Sufficiency in Blackwell s theorem

Transcription

1 Mathematcal Socal Sceces 46 (23) Suffcecy Blacwell s theorem Agesza Belsa-Kwapsz* Departmet of Agrcultural Ecoomcs ad Ecoomcs, Motaa State Uversty, Bozema, MT 59717, USA Receved 1 November 22; accepted 1 Jauary 23 Abstract If oe formato servce s more formatve tha the other, the ay aget values t more. I fact, the opposte mplcato (suffcecy) s also true, ad ths characterzato s the celebrated Blacwell s theorem. We preset here a short proof of suffcecy. Actually, we show a stroger result: t s eough to chec whether the value of the servce s greater tha the value of the other for ay fxed postve a pror probabltes ad a set of termal actos that has oly two elemets. 23 Elsever Scece B.V. All rghts reserved. Keywords: Blacwell; Iformato servce; Iformato theory JEL classfcato: D8 1. Itroducto A mportat problem the theory of formato s how to compare two formato servces. Ths ca be doe ether by seeg a drect relato betwee the servces whereby oe servce s represeted as a trasformato ( garblg) of the other, or drectly by comparg the expected utlty of the servces to the agets usg them. By the celebrated result of Blacwell, the two approaches are equvalet. That the garbled servce s ecessarly of lesser utlty s both smple ad tutvely clear. Less obvous s the other mplcato (the suffcecy): f all agets value oe servce less, the ths servce s a garblg of the other. Here t s crucal that all agets are used comparso, by whch t s usually meat *Tel.: ; fax: E-mal address: awapsz@motaa.edu (A. Belsa-Kwapsz) /3/$ see frot matter 23 Elsever Scece B.V. All rghts reserved. do:1.116/ S (3)4-

2 22 A. Belsa-Kwapsz / Mathematcal Socal Sceces 46 (23) that agets wth arbtrary sets of termal actos ad utlty fuctos are to be cosdered. The questo arses whether oe ca fd a arrower class of agets to dscrmate betwee the servces. I ths paper, we show that oe ca restrct to agets wth the same arbtrarly fxed a pror belefs whose termal acto set cossts of oly two elemets. Addtoally, our 1 proof s shorter ad less volved tha the orgal Blacwell s wor. 2. Blacwell s theorem Cosder a world whch cossts of agets ad Nature. Tomorrow, Nature wll occur oe of states dexed by, 5 1,...,. Today, agets who do ot ow whch state of Nature wll occur must decde whch acto to tae from the set X of avalable termal actos x [ X. Aget s utlty U depeds o today s choce x ad tomorrow s state, U 5 U (x). Aget s a pror belefs about tomorrow s state are represeted by 51 probabltes p., o p 5 1. I order to mae a decso about ther acto, agets use formato servces (e.g. TV, ewspapers, the Iteret). We assume that there are two alteratve formato servces ad Q represeted by stochastc matrces [ ] ad [Q ]. Here we dex the messages of by 5 1,...,m ad s 3m 3mQ m 51 the probablty of recevg message gve state, o occurrece probabltes for dvdual messages are * * 5 1; lewse for Q. The p 5O p, q 5O pq. (1) We shall assume that p *. ad q *. for 5 1,...,m ad 5 1,...,m Q, sce ay message wth zero occurrece probablty ca be removed from the servce. Upo recevg a message from a formato servce, agets revse ther a pror probablty belefs ad we shall deote by * the a posteror probablty of state gve message receved from message servce. Wth aalogous deftos for Q, weget two stochastc matrces [*] ad [Q*] gve by m3 mq3 p pq * 5 ]], Q* 5 ]]. (2) p* q* Defto 1. Iformato servce s more formatve tha Q (KQ) ff there exsts a o-egatve stochastc matrx R 5 [R ], such that m3mq Q 5 R. (3) That s Q 5 o R, so that R ca be terpreted as the codtoal probablty that whe message s receved from Q message was actually set by ( garbled trasmsso). Aget s expected utlty, the absece of formato servces, s gve by 1 We ote that Cremer ˆ (1982), ossard (1975), ad Marscha ad Myasawa (1968) also preset short proofs of Blacwell s theorem wth somewhat dfferet techques.

3 A. Belsa-Kwapsz / Mathematcal Socal Sceces 46 (23) V5max O pu(x), (4) x[x ad t becomes * * V 5O p max O U(x) (5) x[x the presece of formato servce. The utlty ga V 2V s called the value of formato servce. Note that we assume that there s o cost of usg the servce; partcular, V $V, as ca be see from V 5O p max O U(x) $ max O p O U(x) 5max O pu(x) 5V. x[x x[x x[x The followg fudametal result was show by Blacwell Theorem 1. (Blacwell) For formato servce to be more formatve tha Q t s ecessary ad suffcet that the value of formato servce s greater tha the value of formato servce Q for all sets of termal actos, all utlty fuctos, ad all a pror beleves. I symbols KQ (V for all U, p, X). (6) 3. roof of suffcecy The proof of ecessty ca be foud Hrshlefer ad Rley (1992, p. 193). Suffcecy s harder. Blacwell (1951) wors a more advaced settg, whch should mae the followg elemetary argumet useful. I addto, our formulato (below) gves a weaer suffcecy codto ad thus a stroger result. Ideed, stead of assumg that the codto V holds for all sets of termal actos X ad for all p, we show that t s eough to tae oe strctly postve p ad X whch cossts of oly two elemets. I other words, t suffces to chec the codto V oly for those agets who share some fxed a pror belefs ad have ust two possble termal actos. Theorem 2. Let, Q be two formato servces. Fx a pror probabltes p., 5 1,...,, ad a termal acto set X wth cardalty [X $ 2. If, for all utlty fuctos U, we have V $V Q, the formato servce s more formatve tha Q (KQ). The theorem s a mmedate cosequece of the two followg facts. Below, we use y u U(x)l 5 o yu(x), so that V 5O p max O U(x) 5O p max u U(x)l, (7) x[x x[x V 5O q max O Q U(x) 5O q maxq u U(x)l, (8) Q x[x x[x

4 24 A. Belsa-Kwapsz / Mathematcal Socal Sceces 46 (23) where * ad Q * deote the th ad th rows of * ad Q *, respectvely. Also, Cov stads for the covex hull tae the -dmesoal space R. Fact 1. If, for X ad p s as above ad ay U V $V Q, (9) the for 5 1,...,m Q Q * [ Covh * 51,...,m. (1) roof. Suppose Q * [ Covh * 51,...,m for some. The there s a hyperplae L R separatg Q* from all * s. Thus oe has a vector u 5 [u ] 13 [ R orthogoal to L such that l[u u Q* l. ad u u * l # for all 5 1,...,m. We shall use utlty fuctos U, tag for each oly oe of the two values htu,v, where v 5 [v ] 13 s fxed arbtrarly, say v5 1, ad t. s a parameter. Sce * u tul # ad * u vl $, we have ad * V 5O p maxh u tul, u vl 5O p u vl, (11) * * * ± (12) 5 q* lt 1O q* Q* u vl. V 5O q maxhq u tul,q u vl $ q Q u tul 1O q Q u vl Q ± * Therefore, as t 1`, VQ 1` (sce q. ) ad V stays bouded. Ths volates V # V ad fshes the proof. h Q Fact 2. If, for every 5 1,...,m Q Q * # Covh * 51,...,m, (13) the there are stochastc matrces A ad R such that A* 5 Q* ad R 5 Q. (14) roof. The assumpto meas that, for each 5 1,...,m Q, there are umbers A $, m o A 5 1, such that 51 m Q* 5O A *. (15) 51 Thus A* 5 Q* for A 5 [A ] m 3m. I a pror terms, ths traslates to R 5 Q, where Q R 5 [R ] s gve by R 5 A q */p*. m3mq

5 A. Belsa-Kwapsz / Mathematcal Socal Sceces 46 (23) (Ideed, substtutg defto (2) of * ad Q* to (15) yelds m pq /q 5O A p /p, whch rearrages to Q 5O A q /p.) h m Acowledgemets I would le to tha Rub Sapos whose uque lectures attracted my atteto to formato theory. Refereces Blacwell, D., Comparso of expermets. I: roceedgs of the Secod Bereley Symposum o Mathematcal Statstcs ad robablty. Uversty of Calfora ress, pp Cremer, ˆ J., A smple proof of Blacwell s Comparso of Expermets theorem. Joural of Ecoomc Theory 27, Hrshlefer, J., Rley, J.G., The Aalytcs of Ucertaty ad Iformato. Cambrdge Uversty ress, Cambrdge. Marscha, J., Myasawa, K., Ecoomc comparablty of formato systems. Iteratoal Ecoomc Revew 9, ossard, J.., A ote o formato value theory for expermets defed extesve form. Maagemet Scece 22 (4).