ON THE EQUIVALENCE OF ORDINAL BAYESIAN INCENTIVE COMPATIBILITY AND DOMINANT STRATEGY INCENTIVE COMPATIBILITY FOR RANDOM RULES

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1 ON THE EQUIVALENCE OF ORDINAL BAYESIAN INCENTIVE COMPATIBILITY AND DOMINANT STRATEGY INCENTIVE COMPATIBILITY FOR RANDOM RULES Madhuparna Karmokar 1 and Souvk Roy 1 1 Economc Research Unt, Indan Statstcal Insttute, Kolkata September, 2018 Abstract We study random votng mechansms and establsh an equvalence relaton between ordnal Bayesan ncentve compatble (OBIC) random mechansms and domnant strategy ncentve compatble (DSIC) random mechansms. We show that f a random socal choce functon on a doman s lower contour monotonc and locally OBIC wth respect to strct generc prors, then t s also locally domnant strategy ncentve compatble. Strct generc prors are strct subsets of generc prors as defned n Mshra (2016). The Lebesgue measure of the set of such prors s 1, n other words, every pror s almost surely strctly generc. We further show that under OBIC wth strct generc prors, unanmty mples lower contour monotoncty on unrestrcted domans. It follows from our results that almost every (wth probablty 1) OBIC and elementary monotonc (as defned Mshra (2016)) random rule on unrestrcted, sngle-peaked, sngle-crossng, sngle-dpped doman s DSIC. Further, f one defnes approprate noton of localty for mult-dmensonal separable domans, the same holds for such domans when the margnals are unrestrcted, sngle-peaked, sngle-crossng, sngledpped. Thus, our result generalzes the result n Mshra (2016) n two ways: () by consderng random rules, and () by allowng domans that are not connected through adjacent localty. 1. INTRODUCTION In ths paper, we consder two notons of strategc manpulaton, namely, ordnal Bayesan ncentve compatblty (OBIC) and domnant strategy ncentve compatblty (DSIC) and study 1

2 ther equvalence n case of random rules. We say that a votng mechansm s OBIC f for every agent, hs expected outcome probablty vector from truth-tellng frst-order stochastcallydomnates any expected outcome probablty vector obtaned by msreportng. Smlarly, a votng mechansm satsfes DSIC f for every agent, outcome probablty vector from truthtellng frst-order stochastcally-domnates any outcome probablty vector by devatng for every gven collecton of preferences of other agents. d Aspremont and Peleg (1988) ntroduce the noton of OBIC whch s weaker than that of DSIC as they clam that the latter noton s very restrctve. Ths s demonstrated by the classcal Gbbard-Satterthwate theorem (Gbbard (1973) and Satterthwate (1975)) whch states that dctatorshp s the only DSIC and unanmous votng mechansm n the unrestrcted doman. Majumdar and Sen (2004) shows that f the doman of preferences under consderaton s unrestrcted, then for prors that are generc n the set of ndependent belefs, a socal choce functon s OBIC only f t s DSIC. In case of determnstc votng mechansms, the equvalence of OBIC wth generc and ndependent belefs and DSIC for a large class of preference domans (ncludng the unrestrcted doman, the sngle-peaked doman, the sngle-dpped doman, and some sngle-crossng domans) s establshed n Mshra (2016). He shows that f determnstc votng mechansms are OBIC for generc and ndependent prors and also satsfy an addtonal condton called elementary monotoncty, then they are also DSIC. If we consder two preferences where only a par of adjacent alternatves have swtched ther postons, then elementary monotoncty says that the outcome probablty of the alternatve whch after the swap s at a relatvely better poston must not fall. In ths paper we establsh an equvalence relatonshp between OBIC and DSIC n case of random socal choce functons (RSCFs). In case of random votng mechansms, genercty of prors and elementary monotoncty do not guarantee the equvalence between OBIC and DSIC. We show that an RSCF, ϕ that satsfes elementary monotoncty and s OBIC wth generc for ϕ prors s also DSIC for a large class of domans. We consder agents can manpulate only va adjacent preferences and consder rules to prevent such manpulatons. As n the exstng lterature, we use the terms locally OBIC (LOBIC) and locally DSIC (LDSIC) to refer to ths weakened noton of OBIC and DSIC respectvely. Gabrel (2012) and Sato (2013) dentfy domans where local ncentve compatblty s equvalent to ncentve compatblty. Such domans nclude the unrestrcted doman, sngle-peaked, sngle-dpped and many such domans. Thus, for all these domans we establsh an equvalence relatonshp between OBIC and DSIC by drawng an equvalence relatonshp between LOBIC and LDSIC. 2

3 We further show that n case of unrestrcted doman an RSCF, ϕ that s unanmous and OBIC wth generc for ϕ prors also satsfes elementary monotoncty. Hence, under unrestrcted doman, an RSCF, ϕ that s unanmous and OBIC wth generc for ϕ prors s DSIC. Ths result can be vewed as an extenson of Majumdar and Sen (2004). 2. PRELIMINARIES Let A be the set of alternatves and let D be an arbtrary doman. Let G = D, E be a graph over D. Throughout ths paper, we assume that G s arbtrary but fxed. For P, P D such that (P, P) E, we defne δ(p, P) = {(a, b) apb and b Pa} as the set of (ordered) pars of alternatves that change ther relatve orderng from P to P. Defnton 2.1. A random socal choce functon (RSCF) s a map ϕ : D N A. Defnton 2.2. For an agent, a preference P D, a pror µ of, and an RSCF ϕ : D N A, defne nterm expected outcome of agent, denote by ϕ µ (P ), as a probablty dstrbuton over A such that for all a A, ϕ µ (P )(a) = P D µ(p P )ϕ a (P, P ). For P D and a A, we defne U(a, P ) = {b A br a}. For a preference P, by P(k) we denote the k-th rank alternatve, that s P(k) = a f and only f {b A bpa} = k 1. Defnton 2.3. For gven P D and ν, ν A, ν s sad to stochastcally domnate ν, denoted by νr sd ν, f ν(u(a, P )) ν (U(a, P )) for all a A. Defnton 2.4. An RSCF ϕ : D N A s locally domnant strategy ncentve compatble (LDSIC) f for every N, every P, P D such that (P, P ) E, and every P D, ϕ(p, P )R sd ϕ(p, P ), (1) and s domnant strategy ncentve compatble (DSIC) f (1) holds for all P, P D. Defnton 2.5. An RSCF ϕ : D N A s locally ordnal Bayesan ncentve compatble (LOBIC) wth respect to a profle of prors (µ ) N f for every N, every P, P D such that (P, P ) E, we have ϕ µ (P )R sd ϕ µ (P ), (2) 3

4 and s ordnal Bayesan ncentve compatble (OBIC) wth respect to a profle of prors {µ } N f (2) holds for all P, P D. Defnton 2.6. For an RSCF ϕ : D N A, a profle of prors (µ ) N s called generc for ϕ f for all N, all P, P D, and all a A, µ(p P )(ϕ a (P, P ) ϕ a (P, P )) = 0 P mples ϕ a (P, P ) ϕ a (P, P ) = 0 for all P. Note that f µ s generc, then µ s generc for every determnstc SCF f. Defnton 2.7. An RSCF ϕ : D N A s called elementary monotonc f for all N, all P, P D wth (P, P ) E, and all P D, ϕ a (P N ) ϕ a (P, P ) for all a A such that (x, a) δ(p, P ) for some x A. For P, P L(A), we defne P P as the set of alternatves that change ther relatve orderng wth some alternatve from P to P, that s, P P = {x there exsts y A such that [xpy and yp x] or [ypx and xp y]}. Defnton 2.8. An RSCF ϕ : D N A satsfes lower contour monotoncty f for every N, for every P, P D, we have for every P D, ϕ L (P N ) ϕ L (P, P ) for all strct lower contour sets L of P restrcted to P P. 3. RESULTS Theorem 3.1. Let ϕ : D N A satsfy lower contour monotoncty and let (µ ) N be strctly generc for ϕ. Then, ϕ s LOBIC wth respect to (µ ) N mples ϕ s LDSIC. The proof of Theorem 3.1 s relegated to Appendx A. Theorem 3.1 together wth the fact that strctly generc prors have measure 1 yelds the followng corollary. Corollary 3.1. Almost every lower contour monotonc and LOBIC rule s LDSIC. 4

5 Theorem 3.2. Suppose A 3 and ϕ : L(A) n A s LOBIC wth respect to (µ ) N where (µ ) N s strctly generc for ϕ. If ϕ satsfes unanmty, then t satsfes elementary monotoncty. The proof of Theorem 3.2 s relegated to Appendx B. Corollary 3.2. Almost every unanmous and OBIC random rule on the unrestrcted doman s random dctatoral. 4. CONCLUSION In ths paper we have shown that when prors are strctly generc for ϕ, then any RSCF whch s OBIC and satsfes lower contour monotoncty s also LDSIC for a large class of domans ncludng the unrestrcted doman, sngle-peaked doman, sngle-crossng doman, sngle-dpped doman, etc. Also the requrement on prors to be strctly generc for the RSCF s rather weak as such prors occur wth probablty 1. Thus, our result establshes an almost sure equvalence between the OBIC and the DSIC random rules, and thereby they generalze the results n Mshra (2016) and Majumdar and Sen (2004). A. PROOF OF THEOREM 3.1 Defnton A.1. An RSCF ϕ : D N A satsfes block monotoncty f for every N, for every P, P D, where (P, P ) E, we have for every P D, ϕ x (P N ) = ϕ x (P, P ) for all x / P P To prove Theorem 3.1, we frst prove the followng Lemma. Lemma A.1. Let ϕ : D N A be an RSCF and let (µ ) N be generc for ϕ. If ϕ s LOBIC wth respect to (µ ) N, then t must satsfy block monotoncty. Proof. Let ϕ : D N A be an RSCF and let (µ ) N be generc for ϕ. Further, let ϕ be LOBIC wth respect to (µ ) N. We show that ϕ satsfes block monotoncty. Consder an agent N and two preferences P and P be such that P s an (a, b) swap of P. By defnton P (k) = a, P (k + 1) = b and P (k) = b, P (k + 1) = a for some k and P (l) = P (l) for all l / {k, k + 1}. 5

6 Take x A \ {a, b} such that P (k ) = P (k ) = x where k < k. We show that ϕ x (P, P ) = ϕ x (P, P ) for all P D. If k = 1, as P (k ) = P (k ) for all k < k, LOBIC mples that µ(p P )ϕ P (1)(P, P ) µ(p P )ϕ P (1)(P, P ) (3) and µ(p P )ϕ P (1)(P, P ) µ(p P )ϕ P (1)(P, P ) (4) Combnng (3) and (4), we get for all P D µ(p P )ϕ P (1)(P, P ) = µ(p P )ϕ P (1)(P, P ). As (µ ) N s generc for ϕ, ths means ϕ P (1)(P, P ) = ϕ P (1)(P, P ) for all P D. Now suppose the clam s true for all k < k. Note that U(x, P ) = U(x, P ). Applyng LOBIC to the top k alternatves n P and P we get µ(p P )ϕ U(x,P )(P, P ) µ(p P )ϕ U(x,P )(P, P ) and µ(p P )ϕ U(x,P )(P, P ) µ(p P )ϕ U(x,P )(P, P ) Combnng the above nequaltes, we have µ(p P )ϕ U(x,P )(P, P ) = µ(p P )ϕ U(x,P )(P, P ). Usng the nducton hypothess, we have for all k < k, ϕ P (k )(P, P ) = ϕ P (k )(P, P ) for all P D. Hence, we get ϕ x (P, P ) = ϕ x (P, P ) for all P D. Next we show that ϕ {a,b} (P, P ) = ϕ {a,b} (P, P ) for all P D. Applyng LOBIC, we get µ(p P )(ϕ U(b,P )(P, P )) µ(p P )ϕ U(b,P )(P, P )) 6

7 and µ(p P )(ϕ U(a,P )(P, P )) µ(p P )ϕ U(a,P )(P, P )) As U(b, P ) = U(a, P ), we get ϕ {a,b} (P, P ) = ϕ {a,b} (P, P ) for all P D. Fnally, consder an alternatve x A \ {a, b} such that P (k ) = P (k ) = x where k > k + 1. Usng the same argument as n the case when k < k, we obtan ϕ x (P, P ) = ϕ x (P, P ) for all P D. Thus, for every P D, we have ϕ x (P, P ) = ϕ x (P, P ) for all x / {a, b}. Proof of Theorem 3.1. Let ϕ : D N A be an RSCF satsfyng local contour monotoncty and let (µ ) N be generc for ϕ. Further, let ϕ be LOBIC wth respect to (µ ) N. We show that ϕ s also LDSIC. By Lemma A.1, we know that ϕ satsfes block monotoncty. Take N,P D and P, P D such that P s an (a, b)-swap of P, where P (k) = a, P (k + 1) = b and P (k) = b, P (k + 1) = a for some k. Let P (k ) = x for some 1 k A. If k < k or k > k + 1 then U(x, P ) = U(x, P ) = U(x). By block monotoncty we have ϕ U(x)(P, P ) = ϕ U(x) (P, P ). Thus, agent cannot manpulate at (P, P ) va P or at (P, P ) va P. If k = k, then x = a and by block monotoncty ϕ U(P (k 1),P )(P, P ) = ϕ U(P (k 1),P )(P, P ). Moreover, as ϕ satsfes local contour monotoncty, ϕ a (P, P ) ϕ a (P, P ). Ths means ϕ U(a,P )(P, P ) ϕ U(a,P )(P, P ). Thus, cannot manpulate at (P, P ) va P. Now, take k = k + 1. In ths case x = b. By block monotoncty ϕ {a,b} (P, P ) = ϕ {a,b} (P, P ). As P s an (a, b) swap of P, we have U(P (k 1), P ) = U(P (k 1), P ) and U(P (k ), P ) = U(P (k ), P ) = U(P (k 1), P ) {a, b} = U(P (k 1), P ) {a, b}. Hence, ϕ U(x,P )(P, P ) = ϕ U(x,P )(P, P ). Ths means cannot manpulate at (P, P ) va P. Ths completes the proof of the theorem. B. PROOF OF THEOREM 3.2 Frst we show that an RSCF ϕ that s unanmous and LOBIC wth respect to (µ ) N, then ϕ s Pareto effcent. 7

8 Lemma B.1. Let ϕ : L(A) n A be an RSCF and let (µ ) N be generc for ϕ. Then, f ϕ s unanmous and LOBIC wth respect to (µ ) N, then ϕ s Pareto effcent. Proof. Let ϕ : L(A) n A be unanmous and LOBIC wth respect to (µ ) N where (µ ) N s generc for ϕ. We show that ϕ s Pareto effcent. Assume for contradcton that t s not Pareto effcent. For ths, we consder a profle P N L(A) n such that ϕ b (P N ) > 0 and there exsts a A such that ap b for all N. Consder an agent N such that P (k) = a and k = 1. Suppose P (k 1) = x. Consder P L(A) such that P s an (x, a)-swap of P. By Lemma A.1, we know that ϕ satsfes block monotoncty. Ths means ϕ b (P, P ) > 0. Smlarly, by such repeated swaps we reach a preference P for agent such that P (1) = a and ϕ b(p, P ) > 0. Note that as ap b such swaps are ndependent of b. Now we can repeat ths procedure for every agent j such that P j (k) = a and k = 1 and arrve at a profle P N such that P (1) = a for all N and ϕ a(p N ) < 1. Ths contradcts the fact the ϕ s unanmous. Proof of Theorem 3.2. Let ϕ : L(A) n A be unanmous and LOBIC wth respect to (µ ) N where (µ ) N s generc for ϕ. By Lemma B.1, ϕ s Pareto effcent and by Lemma A.1 ϕ satsfes block monotoncty. We show that ϕ satsfes elementary monotoncty. Consder an agent N, a preference profle P D of other agents, and P, P D such that P s an (a, b)-swap of P. Notce that there are some agents n P who prefer a to b and some who prefer b to a. We show that ϕ b (P, P ) = ϕ b ( P, P ). By block monotoncty, ϕ {a,b} (P, P ) = ϕ {a,b} ( P, P ). Assume for contradcton that ϕ b (P, P ) > ϕ b ( P, P ). We carry out the proof n the followng steps. P 1 N a b a x.. b..... b a x... x x b a Table 1 Step 1: We modfy the profle (P, P ) to brng one of the alternatves not n {a, b} just below 8

9 {a, b} for all agents. Let x / {a, b} be some alternatve. If ap j x and bp j x for some j N, then we can do a seres of swaps to lft x up such that t s just below b f ap j b or just below a f bp j a (note that none of these swaps wll nvolve b). By block monotoncty, the probablty that the outcome at the new profle s b s unchanged. Usng a smlar argument, f bp j x and xp j a for some j M, then we can come to a preference where x s just below a mantanng the probablty of the outcome beng b. Now, consder j N, such that xp j b. If x and b are not consecutve n P j, then agan we can do a seres of swaps to come to a preference such that x s just above b (note agan that none of these swaps wll nvolve b). Let us denote ths new profle by PN 1. By block monotoncty, ϕ b (PN 1 ) = ϕ b(p, P ). So, we have reached a profle PN 1, where for every j N, ether x s just above b n Pj 1 or [x s just below b f apj 1 b and x s just below a f bp1 j a]. Table 1 shows the profle P 1. Consder P 2 j D j such that P 2 j s (x, b)-swap of P 1 j for every j N such that x s just above b n P 1 j (Columns 3 and 4 n Table 1) and P2 j D j such that P 2 j = P 1 j for others. By block monotoncty ϕ {x,b} (PN 2 ) = ϕ {x,b}(pn 1 ). But b s preferred to x by all the agents, and hence, Pareto effcency mples that ϕ b (PN 2 ) = ϕ {x,b}(pn 1 ). For every j belongng to Column 4 n Table 1, we then do a sequence of swaps to get x just below a. Denote ths new preference by P 3 j and ths new profle by P 3 N. By block monotoncty, ϕ b(p 3 N ) = ϕ b(p 2 N ). Now, consder the (a, b)-swap of P 3 and denote ths preference as P 3. By an analogous argument ϕ b ( P 3, P3 ) = ϕ {x,b}( P, P ). Ths means, ϕ b ( P 3, P3 ) < ϕ b(p 3, P3 ) The two profles ( P 3, P3 ) and (P3, P3 ) are shown n Table 2. P 3 P 3 P 3 P 3 a b... a... b b... a... b a x a... b... x a... b.... x... x.... x... x... Table 2 Step 2: In ths step, we modfy the profle ( P 3, P3 ) n a partcular way. Frst, we look at an 9

10 agent j N, such that ap 3 j b and bp3 j x. Consder the profle P4 N such that P4 j s (b, x)-swap of P 3 j for each of these agents and P 4 j = P 3 j otherwse. The new profle s shown n Table 3. By block monotoncty, ϕ {b,x} (PN 4 ) = ϕ {b,x}(pn 3 ). But snce a s ranked hgher than x for all the agents, Pareto effcency mples ϕ b (P 4 N ) = ϕ {b,x}(p 3 N ). Agent Other agents a b... a... x b a... x.... x... b... Table 3: New profle n Step 2 (PN 4 ) Step 3: In ths step, we modfy the profle n Table 3 further. In partcular, we lft x just above a. For agent and for all j = such that x s just below a, ths can be done by a (a, x)-swap. For all other agents, ths requres a seres of swaps whch can be done by not nvolvng b. The new profle denoted by PN 5 s shown n Table 4. Snce none of the swaps nvolve b, block monotoncty mples that ϕ b (PN 5 ) = ϕ b(pn 4 ). Agent Other agents x b... x... a b x... a.... a... b... Table 4: New profle n Step 3 (PN 5 ) Step 4: In ths step, we modfy the profle n P 5 N by changng only agent s preference. We do ths by dong an (a, b)-swap of the preference of agent n the profle shown n Table 4. The new profle denoted by PN 6 s shown n Table 5. By block monotoncty, ϕ {a,b}(pn 6 ) = ϕ {a,b}(pn 5 ). But x 10

11 s better than a for all agents, and hence, Pareto effcency mples that ϕ b (P 6 N ) = ϕ {a,b}(p 5 N ). Agent Other agents x b... x... b a x... a.... a... b... Table 5: New profle n Step 4 (PN 6 ) Step 5: In ths step, we modfy the profle PN 6 by changng the preferences of those agents who prefer x to a and a to b(the thrd column of agents n Table 5). We perform a seres of swaps to brng x just one poston above b. The new profle PN 7 s shown n Table 6. By block monotoncty, ϕ b (PN 7 ) = ϕ b(pn 6 ). Agent Other agents x b... a... b a x... x.... a... b... Table 6: New profle n Step 5 (PN 7 ) Step 6: Now, we perform an (x, b)-swap of preferences of those agents who rank x just above b n the profle n Step 5 ths wll be agent and agents n the thrd column n Table 6. The new profle PN 8 s shown n Table 7. By block monotoncty, ϕ {x,b}(pn 8 ) = ϕ {x,b}(pn 7 ). But b s preferred to x for all the agents. Hence, Pareto effcency mples ϕ b (P 8 N ) = ϕ {x,b}(p 7 N ). Step 7: Fnally, we perform a (x, a)-swap for the preferences of all agents n the profle n Step 6 who rank x just above a ths wll nclude agent and agents n the second column of Table 7. The new profle denoted by P 9 N s shown n Table 8. By block monotoncty, ϕ b(p 9 N ) = ϕ b(p 8 N ). But 11

12 Agent Other agents b b... a... x a x... b.... a... x... Table 7: New profle n Step 6 (PN 8 ) the profle shown n Table 8 s exactly the profle ( P 3, P3 )(see Table 2) and we had assumed that ϕ b ( P 3, P3 ) < ϕ b(p 3, P3 ). So we have ϕ b( P 3, P3 ) < ϕ b(p 3, P3 ) ϕ b(pn 9 ) = ϕ b( P 3, P3 ). Ths s a contradcton. Hence, ϕ satsfes elementary monotoncty. Agent Other agents b b... a... a x a... b.... x... x... Table 8: New profle n Step 7 (P 9 N ) Now, applyng Theorem 3.1, we can say that ϕ LDSIC. REFERENCES D ASPREMONT, C. AND B. PELEG (1988): Ordnal Bayesan ncentve compatble representatons of commttees, Socal Choce and Welfare, 5, GABRIEL, C. (2012): When Are Local Incentve Constrants Suffcent? Econometrca, 80, GIBBARD, A. (1973): Manpulaton of Votng Schemes: A General Result, Econometrca, 41,

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