%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: aernomorecr.dvi %%Pages: 27 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips aernomorecr %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 1999.10.04:2031 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 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y(\(in)36 b(our)h(mo)s(del:)51 b(serv)m(e)39 b(the)e(join)m(t)f(surplus)i(maximizing)33 b(set)38 b(of)f(users\))h (or)f(it)f(can)h(balance)g(the)g(budget)h(\(in)-120 3882 y(our)30 b(mo)s(del:)41 b(the)31 b(total)e(pa)m(ymen)m(t)j(collected)e (from)f(users)j(co)m(v)m(ers)h(the)e(total)e(cost)i(exactly\))g(but)g (it)e(can)i(not)g(b)s(e)-120 4052 y(b)s(oth)37 b(e\016cien)m(t)h(and)g (budget)g(balanced)f(\(Green,)i(Kohlb)s(erg,)f(and)f(La\013on)m(t)g ([1976],)h(Rob)s(erts)g([1979]\).)57 b(This)-120 4217 y(tradeo\013,)32 b(b)s(eing)g(particularly)f(clear)h(in)g(our)g(con)m (text,)i(is)e(the)h(cen)m(tral)f(concern)i(of)e(our)h(pap)s(er.)-120 4442 y(When)44 b(the)g(users)g(of)f(the)h(submo)s(dular)e(cost)h(tec)m (hnology)h(are)f(in)f(autarc)m(h)m(y)-8 b(,)47 b(balancing)42 b(the)i(budget)f(is)g(a)-120 4600 y(feasibilit)m(y)37 b(constrain)m(t;)42 b(in)c(this)h(case,)i(the)f(mec)m(hanism)e(design)h (problem)f(is)h(to)f(c)m(haracterize)i(the)f(class)g(of)-120 4769 y(reasonable)46 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y(cost)33 b(\()p Fp(x)418 4396 y Fo(i)474 4381 y Fr(=)28 b(0\))k(is)g(guaran)m(teed)h(to)g(eac)m(h)g(agen)m(t)g(if)e (they)j(rep)s(ort)e(truthfully)-8 b(.)-120 4632 y Fh(Consumer)37 b(So)m(v)m(ereign)m(t)m(y)g(\(CS\))48 b Fr(Ev)m(ery)c(agen)m(t)f(has)g (a)f(message)g Fp(u)2573 4647 y Fo(i)2643 4632 y Fr(guaran)m(teeing)g (that,)j(regardless)d(of)124 4802 y(the)33 b(other)g(rep)s(orted)g(v)-5 b(alues)32 b Fp(u)1286 4817 y Fj(\000)p Fo(i)1369 4802 y Fr(,)g(she)i(gets)f(service)g(\()p Fp(q)2196 4817 y Fo(i)2252 4802 y Fr(=)28 b(1\).)-120 5135 y(No)d(P)m(ositiv)m(e)h(T)-8 b(ransfers)27 b(is)e(a)g(standard)h(assumption)e(in)h(cost)h(sharing)f (situations,)h(where)g(p)s(ositiv)m(e)f(net)h(trans-)-120 5294 y(fers)i(to)f(agen)m(ts)h(are)g(disallo)m(w)m(ed.)41 b(It)27 b(is)g(not)p 1321 5307 141 4 v 27 w(a)h(natural)e(requiremen)m (t)i(for)f(Clark)m(e-Gro)m(v)m(es)h(mec)m(hanisms,)g(whic)m(h)-120 5453 y(do)j(not)f(balance)h(the)g(budget.)44 b(Ho)m(w)m(ev)m(er,)33 b(NPT)f(is)e(not)h(an)f(essen)m(tial)h(assumption)f(for)h(the)g(c)m (haracterization)-120 5617 y(of)h(the)h(MC)g(mec)m(hanism)f(in)g (Section)h(4)f(\(see)h(the)g(Remark)g(follo)m(wing)c(Prop)s(osition)i (3\).)1926 5866 y(7)p eop %%Page: 8 8 8 7 bop -120 197 a Fr(V)-8 b(olun)m(tary)22 b(P)m(articipation)d(is)j (the)g(standard)g(individual)e(rationalit)m(y)f(constrain)m(t;)26 b(agen)m(t)c Fp(i)g Fr(do)s(es)g(not)g(prefer)h(the)-120 361 y(\\status)30 b(quo")f(\()p Fp(q)527 376 y Fo(i)583 361 y Fr(=)f Fp(x)742 376 y Fo(i)798 361 y Fr(=)g(0\))h(allo)s(cation)d (to)j(the)h(allo)s(cation)c(\()p Fp(q)2268 376 y Fo(i)2296 361 y Fp(;)17 b(x)2395 376 y Fo(i)2423 361 y Fr(\))29 b(assigned)h(b)m(y)g(the)g(mec)m(hanism.)42 b(In)30 b(view)-120 531 y(of)25 b(No)g(P)m(ositiv)m(e)g(T)-8 b(ransfers,)28 b(this)c(implies)f(that)i(she)h(pa)m(ys)g(nothing)e(if)g(she)i(gets)g (no)f(service)h(\()p Fp(q)3328 546 y Fo(i)3384 531 y Fr(=)i(0)f Fn(\))g Fp(x)3746 546 y Fo(i)3803 531 y Fr(=)g(0\).)-120 762 y(Consumer)42 b(So)m(v)m(ereign)m(t)m(y)i(means)d(that)h(b)m(y)g (rep)s(orting)f(a)g(high)g(enough)h(willingness)e(to)i(pa)m(y)-8 b(,)44 b(an)m(y)e(agen)m(t)g(is)-120 920 y(guaran)m(teed)f(to)g(get)f (the)h(service.)69 b(F)-8 b(or)40 b(instance,)j(in)d(the)h(mec)m (hanisms)f(discussed)i(b)s(elo)m(w,)h(rep)s(orting)c(an)m(y)-120 1085 y Fp(u)-64 1100 y Fo(i)-6 1085 y Fn(\025)32 b Fp(C)7 b Fr(\()p Fp(i)p Fr(\))34 b(\(their)g(stand)h(alone)e(cost\))i(ensures) i(that)d Fp(q)1962 1100 y Fo(i)2021 1085 y Fr(=)d(1)j(no)g(matter)g (what)g(the)h(other)g(rep)s(orted)f Fp(u)3792 1100 y Fo(j)3863 1085 y Fr(are.)-120 1252 y(Note)k(that)g(w)m(e)h(do)f(not)g (require)g(that)g(rep)s(orting)f Fp(u)1844 1267 y Fo(i)1908 1252 y Fr(=)g(0)h(ensures)i(that)e Fp(q)2716 1267 y Fo(i)2781 1252 y Fr(=)f(0;)j(as)f(long)d(as)j Fp(x)3533 1267 y Fo(i)3598 1252 y Fr(=)e(0)h(when)-120 1416 y Fp(u)-64 1431 y Fo(i)-1 1416 y Fr(=)e(0,)i(w)m(e)h(can)e(ha)m(v)m(e)i Fp(q)830 1431 y Fo(i)894 1416 y Fr(=)d(1.)58 b(If)37 b Fp(C)7 b Fr(\()p Fp(S)f Fr(\))37 b(is)g(strictly)f(increasing)h(in)f Fp(S)6 b Fr(,)39 b(ho)m(w)m(ev)m(er,)i(the)c(later)g(con\014guration) -120 1590 y(\()p Fp(u)-26 1605 y Fo(i)29 1590 y Fr(=)28 b(0)k(but)h Fp(q)436 1605 y Fo(i)492 1590 y Fr(=)28 b(1\))k(will)e(not) j(happ)s(en)g(in)e(an)m(y)j(of)e(the)h(mec)m(hanisms)f(discussed)j(b)s (elo)m(w.)3230 1554 y Fm(10)-120 1821 y Fr(The)h(strategypro)s(of)f (and)h(budget)g(balanced)f(mec)m(hanisms)g(w)m(e)h(consider)g(are)f (deriv)m(ed)h(from)e(an)h(underlying)-120 1980 y Fi(c)-5 b(ost)40 b(sharing)g(metho)-5 b(d)p Fr(.)61 b(A)39 b(cost)g(sharing)g (metho)s(d)f(is)g(a)g(form)m(ula)f Fp(\030)43 b Fr(asso)s(ciating)38 b(to)g(eac)m(h)i(cost)f(function)f Fp(C)-120 2144 y Fr(and)d(to)g(eac)m (h)h(coalition)c Fp(S)40 b Fr(of)35 b(agen)m(ts)h(an)f(allo)s(cation)c (of)k(the)g(total)f(cost)h Fp(C)7 b Fr(\()p Fp(S)f Fr(\))35 b(among)e(these)k(agen)m(ts)e(in)g(the)-120 2318 y(form)f(of)h (nonnegativ)m(e)g(cost)h(shares)g Fp(\030)1314 2333 y Fo(i)1342 2318 y Fr(\()p Fp(S)6 b Fr(\).)1511 2282 y Fm(11)1637 2318 y Fr(These)37 b(cost)e(shares)i(m)m(ust)e(satisfy)g (the)h(budget)g(balancedness)-120 2488 y(constrain)m(t)c Fp(\030)381 2503 y Fo(S)432 2488 y Fr(\()p Fp(S)6 b Fr(\))27 b(=)h Fp(C)7 b Fr(\()p Fp(S)f Fr(\).)-120 2719 y(W)-8 b(e)36 b(no)m(w)h(explain)e(wh)m(y)j(these)f(cost)g(sharing)e(metho)s (ds)h(underlie)g(the)g(relev)-5 b(an)m(t)36 b(strategypro)s(of)g(and)g (budget)-120 2878 y(balanced)30 b(mec)m(hanisms.)43 b(Giv)m(en)30 b(a)g(cost)g(sharing)g(metho)s(d)g(and)g(a)g(pro\014le)g Fp(u)f Fr(of)h(willingness)e(to)i(pa)m(y)-8 b(,)32 b(consider)-120 3042 y(the)48 b(\(normal)e(form\))g Fi(demand)h(game)g Fr(where)i(eac)m(h)f(agen)m(t)g(decides)h(whether)g(or)e(not)g(to)h (request)h(service)-120 3212 y(\(agen)m(t)28 b Fp(i)p Fr('s)g(strategy)g(c)m(hoice)f(is)g Fp(q)1094 3227 y Fo(i)1150 3212 y Fr(=)h(0)f(or)g Fp(q)1487 3227 y Fo(i)1543 3212 y Fr(=)h(1\))f(and)g(costs)h(are)g(shared)g(among)e(those)i(who)g (request)h(service)-120 3376 y(according)k(to)g(the)h(cost)h(sharing)e (metho)s(d)g Fp(\030)5 b Fr(.)46 b(If)33 b(it)g(so)h(happ)s(ens)g(that) g(for)f(all)e(pro\014les)j Fp(u)f Fr(the)h(demand)g(game)-120 3541 y(has)42 b(a)f(unique)g Fp(\013)q Fr(-equilibrium)d(\(where)k Fp(\013)q Fr(-equilibrium)c(can)j(b)s(e)h(either)f(a)g(Nash)h (equilibrium)c(or)j(a)g(strong)-120 3710 y(equilibrium\))26 b(then)k(this)e Fp(\013)q Fr(-equilibrium)e(outcome)i(de\014nes)j(a)e (strategypro)s(of)g(rev)m(elation)f(mec)m(hanism.)42 b(That)-120 3875 y(is,)49 b(this)d(mec)m(hanism)f(elicits)g Fp(u)1088 3890 y Fo(i)1115 3875 y Fr(,)50 b Fp(i)h Fn(2)g Fp(N)10 b Fr(,)50 b(and)c(c)m(ho)s(oses)h(the)g(allo)s(cations)c(of)j (the)g(unique)h Fp(\013)q Fr(-equilibrium)-120 4033 y(outcome)42 b(of)g(the)g(rep)s(orted)h(demand)f(game.)72 b(The)43 b(fact)f(that)g(suc)m(h)i(a)e(mec)m(hanism)g(is)f(strategypro)s(of)h (is)g(a)-120 4198 y(classic)35 b(result)g(of)f(the)i(implemen)m(tation) 31 b(literature:)47 b(see)36 b(Dasgupta,)g(Hammond,)e(and)h(Maskin)h ([1979])e(and)-120 4368 y(Commen)m(t)f(1)h(in)f(Section)h(5.)47 b(The)35 b(k)m(ey)g(to)f(the)g(ab)s(o)m(v)m(e)g(prop)s(ert)m(y)h(\(a)f (unique)g Fp(\013)q Fr(-equilibrium)c(for)j(all)f(pro\014les\))-120 4532 y(is)h(that)g(the)h(cost)g(sharing)f(metho)s(d)f(m)m(ust)i(b)s(e)g Fi(cr)-5 b(oss-monotonic)p Fr(,)31 b(so)j(agen)m(t)g Fp(i)p Fr('s)g(cost)g(share)g(cannot)f(increase)-120 4700 y(when)h(the)f(set)g(of)f(agen)m(ts)h(receiving)g(service)g (expands:)1964 4664 y Fm(12)p -120 4885 1656 4 v -41 4947 a Fl(10)29 4977 y Fs(If)26 b Fk(u)158 4989 y Fg(i)209 4977 y Fs(=)c(0)j(but)h Fk(C)6 b Fs(\()p Fk(S)20 b Fs(+)14 b Fk(i)p Fs(\))23 b Fk(>)f(C)6 b Fs(\()p Fk(S)20 b Fw(\000)13 b Fk(i)p Fs(\))26 b(for)f(all)g Fk(S)j Fw(\022)22 b Fk(N)9 b Fs(,)26 b(then)g Fk(i)f Fs(is)h(not)f(part)g(of)h(the)f(e\016cien)n (t)h(coalition.)35 b(In)26 b(the)g(mec)n(hanisms)-120 5125 y(discussed)e(b)r(elo)n(w,)g(the)h(coalition)e(serv)n(ed)g(is)h 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y(public)j(go)s(o)s(d)g(\()p Fp(C)7 b Fr(\()p Fp(S)f Fr(\))37 b(=)i(1)f(for)g(all)f(nonempt)m(y)j Fp(S)6 b Fr(\),)40 b(Deb)f(and)g(Razzolini)d([1996,)j(1999])f(and)h(Ohseto)h ([1995])-120 5339 y(pro)m(vide)33 b(similar)c(y)m(et)34 b(di\013eren)m(t)e(c)m(haracterization)g(results.)p -120 5420 1656 4 v -120 5511 a Fs(Sprumon)n(t)27 b([1990],)f(Thomson)h ([1983].)-41 5587 y Fl(13)29 5617 y Fs(W)-7 b(e)28 b(will)g(use)g(the)g (notation)962 5596 y(~)948 5617 y Fk(S)t Fs(\()p Fk(u)p Fs(\))g(if,)g(from)g(the)g(con)n(text,)f(the)h(underlying)f(cost)g (sharing)f(metho)r(d)i(is)g(unam)n(biguous.)1926 5866 y Fr(9)p eop %%Page: 10 10 10 9 bop -120 197 a Fr(W)-8 b(e)40 b(do)f(not)g(pro)m(vide)g(a)g(pro)s (of)g(of)f(the)i(\\con)m(v)m(erse")h(statemen)m(t)f(in)f(Prop)s (osition)e(1;)43 b(ho)m(w)m(ev)m(er,)g(the)d(\\direct")-120 356 y(statemen)m(t)33 b(is)f(easy)i(to)e(pro)m(v)m(e)i(and)f(w)m(e)g (rep)s(eat)g(its)f(pro)s(of)g(in)f(App)s(endix)j(A.)-120 581 y Fh(Remark)f(2:)42 b Fr(W)-8 b(e)29 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4887 y Fs([)p Fk(u)2221 4899 y Fg(T)2294 4887 y Fw(\000)2433 4847 y Ft(j)p Fg(T)j Ft(j)p 2391 4868 172 4 v 2391 4915 a(j)p Fg(T)g Ft(j)p Fl(+1)2572 4887 y Fk(C)d Fs(\()p Fk(T)34 b Fs(+)21 b Fk(i)p Fs(\)])32 b(\(see,)i(for)e(example,)h(Green) f(and)-120 5062 y(La\013on)n(t)27 b([1979]\).)35 b(In)27 b(substance,)g(ho)n(w)n(ev)n(er,)e(they)j(are)e(quite)h(di\013eren)n (t;)h(the)g(piv)n(otal)e(mec)n(hanism)h(nev)n(er)f(runs)h(a)f(budget)i (de\014cit)-120 5216 y(\(i.e,)g(it)g(either)g(is)g(budget)g(balanced)f (or)g(runs)g(a)g(surplus\))h(and)f(fails)h(VP)-7 b(,)28 b(whereas)e(the)i(MC)g(mec)n(hanism)f(nev)n(er)g(runs)h(a)f(budget)-120 5369 y(surplus)g(\(i.e,)h(it)g(either)g(is)f(budget)h(balanced)f(or)f (runs)i(a)f(budget)h(de\014cit\))g(and)f(satis\014es)g(VP)-7 b(.)-41 5438 y Fl(19)29 5469 y Fs(W)g(e)33 b(need)f(not)g(require)f(CS) i(since)f(all)f(CG)i(mec)n(hanisms)e(satisfy)h(CS;)g(the)h(set)f(of)g (e\016cien)n(t)g(coalitions)f(is)h(indep)r(enden)n(t)h(of)-120 5617 y(the)28 b(mec)n(hanism,)f(and)h(if)g Fk(u)759 5629 y Fg(i)809 5617 y Fk(>)23 b(C)6 b Fs(\()p Fk(i)p Fs(\))28 b(all)f(e\016cien)n(t)h(coalitions)e(will)i(include)g(agen)n(t)f Fk(i)g Fs(when)h(the)g(cost)f(is)g(submo)r(dular.)1901 5866 y Fr(13)p eop %%Page: 14 14 14 13 bop -120 197 a Fr(The)34 b(pro)s(of)d(of)h(this)h(Prop)s(osition) d(is)j(presen)m(ted)h(in)e(App)s(endix)h(C.)-120 422 y Fh(Remark)48 b(4:)62 b Fr(Both)42 b(assumptions)g(NPT)h(and)f(VP)g (are)g(needed)i(in)d(this)g(c)m(haracterization)g(result.)72 b(If)42 b(w)m(e)-120 586 y(drop)36 b(VP)-8 b(,)36 b(the)g(piv)m(otal)e (mec)m(hanism)h(\(see)i(F)-8 b(o)s(otnote)34 b(18)h(meets)h(all)e (other)i(assumptions.)52 b(If)36 b(w)m(e)g(drop)g(NPT)-120 756 y(then)30 b(w)m(e)g(can)f(mo)s(dify)e(the)i(MC)h(mec)m(hanism)e(b)m (y)i(adding)e(an)h(arbitrary)f(cash)i(pa)m(ymen)m(t)f(\(from)f(the)h (bank)m(er)h(to)-120 926 y(eac)m(h)k(agen)m(t\))f(indep)s(enden)m(t)i (of)e(the)g(rep)s(orted)h(pro\014le.)44 b(The)35 b(pro)s(of)d(of)h 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b(is)g(indi\013eren)m(t)g(to)g(\(1)p Fp(;)17 b(u)2235 4260 y Fo(i)2262 4245 y Fr(\).)43 b(Hence)31 b(the)f(classical)e(domain)g(\(monotonic)-120 4415 y(preferences)38 b({)e(strictly)f(in)g(money)h({)f(con)m(tin)m(uous)i(and)f(con)m(v)m (ex\))i(and)e(that)f(of)h(preferences)i(represen)m(ted)g(b)m(y)-120 4584 y(quasi-linear)c(utilities)f(coincide.)52 b(In)36 b(particular)e(w)m(e)j(can)f(apply)f(Theorem)h(2)g(in)f(Moulin)f ([1999])h(dev)m(elop)s(ed)-120 4754 y(for)c(the)h(classical)e(domain)f (of)i(preferences.)45 b(Moreo)m(v)m(er,)34 b(our)d(domain)f(is)h (monotonically)d(closed)j(\(Dasgupta,)p -120 4856 1656 4 v -41 4917 a Fl(21)29 4948 y Fs(Indeed,)h(assume)e Fk(ac)21 b Fw(\000)f Fk(mc)31 b Fs(increases)e(and)h(so,)h(for)g(all)f Fk(i)e Fs(=)g(1)p Fk(;)14 b(:)g(:)g(:)f(;)h(n)20 b Fw(\000)h Fs(1,)31 b Fk(ac)p Fs(\()p Fk(i)p Fs(\))d Fk(<)g(mc)p Fs(\()p Fk(i)p Fs(\))21 b(+)f Fk(ac)p Fs(\()p Fk(n)p Fs(\))h Fw(\000)f Fk(mc)p Fs(\()p Fk(n)p Fs(\))31 b(whic)n(h)g(then) -120 5096 y(implies:)740 5270 y Fk(\015)p 739 5307 50 4 v 739 5383 a(n)821 5326 y(<)923 5270 y Fs(1)p 919 5307 V 919 5383 a Fk(n)979 5326 y Fs(\()1051 5223 y Fg(n)1011 5248 y Fe(X)1017 5424 y Fg(i)p Fl(=1)1145 5326 y Fk(mc)p Fs(\()p Fk(i)p Fs(\)\))19 b(+)f(\()p Fk(ac)p Fs(\()p Fk(n)p Fs(\))h Fw(\000)f Fk(mc)p Fs(\()p Fk(n)p Fs(\)\))h Fw(\000)f Fk(ac)p Fs(\()p Fk(n)p Fs(\))23 b(=)g Fk(ac)p Fs(\()p Fk(n)p Fs(\))18 b Fw(\000)g Fk(mc)p Fs(\()p Fk(n)p Fs(\))23 b(=)3110 5270 y Fk(\014)p 3110 5307 52 4 v 3111 5383 a(n)-120 5617 y Fs(Other)k(cases)g(are)f(similar.)1901 5866 y Fr(16)p eop %%Page: 17 17 17 16 bop -120 197 a Fr(Hammond)31 b(and)i(Maskin)g([1979]\),)f(a)g (prop)s(ert)m(y)h(that)g(fails)d(for)i(general)g(quasilinear)f (utilities.)-120 428 y(On)k(the)g(other)g(hand,)h(if)e(preferences)j (are)e(not)g(quasi-linear,)e(e\016ciency)j(is)f(no)g(longer)f(equiv)-5 b(alen)m(t)34 b(to)h(maxi-)-120 586 y(mization)30 b(of)i(join)m(t)g (surplus,)h(therefore)g(our)g(Prop)s(ositions)e(2)h(and)h(4)f(are)h(no) 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Fp(\033)n(;)17 b(i)p Fr(\)\))j Fn(\000)g Fp(C)7 b Fr(\()p Fp(E)f Fr(\()p Fp(\033)n(;)17 b(i)j Fn(\000)g Fr(1\)\))-120 3239 y(for)34 b(some)g(ordering)f Fp(\033)38 b Fr(\(where)d Fp(E)6 b Fr(\()p Fp(\033)n(;)17 b(i)p Fr(\))35 b(is)e(the)i(set)g(of)e (the)i(\014rst)g Fp(i)f Fr(agen)m(ts)h(in)e(ordering)h Fp(\033)t Fr(\).)48 b(The)35 b(criterion)d(of)-120 3403 y(minimizing)d(the)34 b(maximal)c(w)m(elfare)k(loss)f(do)s(es)h(not)g (seem)g(to)f(select)h(a)g(clearly)e(iden)m(ti\014able)g(metho)s(d)h(in) g(this)-120 3562 y(family)-8 b(.)49 b(Th)m(us,)37 b(in)d(the)i(sup)s (ermo)s(dular)e(case,)j(the)e(com)m(bination)e(of)i(group)g (strategypro)s(ofness)h(and)f(budget)-120 3721 y(balance)25 b(proscrib)s(es)g(equal)g(treatmen)m(t)g(of)g(equals,)i(whereas)g(the)e (com)m(bination)e(of)i(strategypro)s(ofness)h(and)f(ef-)-120 3885 y(\014ciency)i(\(as)g(in)f(the)g(submo)s(dular)g(case\))h (prescrib)s(es)g(equal)g(treatmen)m(t)f(of)g(equals.)42 b(While)25 b(the)i(former)e(\(group)-120 4055 y(strategypro)s(ofness)32 b(and)g(budget)f(balance\))g(still)e(has)j(sup)s(erior)e(incen)m(tiv)m (e)i(prop)s(erties)f(in)g(the)g(sup)s(ermo)s(dular)-120 4219 y(case,)j(it)d(no)i(longer)e(has)i(claim)d(to)j(greater)f (\015exibilit)m(y)-8 b(.)-120 4745 y Fq(6)161 b(References)-120 5074 y Fr(Aumann,)28 b(R.J.,)h(1959.)41 b Fi(A)-5 b(c)g(c)g(eptable)30 b(Points)g(in)f(Gener)-5 b(al)30 b(Co)-5 b(op)g(er)g(ative)29 b(n-Person)h(Games)p Fr(,)e(Annals)f(of)g(Math-)-120 5233 y(ematics)32 b(Studies)h(series,)g(Princeton,)g(NJ:)g(Princeton)g (Univ)m(ersit)m(y)g(Press.)-120 5458 y(Baumol,)k(W.,)j(J.)e(P)m(anzar,) h(and)f(R.)g(Willig,)d(1982.)59 b Fi(Contestable)38 b(Markets)i(and)f (the)h(The)-5 b(ory)39 b(of)h(Industrial)-120 5617 y(Structur)-5 b(e)p Fr(,)34 b(New)f(Y)-8 b(ork:)44 b(Harcourt)32 b(Brace)h(Jo)m(v)-5 b(ano)m(vic)m(h.)1901 5866 y(18)p eop %%Page: 19 19 19 18 bop -120 197 a Fr(Clark)m(e,)33 b(E.)g(H.,)g(1971.)42 b(\\Multipart)31 b(Pricing)h(of)g(Public)f(Go)s(o)s(ds,")h Fi(Public)j(Choic)-5 b(e)p Fr(,)31 b(11,)i(17-33.)-120 422 y(Dasgupta,)e(P)-8 b(.,)31 b(P)-8 b(.)32 b(Hammond,)e(and)h(E.)g (Maskin,)h(1979.)42 b(\\The)31 b(Implemen)m(tation)e(of)h(So)s(cial)f (Choice)i(Rules,")-120 581 y Fi(R)-5 b(eview)34 b(of)h(Ec)-5 b(onomics)33 b(Studies)p Fr(,)g(46,)f(185-216.)-120 806 y(Deb,)c(R.)e(and)h(L.)f(Razzolini,)f(1996.)41 b(\\Auction)26 b(Lik)m(e)g(Mec)m(hanisms)i(for)d(Pricing)h(Excludable)g(Public)g(Go)s (o)s(ds,")-120 965 y(forthcoming)31 b(in)g Fi(Journal)k(of)g(Ec)-5 b(onomic)33 b(The)-5 b(ory)p Fr(..)-120 1190 y(Deb,)47 b(R.)d(and)g(L.)g(Razzolini,)h(1999.)77 b(\\V)-8 b(olun)m(tary)44 b(Cost)h(Sharing)e(for)h(an)g(Excludable)g(Public)f(Pro)5 b(ject,")-120 1349 y Fi(Mathematic)-5 b(al)34 b(So)-5 b(cial)34 b(Scienc)-5 b(es)p Fr(,)32 b(37,)g(123-138.)-120 1574 y(Deb,)42 b(R.,)f(T.)f(K.)g(Seo,)h(and)f(L.)g(Razzolini,)f(1997.) 63 b(\\The)41 b(Asymptotic)e(W)-8 b(elfare)39 b(Loss)h(for)f(the)i (Serial)d(Sost)-120 1733 y(Sharing)32 b(Rule:)43 b(the)33 b(case)g(of)f(an)h(excludable)f(public)g(pro)5 b(ject,")33 b(Mimeo,)f(Southern)h(Metho)s(dist)f(Univ)m(ersit)m(y)-8 b(.)-120 1958 y(Dutta,)45 b(B.)e(and)h(D.)e(Ra)m(y)-8 b(,)46 b(1989.)74 b(\\A)43 b(Concept)h(of)f(Egalitarianism)c(under)44 b(P)m(articipation)d(Constrain)m(ts,")-120 2116 y Fi(Ec)-5 b(onometric)g(a)p Fr(,)32 b(57,)g(615-635.)-120 2341 y(Dutta,)43 b(B.)f(1990.)70 b(\\The)42 b(Egalitarian)c(Solution)i(and)i (Reduced)h(Game)d(Prop)s(erties)i(in)f(Con)m(v)m(ex)j(Games,")-120 2500 y Fi(International)34 b(Journal)g(of)h(Game)g(The)-5 b(ory)p Fr(,)32 b(19,)g(153-169.)-120 2725 y(Green,)g(J.,)f(E.)h(Kohlb) s(erg,)e(and)h(J.)g(J.)g(La\013on)m(t,)h(1976.)42 b(\\P)m(artial)29 b(Equilibrium)f(Approac)m(h)j(to)g(the)g(F)-8 b(ree)32 b(Rider)-120 2884 y(Problem,")f Fi(Journal)k(of)g(Public)f(Ec)-5 b(onomics)p Fr(,)32 b(6,)g(375-394.)-120 3109 y(Green,)47 b(J.)d(and)g(J.)h(J.)f(La\013on)m(t,)i(1979.)77 b Fi(Inc)-5 b(entives)44 b(in)h(Public)g(De)-5 b(cision)44 b(Making)p Fr(,)j(in)c Fi(Studies)i(in)g(Public)-120 3267 y(Ec)-5 b(onomics)p Fr(,)31 b(V)-8 b(ol.)43 b(1,)32 b(Amsterdam:)43 b(North-Holland.)-120 3493 y(Gro)m(v)m(es,)34 b(T.,)f(1973.)42 b(\\Incen)m(tiv)m(es)35 b(in)d(T)-8 b(eams,")32 b Fi(Ec)-5 b(onometric)g(a)p Fr(,)32 b(41,)g(617-663.)-120 3718 y(Hart,)27 b(S.)f(and)f(A.)h(Mas-Colell,)f(1989.)41 b(\\P)m(oten)m (tial,)25 b(V)-8 b(alue,)27 b(and)f(Consistency)-8 b(,")28 b Fi(Ec)-5 b(onometric)g(a)p Fr(,)26 b(57,)h(589-614.)-120 3943 y(Herzog,)k(S.,)h(S.)e(Shenk)m(er,)j(and)e(D.)f(Estrin,)h(1997.)42 b(\\Sharing)29 b(the)i(`Cost')g(of)f(Multicast)g(T)-8 b(rees:)44 b(an)30 b(axiomatic)-120 4102 y(analysis,")i Fi(T)-7 b(r)i(ansactions)33 b(on)h(Networking)p Fr(,)f(5,)f(847-860.) -120 4327 y(Israelsen,)d(D.,)e(1980.)40 b(\\Collectiv)m(es,)28 b(Comm)m(unes,)g(and)e(Incen)m(tiv)m(es,")k Fi(Journal)f(of)g(Comp)-5 b(ar)g(ative)28 b(Ec)-5 b(onomics)p Fr(,)-120 4481 y(4,)32 b(99-124.)-120 4706 y(La\013on)m(t,)k(J.)f(J.)h(and)f(J.)g(Tirole,)g (1994.)51 b Fi(A)37 b(The)-5 b(ory)37 b(of)g(Inc)-5 b(entives)36 b(in)h(Pr)-5 b(o)g(cur)g(ement)38 b(and)e(R)-5 b(e)g(gulation)p Fr(,)36 b(Cam-)-120 4865 y(bridge:)43 b(MIT)34 b(Press.)-120 5090 y(Lo)s(ehman,)43 b(E.)e(and)g(A.)h(Whinston,)h(1974.)68 b(\\An)42 b(Axiomatic)d(Approac)m(h)j(to)e(Cost)i(Allo)s(cation)c(for)j (Public)-120 5249 y(In)m(v)m(estmen)m(t,")35 b Fi(Public)f(Financ)-5 b(e)34 b(Quarterly)p Fr(,)f(2,)f(236-251.)-120 5474 y(McAfee,)46 b(P)-8 b(.)42 b(1993.)71 b(\\A)42 b(Dominan)m(t)e(Strategy)j(Double)e (Auction")g Fi(Journal)j(of)f(Ec)-5 b(onomic)42 b(The)-5 b(ory)p Fr(,)44 b(56:2,)1901 5866 y(19)p eop %%Page: 20 20 20 19 bop -120 197 a Fr(434-450.)-120 403 y(Monderer,)37 b(D.,)f(and)f(L.)h(S.)f(Shapley)-8 b(,)37 b(1993.)51 b(\\P)m(oten)m(tial)34 b(Games,")h Fi(Games)i(and)g(Ec)-5 b(onomic)37 b(Behavior)p Fr(,)d(14,)-120 556 y(124-143.)-120 762 y(Moulin,)d(H.,)h(1994.)42 b(\\Serial)30 b(Cost-Sharing)h(of)g (Excludable)h(Public)f(Go)s(o)s(ds,")g Fi(R)-5 b(eview)33 b(of)h(Ec)-5 b(onomics)33 b(Stud-)-120 916 y(ies)p Fr(,)f(61,)g (305-325.)-120 1142 y(Moulin,)53 b(H.,)h(1999.)94 b(\\Incremen)m(tal)49 b(Cost)h(Sharing;)58 b(c)m(haracterization)48 b(b)m(y)j(strategypro)s (ofness,")k Fi(So)-5 b(cial)-120 1300 y(Choic)g(e)34 b(and)g(Welfar)-5 b(e)p Fr(,)32 b(16,)h(279-320.)-120 1526 y(Moulin,)e(H.,)i(and)g(S.)g(Shenk)m(er,)h(1992.)43 b(\\Serial)31 b(Cost)i(Sharing,")e Fi(Ec)-5 b(onometric)g(a)p Fr(,)32 b(50,)g(1009-1039.)-120 1751 y(Mutusw)m(ami,)53 b(S.,)g(1997.)91 b(\\Strategypro)s(of)47 b(Mec)m(hanisms)j(for)e(Cost)h (Sharing,")j(w)m(orking)c(pap)s(er,)53 b(Indian)-120 1910 y(Statistical)30 b(Institute.)-120 2115 y(Ok)-5 b(ada,)41 b(N.,)h(T.)e(Hashimoto,)h(and)e(P)-8 b(.)41 b(Y)-8 b(oung,)41 b(1982.)64 b(\\Cost)40 b(Allo)s(cation)d(in)i(W)-8 b(ater)40 b(Resources)h(Dev)m(elop-)-120 2274 y(men)m(t,")32 b Fi(Water)k(R)-5 b(esour)g(c)g(es)34 b(R)-5 b(ese)g(ar)g(ch)p Fr(,)32 b(18,)g(361-373.)-120 2499 y(Ohseto,)38 b(S.,)g(1995.)54 b(\\Strategy-Pro)s(of)35 b(Cost)i(Sharing)e(for)h(an)g(Excludable)h (Public)e(Pro)5 b(ject,")38 b(mimeo,)e(Rit-)-120 2658 y(sumeik)-5 b(an)32 b(Univ)m(ersit)m(y)-8 b(.)-120 2883 y(Ostrom,)32 b(E.,)h(1991.)43 b Fi(Governing)34 b(the)h(Commons)p Fr(,)c(New)i(Y)-8 b(ork:)44 b(Cam)m(bridge)32 b(Univ)m(ersit)m(y)h (Press.)-120 3109 y(Rob)s(erts,)44 b(K.)d(1979.)70 b(\\The)42 b(Characterization)e(of)h(Implemen)m(table)f(Choice)i(Rules,")h(in)e Fi(A)-5 b(ggr)g(e)g(gation)43 b(and)-120 3267 y(R)-5 b(evelation)46 b(of)h(Pr)-5 b(efer)g(enc)g(es)p Fr(,)49 b(J.J.)d(La\013on)m(t,)k(ed.)84 b(Studies)46 b(in)g(Public)f (Economics,)k(Amsterdam:)70 b(North)-120 3426 y(Holland.)-120 3632 y(Sc)m(hmeidler,)48 b(D.,)g(1969.)81 b(\\The)46 b(Nucleolus)f(of)f(a)h(Characteristic)g(F)-8 b(unction)45 b(Game,")i Fi(SIAM)g(Journal)f(of)-120 3791 y(Applie)-5 b(d)34 b(Mathematics)p Fr(,)f(17,)f(1163-1170.)-120 4016 y(Sen,)38 b(A.K.)e(1966.)54 b(\\Lab)s(our)36 b(Allo)s(cation)d(in)i(a)h (Co)s(op)s(erativ)m(e)g(En)m(terprise,")i Fi(R)-5 b(eview)38 b(of)g(Ec)-5 b(onomic)37 b(Studies)p Fr(,)-120 4170 y(33,)32 b(361-371.)-120 4395 y(Seriza)m(w)m(a,)h(S.,)f(1996.)42 b(\\Strategy-Pro)s(of)31 b(and)h(Individually)e(Rational)g(So)s(cial)g (Choice)i(F)-8 b(unctions)31 b(for)h(Public)-120 4554 y(Go)s(o)s(ds)g(Economies,")g Fi(Ec)-5 b(onomic)33 b(The)-5 b(ory)p Fr(,)33 b(7,)f(501-512.)-120 4779 y(Seriza)m(w)m(a,)50 b(S.,)g(1999.)84 b(\\Strategypro)s(of)45 b(and)i(Symmetric)e(So)s(cial) f(Choice)j(F)-8 b(unctions)46 b(for)f(Public)h(Go)s(o)s(ds)-120 4938 y(economies,")32 b Fi(Ec)-5 b(onometric)g(a)p Fr(,)32 b(67,)g(1,)g(121-146.)-120 5168 y(Shapley)-8 b(,)35 b(L.)g(S.,)g(1953.) 49 b(\\A)34 b(V)-8 b(alue)34 b(for)g(n-P)m(erson)i(Games")e(in)g(Kuhn,) h(H.,)h(and)e(W.)h(T)-8 b(uc)m(k)m(er)37 b(\(eds.\))50 b Fi(Contri-)-120 5333 y(butions)35 b(to)g(the)g(The)-5 b(ory)34 b(of)h(Games)p Fr(,)d(Princeton:)44 b(Princeton)32 b(Univ)m(ersit)m(y)i(Press.)-120 5558 y(Shapley)-8 b(,)33 b(L.)f(S.,)h(1971.)43 b(\\Core)33 b(of)f(Con)m(v)m(ex)j(Games,")c Fi(International)j(Journal)h(of)f(Game)h(The)-5 b(ory)p Fr(,)32 b(1,)h(11-26.)1901 5866 y(20)p eop %%Page: 21 21 21 20 bop -120 197 a Fr(Shark)m(ey)-8 b(,)34 b(W.,)f(1982.)43 b Fi(The)34 b(The)-5 b(ory)34 b(of)h(Natur)-5 b(al)36 b(Monop)-5 b(oly)p Fr(,)32 b(New)h(Y)-8 b(ork:)44 b(Cam)m(bridge)32 b(Univ)m(ersit)m(y)h(Press.)-120 422 y(Shark)m(ey)-8 b(,)39 b(W.,)g(1995.)55 b(\\Net)m(w)m(ork)38 b(Mo)s(dels)e(in)g (Economics")g(Chapter)i(9)e(in)g Fi(Network)j(R)-5 b(outing)p Fr(,)38 b(M.)f(O.)g(Ball)-120 586 y(et)i(al.)60 b(\(eds.\),)42 b(Handb)s(o)s(ok)c(in)g(Op)s(erations)g(Researc)m(h)i(and)f(Managemen)m (t)g(Science,)i(V)-8 b(ol.)61 b(8,)40 b(Amsterdam:)-120 751 y(North-Holland.)-120 956 y(Shenk)m(er,)34 b(S.,)e(1992.)42 b(\\On)31 b(the)h(Strategy-Pro)s(of)f(and)h(Smo)s(oth)e(Allo)s(cation)e (of)j(Priv)-5 b(ate)31 b(Go)s(o)s(ds)g(in)g(a)g(Pro)s(duc-)-120 1115 y(tion)h(Econom)m(y)-8 b(,")32 b(mimeo,)f(Xero)m(x)i(P)-8 b(AR)m(C.)-120 1340 y(Shenk)m(er,)35 b(S.,)d(1995.)43 b(\\Making)32 b(Greed)h(W)-8 b(ork)32 b(in)g(Net)m(w)m(orks:)46 b(a)32 b(game)f(theoretic)i(analysis)f(of)g(switc)m(h)h(service)-120 1499 y(disciplines,")e Fi(T)-7 b(r)i(ansactions)33 b(on)i(Networking)p Fr(,)d(3,)g(819-831.)-120 1724 y(Sh)m(ubik,)d(M.,)f(1962.)41 b(\\Incen)m(tiv)m(es,)30 b(Decen)m(tralized)c(Con)m(trols,)i(the)g (Assignmen)m(t)f(of)f(Join)m(t)g(Costs)i(and)f(In)m(ternal)-120 1883 y(Pricing,")k Fi(Management)j(Scienc)-5 b(e)p Fr(,)32 b(8,)g(325-343.)-120 2108 y(Sonmez,)38 b(T.,)h(1993.)55 b(\\P)m(opulation)36 b(Monotonicit)m(y)g(of)g(the)i(Nucleolus)e(on)h(a) g(Class)g(of)f(Public)h(Go)s(o)s(d)e(Prob-)-120 2267 y(lems,")d(mimeo,)e(Univ)m(ersit)m(y)j(of)f(Ro)s(c)m(hester.)-120 2492 y(Sprumon)m(t,)46 b(Y.,)h(1990.)75 b(\\P)m(opulation)42 b(Monotonic)h(Allo)s(cation)e(Sc)m(hemes)k(for)e(Co)s(op)s(erativ)m(e)g (Games)g(with)-120 2651 y(T)-8 b(ransferable)33 b(Utilit)m(y)-8 b(,")30 b Fi(Games)35 b(and)f(Ec)-5 b(onomic)33 b(Behavior)p Fr(,)f(2,)g(378-394.)-120 2876 y(Stra\016n,)46 b(P)-8 b(.)43 b(D.,)i(and)e(J.)h(P)-8 b(.)43 b(Heaney)-8 b(,)47 b(1981.)73 b(\\Game)42 b(Theory)i(and)f(the)h(T)-8 b(ennessee)46 b(V)-8 b(alley)41 b(Authorit)m(y)-8 b(,")-120 3035 y Fi(International)34 b(Journal)g(of)h(Game)g(The)-5 b(ory)p Fr(,)32 b(10,)g(35-43.)-120 3260 y(Thomas,)37 b(A.)g(1980.)54 b Fi(A)39 b(Behavior)-5 b(al)37 b(A)n(nalysis)g(of)i(Joint)f(Cost)g(A)n (l)5 b(lo)-5 b(c)g(ation)38 b(and)f(T)-7 b(r)i(ansfer)38 b(Pricing)p Fr(,)e(Arth)m(ur)-120 3419 y(Andersen)e(and)f(Compan)m(y)g (Lecture)h(Series,)f(Stip)s(es)f(Publishing)f(Compan)m(y)-8 b(.)-120 3644 y(Thomson,)34 b(W.)g(1983.)46 b(\\The)34 b(F)-8 b(air)32 b(Division)f(of)i(a)h(Fixed)f(Supply)h(among)e(a)i(Gro) m(wing)e(P)m(opulation,")h Fi(Math-)-120 3803 y(ematics)h(of)h(Op)-5 b(er)g(ations)34 b(R)-5 b(ese)g(ar)g(ch)p Fr(,)31 b(8,)i(319-326.)-120 4028 y(W)-8 b(eitzman,)43 b(M.)g(1974.)71 b(\\F)-8 b(ree)42 b(Access)i(vs)f(Priv)-5 b(ate)41 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Fn(62)51 b Fp(T)14 b Fr(\))45 b(and)h(set)g Fp(S)2913 5581 y Fj(0)2986 5617 y Fr(=)3128 5592 y(~)3112 5617 y Fp(S)6 b Fr(\()p Fp(\030)f(;)17 b(u)3364 5581 y Fj(0)3385 5617 y Fr(\).)83 b(Finally)-8 b(,)46 b(set)1901 5866 y(21)p eop %%Page: 22 22 22 21 bop -120 197 a Fp(S)-54 161 y Fj(00)16 197 y Fr(=)27 b Fp(S)c Fn([)18 b Fp(S)352 161 y Fj(0)405 197 y Fr(and)30 b(denote)h(b)m(y)g Fp(q)j Fr(\(resp)s(ectiv)m(ely)-8 b(,)32 b Fp(q)1748 161 y Fj(0)1801 197 y Fr(and)e Fp(q)2035 161 y Fj(00)2077 197 y Fr(\))g(the)h(binary)f(inclusion)e(v)m(ector)j (corresp)s(onding)f(to)-120 367 y Fp(S)38 b Fr(\(resp)s(ectiv)m(ely)-8 b(,)34 b Fp(S)633 330 y Fj(0)688 367 y Fr(and)f Fp(S)944 330 y Fj(00)986 367 y Fr(\).)44 b(Chec)m(k)34 b(\014rst)1017 620 y Fp(f)11 b(or)34 b(al)r(l)i(i)28 b Fn(2)g Fp(T)41 b Fr(:)60 b Fp(u)1744 635 y Fo(i)1772 620 y Fp(q)1819 579 y Fj(00)1815 645 y Fo(i)1883 620 y Fn(\000)23 b Fp(\030)2026 635 y Fo(i)2054 620 y Fr(\()p Fp(S)2158 579 y Fj(00)2200 620 y Fr(\))28 b Fn(\025)g 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y(sharing)i(metho)s(d)g Fp(\030)37 b Fr(and)c(a)f(utilit)m(y)e (pro\014le)j Fp(u)p Fr(.)43 b(W)-8 b(e)33 b(write)f Fp(S)h Fr(=)2275 4445 y(~)2259 4470 y Fp(S)6 b Fr(\()p Fp(\030)f(;)17 b(u)p Fr(\))31 b(and)h Fp(T)42 b Fr(=)27 b Fp(S)3037 4434 y Fj(\003)3076 4470 y Fr(\()p Fp(u)p Fr(\).)-120 4700 y Fh(Step)38 b(1:)43 b Fp(S)34 b Fn(\022)28 b Fp(T)-120 4931 y Fr(Assume,)35 b(to)e(the)g(con)m(trary)-8 b(,)35 b(that)e Fp(S)28 b Fn(\000)c Fp(T)47 b Fr(is)33 b(nonempt)m(y)-8 b(.)46 b(Observ)m(e)35 b(that)f(for)e(all)g Fp(i)d Fn(2)h Fp(S)6 b Fr(,)33 b(w)m(e)h(ha)m(v)m(e)h Fp(u)3644 4946 y Fo(i)3701 4931 y Fn(\025)30 b Fp(\030)3851 4946 y Fo(i)3878 4931 y Fr(\()p Fp(S)6 b Fr(\))-120 5117 y(\(b)m(y)33 b(construction)g(of)740 5092 y(~)723 5117 y Fp(S)39 b Fr(as)32 b(in)g(\(3\)\).)43 b(Hence,)34 b(w)m(e)g(compute:)-109 5367 y(\()p Fp(u)-15 5382 y Fo(S)t Fj([)p Fo(T)155 5367 y Fn(\000)23 b Fp(C)7 b Fr(\()p Fp(S)28 b Fn([)22 b Fp(T)14 b Fr(\)\))22 b Fn(\000)h Fr(\()p Fp(u)909 5382 y Fo(T)985 5367 y 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b(the)h(largest)f(e\016cien)m(t)h (coalition.)-120 422 y(The)f(inclusion)d(just)i(established)g(implies)e (in)h(particular)f(the)j(follo)m(wing)c(expression)k(for)e(the)i (e\016ciency)g(loss:)1251 693 y Fp(\015)5 b Fr(\()p Fp(\030)g(;)17 b(u)p Fr(\))26 b(=)i Fp(w)s Fr(\()p Fp(N)5 b(;)17 b(u)p Fr(\))k Fn(\000)h Fp(w)s Fr(\()2240 668 y(~)2224 693 y Fp(S)5 b Fr(\()p Fp(\030)g(;)17 b(u)p Fr(\))p Fp(;)g(u)p Fr(\))-120 970 y(T)-8 b(o)33 b(see)g(this,)g(apply)f(Step)h(1)f(to)h (the)g(restriction)e(of)h(the)h(game)f(to)2394 945 y(~)2377 970 y Fp(S)6 b Fr(\()p Fp(\030)f(;)17 b(u)p Fr(\).)-120 1206 y Fh(Step)38 b(2:)43 b Fi(F)-7 b(or)34 b(al)5 b(l)35 b Fp(R)28 b Fn(\022)g Fp(N)10 b Fi(,)36 b Fp(R)23 b Fn(\\)f Fp(S)34 b Fr(=)28 b Fn(;)p Fi(,)34 b(ther)-5 b(e)35 b(exists)f Fp(i)28 b Fn(2)g Fp(R)36 b Fi(such)f(that)g Fp(u)2698 1221 y Fo(i)2754 1206 y Fp(<)27 b(\030)2900 1221 y Fo(i)2928 1206 y Fr(\()p Fp(S)h Fn([)22 b Fp(R)q Fr(\))-120 1443 y(This)33 b(follo)m(ws)f(from)f(the)i(de\014nition)f(of)h Fp(S)6 b Fr(;)32 b(if)g(w)m(e)i(ha)m(v)m(e)g(for)e(all)f Fp(i)d Fn(2)h Fp(R)q Fr(,)k Fp(\030)2571 1458 y Fo(i)2599 1443 y Fr(\()p Fp(S)28 b Fn([)23 b Fp(R)q Fr(\))28 b Fn(\024)g Fp(u)3116 1458 y Fo(i)3177 1443 y Fr(then)33 b(ev)m(ery)i(coalition)-120 1615 y Fp(S)-54 1579 y Fo(t)19 1615 y Fr(in)43 b(the)g(sequence)k(de\014ned)e(b)m(y)f(\(3\))f(m)m(ust) h(con)m(tain)f Fp(S)35 b Fn([)30 b Fp(R)44 b Fr(\(b)m(y)h(cross)f (monotonicit)m(y)e(and)h(an)h(ob)m(vious)-120 1785 y(induction)32 b(argumen)m(t\),)g(hence)i(a)e(con)m(tradiction.)-120 2021 y Fh(Step)38 b(3:)43 b Fi(Bounding)34 b Fp(\015)5 b Fr(\()p Fp(\030)g(;)17 b(u)p Fr(\))-120 2257 y(W)-8 b(e)30 b(represen)m(t)h(an)e(ordering)f(of)h Fp(N)38 b Fr(=)28 b Fn(f)p Fr(1)p Fp(;)17 b Fr(2)p Fp(;)g Fr(3)p Fp(;)g(:)g(:)g(:)d(;)j(n)p Fn(g)29 b Fr(b)m(y)h(a)f(p)s(erm)m(utation)f Fp(\033)33 b Fr(of)c Fp(N)10 b Fr(,)30 b(with)f(the)h(in)m (terpretation)-120 2431 y(that)g Fp(\033)t Fr(\(1\))g(is)g(the)h (\014rst)g(in)f(the)h(ordering,)f Fp(\033)t 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Fp(\016)t Fr(\()p Fp(\030)5 b Fr(\))30 b(for)h(the)h(arbitrary)e(pro\014le)g Fp(u)h Fr(\014xed)h(at)f(the)h(b)s(eginning)e(of)g(the)i(pro)s(of.)42 b(If)-120 3674 y Fp(S)33 b Fr(=)28 b Fp(T)14 b Fr(,)31 b(w)m(e)i(ha)m(v)m(e)f Fp(\015)5 b Fr(\()p Fp(\030)g(;)17 b(u)p Fr(\))26 b(=)i(0)j(and)g(the)h(inequalit)m(y)e(follo)m(ws)g(from) g(the)h(nonnegativit)m(y)g(of)g Fp(\016)k Fr(\(a)c(consequence)-120 3844 y(of)38 b(cross)i(monotonicit)m(y\).)59 b(F)-8 b(rom)37 b(no)m(w)j(on,)g(assume)f Fp(S)44 b Fn(6)p Fr(=)38 b Fp(T)52 b Fr(and)39 b(so,)h(b)m(y)f(Step)h(1,)g(w)m(e)f(m)m(ust)g(ha)m (v)m(e)h Fp(S)k Fn(\032)38 b Fp(T)14 b Fr(.)-120 4014 y(W)-8 b(e)32 b(lab)s(el)e(\(arbitrarily\))f(the)j(agen)m(ts)h(in)e Fp(N)g Fn(\000)21 b Fp(T)45 b Fr(as)32 b Fn(f)p Fr(1)p Fp(;)17 b Fr(2)p Fp(;)g(:)g(:)g(:)e(;)i(k)s Fn(g)31 b Fr(and)h(use)h(cross)g(monotonicit)m(y)d(to)h(see)i(that)-120 4184 y Fp(C)7 b Fr(\()p Fp(N)j Fr(\))22 b Fn(\000)h Fp(C)7 b Fr(\()p Fp(T)14 b Fr(\))27 b Fn(\024)h Fp(\030)642 4199 y Fo(N)7 b Fj(\000)p Fo(T)815 4184 y Fr(\()p Fp(N)j Fr(\))33 b(and:)1058 4532 y Fp(C)7 b Fr(\()p Fp(N)j Fr(\))23 b Fn(\000)f Fp(C)7 b Fr(\()p Fp(T)14 b Fr(\))27 b Fn(\024)1818 4424 y Fo(k)1778 4449 y Fe(X)1781 4631 y Fo(i)p Fm(=1)1914 4532 y Fp(\030)1957 4547 y Fo(i)1985 4532 y Fr(\()p Fp(N)33 b Fn(\000)22 b(f)p Fr(1)p Fp(;)17 b(:)g(:)g(:)f(;)h(i)22 b Fn(\000)g Fr(1)p Fn(g)p Fr(\))-120 4874 y(Next)30 b(w)m(e)g(lab)s(el) e(the)i(elemen)m(ts)f(in)g Fp(T)g Fn(\000)15 b Fp(S)36 b Fr(suc)m(h)31 b(that)e Fp(T)g Fn(\000)15 b Fp(S)34 b Fr(=)28 b Fn(f)p Fp(k)18 b Fr(+)d(1)p Fp(;)i(k)i Fr(+)c(2)p Fp(;)i(:)g(:)g(:)f(;)h(k)h Fr(+)d Fp(k)3180 4838 y Fj(0)3203 4874 y Fn(g)p Fr(.)43 b(W)-8 b(e)29 b(apply)g(Step)h(2)-120 5044 y(rep)s(eatedly)-8 b(,)26 b(\014rst)f(to)f Fp(R)747 5059 y Fm(1)815 5044 y Fr(=)j Fp(T)20 b Fn(\000)6 b Fp(S)30 b Fr(to)24 b(pic)m(k)h(an)f(agen)m(t)h(lab)s(eled)e Fp(k)9 b Fr(+)d(1)23 b(in)h Fp(R)2578 5059 y Fm(1)2618 5044 y Fr(,)i(then)f(to)f Fp(R)3070 5059 y Fm(2)3137 5044 y Fr(=)k Fp(T)19 b Fn(\000)6 b Fr(\()p Fp(S)11 b Fn([)6 b(f)p Fp(k)j Fr(+)d(1)p Fn(g)p Fr(\))23 b(to)-120 5214 y(pic)m(k)j(an)g(agen)m(t)g Fp(k)12 b Fr(+)d(2)25 b(in)g Fp(R)867 5229 y Fm(2)907 5214 y Fr(,)j(and)e(so)g(on,)h(with)f(agen)m (t)g Fp(k)12 b Fr(+)d Fp(i)25 b Fr(selected)i(in)e Fp(R)2631 5229 y Fo(i)2688 5214 y Fr(=)i Fp(T)22 b Fn(\000)9 b Fr(\()p Fp(S)14 b Fn([)9 b(f)p Fp(k)j Fr(+)d(1)p Fp(;)17 b(:)g(:)g(:)f(;)h(k)12 b Fr(+)d Fp(i)g Fn(\000)g Fr(1)p Fn(g)p Fr(\))-120 5378 y(suc)m(h)34 b(that)6 5617 y Fp(u)62 5632 y Fo(k)r Fm(+1)222 5617 y Fp(<)27 b(\030)368 5632 y Fo(k)r Fm(+1)501 5617 y Fr(\()p Fp(T)14 b Fr(\))p Fp(;)j(:)g(:)g(:)e (;)i(u)922 5632 y Fo(k)r Fm(+)p Fo(i)1071 5617 y Fp(<)27 b(\030)1217 5632 y Fo(k)r Fm(+)p Fo(i)1338 5617 y Fr(\()p Fp(T)36 b Fn(\000)23 b(f)p Fp(k)i Fr(+)d(1)p Fp(;)17 b(:)g(:)g(:)e(;)i(k)25 b Fr(+)d Fp(i)h Fn(\000)f Fr(1)p Fn(g)p Fr(\))p Fp(;)17 b(:)g(:)g(:)f(;)h(u)2801 5632 y Fo(k)r Fm(+)p Fo(k)2934 5613 y Fa(0)2986 5617 y Fp(<)28 b(\030)3133 5632 y Fo(k)r Fm(+)p Fo(k)3266 5613 y Fa(0)3291 5617 y Fr(\()p Fp(S)g Fn([)22 b(f)p Fp(k)j Fr(+)d Fp(k)3783 5576 y Fj(0)3806 5617 y Fn(g)p Fr(\))1901 5866 y(23)p eop %%Page: 24 24 24 23 bop -120 197 a Fr(Finally)-8 b(,)45 b(w)m(e)h(lab)s(el)c (\(arbitrarily\))g(the)j(agen)m(ts)g(in)f Fp(S)51 b Fr(as)44 b Fn(f)p Fp(k)34 b Fr(+)c Fp(k)2362 161 y Fj(0)2416 197 y Fr(+)g(1)p Fp(;)17 b(:)g(:)g(:)e(;)i(n)p Fn(g)45 b Fr(and)f(rep)s(eatedly)h(use)h(cross)-120 361 y(monotonicit)m(y)31 b(to)h(sho)m(w:)966 705 y Fp(C)7 b Fr(\()p Fp(S)f Fr(\))27 b Fn(\024)1458 597 y Fo(n)1419 622 y Fe(X)1317 807 y Fo(i)p Fm(=)p Fo(k)r Fm(+)p Fo(k)1529 788 y Fa(0)1551 807 y Fm(+1)1657 705 y Fp(\030)1700 720 y Fo(i)1728 705 y Fr(\()p Fp(S)h Fn(\000)23 b(f)p Fp(k)i Fr(+)d Fp(k)2232 664 y Fj(0)2277 705 y Fr(+)g(1)p Fp(;)17 b(:)g(:)g(:)f(;)h(i)22 b Fn(\000)h Fr(1)p Fn(g)p Fr(\))-120 1071 y(Summing)31 b(the)i(ab)s(o)m(v)m(e)g(inequalities)e(yields)919 1415 y Fp(C)7 b Fr(\()p Fp(N)j Fr(\))23 b Fn(\000)g Fp(C)7 b Fr(\()p Fp(T)14 b Fr(\))21 b(+)h Fp(u)1682 1430 y Fo(T)10 b Fj(\000)p Fo(S)1860 1415 y Fr(+)22 b Fp(C)7 b Fr(\()p Fp(S)f Fr(\))28 b Fn(\024)2348 1307 y Fo(n)2310 1332 y Fe(X)2313 1514 y Fo(i)p Fm(=1)2446 1415 y Fp(\030)2489 1431 y Fo(\033)r Fm(\()p Fo(i)p Fm(\))2615 1415 y Fr(\()p Fp(E)6 b Fr(\()p Fp(i;)17 b(\033)t Fr(\)\))-120 1777 y(where)34 b Fp(\033)i Fr(is)c(the)h(ordering)f Fp(\033)t Fr(\()p Fp(i)p Fr(\))c(=)g Fp(i)p Fr(,)33 b Fp(i)27 b Fr(=)h(1)p Fp(;)17 b(:)g(:)g(:)f(;)h(n)p Fr(.)43 b(The)33 b(lefthand)g(term)f(in)f(the)i(inequalit)m(y)f(ab)s(o)m(v)m(e)h(equals) 796 2057 y(\()p Fp(u)890 2072 y Fo(T)966 2057 y Fn(\000)23 b Fp(C)7 b Fr(\()p Fp(T)14 b Fr(\)\))21 b Fn(\000)i Fr(\()p Fp(u)1543 2072 y Fo(S)1615 2057 y Fn(\000)g Fp(C)7 b Fr(\()p Fp(S)f Fr(\)\))22 b(+)g Fp(C)7 b Fr(\()p Fp(N)j Fr(\))28 b(=)f Fp(\015)5 b Fr(\()p Fp(\030)g(;)17 b(u)p Fr(\))k(+)h Fp(C)7 b Fr(\()p Fp(N)j Fr(\))-120 2336 y(hence)34 b(the)f(desired)g(conclusion)f Fp(\015)5 b Fr(\()p Fp(\030)g(;)17 b(u)p Fr(\))26 b Fn(\024)j Fp(\016)t Fr(\()p Fp(\030)5 b Fr(\))31 b(follo)m(ws.)-120 2572 y Fh(Step)38 b(4:)43 b Fr(sup)414 2596 y Fo(u)p Fj(2<)553 2573 y Fd(n)553 2618 y Fc(+)624 2572 y Fp(\015)5 b Fr(\()p Fp(\030)g(;)17 b(u)p Fr(\))27 b(=)g Fp(\016)t Fr(\()p Fp(\030)5 b Fr(\))-120 2831 y(In)33 b(view)g(of)f(Step)h(3,)f(it)g(is)g(enough)h(to)f(sho)m(w) i(that)e(for)g(an)m(y)i(ordering)e Fp(\033)k Fr(of)c Fp(N)10 b Fr(,)33 b(w)m(e)h(ha)m(v)m(e)992 3175 y(sup)971 3254 y Fo(u)p Fj(2<)1110 3231 y Fd(n)1110 3276 y Fc(+)1177 3175 y Fp(\015)5 b Fr(\()p Fp(\030)g(;)17 b(u)p Fr(\))26 b Fn(\025)j Fp(\016)t Fr(\()p Fp(\030)5 b(;)17 b(\033)t Fr(\))26 b(=)2031 3067 y Fo(n)1993 3092 y Fe(X)1996 3274 y Fo(i)p Fm(=1)2129 3175 y Fp(\030)2172 3190 y Fo(i)2200 3175 y Fr(\()p Fp(E)6 b Fr(\()p Fp(i;)17 b(\033)t Fr(\)\))22 b Fn(\000)h Fp(C)7 b Fr(\()p Fp(N)j Fr(\))-120 3546 y(W)-8 b(e)41 b(pic)m(k)g(an)g(arbitrary)f(ordering)g Fp(\033)t Fr(,)j(denoted)f(for)e(simplicit)m(y)e Fp(\033)t Fr(\()p Fp(i)p Fr(\))k(=)g Fp(i)p 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Fr(])382 b(\(16\))-120 5450 y(T)-8 b(o)40 b(see)g(this,)i(sum)d(up)h(the)g(inequalities)e Fp(\030)1521 5465 y Fo(j)1557 5450 y Fr(\()p Fp(S)6 b Fr(\))39 b Fn(\024)i Fp(\030)1899 5465 y Fo(j)1935 5450 y Fr(\()p Fp(S)32 b Fn(\000)c Fp(i)p Fr(\))40 b(and)f(in)m(v)m(ok)m(e)i (the)f(monotonicit)m(y)e(of)h(the)h(cost)-120 5617 y(function.)1901 5866 y(24)p eop %%Page: 25 25 25 24 bop -120 197 a Fr(If)-6 172 y(~)-22 197 y Fp(S)33 b Fr(=)28 b Fn(;)k Fr(w)m(e)h(ha)m(v)m(e)724 416 y Fp(\015)5 b Fr(\()p Fp(\030)g(;)17 b(u)p Fr(\))26 b(=)i Fp(w)s Fr(\()p Fp(N)5 b(;)17 b(u)p Fr(\))26 b Fn(\025)i Fp(u)1653 431 y Fo(N)1742 416 y Fn(\000)23 b Fp(C)7 b Fr(\()p Fp(N)j Fr(\))28 b(=)g Fp(\016)t Fr(\()p Fp(\030)5 b(;)17 b(\033)t Fr(\)\(1)j Fn(\000)j Fp(\017)p Fr(\))f Fn(\000)h Fp(\017C)7 b Fr(\()p Fp(N)j Fr(\))-120 672 y(and)35 b(so)g(w)m(e)h(are)f(done.)51 b(Ho)m(w)m(ev)m(er,)37 b(as)e(w)m(e)h(noted)g(ab)s(o)m(v)m(e,)2070 646 y(~)2053 672 y Fp(S)41 b Fr(need)35 b(not)g(b)s(e)g(empt)m(y)-8 b(.)51 b(W)-8 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Fr(\))28 b(=)f Fp(\016)t Fr(\()p Fp(\030)5 b(;)17 b(\033)t Fr(\))27 b(=)g Fp(\030)1108 5152 y Fo(i)1136 5137 y Fr(\()p Fp(N)10 b Fr(\))23 b(+)f Fp(\016)t Fr(\()p Fp(\030)5 b(;)17 b(\033)1653 5152 y Fj(\000)p Fo(i)1735 5137 y Fr(\))22 b(+)g(\()p Fp(C)7 b Fr(\()p Fp(N)33 b Fn(\000)22 b Fp(i)p Fr(\))h Fn(\000)f Fp(C)7 b Fr(\()p Fp(N)j Fr(\)\))28 b(=)g Fp(\030)2903 5152 y Fo(i)2931 5137 y Fr(\()p Fp(N)10 b Fr(\))22 b(+)g Fp(P)14 b Fr(\()p Fp(N)32 b Fn(\000)23 b Fp(i)p Fr(\))f Fn(\000)h Fp(C)7 b Fr(\()p Fp(N)j Fr(\))-120 5447 y(so)34 b Fp(\030)44 5462 y Fo(i)72 5447 y Fr(\()p Fp(N)10 b Fr(\))30 b(=)g Fp(P)14 b Fr(\()p Fp(N)c Fr(\))22 b Fn(\000)i Fp(P)14 b Fr(\()p Fp(N)33 b Fn(\000)23 b Fp(i)p Fr(\).)47 b(The)35 b(conclusion)e Fp(\030)i Fr(=)29 b Fp(\030)2113 5411 y Fj(\003)2186 5447 y Fr(then)34 b(follo)m(ws)f(from)f(the)i(p)s(oten)m (tial)f(form)m(ula)e(for)-120 5617 y(the)i(Shapley)g(v)-5 b(alue)32 b(\(18\).)p Fb(2)1901 5866 y Fr(26)p eop %%Page: 27 27 27 26 bop -120 197 a Fq(C)161 b(Pro)t(of)54 b(of)g(Prop)t(osition)g(3) -120 529 y Fr(T)-8 b(o)42 b(c)m(hec)m(k)j(that)d(MC)h(satis\014es)f (NPT,)i(\014rst)e(pic)m(k)h(a)f(pro\014le)f Fp(u)h Fr(and)g(an)g(agen)m (t)h Fp(i)f Fr(in)g(an)g(e\016cien)m(t)h(coalition,)-120 693 y Fp(i)28 b Fn(2)g Fp(S)101 657 y Fj(\003)140 693 y Fr(\()p Fp(u)p Fr(\);)k(nonnegativit)m(y)g(of)g Fp(x)1112 657 y Fj(\003)1112 718 y Fo(i)1152 693 y Fr(\()p Fp(u)p Fr(\))g(amoun)m(ts)h(to)f Fp(w)s Fr(\()p Fp(N)5 b(;)17 b(u)p Fr(\))26 b Fn(\024)i Fp(u)2348 708 y Fo(i)2398 693 y Fr(+)22 b Fp(w)s Fr(\()p Fp(N)32 b Fn(\000)23 b Fp(i;)17 b(u)p Fr(\),)32 b(whic)m(h)h(follo)m(ws)e(from)271 963 y Fp(w)s Fr(\()p Fp(N)5 b(;)17 b(u)p Fr(\))26 b(=)h Fp(u)788 978 y Fo(S)835 960 y Fa(\003)871 978 y Fm(\()p Fo(u)p Fm(\))993 963 y Fn(\000)22 b Fp(C)7 b Fr(\()p Fp(S)1273 922 y Fj(\003)1312 963 y Fr(\()p Fp(u)p Fr(\)\))27 b Fn(\024)i Fp(u)1671 978 y Fo(i)1720 963 y Fr(+)22 b Fp(u)1874 978 y Fo(S)1921 960 y Fa(\003)1957 978 y Fm(\()p Fo(u)p Fm(\))p Fj(\000)p Fo(i)2157 963 y Fn(\000)h Fp(C)7 b Fr(\()p Fp(S)2438 922 y Fj(\003)2477 963 y Fr(\()p Fp(u)p Fr(\))22 b Fn(\000)g Fp(i)p Fr(\))28 b Fn(\024)g Fp(u)2990 978 y Fo(i)3040 963 y Fr(+)22 b Fp(w)s Fr(\()p Fp(N)32 b Fn(\000)23 b Fp(i;)17 b(u)p Fr(\))-120 1240 y(VP)44 b(is)e(a)h(straigh)m(tforw)m(ard)g(consequence)k(of)42 b Fp(w)s Fr(\()p Fp(N)e Fn(\000)29 b Fp(i;)17 b(u)p Fr(\))46 b Fn(\024)g Fp(w)s Fr(\()p Fp(N)5 b(;)17 b(u)p Fr(\).)74 b(CS)44 b(follo)m(ws)e(\(as)h(it)g(do)s(es)g(for)g(all)-120 1409 y(CG)d(mec)m(hanisms\))f(from)f(the)i(fact)g(\(due)g(to)f(submo)s (dularit)m(y\))g(that,)i(for)e(a)g(giv)m(en)h Fp(i)p Fr(,)i(if)c Fp(u)3320 1424 y Fo(i)3388 1409 y Fn(\025)i Fp(C)7 b Fr(\()p Fn(f)p Fp(i)p Fn(g)p Fr(\))39 b(then)-120 1584 y Fp(u)-64 1599 y Fo(S)t Fm(+)p Fo(i)87 1584 y Fn(\000)23 b Fp(C)7 b Fr(\()p Fp(S)28 b Fr(+)22 b Fp(i)p Fr(\))28 b Fn(\025)g Fp(u)748 1599 y Fo(S)820 1584 y Fn(\000)23 b Fp(C)7 b Fr(\()p Fp(S)f Fr(\))32 b(for)g(all)f Fp(S)i Fn(\022)28 b Fp(N)43 b Fr(and)33 b(all)d Fp(u)2156 1599 y Fj(\000)p Fo(i)2267 1584 y Fn(2)e(<)2433 1547 y Fo(n)p Fj(\000)p Fm(1)2433 1608 y(+)2570 1584 y Fr(.)-120 1826 y(Con)m(v)m(ersely)-8 b(,)38 b(c)m(ho)s(ose)e(a)f(mec)m(hanism)f Fp(M)46 b Fr(as)35 b(in)g(the)g(premises)h(of)e(the)i(\014rst)g (statemen)m(t.)51 b(As)36 b(in)e(equation)h(\(6\))-120 1996 y(w)m(e)f(kno)m(w)f(that)g Fp(x)p Fr(\()p Fp(u)p Fr(\))f(m)m(ust)h(ha)m(v)m(e)h(the)f(form:)1159 2266 y Fp(x)1214 2281 y Fo(i)1243 2266 y Fr(\()p Fp(u)p Fr(\))27 b(=)g Fp(u)1561 2281 y Fo(i)1589 2266 y Fp(q)1632 2281 y Fo(i)1661 2266 y Fr(\()p Fp(u)p Fr(\))21 b Fn(\000)i Fr(\()p Fp(w)s Fr(\()p Fp(N)5 b(;)17 b(u)p Fr(\))j Fn(\000)j Fp(h)2460 2281 y Fo(i)2488 2266 y Fr(\()p Fp(u)2582 2281 y Fj(\000)p Fo(i)2665 2266 y Fr(\)\))-120 2540 y(Consider)39 b(an)g(arbitrary)e(\()p Fp(N)g Fn(\000)27 b Fp(i)p Fr(\))38 b(pro\014le)h Fp(u)1582 2555 y Fj(\000)p Fo(i)1703 2540 y Fr(and)f(denote)i(b)m(y)f Fp(u)2416 2503 y Fm(0)2494 2540 y Fr(its)f(completion)f(to)h(an)g Fp(N)10 b Fr(-pro\014le)38 b(suc)m(h)-120 2714 y(that)33 b Fp(u)148 2677 y Fm(0)148 2738 y Fo(i)216 2714 y Fr(=)c(0.)46 b(W)-8 b(e)34 b(m)m(ust)f(ha)m(v)m (e)i Fp(x)1135 2729 y Fo(i)1163 2714 y Fr(\()p Fp(u)1257 2677 y Fm(0)1296 2714 y Fr(\))29 b(=)g(0)k(\(b)m(y)h(NPT)h(and)e(VP)-8 b(,)34 b(as)g(noted)f(at)g(the)h(b)s(eginning)e(of)h(Section)g(2\))-120 2878 y(and)g(the)g(ab)s(o)m(v)m(e)g(equation)f(yields)908 3217 y Fp(h)964 3232 y Fo(i)992 3217 y Fr(\()p Fp(u)1086 3232 y Fj(\000)p Fo(i)1169 3217 y Fr(\))27 b(=)h Fp(w)s Fr(\()p Fp(N)5 b(;)17 b(u)1632 3176 y Fm(0)1670 3217 y Fr(\))28 b(=)f Fp(w)s Fr(\()p Fp(N)32 b Fn(\000)23 b Fp(i;)17 b(u)2293 3176 y Fm(0)2332 3217 y Fr(\))27 b(=)h Fp(w)s Fr(\()p Fp(N)k Fn(\000)23 b Fp(i;)17 b(u)p Fr(\))-120 3487 y(for)27 b(an)m(y)h(pro\014le)f Fp(u)f Fr(with)h(\()p Fp(N)22 b Fn(\000)11 b Fp(i)p Fr(\))28 b(pro)5 b(jection)27 b Fp(u)1634 3502 y Fj(\000)p Fo(i)1717 3487 y Fr(.)42 b(Th)m(us,)30 b(the)d(mec)m(hanism)g Fp(M)38 b Fr(tak)m(es)29 b(the)e(same)h(form)e(as)h(MC,)-120 3657 y(except)38 b(for)f(the)g(fact)f(that)h Fp(S)6 b Fr(\()p Fp(u)p Fr(\))36 b(can)h(b)s(e)g(an)m(y)g(e\016cien)m(t)h (coalition)c(\(not)i(necessarily)h(the)h(largest)e(one)h(as)g(in)-120 3827 y(MC\).)32 b(In)g(order)g(to)f(c)m(hec)m(k)j(that)e(the)g(t)m(w)m (o)g(mec)m(hanisms)f(are)h(w)m(elfare)f(equiv)-5 b(alen)m(t,)32 b(w)m(e)h(need)f(only)f(to)h(consider)-120 3997 y(an)j(agen)m(t)g Fp(i)h Fr(in)e Fp(S)534 3960 y Fj(\003)573 3997 y Fr(\()p Fp(u)p Fr(\))24 b Fn(\000)g Fp(S)6 b Fr(\()p Fp(u)p Fr(\))34 b(b)s(ecause)j(b)s(oth)e(mec)m(hanisms)g(giv)m(e)g(the)g(precisely)h (same)f(allo)s(cation)c(to)k(ev)m(ery)-120 4166 y(agen)m(t)29 b(in)g Fp(S)6 b Fr(\()p Fp(u)p Fr(\))28 b(or)h(outside)g(of)f Fp(S)1098 4130 y Fj(\003)1137 4166 y Fr(\()p Fp(u)p Fr(\).)42 b(F)-8 b(or)28 b(an)m(y)i(agen)m(t)f Fp(i)h Fr(in)e Fp(S)2188 4130 y Fj(\003)2227 4166 y Fr(\()p Fp(u)p Fr(\))15 b Fn(\000)g Fp(S)6 b Fr(\()p Fp(u)p Fr(\),)29 b(w)m(e)h(ha)m(v)m(e)g Fp(w)s Fr(\()p Fp(N)5 b(;)17 b(u)p Fr(\))27 b(=)g Fp(w)s Fr(\()p Fp(N)e Fn(\000)15 b Fp(i;)i(u)p Fr(\))-120 4336 y(hence)27 b(she)g(receiv)m(es)h(from)c(the)i(t)m(w)m(o)h(mec)m (hanisms)e(the)i(\()p Fp(q)1996 4351 y Fo(i)2024 4336 y Fp(;)17 b(x)2123 4351 y Fo(i)2151 4336 y Fr(\))26 b(allo)s(cations)d (of)i(\(0)p Fp(;)17 b Fr(0\))25 b(and)h(\(1)p Fp(;)17 b(u)3411 4351 y Fo(i)3438 4336 y Fr(\))26 b(resp)s(ectiv)m(ely)-8 b(,)-120 4501 y(whic)m(h)33 b(are)g(w)m(elfare)f(equiv)-5 b(alen)m(t.)p Fb(2)1901 5866 y Fr(27)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF