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% HISTORY % Log:psnup.pro,vLog: psnup.pro,v Log:psnup.pro,v % Revision 1.1.2.1 1995/11/01 13:33:33 moore % Created from Transcript 4.0 sources % [1995/10/31 15:23:42 moore] % % EndLogEndLogEndLog % Psnup prolog % Copyright (c) 1990,1992 Adobe Systems Incorporated. All Rights Reserved. % GOVERNMENT END USERS: See Notice file in TranScript library directory % -- probably /usr/lib/ps/Notice % RCS: Header:/afs/cs.cmu.edu/misc/transcript/rcs/lib/psnup.pro,v1.1.2.11995/11/0113:33:33mooreExpHeader: /afs/cs.cmu.edu/misc/transcript/rcs/lib/psnup.pro,v 1.1.2.1 1995/11/01 13:33:33 moore Exp Header:/afs/cs.cmu.edu/misc/transcript/rcs/lib/psnup.pro,v1.1.2.11995/11/0113:33:33mooreExp /PNSP { % scalefactor PNSP /PN save def dup scale } bind def /PNEP { PN restore PNshowpage } bind def /PNBOX { % width height PNBOX gsave 1 setlinewidth exch /width exch def 0 0 moveto width 0 rlineto 0 exch rlineto 0 width sub 0 rlineto closepath stroke grestore } bind def /PNLS { % scalefactor -width PNLS /PN save def 90 rotate 0 exch translate dup scale } bind def %%EndProlog %%BeginSetup %%Feature: *Resolution 300dpi TeXDict begin %%IncludeFeature: *PageSize Letter %%EndSetup %%Page: ? 1 0.4718 PNSP /PNP save def 31.7922 839.3143 translate 0 0 bop Black Black BrickRed 37 197 a Fv(Wh)n(y)33 b(do)g(we)g(use)g (this)f(\223Competitive)336 397 y(Ratio\224)h(thing)f(an)n(ywa)n(ys?)p Black 922 596 a Fu(or)p BrickRed 446 737 a Ft(Decision)26 b(Theories)g(and)733 936 y(Optimality)p Black 570 1278 a Fs(Class)d(Presentation)g(f)n(or)139 1477 y(15-854:)28 b(Appro)n(ximation)23 b(and)f(Online)i(Algor)q(ithms)p BrickRed Black RoyalBlue 413 2201 a Fr(P)n(atric)n(k)f(Rile)o(y)p Black 22 w Fs(and)p RoyalBlue 23 w Fr(Ell)o(y)g(Winner)p Black 742 2359 a Fs(Apr)q(il)h(5,)f(2000)p RoyalBlue Black Black 934 2553 a Fq(0)466 b(15-854)15 b(Presentation)p Black eop PNP restore /PNP save def 648.5610 839.3143 translate 0 1 bop Black Black BrickRed Black BrickRed 272 125 a Fv(Wh)n(y)32 b(Competitive)g(Ratio?)p Black Black 2 211 1947 2 v 2 213 V 2 215 V Black 2 862 a Fs(The)18 b(questions)i(y)n(ou)f (should)g(ha)n(v)n(e)f(been)h(asking,)h(b)n(ut)f(prob-)2 978 y(ab)n(ly)k(w)o(eren't:)p Black 74 1157 a Fp(\017)p Black 41 w Fs(Wh)n(y)36 b(do)f(w)o(e)g(use)h(the)f(competitiv)n(e)h(r)o (atio)g(to)f(sa)n(y)h(an)159 1273 y(algor)q(ithm)23 b(is)p RoyalBlue 24 w(good)p Black 23 w(or)p RoyalBlue 22 w(bad)p Black(?)p Black 74 1452 a Fp(\017)p Black 41 w Fs(What)32 b(if)g(I)g(ha)n(v)n(e)h(a)f(g)o(r)q(udge)f(against)i(the)f(competitiv)n (e)159 1568 y(r)o(atio?)d(Could)22 b(I)h(reasonab)n(ly)p RoyalBlue 24 w(use)g(something)g(else)p Black(?)p Black 74 1746 a Fp(\017)p Black 41 w Fs(W)n(e)17 b(alw)o(a)n(ys)h(consider)f (the)h(w)o(orst)f(case)h(adv)n(ersar)r(y)-8 b(.)26 b(What)159 1863 y(if)40 b(I)g(kno)o(w)g(the)f(adv)n(ersar)r(y)p RoyalBlue 40 w(probab)n(ly)p Black 40 w(w)o(on't)h(be)g(that)159 1979 y(bad?)p Black 934 2553 a Fq(1)466 b(15-854)15 b(Presentation)p Black eop PNP restore /PNP save def 31.7922 42.5454 translate 0 2 bop Black Black BrickRed Black BrickRed 754 125 a Fv(Outline)p Black Black 2 188 1947 2 v 2 190 V 2 192 V Black Black 74 710 a Fp(\017)p Black RoyalBlue 41 w Fs(Basic)24 b(De\002nitions)p Black Black 74 889 a Fp(\017)p Black 41 w Fs(Str)q(ict)g(Uncer)s(tainty)p Black 196 1036 a Fr(\226)p Black 42 w Fs(Other)e(optimality)i(cr)q(iter)q(ia)p Black 196 1183 a Fr(\226)p Black 42 w Fs(\223Bad\224)e(e)n(xamples)p Black 196 1331 a Fr(\226)p Black 42 w Fs(Axiomatic)i(char)o(acter)q (ization)g(of)e(cr)q(iter)q(ia)p Black 74 1509 a Fp(\017)p Black 41 w Fs(Pure)h(Risk)p Black 196 1657 a Fr(\226)p Black 42 w Fs(Absolute)g(Utility)h(Functions)p Black 196 1804 a Fr(\226)p Black 42 w Fs(P)l(ersonalization)g(and)e(Utility)i (Theor)r(y)p Black 196 1951 a Fr(\226)p Black 42 w Fs(Other)e (Approaches)p Black 74 2130 a Fp(\017)p Black 41 w Fs(Conclusion)p Black 934 2553 a Fq(2)466 b(15-854)15 b(Presentation)p Black eop PNP restore /PNP save def 648.5610 42.5454 translate 0 3 bop Black Black BrickRed Black BrickRed 26 125 a Fv(Simple)33 b(Decision)g(Theoretic)g(Model)p Black Black 2 211 1947 2 v 2 213 V 2 215 V Black Plum 1077 636 a Fp(S)p Black OliveGreen 543 736 a Fo(s)584 753 y Fn(1)p Black OliveGreen 890 736 a Fo(s)931 753 y Fn(2)p Black OliveGreen 1238 736 a Fo(s)1279 753 y Fn(3)p Black OliveGreen 1585 736 a Fo(s)1626 753 y Fn(4)p Black 412 767 1390 3 v Plum 43 885 a Fp(A)p Black RoyalBlue 276 835 a Fo(a)323 852 y Fn(1)p Black 409 864 3 100 v 533 835 a Fm(70)p 757 864 V 243 w(80)p 1104 864 V 244 w(90)p 1452 864 V 269 w(1)p 1799 864 V 412 867 1390 3 v RoyalBlue 276 935 a Fo(a)323 952 y Fn(2)p Black 409 964 3 100 v 559 935 a Fm(1)p 757 964 V 295 w(1)p 1104 964 V 295 w(1)p 1452 964 V 191 w(10)p Fo(;)14 b Fm(000)p 1799 964 V 412 967 1390 3 v Black 74 1307 a Fp(\017)p Black 41 w Fs(One-shot)23 b(prob)n(lem)p Black 74 1486 a Fp(\017)p Black 41 w Fs(Set)p Plum 23 w Fp(A)p Black 23 w Fs(of)g(actions)h(\(assume)f(\002nite)g(f)n (or)f(simplicity\))p Black 74 1664 a Fp(\017)p Black 41 w Fs(Set)p Plum 23 w Fp(S)p Black 30 w Fs(of)g(states)i(\(assume)f (\002nite)g(f)n(or)f(simplicity\))p Black 74 1843 a Fp(\017)p Black 41 w Fs(Cost)h(function)p Plum 24 w Fo(C)p Black 29 w Fm(:)h Fp(A)c(\002)g(S)31 b(!)24 b Fl(R)p Black 74 2021 a Fp(\017)p Black RoyalBlue 41 w Fs(W)n(e)p Black 23 w(choose)f(an)p RoyalBlue 23 w(action)p Black(,)p OliveGreen 23 w(nature)p Black 23 w(chooses)g(a)p OliveGreen 23 w(state)p Black Black 74 2200 a Fp(\017)p Black 41 w Fs(W)n(e)p RoyalBlue 22 w(pa)n(y)p Black 22 w(the)f(amount)g (speci\002ed)i(b)n(y)f(the)f(cost)h(function)p Black 934 2553 a Fq(3)466 b(15-854)15 b(Presentation)p Black eop PNP restore PNEP %%Page: ? 2 0.4718 PNSP /PNP save def 31.7922 839.3143 translate 0 4 bop Black Black BrickRed Black BrickRed 367 125 a Fv(Pref)o(erence)34 b(Relations)p Black Black 2 187 1947 2 v 2 189 V 2 191 V Black 2 572 a Fs(In)25 b(order)g(to)g(be)h(as)f (gener)o(al)h(as)g(possib)n(le)o(,)h(w)o(e'll)g(talk)f(about)2 689 y(a)p RoyalBlue 22 w(pref)n(erence)c(relation)h Fk(4)p Black OliveGreen 2 846 a Fo(a)h Fk(4)g Fo(b)p Black 22 w Fs(means)f(that)g Fo(a)g Fs(is)g(pref)n(erred)e(to)i Fo(b)p Fs(.)2 1004 y Fj(Notice)28 b(the)f(similar)q(ity)j(to)e Fp(\024)p Fj(,)h(re\003ecting)f(that)g(smaller)g(cost)2 1120 y(is)23 b(usually)h(pref)n(erred)2 1461 y Fs(Impor)s(tant)e (notes:)p Black 74 1639 a Fp(\017)p Black 41 w Fk(4)h Fs(m)o(ust)g(be)p RoyalBlue 23 w(Complete)p Black 23 w(and)p RoyalBlue 23 w(T)-10 b(r)o(ansitiv)n(e)p Black Black 74 1818 a Fp(\017)p Black OliveGreen 41 w Fo(a)48 b Fp(\036)g Fo(b)p Black Fs(:)54 b(denotes)36 b Fo(a)f Fs(is)p RoyalBlue 37 w(str)q(ictly)j(pref)n(erred)p Black 33 w(to)e Fo(b)g Fs(\(i.e)o(.)159 1934 y Fo(b)24 b Fp(6)p Fk(4)h Fo(a)p Fs(\))p Black 74 2112 a Fp(\017)p Black OliveGreen 41 w Fo(a)g Fp(\031)g Fo(b)p Black Fs(:)k(denotes)23 b Fo(a)g Fs(is)p RoyalBlue 24 w(indiff)n(erent)p Black 22 w(to)h Fo(b)e Fs(\(i.e)o(.)29 b Fo(a)c Fk(4)g Fo(b)e Fs(and)159 2229 y Fo(b)h Fk(4)h Fo(a)p Fs(\))p Black 934 2553 a Fq(4)466 b(15-854)15 b(Presentation)p Black eop PNP restore /PNP save def 648.5610 839.3143 translate 0 5 bop Black Black BrickRed Black BrickRed 232 125 a Fv(Cer)r(tainty)33 b(and)g(Uncer)r(tainty)p Black Black 2 211 1947 2 v 2 213 V 2 215 V Black 2 415 a Fs(Se)n(v)n(er)o(al)20 b(cases)i(based)f(on)g(the)f(amount)h(of)f(inf)n(or)r(mation)h(w)o(e)2 531 y(ha)n(v)n(e:)p Black 74 709 a Fp(\017)p Black RoyalBlue 41 w Fs(Under)k(Cer)s(tainty)p Black(:)35 b(W)n(e)26 b(kno)o(w)f(what)h(state)g(nature)g(will)159 826 y(pic)n(k.)k(Classic) 24 b(optimization.)p OliveGreen 159 962 a(Not)f(what)f(w)o(e'll)i(do)f (here)o(.)p Black Black 74 1141 a Fp(\017)p Black RoyalBlue 41 w Fs(Under)f(Uncer)s(tainty)p Black(:)p Black 196 1288 a Fr(\226)p Black Plum 42 w Fs(Str)q(ict)c(Uncer)s(tainty)p Black(:)26 b(W)n(e)17 b(ha)n(v)n(e)h Fj(no)h Fs(inf)n(or)r(mation)e (about)284 1404 y(lik)n(elihood)25 b(of)d(the)h(states)h(of)f(nature)p OliveGreen 284 1531 a(W)n(e'll)g(do)g(this)h(\002rst)p Black Black 196 1678 a Fr(\226)p Black Plum 42 w Fs(Pure)29 b(Risk)p Black(:)41 b(Probabilities)30 b(of)f(the)g(v)n(ar)q(ious)h (states)284 1794 y(are)22 b(kno)o(wn)p OliveGreen 284 1921 a(W)n(e'll)h(do)g(this)h(second)p Black Black 196 2068 a Fr(\226)p Black Plum 42 w Fs(P)m(ar)s(tial)h(Inf)n(or)r(mation)p Black(:)32 b(Some)n(where)24 b(in)h(the)g(middle)284 2184 y(of)e(the)g(pre)n(vious)f(tw)o(o)m(.)28 b(Most)23 b(real-w)o(or)q(ld)f(cases)o(.)p OliveGreen 284 2311 a(This)34 b(is)g(really)g(hard,)h(so)f(w)o(e)f(w)o(on't)g(talk)h(about) f(it)284 2427 y(m)o(uch)p Black Black 934 2553 a Fq(5)466 b(15-854)15 b(Presentation)p Black eop PNP restore /PNP save def 31.7922 42.5454 translate 0 6 bop Black Black BrickRed Black BrickRed 492 125 a Fv(Pr)n(ogress)34 b(Repor)r(t)p Black Black 2 211 1947 2 v 2 213 V 2 215 V Black Black 74 722 a Fp(\017)p Black 41 w Fs(Basic)24 b(De\002nitions)p Black 74 900 a Fp(\017)p Black RoyalBlue 41 w Fs(Str)q(ict)g(Uncer)s(tainty)p Black Black 196 1048 a Fr(\226)p Black 42 w Fs(Other)e(optimality)i(cr) q(iter)q(ia)p Black 196 1195 a Fr(\226)p Black 42 w Fs(\223Bad\224)e(e) n(xamples)p Black 196 1342 a Fr(\226)p Black 42 w Fs(Axiomatic)i(char)o (acter)q(ization)g(of)e(cr)q(iter)q(ia)p Black 74 1521 a Fp(\017)p Black 41 w Fs(Pure)h(Risk)p Black 196 1668 a Fr(\226)p Black 42 w Fs(Absolute)g(Utility)h(Functions)p Black 196 1815 a Fr(\226)p Black 42 w Fs(P)l(ersonalization)g(and)e (Utility)i(Theor)r(y)p Black 196 1963 a Fr(\226)p Black 42 w Fs(Other)e(Approaches)p Black 74 2141 a Fp(\017)p Black 41 w Fs(Conclusion)p Black 934 2553 a Fq(6)466 b(15-854)15 b(Presentation)p Black eop PNP restore /PNP save def 648.5610 42.5454 translate 0 7 bop Black Black BrickRed Black BrickRed 503 125 a Fv(V)-7 b(alue)33 b(Functions)p Black Black 2 187 1947 2 v 2 189 V 2 191 V Black 2 598 a Fs(And)23 b(y)n(ou)g(thought)g(w)o(e) f(w)o(ere)h(done)f(with)h(de\002nitions!)2 963 y(A)p Plum 34 w(V)-6 b(alue)35 b(Function)g Fo(V)p Black 54 w Fs(is)g(just)g(a)g(con)n(v)n(enient)g(w)o(a)n(y)f(to)g(e)n(x-)2 1079 y(press)23 b(a)p RoyalBlue 22 w(pref)n(erence)f(relation)p Black Black 74 1257 a Fp(\017)p Black Plum 41 w Fo(V)p Black 43 w Fs(assigns)i(a)f(real)g(n)o(umber)f(to)h(each)g(action)p Black 74 1436 a Fp(\017)p Black 41 w Fs(Pref)n(erred)p RoyalBlue 159 1573 a Fo(a)h Fk(4)h Fo(b)p Black 22 w Fs(if)f(and)e(only)i(if)p Plum 23 w Fo(V)c Fm(\()p Fo(a)p Fm(\))k Fp(\024)h Fo(V)19 b Fm(\()p Fo(b)p Fm(\))p Black Black 74 1751 a Fp(\017)p Black 41 w Fs(Str)q(ictly)24 b(Pref)n(erred)p RoyalBlue 159 1888 a Fo(a)g Fp(\036)h Fo(b)p Black 23 w Fs(if)e(and)g(only)g(if)p Plum 23 w Fo(V)d Fm(\()p Fo(a)p Fm(\))k Fo(<)h(V)20 b Fm(\()p Fo(b)p Fm(\))p Black Black 74 2067 a Fp(\017)p Black 41 w Fs(Indiff)n(erent)p RoyalBlue 159 2204 a Fo(a)k Fp(\031)h Fo(b)p Black 23 w Fs(if)e(and)g(only)g(if)p Plum 23 w Fo(V)d Fm(\()p Fo(a)p Fm(\))k(=)h Fo(V)20 b Fm(\()p Fo(b)p Fm(\))p Black Black 934 2553 a Fq(7)466 b(15-854)15 b(Presentation)p Black eop PNP restore PNEP %%Page: ? 3 0.4718 PNSP /PNP save def 31.7922 839.3143 translate 0 8 bop Black Black BrickRed Black BrickRed 4 125 a Fv(Some)32 b(Candidate)g(Decision)h(Criteria)p Black Black 2 188 1947 2 v 2 190 V 2 192 V Black Black 74 383 a Fp(\017)p Black RoyalBlue 41 w Fs(Competitiv)n(e)g(Ratio)p Black(:)49 b(Minimiz)o(e)33 b(m)o(ultiplicativ)n(e)i(times)159 486 y(w)o(orse)23 b(than)g(OPT)-10 b(.)488 676 y Fo(V)19 b Fm(\()p Fo(a)p Fm(\))25 b(=)f(max)885 719 y Fi(s)1004 551 y Fh(\()1175 619 y Fo(C)6 b Fm(\()p Fo(a;)14 b(s)p Fm(\))p 1067 657 491 3 v 1067 738 a(min)1222 748 y Fi(x)1280 738 y Fo(C)6 b Fm(\()p Fo(x;)15 b(s)p Fm(\))1562 551 y Fh(\))p Black 74 884 a Fp(\017)p Black RoyalBlue 41 w Fs(Minimax)24 b(Cost)p Black(:)k(Minimiz)o(e)c(w)o(orst-case)e(cost.) 658 1039 y Fo(V)e Fm(\()p Fo(a)p Fm(\))k(=)h(max)1055 1082 y Fi(s)1175 1039 y Fo(C)6 b Fm(\()p Fo(a;)14 b(s)p Fm(\))p Black 74 1194 a Fp(\017)p Black RoyalBlue 41 w Fs(Minimax)21 b(Reg)o(ret)p Black(:)27 b(Minimiz)o(e)21 b(additiv)n(e)g(amount)f(w)o(orse)159 1297 y(than)j(OPT)-10 b(.)333 1452 y Fo(V)20 b Fm(\()p Fo(a)p Fm(\))k(=)h(max)730 1495 y Fi(s)849 1370 y Fh(n)897 1452 y Fo(C)6 b Fm(\()p Fo(a;)14 b(s)p Fm(\))20 b Fp(\000)g Fm(min)1336 1495 y Fi(x)1449 1452 y Fo(C)6 b Fm(\()p Fo(x;)15 b(s)p Fm(\))1727 1370 y Fh(o)p Black 74 1607 a Fp(\017)p Black RoyalBlue 41 w Fs(P)l(essimism/Optimism)39 b(Inde)n(x)p Black(:)56 b(Balance)37 b(w)o(orst)g(and)159 1710 y(best)23 b(cases)o(.)29 b(Choose)23 b Fo(\013)i Fp(2)f Fm([0)p Fo(;)15 b Fm(1])186 1865 y Fo(V)k Fm(\()p Fo(a)p Fm(\))25 b(=)f Fo(\013)15 b Fm(max)654 1908 y Fi(s)773 1865 y Fo(C)6 b Fm(\()p Fo(a;)14 b(s)p Fm(\))19 b(+)h(\(1)g Fp(\000)f Fo(\013)p Fm(\))d(min)1538 1908 y Fi(s)1647 1865 y Fo(C)6 b Fm(\()p Fo(a;)14 b(s)p Fm(\))p Black 74 2020 a Fp(\017)p Black RoyalBlue 41 w Fs(Pr)q(inciple)25 b(of)f(Insuf\002cient)h(Reason)p Black(:)30 b(Assume)24 b(unif)n(or)r(m)159 2123 y(probability)g(distr)q (ib)n(ution.)689 2321 y Fo(V)c Fm(\()p Fo(a)p Fm(\))k(=)1016 2250 y Fh(X)1051 2401 y Fi(s)1139 2264 y Fo(C)6 b Fm(\()p Fo(a;)14 b(s)p Fm(\))p 1139 2302 276 3 v 1222 2383 a Fp(jS)6 b(j)p Black 934 2553 a Fq(8)466 b(15-854)15 b(Presentation)p Black eop PNP restore /PNP save def 648.5610 839.3143 translate 0 9 bop Black Black BrickRed Black BrickRed 360 125 a Fv(Minimax)33 b(Cost)f(is)h(bad!)p Black Black 2 188 1947 2 v 2 190 V 2 192 V Black OliveGreen 507 732 a Fo(s)548 749 y Fn(1)p Black OliveGreen 908 732 a Fo(s)949 749 y Fn(2)p Black OliveGreen 1308 732 a Fo(s)1349 749 y Fn(3)p Black OliveGreen 1708 732 a Fo(s)1749 749 y Fn(4)p Black 350 763 1602 3 v RoyalBlue 214 835 a Fo(a)261 852 y Fn(1)p Black 347 864 3 104 v 523 835 a Fm(1)p 748 864 V 348 w(1)p 1148 864 V 349 w(1)p 1549 864 V 217 w(10)1697 806 y Fn(6)1758 835 y Fm(+)18 b(1)p 1949 864 V 350 867 1602 3 v 43 939 a Fp(!)p RoyalBlue 83 w Fo(a)261 956 y Fn(2)p Black 347 968 3 104 v 474 939 a Fm(10)578 910 y Fn(6)p 748 968 V 875 939 a Fm(10)979 910 y Fn(6)p 1148 968 V 1275 939 a Fm(10)1379 910 y Fn(6)p 1549 968 V 1676 939 a Fm(10)1780 910 y Fn(6)p 1949 968 V 350 971 1602 3 v Black 74 1429 a Fp(\017)p Black 41 w Fs(Minimax)24 b(Cost)f(chooses)p RoyalBlue 23 w Fo(a)1060 1446 y Fn(2)p Black Black 74 1607 a Fp(\017)p Black 41 w Fs(Ho)o(w)o(e)n(v)n(er)l(,)p RoyalBlue 38 w Fo(a)584 1624 y Fn(1)p Black 664 1607 a Fs(is)p Plum 37 w(m)o(uch)p Black 36 w(better)35 b(in)h(most)g(cases) h(and)159 1724 y(only)24 b(slightly)g(w)o(orse)f(in)g(one)g(case)p Black 74 1902 a Fp(\017)p Black 41 w Fs(W)n(e'll)h(ha)n(v)n(e)f(lots)h (of)f(reg)o(ret)f(if)h(w)o(e)g(use)g(this!)p Black 74 2081 a Fp(\017)p Black 41 w Fs(Minimax)h(Reg)o(ret)e(and)h(Competitiv)n (e)g(Ratio)g(do)g(better)p Black 934 2553 a Fq(9)466 b(15-854)15 b(Presentation)p Black eop PNP restore /PNP save def 31.7922 42.5454 translate 0 10 bop Black Black BrickRed Black BrickRed 303 125 a Fv(Minimax)33 b(Regret)g(is)h(bad!)p Black Black 2 211 1947 2 v 2 213 V 2 215 V Black OliveGreen 488 771 a Fo(s)529 788 y Fn(1)p Black OliveGreen 851 771 a Fo(s)892 788 y Fn(2)p Black OliveGreen 1214 771 a Fo(s)1255 788 y Fn(3)p Black OliveGreen 1577 771 a Fo(s)1618 788 y Fn(4)p Black 350 803 1452 3 v RoyalBlue 214 871 a Fo(a)261 888 y Fn(1)p Black 347 900 3 100 v 504 871 a Fm(1)p 710 900 V 311 w(1)p 1073 900 V 311 w(1)p 1436 900 V 198 w(11)p Fo(;)14 b Fm(000)p 1799 900 V 350 902 1452 3 v 43 971 a Fp(!)p RoyalBlue 83 w Fo(a)261 988 y Fn(2)p Black 347 1000 3 100 v 407 971 a Fm(1)p Fo(;)h Fm(000)p 710 1000 V 116 w(1)p Fo(;)g Fm(000)p 1073 1000 V 116 w(1)p Fo(;)g Fm(000)p 1436 1000 V 100 w(10)p Fo(;)f Fm(000)p 1799 1000 V 350 1002 1452 3 v Black 74 1476 a Fp(\017)p Black 41 w Fs(Minimax)24 b(Reg)o(ret)e(chooses)p RoyalBlue 24 w Fo(a)1138 1493 y Fn(2)p Black Black 74 1654 a Fp(\017)p Black 41 w Fs(Ho)o(w)o(e)n(v)n(er)l(,)p RoyalBlue 28 w Fo(a)574 1671 y Fn(1)p Black 647 1654 a Fs(pa)n(ys)29 b(almost)f(nothing)h(in)f(three)g(states)159 1770 y(and)23 b(in)g(the)g(other)l(,)f(increases)i(cost)f(b)n(y)h(only)g Fm(10\045)p Fs(.)p Black 74 1949 a Fp(\017)p Black 41 w Fs(Diminishing)e(Marginal)f(Utility:)29 b Fm($10)20 b Fs(matters)g(more)h(to)159 2065 y(a)i(pauper)f(than)h(to)g(Bill)h (Gates)p Black 902 2553 a Fq(10)466 b(15-854)15 b(Presentation)p Black eop PNP restore /PNP save def 648.5610 42.5454 translate 0 11 bop Black Black BrickRed Black BrickRed 138 125 a Fv(P)l(essimism/Optimism)33 b(Inde)n(x)g(is)829 241 y(bad!)p Black Black 2 303 1947 2 v 2 305 V 2 307 V Black Black 74 580 a Fp(\017)p Black 41 w Fs(First,)24 b(there')l(s)f(the)g (question)g(of)g(ho)o(w)f(to)h(choose)g Fo(\013)p Black 74 758 a Fp(\017)p Black 41 w Fs(But)g(there)g(is)g(a)g(more)f(ser)q (ious)i(prob)n(lem)p Fo(:)15 b(:)g(:)p OliveGreen 337 969 a(s)378 986 y Fn(1)p Black OliveGreen 740 969 a Fo(s)781 986 y Fn(2)p Black OliveGreen 1150 969 a Fo(s)1191 986 y Fn(3)p Black Plum 1566 969 a Fo(V)p Black 179 1001 1223 3 v RoyalBlue 43 1069 a(a)90 1086 y Fn(1)p Black 176 1098 3 100 v 353 1069 a Fm(1)p 579 1098 V 351 w(0)p 981 1098 V 350 w(1)p 1384 1098 V 1399 1098 V 318 w(1)p Fo(=)p Fm(2)p 179 1100 1223 3 v RoyalBlue 43 1168 a Fo(a)90 1185 y Fn(2)p Black 176 1198 3 100 v 353 1168 a Fm(0)p 579 1198 V 351 w(1)p 981 1198 V 350 w(1)p 1384 1198 V 1399 1198 V 318 w(1)p Fo(=)p Fm(2)p 179 1200 1223 3 v RoyalBlue 67 1277 a Fo(z)p Black 176 1309 3 112 v 357 1243 a Fn(1)p 357 1258 44 3 v 357 1309 a(2)p 579 1309 3 112 v 760 1243 a(1)p 760 1258 44 3 v 760 1309 a(2)p 981 1309 3 112 v 1158 1277 a Fm(1)p 1384 1309 V 1399 1309 V 318 w(3)p Fo(=)p Fm(4)p 179 1312 1223 3 v 2 1491 a Fs(Let)22 b Fo(\013)j Fm(=)330 1457 y Fn(1)p 330 1471 44 3 v 330 1523 a(2)378 1491 y Fs(.)p Black 74 1669 a Fp(\017)p Black 41 w Fs(actions)p RoyalBlue 24 w Fo(a)491 1686 y Fn(1)p Black 559 1669 a Fs(and)p RoyalBlue 23 w Fo(a)767 1686 y Fn(2)p Black 834 1669 a Fs(are)e(both)f(optimal)p Black 74 1848 a Fp(\017)p Black 41 w Fs(action)p RoyalBlue 26 w Fo(z)p Black 29 w Fs(is)k(just)f(a)h(probability)f(mixture)g(of)p RoyalBlue 26 w Fo(a)1624 1864 y Fn(1)p Black 1694 1848 a Fs(and)p RoyalBlue 25 w Fo(a)1904 1864 y Fn(2)p Black 159 1964 a Fo(z)j Fm(:=)c(\(1)p Fo(=)p Fm(2\))p Fo(a)636 1981 y Fn(1)700 1964 y Fm(+)c(\(1)p Fo(=)p Fm(2\))p Fo(a)1075 1981 y Fn(2)p Black 74 2142 a Fp(\017)p Black 41 w Fs(A)32 b(probability)h(mixture)e(will)i(e)n(v)n(entually)f(choose)h(1)f(of)159 2258 y(the)h(optimal)f(actions)o(,)k(so)d(ho)o(w)e(could)i(it)g(be)g (subopti-)159 2375 y(mal?)p Black 902 2553 a Fq(11)466 b(15-854)15 b(Presentation)p Black eop PNP restore PNEP %%Page: ? 4 0.4718 PNSP /PNP save def 31.7922 839.3143 translate 0 12 bop Black Black BrickRed Black BrickRed 20 125 a Fv(Principle)33 b(of)g(Insuf\002cient)e(Reason)i(is)829 241 y(bad!)p Black Black 2 303 1947 2 v 2 305 V 2 307 V Black Black 74 569 a Fp(\017)p Black 41 w Fs(Extremely)24 b(v)n(ague:)k(What)23 b(are)f(the)h(basic)h(e)n(v)n(ents?)p Black 74 748 a Fp(\017)p Black 41 w Fs(Is)g(one)e(par)s(tition)i (better)e(than)h(another?)2 969 y(An)g(e)n(xample:)p Black 74 1147 a Fp(\017)p Black 41 w Fs(Y)-12 b(ou)25 b(ha)n(v)n(e)f(an)f(ur)r(n)h(with)f(tw)o(o)h(balls)o(,)h(both)e(of)h (which)g(are)159 1263 y(either)p RoyalBlue 23 w(b)n(lue)p Black 23 w(or)p Plum 23 w(pur)r(ple)p Black Black 74 1442 a Fp(\017)p Black 41 w Fs(What)f(is)h(the)f(probability)h(of)f(dr) o(a)n(wing)g(tw)o(o)g(balls)h(of)f(dif-)159 1558 y(f)n(erent)f(colors?) p Black 74 1737 a Fp(\017)p Black 41 w Fs(One)h(par)s(tition)h(of)f (the)f(outcome)h(space:)159 1874 y Fg(f)p RoyalBlue Fs(b)n(lue)p Black(,)p RoyalBlue(b)n(lue)p Black Fg(g)h(f)p RoyalBlue Fs(b)n(lue)p Black(,)p Plum(pur)r(ple)p Black Fg(g)g(f)p Plum Fs(pur)r(ple)p Black(,)p Plum(pur)r(ple)p Black Fg(g)p Black 74 2052 a Fp(\017)p Black 41 w Fs(Another)f(par)s(tition)g (of)g(the)g(outcome)f(space:)159 2189 y Fp(h)p RoyalBlue Fs(b)n(lue)p Black(,)p RoyalBlue 24 w(b)n(lue)p Black Fp(i)c(h)p RoyalBlue Fs(b)n(lue)p Black(,)p Plum 24 w(pur)r(ple)p Black Fp(i)g(h)p Plum Fs(pur)r(ple)p Black(,)p RoyalBlue 23 w(b)n(lue)p Black Fp(i)g(h)p Plum Fs(pur)r(ple)p Black(,)p Plum 23 w(pur)r(ple)p Black Fp(i)p Black 74 2368 a(\017)p Black 41 w Fs(Which)24 b(one)f(is)g(better/r)q(ight?)p Black 902 2553 a Fq(12)466 b(15-854)15 b(Presentation)p Black eop PNP restore /PNP save def 648.5610 839.3143 translate 0 13 bop Black Black BrickRed Black BrickRed 241 125 a Fv(Competitive)32 b(Ratio)g(is)h(bad!)p Black Black 2 211 1947 2 v 2 213 V 2 215 V Black Black 74 755 a Fp(\017)p Black 41 w Fs(Can't)23 b(handle)p RoyalBlue 23 w(z)o(ero)p Black 23 w(or)p RoyalBlue 22 w(negativ)n(e)p Black 23 w(costs!)p OliveGreen 526 1234 a Fo(s)567 1251 y Fn(1)p Black OliveGreen 964 1234 a Fo(s)1005 1251 y Fn(2)p Black OliveGreen 1402 1234 a Fo(s)1443 1251 y Fn(3)p Black OliveGreen 1840 1234 a Fo(s)1881 1251 y Fn(4)p Black 350 1266 1752 3 v 43 1334 a Fp(!)p RoyalBlue 83 w Fo(a)261 1351 y Fn(1)p Black 347 1363 3 100 v 516 1334 a Fm(70)p 785 1363 V 334 w(80)p 1223 1363 V 334 w(90)p 1661 1363 V 209 w(1)p Fo(=)p Fm(10)p Fo(;)14 b Fm(000)p 2099 1363 V 350 1365 1752 3 v RoyalBlue 214 1433 a Fo(a)261 1450 y Fn(2)p Black 347 1462 3 100 v 542 1433 a Fm(1)p 785 1462 V 386 w(1)p 1223 1462 V 386 w(1)p 1661 1462 V 386 w(1)p 2099 1462 V 350 1465 1752 3 v Black 74 1913 a Fp(\017)p Black 41 w Fs(Competitiv)n(e)23 b(r)o(atio)g(choose)p RoyalBlue 24 w Fo(a)1136 1930 y Fn(1)p Black Black 74 2092 a Fp(\017)p Black 41 w Fs(Is)h(that)e(what)h(y)n(ou)g(w)o(ould)g (tak)n(e?)p Black 902 2553 a Fq(13)466 b(15-854)15 b(Presentation)p Black eop PNP restore /PNP save def 31.7922 42.5454 translate 0 14 bop Black Black BrickRed Black BrickRed 281 125 a Fv(Ar)n(gh!)40 b(Ever)q(ything)33 b(is)g(bad!)396 241 y(Should)f(we)h(give)g(up?)p Black Black 2 328 1947 2 v 2 330 V 2 332 V Black Black 74 981 a Fp(\017)p Black RoyalBlue 41 w Fs(Nothing)23 b(is)h(perf)n(ect!)p Black 27 w(W)n(elcome)f(to)g(the)g(real)f(w)o(or)q(ld.)p Black 74 1160 a Fp(\017)p Black 41 w Fs(Can)17 b(w)o(e)h(sa)n(y)f(that)h (some)f(of)h(these)f(are)p RoyalBlue 17 w(better)g(or)g(w)o(orse)p Black 159 1276 a(than)23 b(others?)p Black 74 1455 a Fp(\017)p Black 41 w Fs(Ho)o(w)e(can)i(w)o(e)p RoyalBlue 22 w(systematically)p Black 24 w(compare)e(decision)i(cr)q(i-)159 1571 y(ter)q(ia?)p Black 74 1749 a Fp(\017)p Black 41 w Fs(One)36 b(possibility)i(is)e(to)g(giv)n(e)p OliveGreen 36 w(axioms)p Black 36 w(which)p OliveGreen 36 w(char)o(ac-)159 1866 y(ter)q(iz)o(e)p Black 32 w(each)d(of)f(the)g(cr)q(iter)q(ia.)56 b(Depending)31 b(on)h(which)159 1982 y(axioms)o(,)24 b(y)n(ou)f(lik)n(e)o(,)h(y)n(ou)g(can)f(choose)g(y)n(our)g(cr)q(iter)q (ion.)p Black 902 2553 a Fq(14)466 b(15-854)15 b(Presentation)p Black eop PNP restore /PNP save def 648.5610 42.5454 translate 0 15 bop Black Black BrickRed Black BrickRed 168 125 a Fv(Generaliz)q(ed)33 b(Decision)g(Model)p Black Black 2 188 1947 2 v 2 190 V 2 192 V Black 2 439 a Fs(This)19 b(slide)h(could)f(be)g(\002lled)h(with)f(lots)g(of)g(symbols)g(and)g (other)2 555 y(stuff)k(y)n(ou)g(w)o(ould)g(ha)n(v)n(e)h(to)f(stare)f (at)h(f)n(or)f(a)h(while)o(.)2 712 y(Instead,)g(I'll)h(just)f(sa)n(y)g (this:)p Black 74 891 a Fp(\017)p Black 41 w Fs(Consider)k(the)p RoyalBlue 27 w(algor)q(ithmic)g(decision)h(prob)n(lem)p Black Plum 27 w Fo(P)p Black 43 w Fm(:=)159 1007 y(\()p Plum Fp(A)p Black 1 w Fo(;)p Plum 15 w Fp(S)p Black 6 w Fo(;)p Plum 15 w(C)p Black 5 w Fm(\))p Black 74 1186 a Fp(\017)p Black Plum 41 w(A)p Black 23 w Fs(is)c(a)f(non-empty)f(set) i(of)p RoyalBlue 22 w(algor)q(ithms)p Black 159 1323 a(There)29 b(could)f(be)h(in\002nitely)h(man)o(y)-8 b(.)44 b(The)n(y)29 b(can)g(be)f(on-)159 1439 y(line)c(or)e(of\003ine)o(.)p Black 74 1617 a Fp(\017)p Black Plum 41 w(S)p Black 44 w Fs(is)38 b(a)f(non-empty)g(set)g(of)h(\002nite)f(\(b)n(ut)g (unbounded\))159 1734 y(length)p RoyalBlue 23 w(request)23 b(sequences)p Black 159 1871 a(There)g(could)h(be)e(in\002nitely)j(man) o(y)-8 b(.)p Black 74 2049 a Fp(\017)p Black Plum 41 w Fo(C)p Black 30 w Fm(:)24 b Fp(A)c(\002)g(S)30 b(!)25 b Fl(R)159 2186 y Fs(This)f(is)g(the)p RoyalBlue 23 w(cost)f(function)p Black Black 74 2365 a Fp(\017)p Black 41 w Fs(Still)h(compare)f(to)g (OPT)-10 b(,)24 b(which)f(ma)n(y)f(not)h(be)g(in)g Fp(A)p Fs(.)p Black 902 2553 a Fq(15)466 b(15-854)15 b(Presentation)p Black eop PNP restore PNEP %%Page: ? 5 0.4718 PNSP /PNP save def 31.7922 839.3143 translate 0 16 bop Black Black BrickRed Black BrickRed 382 125 a Fv(Ax)33 b Ff(0)p Fv(:)41 b(Completeness)p Black Black 2 211 1947 2 v 2 213 V 2 215 V Black 2 849 a Fs(There)22 b(is)i(a)f(pref)n(erence)e(relation)p OliveGreen 23 w Fk(4)1168 867 y Fi(P)p Black 1250 849 a Fs(on)i(the)g(algor)q(ithms)o (.)p 2 1303 V 2 1840 a(Pref)n(erence)e(relation)p OliveGreen 23 w Fk(4)787 1858 y Fi(P)p Black 869 1840 a Fs(o)o(v)n(er)p Plum 22 w Fp(A)p Black Fs(.)2 1997 y(Must)i(be)p RoyalBlue 22 w(complete)p Black 24 w(and)p RoyalBlue 23 w(tr)o(ansitiv)n(e)p Black Black 902 2553 a Fq(16)466 b(15-854)15 b(Presentation)p Black eop PNP restore /PNP save def 648.5610 839.3143 translate 0 17 bop Black Black BrickRed Black BrickRed 197 125 a Fv(Ax)33 b Ff(1)p Fv(:)41 b(Algorithmic)32 b(Labeling)p Black Black 2 211 1947 2 v 2 213 V 2 215 V Black 2 779 a Fs(It)22 b(doesn't)i(matter)e(what)h(y)n(ou)g(call)h(the)f(algor)q (ithms)o(.)p 2 1164 V OliveGreen 2 1630 a Fk(4)69 1648 y Fi(P)p Black 150 1630 a Fs(is)h(in)n(v)n(ar)q(iant)g(under)e(algor)q (ithm)h(labeling)2 1788 y(Let)f Fo(P)207 1758 y Fe(0)254 1788 y Fm(=)i(\()p Fp(A)468 1758 y Fe(0)491 1788 y Fo(;)15 b Fp(S)6 b Fo(;)15 b(C)697 1758 y Fe(0)718 1788 y Fm(\))p Fs(.)2 1945 y(If)22 b(there)h(e)n(xists)g(an)g(isomor)r(phism)h Fo(h)g Fm(:)g Fp(A)h(!)f(A)1527 1916 y Fe(0)1549 1945 y Fs(,)f(then)2 2061 y Fo(a)h Fk(4)140 2080 y Fi(P)223 2061 y Fo(b)e Fs(if)h(and)g(only)h(if)f Fo(h)p Fm(\()p Fo(a)p Fm(\))h Fk(4)1016 2085 y Fi(P)1072 2065 y Fd(0)1118 2061 y Fo(h)p Fm(\()p Fo(b)p Fm(\))p Black 902 2553 a Fq(17)466 b(15-854)15 b(Presentation)p Black eop PNP restore /PNP save def 31.7922 42.5454 translate 0 18 bop Black Black BrickRed Black BrickRed 147 125 a Fv(Ax)33 b Ff(2)p Fv(:)41 b(Request)32 b(Seq.)41 b(Labeling)p Black Black 2 211 1947 2 v 2 213 V 2 215 V Black 2 671 a Fs(It)19 b(doesn't)i(matter)e(what)g(y)n(ou)h(call)h(the)f(request)f (sequences)o(.)p Black 74 850 a Fp(\017)p Black 41 w Fs(This)37 b(is)g(a)f(little)h(str)o(ange)f(because)g(the)g(request)g (se-)159 966 y(quences)c(could)f(ha)n(v)n(e)g(diff)n(erent)f(lengths)o (,)j(b)n(ut)e(this)h(is)159 1082 y(hard)23 b(to)g(argue)f(against.)p 2 1380 V OliveGreen 2 1738 a Fk(4)69 1756 y Fi(P)p Black 150 1738 a Fs(is)i(in)n(v)n(ar)q(iant)g(under)e(request)g(sequence)i (labeling)2 1895 y(Let)e Fo(P)207 1866 y Fe(0)254 1895 y Fm(=)i(\()p Fp(A)p Fo(;)16 b Fp(S)567 1866 y Fe(0)589 1895 y Fo(;)f(C)697 1866 y Fe(0)718 1895 y Fm(\))p Fs(.)2 2053 y(If)22 b(there)h(e)n(xists)g(an)g(isomor)r(phism)h Fo(g)j Fm(:)d Fp(S)31 b(!)24 b(S)1499 2024 y Fe(0)1521 2053 y Fs(,)f(then)2 2169 y Fo(a)h Fk(4)140 2188 y Fi(P)223 2169 y Fo(b)e Fs(if)h(and)g(only)h(if)f Fo(g)s Fm(\()p Fo(a)p Fm(\))h Fk(4)1010 2192 y Fi(P)1066 2173 y Fd(0)1112 2169 y Fo(g)s Fm(\()p Fo(b)p Fm(\))p Black 902 2553 a Fq(18)466 b(15-854)15 b(Presentation)p Black eop PNP restore /PNP save def 648.5610 42.5454 translate 0 19 bop Black Black BrickRed Black BrickRed 291 125 a Fv(Ax)34 b Ff(3)p Fv(:)40 b(W)l(eak)34 b(Domination)p Black Black 2 187 1947 2 v 2 189 V 2 191 V Black 2 771 a Fs(If)p OliveGreen 22 w Fo(a)p Black 23 w Fs(is)24 b(alw)o(a)n(ys)f(at)f(least)i(as)f(cheap)g(as)p OliveGreen 23 w Fo(b)p Black Fs(,)g(then)p OliveGreen 23 w Fo(a)h Fk(4)1658 790 y Fi(P)1741 771 y Fo(b)p Black Black 74 950 a Fp(\017)p Black 41 w Fs(An)o(y)f(r)o(ational)h(decision)g(cr)q (iter)q(ion)f(should)g(satisfy)h(this)p 2 1370 V 2 1852 a Fo(a)e Fs(w)o(eakly)i(dominates)f Fo(b)g Fp(\))p Fo(a)g Fk(4)1036 1871 y Fi(P)1119 1852 y Fo(b)p Black 74 2030 a Fp(\017)p Black 41 w Fo(a)17 b Fs(w)o(eakly)i(dominates)e Fo(b)g Fs(iff)h Fp(8)p Fo(s)24 b Fp(2)g(S)6 b Fo(;)15 b(C)6 b Fm(\()p Fo(a;)14 b(s)p Fm(\))25 b Fp(\024)f Fo(C)6 b Fm(\()p Fo(b;)14 b(s)p Fm(\))p Black 902 2553 a Fq(19)466 b(15-854)15 b(Presentation)p Black eop PNP restore PNEP %%Page: ? 6 0.4718 PNSP /PNP save def 31.7922 839.3143 translate 0 20 bop Black Black BrickRed Black BrickRed 252 125 a Fv(Ax)33 b Ff(4)p Fv(:)41 b(Unif)n(orm)31 b(Contin)o(uity)p Black Black 2 211 1947 2 v 2 213 V 2 215 V Black 2 508 a Fs(T)-10 b(ak)n(e)18 b(a)g(sequence)g(of)f(algor)q(ithmic)h(decision) h(prob)n(lems)e Fp(f)p Plum Fo(P)1911 527 y Fi(k)p Black 1953 508 a Fp(g)2 624 y Fs(whose)31 b(\(cost)g(functions\))h(con)n(v)n (erge)f(\(unif)n(or)r(mly\))g(to)p Plum 31 w Fo(P)p Black 51 w Fm(=)2 741 y(\()p Plum Fp(A)p Black Fo(;)p Plum 15 w Fp(S)p Black 6 w Fo(;)p Plum 15 w(C)p Black 6 w Fm(\))2 898 y Fs(If)p OliveGreen 22 w Fo(a)25 b Fk(4)209 917 y Fi(P)255 933 y Fc(k)317 898 y Fo(b)p Black 22 w Fs(f)n(or)d(all)i Fo(k)r Fs(,)f(then)p OliveGreen 22 w Fo(a)h Fk(4)1015 917 y Fi(P)1098 898 y Fo(b)p Black Black 74 1077 a Fp(\017)p Black 41 w Fs(Note)19 b(that)f(cost)i (functions)f(are)f(only)i(par)s(t)e(of)h(sequence)159 1193 y(that)k(changes)p Black 74 1371 a Fp(\017)p Black 41 w Fs(W)n(e)i(need)h(this)g(technical)h(point)e(of)h(\223unif)n(or)r (m\224)e(con)n(v)n(er-)159 1488 y(gence)k(because)h(otherwise)e(it')l (s)i(not)f(consistent)h(with)159 1604 y(the)23 b(Competitiv)n(e)g (Ratio)p 2 1736 V 2 1932 a(Let)35 b Fp(f)p Fo(P)252 1952 y Fi(k)340 1932 y Fm(=)47 b(\()p Fp(A)p Fo(;)15 b Fp(S)6 b Fo(;)15 b(C)777 1952 y Fi(k)819 1932 y Fm(\))p Fp(g)36 b Fs(be)f(a)g(sequence)h(of)f(decision)2 2049 y(prob)n(lems)22 b(such)h(that)f Fp(f)p Fo(P)819 2068 y Fi(k)861 2049 y Fp(g)g Fs(unif)n(or)r(mly)h(con)n(v)n(erges)f(to)h Fo(P)36 b Fm(=)2 2165 y(\()p Fp(A)p Fo(;)15 b Fp(S)6 b Fo(;)15 b(C)6 b Fm(\))p Fs(.)2 2322 y(If)22 b Fo(a)j Fk(4)209 2341 y Fi(P)255 2357 y Fc(k)317 2322 y Fo(b)d Fs(f)n(or)g(all)i Fo(k)r Fs(,)f(then)f Fo(a)i Fk(4)1015 2341 y Fi(P)1098 2322 y Fo(b)p Black 902 2553 a Fq(20)466 b(15-854)15 b(Presentation)p Black eop PNP restore /PNP save def 648.5610 839.3143 translate 0 21 bop Black Black BrickRed Black BrickRed 114 125 a Fv(Ax)33 b Ff(5)p Fv(:)41 b(Independence)32 b(of)g Fb(W)l(eakl)n(y)289 241 y(Dominated)42 b Fv(Alternatives)p Black Black 2 303 1947 2 v 2 305 V 2 307 V Black 2 662 a Fs(If)24 b(w)o(e)h(add)g(an)g(algor)q(ithm)g(which)p RoyalBlue 25 w(does)g(not)g(perf)n(or)r(m)e(better)2 778 y(than)34 b(the)g(pre)n(vious)g(best)g(at)g(an)o(y)g(single)h (point)p Black(,)j(then)c(the)2 894 y(r)o(anking)23 b(of)g(the)f(or)q (iginal)i(algor)q(ithms)f(is)h(unchanged.)p Black 74 1073 a Fp(\017)p Black 41 w Fs(Wh)n(y)f(this)h(bit)f(about)g(not)f (perf)n(or)r(ming)g(better?)p Black 74 1251 a Fp(\017)p Black 41 w Fs(Because)d(it')l(s)g(not)g(consistent)g(with)f(Competitiv) n(e)h(Ratio)159 1368 y(otherwise!)p 2 1547 V 2 1811 a(Let)35 b Fo(P)220 1781 y Fe(0)288 1811 y Fm(=)47 b(\()p Fp(A)29 b([)e Fo(E)5 b(;)15 b Fp(S)6 b Fo(;)16 b(C)6 b Fm(\))35 b Fs(be)g(an)o(y)g(algor)q(ithmic)h(e)n(xten-)2 1927 y(sion)c(of)g Fo(P)51 b Fm(=)41 b(\()p Fp(A)p Fo(;)15 b Fp(S)6 b Fo(;)15 b(C)6 b Fm(\))32 b Fs(such)g(that)f Fp(8)p Fo(a)40 b Fp(2)f(A)p Fs(,)34 b Fp(8)p Fo(s)40 b Fp(2)f(S)6 b Fs(,)2 2043 y Fo(C)71 2014 y Fe(0)92 2043 y Fm(\()p Fo(a;)15 b(s)p Fm(\))41 b(=)h Fo(C)6 b Fm(\()p Fo(a;)14 b(s)p Fm(\))32 b Fs(and)g Fp(8)p Fo(b)41 b Fp(2)g Fo(E)5 b Fs(,)35 b Fp(8)p Fo(s)41 b Fp(2)f(S)6 b Fs(,)36 b Fp(9)p Fo(a)k Fp(2)h(A)2 2159 y Fs(such)27 b(that)f Fo(C)6 b Fm(\()p Fo(a;)15 b(s)p Fm(\))30 b Fp(\024)i Fo(C)6 b Fm(\()p Fo(b;)14 b(s)p Fm(\))p Fs(.)39 b(Then)28 b Fk(4)1384 2182 y Fi(P)1440 2163 y Fd(0)1487 2159 y Fs(is)g(equiv)n(alent)2 2275 y(to)22 b Fk(4)160 2294 y Fi(P)242 2275 y Fs(o)o(v)n(er)g Fp(A)p Fs(.)p Black 902 2553 a Fq(21)466 b(15-854)15 b(Presentation)p Black eop PNP restore /PNP save def 31.7922 42.5454 translate 0 22 bop Black Black BrickRed Black BrickRed 406 125 a Fv(Illustrating)33 b(Axiom)f(5)p Black Black 2 211 1947 2 v 2 213 V 2 215 V Black Black Black 701 660 1025 3 v RoyalBlue 609 728 a Fo(a)p Black 698 757 3 100 v OliveGreen 274 w Fm(1)p Black 1210 757 V 460 w(3)p 1723 757 V 701 760 1025 3 v 267 828 a Fs(OPT)p Fp(!)p RoyalBlue 89 w Fo(b)p Black 698 857 3 100 v 279 w Fm(2)p 1210 857 V OliveGreen 460 w(1)p Black 1723 857 V 701 860 1025 3 v 701 959 V RoyalBlue 607 1027 a Fo(x)p Black 698 1056 3 100 v OliveGreen 274 w Fm(1)p Black 1210 1056 V OliveGreen 408 w(100)p Black 1723 1056 V 701 1059 1025 3 v Black 74 1187 a Fp(\017)p Black RoyalBlue 41 w Fo(x)p Black 24 w Fs(is)23 b(a)g(v)n(alid)h(addition)f(according)h(to)f(Axiom)g(5.)p Black Black 871 1581 855 3 v Plum 267 1649 a(ne)n(w)f(OPT)p Black Fp(!)p RoyalBlue 84 w Fo(a)p Black 868 1678 3 100 v Plum 230 w Fm(1)p Black 1295 1678 V 375 w(3)p 1723 1678 V 871 1680 855 3 v 439 1749 a Fs(OPT)p Fp(!)p RoyalBlue 89 w Fo(b)p Black 868 1778 3 100 v 234 w Fm(2)p 1295 1778 V OliveGreen 375 w(1)p Black 1723 1778 V 871 1780 855 3 v 871 1880 V RoyalBlue 780 1948 a Fo(y)p Black 868 1977 3 100 v Plum 186 w Fm(1)p Fo(=)p Fm(2)p Black 1295 1977 V OliveGreen 276 w(100)p Black 1723 1977 V 871 1979 855 3 v Black 74 2107 a Fp(\017)p Black RoyalBlue 41 w Fo(y)p Black 27 w Fs(is)h(a)p RoyalBlue 23 w(not)p Black 23 w(a)g(v)n(alid)h(addition)f(according)g(to)g(Axiom)h(5.)p Black 902 2553 a Fq(22)466 b(15-854)15 b(Presentation)p Black eop PNP restore /PNP save def 648.5610 42.5454 translate 0 23 bop Black Black BrickRed Black BrickRed 35 125 a Fv(Ax)33 b Ff(5)294 83 y Fa(0)325 125 y Fv(:)40 b(Independence)32 b(of)h Fb(Irrele)n(v)n(ant)618 241 y Fv(Alternatives)p Black Black 2 303 1947 2 v 2 305 V 2 307 V Black 2 508 a Fr(Not)22 b(Consistent)h(with)g(the)g(Competitive)g(Ratio)2 712 y Fs(If)f(w)o(e)h(add)g(an)f(algor)q(ithm)h(then)g(the)f(r)o (anking)h(of)g(the)g(or)q(iginal)2 828 y(algor)q(ithms)g(is)h (unchanged.)p Black 74 1006 a Fp(\017)p Black 41 w Fs(T)-10 b(o)41 b(some)f(people)o(,)45 b(this)c(seems)f(easier)g(to)h(suppor)s (t)159 1122 y(than)23 b(Axiom)h Fm(5)p Fs(.)p Black 74 1301 a Fp(\017)p Black 41 w Fs(What)f(about)g(the)g(man)f(in)i(the)f (restaur)o(ant?)p Black 74 1479 a Fp(\017)p Black 41 w Fs(Ho)o(w)o(e)n(v)n(er)l(,)29 b(if)h(w)o(e)f(require)g Fp(A)g Fs(to)g(be)p RoyalBlue 30 w(w)o(ell-represented)p Black 159 1596 a(at)24 b(e)n(v)n(er)r(y)f(point)h(\(i.e)o(.)29 b(f)n(or)23 b(e)n(v)n(er)r(y)g(state)h(of)f(nature)o(,)g(there)159 1712 y(e)n(xists)32 b(an)f(optimal)h(algor)q(ithm\),)h(Axiom)f Fm(5)f Fs(is)h(equiv)n(a-)159 1828 y(lent)23 b(to)g(Axiom)h Fm(5)712 1799 y Fe(0)733 1828 y Fs(.)p 2 1898 V 2 2053 a(Let)35 b Fo(P)220 2024 y Fe(0)288 2053 y Fm(=)47 b(\()p Fp(A)29 b([)e Fo(E)5 b(;)15 b Fp(S)6 b Fo(;)16 b(C)6 b Fm(\))35 b Fs(be)g(an)o(y)g(algor)q(ithmic)h(e)n(xten-)2 2169 y(sion)c(of)g Fo(P)51 b Fm(=)41 b(\()p Fp(A)p Fo(;)15 b Fp(S)6 b Fo(;)15 b(C)6 b Fm(\))32 b Fs(such)g(that)f Fp(8)p Fo(a)40 b Fp(2)f(A)p Fs(,)34 b Fp(8)p Fo(s)40 b Fp(2)f(S)6 b Fs(,)2 2285 y Fo(C)71 2256 y Fe(0)92 2285 y Fm(\()p Fo(a;)15 b(s)p Fm(\))42 b(=)h Fo(C)6 b Fm(\()p Fo(a;)14 b(s)p Fm(\))p Fs(.)59 b(Then)34 b Fk(4)1110 2308 y Fi(P)1166 2289 y Fd(0)1220 2285 y Fs(is)g(equiv)n(alent)g(to)f Fk(4)1890 2304 y Fi(P)2 2402 y Fs(o)o(v)n(er)22 b Fp(A)p Fs(.)p Black 902 2553 a Fq(23)466 b(15-854)15 b(Presentation)p Black eop PNP restore PNEP %%Page: ? 7 0.4718 PNSP /PNP save def 31.7922 839.3143 translate 0 24 bop Black Black BrickRed Black BrickRed 507 125 a Fv(Ax)34 b Ff(6)p Fv(:)40 b(Con)-5 b(ve)n(xity)p Black Black 2 211 1947 2 v 2 213 V 2 215 V Black 2 720 a Fs(An)o(y)p RoyalBlue 28 w(probability)28 b(mixture)p Black 28 w(of)g(indiff)n (erent)f(algor)q(ithms)h(is)p RoyalBlue 29 w(no)2 836 y(w)o(orse)22 b(than)p Black 23 w(the)h(or)q(iginal)g(algor)q(ithms)p Black 74 1015 a Fp(\017)p Black 41 w Fs(An)o(y)g(probability)f(mixture) h(will)f(e)n(v)n(entually)h(lead)f(to)h(one)159 1131 y(of)g(the)g(algor)q(ithms)o(,)g(so)g(ho)o(w)g(could)g(it)g(be)g(w)o (orse?)p 2 1478 V 2 1888 a(Let)j Fo(a)190 1905 y Fn(1)235 1888 y Fo(;)15 b(a)321 1905 y Fn(2)397 1888 y Fp(2)31 b(A)d Fs(with)e Fo(a)805 1905 y Fn(1)882 1888 y Fp(\031)31 b Fo(a)1028 1905 y Fn(2)1073 1888 y Fs(.)41 b(F)n(or)26 b(an)o(y)h Fo(p)32 b Fp(2)f Fm([0)p Fo(;)14 b Fm(1])p Fs(,)28 b(let)2 2004 y Fo(b)36 b Fm(=)h Fo(pa)284 2021 y Fn(1)353 2004 y Fm(+)25 b(\(1)g Fp(\000)f Fo(p)p Fm(\))p Fo(a)798 2021 y Fn(2)843 2004 y Fs(.)50 b(Use)30 b Fo(P)1161 1975 y Fe(0)1219 2004 y Fm(=)37 b(\()p Fp(A)25 b([)f(f)p Fo(b)p Fp(g)p Fo(;)15 b Fp(S)6 b Fo(;)15 b(C)6 b Fm(\))p Fs(.)2 2120 y(Then)23 b Fo(b)h Fk(4)342 2143 y Fi(P)398 2124 y Fd(0)443 2120 y Fo(a)490 2137 y Fn(1)558 2120 y Fs(and)f Fo(b)h Fk(4)848 2143 y Fi(P)904 2124 y Fd(0)949 2120 y Fo(a)996 2137 y Fn(2)p Black 902 2553 a Fq(24)466 b(15-854)15 b(Presentation)p Black eop PNP restore /PNP save def 648.5610 839.3143 translate 0 25 bop Black Black BrickRed Black BrickRed 432 125 a Fv(Ax)33 b Ff(7)p Fv(:)41 b(Redundanc)o(y)p Black Black 2 211 1947 2 v 2 213 V 2 215 V Black 2 590 a Fs(W)n(e)22 b(can)h(eliminate)h(redundant)e(states)h(of)g(nature)2 748 y Fj(in)g(other)f(w)o(ords)2 905 y Fs(Redundant)g(columns)h(can)h (be)e(remo)o(v)n(ed)p OliveGreen 217 1027 a Fo(s)258 1044 y Fn(1)p Black OliveGreen 385 1027 a Fo(s)426 1044 y Fn(2)p Black OliveGreen 554 1027 a Fo(s)595 1044 y Fn(3)p Black OliveGreen 723 1027 a Fo(s)764 1044 y Fn(4)p Black OliveGreen 892 1027 a Fo(s)933 1044 y Fn(5)p Black 175 1059 844 3 v RoyalBlue 85 1127 a Fo(a)p Black 173 1156 3 100 v 102 w(x)p 341 1156 V 111 w(w)p 510 1156 V 106 w(w)p 679 1156 V 106 w(w)p 848 1156 V 106 w(w)p 1016 1156 V 175 1158 844 3 v RoyalBlue 85 1226 a(a)p Black 173 1256 3 100 v 103 w(y)p 341 1256 V 127 w(z)p 510 1256 V 128 w(z)p 679 1256 V 127 w(z)p 848 1256 V 128 w(z)p 1016 1256 V 175 1258 844 3 v 1131 1127 a Fp(,)p OliveGreen 1490 1027 a Fo(s)1531 1044 y Fn(1)p Black OliveGreen 1686 1027 a Fo(s)1727 1044 y Fn(2)p Black 1434 1059 393 3 v RoyalBlue 1344 1127 a Fo(a)p Black 1432 1156 3 100 v 116 w(x)p 1628 1156 V 139 w(w)p 1825 1156 V 1434 1158 393 3 v RoyalBlue 1344 1226 a(a)p Black 1432 1256 3 100 v 117 w(y)p 1628 1256 V 155 w(z)p 1825 1256 V 1434 1258 393 3 v Black 74 1402 a Fp(\017)p Black 41 w Fs(Remember)g(w)o(e)h(are)f(in)p RoyalBlue 23 w(str)q(ict)i(uncer) s(tainty)p Black 2 1618 1947 2 v 2 1902 a(Let)j Fo(P)212 1873 y Fe(0)267 1902 y Fm(=)34 b(\()p Fp(A)491 1873 y Fe(0)513 1902 y Fo(;)15 b Fp(S)611 1873 y Fe(0)634 1902 y Fo(;)g(C)742 1873 y Fe(0)763 1902 y Fm(\))p Fs(,)29 b(where)e Fo(S)1166 1873 y Fe(0)1222 1902 y Fm(=)33 b Fo(S)28 b Fp(\000)c Fo(E)33 b Fs(and)27 b Fo(C)1839 1873 y Fe(0)1889 1902 y Fs(is)2 2018 y(identical)k(to)g Fo(C)36 b Fs(at)30 b(all)i Fo(a)37 b Fp(2)h(A)p Fo(;)15 b(s)37 b Fp(2)h(S)1271 1989 y Fe(0)1293 2018 y Fs(.)51 b(If)31 b(f)n(or)e(all)i Fo(s)38 b Fp(2)f Fo(E)5 b Fs(,)2 2134 y(there)34 b(e)n(xists)h(some)g Fo(s)746 2105 y Fe(0)812 2134 y Fp(2)45 b(S)975 2105 y Fe(0)1032 2134 y Fs(such)35 b(that)g(f)n(or)e(all)j Fo(a)45 b Fp(2)f(A)p Fs(,)2 2251 y Fo(C)71 2222 y Fe(0)92 2251 y Fm(\()p Fo(a;)15 b(s)259 2222 y Fe(0)280 2251 y Fm(\))25 b(=)g Fo(C)6 b Fm(\()p Fo(a;)14 b(s)p Fm(\))p Fs(,)23 b(then)f Fk(4)1021 2269 y Fi(P)1103 2251 y Fs(is)h(equiv)n(alent)h(to)f Fk(4)1742 2274 y Fi(P)1798 2254 y Fd(0)p Black 902 2553 a Fq(25)466 b(15-854)15 b(Presentation)p Black eop PNP restore /PNP save def 31.7922 42.5454 translate 0 26 bop Black Black BrickRed Black BrickRed 365 125 a Fv(Ax)34 b Ff(8)p Fv(:)40 b(Pr)n(opor)r(tionality)p Black Black 2 211 1947 2 v 2 213 V 2 215 V Black RoyalBlue 2 430 a Fs(Scaling)22 b(an)o(y)g(column)p Black 23 w(does)g(not)g(aff)n (ect)f(the)h(pref)n(erence)e(re-)2 546 y(lation.)p Black 74 725 a Fp(\017)p Black 41 w Fs(This)29 b(is)f(the)f(hardest)g(axiom)g (to)h(justify)-8 b(.)42 b(An)o(y)27 b(sugges-)159 841 y(tions?)p Black 74 1020 a Fp(\017)p Black 41 w Fs(If)22 b(w)o(e)g(assume)h(there)e(is)i(a)f(\002x)n(ed)g(unkno)o(wn)g (probability)159 1136 y(distr)q(ib)n(ution,)i(this)g(might)f(mak)n(e)g (some)g(sense)o(.)p Black 74 1314 a Fp(\017)p Black 41 w Fs(W)n(e)g(are)f(only)i(interested)e(in)i(relativ)n(e)f(perf)n(or)r (mance)o(...)p 2 1369 V 2 1489 a(Let)30 b Fo(P)51 b Fm(=)38 b(\()p Fp(A)p Fo(;)16 b Fp(S)6 b Fo(;)15 b(C)6 b Fm(\))31 b Fs(be)f(a)h(prob)n(lem)g Fo(\017)1322 1507 y Fi(P)1381 1489 y Fs(-bounded)f(a)n(w)o(a)n(y)2 1605 y(from)22 b(z)o(ero)h(\(i.e)o (.)30 b Fm(inf)653 1615 y Fi(a;s)763 1605 y Fo(C)6 b Fm(\()p Fo(a;)14 b(s)p Fm(\))25 b Fp(\025)g Fo(\017)1191 1624 y Fi(P)1276 1605 y Fo(>)g Fm(0)p Fs(\).)k(Let)24 b Fo(c)f Fs(be)g(an)o(y)2 1721 y(positiv)n(e)f(real)g(n)o(umber)l(,)f (and)g Fo(s)981 1692 y Fe(\003)1041 1721 y Fs(be)h(an)o(y)f(element)h (of)f Fp(S)6 b Fs(.)28 b(Set)2 1838 y Fo(P)70 1808 y Fe(0)116 1838 y Fm(=)c(\()p Fp(A)p Fo(;)16 b Fp(S)6 b Fo(;)15 b(C)537 1808 y Fe(0)558 1838 y Fm(\))24 b Fs(such)f(that)319 2054 y Fo(C)388 2019 y Fe(0)410 2054 y Fm(\()p Fo(a;)14 b(s)p Fm(\))25 b(=)745 1928 y Fh(\()836 2004 y Fo(c)20 b Fp(\001)g Fo(C)6 b Fm(\()p Fo(a;)14 b(s)p Fm(\))66 b Fs(if)24 b Fo(s)g Fm(=)g Fo(s)1554 1975 y Fe(\003)836 2103 y Fo(C)6 b Fm(\()p Fo(a;)15 b(s)p Fm(\))167 b Fs(o)m(.w)-5 b(.)2 2299 y(Then)17 b(if)h Fo(P)334 2270 y Fe(0)374 2299 y Fs(is)g Fo(\017)486 2318 y Fi(P)545 2299 y Fs(-bounded)f(a)n(w)o (a)n(y)g(from)f(z)o(ero)m(,)i Fk(4)1564 2322 y Fi(P)1620 2303 y Fd(0)1659 2299 y Fs(is)g(equiv-)2 2415 y(alent)23 b(to)g Fk(4)363 2434 y Fi(P)421 2415 y Fs(.)p Black 902 2553 a Fq(26)466 b(15-854)15 b(Presentation)p Black eop PNP restore /PNP save def 648.5610 42.5454 translate 0 27 bop Black Black BrickRed Black BrickRed 508 125 a Fv(Illustrating)32 b(Ax)h Ff(8)p Black Black 2 211 1947 2 v 2 213 V 2 215 V Black OliveGreen 358 622 a Fo(s)399 639 y Fn(1)p Black OliveGreen 801 622 a Fo(s)842 639 y Fn(2)p Black OliveGreen 1244 622 a Fo(s)1285 639 y Fn(3)p Black OliveGreen 1687 622 a Fo(s)1728 639 y Fn(4)p Black 179 654 1773 3 v RoyalBlue 43 722 a Fo(a)90 739 y Fn(1)p Black 176 751 3 100 v 348 722 a Fm(70)p 620 751 V 339 w(80)p 1063 751 V 339 w(90)p 1506 751 V 365 w(1)p 1949 751 V 179 753 1773 3 v RoyalBlue 43 821 a Fo(a)90 838 y Fn(2)p Black 176 850 3 100 v 373 821 a Fm(1)p 620 850 V 392 w(1)p 1063 850 V 391 w(1)p 1506 850 V 268 w(10)p Fo(;)14 b Fm(000)p 1949 850 V 179 853 1773 3 v OliveGreen 358 1167 a Fo(s)399 1184 y Fn(1)p Black OliveGreen 801 1167 a Fo(s)842 1184 y Fn(2)p Black OliveGreen 1244 1167 a Fo(s)1285 1184 y Fn(3)p Black OliveGreen 1687 1167 a Fo(s)1728 1184 y Fn(4)p Black 179 1199 V RoyalBlue 43 1267 a Fo(a)90 1284 y Fn(1)p Black 176 1296 3 100 v 348 1267 a Fm(70)p 620 1296 V 339 w(80)p 1063 1296 V 339 w(90)p 1506 1296 V 212 w(1)p Fo(=)p Fm(10)p Fo(;)g Fm(000)p 1949 1296 V 179 1298 1773 3 v RoyalBlue 43 1366 a Fo(a)90 1383 y Fn(2)p Black 176 1395 3 100 v 373 1366 a Fm(1)p 620 1395 V 392 w(1)p 1063 1395 V 391 w(1)p 1506 1395 V 391 w(1)p 1949 1395 V 179 1398 1773 3 v Black 74 1919 a Fp(\017)p Black 41 w Fs(Axiom)35 b(8)f(sa)n(ys)g(that)g(these)g(tw)o(o)f(decision)i(prob)n(lems)159 2035 y(are)23 b(essentially)h(the)f(same)o(.)p Black 74 2213 a Fp(\017)p Black 41 w Fs(Do)g(y)n(ou)g(ag)o(ree?)p Black 902 2553 a Fq(27)466 b(15-854)15 b(Presentation)p Black eop PNP restore PNEP %%Page: ? 8 0.4718 PNSP /PNP save def 31.7922 839.3143 translate 0 28 bop Black Black BrickRed Black BrickRed 287 125 a Fv(Ax)33 b Ff(8)546 83 y Fa(0)577 125 y Fv(:)41 b(Column)32 b(Linearity)p Black Black 2 211 1947 2 v 2 213 V 2 215 V Black 2 415 a Fr(Not)22 b(Consistent)h(with)g(the)g(Competitive)g (Ratio)p RoyalBlue 2 583 a Fs(Shifting)f(an)o(y)f(column)p Black 22 w(does)g(not)g(aff)n(ect)g(the)h(pref)n(erence)d(re-)2 699 y(lation.)p Black 74 877 a Fp(\017)p Black 41 w Fs(Imagine)35 b(someone)f(told)g(y)n(ou)h(the)n(y)g(w)o(ould)f(giv)n(e)h(y)n(ou)159 993 y Fm($10)29 b Fs(e)n(xtr)o(a)g(if)h(nature)f(happened)g(to)h(pic)n (k)h Fo(s)1601 1010 y Fn(3)1645 993 y Fs(.)49 b(W)n(ould)159 1110 y(that)23 b(change)g(y)n(our)g(pref)n(erences?)p Black 74 1288 a Fp(\017)p Black 41 w Fs(What)g(about)g Fm($1)p Fo(;)14 b Fm(000)p Fo(;)g Fm(000)p Fs(?)p Black 74 1467 a Fp(\017)p Black 41 w Fs(This)28 b(char)o(acter)q(iz)o(es)e (another)g(notion)g(of)g(relativ)n(e)h(per-)159 1583 y(f)n(or)r(mance)p 2 1616 V 2 1730 a(Let)i Fo(P)47 b Fm(=)36 b(\()p Fp(A)p Fo(;)16 b Fp(S)6 b Fo(;)15 b(C)6 b Fm(\))29 b 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Fm(66)p Fo(U)9 b Fm(\(2400\))365 1341 y Fp(\))25 b Fo(:)p Fm(34)p Fo(U)9 b Fm(\(2400\))23 b Fo(>)h(:)p Fm(33)p Fo(U)9 b Fm(\(2500\))p Fo(:)2 1540 y Fs(But)23 b(the)f(second)i(pref)n (erence)d(\()p Fo(L)1079 1559 y Fi(C)1162 1540 y Fp(\037)k Fo(L)1315 1559 y Fi(D)1379 1540 y Fs(\))d(speci\002es:)2 1718 y Fo(:)p Fm(33)p Fo(U)9 b Fm(\(2500\))j(+)g Fo(:)p Fm(67)p Fo(U)d Fm(\(0\))23 b Fo(>)i(:)p Fm(34)p Fo(U)9 b Fm(\(2400\))j(+)g Fo(:)p Fm(66)p Fo(U)d Fm(\(0\))385 1918 y Fp(\))25 b Fo(:)p Fm(33)p Fo(U)9 b Fm(\(2500)23 b Fo(>)h(:)p Fm(34)p Fo(U)9 b Fm(\(2400\))p Fo(:)p Black 902 2553 a Fq(43)466 b(15-854)15 b(Presentation)p Black eop PNP restore PNEP %%Page: ? 12 0.4718 PNSP /PNP save def 31.7922 839.3143 translate 0 44 bop Black Black BrickRed Black BrickRed 213 125 a Fv(Other)33 b(Pr)n(ob)o(lems)f(with)g(Utility)760 241 y(Theor)q(y)p Black Black 2 327 1947 2 v 2 329 V 2 331 V Black 2 531 a Fs(It')l(s)j(dif\002cult)g(to)g(get)f(and)g(use)p OliveGreen 35 w(personal)p Black 34 w(utility)i(functions)o(.)2 647 y(Often)29 b(assume)h(utility)g(function)g(is)g(some)f (\223nice\224)g(conca)n(v)n(e)2 763 y(\(lik)n(e)23 b(log\))g(or)g (linear)l(.)2 921 y(Expected)32 b(Utility)i(Theor)r(y)e(widely)h (accepted)g(as)f(descr)q(ip-)2 1037 y(tiv)n(e)22 b(of)g(r)o(ational)h (choice)g(under)e(r)q(isk,)i(b)n(ut)f(peoples')h(choices)2 1153 y(often)f(violate)i(it.)2 1311 y(Another)c(violation:)28 b(Risk)22 b(a)n(v)n(erse)f(in)g(positiv)n(e)h(domain)f(tend)2 1427 y(to)35 b(be)g(r)q(isk)h(seek)n(ers)g(in)g(negativ)n(e)g(domain.) 66 b(E.g.,)38 b(pref)n(er)2 1544 y Fm(80\045)22 b Fs(r)q(isk)i(of)f (losing)g(F2psEndrs4000)g(to)g(sure)g(loss)h(of)e($3000.)p Black 902 2553 a Fq(44)466 b(15-854)15 b(Presentation)p Black eop PNP restore /PNP save def 648.5610 839.3143 translate 0 45 bop Black Black BrickRed Black BrickRed 226 125 a Fv(Other)33 b(Appr)n(oac)o(hes)f(to)g(Risk)p Black Black 2 211 1947 2 v 2 213 V 2 215 V Black 2 414 a Fs(Man)o(y)h(non-v)n (on)f(Neumann-Morganster)r(n)f(optimality)j(cr)q(i-)2 530 y(ter)q(ia.)2 688 y(One:)51 b(utility)36 b(function)f Fo(U)9 b Fm(\()p Fo(k)933 705 y Fn(1)978 688 y Fm(\()p Fo(L)p Fm(\))p Fo(;)15 b(k)1203 705 y Fn(2)1248 688 y Fm(\()p Fo(L)p Fm(\)\))34 b Fs(where)g Fo(k)1768 705 y Fi(i)1829 688 y Fs(are)2 804 y Fj(char)o(acter)q(istic)20 b(par)o(ameters)h Fs(of)f(the)f(under)q(lying)h(probability)2 920 y(distr)q(ib)n(ution)k(of)f Fo(L)p Fs(.)2 1078 y(Usually)-8 b(,)26 b Fo(k)359 1095 y Fn(1)404 1078 y Fm(\()p Fo(L)p Fm(\))f Fs(is)h(the)f(e)n(xpected)g(pa)n(y)n(off)g(and)g Fo(k)1607 1095 y Fn(2)1652 1078 y Fm(\()p Fo(L)p Fm(\))h Fs(is)f(a)2 1194 y(measure)d(of)h(r)q(isk.)2 1352 y(Most)39 b(common)g Fo(k)622 1369 y Fn(2)705 1352 y Fs(is)h(standard)f(de)n (viation.)76 b(Then)40 b(this)2 1468 y(method)17 b(is)h(called)p OliveGreen 18 w(the)g(Mean-V)-6 b(ar)q(iance)17 b(cr)q(iter)q(ion)p Black 18 w(and)g Fo(U)9 b Fm(\()p Fo(L)p Fm(\))25 b(=)2 1584 y Fr(E)p Fm(\()p Fo(L)p Fm(\))19 b Fp(\000)h Fo(\016)j Fp(\003)c Fo(\033)s Fm(\()p Fo(L)p Fm(\))p Fs(.)p Black 902 2553 a Fq(45)466 b(15-854)15 b(Presentation)p Black eop PNP restore /PNP save def 31.7922 42.5454 translate 0 46 bop Black Black BrickRed Black BrickRed 104 125 a Fv(Between)33 b(Uncer)r(tainty)g(and)f(Risk)p Black Black 2 211 1947 2 v 2 213 V 2 215 V Black 2 414 a Fs(Ho)o(w)22 b(can)h(w)o(e)g(use)g(par)s(tial)g(inf)n(or)r(mation?)2 572 y(Ba)n(y)n(esian)d(approach:)27 b(tr)r(y)20 b(to)g(tr)o(ansf)n(or)r (m)f(uncer)s(tainties)i(into)2 688 y(r)q(isks)j(\(probabilities\).)2 846 y Fr(Prior)f(Distrib)n(ution:)29 b Fs(Assigned)24 b(probability)g(model.)2 1004 y(When)17 b(ne)n(w)g(sample)h(is)g (obtained,)h(update)e(pr)q(ior)g(using)h(Ba)n(y)n(es')2 1120 y(r)q(ule)23 b(f)n(or)e(conditional)j(probabilities)o(.)2 1278 y Fr(P)m(osterior:)k Fs(The)23 b(result)g(of)g(such)h(an)f(update) o(.)p Black 902 2553 a Fq(46)466 b(15-854)15 b(Presentation)p Black eop PNP restore /PNP save def 648.5610 42.5454 translate 0 47 bop Black Black BrickRed Black BrickRed 466 125 a Fv(Assigning)32 b(a)h(Prior)p Black Black 2 211 1947 2 v 2 213 V 2 215 V Black 2 415 a Fs(W)n(e)17 b(don't)g(ha)n(v)n(e)h (complete)g(inf)n(or)r(mation.)26 b(Ho)o(w)16 b(do)i(w)o(e)f(come)2 531 y(up)27 b(with)g(a)g(pr)q(ior?)40 b(T)-10 b(w)o(o)27 b(main)g(schools)h(of)f(probability)h(the-)2 648 y(or)r(y:)2 805 y Fr(Objectivists:)47 b Fs(Sample)34 b(the)f(\223tr)q(ue\224)e (under)q(lying)j(distr)q(ib)n(u-)2 922 y(tion,)g(then)f(use)f(empir)q (ical)h(distr)q(ib)n(ution,)j(maxim)o(um)d(lik)n(eli-)2 1038 y(hood,)22 b(maxim)o(um)h(entrop)n(y)-8 b(,)22 b(...)29 b(to)23 b(reco)o(v)n(er)f(a)h(pr)q(ior)l(.)2 1196 y(If)39 b(no)g(samples)o(,)44 b(star)s(t)c(with)f(e)n(v)n(en)g(or)g(r)o(andom)g (probabil-)2 1312 y(ity)33 b(distr)q(ib)n(ution\227fur)s(ther)g (sampling)h(will)f(lead)h(to)f(con)n(v)n(er-)2 1428 y(gence)23 b(to)g(\223tr)q(ue\224)e(distr)q(ib)n(ution.)2 1586 y Fr(Subjectivists:)26 b Fs(Set)19 b(of)h(qualitativ)n(e)g(beliefs)g (based)g(on)g(per-)2 1702 y(sonal)27 b(e)n(xper)q(ience)f(and)h (consistent)g(with)g(a)g(cer)s(tain)g(set)g(of)2 1818 y(axioms)o(.)2 1976 y(P)l(ersonal)17 b(pref)n(erence)f(relation)i(o)o (v)n(er)e(set)i(of)g(probability)g(dis-)2 2092 y(tr)q(ib)n(utions)o(,) 23 b(probability)h(pr)q(ior)e(and)h(utility)h(function.)p Black 902 2553 a Fq(47)466 b(15-854)15 b(Presentation)p Black eop PNP restore PNEP %%Page: ? 13 0.4718 PNSP /PNP save def 31.7922 839.3143 translate 0 48 bop Black Black BrickRed Black BrickRed 474 125 a Fv(Deep)33 b(Conc)n(lusion)p Black Black 2 211 1947 2 v 2 213 V 2 215 V Black 2 414 a Fs(The)23 b(mor)o(al)g(of)g(the)f (stor)r(y:)p Black 74 593 a Fp(\017)p Black 41 w Fs(There)k(are)e(a)i (lot)f(of)h(w)o(a)n(ys)f(to)g(measure)g(the)g(quality)i(of)159 709 y(choices)f(in)f(str)q(ict)g(uncer)s(tainty)-8 b(,)25 b(pure)f(r)q(isk,)i(and)f(mix)n(ed)159 825 y(en)n(vironments)o(.)p Black 74 1004 a Fp(\017)p Black 41 w Fs(No)h(one)g(w)o(a)n(y)f(is)i (better)f(in)g(an)o(y)g(of)g(the)g(en)n(vironments)o(.)159 1120 y(Y)-12 b(ou)28 b(ha)n(v)n(e)h(to)e(mak)n(e)h(an)g(educated)f (choice)i(based)f(on)159 1236 y(the)23 b(char)o(acter)q(istics)i(of)d (the)h(cr)q(iter)q(ia.)p Black 74 1415 a Fp(\017)p Black 41 w Fs(Lif)n(e)g(is)h(tough!)p Black 902 2553 a Fq(48)466 b(15-854)15 b(Presentation)p Black eop PNP restore PNEP %%Trailer end userdict /end-hook known{end-hook}if