Dynamic Pricing in the Airline Industry

Transcription

1 Dyam Prg h Arl Idury R. Pro MAf ad Vra Vld Calfora Iu of hology Abra: Dyam r drmao adu r bad o h oo alu of fuur al, whh ar wh m ad u aalabl. h ar ury h horal lraur o dyam r drmao, ad ofro h hor wh w daa from arl rg bhaor. Corrod o: R. Pro MAf, ar Hall, Calfora Iu of hology, Paada, CA 95, ro@maf..

2 Comurzd rrao ym wr dlod h 95 o ra of arl a boog ad far formao. Ially h wr ral ym, bu wr oo mad aalabl o ral ag. Drgulao of arl rg 978 rmd muh mor u of h ym for oom ay, ally rg. h al dlom of dyamally adud rg of rdd o Amra Arl Robr Cradall, a a ro o h r of dou arl Pol r h arly 98. h omly ad oau of arl rg ha grow or m. A a rul, h yld maagm ym mloyd by arl for rg ha bom o of h mo ara ad oml formao ym o h la, ad o wh a ry larg oom omo. Arl rg rr a gra hallg for modr oom aaly bau o da from h law of o r ll of aaly. h ar ury h horal lraur, whh moly foud orao rarh oural, dlo om w hory, a h hol our owldg, ad drb om rul from a w daaba of arl r. Dyam rg, whh alo ow a yld maagm or ru maagm, a of rg rag amd a rag rof. h hu ar mo uful wh wo rodu harar o-. Fr, h rodu r a a o m, l hol room, arl flgh, grad lry, or m-dad ll bfor rodu. Sod, aay fd wll ada ad a b augmd oly a a rlaly hgh margal o. h harar ra h oal for ry larg wg h ooruy o of al, bau h ooruy o of al a oal forgo ubu al. h alu of a u a horag uao h hgh alu of a urd uomr. Forag h alu g urr al ad aalabl aay rr dyam rg. Yld maagm hu ar rordly u aluabl. O ma ugg ha Amra Arl mad a ra $5 mllo r yar bad o yld maagm hu Da 994. h umbr may b flad for ral rao. Fr, lud al of yld maagm ragy o ohr, a ood o Amra ow u of h hu, alhough h alu of Amra ral u u a u lghly l. Sod, orora damagd good odrao h form of Saurday-gh ayor rro, a wll dyam rg. Suh rro fala a r drmao, ad ar raoably wll-udrood ohr o Dr ad MAf 996. Nrhl, hr ll doub ha dyam r drmao oomally mora. h rg ym ud by mo maor arl ar rmarably oau o h oumr, whh o urrg g o ma ha Amra Arl hag half a mllo r r day. h mld fruy of r hag m ally larg g ha Amra arr aroud 5, agr r day. hr urrgly ll rarh oom oural org yld maagm, g rom rg gfa dur ad h oom mora aahd o. h ar orbu o our udradg of yld maagm f way. Fr, rod a ury of yld maagm rarh orao rarh oural. Sod, w lor a g modl of Gallgo ad a Ryz 994 ha ha a umbr of drabl ror, ludg lod form oluo ad har rdo, o addr dyam rg odrao. hrd, mo of h lraur aum dmad a a o bu ully form. W odr h mlao of oa lay of dmad ad dmora om w ual org h mor adard a. W am h a h o of a ff alloao, rahr ha h rof-mamzg alloao, ad how ha may of h oluo arbud o rof-mamzao ar aually ou of h dyam ffy. Fourh, w a a w loo a dyam rg from h r of llg oo. A roblm arl fa ha la arral may ha gfaly hghr alu ha arly arral, uggg h arl ough o ll wo d of : a guarad u ad a ha a b dlayd a h arl ru. Ffh, w olld arl rg daa ad grad ylzd fa abou h drma of rg, falag h aluao of modl.

3 . Arl Prg Arl rg h Ud Sa oau. I o uommo for o-way far o d roud-r r. h dffr r bw rfudabl ad o-rfudabl of a faor of four or f. Pr hag fruly, wh low far o a arular flgh bg aalabl, h o, ad h aalabl aga. Arag r for roud-r bw Pho ad Lo Agl dffr ddg o whhr hy orga Lo Agl or Pho. h arularly myrou ha h am arl fly h roud-r wh h am of offrg. W olld daa o far for,6 flgh from Amra Arl, Orbz ad raloy. Ially w olld daa o ah flgh gh m daly, bu wh Amra Arl obd o h olum of arh, w od ug Amra ad ald ba h ohr wo o o r day r flgh. A w omld om of h arh, w ald h fruy u. Nrhl, h rr a mor loo a dyam rg ha aalabl from ay ohr our o our owldg. W wll a u our fdg afr rg h hory o u h fdg r. W wll u h followg rmology. W wll u r dro o rfr o a radomzao by frm rg, whh ah uomr ay a dal r, bu ha r drmd by a radom ro. W wll u dyam r drmao o rfr o hargg dffr uomr d maru or margal o bad o h m of urha; wh uh rg ff mamz h d r alu of h ga of rad, w ll all dyam rg rahr ha dyam r drmao. Rro l Saurday-gh ayor, ha ra l aluabl rodu, ol a r drmao.. g Lraur Radr g a gral duo of yld maagm ar rfrrd o Wllam 999, roo ad uo 994, ad Km 989. Km 989 du uao arora for yld maagm oluo, arular u old uh a dmad forag ad orboog, oluo hu, ad maagral mlao of yld maagm ym. roo ad uo 994 du h r of yld maagm durg ad afr drgulao of h 97 ad 98, ug h hg dury a a dald aml, ad Wllam 999 du yld maagm rm of h rao bw h frm, rour, rodu, uomr, ad omor. allur ad a Ryz 4 ha rad a horough boo for h ud of yld maagm. Yld maagm alao h mad-o-ordr MO maufaurg dury lud Harr ad Pdr 995, Srdhara 998, ad aru ad Srdhara 4. oh MO frm ad r rodr uh a arl fa h roblm of ffly ulzg a fd aay udr ura or hgh dmad ordr o mamz ru, ad hu may yld maagm rul ar alabl o h MO maufaurg dury. Howr, MO maufaurg dffr o h y o of f m horzo ad uhagg aay. h horzo f, h faory r o all orao a a f m or a ommo dadl for all ay, ad h aay o fd, ha a ordr ar omld, aay rlhd. hu, h MO roblm mor of a o ou roblm ha a yld maagm roblm. Harr ad Pdr 995 du h alably of radoal yld maagm o MO maufaurg ad maagral mlao ad dlo a rla horal framwor ug r la bad o u-aay ra. Srdhara 998 drb h u of yld maagm maufaurg uao wh hghr dmad ha aay, dug hr mhod of A horough lraur rw oad MGll ad Va Ryz 999, alog wh a oml gloary of rm alabl o yld maagm, arularly h arl dury. h lraur rw largly d from hr ad may b odrd adm.

4 rag ffy ad ru: aay raog bad o r la, rad oordao bw marg ad maufaurg, ad uborag. aru ad Srdhara 4 furhr lor aay raog by dlog a dyam aay aorom rodur bad o drmag faor of ro wh a hghr d rof marg. Rahr ha a ouou ram of o-m maufaurg ru, Carr ad Looy odr a o-om frm ha agr o ommm of a ormally drbud radom aual dmad. h r wdor roblm mah a ow aay wh a drd aggrga dmad drbuo. hy alo odr h ff of mull r la. F ad Wag 998 udy a mlar aual aay maagm roblm: dlog a omal harg oly of a rwabl rour. h rour oulao modld a a m-dd log uao wh rod off, ad h mamum aual-uaabl yld drmd alog wh h orrodg harg oly. h mo ommo g for yld maagm rarh moad by h arl ad hol dur. h rmadr of h lraur rw fou o h alao, ad h r of h ar dal wh h arl dury luly. omr 996 ad lobaba ad Wlo 997 ga ff of yld maagm ral o h frm ug. omr 996 r argum for ffy of yld maagm rg h arl dury, ad lobaba ad Wlo 997 ga h ma of yld maagm roduo om arl mar. Mo yld maagm rarh, howr, dal wh how o aually mamz ru. O aroah o aum ha uomr arr o ru a flgh, a h r hy wll ay, ad h h frm dd whhr or o o r hm. Va Sly ad Youg udy h uao rm of fllg a aa of fd aay wh ob of ow wgh or or wgh, o fll mul-dmoal aa ad alu, ah y whh arr a a m-dd Poo ro. Wh h goal of mamzg alu, ah ob ad or rd a h m of arral. h a of ual wgh ald o arl a ory orol, ah uomr u o a, ad h mul-dmoal aa ald o h roblm of alloag a a mull org ad dao flgh wor. Sawa 3 addr a mlar uo: Cuomr arr hroughou h ouou m horzo ad a hr r ad dmad z, ad h frm dd whhr o a h ru. A m-maro do ro ud, ad a omal oly ad aalyal ror ar foud wh dmad arr a a m-maro ro. A mor ral way o ra uomr r rfr wh r o h arl dury, howr, o aum ha h uomr wllg o ay uow wh hy ru a. Gallgo ad a Ryz 994 u y orol hory orollg h dmad y wh r hag o udy dyam rg udr mrf omo., dmad r- ad oha dmad modld a a Poo ro. A lod-form oluo foud for oal dmad fuo ad a ur-boud o ru udr gral dmad fuo foud ug a drm hur. Alo, moooy of h omal r oly how, a wll a aymo omaly of a fd-r hur wh mar z. h rul ar h dd o allow omoud Poo dmad, dr r, m-dd dmad, orboog, ruly ad allao, holdg o ad doug, ad arabl al aay. Fg ad Xao a odr h roblm from h r of whg bw a rdrmd f of r a alulad m hrhold ddg o rmag m ad o. Dmad a Poo ro a ah r ll, ad hy fd h omal ouou m whg oly of a arbrary umbr of hr r mar-u or mar-dow. Fg ad Xao b d h o allow rrbl r hag. hy how ha ay ub of r h omal oluo ar of h 3

5 mamum oa lo oad h full of allowd r. Fg ad Gallgo alo addr h dr r m-hrhold roblm, bu allow dmad o b Maroa,.. far ad dmad ar o oly dd o rmag m bu alo o ror al. Chaw addoally allow dmad Poo o b m ad rod h oo of r-og a a u o. Zhao ad Zhg fd ruural ror of h omal r-whg oly from a oma, bu o arly f, r wh dmad a ohomogou Poo ro ad ga oluo mhod h a of dr r. adll odr rodu-dffrad mar gm, ah of whh allowd o adrd r a a m, ad formula a ffly omud dyam rogrammg oluo allowg gral dmad fuo. I h modl, uomr ru a hol room or flgh wh a arular of arbu ad hy ar g a r uo bad o rmag m ad aalably of all rla ommod. Rahr ha dyamally hagg r o mamz ru, om auhor rao aay wh r la o ur ha hgh-ayg uomr ar rd, ffly mlmg a mar-u oly bad o rmag aay ad, f a alloao bw la dyamally orolld, rmag m. Laday 996, aumg drm o-lar dmad ad a fd o for rag ah r la, dlo a dyam-rogrammg oluo for fdg h omal umbr of r la, h omal umbr of aay u fally, hol room alload o ah r la, ad h omal r a ah la. odly ad Wahrford 995 allow ura dmad, ad udy wh o ural low-r al bad o h robably of olag ad d margal ru. hy do allow r-og of r la, bu do o aum ha all low-far uomr wll arr bfor hgh-far uomr hrby forg all hgh-far uomr o ay hr full aabl r. L aga odr drm dmad, ad ud h u of al rro uh a ada urhag, or mmum r durao for gmg dmad. H dlo ral y ror of h omal rro ol, ad al h rul o arl rg, whr lur ralr rla r lay omard o bu ralr rm h ff u of rro. Daa 999a,b ud a of mar gmao ohr ha mhod of omal alloao. Daa 999a ud h homo of ulbrum r dro, howg ha h omal rg ym udr ura dmad r drd moool, mrfly ad rfly om mar. H how ha h amou of r dro ra wh h ll of omo, ug h o la h obrao ha rou rd by mor arl hb mor r dro. Daa 999b dmora how g mull r ll o flgh a dffr m a hf dmad from h hghr-dmad darur m o h alra flgh, wh uow whh m h a. H addr h om ad moooly a ad u h modl o how ha hgh-far uomr may bf from r dro a wll a low-far uomr. Mog byod gl flgh-lg aaly, ral auhor odr org-dao wor. Fg ad Xao odr a wor of mull org, o hub, ad o dao. Pr for ah flgh lg ar d, ad dmad a ah org a Poo ro. hy u a d far ruur for ory orol ad fd h omal m hrhold for log org-hub flgh. hy h d hr rul o allow mull far o ah org-dao flgh ad m-dd dmad. Kuyumu ad Gara-Daz ad Gara-Daz ad Kuyumu 997 u a grah-hory aroah for alloag a a flgh wor. Gara-Daz ad Kuyumu 997 aum a o-d a alloao ym, ormally drbud radom dmad ha dd bw r ll, ad a fd umbr of far la. hy dlo a algorhm ulzg ug-la for alloag a hroughou h org-dao wor ad ga omuaoal m. Kuyumu ad Gara- Daz u h am aumo ha dmad o ah day of h w may b dffr, ad arl aa ar odrd. hy dlo hr modl for h wor rg ad a alloao roblm, h hrd bg a olyhdral grah hory aroah, ad a oluo rodur ha hy omuaoally. h dooral h of Wllamo 99 omar h u of ral wor-ll a 4

6 ory orol hur mulao. Sh olud ha odrg wor-wd ff may ra ru by -4%, bu oly wh h load faor ry hgh. d or, Frlg ad Prma d hr rarh by dug h fdg ha drm aromao mhod ourform robabl hur whh hy lam du o a dffr far la ruur modlg ad mulao. I h ud of dyam r drmao, hr a o bw h raal ad h ghful. Comuao-bad aaly rl ould b ud o ol ral-world roblm of omal rg, whl mlr hor luda oal rl for udradg rg grally. A ommo aumo h lraur o a dr r grd, or u wo r, whh mlf h roblm o o of ddg wh o wh o aohr r Fg ad Xao a, b, Fg ad Gallgo, Sawa 3. h aumo of a dr r grd do m ufd, hr o horal groud, hr o oomy of ug fw r, or o raal groud, arl ad hol fa mloy a larg of r. hu, h df of h aumo rl o h mlfao of h mahmaal roblm. h ma adaag of h aumo ha rog of a omal oluo ral, bu ll l gad. A o Chaw 999, whh lo h f grd o dlo a uo for wh r r ad fall. h u of a f fuoal form for dmad ommo. h mo ommo aumo oal dmad, a -b, for oa a, ad b. h form of dmad uful o aum bau a oa, ad hu margal ru r mu a oa. A a rul, a moool would hoo o harg a r whh margal o lu a oa. h faur of oal dmad ma h oluo o h moooly roblm rdu o h roblm of alulag margal o, a mlr hough oral roblm. Gallgo ad a Ryz 994 h b ar of h of oal dmad ar, ad w du ly blow ad dd a om addoal uo of h hory. A ary of mham for brgg uomr o h llr ar odrd h lraur. h mo ommo aumo a oa Poo ro bu obly wh m-aryg arral ra Zhao ad Zhg. A rg ara o h arral ra ro a maro whg modl, whh ol a gal rao roblm: dg from h bhaor of buyr whhr dmad odo ha hagd Fg ad Gallgo. h oo bw h la a-load rg roblm ad dyam rg lord Daa 999b. h ar olud ha r hould b rag o arag, bau of a orrlao bw hgh dmad ad hgh r. A Daa aaly, ral auhor o mull la of uomr, who may arr a d ro. Whl h mo ommo ob fuo o mamz d ru, Carr ad Looy odr h alra aumo of rg o ll ou. Prg o ll ou a bad ragy for ral rao. Fr, a rof mamzg frm would o r o ll ou. Howr, mor alarmgly, h ff oluo ha mamz h ga from rad do r o ll ou. Mo of h lraur fou o h roblm of a gl flgh, rag omo ad alra flgh a fd, bu ral auhor ha mad hadway o h roblm of mull flgh ad rou. Fg ad Xao am a ml Y-had ar hrough a hub. Nwor u ar alo amd d or, Frlg ad Prma. Daa 999a h oly auhor o dlo a full hory of omo bw dyamally rg llr. h hory, olg wo frm rg or wo rod, mhaz ha r dro may rul from uh omo. h rul alo aalabl a, o-rod roblm. 5

7 A mora orgh of h lraur h ab of doug. Vrually h r lraur rum o doug. I h hol o, zro doug ma, bau f o boo a hol ada, grally aym mad ul h m o ay h hol, whh ml h am dou faor whhr o boo h room arly or lar. Wh arl, howr, grally aym mad a h m of boog, o a h m of darur. h mar bau h m ral ar log ough for doug o b gfa, g h may b bood moh ada. 3. Dyam Pr Drmao wh Pr Commm h of r hag foud aual arl rg myrou bau a moool wh ommm ably, a adard framwor, do wa o gag a all! o dlo h oluo fr rod by Soy 979, w ar wh a mlfd ro of hr aaly. h aaly wa dramaally dd by oard 4. h llr ll hr a durabl good or a o-m u good l a arla r. m omod of dr rod,, ad buyr ad h llr ha a ommo dou faor. hr ar a ouum of oal buyr who ha alu rrd by a dmad, o ha g h maur of buyr wllg o ay. W aum ha a oumr dmad r ul h oumr urha. h moool hoo a r u,, whh a b a o b o-rag whou lo of graly. A oumr wh a alu wll rfr m o m f > [] h uao rodug dffr,, df a u of ral alu ha ma h buyr dffr bw urhag a ad urhag a. No ha h ora o buyr how ha, f a buyr wh alu hoo o buy bfor m, h all buyr wh alu dg ha alo urhad by h m. [] h of uao a b old for rm of h ral alu:... [3]. h moool ll - - rod, whr dfd o ha. h moool rof ar π [4] 6

8 7. hu, h omum ll of oa a h o-ho rof mamzg ll, whh rur h rof aoad wh a a moooly. h ably o dyamally drma do o ra h ably of h moool o ra r from h buyr. hr a mora lo o b draw from Soy horm. If dyam r drmao layg a rol, bau of w uomr arrg, rahr ha a am o ra mor rof from a g of uomr by hrag low alu uomr wh dlayd urha. ha, dyam r drmao dr by uomr dyam rahr ha r drmao or a g of uomr. 4. Couou m hory Gallgo ad a Ryz 994 rodud a lod form modl of dr m omal rg. h modl ry uful for raably, ad w rrodu om of hr aaly hr, a wll a d by grag rdo abou h arag ah of r. L λ b h arral robably of uomr r u of m, aumd oa. A oa arral a b aumd whou lo of graly by dg m rm of uomr arral. m wll ar a ad d a. If o old, h rodu rh a, whh mgh our bau a flgh a off or h rodu m-dad l a hol room or m-har odomum, whr wha for al h u of h rodu o da. For hal rao, o doug odrd h o. Howr, om alao, o doug h rgh aumo. For aml, hol grally do o harg ul h uomr arr, ddly of h m h room bood, a uao whh orrod o o doug bad o la boog. h margal o of h rodu, a alu whh mgh lud ormal margal o lag a hol room, a mal rd o a arl bu ould alo lud lo

9 bu h uomr ha a a arular flgh l lly o a a alra flgh by h am arl. Poal uomr dmad a gl u, ad hr wllg o ay g by a umula drbuo fuo F. h alu of hag m for al a m dod by. Clarly hag ohg o ll oy zro alu. Moror, f o old by, a ory of m alo ha zro alu, yldg:. [5] Codr a mall rm of m,, byod a urr m. Wh robably -λ, o uomr arr, o ha h urr alu bom. Alraly, wh robably λ, a uomr arr ad h frm hr ma a al or do o. For r, h al our wh robably -F. Wh a al our, h alu bom - -, bau h ory drad by o. Summarzg: or F λ F ma λ λ λ λ [6] λ λ ma F λ λ [7] hrfor, ddg by ad dg o zro, λ ma F [8] h ro for omod of wo rm. Fr, hr ar rof from a al, -. Sod, hr h lo oo of llg h u h fuur, a oo ha ha alu. I o obl o ol h dffral uao for a arbrary dmad fuo F. Howr, wh a o ho of F, obl o rod a l oluo. L a F. [9] No ha - -a -m mamzd a * m. a h L h - -a a -m. h a h ml h. hu, ry rm o a mamum, ad o f hr a rm o, h global mamum. Moror, * a rm o. 8

10 9, * a [] ad. a a a λ [] h mulla oa a λ β rr h arral ra of buyr wllg o ay h a moooly r. a hu, a m, h d umbr of buyr wllg o ay h moooly r β-. h obrao hl la why! β [] wll aar h oluo. h fr rul hararz h alu fuo ad r a lod-form mar. Lmma Gallgo & a Ryz 994: Log a ad. * Log a β λ A m zro, uo hr a al aay. L b h robably ha hr ar u lf for al a m. horm :! β. h d umbr of a old β Lmma ad horm g a oml, lod form dro of rof, r ad al for h dyam moool rag yld maagm. For aml, h robably ha hr o aalabl aay a m.!! β β [3]

11 h formula ur ha, wh uff m ad a g aay, h flgh ll ou, bau lm. [4] How do a h mar aff h ouom? o modl h, odr rag boh h aay ad h arral ra of uomr, λ, rooroally. L γβ/, o ha! β!! γ! γ γ f γ > [5] f γ Wha rg abou h ro ha h robably of llg all h aay,, org o h am ll a would ar f h r wa u oa a h moooly r. S h a r d h ll alway, bau h oo alu o, h r mu g ry lo o h a moooly r mo of h m ordr for h lmg robabl o od. Pr u rof of h dyamally drmag frm ar a Log a Log γ! a γ Log γ f γ < [6] f γ A a alra o yld maagm ad dyam r drmao, odr a frm ha a r ad la fd. Suh a frm, a o-r moool, wll ar lowr rof ha h yldmaagm frm. I h rof rduo gfa? A moool who oly offr o r wll ha a flow-ra of al of λ-f. If η h umbr of fuur al g aay a m, h η ad η η d. [7] Lmma 3: η! h rof aoad wh a gl r a b umrally mamzd.

12 Fgur rod a aml of dyam r drmao, arg wh u for al. h aramr lud zro margal o, a 365 day rod wh a arag of o uomr ry ohr day, ad dmad h form F -. h r aoad wh arou umbr of u o b old ar llurad, for aml, 3 how h omal r aoad wh hr u o b old. h arag r, odoal o u rmag for al, dod by ; h rr h arag of od r. No ha u may o b aalabl for al, o ha h d r a b of a fo, alhough h omaro o h o r moool bl ha r may alo o b offrd. h omal oa r dod. Prof ar 5.45% hghr udr dyam r drmao ha hy ar wh a oa r. I h fgur, h moooly r h horzoal a, a, ad hr hm mro ubaally o h a moooly r Fgur : Pr, ad d r,, λ½, α,, 365. Fgur llura a ommo faur of mulao: h d r r, h fall. h for old ar a follow. Fr, r mu ually fall bau hr a o oo alu ror o m, ad h oo alu zro a m. hu, ror o, r ar rly largr ha h a moooly r ad org o h a moooly r a. Whl obl ha r fall for h r m ral, hy may ally r bau arly al, by rdug aalabl aay, dr u h r. Fgur llura h robably ha all u ll udr dyam r drmao. h org o lghly or 85.35% by h d of h rod.

13 Fgur : h Probably of Zro Caay I dyam r drmao rofabl wh ad λ ar boh ry larg? L θ rr λ/. I h a, h oluo for dmad η m { θ a, } [8] h mamzd a h a moooly r,, a rodd γθ--a <. Ohrw, h omal γ f γ < r af θ -a. I follow ha rof ar ad agr wh a Log γ f γ drmaory rof h lm for larg. ha, r u ga rof of dyam r drmao or a omally ho oa r org o zro, alhough h oal ga wll ll b o. h ha bau mo al a la a a aromaly oa r; dyam r drmao adaagou oly a h robably of aually llg ou hag, for a rlaly mall oro of h ry larg umbr of al. O a raoably rr h rul o ay ha dyam r drmao oly mar o h la wy or o al, o wh a larg umbr of u ar old, dyam r drmao do mar ry muh. Dyam r drmao ha a rlaly mod ff wh hr ar or mor a aalabl. h d of rof rdd, for raoabl aramr alu, udr dyam r drmao ar o ry larg, l ha %, wh omard o a aroraly oa r. A mora a of h oluo ha dyam r drmao do o aar o aou for h d of alu lamd for Amra Arl u of yld maagm. 5. ffy h Gallgo & a Ryz Modl A ff oluo h modl ha h rory ha h alu fuo mamz h ga from rad rahr ha h rof. h alu fuo, h, af

14 F C λ F ma λ λ λ λ [9] whr C h oumr alu, lu llr rof, odoal o h oumr alu dg. I h modl, F C F d F [] a d a a a. hu h ff oluo h oluo a moooly who o ar rdud by /a, h a moooly rof, would hoo. h aumd ruur of dmad ur ha all h uala oluo draw aly ually o ffy a o moooly. I arular, h ha of h m ah of r ad h oluo ha for larg roblm h ga of dyam r drmao ar mall aly ually o ff oluo a hy do o moooly oluo. I our w, h Gallgo & a Ryz modl o a ry uful rum for amg ffy, bau of h mlary of h moooly r ad ff r. Rdug a moool margal o by h moool maru do o rodu h ff r, bu a hghr r udr h adard rgulary odo l log oay. Howr, worh mhazg ha h ff oluo ad h moooly oluo lad o h am of r ah, o ha f o ar uobrabl, h wo modl ar obraoally ual. 6. ffly Alloag Lmd Caay udr Uray h lod form of Gallgo a Ryz fala omuao bu obur oom odrao bau of h aur of h dmad. I h o, w odr ffly olg h alloao roblm, rahr ha mamzg rof. ffy o uraoabl, arl fa omo ad a rfly om modl would ugg h ff alloao of h aalabl uay. I m lly ha hr h moooly or h fully ff oluo auraly rr h ral world, whh robably fall omwhr bw h wo rm. or ad Ro 994 rha h formo aml of a am o a arl r omo, bu a h ll of arag far. Dlog a hory ha rm ag h dgr of omo a h dald ll of h r ah, ora, r a formdabl hallg. Obrao of ua ar gog o b ral o h am of h dgr of omo, bau r obrao alo ar o uff o dfy h modl. h d of ff alloao roblm wa fr udd h o of lry grao. Co of aay du a a load roblm oau 949, Wllamo 966, o amd bau h margal o of lry grao muh hghr wh h la aay rahd ha blow ha ll, o ha h a load mar a lo o h oom of lry grao. h adard aumo a load roblm ha dmad ah rod ar dd, ad ha h full uly aalabl ah rod. h a-load roblm rally a aay lag roblm, whr h 3

15 dd aay dd o r. For a wo rod modl, l b h dmad rod. h frm rof ar g by π β ma {, } m, [] whr β h margal o of aay ad m h margal o of roduo. I h arl ad hol o, h adard a load modl oorly ud, bau aay old o rod aalabl h ubu rod. hu, h adard modl of h a load roblm a how larg a la ha fl roud-r bw Dalla ad Chago hould b, g arao oal dmad from day o day or from w o w. Whl h roblm rr a mora logal roblm, ha ll or ohg o do wh dyam rg. I ora, arl alo fa h roblm ha a o a arular flgh a b oud by wo agr, ad h roblm of alloag lmd aay ugg a u d a load modl. Moror, h fa ha fuur dmad ow a h m of orag h fr rod a rual a of h a load roblm fag arl ad hol. ha, arl ad hol ora for al al, o owg h ralzao of orall dmad. W rodu a w modl of radom arral ha ha om ara ror. h radom om h form of radom arabl rod, wh rod dmad bg, h, for r. W alo aum ha obrabl a h bgg of rod, o ha h frm a hr r or uay ha rod; wha ura h fuur dmad, o h urr dmad. 3 W wll fou o h oa lay of dmad a, α -, bau h a adard mral modl ad a raably. W wll rfr o a r aa al, ad a oal al, bu h rrao a h arl o. I h arl o, f h umbr of uomr, would ha o b h robably of al, whh a hould b o grar ha o, whh o wh h oa lay aumo. A b, h modl rr a aromao for h arl o. h o odr oly h ff alloao. h ff alloao rg for ral rao. Fr, omo wh ohr rodr wll grally uh frm rodu offrg oward ffy ad away from moooly, o wh gorou omo, ffy mamzao robably a br modl ha moooly rof mamzao. Sod, h moooly modl ha b br udd ha h om, ff modl. hrd, om of h rg bhaor h moooly modl do o ar from r drmao bu bau of h da of ffy. h our of h bhaor hallgg o whou udyg h ff alloao. ha, wha aard o b r drmao mrly arao margal o. h llr ha a aay K. W wll fou o h wo-rod a hroughou h o, alhough h radly gralz o mor rod. If r aa al ar of h fr rod dmad, h umbr of a aalabl for al h od rod K. h ar alud by oumr a a r afyg K [] 3 Daa 998 modl a al a, wh ag o wo o alu, rrd a a wllg o ay for lur ad bu ralr. 4

16 5 or K [3] h o of fr rod al r u dramaally h umbr of al. Suo ha oa lay of dmad, α -. W aum > o ha h oumr urlu f. For ay g, α K [4] h r ha lar h mar rod af. h uomr alu of uay α. hu h orall ga from rad ar α α K W [5] α K h ga from rad ar mamzd a. K [6] h r aa od rod al ar K K [7]

17 6 au α, a rudmary alulao o how ha h d r h fr rod ad od rod od. ualy of h r a ary odo for mamzg h ga from rad; ohrw would ra h ga from rad o alloa mor of h a o h hgh-rd m. Som a ar malload, bau orag wh h arly uomr our ror o owg h ralzao of umbr of lar uomr. Moror, worh mhazg ha h malloao ff mamz h ga from rad ad o a ou of moooly rg. I our rmology, dyam rg bu o dyam r drmao. How larg h ff? h arag rduo h har of rod wo uomr af. [8] horm 4: h od rod har largr o arag ha h fr rod har, ha,. Proof Ad. horm 4 how ha h har of uomr rd h od rod a la a gra a ar rd h fr rod. h a ou of h obly of mall domaor ladg o larg har; wh fw uomr arr, h r low ough o ur h la ll ou. h al a a frao of h arag alu of ar l ha rod. ha,.

18 7 Igh o h magud of h ff malloao g by h rul. L Var CV b h off of arao of. horm 5: CV Proof Ad. h har of a alload o h fr rod may da from h ff ual har by a muh a CV ; for h a of ad CV, h ould b 5%. h boud horm 5 a, h ha for a bomal radom arabl wh o alu ual o zro, h rgh ualy hold wh ualy, ad a, h lf ualy hold wh ualy. h ga from rad grad ar α α K K K W. h frm ru α α α K K K K K hu, frm ru rooroal o wlfar udr oa lay of dmad, ad h aaly of ru mamzao udr ffy, of our aalyz wlfar mamzao. o rform h aaly, w ur o h log ormal a. h ffy h modl ar bau orag wh arly uomr our ror o h m ha rod wo dmad ralzd. How larg h ffy? o awr ha uo, w odr h a whr log-ormally drbud. I h log ormal drbuo a, h wlfar lo aoad wh h mg mar a b ry larg.

19 8 7. h Log Normal Ca If h drbuo of log-ormal, o ha h log ormally drbud wh ma ad ara σ, may of h ro ha lod form, graly falag omuao. I arular, ½ σ ad σ σ [9] h off of arao for. ½ σ σ σ Var CV [3] hu, σ CV, ad [3] σ CV. [3] h d ga from rad ar σ α α K K W [33] How muh do ual orag o rm of ru or wlfar? W ra a h d alu of a logormal wh h am ara bu dffr ma ha. h rooro of h ga from rad rrd hrough orag or wo rod dod % W. I ha h alu

20 % W [34] W how by aml ha % W a b a low a 64%, ha, h ably o ora ada rdu wlfar by a hrd rla o mulaou orag. 4 h aml ha a muh largr fuur oulao of dmad ha fr rod dmad. Howr, h ara of dmad larg oo, ad h au h lo aoad wh uray o b ry larg. I arular, a glgbl frao of { } h aml, a dffr whh aou for mo of h ffy lo. for mor raoabl aramr, lo a b 5% or mor. A aml, wh u o % lo, llurad Fgur 3. I h aml, h a o h alu.6, ad 88, whl h d alu of a o h alu 9, 98 ad σ 3.9 σ 5 σ W Fgur 3: ffy lo rla o mulaou orag. Paramr: -5,. 4 Sfally, o of aramr 3.474, σ.65, 45.53, ad.8 8. h of aramr g h followg d alu for. { }. 49, ad

21 h ma oluo ha h mora ff dyam rg ar o from a am o ra mor moy from h oumr, bu from oml mar, ad arular from mobly of mulaou orag wh all oal buyr. Dyam rg ud rmarly o mga h o of h mg mar. Moror, wlfar o of h mg mar ar oally u larg. h oluo ugg ha oo, whh ra mar for ada orag, ar a mora a of boh mamzg ru ad of ffly alloag rour. 8. Oo ad Irrubl Sal h wlfar lo dfd h rou o ar bau of oml mar. I arular, o obl o ora wh h rod ag a h m ha h rod mar lar. h la of orag lad o ffy bau omm oo may a ar old h fr rod, wh h dmad h od rod uuually hgh, whl omm oo fw ar old, bau dmad wa udly low. A oluo o h roblm grally o ll og a, ha, o ll a oo. I h a, h ma alu of h oo o rm h llr o ll h a o a hghr alu buyr h od rod. h d of oo ll ud hol ad arl, bu u ommo l raor whr go by h am of rrubl ramo rgh, a ood o frm guarad rgh. Prl.om old a d of rrubl r, whr hy old h wll ada bu dd fy h m of h flgh ul a day or o ada. o ga om gh o h al of rrubl r, odr fr llg wo ual of r, >, whr h robably of r, ad h ar old a r >. A oumr wh y alu good a. A y oumr rfr y f, [35] or. [36] I addo, a oumr rfr good o ohg a all f. If, h o oumr r buy good, o w mo h odo, whou lo of graly, ha. L F h umula drbuo of alu. h dmad for good F ad h dmad for good F F. L b h margal o of r y. h llr rof ar

22 , π F F F Ma. [37] I hlful o rodu h oao F Ma R, whh h rof mamzao aoad wh o good. L arg * F Ma b h oluo o h o good mamzao roblm. Margal ru drag f * a rag fuo, o ha a ra o rdu h uay ad ra h r hargd by h moool. h a adard aumo. horm 6: Suo margal ru drag. A rof-mamzg moool ll h low ualy good f ad oly f, whh a π R R, * * ad *. Ohrw, π R ad *. h roof h ad. h rrubl good roblm bra u o wo ara mamzao roblm, o for h low ualy good, ad o for h dffr of h low ualy good ad h hgh ualy good. h y of rul d from h uual rul whr a moool llg o wo y of oumr offr wo ual; hr h moool llg wo good o a ouum of oumr. h r rul rm of Mua ad Ro 978 aaly, howr. Codr a llr ha dd ll good, ad h bg llg good. Do ha llr r r, or fall? h r fall f < * * * [38] hu, f * oa, h r fall wh h od good rodud, ad h r r wh * o. I h a of uform drbuo of alu, or oal drbuo, or oa lay, * lar o, ad hu roduo of h od good do hag h r. Mor grally, may ra or dra.

23 I h arl o, h o of a al grally a hadow o how muh ru forgo bau h a a? Couly, h odo auomaally afd. h o of a robably of r al a lo flbly rla o h r ll. ha lo flbly ma o rhdulg h r ruma whh would b drabl o do o. hu h o hould b a arag of or h flgh wh h rrubl r offrd a hduld, lu h o ruma whr wa rofabl o rru. S h lar hould d h formr g omal rruo, h o of r for a hould d h o a, whh ma alway omal o offr h rrubl r. A a raal mar, h rgh rrubl r robably a ha l h arl hoo h m h agr fl, bu h flgh a day. h d o oorda ar ral wh hol ad ohr dao a rr h ably of h mo agr o b flbl or mull day. Nrhl, flbly wh r o h m of ral oally rodu gfa ga h d alu of ar ral, boh h ably of h r ym o dfy ad alloa a o h hgh alu agr, ad h ably o ulz mor aay. h ga from radg oo ar largr wh h frm o a r ha good for a arabl amou of m. h homo wa udd by Daa 999a. Radom dmad wll grally ra a malloao, whh w modld by alloag fr rod a whou y owg od rod dmad. Howr, udr a od r, a wll b malload bau h od r wll grally o b h orr r o lar h mar, hr rag a urlu or a horag. Codr h modl dlod h rou o, wh a radom dmad h form ad oa lay α -. L K b h aay ad uo ha wh aay bd, h uomr ar a radom lo of ho wllg o urha. h ga from rad udr a oa r ar W M d [39] α { K, } M{ K, α} h uao ar bau, wh K<, h aay ora bd, ad h ga from rad ar h arag of h alu of a or h of oumr wllg o buy ho a. O rg faur of h uao ha ffy rur a o robably of my a. If hr a zro robably of my a, h h r lowr ha h mar larg r. Low r ra radom agm wh dmad larg, whh al a lo of ffy; h lo balad aga h ra a old wh dmad low. Idd, h fr ordr odo for mamzg W a b rd a: W K Pr{ K } [4] hu, h robably ha aay bd, K, o a h r mamzg h ga from rad.

24 9. Aual Arl Prg Par I ordr o a h of dyam r drmao, w olld daa o four y ar: Lo Agl LAX, UR, SNA o La Vga LAS ay Ara SFO, SJC, OAK o Porlad PDX Dalla DFW o Olahoma Cy OKC Dalla DFW o ula, OK UL W olld daa o r offr from Orbz, raloy ad for ar of h rod, from AA.om. Uforualy, Souhw Arl o faurd o ay of h our. Of h aror ar, four LAX ad UR o LAS, SJC ad OAK o PDX ar alo rd by Souhw. O would Souhw o r a modrag flu, g rlaly low ara r, alhough Souhw ha rodud a far ruur whh may rul mor dro, rahr ha l. 5 I ay a, wh h daa aalabl, h ff of Souhw r wll ha o rma a o for furhr rarh. Purhag a Amra Arl o AA.om grally $5 har, bu howd o ohr dffr. h fgur blow oly how far uod by Orbz, whh had h mo offrg. W olld daa o flgh, all h yar 4, darg o 8/6, 8/8 8/9, 9/9, 9/, 9/, 9/3, 9/5, 9/6, /7, /9, /, / /3 ad /4. S arl rr or 99% of h daa, wh h followg rooro of h flgh. Arag r ad h fruy of arl ar rd abl. abl : Arag Pr h Daa Arl AA Alaa Dla Ud AmW USAr NW Prooro of Flgh 5.5%.7% 6.% 4.6% 8.9% 6.% 6.% Arag Pr $8.5 $6.59 $79.3 $44.6 $66.7 $39. $9.48 hr ar f maor rooo ha a b d wh h daa.. Pr fall a aoff aroah. Pr ar rg ally 3. Como rdu ara of r 4. Pr hag wh ah a old 5. Pr of ubu ar orrlad a. Subu m/day b. Subu aror h fr rooo ha r ar fallg a aoff aroah. h a rmarably robu rdo of hor ha ha dal uomr y arrg or m. h robu follow from h followg obrao. h o of llg a a omod of hr omo. Fr, hr h o of r. Sod, hr h o of o llg ha a o a ubu flgh. h od omo lud ay yhologal ff o oumr ao oumr may rao ha urr low r ar daor of lly low fuur r, whh d o ma dmad mor la, o h arl drm. hrd, hr h forgo oo of llg h a lar o h am flgh. h fr wo o ar roughly oa or m, ad raly aromaly oa a h flgh 5 h la, wh aml far UR o LAS ar Rfudabl Aym $94, Rrd $86, Ada Purha $7, Fu Far $49, Promooal $39, ad Ir O-way $39. 3

25 aroah. h hrd o, howr, go o zro a h flgh aroah. hu, mo hor wll ugg ha r fall h la mom bfor a flgh, o arly o zro bu o om omal r rflg h mar owr of h arl ad h ff of low-r al o al of fuur flgh. u h r hould ll fall. A mor alzd rdo ha h arag r r ally. h a faur of h Gallgo-a Ryz modl alrady dud, ad ohr. I mor u modl whh oumr a arh ha h mor ommo modl whr oumr who do urha daar; f r ar fallg o arag, oumr wll d o wa o urha. La fallg r ar o obouly loabl bau of h gfa llhood of h la fllg u. hu, oumr mu wgh h d ga from dlay aga h o of falg o aur a a a all; h lar o glgbl wh a uffly log m-horzo. o a h rooo, w ra h followg rgro o rd h r. W ha dummy arabl for day wh o, wo, hr ad four w rmag, arl-f dumm Amra omd, hour of aoff orgh omd, ad y-ar dfr LAX o LAS omd. W omd urba, Log ah ad SNA aror bad o fw flgh or h rod. W oly amd o-way oomy far, ad ha 3837 obrao. h 38 arabl aou for 73.5% of h arao. Pr r $5 h w bfor aoff, whh o o of a r of $8. h rou w. h ulma forgh bfor aoff aou for $6 r ra. hr a lgh,.9 r day ra ror o ha. hu, h ma m-ah rdo of h adard hory aar o b mrally fal. h ff ar u gfa, ha o h z of h daaba. W ha o amd o orol for orrlao obrao, o ha h -a, whh ra obrao a dd, lly ora h gfa. h mora oluo ha hagg dmad a al faur of h daa, ad modl ha aum ha h ha of dmad oa or m ar mrally ald. A fw ohr rmar follow from h rgro. Alhough Amra Arl a low rd arl h daa, hgh-rd adug for fd ff, hough o a a US Ar. Pr ar hgh md-day ad arly g. LAX a rlaly aror, whl Sa Frao hghrd. h ugg ha omo ad h mar do o rod a good aou of rg, Sa Frao ha a larg umbr of flgh. Souhw r a LAX, OAK ad SJC mgh hl la h arag rg. No, howr, ha aror dumm ar alo aoug for h lgh of flgh, o omaro hould oly b mad amog d aror ar for a g y-ar. 4

26 abl : Pr Rgro ma S Sa PValu Coa X - 64 Day WK X - 44 WK X WK WK Alaa X - Dla X Ud X - 6 Amw USAr X NW X - 6 AM X - 79 AM X - 4 AM AM AM AM X - 6 Noo X - PM PM PM PM PM PM PM PM X - PM ULDFW X - 6 OKCDFW X DFWUL X - 57 DFWOAK X - 33 SFOPDX X SJCPDX X - 63 OAKPDX X - 3 PDXSFO X PDXSJC X - 65 PDXOAK X LASLAX X - 45 I aohr ar Carboau, MAf, Malo & Malo 4, how ha h hrd rooo o uord a larg daa of arl r. Idd, mor omo wa waly orrlad wh mor dro, o l. h fdg o wh or ad Ro 994 fdg. 5

27 Som hor o a wo r ruur, ohr rm r o ary ououly. I h daa, om arl larly la oward a wo r ruur. For aml, Amra Arl r o OAK o PDX how d of wo ma r, wh a hrd u ror o aoff. h llurad Fgur 4. 6 Mo of Norhw far how u wo r. Arag of Pr Flgh AA_685_9/3 AA_69_9/3 AA_6966_9/3 AA_748_9/ Day l Fgur 4: Amra OAK o PDX, Smbr 3. Howr, ohr flgh how d of almo ouou adum, ad dd omm Amra aar almo o radomz offr. Fgur 5 how a aml of wo Amra Arl flgh, AA 38 o S 3 ad S 5, wh rmarabl arao. So h d faor of h rooo ouou adum of r md. Why do arl u wo r? A adard oom laao ha hr a alu r ommm o a oumr lag. h h rao g for raura, mo, ad h l o maa a oa r, or wo r, h fa of rdably fluuag dmad. Howr, r do m arly rdabl ough for rdably o b h rao for ug oly wo r. A mor laubl hory ha arl u wo r bau h hory br dlod for wo r ym. h laubl for a o arl l Norhw, bu mlaubl for Amra Arl, whh ha raordary r adum o ohr rou. 6 h fgur dfy flgh Arl_Flgh Numbr_Darur Da. All daa h fgur om from Orbz wb. 6

28 Arag of Pr Flgh AA_38_9/3 AA_38_9/ Day l Fgur 5: Amra 38 r lod aga day ror o 9/5 h fal rdo of h hor ha h r of ubu hould b orrlad. h, aga, a robu rdo ad rur oly oumr ubuo a a hyoh. o a h, w lood a h orrlao bw r of flgh from h bay ara o Porlad, ad Porlad o h bay ara. ad o ral m, m ha SFO ad SJC hould b h lo ubu, followd by SFO ad OAK, followd by SJC ad OAK. I fa, howr, h orrlao ar almo aly h rr. abl 3: Corrlao Coff Flgh Par ρ Flgh Par ρ SFOPDX, SJCPDX.67 PDXSFO, PDXSJC.596 SFOPDX, OAKPDX.9 PDXSFO, PDXOAK SJCPDX, OAKPDX.867 PDXSJC, PDXOAK.954 abl 3 how h orrlao off of arag r, wh h orrlao bw ma r offr o ah rou a a fuo of h umbr of day ror o darur. h rdo ha h fr row of h abl hould ha h hgh orrlao, followd by h od, followd by h hrd. Howr, aromaly h oo ar. h r of Souhw a Sa Jo ad Oalad, bu o a Sa Frao, may b h laaory faor, ally lgh of h fa ha h r ll a SFO muh hghr ha a h ohr wo aror. I mor hallgg o a whhr hr rog orrlao bw ubu flgh o h am rou. h followg ur how om orrlao or m of h ubu flgh bw Porlad 7

29 ad Oalad a 6PM, 7PM ad 8PM ± ½ hour. h flgh ha a arag orrlao off of abou.8 orall. Arag of Pr Hour Day l Fgur 6: PDX o OAK, rur, a arou m O h ohr had, Fgur 7 r a mlar amao of oomy flgh from LAX o LAS. No uh orrlao radly aar ad h flgh ar waly orrlad. Afroo flgh, rd Fgur 8, ar mlar. 8

30 Arag of Pr Hour Day l Fgur 7: LAX o LAS oomy la, 6AM o 9AM Orall, h maor rdo ludg h mo robu rdo of h hor aar o b olad h daa. Couly, w hor, ad robably a w aroah o h aaly, ar dd.. Rarh Pro ad Myr hr ar may rarh ro ha flow from h aaly of h ar. Fr, muh of h g wor foudd o a moooly dro. h aaly of h ar ugg ha rag h modl a a om modl may ra h raably of h modl whou arfg hr mral rla or omly of bhaor. Sod, h al of oo r h obly of hag h ffy of ada orag, a obly ha ha b ll lord. Couou m hu from h fa lraur may lay a mora rol dlog a hory of oo for rhabl good. h m a fruful aroah, bau oo hml ar rhabl, ad ouou m oha alulu hu ha layd a mora rol mlfyg mlar roblm. h g hory fard oorly wh ofrod wh h daa. I arular, h falur of r o fall a aoff aroah daag o hor, lag adg oly ho hor whh la arrg oal agr ha a rlaly hgh wllg-o-ay. Whl h a raoabl hyoh, ohl d furhr dlom, ad hallgg o h of aural aumo o mo ordr o h roblm, o dmad a b almo ayhg. 9

31 h daa r a ary of myr. h ga o arhg ar oaoally ormou. Whl r r a aoff aroah, oaoally hy bou aroud odrably. h fdg mrrord zo al, whh mrally amd h ga o arhg for flgh. hy fd a arag ag of 7% by ug a arh algorhm, rla o u boog h flgh h fr m hd. Arag of Pr Hour Day l Fgur 8: Afroo LAX o LAS Flgh by m of darur Wha rha mor myrou, ad llumag, a d ha ourrd md-july o Alaa, a flgh from OAK o PDX. For aromaly a w, h r of AL darg 9/3 arly doubld, whl h r of h 9/5 ad 9/6 darur wr aromaly uhagd. h 9/6 darur had b mor. I fa, h r of h 9/6 darur h fll brfly. h llurad Fgur 9. 3

32 Arag of Pr Flgh AL 9/3 AL 9/5 AL 9/ Day Fgur 9: Pr of AL, darg 9/3, 9/5 ad 9/6 G darur m, Amra Arl 685 a maor ubu for AL. Durg h m, AA 685 flgh rd oly a ry mod ra r, whh llurad Fgur. h r of AA685 had b ubaally hghr ha h r of AL, bu wh AL w u, h r of AL wa muh hghr. h r of h AA 685 w u lghly, h am ba dow o h rouly ralg ll. Wha ma h o myrou ha AA 685 ad AL rr h am arla AA 685 a od-har flgh orad by Alaa arl. ha, h ral myry why h wo dal fgh ar bg mard a uh dffr r, wh h fa ha h flgh ar od-hard larly dad o h wb. o ma mar mor myrou, h bl r of AL od wh a lgh ra h r of AA 685 for flgh darg o ohr day, bu o h r of AL o ohr day. h llurad Fgur. 3

33 Arag of Pr Flgh AA_685_9/3 AL 9/ Day Fgur : Pr of AL, AA 685 Arag of Pr Flgh AA_685_9/5 AL 9/ Day Fgur : AL, AA 685 darg 9/5 3

34 I m lly ha Amra rad o h r of ral. h fa ha h dal flgh wa bg offrd for o muh mor wh bood a a od-har o Amra rahr ha o Alaa, Alaa r ra wa rfld h r of alra day, h r of Amra dal flgh oly ro lghly ad alo ro o alra day, ugg ha oumr brg a ubaal brad loyaly o h flgh, arl u a far rogram ha ra o rg by ohr aro a of ubu flgh, ad 3 h ro oorly udrood by rarhr. h a of ollg daa for o-ommral u ugg ha rr-grg h rg mham of arl a fabl ad oally ry rg rarh ro.. Coluo h ar ha odrd o may dffr hor ad daa ha worh mhazg a fw hghlgh. Dyam r drmao rmarly dr by uomr dyam rahr ha r drmao or a g of uomr. Wh a larg umbr of u o ll, h r u ga rof of dyam r drmao or a oa r mall, alhough h oal ga wll ll b o. Mo al a la a a aromaly oa r; dyam r drmao adaagou oly a h robably of aually llg ou hag, for a rlaly mall oro of a larg umbr of al. O way o ummarz h oluo ha dyam r drmao oly mar gfaly o h la wy or o al. Moooly ad ff dyam rg may b obraoally ual, ad ar h mora modl of Gallgo ad a Ryz 994. Drly olg for ff oluo r a alra aroah wh oal mral mr ad rla raably. h mo mora ff dyam r drmao ar o from a am o ra mor moy from h oumr, bu from addrg oml mar, ad arular from h alu ad o of ada orag. Oo, whh ra mar for ada orag, ar a mora a of boh mamzg ru ad of ffly alloag rour. h rrubl good roblm bra u o wo ara mamzao roblm, o for h low ualy good, ad o for h dffr of h low ualy good ad h hgh ualy good. I h arl o, a rrubl good o ha rod h arl grar flbly wh r o h m of flgh. h o of dlrg a a rrg grar arl flbly auomaally lowr, ad hu ar of boh rof mamzao ad ff roo of r. ffy rur a o robably of my a. Prg o ll ou ff. Syma hag dmad ar a al faur of h daa, ad modl ha aum ha arly ad la arral ar dal ar mrally ald. 33

35 Rfr adll, Ralh D., A omal, dyam oly for hol yld maagm, uroa Joural of Oraoal Rarh, ol., o. 3, aru, M., ad Srdhara, V. 4, Dg ad aluao of a dyam aay aorom rodur, uroa Joural of Oraoal Rarh, ol. 55, o., lobaba, Pr P., ad Wlo, Joh L. 997, Ima of yld maagm om arl mar, Joural of Ar raor Maagm, ol. 3, o., oard, Smo Dyam Moooly, Saford Ury, uublhd maur. odly, S.., ad Wahrford, L.R. 995, Prhabl-a ru maagm: gr ad mullr yld maagm wh dro, Iraoal Joural of Maagm S, ol. 3, o., oau, Marl 949, La arfao d dmad o: alao d la hor d la au ou margal, Ru Gral d l lr, ol. 58 Augu, glh ralao aalabl J u 33, Arl 96. or, Sr, ad Ro, Nay 994, Como ad r dro h U.S. arl dury, Joural of Polal oomy, ol. 3 Augu, omr, hodor C. 996, ffy odrao arl rg ad yld maagm, raorao Rarh Par A, ol. 3, o. 4 July, roo, M., ad uo, K.J. 994, Yld maagm: a homo of h 98 ad 99?, Iraoal Joural of raor oom, ol., o. Ju, Carboau, Sha, MAf, R. Pro, Malo, Hugo ad Malo, Su 4, Pr drmao ad mar owr, uublhd. Carr, So, ad Looy, Wllam, h r wdor roblm: hoog a omal dmad orfolo for aaad rour, Maagm S, ol. 46, o. 7 July, Chaw, Rhard., Omal dyam rg of rhabl rodu wh oha dmad ad a f of r, uroa Joural of Oraoal Rarh, ol. 5, o., Daa, Jr., Jam D. 999a, ulbrum r dro udr dmad uray: h rol of oly aay ad mar ruur, RAND Joural of oom, ol. 3, o. 4 Wr, Daa, Jr., Jam D. 999b, Ug yld maagm o hf dmad wh h a m uow, RAND Joural of oom, ol. 3, o. 3 Auum, Da, Paul 994, Arl rofably yld o maagm, SIAM Nw, ol. 7, o. 5 May/Ju. d or, Sa V., Frlg, Rhard, ad Prma, Nada, Mahmaal rogrammg for wor ru maagm rd, uroa Joural of Oraoal Rarh, ol. 37, o.,

36 d. Harr, Frdr H., ad Pdr, Joaha P. 995, A ru maagm aroah o dmad maagm ad ordr boog ambl-o-ordr maufaurg, Joural of Orao Maagm, ol. 3, o. 4, Dr, Raymod ad MAf, R. Pro 996, Damagd good, Joural of oom ad Maagm Sragy, ol. 5, o. Summr, zo, Or, Koblo, Crag A., uhda, Raaoom, ad Ya, Aladr 3, o buy or o o buy: mg arl far daa o mmz urha r, Prodg of h 9h ACM SIGKDD Iraoal Cofr o Kowldg Dory ad Daa Mg, Wahgo DC. Fa, Mg, ad Wag, K 998, Omal harg oly for gl oulao wh rod off, Mahmaal o, ol. 5, o., Fg, Youy, ad Gallgo, Gullrmo, Prhabl a ru maagm wh maroa m dd dmad, Maagm S, ol. 46, o. 7 July, Fg, Youy, ad Xao, ahu a, Omal ol of yld maagm wh mull rdrmd r, Orao Rarh, ol. 48, o. Marh-Arl, Fg, Youy, ad Xao, ahu b. A ouou-m yld maagm modl wh mull r ad rrbl r hag, Maagm S, ol. 46, o. 5 May, Fg, Youy, ad Xao, ahu. A dyam arl a ory orol modl ad omal oly, Orao Rarh, ol. 49, o. 6 Nombr-Dmbr, Gallgo, Gullrmo, ad a Ryz, Garr 994, Omal dyam rg of or wh oha dmad or f horzo, Maagm S, ol. 4, o. 8 Augu, Gara-Daz, Albro, ad Kuyumu, Ahm 997, A ug-la rodur for mamzg ru yld maagm, Comur ad Idural grg, ol. 33, o. -, Km, Shryl. 989, Yld maagm: a ool for aay-orad r frm, Joural of Orao Maagm, ol. 8, o. 4 Oobr, Kuyumu, A., ad Gara-Daz, A., A olyhdral grah hory aroah o ru maagm h arl dury, Comur & Idural grg, ol. 38, o. 3, Laday, S.P. 996, Omal mar gmao of hol room-h o-lar a, Iraoal Joural of Maagm S, ol. 4, o., La, K-Kug, ad Ng, Wa-Lug 3, A oha aroah o hol ru omzao, Comur & Orao Rarh, Arl Pr. L, Mhal Z.F., Prg o-orabl rhabl good by ug a urha rro wh a alao o arl far rg, uroa Joural of Oraoal Rarh, ol. 34, o. 3, MAf, R. Pro, Coar mahg, oomra, ol. 7, o. 5,

37 MGll, Jffry I., ad Va Ryz, Garr J. 999, Ru maagm: rarh orw ad ro, raorao S, ol. 33, o. May, Mua, Mhal ad Ro, Shrw 978, Moooly ad rodu ualy, Joural of oom hory, ol. 8, Or, Shmul, Smh, Sh, ad Wlo, Robr 985, Caay rg, oomra, ol. 53 May, Sawa, K. 3, Omal ol ouou m ory orol modl wh lmd uly, Comur & Mahma wh Alao, ol. 46, o. 7, Srdhara, Sr V. 998, Maagg aay ghly orad ym, Iraoal Joural of Produo oom, ol , Soy, Nay L. 979, Irmoral r drmao, h Quarrly Joural of oom, ol. 93, o. 3 Augu, allur, Kalya. ad a Ryz, Garr J. 4, h hory ad Pra of Ru Maagm, Kluwr Aadm Publhr, Dordrh, Nhrlad. Va Sly, Rhard, ad Youg, Y, F horzo oha aa wh alao o yld maagm, Orao Rarh, ol. 48, o. Jauary-Fbruary, Wllam, Lor 999, Ru maagm: mrooom ad bu modlg, u oom Arl, Wllamo, Olr 966, Pa load rg ad omal aay udr dbly ora, Amra oom Rw, ol. 56 Smbr, Wlo, Robr 989, ff ad om raog, oomra, ol. 57 Jauary. Zhao, W, ad Zhg, Yu-Shg, Omal dyam rg for rhabl a wh ohomogou dmad, Maagm S, ol. 46, o. 3 Marh,

38 37 Ad Proof of horm : h oluo of h robably ha hr ar aly uold a gord by h robably of llg a a wh hr ar uold, ad h robably of llg a a wh hr ar uold, o ha F F λ λ λ λ G a aay a m,,, ad. h ud a h ba of a duo o ablh h horm. Suo h horm ru a. h [4]! β h lar dffral uao, wh h boudary odo -, g d! β [4]!! d β β β [43]

39 h d umbr of uold m, -, af β! [44] β β! β β! β β. Q..D. Lmma A:!! f f > <! Proof of Lmma A: No d L ψ. If <, ψ for [,/], o! d. Now uo >. For, ψ drag. Loally, aroud, ψ. hu, d d d. [45] 38

40 39 h roof for a b hadld by obrg ha!! o-rag. Q..D. Lmma A: <! f Log f Log Proof of Lmma A: d Log d Log Log!!! d Log! For >, ad larg, h rm - d h gral drag ad hu aromaly ual o fr ordr aylor ao y y. hrfor, for >, d y Log d Log!! [46] Log d y Log!! Log Log!!. Log Log π h la aromao o Srlg Formula, π!

41 4 For <, h roof mlar, alyg Srlg formula ad obag d Log d Log π! [47] ½ ½ d Log d Log π π Log Γ Γ ½ Q..D. Jufao for om aro h : If h r oa a a, h robably of a al g ha a uomr ha arrd a. hu, l b h robably of mag a la al h rod [, ]., ad for >. d a a λ λ [48] a a λ λ β β h ol for! β β [49] hu, h robably ha h flgh ll ou < > ½!! γ γ γ γ β γ β f f f [5]

42 4 Proof of Lmma 3: Fr o ha f aay zro, h al ar zro, rfyg h ba of a duo. Now uo Lmma 3 ru for -. d η η d! d!!!! Q..D. Proof of horm 4: L b a oga radom arabl, ad,. ad ar ouga o. L f ad g. h from h Holdr Iualy, { } { } { } fg g f or { }.

43 4 or Q..D. Proof of horm 5: L b a oga radom arabl, ad,. ad ar ouga o. L f ad g. h from h Holdr Iualy, { } { } { } fg g f or { } } { or { } } { { } } { } { CV Q..D. Proof of horm 6:, π F F F Ma

44 43, F F Ma, F F Ma R R W aum h adard odo ha margal ru drag, whh a hr ar al of good f ad oly f f ad oly f f ad oly f f ad oly f.