρ τεa τ } τ T ; λ ρ (T εa ) = εc ρ a. 6 c ρ + + UXSlRSQYaXS Z 999! $ c ρ (1,...,1) = t = tλ ρ (I) $ λ ρ C(v) + $ 8 UXSlPQRS! v : T t + T " 992! v(a) = min ρ R c ρ a ; v(1,...,1) = t # T εa 993! v(t εa ) = v(εa) = εv(a). )" $ # UXSl]RUa^Q^]\_Y +"(c!(c 9:O! J(v) = { x T t + } x τ = t UXSl[\XS τ T +"#,+"!(c 9:9! $ 999! C(v) := {x J(v) x v}

ε!"#$&'"+" ε 7 " # " $ ( 8 # # # 5 4 4 5 + "' >?@ " $+ $ $ # >?@ # λ ρ + :9! c ρ : T t T, c ρ (a) := τ T c ρ τa τ = c ρ a. c ρ ( )? λ ρ "+ # a # ε > 0 #" " T εa 991!!+ $ ::! λ ρ (T εa ) = c ρ (εa) = εc ρ (a) = εc ρ a. ; + "# v! # T εa v(a) = min ρ R cρ (a). εv(a) = min ρ R εcρ (a) = min ρ R εcρ (a) = min ρ R λρ (T εa) = v(t εa ). "f"fg rstuvwvxu z y z { }~ :E! A : { a T t + c ρ (a) 1 (ρ R) }. A e ~ Ž ~ Œ ~ ~ Ž ~ Š~ƒ Ž Ž A Œ ~ ~ ~ƒ~ Ž A e Š~ ~ Œ ~ XSVSiQ_^ is[^\xy ŽŠ XSVSiQ_^ UXSl[\QV]^]\_Y ŽŠ ε > 0 Ž Ž T ~ ε lxsvsiq_^ Œ ~ Š~ Š~ ~ ~ ŽŠ a A e Œ Œ λ(t) = εa ~ Œ ~ Š ~ T = T εa Žƒ~ ƒ~ ~ Ž εa lxsvsiq_^ Ž Ž $+ ε + + # a "# min{c ρ (a) ρ R} = v(a). $ #' a " a 0 := a v(a) = λ(t) v(t),

ε v(a 0 ) = min{c ρ (a 0 ) ρ R} = 1 v(a) v(a) = 1. + a 0 + # A ε > 0 # ε := ε v(a0 ) v(a) ε := ε v(a0 ) = ε 1 $ ε v(a) v(a) v(t ε a 0 ) = ε v(a 0 ) = ε v(a0 ) v(a) v(a) = εv(a0 ) v(t ε a 0 ) = ε v(a 0 ) = ε v(a0 ) v(a) v(a) = εv(a0 ) $ T ε a 0 + " # + $ A + " a 0 A + ( # A 6$+ $ 8 ( # A e a 0 # # $ ε * 4 # 5! " # # $ # # & # Š -./ " ; " * $ $+ $ $ $ #9" # 8 ## α = {α ρ } ρ R [\_is α ρ 0, (ρ R) ρ Rα ρ = 1, f!f# $+ 8 b## "# + #?" s z z Ž Š â A Œ ~ Š~ ~ ā A Œ Œ { ā â Œ ~ Š~ ~ E A e E = { a (k) k K } A e Œ K := {1,...,K} Ž Š~ ~ ~ ŽŠ ~ ~ Ž Ž ~ Ž ~ ~ {γk } k K Œ Œ :! ā = k K γ k a (k) ŒŽ Š ~

ε min{c ρ â ρ R 0 } = 1 Œ ~ Œ ~Š~ Žƒ~ ρ R Œ Œ ŽŠ k K ŒŽ Š ~ c ρ (â) = c ρ (ā) = c ρ (a (k) ) = 1 ()d"+** UX\\` b## E: Š >?@ & $ # # sx s z s u s vw u s sx s z }~ ϑ,η ~ ƒ Ž ~ S ~ Ž Ž Œ Œ ϑdoms η Œ ~ Œ ~Š~ ε0 > 0 Œ Œ ŽŠ 0 < ε < ε 0 Œ ~ Š~ Š~ ~ ~ ŽŠ a A e ε Š~ ~ Ž Ž T = T a ε S "!!c()" :0! λ(t) = εa ϑdom T η. ŽŒ ~Š ŽŠ Œ Š~ ~ Ž Žƒ Ž ~ ~ ~ Š Ž Ž ~Š ε Š~ ~ Ž Ž Ž UX\\` " 9 # Œ ~ ŽŠ~ƒ Š >?@ -./! ; " "# $" 8 # rstuvwvxu z z { }~ x ~ Š~ ƒ Ž ~ y T t Ž ~ a ~ Š~ Ž Ž ~ Œ Œ x h\r]_q^sy i]q y a :.! xa v(a) xτ > y τ ŽŠ τ Œ aτ > 0. ~ Š ~xdoma y Ž ~ Žƒ Ž Žƒ Œ ~ a Œ ~ ˆ Ž ~ ~Š }~ v ~ Š~ ƒ~ ~ Ž Š~ ƒ Ž ~ ijklm^qnvs Œ ~Š~ Ž Š x,y Œ Œ xdomy ŒŽ Š ~ 4 5 Ž Š ~ ~Š Š~ ƒ Ž y / Œ ~ Š~ ~ x Œ Œ xdomy ŒŽ Š ~ 4 ( 5 * * #5 4 $ # # $+ + # # + + (# A rstuvwvxu z z Ž Š Ž ~ ~ ƒ~ Š ~ Ž ϑ ~ m τ := ess inf D τ (τ T) m := (m τ ) τ T. m Œ ~ ~ ŽŠ Ž ~ ~ R]_]RQ Ž ϑ

ε +",c,c s z z }~ x ~ Š~ ƒ Ž ~ ϑ ~ Ž ~ ~ ƒ~ ˆ Š ~ Ž }~ m ~ Ž ~ Œ ~ ~ ŽŠ Ž ƒ ƒ Ž ϑ Ž Š Žƒ~ Š~ Ž Ž a ~ Œ ~ xdoma m Œ ~ Œ ~Š~ ε0 > 0 Œ Œ ŽŠ ε Œ 0 < ε < ε0 Œ ~ Š~ ε Š~ ~ Ž Ž T ε = T εa Œ Œ :1! ŒŽ Š ~ ϑ x dom T εa ϑ xx # -./+ " #$9 >?@ z s z z +"!*f!*f x x z z }~ ~ ~ ~ Ž Š~ ƒ Ž Œ ~ ~ ~ H := {ϑ x x } xx 1 st STEP : b η / H b m # # η $+ m #+ η #+ η = ϑ m m / * m #+ # m / * x a 7" xdoma m b## :0 $ 9$ + $ ε > 0 ϑ x dom T εa η + H ( 2 nd STEP : b ϑ x,ϑ y H # # T ϑ x dom T ϑ Z y * # $ # + a A e + $ ε > 0 ϑ x dom T εa ϑ y Z ϑ x,ϑ y F # # xdoma y + " H + $+ ( z s z z { ###" :2! := x T T α = {α ρ } ρ R 4 ( 5 x ρ R α ρ c ρ }. )" $ x ( # # λ ρ "" ( "

ε ###+"c s z z & $ ; # # {c ρ } ρ R _\_lhsps_sxq^s # # " # # 9 # # " * # " ;# 8 4 # $ 5 " # + 8 + $" {c ρ (# <+ # } ρ R σ R c σ = α r c ρ ρ R\{σ} $ 4 ( 5 {αρ } ρ R\{σ} + a T T c σ a = ρ R\{σ} α ρ c ρ a min{c ρ a ρ R\{σ}}, v(a) = min ρ R cρ (a) = min ρ R\{σ} cρ (a). 6+ c σ # # "" $ " " _\_lhsps_sxq[w $ 9 # {c ρ } ρ R ~ Š ~ ~ ~ $ 8 # # ;# # " " < ( #+ + + ( # a A e b :3! T := {τ T a τ > 0} R := {ρ R λ ρ a = 1 }. & T R [ Q XQ[^SX]Y^][Y a < + $ $ $ ' a ; # ; {aτ } τ T #df+" f*(,) :9O! c ρ a = 1 ρ R a τ = 0 τ T\T. 7"+ 8 (# A e # {c ρ } ρ R " + $ 9" # " $ 8 # {c ρ } ρ R * + # + T, R # a + $ T = R > 0 7" + $.0! [ QXQ[^SX] ]_P YWY^SR a + :9O!? # T = R ; T = R a τ (τ T ) + # $ $ $ $ # # {c ρ } ρ R " (

ε #)" f" < A ( # $ ε! $ 8 :2! Z 8 A $ xa 1 x A _\XRQVY #'" # # A A A e 6+ # a A " {x xa = 1} * + ; "+ + (# A + (t 1) #! " + + `Q[S^ $ s z z #)" " :99! { = x T t + xa 1(a A) } = { x T t + xa 1(a A e ) } ~ f)" " :9:! A = { a T t + xa 1 (x ) } = { a T t + c ρ a 1 (ρ R) } #" A # 9

!"#$&' +"#, * $ _\_lhsps_sxq[w ( # a A e ; 8 [ QXQ[^SX]Y^][Y+ E9! T := {τ T a τ > 0} R := {ρ R λ ρ a = 1 } $ T = R > 0 $ ' a ; '" #+ + # ; {aτ } τ T #df+" f*( E:! c ρ a = 1 ρ R a τ = 0 τ T\T E:!? # R ; T = R a τ (τ T ) a A e 7 (# + A a " " ( # " * + # " " (# #" ; 8 E:!+ ( " 8 R > *d",!( sx s z y z }~ a A e ~ Š~ ~ Š~ Ž Ž ~ T, R ~ Œ ~ Œ Š ~ Š Œ T = R =: s 2 }~ π R Œ ~ Œ ~ Š~ ~ ~ a = a π A e Œ Œ Š ~ Š T R Œ Œ Œ ~ ~Š (#!( EE! [a,a ] := {ta +(1 t)a 0 t 1} ~ ~ Ž A Ž ~ Ž Œ ~ Ž Ž ~Š ~ ŒŽ Š ~ { Œ ~ Š Œ ~ Š~ ~ τ T Œ Œ (#!( g E! T = T \{τ}, R = R \{π}. Š ~ ~ Œ ~Š~ ~ Žƒ~ σ / R Œ Œ (#!( E.! T = T, R = (R \{π}) {σ}. xx 1 st STEP : '" #!(()" E0! c ρ a = 1 (ρ R ) a τ = 0 (τ / T ).

"! #" c π # #+ +!( () E1! R \{π} T :=!( (## E2! { } a T t c ρ a = 1 (ρ R \{π}) a τ = 0 (τ / T ) 7 a (# A + A A 8? $ A # (# a A $ > a 8 ; aτ O # τ T ; a 8 ; c σ a = 1 1 st STEP : & 8 $ T = T \{τ}, R = (R \{π}). a ;! #.!( ()*! E3! c ρ a = 1 (ρ R ) a τ = 0 (τ / T ). 2 nd STEP : * a #!( ()! E9O! $ E99! c ρ a = 1 (ρ (R \{π}) {σ}) a τ = 0 (τ / T ). T = T, R = (R \{π}) {σ}. z s z z *d",!"$") sx s z z }~ a A e ~ Š~ ~ ~ ŽŠ ~ T, R ~ Œ ~ Œ Š ~ Š Œ T = R =: s }~ τ / T Œ ~ Œ ~Š~ ~ ~ a = a τ A e Œ Œ Š ~ Š T R Œ Œ Œ ~ ~Š (#!"$") E9:! [a,a ] := {ta +(1 t)a 0 t 1} ~ ~ Ž A Ž ~ Ž Œ ~ Ž Ž ~Š ~ ŒŽ Š ~ { Œ ~ Š Œ ~ Š~ ~ σ / R Œ Œ (#!"$")g E9E! (#!"$") E9! T = T {τ}, R = R {σ}. Š ~ ~ Œ ~Š~ ~ Žƒ~ τ T Œ Œ T = (T \{ τ}) {τ}, R = R.

xx (# a $ " " τ #+ + #" ; aτ = 0! {!"$")() E9.! R T {τ} := a T t c ρ a = 1 (ρ R ) a τ = 0 (τ / (T {τ})) $ " + A A ( $ a }. z s z z #,+"(ed* x x z z }~ a A e ~ Š~ ~ Š~ Ž Ž ~ T, R ~ Œ ~ Œ Š ~ Š Œ ~ a Ž ~ Ž ~ ~ (#"(ed* E90! [a,a ] := {ta +(1 t)a 0 t 1} Œ ~ Œ Ž Š ~Š ~ a ρ (ρ R ) a τ (τ / T ) ŽŠ Ž Œ ~ Žˆ Š~ƒ { Œ ~ Ž Š~ƒ Ž Š~ Ž ~Š { s = T = R = 1 R = {π} T = { τ} Œ ~ a = e τ ~ ŽŠ Ž T t Ž ~ Ž ~ A π Œ ~ ~Š~ 1 Š~ ~ t 1 ~ ~ ~ [a,a τ ] Ž Š τ T \ { τ} Œ ~ Œ ~ Š T = (T \ { τ}) {τ}, R = R ŽŠ Ž E9E! Ž Š ~ ~ T = (T \ { τ}) {τ}, R = R ŽŠ Ž E9! Œ Œ ~ Š~ ~ Š Š~ ~ ~ Ž Š ~ Œ Ž a τ Š ~ Œ τ τ s 2 Œ ~ Œ ~Š~ Š~ ~ t ~ ~ ~ [a,a ] Œ ~ ~ Š~ ~ Œ ~ Š Ž ~ ~ Š Š~ Ž Ž Žƒ~ τ Œ ~ Ž Š~ƒ ŽŠ ~ Žƒ~ A π 1 { ~ Š~ƒ a Ž A Œ ƒž r Ž ~ ŽŽŠ ~ "!c() rstuvwvxu z z }~ a A e ~ Š~ ~ Š~ Ž Ž ~ T, R ~ Œ ~ Œ Š ~ Š }~ τ T,π R { ~ Œ a XSha[Sh Ž a (τ,π) T R E! Œ ŽŠ Œ ~ ~ Ž ~ ~ a ~ Š ~ EE! Œ ~ ŽŠŠ~ Ž Œ Š ~ Š ~ Œ ~ (#!( E91! T = T \{τ}, R = R \{π}. ~ Š ~ a τ,π a. Š~ ~Š Ž Œ ~ Ž ~Š Ž ~ ŠŽ ~ Ž Š~ Ž ŽŠ XSha[^]\_ Š~ ~ Š~ Ž Ž a A e Ž ~ ] XXSha[]nVS Œ ~ Š~ Ž (τ,π,a ) T R A e Œ Œ ŒŽ Š ~ a τ,π a

ff#,)* s z z }~ a A e ~ Š~ ~ Š~ Ž Ž ~ T, R ~ Œ ~ Œ Š ~ Š Œ ~ Œ ~Š~ ~ ŠŠ~ ~ Š~ ~ Š~ Ž Ž a A e Œ Œ Š ~ Š T T R R xx ;+ a 8 # z s z z ", s z z (# ( ; " $ $ "' & " $ * ( >?@ # {0,1,...,r} {1,...,r} # "! # 7" # $ R = {0,1,...,r} Q = {1,...,r} & $ a,a a ( QuO 1 st STEP : < + # σ Q # 4 95! a 0 σ # ; " ",gg E92! ",ggg E93! c ρ (ρ (Q\{σ}) {0}) = R\{σ} # # 4 "5 ; $ τ = ( τ 1,..., τ r ) c = l 0 τρ ρ 0 = min C c0 ρ ρ (ρ Q\{σ}). + 9 τρ ## c 0 C ρ 6+ (ρ Q\{σ}) T = { τ 1,..., τ r } = { τ ρ } ρ Q R = R\{σ}. Z # QuO + 4 "5 ; 8 ", ( E:O! c 0 τ = r c0 (T ) = l ρ 0 = l ρ 0 < 1, τ T ρ Q ρ=1 a 0 σ a 0 σ = (...1...,...1,...,... 1 ρ Q\{σ} c0 τ ρ c 0 τ σ...,...1...,...1...). + ; 8" (()"f+" E:9! c ρ a 0 σ = 1 (ρ Q\{σ}), c 0 a 0 σ = 1 a τ = 0 (τ / T ).

$ c σ a 0 σ = 1 ρ Q\{σ} c0 τ ρ = 1 c 0 τ σ * a 0 σ τ T \{ τ σ} c0 τ c 0 τ σ > 1. ( 2 nd $ 95 $ STEP : 4 $ ; $ c ρ (ρ R) 6$+ " 8 $ $ σ " σ = r a 0 r 95 " 4 ; τ = ( τ 1,..., τ r,τ r ) 7" $ $ τρ = l ρ 0 (ρ Q) τr C r. E::! T := { τ 1,..., τ r,τ r }, R = R = {0,1,...,r}. $ # ", ( E:E! r r r 1 c 0 (T \{τ r }) = c 0 τ ρ = l ρ 0 < 1 < c 0 τ ρ +c 0 τ r = c 0 (T ). ρ=1 ρ=1 ρ=1 τρ 9 ## c 0 C ρ+ c 0 τ r 4 "5 # &" l r := ρ R {r} c0 τ ρ + a 0 r $!df "fc+ E:! a 0 r = (...1...;...1,...; 6 α β #..., l r +c0 τ r 1 c 0 τ r c 0 τ r,..., 1 (l r +c0 τ r ) c 0 τ r c 0 τ r,...) =: (...1...;...1,...;...,α,...,β,...). E:.! α+β = 1 αc 0 τ r +βcτ 0 r = 1 lr. a 0 r (()"f+" E:0! c ρ a 0 r = 1 (ρ R = R), a τ = 0 (τ / T ). $ " #" c r # #+ + { } "()"" E:1! R \{r} T := a T t c ρ a = 1 (ρ R \{r}), π a τ = 0 (τ / T. := { τ 1,..., τ r,τ r }).

& #' t a t $ f+f f+f E:2! f+f f+ff E:3! a t := (...1...;...1,...;..., t,..., 1 (l r +t)...) (t T). c 0 τ r cτ 0 r < t = αh τr #' # # a αh τr = (...1...;...1,...;...,α,...,β,...) = a 0 r, t = h τr $ ## EEO! a h τr = (...1...;...1,...;...,1,..., 1 (l r +h τ r ) h τr...) = a 0 r, $ a 0 r 1 st STEP $ # a 0 r " " [a 0 r,a 0 r ] " a 0 r + + $+ $ 0 r r,0 a a 0 r. 3 rd STEP : & " $ # a 4 8 95 # # " c ρ (ρ Q)

& ( # A "!" (++ 9! 7 $ 8 A := { a A e a }. :! "+!*d f+#f)" H := { x J(v) xa v(a ) = 1 (a A ) } J(v). $ # H + # > rstuvwvxu z y z }~ ρ R Š~ ƒ Ž x ρ #f)(" E! x ρ H, x ρ c ρ Ž ~ ^Xa_[Q^]\_ Ž c ρ Œ ~ ƒ Ž ŽŠŠ~ Ž Ž x ρ Œ ~ ƒ~ Š~ "!*f)*! ξ ρ = ξ xρ := τ T x ρ τ λ D τ Œ ~ ξ ρ = τ Tx ρ τ D τ. ξ ρ ^Xa_[Q^]\_ RSQYaXS ~ λ ρ b + { x ρ } ρ R # + (.! := xu { x ρ } ρ R $ ~ + $ ρ Q + + $ c ρ #+ E! # x ρ = c ρ 6+! #! # * ( $+ + ρ R\Q + ρ R\Q # x ρ $ x ρ a = c ρ a = 1 $ c ρ a = 1 7" A + $ # σ R + [ QXQ[ ^SX]Y^][Y c σ 7 $ $ ( # A $ > T σ := {τ T c σ τ > 0}, Aσ := {a A e c σ a = 1}. + $ " + $ T σ = A σ = C σ > 0 c σ ; [ QXQ[^SX]Y^][ YWY^SR $ # ; {xτ } τ T " #df+" # 0! xa = 1 a A σ x τ = 0 τ T\T σ.

(+ $ A σ := {a A σ a } x σ x σ a = 1 a σ Z A ;+ $" "df*" rstuvwvxu z z ~ ~ ~ 1! E σ := { τ R a A, τ T,σ R } E σ := C σ \ E σ. $" # { x ρ } ρ R ( # # f+ (" rstuvwvxu z z }~ σ R\Q Š~ ƒ Ž x σ Ž ~ Y^Q_ hqxh ^Xa_[Q^]\_ is[^\x ŽŠŠ~ Ž Ž c σ Œ ~ Ž Ž ŒŽ Š ~ { f+ (*f 2! x σ τ := cσ τ (τ E σ ) f+ (*f 3! x σ τ cσ τ (τ E σ ) 9O! x σ τ = 0 (τ T\( E σ E σ )) Œ Œ x σ ( σ ) = x σ (I) = t. Œ ~ Y^Q_hQXh ^Xa_[Q^]\_ RSQYaXS ξσ ~ λ σ Œ ~ ŽŠŠ~ Ž ƒ Ž ~ ~ Š ~ ~ Ž ~ x σ Y^Q_hQXh ^Xa_[Q ^]\_ is[^\x ŽŠŠ~ Ž Ž c σ Œ ~ ƒ Ž ~ ~ Š ~ Œ ~ ƒ~ Š~ "!*f)* 99! ξ σ = τ T x σ τλ D τ Œ ~ ξ σt = ξ xσ := τ T (t) x σ τ D τ, ( f+""#*($" # x σ ξ σ! ]RUa^Q^]\_+ x σ ( E σ ) t sx s z z }~ σ R\Q ~ x σ ~ Š Š Ž }~ a A σ ~ Š~ ~ Š~ Ž Ž

{ a = a ŠŠ~ ~ ~ a A σ Œ ~ 9:! x π a = 1 = v(a ). a = a Š~ ~ ~ a A σ \A σ Œ ~ Œ ~Š~ a Œ Œ Š ~ Š T T R R a = a Š~ ~ Œ ~ 9E! x σt a 1 = v(a σ ). xx 1 st STEP : b a A σ + τ T τ E σ x σ τ = c σ τ 2! ; x σ a = c σ a = v(a ) = 1. $ 2 nd STEP : ( a A σ \ A π ; " $ a " # a + " " # ; + T T + R R 3 rd STEP : + $ 2! 3! $ x σ τ + cσ τ τ T + + σ R x σ a c σ a = 1 ( f+"#f) sx s z z ƒ~ Œ Ž Š Žƒ~ σ R x σ ( E σ ) t. Œ ~ Œ ~Š~ ~ ƒ Ž x σ Œ Œ z s z z { "#+, 9! x σ c σ, ()d 9.! x σ H. Œ ~ ~ ~Š σ Q ~ c σ Ž Š ~ ~ λ σ ŽŠ~ ~ ~ƒ~ Œ ~ ~ e#+, 90! x σ = c σ

xx # # c ρ # 8 c ρ a 1 a A e < H + $ σ R \ Q # ; # z s z z " s z z 7" $ ; " >?@ # "' ( # E0 < σ = 0 R \Q + + c σ = c 0 $ $ ( x 0 1 st STEP : < $ # E 0 = E σ # 4 95 a 0 σ 8 ( # E0+ $ a 0 σ = (...1...,...1,...,... 1 ρ Q\{σ} c0 τ ρ c 0 τ σ...,...1...,...1...). 7 a 0 σ $ T E 0 7 " ; T $ σ $ E 0 = T undercutting T. 6+ $ x 0 $ c 0 + x 0 = c 0 = E σ E + 0 x 0 $ c 0 " ; 2 nd STEP : (+ " c σ = c 0+ $ 4 95 $ 2 nd STEP ( # E0 $ 0 r r,0 a a 0 r. 7 a 0 r + $ ", ( 91! τ1,..., τ r E 0, τ r E 0, $ T = T \{τ r } E 0, {τ r } E 0. 92! #" $ R = {0,1,...,(r 1)} R = R. 93! E 0 = T T, E 0 = T\ E 0.

x 0 E 0 = c0 E 0 Š >?@+ Œ ~ ŽŠ~ƒ ~ Ž + x σ ( E σ ) t σ = 0 R \ Q + $ 9$ x σ = x 0 (# & (+ H x σ ( E σ ) > t, + +! c 0 λ 0 * " (+ + $ $ 9 # x σ ( E σ ) > t, # $ c σ # # + $+ $ "!"#$&',c ( # A = { a T t + c ρ a 1 (ρ R) } $ * # # :E! ε "" # # A # # ( {c ρ } ρ R $ { ###)" ".9! = x T T α 4 ( 5 :2! 6+ x ρ R α ρ c ρ } #)" ".:! { = x T t + xa 1 (a A) } = { x T t + xa 1 (a A e ) }. * $ #9 :2 c ρ (# $+ < a e " A ".E! H a := { x R t xa = 1 }. 7 a (# $ "+ Ha #(# # $ $ rstuvwvxu z y z }~ a A e Œ ~ Œ ~ ~ (t 1) ƒ~ Ž `Q[S^ ŽŠŠ~ Ž Ž a.! F a := H a.

+"cf#"*)" s z z & #" " # $ # r " {c ρ } ρ R # + " # # ( (# <".9 A 2 #+ # " <".: # E # $ $ #" # " # c 1 F a c 2 a a 1 <".9> A $ # f#",# "!

T t+ T\T a 2 c 2 F a c 1 a a 1 <".:> A 6" # f#",## "! + T R a + F $ a f#"*f+!*..! F a = x = x1 +x 2 α 4 ( 5 : x 1 = = xu { c ρ ρ R } + T t+ T\T. ρ R α ρ c ρ, x 2 T t+ < ρ R $ c ρ + H a {c ρ } ρ R (# F a = H a T\T * {c ρ } ρ R # a '" # #df+" f*.0! c ρ a = 1 ρ R a τ = 0 τ T\T.

! ee"!*($".1! #$ # "" $+ # x Fa x = x T + x T\T = α ρ c ρ + x T\T, ρ R $ 4 ( 5 " α = {α ρ } ρ R x T\T T + T\T $ $ # " # $ # 7 ;+ $ " {c ρ } ρ R $ " # # 9 # *d" f!(#,c*d sx s z z { }~ x 0 / Œ ~ Œ ~Š~ ~ Žƒ~ a A e Œ Œ Š ~ Š T R ~ x Ha Œ Œ!*f+,c.2! x T > x 0, T xa = 1 = v(a ), ŽŠ~ Ž ~Š Œ ~ Š~ Ž ~ Žƒ Ž α = {α ρ } Œ Œ ρ R!*f+,c.3! x T = ρ R α ρ c ρ T ŒŽ Š ~ }~ x 0 Œ ~ Œ ~Š~ ~ Žƒ~ a A e Œ Œ Š ~ Š T R ~ x Ha Œ Œ!*f+,cg.9O! xx x x 0. x Ž ~ Žƒ Ž Ž Œ ~c ρ (r R ) ~ x T\T = 0 1 st STEP : < x 0 7 / C ( $ 8 # x 00 / x 0 < x00 + ##'" # x 00 + 8 # x x x 00 # #+ a + " &" $ # a (#+ + + # " x Ha a + $ 4 ( 5 > α + $ x T = ρ R α ρ c ρ T.!*f+,cgg.99! x x 00 > x 0, x T > x 0 T.

$!*f+,c.9:! xa = x T a = $.99!.99!.2! ρ R α ρ c ρ a = ρ R α ρ = 1 = v(a ). 2 nd STEP : $ x 0 $ x 0 / C x x < x 0 7"+ # $ = Fa x Fa T + R =!*f+,cg.9e! x = ρ R α ρ c ρ + x T\T < x 0 x T < x 0 T. $+ #.9:! $!*f+,c.9! xa = α ρ c ρ a = ρ R " # ρ R α ρ = 1 = v(a ) 3 rd STEP : x 0 #+ # z s z z #,c)(#d*,c s z z " #.2! # + x # 6$+ $ ( + + # # # c ρ (T) = r λ ρ (I) = 1! ρ R + #.E # + ( c ρ (r R)! ( < # # $ x 0 $ x 0 C(v)! 7 x 0 #+ $.9E! t = τ T x 0 τ τ T x τ = α ρ ρ R τ Tc ρ τ + τ T x T\T τ = t+ τ T x T\T τ $ # x 0 = x x T\T = 0 + x 0 = ρ R α ρ c ρ C(v). $ $ x ρ " # ξρ $ # $

rstuvwvxu z z }~ { x ρ } ρ R ~ ƒ Ž Š Ž ~ ŽŠ ~ { ξρ} ~ Œ ~ ŽŠŠ~ Ž ƒ Ž Š Ž ƒ~ Š~ Œ ~!*f)#f).9.! = \_i { x ρ } ρ R H = \_i { ξρ}, ρ R Š~ ~ ~ ŽŠ Y^Q_hQXh ijklm^qnvs ms^ & $ H 7 " $ $ >?@+ # 9-0/= " ( <$" /KC- /+ $ # (C ρ,λ ρ ) ##" ρ *#" "" $ "> " #' # 4 5 $ $ " $ # $ $ # " ( # 4 ( 5 " # " $ $ "" > 4 (5 # $ ##!! " $ + " 4 ( 5 " α 4 5 " $ 4 5! $ #'+ + #! "' $ " # '+ # # ; 4 #95 # " < # < " + " # # $ $ $ $ * ; " ' # -9O/+-99/! < = ; " $ # >?@ # + # ;#+ " $ " #9+ " 7112K/ /BL F//B/B -:/ 5I/BB /BL -./0123-9/! $ # '?" 4 $ 5 $ " "" " 4 $ 5> " " $ " " #9 ρ R

& $ 5? 4 + $ 4 5? ( *+ # # $ + $ *+ " " $ " # < " # $ ; # 8 # # # Z $ $ 8 $ # # +?" 4 $5 "# ( ; # " "# $ +,+ # ;# # " #9 7 ; $ $?" + 5 "? 4 (C ρ,λ ρ < $+ " ) >?@+ # ## " *"+)",c sx s z ( # )! z }~ ~ ˆ Š ~ }~ x 0 J(v) ~ Š~ ƒ Ž x 0 / Œ ~ Œ ~Š~ x Š~ ~ ŠŠ~ ~ Š~ Ž Ž a Œ Œ xdoma x ŒŽ Š ~ 0 Œ ~ ~ Š ~ xx 1 st STEP : < # x 0 / + " #.E+ a A e $ T R $ x + $ H a ( + $ α +"( "(.90! "()" f+.91! x T > x 0, T x T = α ρ c ρ, T ρ R xa = 1 = v(a ). 8+ $ # α ".90!+ x := α ρ x ρ ρ R Z # b## E. $ 8 a A e $ T R T T R R.

Z 8 E+ $ x ρ T = c ρ T (ρ R ) ; + x T = x T ")(#d,cg.92! x T = x T > x 0 T, ")(#d,c.93! xa = ρ R α ρ x ρ a = $+.92! 0E! $ ρ R α ρ c ρ a = ρ R α ρ = 1. xdom a x 0 2 nd STEP : $ # x 0 $ 2 nd item #.E + $ 9 # a A e $ T R $ x H a!*f+,cg.:o! x x 0. $+ #9.:+ #.1!+ # x x = α ρ c ρ + x T\T. ρ R 7 x ρ (ρ R ) # $ τ T xρ τ = t (ρ R ) t = ρ R α ρ = τ T ρ R α ρ ( τ T ( τ T x ρ τ c ρ τ ) ) + τ T x τ τ Tx 0 τ = t ρ R α ρ x T\T τ ( x T\T = 0 ; ;+ + τ T c ρ τ ) x 0 = ρ R α ρ x ρ, z s z z

< $ "" $ & " $ 9$+ $ " "" v 8 # # #.E $ # c ρ (ρ R) * $ $+ $ 9 # ( x x # sx s z y z Œ ~Š~ Ž Š ~ x, x Œ Œ x > x x a = 1 \_i {c ρ ρ R} a ina e xx 7# +$ 4 ( 5 α β + $ x = ρ R α ρ c ρ + x = ρ R β ρ c ρ $ x > x x a = 1. ()*,c +" 1 = α ρ c ρ a = x a > xa = ρ R ρ R z β ρ c ρ a ρ Rβ ρ = 1 sx s z * )! z Š ~ ~Š ~ Œ ~ ~ ~ s z z ")(#d#,)*+ 09! xx b x, x a $ T + R xdom a x. 7# + 4 ( 5 α β + $ x = ρ R α ρ x ρ x = β ρ x ρ. 7" b## E. $ 8 a A e $ T R T T R R. Z 8 E+ $ x ρ T = c ρ T (ρ R) $ xa > xa. $ # # ; ρ R ")(#d,c 0:! xa = ρ R α ρ x ρ a = ρ R α ρ c ρ a = ρ Rα ρ = 1

$ ")(#d,c 0E! xa = ρ R β ρ x ρ a = ρ R β ρ c ρ a = ρ Rβ ρ = 1, z s z z $)c$)c!*f " sx s z z { x ρ } ρ R ~ ƒ Ž Š Ž ~ { ξρ}ρ R ~ Œ ~ ŽŠŠ~ Ž ~ Ž ƒ Ž ŽŠ v Œ ~ 0! ~ ~ H = \_i { ξρ}ρ R xx <$ # #.0+ # 0:+ :1 z s z z s z z & H Y^Q_hQXh ; $ -9O/ -99/ <+ ξρ $ λ ρ $ 9 H ^Xa_[Q^Sh (# # " " # λ ρ (ρ R)

' -9/ 7# b + ) )+ )+ 931 ' g ~ Ž Ž Žƒ ƒ~ + -:/ b Z > + ŽŠ~ Ž Ž Ž ƒ ~ Š ŠŽ Ž # ; ƒ~ + 9329!+ :O :E & -E/ + 6'#+ + Z '+ ŽŠ~ ~ ~ Ž Š ~ ƒ~ Š ~ Ž Žƒ + # 9330!+ :OO :99 ' & g - / 6+ ŽŠƒ Ž Ž Š ~ Š ~ ƒ Š ~ + # 931!+.E 00 -./ # + Ž ~ ~ ~ ŽŠ ~ƒ ŽŠŒŽ Ž ƒ~ Š ε Š~ ~ Ž Ž + &9" )+ *&+ Z :O9E!+ O :2 & g -0/ + Ž ~ ~ ~ ŽŠ ~ƒ ŽŠŒŽ Ž ƒ~ + &9" Z :O9!+ E9 + ) **> )+ *&+ ## & g -1/ + Ž ~ ~ ~ ŽŠ ~ƒ ŽŠŒŽ Ž ƒ~ + * & :O9!+ E1 + ) ***> 7 # #> ; & g -2/ + Ž ~ ~ ~ ŽŠ ~ƒ ŽŠŒŽ Ž ƒ~ + * & :O9!+ 9 + ) *,> b" #> ( # & g -3/ + Ž ~ ~ ~ ŽŠ ~ƒ ŽŠŒŽ Ž ƒ~ + * & :O90!+.3 + ),> 7 # -9O/ # Z '+ Œ Š ~ Š Ž Ž ˆ ~ ~ ŽŠ ~ Š ŠŽ Ž ƒ~ + * # :OOO!+ E2 09-99/ + Ž ~ ~ ~ ŽŠ ~ Š ŠŽ Ž ƒ~ + * # :OO2!+ E99 E92 $& -9:/ b 9+ ƒ Š ~ ƒ~ + # y 9303!+ 3 :. -9E/ # ;9 "+ Œ ~ Ž Š Ž ƒ~ ~ Žˆ Ž ƒ ~ Œ ŽŠ+ ) )+ )+ 93 + 93 1+ 93.E