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... individual n + 1 cast ballot B,+l = (x y z). Let B’ = (Bl ,...,sBil,..., B,). This implies, based on (6) thatsvn+ls3(B, B,,,) = vFs3(B)sandsc~+~,~(B’, B,,,) = z13”*~(B’).sSubstitution into (7) givess=-=(8)-=-s(9)szP+~*~(B’, B,+J Bi v”+~*~(B, B,,,) (10)swhich shows that t~+l,~ is manipulable at (B, B,,,). This contradicts thesassumption that the lemma’s necessary part is false.sSuppose the sufficient part ...

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... sense thatsBi E Ri . By substituting ballot Bi’ for Bi individual i can improve thesoutcome of the vote according to his own preferences, i.e.,sv”“(B, )...) Bit,..., B,) aivnm(B1 )..., Ri )...) B,). =-=(2)-=-sThe ballot Bi 3 Ri is the individual’s sincere strategy and the ballotsBi # Ri is a sophisticated strategy.s4 Set valued decision functions can give unambiguous choices if they are coupledswith a lot...

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...t andFarquharson [3]conjectured in passing thatfor thescase of three or more alternatives no nondictatorial strategy-proof votingsprocedure exists. By means of distinctly different techniques Gibbard =-=[7]-=-andsSatterthwaite [ 131 independently formalized and proved this conjecture.2sIn addition Zeckhauser [19] proved a similar existence theorem. Vickerys[IS] and Gibbard [7] speculated about, but did not...

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...atorial and strong alternative-sexcluding voting procedures:sv”*~(B) = f&B) where T = S, and i E I,, (14)svns3(B) = h2’(B) = x, (15)sv”*“(B) = h;3(B) = y, (16)svnS3(B) = h%3(B) = z,sv”‘~(B) = h$3(B),s=-=(17)-=-s(18)s200 MARK ALLEN SATTERHWAITEs?P3(B) = hym3(B), and (19)sP3(B) = @3(B), (20)swhere the notation h>3 represents a strong alternative-excluding votingsprocedure with range U and where B E pmn, S, = ...

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...functions can give unambiguous choices if they are coupledswith a lottery mechanism that randomly selects one alternative from among any setssof tied alternatives. This is the approach which Fishburn =-=[6]-=- and Zeckhauser [19]sadopted. I reject this approach here because I think that the use of decision mechanismsswith a random element would be politically unacceptable to almost all committees.sGibbard ...

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...of Wn+l depends on the fact thatsXn contains seven classes of fully dictatorial and strong alternative-sexcluding voting procedures:sv”*~(B) = f&B) where T = S, and i E I,, (14)svns3(B) = h2’(B) = x, =-=(15)-=-sv”*“(B) = h;3(B) = y, (16)svnS3(B) = h%3(B) = z,sv”‘~(B) = h$3(B),s(17)s(18)s200 MARK ALLEN SATTERHWAITEs?P3(B) = hym3(B), and (19)sP3(B) = @3(B), (20)swhere the notation h>3 represents a strong alte...

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...m indebted to Jean-Marie Blin, Richard Day, Theodore Groves, Rubin Saposnik,sMaria Schmundt, Hugo Sonnenschein, and an anonymous referee for their help in thesdevelopment of this paper.s1 Farquharson =-=[4]-=- introduced the terms sophisticated strategy and sincere strategy.s187sCopyright 0 1975 by Academic Press, Inc.sAU rights of reproduction in any form reserved.s188 MARK ALLEN SATTERTHWAITEsthat under ...

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...11seach commented on different aspects of the manipulability of non-sdictatorial voting procedures.sThis paper has six sections. In Section 2 I formulate the problem andsz In my doctoral dissertation =-=[13]-=- I stated Theorem 1 (existence of a strategy proofsvoting procedure) and proved it using the constructive proof presented in Section 3sof this paper. This work was done independently of Gibbard. Subse...

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...ividual n + 1 cast ballot B,+l = (x y z). Let B’ = (Bl ,...,sBil,..., B,). This implies, based on (6) thatsvn+ls3(B, B,,,) = vFs3(B)sandsc~+~,~(B’, B,,,) = z13”*~(B’).sSubstitution into (7) givess(8)s=-=(9)-=-szP+~*~(B’, B,+J Bi v”+~*~(B, B,,,) (10)swhich shows that t~+l,~ is manipulable at (B, B,,,). This contradicts thesassumption that the lemma’s necessary part is false.sSuppose the sufficient part is f...

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...+~*~ is not strategy proof for somesindividual i E I, . This implies that a ballot set (B, B,+l) = (Bl ,..., Bi ,..,,s4 3 Bn,,) E P:+’ and ballot B,’ exist such thatsv”+~*~(B’, B,,,) &v”+~*~(B, B,,,) =-=(11)-=-swhere (B’, B,,,) = (B, ,..., Bi’y..., B, , B,+l). Assume without loss ofsgenerality that Bnfl = (X y z). Equations (8) and (9) hold and thereforesvF*” may be substituted for zP+~*~ :sv;*~(B’) &vns3(B...

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...itically unacceptable to almost all committees.sGibbard [7] argued in detail in favor of this paper’s approach.s5 I have adapted this definition of strategy-proofness from Schmeidler andsSonnenschein =-=[14]-=-. My earlier definition in [13] is equivalent, but more awkward tosuse in proofs.sSTRATEGY-PROOFNESS AND ARROW'S CONDITIONS 191sThe second interpretation relates to the theory of games. If a votingspr...

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...ial strategy-proof votingsprocedure exists. By means of distinctly different techniques Gibbard [7]andsSatterthwaite [ 131 independently formalized and proved this conjecture.2sIn addition Zeckhauser =-=[19]-=- proved a similar existence theorem. Vickerys[IS] and Gibbard [7] speculated about, but did not definitively establishsthe relationship between strategy-proofness and Arrow’s four require-sments. Fina...

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...al and strong alternative-sexcluding voting procedures:sv”*~(B) = f&B) where T = S, and i E I,, (14)svns3(B) = h2’(B) = x, (15)sv”*“(B) = h;3(B) = y, (16)svnS3(B) = h%3(B) = z,sv”‘~(B) = h$3(B),s(17)s=-=(18)-=-s200 MARK ALLEN SATTERHWAITEs?P3(B) = hym3(B), and (19)sP3(B) = @3(B), (20)swhere the notation h>3 represents a strong alternative-excluding votingsprocedure with range U and where B E pmn, S, = (x, y...

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...social ordering Al, alternative x cannot have movedsdown in relation to its position within the original social ordering A, .sThe reasonableness of the ND, CS, and NNR conditions is obvious.sFishburn =-=[5]-=- and Plott [12] contain excellent discussion of the reasonable-sness of rationality and IIA. Conditions CS, NNR, and IIA, as Arrows[l, pp. 971 has noted, imply the condition of Pareto optimality.sPare...

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...). Let B’ = (Bl ,...,sBil,..., B,). This implies, based on (6) thatsvn+ls3(B, B,,,) = vFs3(B)sandsc~+~,~(B’, B,,,) = z13”*~(B’).sSubstitution into (7) givess(8)s(9)szP+~*~(B’, B,+J Bi v”+~*~(B, B,,,) =-=(10)-=-swhich shows that t~+l,~ is manipulable at (B, B,,,). This contradicts thesassumption that the lemma’s necessary part is false.sSuppose the sufficient part is false. Therefore a v”*ls3 with its set of...

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...g Al, alternative x cannot have movedsdown in relation to its position within the original social ordering A, .sThe reasonableness of the ND, CS, and NNR conditions is obvious.sFishburn [5] and Plott =-=[12]-=- contain excellent discussion of the reasonable-sness of rationality and IIA. Conditions CS, NNR, and IIA, as Arrows[l, pp. 971 has noted, imply the condition of Pareto optimality.sPareto Optimality (...

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... thatsXn contains seven classes of fully dictatorial and strong alternative-sexcluding voting procedures:sv”*~(B) = f&B) where T = S, and i E I,, (14)svns3(B) = h2’(B) = x, (15)sv”*“(B) = h;3(B) = y, =-=(16)-=-svnS3(B) = h%3(B) = z,sv”‘~(B) = h$3(B),s(17)s(18)s200 MARK ALLEN SATTERHWAITEs?P3(B) = hym3(B), and (19)sP3(B) = @3(B), (20)swhere the notation h>3 represents a strong alternative-excluding votingspr...