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Outcomes of admissible Nash equilibria and sophisticated voting when decisions are based on pairwise comparisons
Mathematical Social Sciences 2 (1981) 39-54 North-Holland
Publishing Company
OUTCOMES OF ADMISSIBLE NASH EQUILIBRIA AND SOPHISTICATED VOTING WHEN DECISIONS ARE BASED ON PAIRWISE COMPARISONS Prasanta K. PATTANAIK Departnrent of Economics, Uwivetsit_vof Birrningharn, Birntingharn Bi5 2%T. 0. A’.
Manimay SENGUPTA Department of Economics, University of New South CCb/es,Kensington. A’..% 11’., 1 ustralia 20.33 Communicated by K.H. Kim Received 9 October 1980 Revised 20 November 1980
Given the absence of non-trivial decision rules which are strategy-proof in the GibbardSatterthwaite sense, Peleg and Sen have suggested that one might search for decision rules under which there will be always some Nash equilibrium yielding the same outcome as would be yielded by honest voting. This paper shows that most decision rules based on pairwise comparison will fail to satisfy the requirement of Peleg and Sen if one imposes the additional condirion that the relevant Nash equilibrium
should consist of undominated
voting strategies only (a mosl
reasonable condition in the absence - cooperation). it is also shown that under most decision rules based on pairwise comparisons, it will not be possible either to ensure that the outcome of honest voting will be always an outcome of sophisticated voting (in Farquharson’s sense) or to ensure that the outcome of honest voting will be always a core outcome. Key words: Nash equilibrium; Dominated strategy; Sophisticated \ oting; Core.
1. Introduction Since it is impossible to find a non&Gal decision rule under which every possible profile of sincere individual orderings will always constitute a I%ashequilibrium,’ Peleg (1978) and A.K. Sen (in private communication) have suggested that one might search for reasonable decision procedures under which for every profile of sincere preferences there would be at least one equilibrium yielding the same outcome as the sincere preferences. 2 This attractive property (which can be formulated in terms of any notion of equilibrium) has been called ‘exact consistency’ by Peleg. Considering the voting process as a non-cooperative game and using Nash equilibrium as the relevant concept, it is indeed possible to show * See Gibbard (1973, 1977). Satterthwaite (1975). Barber3 (1977a. b). Pattanaik (197%
Pawner and
Wesley (1977), Schmeidler and Sonneschein tforthcoming) and Sengupta ( 1979) among others. * See also Dutta and Pattanaik (1978).
01654896/81/0000-0000/$02.75
0 1981 North-Holland
P. K. Partanaik, M. Sengupta / Admissible Naslt equilibria
that most decision rules will be exactly consistent. HoNever, it is well known that there can be a plethora of Nash equilibria, some of INhich may lack all intuitive ibility in predicting the outcome? So there exists the need for going beyond the requirement of a Nash equilibrium yielding the same outcome as the profile of incere preferences: one would presumably want the equilibrium to be intuitively plausible. A minimal test of such intuitive plausibility would be to require that the Nash equilibrium be composed of undominated voting strategies only.4 One objective of this paper is to show that under most decision rules based on pairwise comparisons, it will be often impossible to find a Nash equlibrium which yields the me outcoms as the profile of sincere preferences and in which each voting strategy 1s undominated. Repeated applications of the principle of eliminating dominated voting strategies one to the concept of sophisticated voting first introduced by Farquharson ). Within this framework we show that under every decision rule belonging to the class referred to earlier, it will be often impossible to realise the outcome of sincere preferencesas an outcome of sophisticated voting. is well known, the requirement that the strategies be undominated loses its alppeal in the co-operative context where non-trivial coalitions are permitted. 5 Within the co-operative framework we show that under no decision rule belonging to evat a wider class than the one consideredin the non-cooperative context, will it be !Ale to ensure that the outcome of sincere preferences would necessarily figure among the core outcomes.
1. Notation and some preliminary definitions Let Y be the set of alternatives (1q ~3) and let :V= 2 y- {@}; the elements of cc/ ate calLed issues. N= { 1,2, . . . ,n) is the set of individuals, assumed to be finite (a non-empty subset of JV is called a coali!ion). ,9 is the set of all asymmetric, connected and transitive binary relations defined over Y. The elements (P~ga*apPi**-*,Pn)g (P{p***pP~q*..p P&)etc. of .9 n will be called situations and will be iindicated by S, s’, etc. Pi, Pi’, etc. have the usual interpretation as the strict prefctrence relations that can be expressed by individual i. The true or sincere strict preference relation of individual i will be indicated by pi (Pi E 9). s = (PI, . . . , P,,) will be calfed the sincere situation. For ali in N and all x,y E Y, x&y iff -Upix. Similarly we have Rj, R,i,etc. TWOsituations s and S'are said to be i-variant (i EN) iff Pi # Pi’, and for all j# i, Pj = Pj. ’ Cf. Fivquhar5on (1%9, ’ ?MWFaryuharwn
(1969,
p_28). p. 28).
’ For m!mpie, in the well known gameof Prisoner’s Dilemma, if we change the rules of the game and
permit ca-qeration, then the simultaneous adoption of dominated strategies by the two players would make bath af them better off compared to the situation where both of them adopt undominated kc. !&CCtuce and Haiffa (1957, pp. 94-97).
P. K. Pattauaik,
M. Sengupta
i Adnrissible
Nash equilibria
41
A group decision function (GDF) is a function f: +“x.IQ --)Y such that for all (S&E .4W’r/, f(s,A)eA. For all SE 4” and all x,y~ Y, XPY iff sf v and .v= f(s, {x,y)). (Similarly P’ corresponds to s’, P” corresponds to s”i and so on.) Clear!), P is asymmetric and connected. In this paper we shall focus on an important class of GDFs, namely GDFs based on pairwise comparisons. Definition 1. A DGF f is based on pairwise comparisons
(or, equivalently, satisfies the property PC) iff: (i) for all (s,A) E .r’“x !+ and for all XE Y, if XEA and xP_vfor all y E ( Y- {x}), then f(s, A ) = x; and (ii) for all A E .+ and for all s, s’ E I) “, if P and P’ coincide over A, then f(s, A ) = f(s’, A ). The above definition of GDFs based on pairwise comparisons does not impose any restriction on f(s,A) when there is no ‘best’ alternative in A, i.e., when there is no P-maximal element in A. Judged in terms of the method in which such ‘deadlocked’ situations are handled, most GDFs based on pairwise comparisons come in one of the following two categories which we call Type 1 and Type II GDFs based on pairwise comparisons. Under Type I GDFs based o:i pairwise comparisons, a prespecified alternative (‘status quo’) in the issue gets chosen if there is no ‘best’ alternative. Formally f is a Type I GDF based on pairwise comparisons iff f is a GDF which, in addition to PC, satisfies the following property: for all A E .@,there exists a EA such that for all SE 3’“, if A has no P-maxima1 element then f(s,A) = a. Procedures (j?) and (y) of Black (1958, pp. 22-24) and the GDFs discussed by Sengupta (1978) are of this type. Much more commonly used are what we call Type II GDFs based on pairwise comparisons. Under these decision rules the alternatives in the issues are taken up in a predetermined sequence (the sequence in the ‘agenda’). The first alternative in the sequence is compared with the second and the winner is carried over to be compared with the third and so on until the end of the sequence is reached. An alternative defeated at an earlier stage in the sequence is never taken up again at a later siage. For example, if the issue is {x,y, z, w}, the sequence may be (t,y, .Y,w). Then if we have (xPy & yPz & tpw & wPx), the final winner will be w. On the other hand, if we had (xP’y &yP’z & zP’x& xP’w), x :Nill be the final winner. Thus even though both cases are characterised by the absencle of a best alternative in A, the chosen alte.:native is different in the two cases. More formally, f is a Type 11GDF based 0~ pairwise comparisons iff, in addition to PC, f satisfies the following property: for all A E ;‘Y,there exists a one-to-one function (z from the set of numbers { 1.2, . . . ,A ) to A such that for all s E .4 “, f(s, A) = as*“i where 4’ = a(1) and for all integers k (/A( -.-I Ikl I), if a(k+ l)Pa”, then ak+i=o(k+ 1) and if a”Pa(k+ 1). then (f+l :c aA, Procedure (Yof Black”(l958, p. 21) is of this type. Though the two types of GDFs based on pairwise comparisons that we have discussed above are structurally very different, they share a propert: which is important from our point of view. l
42
P. K. Pattanaik, M. Sengupta / Admissible Nash @quilibria
1. Let J be a Typ I or Type II GDF based on pairwise comparisons. 7Reaf sxtt&$iesthefollowing property: (0) for all A E ‘!I,iy 1A I= 3, then there exists a (EA such that for all s E .dn, if there is no P-maximal el -went in A, then f (s, A ) = a. pe I GDF based on pairwise comparisons, then it is obvious that f Consider the case where f is a Type II GDF based on pairwise t A = {x~YJ). There exists a one-to-one function o from { 1,2,3} to A such that for all SE :-i”“, j’(s,A) = alAfwhere alAlis specified as in the definition of Type II GQFs based on pabwise comparisons. Without loss of generality assume that a( 1) =x; n(2) = y; and (x(3)= t., Consider any s E :y n such that A has no Pimal element. Then either (xPy&yPz & Zpx) or (yP’& XPZ& Zpu). In either it can be e&ly checked that f(s, A) = aIAl= z. Hence f satisfies (D). In the following sections, we mainly consider GDFs which satisfy property (D) in ition to PC. #Miaitbn 2. A GDF satisfies PC’ iff it satisfies D in tiddition to PC. We now define several well-known properties of a GDF. grlit&ian 3. Let f be the GDF; let x and y be any two distinct alternatives; and let s and s’ te any two situations. (3.1). Limited Independence of Irrelevant Alternatitx (LIIA). If for all i E N, Pi and Pi coincide over {x,y), then P and P’ coincide over (x,y).
(3.21. Limited Monotonicity(LM). If for all i EN, [xPia*xP,!a, for all a E Y] and [uP,,b-aP;b, for all a, be (Y - (x})], then [xPy+xP’yJ. (3.3). Limited Absence of Vetoes(LAV). If I(ie N IxPiyj1z n - 1, then xPy.
(3.4). Limited Relevance of Individuals (LRI). For all k N, there exist i-variant situations so and so0 such that xpy and yPoox. (3.9. LRI’. For ail iE N, if there exist a, b E Y and i-variant s”,#% :S n such that uP@band bP%, then for all distinct c,d E Y, there exist i-variant s*,s**E :pn, such that cP+d and dP+*c l
All these properties are familiar with thlepossible exception of LRI and LRI’. LRI is a very weak and acceptable property originally introduced by Farquharson (1969). What it says is that for every pair of alternatives, every individual’s vote matters sometimes. LRI’ is even weaker than LR’I. What LRI’ requires is that either for every pair of alternatives an individual’s vote matters sometimes, or his vote does not matter for any pair at all. (LAV together with the well known property of pairwise anonymity implies LRI while the familiar property of pairwise neutrality implies LRI’,)
P.K. Puttanaik, M. Sertgtryru /’ Adrnissibie Nash equilibria
43
Let F be the class of all GDFs which satisfy PC’, LIIA, LM, LRI and LAW; lu:tG be the class of all GDFs which satisfy PC’, LIIA, LM, ERI’ and LAV; and let li be the class of all GDFs which satisfy PC, LIIA, LM and LAV. Clearly Fc G c H.
3. Outcomes of admissible Nash equilibria
In this section we assume that no co-operation among individuals is possible. We first introduce the notion of admissible Nash equilibria. Next we prove our main results which show that if one is considering a GDF belonging to For G‘, then it is impossible to ensure that for every possible issue and for every possible sincere preference profile, there will be a Nash equilibrium which would consist of only undominated voting strategies and which would yield the same outcome as the sincere preference profile. Definition 4 (4.1). Given an issue A and given the sincere situation S, a situation s is a Nash equilibrium iff there does not exist k .N and situation s’ such that s’ and s are ivariant and f(s’, A)Pif(S, A). (4.2). Let A be the given issue and let Pi be the sincere preference of k IV. Let Pi, P: E 9. For individual i, Pi weakly dominates P; iff for all i-variant s*, s** E 7”’ Pi dominaies Pj iff P, weakly such that P,*= Pi and P,?* = P:, f(s*,A)Rif(s**,A); dominates Pj and for some i-variant s*,s**E .Y” such that P,*= P, and P,**= PI, f(s*,A)Pif(s**,A); Pi is admissible (or, equivalently, undominated) iff for all r’iE (9, Pi does not dominate Pi. 6 (4.3). Given the issue A and the sincere situation S, a situation s is admissible iif for all ic IV, Pi is admissible. A major problem with the notion 0: Nash equilibrium is that eFc:n most implausible situations can qualify as Nash equilibria. For example, given the issue {X,JJ,z), if for all k IV, XpyPiz, then under every f E H a situation s such that for all iEN, ZPiyPiX, will constitute a Nash equilibrium.’ As a minimal test of the plausibility of Nash equilibria, one should ask whether the preferer?
’ Note that the notions of admissibiiity, weak domination, and domination are relatiw to the piben issue and the sincere preference of the individual under consideration. -r See Farquharson (1969, p. 28) and also Dutta anri Pattanaik (1978).
P. K. Pattanaik, M. Sengupta / Admissible Aiash equilibria
GINS belongingto F, cDnecould ask whether it is possible to find a GDF@ F such that for every possible issue A and every possible siilcere situation 8, there would exist an admissible Nash equilibrium s such that j&b\) =&A). Unfortunately, as shown by tRe following theorem, the answer has to be in the negative. mm 1. Let f E F. Then there exists an issue A,, an individual i, a sincere sitwtion *s,and 4% .9 such that P is admissiblefor i given the issue A and the sincere prcefetvmv&, and for aI1admissiblesituations S, ifPi =:F, then f (s,A )Pif (S,A ).
To prove this theorem we first introduce some additional notation, and derive 1 preliminaryresults. n most of the proofs we shall USC, a three element issue {x,y, z}. Since the GDFs that we consider always satisfy LIIA and PC, without loss of generality any idual k can be regardedas having six possible voting strategies given the issue z): (1) ,&y&; (2) x&z&y; (3) y&z&x; (4) Y~~x&z; (5) z&.x&y; and (6) fl&~x. To economise notation we shall refer to these preference relations or voting stradegiesas (l), (2), . . . , (6) respectively. Given the issue {x,-y,z} and given any 8% 9, x i;s said to be P-best iff (xpy &x&); x is said to be p-medium iff [&!!St x/%) or (#x dkxpy)]; and x is said to be P-worst iff (y#!x & zpx). Similarly, we speak of y or z being P-best or P-worst or &medium. 1st T !.eowm 1,and also in Theorem 2 later, we are concerned with GDFs which satisfy D. Under each one of these GDFs, for every three-element issue A, there mists a firredalternative in A which gets chosen from the issue A whenever there is no P-maximalelement in A. This fixed element of A will be indicated by g(A) in the , following proofs in this section. ~WBIB 1. Let i be any individual. Given any issue and any sincere preference PIE ~9. { 1.t). For all Pi, Pi, Pi”E 9, if Pi weakb dominates Pi, and Pi werrklydominates Py, thw P, weak&y dominates Pi”. (t.2). For all Pie .@, lf Pi is dominated or inadmissible, rhen there exists undominatedPi E .Y’such that Pi’ ,dominates Pi. The proof of this lemma is obvious and is omitted. timma 2, Let t,he GDF belong to F. Let ie N and let A be the issue where IAl = 3. Let aeA. Consider any two i-variant situations s and s’ such that for all b,CE(A - {a)), (bPiC (iff bP#); tied for all bE(A - (a))* aPib+aPjb. If g(A) =a andf(s,A)=a, then f(s’,A)=a. Proof, Let A = {a,~, y ) .. Since f(s, A) = a, either [aPx & aPy] or [(aPx & xPy & yPa) OPtiPa& aPy&yPx)J. If (aPx& aPy) by LILA and LM, (aP’x& aP*Lv)and hence
P.h’. Patranaik, M. Sengupta / .4dmissible Nmh equilibrirl
JS
f(s', A)
=a. Suppose (aPx& xPy & yPa). Then by LllA and LM, (aP’x& xPy). Now either yP’a or aP’y. Given g(A) = a and (aP’x& xP’y), it is clear that irrespective of whether yP’a or aP’y we have f(s’, A) L-a. Similarly, it can be shown that if (xPa & aPy & yPx) we would have f(s’,A) = a. This completes the proof. Lemma 3. Let the GDF f belong to F. Let k N and let A = (x, y, 2) be the issue. Let pi and Pi be any two elements of .9. Then for individual i : (3.1). Pi is undominated. (3.2). If g(A ) is Pi-best in A, then Pi is undominated on/y if g(A ) is &best in A. (3.3). If g(A) is Pi-Worst in A, then Pi is undominated only
ifg(A
) is &worst
in A.
(3.4). If g(A) is Pi-mediutnin A, then Pi is the om’y undomkiated strategy. Proof. Without loss of generality assume XPiyPiZ. We first show that for individual if g(A) =x,
then (1) weakly dominates (3) and (4) and (2) weakly dominates (5) and (6);
if
then (1) weakly dominates (2) and (5) and (4) weakly (a) dominates (3) and (6);
g(/l ) =
2,
and if g(A) = y,
then (1) weakly dominates (2>, (3). (4), (5) and (6).
Proof of (a). To prove (a) it is clearly sufficient to show that for any two i-variant situations, s, s’ E 9”, f(s, A)&f(s’, A) in all the following cases: (i) [g(A)=x] & [Pi=(l)] & [PI ~(3) or (4)]; (ii) [g(A) =x] & [Pi = (2)] & [Pl = (5) or (Q]; (iii) (g(A)=z] & [Pi=(l)] & [P;=(2) or (5)j; (iv) [g(A)=21 & [Pi=(4)] & [P;=(3) or (6)]; and (V) [g(A)=y] & [Pi=Pi=(l)] & [P:*(l)]. Consider any two i-variant situations, S,S’E . Y)? We have altogether four possibilities: [xPy & xPz], or [ yPz & yPx], or [ZPX& zPy] or [(xP_v& yP: & $5) or (yPx & XPZ & zPy)] . Suppose (XPY& xPz). Then f(s, A) =x. Given that s is P,-best, it is
P. K. Pattanaik, hi. Sengupta / Adtnissible Nash equilibria
(@!!& zP_v). Then f(s,, A ) = z. Given LIN and LM, it can be easily ait in cases(i) through (v), we have (EP’x& zP’y) and hence f(s’, A) = 2.
Ch Hence f(s, A&f@‘, A). NOW ISUP~H
[(~Py&yPz& z??x)or (yPx&xPz dkzPy)). Consider first cases (ii)
and @I). Sincef(s’,A)=x in these cases, whatever f(s’,A) may be, it is clear that f(s,A)&/@‘,,4). Now consider cases (iii) and (iv). Since in cases (iii) and (iv) j@, A) = Z, by Lemma 2, we must have@‘, A) = z and hence_& A)WJ(s’, A). Finally consider case (v), We know that (yPx or zPx). In case (v), if yPx then by LIIA ar,d ‘x; and if ZPXthen again by LIIA and LM, zP’x. In both these cases x is not Hence given that g(A) =y, we cannot havef(s”,A) =x. Thus in case (v), we have,/& A) =y and/&‘, A)+x, which impliesf(s,A)~&s’, A). We now show that irrespective of the values of g(A), for individual i:
(I) is not weakly dominated by any other strategy; (2) is not weakly dominated by (3) or (4) or (5) or (6);
(In
and
(4) is not weakly dominated by (2) or (3) or (5) or (6). B-f
a~?t/p). We first shaw that for individual i, (2) is not weakly dominated by (3) irrespectiveof the value of g(A). Note that over (x,y}, (2) coincides with fii but (3) does not. By LRI, LIIA and LM there exist i-variant s”,s% .P such that (yPfx&yP%&x.y&xPmy). Construct s and S’ such that for all j+ i, ((P’= fj) & (xPjz & yPjz) & (xPjy iff XP$) & (yPjX iff YeX)y and [(Pi = (3), i.e. yPiZPiX)& (Pi’= (2), i.e. xP~zP;~)]. By LAV, we have (xPz& yPz&xP’z &yP’z). Comparing s with so, and s’ with sou, we have (JP” & xP~). Hence f(s, A) = y and f(s’, A) =x. Since XBiy, it is clear that (3) does not weakly dominate (2). The proof for all other cases can be completed by following exactly the same method as adopted in the preceding paragraph. Combining (a) and !j?) we have: (I) is undominated in all cases; If g(A) =x, If g(A) = z, If g(A) =y,
then (3), (4), (5) and (6) are dominated; then (2), 43), (5) and (6) are dominated; then al1 strategies excepting (1) are dominated.
This compIetes the proof’ of the lemma. We can now prove Theorem I
Let A = (my, z} be the given issue throughout this proof. Since ftz F and IAI= 3, g&4) is defined. Without loss of generality, assume g(,Q) =x. sing Arrow’si (1964, p. 98) berminology, a coalition\ t’ is said to be decisive for (meA against trE A 8dt”f (a# b) and for all s E :P such that afib for all i E L’, and bPia
for all k (N- L’), we have afb. We write aD,.,b to indicate that L’ is decisive for ,a against b. Let Y be the set of all coalitions which are decisive for some ae A against some kA. Let LE I/ be such that for all L’E Y, IL’lzlLI. By LAV, j~?.,(>l. Let h,kEL.
There are six possibilities: (i) xDLy; (ii) yDl. z; (iii) zD/_x; (iv) yD,_x; (v) xDl.z; and (vi) zDLy. I. Suppose xDLy. Let t be a smallest subset of (N - L) such that yDl-, iAI z. Since yDqN_L,uIkIz (by LIIA and the fact that L is the smallest coalition in _I) and since -yD,z (by LAV), it is clear that such t must exist and must be non-empty. Let I E L‘. Consider a sincere situation s and any situation s, such that:
forall i E [N -- (LU (k ))], for all k$--
(I}),
; ypiZPiX & P, = (3) or (6) SO that YPiZPiX or ZP,_VP,X, zPixP,y
& zPixP,y
y&Prx & zP,yP,x, x&y&z & Pk = (1) or (2) so that xP,yPg or xPgP,y.
Note that since & (N- L), [L - (k}] c [N- (LU {k})]. Since sD,y, we have (xpy dkxfy). By LAV, we have (zpx & zPx). Since L’is the smallest subset of (IV- L) such that (LU (k}) is decisive for y against z, we have yk, and -yPz which implies zPy. Thus we have (xpy & y& & zpx) & (xPy & zPy & zPx). Hence f(!&A ) = s and j&A) =z. Since z&x, it is clear that P, does not dominate P,. By Lemma (1.2). either P, is undominated or P, is dominated by an admissible voting strategy of I. Since by Lemma 3, every voting strategy in (.d - (P,, P,}) is dominated and since P, does not dominate P,, by L,emma (1.2) it follows that P, is admissible. Note that for all k [IV- (LU (k))], Pi is the only admissible voting strategy; for all ie (t - (I)), the set of admissible voting strategies is a subset of {(3), (6)); and for k, the set of admissible voting strategies is a subset of {(l), (2)). Hence it follows that for all admissible S*E .P, if PF = PI then z = f(s, A) = f(s*, A )P,f($ A) =x. II. Now suppose yDLz. Note that by assumption h, kc L. Consider sincere situation s and any situation s such that: yjshz&x & ZPhyPhX; for all ie (L - {h, k}),
for ali &(N-
L),
yPiZ,PiX & P, = (3) or (6) SO that yPiZPiX or ZP,_VP,X; ZPiXPiy & tP,XPi_Y* xpky&z & Pk = (1) or (2) so that xP,yP,z or xPnzPc_V.
It can be easily seen that given that yD‘2 and L is the smallest coalition in I, and given LAV, we have (y& & #x & xpy) & (zPy & zPx & xPy). Hence f($ A) =x and f(s, A) = E. Since z&,x, it is clear that li,, does not dominate PII. Since the set of admissible voting strategies for h is a subset of {&, P,,), by Lemma (1.2) and by the fact that &, does not dominate P,, it follows that PI, is admissible for, h. NOW
P. K. Pattanaik, M. Senguptu / Admissible Nash equilibria
48
f’owing reasoning analogous to that in Step I, it can be shown that for all admissible S+E V, if Ph* = P,,, then z =f(s, A) =f(s*, A)&f($, A) =x. 111.Suppse @Lx. There are two possibilities: either there exists L’ C,(A’- L) such that y&g QICthere does not exist any such coalition L’. 1lJ.t. Sappose ?here exist L’ c (N- L) such tha.tyDrz. Then let t’ be a smallest subset of (k - 1,) such that y&z. Let t E t. Construct s and s such that
for all i E(N - L), for all it: (i- (t}),
zPiXp5ir& tpiXPiy; (XPiyPizor XPizPiy), y&.P*x 6%zP,yP,x. XP#PiZ
&
Since t c (N- L), it is clear that L c (iy - c). Since zDLx, we have (#x & z,Px). By LAV, we have (xpy & xPy). By the fact that L’is the smallest subset of (N-- L) such that yDcz, we have yk and -yPz (which implies zPy). Thus we have (z/k& #y & yk) & (~9%& xPy & zPy). Hence j@, A ) F x and f (s, A) = z. Since &x, p)r does not dominate P,. Now following reasoning analogous to that in Step I, it can be shown that P, is admissible for t and. that for all admissible 5% M”, if PF= P,, then z=f(s,A) =f(s*,A)P,f(s,A) =x. IfL2. Suppose there does not exist L’ c (N- L) stachthatyDLfz.Then construct 8,s E ,fn such that
forall
i E (L - (k i)), ZPixPiy& ZpixpiVE for all i@V- L), xpiypjz & ZPkYPkX. (XPJypJ
YP&X
?
or
xPid’iy)#
&
Since Z.&X, we have (tix& Px). By LAV, we have (xpy & xJ?y). Since L is the smallestcoalition in Y; given LIIA we have -fly which implies y/k Since there does not exist any coalition L’ s (N-L) such that yDLz, we have @y. (For, if -Ipy thenyl?z and then given LlIA and LM, some subset of (N--L) will become decisive for y against z.) Thus we have (xPy&y&& fix) & (xPy& zPy & zPx). Hence $@,A) =x and f(& A) =z. Since zp/Ix, again following reasoning anaiqpus to that in Step I, it cart be shown that Pk is admissible for k and that for all admissible S*E .Y if Pt=Pk, then t=f(s,A)=f(s*,A)&f(S,A)=x. The proofs for the cases where x&z, z&y and yDLx, are respectively -,iralogous 80 the proofs for the cases where xDLy, yDLz and z&x, except for the fact that y and t are to be interchanged everywhere in the proofs. It is possible to relax LRI to LRI’ in Theorem 1, thereby extending the cla.ss of GDFs covered, from F to G. W 2. Let f E G. Then there exists an issue A, an individual i, a sincere s&u&~ 3, and l% :@such thatP is admissiblefob*i given the issueA and the sincere pfleferemv p,, md for all admissiblesituationss, if Pi = p, thenf (s,A )Pif (39A ).
P. I :. ?WtunaiX
, hf. Sen,qtcpta / Mtnissi3le
Msh
equilibria
49
Proof. Since fe G, either fe F or f e(G- F). If f ts F, then Theorem 1 can be
applied to prove the rerluired conclusion. If& (G -F), LRI. Then N can be partitioned into:
then f satisfies LRI’ but not
N, = (& N 1there do not exist distinct x,y E Y and i-variant s”,#% such that f(s”, (x,y}) =x and f(soo,{x,y)) =y},
+I”
and &= {iENifol: all d ist inct x,y E Y, there exist i-variant so,s% .4” such thatf(sq{x,Y})=x andf(d? (X,Y})=Y}= Without loss of generality assume that &= (1, . . ..m} and N, = (m + 1, . . ..I?} where 15 m < n. Since f satisfies PC’, it is clear that for any given issue, the outcome under f is determined only by the preferences of individuals in &, i.e. there exists function f’ : .@“Ix ~!lr-+ Y such that for all ((P,, . . . , P,,,),A) E Yfl x .+,f’[(P,, . . . , P,,,), A] E A and for all SAE.P” if pi= P; for all ~EAJ~,thenfl((PE, . . . , PJ, A] =f(S, A). It is clear that f’ can be considered to be a ‘GDF’ for the smaller ‘society’ Nr, and that since f satisfies LIIA, LM, PC’, LAV and LRI’, f’ satisfies the properties of LIIA, LM, PC’, LAV and LRI with reference to the smaller society A$. Hence Theorem 1 can be applied to f’ to show that there exists an issue A, an individual !E &, (PI, . . . , P,,,) E F1, and PE .4 such that under f’, p is admissible for i given the issue A and the sincere preference .&, and for all admissible Given the (P ,, . . . , P,,) E Y”, if Pi=Pi, then f[(P,,...,P,,,),A]Pi[I(P,,...,P,,,),A]. relation between f’ and f, the required conclusion now follows immediately. The following result follows as an immediate corollary of Theorem 2. Corollary of Theorem 2. Let f E G. Then for some issue A, and some sincere
situation 3, there does not exist arty admissible situakiorl which is a Nash equilibrium and yields the same outcom4v as S. In this section we have mainly concentrated on the notion of admissibility of voting stragegies so as to mark out those Nash equilibria which are obviously implausible; the main impact of Theorem 2 is to show that for every GDF belonging to G, there exists a sincere situation s and “In issue A such that all Nash equilibria which yield the outcome f(gA) must be in this implausible category. It is, however, possible to pursue the notion of admissib;lity to analyse the likely outcomes, independently of the notion of Nash equilibrium. Successive applications of the notion of domination among voting strategie.5 leads one to Farquharson’s (1969) concept of sophisticated voting which we discu.;s in the next section.
4. Outcomes of sophisticated voting under GDFs belonging to F Continuing the assumption, that no co-operation mtrong individuals is possible, in
w
P. K. Paltanaik, M. Sengupta / Admissibk Nash equilibria
n we discussthe problem of ‘sophisticated voting’ * in the context of GDFs g to F. We first define the concept of sophisticated voting strategies. Next hat under every GDF belonging to1F, the outcome of sophisticated voting always determinate for issues with not more than three alternatives, and that for every such GDF in F and for every issue with three distinct elements, one can find a sincere situation such that the determinate outcome of sophisticated voting will be different from the outcome of the sincere situation. Let _f be the given GDF and let s be any given sincere situation. Then for every issue Ca, and for every k N, let Sy($ A) = .Jo and for all positive integers r, let fulfilling the following property: A) be theset of all kS:-’
There does not exist P’E S:‘* ‘(8,A ) such that for all S&E S[- ‘(s,A) x S; ‘(3,A ) x . . . x SL ‘(4 A), if P, = 9 and P; = p’, then f(s’, A)&j(.s, A), and for seine i-?;oriant ~$8 S; ‘(s,A) x 0.0xS;-‘($A), [(Pi=P)&(P;*=P’)&(f(S’,A)Pif(SgA))j. kor all positive integers r, S’($A) =:S#,A) x 5. x Sk@,A). Let S be the smallest E integer such that for all & JV, S;(& A) = S{’ ‘(g,A). Then for all i E Iv, ) is called the set of sophisticuted voting strategies of individual i. We say that the outcome of sophisticated voting is determinate for the issue A iff there exists A suchthat for all SE S@, A), f(s, A) =x. For the motivation of these definitions, the rea kr may refer to Farquharson (1969). l
TICllrorrtrr, 3. Let the GDF f belong to F. (3.1). Fora// SE .fn and for every issue A such that /A 1I 3 the outcome of sophisti-
mted voting is determinate. (3.2). For every issue A such that ;AI= 3, there exists a sincere situation s E .Yn such that f&A) is not an outcome of sophisticated voting, i.e. for all SE Si($,A),
f&4#fWO.
~2, the proof is obvious, given LIL4 and LM. We have only to case where IAl = 3. Let A = (x,y, z}. Since f E F and IAl = 3, there exists an alternative g(A) in A such that for all SE .9”, this alternative emerges as the outcome in the absenceof a P-maximal alternative in A. Without loss of generality assumethat g(A) =x. Let S be any sincere situation. Given LIIA, any individual i can be considered to have six alternative voting strategies numbered as in section 2. Partition N into x is &best}, N2={.ieNjx is Pi-worst}, and N3={idVlx is Pimedium}. Clearly, by Lemma 3, for all SE S’($ A), .Yis Pi-best for all i EN, ; x is Pi-worst
for all id Nl; and Pi = P, for all in NJ. Then given LIIA, one of the fdlowii:g possibilities: must hold: (i) for all s&‘&4), (xP_v&~f%); (ii) for all SE S’@,A), (yfxdk zPx); (iii) for all SE S’(S,A), (yPx& xPz); (iv) for all s E S ‘(s,A ), (zPx & spy). (i) Suppose (i) holds. Then by PC’, for all s E S’($ A), f(s, A j =x. Hence JC is clearly the determinate outcome of sophisticated voting. (ii) Suppose (ii) holds. Then for all SE S’($A), f(s,A)~ (y,2}. If either for all SE S’WQ, .f@,A) =y, or for all SES’(S,A), &A) =z, then the outcome of sophisniaated voting is determinate. Suppose for some s’ E S’(S,.A), f(s’, A ) = y and for some s” E S’(&A ), f(s”, A ) = 7, Then given that for all s E S ‘(3,.4 ), f(s, A ) E ( _Y,,z } and given LIIA, LM and LRI, it is easy to check that for all id (N, U AL) if _vP,z, then yP;z for all P; E $($A) and if ZPiy, thera ZPiy for all P, E Sf(s, A ). Since for all i E N’, x is Pi-best for all Pi E S,!(&A); for all iE Nz, x is Pi-worst for a.11Pi E S,‘(S,4 ); and for all i E NJ, ?#@,A) = {ai) 9 it follows from what we have said above that, for all ie N, $@,A) = I&}. Hence the outcome of sophisticated voting is determinate. (iii) Suppose (iii) holds. Then either for all SE S’(S,A), yPz; or for all SE S’(S, A), zPy; or [for some SES~(S,A), yPz and for some s%S’($A), zP’y). If for all SE S’($A), yPz then given that yPx for all SE Y&A), by PC we have Y =Jjs,A ) for all SE S’(&4). Outcome of sophisticated voting is clearly determinate in this case. Similarly, the outcome of sophisticated voting will be determinate if we have zPy for all s E S’(s, A). Now consider the case where yPz for some s t: S’(S, A) and zP ‘y for some s’ E S ‘(3,A ). We know that (yp~ & XPZ) for all s E S ‘(s,A ). Hence for all SE S ‘(3,14), if y&z, then f&4) =y, and if tpy, thenJ’I$A) =g(A) -xx’. (Clearly, there does not exist SE Sl($A) such thatf($A) = z.) Hence, for all ieArl, zP,y for all P, E S&J); and for all ie IQ, yPiZ for all P, E &,A). Hence for all i E Nl, S,‘(S,A ) = {(2)) and for all i e Nz, S&, A) = {(3)). As we have seen already, for all i E Nj, S,‘(S,A I= (P,) . Hence the outcome of sophisticated voting is determinate. The proof for case (iv) is exactly similar to that for case (iii). (3.2). Let A be any issue with exactly three elements. Let A = {.l;y, z). Without loss of generality assume that g(A)=x (where g(A) is specified as in section 2). Proceeding as in Theorem 1 we can show the following: there exists SE 9” and ic R such that for some &% 9, p is admissible for i [i.e. PE S,‘&4)] and for all SE +“. if s is admissible [i.e. SES’(S,A)], and if Pi=1’, then f(s,A)P,f($A). Throughout the rest of this proof we consider this specific sincere situation S. Note that as in the proof of Theorem (3. I), here also we have four !ogical possibilities: (i) for all sG’@,A), xPy & XPZ; (ii) for all s E S’@, A), yPx & ZPX; (iii) for all s E S ‘(3,A ), yRx & -4-P z; (iv) for all s E S ‘(3,A ), zp.U& xPy.
P. K. Pattanaik, M. Senguptu / Admissible Nash equilibria
$2
If (i) holds then f[S’@,A)] = (x}; if (ii) holds, then f[Sr(&4)] c (y,z); if (iii) holds, then given that g(A)=x, f[Sl($A)) c (x,y}; and lastly, if (iv) holds, then given that &4)=x, f[S’(&A)] G (x,z). Thus in all cases f [SI(.$A)] has no more thun two distinct elements. If f(&4) does not belong tof[Sr(S, A)], then the required conclusion follows directly (since Si(s,A) E Sr@,A)). Suppose f&A) Ef [Sl(s,A)J. Letf(S,A)=a and f [S’($A)1= {a,b} where a&A. By what we have said earlier, for some &N and some PE 9, kS#,A) and for all SE Sr@,A), if Pi= p, then
b==f(s,A)PJ(s,A)=a. Since _f[S1(s,A)]={a,6) it follows that for all F’E ,f, I% Sf(8.A) only if for all s E S,!(g,~1) such that Pi = p’, f(s, A )= 6.Hence it is clear that for all s E S&A), f(s, A) = 6. Hence for all s E S?(&A), f(s, A) = b # f(s, A). in all easesf($ A) is not an outcome of sophisticated voting. 3. The con under GDFs belongingto W In the previous sections we assumed that no co-operation among individuals was aibte. Given this assumption, it was natural to expect that no individual would adopt a dominated preference relation as his voting strategy. If this assumption Red, then the requirement of admissibility loses its appeal. However, the hope of atUning the outcome of the sincere situation through some ‘equilibrium’ does not become much brighter if one is interested in GDFs belonging to H, and if the equilibrium concept is that of a situation belonging to the core. This is what we show in our Theorem 4. Firsi i5.eintroduce the definition of the core.
5. Let f be the given GDF. For any given issue A and any sincere situation 5, the core - to be indicated by Co(J $A) - is defined to be the set of situations s fulfilling the fokiowing property: there does not exist coalition L and preferencerelations pi E 9 (r”E L) such that for all S’E 9” if Pi’= pi for all i E L, then f(s’,A]&fi(s,A) for all ie L. #kfkilior
Tk~mn 4. Let f E H. Then for some issue A and some sincere situation $, there doersnot exist amySE CocI;&A) such that f(s, A) =f(S, A)q Proof. Let A = {qy, t} be the issue. Let L, be the smallest decisive coalition defined as in the proof of Theorem 1. Without loss of generality assume that x&y (where DI. is defined as in the proof of Theorem 1). Let k L. Construct S such that for all iE (L - {k)), for al1 in (N- L),
XFiyFiz; yPi&iX, *OkXP&J.
we have xpy (since x&y), J&T (by LAV) and @x (by the fact that L is the smallest decisive coalition). There are three possibilities: (i) f(S, A) = 2; (ii)
ckriy
A) =x; or (iii) f($A) =y. ’ %otc thatfdcm
two wcescarily satisfy PC’ or LRI’.
(i) Suppose f(S, A) = E. Then assume S= S. Now consider any s E .y)flsuch that J(s, A ) = Z. Clearly -sECOW;S,A) since for all S’E Y such that _yP;#s for all k(N{k)), we have f(s’,A)=y (by LAV and PC), and for all &(N- (k)), pii,:. (ii) Suppose f(S, A) =x. Then assume S= S. Now consider any s E . e such that f(s,A)=x. We sha II show that -s E COW;&A). Let S’E P be such that for all in ((N- t)U (k}), WxP’y. Since L is the smallest decisive set, given LIIA and LM, we have -xP’z & -yP’z. Hence (zP’x& zP’y). Therefore by PC, f(s’, A) = z. Since for all in ((IV- t) U (k}), z&x, it follows that --s E COW;S, A). (iii) Suppose f(S; A) =y. Given LIIA and LM, there are two possibilities: either (i) xD,z, or (ii) z&- L~~. (iii.1) Suppose xD,z. Then let S=S. Let s be any situation such that .f(s, ,4) -= f($A)=y. For all S’E .+” such that yP;zP/y for all k L, we have .YP’z (since xDLz) and xP’y (since x&y) and hence f(s’, A) = x. Given that X&V for all i E I_, and given that f(s’, A) -x for ail S’E .+” such that xP;zP;_vfor all ic L, it is clear that -s E COW;S,A). (iii.2) Suppose zDN_ I_x Let Lo be the smallest subset of (N-L) such that zDLox. (It is easy to check that such Lo exists and ILo1> 1.) Clearly L E (IV- L”.) Let &Lo. Construct SE .+‘Isuch that for all i fz(N- LO), for all ie (LO- {t}),
XPiyPiZ;
yp,zJsiX; &x&y.
Since L E (N- Lo) and since xDLy, we have ,s.-&. By LAV, yp:. Lastly, by the assumption that zDLox, we have 3x. Thus (x& &_I?&& $x). Hence given PC, f($A)=f&4)=y. Now consider any& .+SNsuchthat for all iE[(,K-LO)U(~)], xP$P$ Since L c (N- Lo) and since xDi_y, clearly we have sP’y. Since L” is the sma&st subset of (N- L) such that Z&OX, -zDLo_ III~.. Hence given LIIA and LM, it is clear that xD(+ Lo,uIip. Hence xP’z. Since (.~P’y& xP’z), by PC, f(s’,A)=x. Thus it is clear that for all SE Y such tkt for all k [(N- L”)U (r)], xP;yP$, we have f(s’, A)&f($ A) for all k [(N- L”)U it}]. Hence it is clear that there does not exist s E Codf, 3,A) such that f(s, A) = f(g A ). Thus in all cases there exists SE .F such that for no s E Codf, S,..-I) ate haie f(s, A) = f(S, A). This completes the proof.
6. Concluding remarks The main conclusions of this paper are essentially negative in their impact. The) show that irrespective of whether the decision making process is visualized as a noncooperative or as a co-operative game, under most group decision procedures based on pairwise comparisons it will be impossible to ensure that there will be always some plausible equilibrium yielding the same outcome as the profile of sincere individual preferences. In the non-cooperative contest, the negative conclusion was
P. K. Pattanaik, M. Sengupta / R dmis.rible Nash equilibria
34
atdished using the notion of admissible Nash equilibria while in the co-operative context we used the nMion of the core. In the non-cooperative framework it was also shown that the negative conclusion remains intact even when we give up the notian of Nash equilibrium altogether and switch to the concept of sophisticated vatin@. ‘Ilwsin general, it appears that within the class of group decision procedures on pain&e comparisons, the search for an exactly consistent decision procedure (in the Peleg-Sen sense)is not likely to succeed if exact consistency is definedin terms of certain plausible notions of equilibrium.
K,J, Arrow. Social Choice and Individual Values (Wiley, Kew York, 1964) 2nd ed. 0, Black, ‘fhe Theory of Committees and Elections (Cambridge University Press, London, 1958). % &~bera,
Manipulation of social decision functions, J. Economic Theory I5 (1977a) 173-179.
S, &&era.
The manipulation of social choice mechanisms that do not. leave ‘too much’ to chance.
ldonomrica
45 ( I977b) lS73- 1388.
1. Du~ta and P.K. Bartanaik, On nicely consistent voting systems, Econometrica 46 (1978) 163-170. 8, Farquhar~n,
Theory of Voting (Yale University Press, New Haven, 1969).
rd, Manipularion of voting schemes: a general result, Econometricn 41 ( 1973) 587-601.
A, G
A, Gibbard, Manipulation of schemes that mix voting with chance, Econometrica 45 (1977) 665-681. #.D. I,wc and H, Raiffa, Games and Decisions (Wiley, New York, 1957). PA.
Pattch4k.
F*.
Strategy and Group Choice (North-Holland,
Amsterdam,
1978).
ctcr and E. Wesley, Stability of social choices in infinitely large societies, J. Economic Theory 14 (IW7)
II. P&g,
232-262. Consistent voting systems, Econometrica 46 (1978) 153-163.
D. Schmcidlerand H. Sonnenschein, Two proofs of the Gibbard-Satterthwaite
ofa strategy-proof
theorem on the possibility
social choice function, Proc. Conf. on Decision Theory and Social Ethics (Reidel,
Dordrcchr), IO appear. MA.
Satterthwaite, Strategy-proofness and Arrow’s conditions: existence and correspondence theorems
for vating procedures and social welfare functions, J. Economic Theory 10 (1975) 187-217. &#, Sengupta, On a difficulty in the analysis of strategic voting, Econometrica 46 (I 978) 33 l-343. M. Sengupta, Monotonicity,
independence of irrelevant alternatives and strategy-proofness of group
decision functions, Rev. Economic Studies (1979).
Publishing Company
OUTCOMES OF ADMISSIBLE NASH EQUILIBRIA AND SOPHISTICATED VOTING WHEN DECISIONS ARE BASED ON PAIRWISE COMPARISONS Prasanta K. PATTANAIK Departnrent of Economics, Uwivetsit_vof Birrningharn, Birntingharn Bi5 2%T. 0. A’.
Manimay SENGUPTA Department of Economics, University of New South CCb/es,Kensington. A’..% 11’., 1 ustralia 20.33 Communicated by K.H. Kim Received 9 October 1980 Revised 20 November 1980
Given the absence of non-trivial decision rules which are strategy-proof in the GibbardSatterthwaite sense, Peleg and Sen have suggested that one might search for decision rules under which there will be always some Nash equilibrium yielding the same outcome as would be yielded by honest voting. This paper shows that most decision rules based on pairwise comparison will fail to satisfy the requirement of Peleg and Sen if one imposes the additional condirion that the relevant Nash equilibrium
should consist of undominated
voting strategies only (a mosl
reasonable condition in the absence - cooperation). it is also shown that under most decision rules based on pairwise comparisons, it will not be possible either to ensure that the outcome of honest voting will be always an outcome of sophisticated voting (in Farquharson’s sense) or to ensure that the outcome of honest voting will be always a core outcome. Key words: Nash equilibrium; Dominated strategy; Sophisticated \ oting; Core.
1. Introduction Since it is impossible to find a non&Gal decision rule under which every possible profile of sincere individual orderings will always constitute a I%ashequilibrium,’ Peleg (1978) and A.K. Sen (in private communication) have suggested that one might search for reasonable decision procedures under which for every profile of sincere preferences there would be at least one equilibrium yielding the same outcome as the sincere preferences. 2 This attractive property (which can be formulated in terms of any notion of equilibrium) has been called ‘exact consistency’ by Peleg. Considering the voting process as a non-cooperative game and using Nash equilibrium as the relevant concept, it is indeed possible to show * See Gibbard (1973, 1977). Satterthwaite (1975). Barber3 (1977a. b). Pattanaik (197%
Pawner and
Wesley (1977), Schmeidler and Sonneschein tforthcoming) and Sengupta ( 1979) among others. * See also Dutta and Pattanaik (1978).
01654896/81/0000-0000/$02.75
0 1981 North-Holland
P. K. Partanaik, M. Sengupta / Admissible Naslt equilibria
that most decision rules will be exactly consistent. HoNever, it is well known that there can be a plethora of Nash equilibria, some of INhich may lack all intuitive ibility in predicting the outcome? So there exists the need for going beyond the requirement of a Nash equilibrium yielding the same outcome as the profile of incere preferences: one would presumably want the equilibrium to be intuitively plausible. A minimal test of such intuitive plausibility would be to require that the Nash equilibrium be composed of undominated voting strategies only.4 One objective of this paper is to show that under most decision rules based on pairwise comparisons, it will be often impossible to find a Nash equlibrium which yields the me outcoms as the profile of sincere preferences and in which each voting strategy 1s undominated. Repeated applications of the principle of eliminating dominated voting strategies one to the concept of sophisticated voting first introduced by Farquharson ). Within this framework we show that under every decision rule belonging to the class referred to earlier, it will be often impossible to realise the outcome of sincere preferencesas an outcome of sophisticated voting. is well known, the requirement that the strategies be undominated loses its alppeal in the co-operative context where non-trivial coalitions are permitted. 5 Within the co-operative framework we show that under no decision rule belonging to evat a wider class than the one consideredin the non-cooperative context, will it be !Ale to ensure that the outcome of sincere preferences would necessarily figure among the core outcomes.
1. Notation and some preliminary definitions Let Y be the set of alternatives (1q ~3) and let :V= 2 y- {@}; the elements of cc/ ate calLed issues. N= { 1,2, . . . ,n) is the set of individuals, assumed to be finite (a non-empty subset of JV is called a coali!ion). ,9 is the set of all asymmetric, connected and transitive binary relations defined over Y. The elements (P~ga*apPi**-*,Pn)g (P{p***pP~q*..p P&)etc. of .9 n will be called situations and will be iindicated by S, s’, etc. Pi, Pi’, etc. have the usual interpretation as the strict prefctrence relations that can be expressed by individual i. The true or sincere strict preference relation of individual i will be indicated by pi (Pi E 9). s = (PI, . . . , P,,) will be calfed the sincere situation. For ali in N and all x,y E Y, x&y iff -Upix. Similarly we have Rj, R,i,etc. TWOsituations s and S'are said to be i-variant (i EN) iff Pi # Pi’, and for all j# i, Pj = Pj. ’ Cf. Fivquhar5on (1%9, ’ ?MWFaryuharwn
(1969,
p_28). p. 28).
’ For m!mpie, in the well known gameof Prisoner’s Dilemma, if we change the rules of the game and
permit ca-qeration, then the simultaneous adoption of dominated strategies by the two players would make bath af them better off compared to the situation where both of them adopt undominated kc. !&CCtuce and Haiffa (1957, pp. 94-97).
P. K. Pattauaik,
M. Sengupta
i Adnrissible
Nash equilibria
41
A group decision function (GDF) is a function f: +“x.IQ --)Y such that for all (S&E .4W’r/, f(s,A)eA. For all SE 4” and all x,y~ Y, XPY iff sf v and .v= f(s, {x,y)). (Similarly P’ corresponds to s’, P” corresponds to s”i and so on.) Clear!), P is asymmetric and connected. In this paper we shall focus on an important class of GDFs, namely GDFs based on pairwise comparisons. Definition 1. A DGF f is based on pairwise comparisons
(or, equivalently, satisfies the property PC) iff: (i) for all (s,A) E .r’“x !+ and for all XE Y, if XEA and xP_vfor all y E ( Y- {x}), then f(s, A ) = x; and (ii) for all A E .+ and for all s, s’ E I) “, if P and P’ coincide over A, then f(s, A ) = f(s’, A ). The above definition of GDFs based on pairwise comparisons does not impose any restriction on f(s,A) when there is no ‘best’ alternative in A, i.e., when there is no P-maximal element in A. Judged in terms of the method in which such ‘deadlocked’ situations are handled, most GDFs based on pairwise comparisons come in one of the following two categories which we call Type 1 and Type II GDFs based on pairwise comparisons. Under Type I GDFs based o:i pairwise comparisons, a prespecified alternative (‘status quo’) in the issue gets chosen if there is no ‘best’ alternative. Formally f is a Type I GDF based on pairwise comparisons iff f is a GDF which, in addition to PC, satisfies the following property: for all A E .@,there exists a EA such that for all SE 3’“, if A has no P-maxima1 element then f(s,A) = a. Procedures (j?) and (y) of Black (1958, pp. 22-24) and the GDFs discussed by Sengupta (1978) are of this type. Much more commonly used are what we call Type II GDFs based on pairwise comparisons. Under these decision rules the alternatives in the issues are taken up in a predetermined sequence (the sequence in the ‘agenda’). The first alternative in the sequence is compared with the second and the winner is carried over to be compared with the third and so on until the end of the sequence is reached. An alternative defeated at an earlier stage in the sequence is never taken up again at a later siage. For example, if the issue is {x,y, z, w}, the sequence may be (t,y, .Y,w). Then if we have (xPy & yPz & tpw & wPx), the final winner will be w. On the other hand, if we had (xP’y &yP’z & zP’x& xP’w), x :Nill be the final winner. Thus even though both cases are characterised by the absencle of a best alternative in A, the chosen alte.:native is different in the two cases. More formally, f is a Type 11GDF based 0~ pairwise comparisons iff, in addition to PC, f satisfies the following property: for all A E ;‘Y,there exists a one-to-one function (z from the set of numbers { 1.2, . . . ,A ) to A such that for all s E .4 “, f(s, A) = as*“i where 4’ = a(1) and for all integers k (/A( -.-I Ikl I), if a(k+ l)Pa”, then ak+i=o(k+ 1) and if a”Pa(k+ 1). then (f+l :c aA, Procedure (Yof Black”(l958, p. 21) is of this type. Though the two types of GDFs based on pairwise comparisons that we have discussed above are structurally very different, they share a propert: which is important from our point of view. l
42
P. K. Pattanaik, M. Sengupta / Admissible Nash @quilibria
1. Let J be a Typ I or Type II GDF based on pairwise comparisons. 7Reaf sxtt&$iesthefollowing property: (0) for all A E ‘!I,iy 1A I= 3, then there exists a (EA such that for all s E .dn, if there is no P-maximal el -went in A, then f (s, A ) = a. pe I GDF based on pairwise comparisons, then it is obvious that f Consider the case where f is a Type II GDF based on pairwise t A = {x~YJ). There exists a one-to-one function o from { 1,2,3} to A such that for all SE :-i”“, j’(s,A) = alAfwhere alAlis specified as in the definition of Type II GQFs based on pabwise comparisons. Without loss of generality assume that a( 1) =x; n(2) = y; and (x(3)= t., Consider any s E :y n such that A has no Pimal element. Then either (xPy&yPz & Zpx) or (yP’& XPZ& Zpu). In either it can be e&ly checked that f(s, A) = aIAl= z. Hence f satisfies (D). In the following sections, we mainly consider GDFs which satisfy property (D) in ition to PC. #Miaitbn 2. A GDF satisfies PC’ iff it satisfies D in tiddition to PC. We now define several well-known properties of a GDF. grlit&ian 3. Let f be the GDF; let x and y be any two distinct alternatives; and let s and s’ te any two situations. (3.1). Limited Independence of Irrelevant Alternatitx (LIIA). If for all i E N, Pi and Pi coincide over {x,y), then P and P’ coincide over (x,y).
(3.21. Limited Monotonicity(LM). If for all i EN, [xPia*xP,!a, for all a E Y] and [uP,,b-aP;b, for all a, be (Y - (x})], then [xPy+xP’yJ. (3.3). Limited Absence of Vetoes(LAV). If I(ie N IxPiyj1z n - 1, then xPy.
(3.4). Limited Relevance of Individuals (LRI). For all k N, there exist i-variant situations so and so0 such that xpy and yPoox. (3.9. LRI’. For ail iE N, if there exist a, b E Y and i-variant s”,#% :S n such that uP@band bP%, then for all distinct c,d E Y, there exist i-variant s*,s**E :pn, such that cP+d and dP+*c l
All these properties are familiar with thlepossible exception of LRI and LRI’. LRI is a very weak and acceptable property originally introduced by Farquharson (1969). What it says is that for every pair of alternatives, every individual’s vote matters sometimes. LRI’ is even weaker than LR’I. What LRI’ requires is that either for every pair of alternatives an individual’s vote matters sometimes, or his vote does not matter for any pair at all. (LAV together with the well known property of pairwise anonymity implies LRI while the familiar property of pairwise neutrality implies LRI’,)
P.K. Puttanaik, M. Sertgtryru /’ Adrnissibie Nash equilibria
43
Let F be the class of all GDFs which satisfy PC’, LIIA, LM, LRI and LAW; lu:tG be the class of all GDFs which satisfy PC’, LIIA, LM, ERI’ and LAV; and let li be the class of all GDFs which satisfy PC, LIIA, LM and LAV. Clearly Fc G c H.
3. Outcomes of admissible Nash equilibria
In this section we assume that no co-operation among individuals is possible. We first introduce the notion of admissible Nash equilibria. Next we prove our main results which show that if one is considering a GDF belonging to For G‘, then it is impossible to ensure that for every possible issue and for every possible sincere preference profile, there will be a Nash equilibrium which would consist of only undominated voting strategies and which would yield the same outcome as the sincere preference profile. Definition 4 (4.1). Given an issue A and given the sincere situation S, a situation s is a Nash equilibrium iff there does not exist k .N and situation s’ such that s’ and s are ivariant and f(s’, A)Pif(S, A). (4.2). Let A be the given issue and let Pi be the sincere preference of k IV. Let Pi, P: E 9. For individual i, Pi weakly dominates P; iff for all i-variant s*, s** E 7”’ Pi dominaies Pj iff P, weakly such that P,*= Pi and P,?* = P:, f(s*,A)Rif(s**,A); dominates Pj and for some i-variant s*,s**E .Y” such that P,*= P, and P,**= PI, f(s*,A)Pif(s**,A); Pi is admissible (or, equivalently, undominated) iff for all r’iE (9, Pi does not dominate Pi. 6 (4.3). Given the issue A and the sincere situation S, a situation s is admissible iif for all ic IV, Pi is admissible. A major problem with the notion 0: Nash equilibrium is that eFc:n most implausible situations can qualify as Nash equilibria. For example, given the issue {X,JJ,z), if for all k IV, XpyPiz, then under every f E H a situation s such that for all iEN, ZPiyPiX, will constitute a Nash equilibrium.’ As a minimal test of the plausibility of Nash equilibria, one should ask whether the preferer?
’ Note that the notions of admissibiiity, weak domination, and domination are relatiw to the piben issue and the sincere preference of the individual under consideration. -r See Farquharson (1969, p. 28) and also Dutta anri Pattanaik (1978).
P. K. Pattanaik, M. Sengupta / Admissible Aiash equilibria
GINS belongingto F, cDnecould ask whether it is possible to find a GDF@ F such that for every possible issue A and every possible siilcere situation 8, there would exist an admissible Nash equilibrium s such that j&b\) =&A). Unfortunately, as shown by tRe following theorem, the answer has to be in the negative. mm 1. Let f E F. Then there exists an issue A,, an individual i, a sincere sitwtion *s,and 4% .9 such that P is admissiblefor i given the issue A and the sincere prcefetvmv&, and for aI1admissiblesituations S, ifPi =:F, then f (s,A )Pif (S,A ).
To prove this theorem we first introduce some additional notation, and derive 1 preliminaryresults. n most of the proofs we shall USC, a three element issue {x,y, z}. Since the GDFs that we consider always satisfy LIIA and PC, without loss of generality any idual k can be regardedas having six possible voting strategies given the issue z): (1) ,&y&; (2) x&z&y; (3) y&z&x; (4) Y~~x&z; (5) z&.x&y; and (6) fl&~x. To economise notation we shall refer to these preference relations or voting stradegiesas (l), (2), . . . , (6) respectively. Given the issue {x,-y,z} and given any 8% 9, x i;s said to be P-best iff (xpy &x&); x is said to be p-medium iff [&!!St x/%) or (#x dkxpy)]; and x is said to be P-worst iff (y#!x & zpx). Similarly, we speak of y or z being P-best or P-worst or &medium. 1st T !.eowm 1,and also in Theorem 2 later, we are concerned with GDFs which satisfy D. Under each one of these GDFs, for every three-element issue A, there mists a firredalternative in A which gets chosen from the issue A whenever there is no P-maximalelement in A. This fixed element of A will be indicated by g(A) in the , following proofs in this section. ~WBIB 1. Let i be any individual. Given any issue and any sincere preference PIE ~9. { 1.t). For all Pi, Pi, Pi”E 9, if Pi weakb dominates Pi, and Pi werrklydominates Py, thw P, weak&y dominates Pi”. (t.2). For all Pie .@, lf Pi is dominated or inadmissible, rhen there exists undominatedPi E .Y’such that Pi’ ,dominates Pi. The proof of this lemma is obvious and is omitted. timma 2, Let t,he GDF belong to F. Let ie N and let A be the issue where IAl = 3. Let aeA. Consider any two i-variant situations s and s’ such that for all b,CE(A - {a)), (bPiC (iff bP#); tied for all bE(A - (a))* aPib+aPjb. If g(A) =a andf(s,A)=a, then f(s’,A)=a. Proof, Let A = {a,~, y ) .. Since f(s, A) = a, either [aPx & aPy] or [(aPx & xPy & yPa) OPtiPa& aPy&yPx)J. If (aPx& aPy) by LILA and LM, (aP’x& aP*Lv)and hence
P.h’. Patranaik, M. Sengupta / .4dmissible Nmh equilibrirl
JS
f(s', A)
=a. Suppose (aPx& xPy & yPa). Then by LllA and LM, (aP’x& xPy). Now either yP’a or aP’y. Given g(A) = a and (aP’x& xP’y), it is clear that irrespective of whether yP’a or aP’y we have f(s’, A) L-a. Similarly, it can be shown that if (xPa & aPy & yPx) we would have f(s’,A) = a. This completes the proof. Lemma 3. Let the GDF f belong to F. Let k N and let A = (x, y, 2) be the issue. Let pi and Pi be any two elements of .9. Then for individual i : (3.1). Pi is undominated. (3.2). If g(A ) is Pi-best in A, then Pi is undominated on/y if g(A ) is &best in A. (3.3). If g(A) is Pi-Worst in A, then Pi is undominated only
ifg(A
) is &worst
in A.
(3.4). If g(A) is Pi-mediutnin A, then Pi is the om’y undomkiated strategy. Proof. Without loss of generality assume XPiyPiZ. We first show that for individual if g(A) =x,
then (1) weakly dominates (3) and (4) and (2) weakly dominates (5) and (6);
if
then (1) weakly dominates (2) and (5) and (4) weakly (a) dominates (3) and (6);
g(/l ) =
2,
and if g(A) = y,
then (1) weakly dominates (2>, (3). (4), (5) and (6).
Proof of (a). To prove (a) it is clearly sufficient to show that for any two i-variant situations, s, s’ E 9”, f(s, A)&f(s’, A) in all the following cases: (i) [g(A)=x] & [Pi=(l)] & [PI ~(3) or (4)]; (ii) [g(A) =x] & [Pi = (2)] & [Pl = (5) or (Q]; (iii) (g(A)=z] & [Pi=(l)] & [P;=(2) or (5)j; (iv) [g(A)=21 & [Pi=(4)] & [P;=(3) or (6)]; and (V) [g(A)=y] & [Pi=Pi=(l)] & [P:*(l)]. Consider any two i-variant situations, S,S’E . Y)? We have altogether four possibilities: [xPy & xPz], or [ yPz & yPx], or [ZPX& zPy] or [(xP_v& yP: & $5) or (yPx & XPZ & zPy)] . Suppose (XPY& xPz). Then f(s, A) =x. Given that s is P,-best, it is
P. K. Pattanaik, hi. Sengupta / Adtnissible Nash equilibria
(@!!& zP_v). Then f(s,, A ) = z. Given LIN and LM, it can be easily ait in cases(i) through (v), we have (EP’x& zP’y) and hence f(s’, A) = 2.
Ch Hence f(s, A&f@‘, A). NOW ISUP~H
[(~Py&yPz& z??x)or (yPx&xPz dkzPy)). Consider first cases (ii)
and @I). Sincef(s’,A)=x in these cases, whatever f(s’,A) may be, it is clear that f(s,A)&/@‘,,4). Now consider cases (iii) and (iv). Since in cases (iii) and (iv) j@, A) = Z, by Lemma 2, we must have@‘, A) = z and hence_& A)WJ(s’, A). Finally consider case (v), We know that (yPx or zPx). In case (v), if yPx then by LIIA ar,d ‘x; and if ZPXthen again by LIIA and LM, zP’x. In both these cases x is not Hence given that g(A) =y, we cannot havef(s”,A) =x. Thus in case (v), we have,/& A) =y and/&‘, A)+x, which impliesf(s,A)~&s’, A). We now show that irrespective of the values of g(A), for individual i:
(I) is not weakly dominated by any other strategy; (2) is not weakly dominated by (3) or (4) or (5) or (6);
(In
and
(4) is not weakly dominated by (2) or (3) or (5) or (6). B-f
a~?t/p). We first shaw that for individual i, (2) is not weakly dominated by (3) irrespectiveof the value of g(A). Note that over (x,y}, (2) coincides with fii but (3) does not. By LRI, LIIA and LM there exist i-variant s”,s% .P such that (yPfx&yP%&x.y&xPmy). Construct s and S’ such that for all j+ i, ((P’= fj) & (xPjz & yPjz) & (xPjy iff XP$) & (yPjX iff YeX)y and [(Pi = (3), i.e. yPiZPiX)& (Pi’= (2), i.e. xP~zP;~)]. By LAV, we have (xPz& yPz&xP’z &yP’z). Comparing s with so, and s’ with sou, we have (JP” & xP~). Hence f(s, A) = y and f(s’, A) =x. Since XBiy, it is clear that (3) does not weakly dominate (2). The proof for all other cases can be completed by following exactly the same method as adopted in the preceding paragraph. Combining (a) and !j?) we have: (I) is undominated in all cases; If g(A) =x, If g(A) = z, If g(A) =y,
then (3), (4), (5) and (6) are dominated; then (2), 43), (5) and (6) are dominated; then al1 strategies excepting (1) are dominated.
This compIetes the proof’ of the lemma. We can now prove Theorem I
Let A = (my, z} be the given issue throughout this proof. Since ftz F and IAI= 3, g&4) is defined. Without loss of generality, assume g(,Q) =x. sing Arrow’si (1964, p. 98) berminology, a coalition\ t’ is said to be decisive for (meA against trE A 8dt”f (a# b) and for all s E :P such that afib for all i E L’, and bPia
for all k (N- L’), we have afb. We write aD,.,b to indicate that L’ is decisive for ,a against b. Let Y be the set of all coalitions which are decisive for some ae A against some kA. Let LE I/ be such that for all L’E Y, IL’lzlLI. By LAV, j~?.,(>l. Let h,kEL.
There are six possibilities: (i) xDLy; (ii) yDl. z; (iii) zD/_x; (iv) yD,_x; (v) xDl.z; and (vi) zDLy. I. Suppose xDLy. Let t be a smallest subset of (N - L) such that yDl-, iAI z. Since yDqN_L,uIkIz (by LIIA and the fact that L is the smallest coalition in _I) and since -yD,z (by LAV), it is clear that such t must exist and must be non-empty. Let I E L‘. Consider a sincere situation s and any situation s, such that:
forall i E [N -- (LU (k ))], for all k$--
(I}),
; ypiZPiX & P, = (3) or (6) SO that YPiZPiX or ZP,_VP,X, zPixP,y
& zPixP,y
y&Prx & zP,yP,x, x&y&z & Pk = (1) or (2) so that xP,yPg or xPgP,y.
Note that since & (N- L), [L - (k}] c [N- (LU {k})]. Since sD,y, we have (xpy dkxfy). By LAV, we have (zpx & zPx). Since L’is the smallest subset of (IV- L) such that (LU (k}) is decisive for y against z, we have yk, and -yPz which implies zPy. Thus we have (xpy & y& & zpx) & (xPy & zPy & zPx). Hence f(!&A ) = s and j&A) =z. Since z&x, it is clear that P, does not dominate P,. By Lemma (1.2). either P, is undominated or P, is dominated by an admissible voting strategy of I. Since by Lemma 3, every voting strategy in (.d - (P,, P,}) is dominated and since P, does not dominate P,, by L,emma (1.2) it follows that P, is admissible. Note that for all k [IV- (LU (k))], Pi is the only admissible voting strategy; for all ie (t - (I)), the set of admissible voting strategies is a subset of {(3), (6)); and for k, the set of admissible voting strategies is a subset of {(l), (2)). Hence it follows that for all admissible S*E .P, if PF = PI then z = f(s, A) = f(s*, A )P,f($ A) =x. II. Now suppose yDLz. Note that by assumption h, kc L. Consider sincere situation s and any situation s such that: yjshz&x & ZPhyPhX; for all ie (L - {h, k}),
for ali &(N-
L),
yPiZ,PiX & P, = (3) or (6) SO that yPiZPiX or ZP,_VP,X; ZPiXPiy & tP,XPi_Y* xpky&z & Pk = (1) or (2) so that xP,yP,z or xPnzPc_V.
It can be easily seen that given that yD‘2 and L is the smallest coalition in I, and given LAV, we have (y& & #x & xpy) & (zPy & zPx & xPy). Hence f($ A) =x and f(s, A) = E. Since z&,x, it is clear that li,, does not dominate PII. Since the set of admissible voting strategies for h is a subset of {&, P,,), by Lemma (1.2) and by the fact that &, does not dominate P,, it follows that PI, is admissible for, h. NOW
P. K. Pattanaik, M. Senguptu / Admissible Nash equilibria
48
f’owing reasoning analogous to that in Step I, it can be shown that for all admissible S+E V, if Ph* = P,,, then z =f(s, A) =f(s*, A)&f($, A) =x. 111.Suppse @Lx. There are two possibilities: either there exists L’ C,(A’- L) such that y&g QICthere does not exist any such coalition L’. 1lJ.t. Sappose ?here exist L’ c (N- L) such tha.tyDrz. Then let t’ be a smallest subset of (k - 1,) such that y&z. Let t E t. Construct s and s such that
for all i E(N - L), for all it: (i- (t}),
zPiXp5ir& tpiXPiy; (XPiyPizor XPizPiy), y&.P*x 6%zP,yP,x. XP#PiZ
&
Since t c (N- L), it is clear that L c (iy - c). Since zDLx, we have (#x & z,Px). By LAV, we have (xpy & xPy). By the fact that L’is the smallest subset of (N-- L) such that yDcz, we have yk and -yPz (which implies zPy). Thus we have (z/k& #y & yk) & (~9%& xPy & zPy). Hence j@, A ) F x and f (s, A) = z. Since &x, p)r does not dominate P,. Now following reasoning analogous to that in Step I, it can be shown that P, is admissible for t and. that for all admissible 5% M”, if PF= P,, then z=f(s,A) =f(s*,A)P,f(s,A) =x. IfL2. Suppose there does not exist L’ c (N- L) stachthatyDLfz.Then construct 8,s E ,fn such that
forall
i E (L - (k i)), ZPixPiy& ZpixpiVE for all i@V- L), xpiypjz & ZPkYPkX. (XPJypJ
YP&X
?
or
xPid’iy)#
&
Since Z.&X, we have (tix& Px). By LAV, we have (xpy & xJ?y). Since L is the smallestcoalition in Y; given LIIA we have -fly which implies y/k Since there does not exist any coalition L’ s (N-L) such that yDLz, we have @y. (For, if -Ipy thenyl?z and then given LlIA and LM, some subset of (N--L) will become decisive for y against z.) Thus we have (xPy&y&& fix) & (xPy& zPy & zPx). Hence $@,A) =x and f(& A) =z. Since zp/Ix, again following reasoning anaiqpus to that in Step I, it cart be shown that Pk is admissible for k and that for all admissible S*E .Y if Pt=Pk, then t=f(s,A)=f(s*,A)&f(S,A)=x. The proofs for the cases where x&z, z&y and yDLx, are respectively -,iralogous 80 the proofs for the cases where xDLy, yDLz and z&x, except for the fact that y and t are to be interchanged everywhere in the proofs. It is possible to relax LRI to LRI’ in Theorem 1, thereby extending the cla.ss of GDFs covered, from F to G. W 2. Let f E G. Then there exists an issue A, an individual i, a sincere s&u&~ 3, and l% :@such thatP is admissiblefob*i given the issueA and the sincere pfleferemv p,, md for all admissiblesituationss, if Pi = p, thenf (s,A )Pif (39A ).
P. I :. ?WtunaiX
, hf. Sen,qtcpta / Mtnissi3le
Msh
equilibria
49
Proof. Since fe G, either fe F or f e(G- F). If f ts F, then Theorem 1 can be
applied to prove the rerluired conclusion. If& (G -F), LRI. Then N can be partitioned into:
then f satisfies LRI’ but not
N, = (& N 1there do not exist distinct x,y E Y and i-variant s”,#% such that f(s”, (x,y}) =x and f(soo,{x,y)) =y},
+I”
and &= {iENifol: all d ist inct x,y E Y, there exist i-variant so,s% .4” such thatf(sq{x,Y})=x andf(d? (X,Y})=Y}= Without loss of generality assume that &= (1, . . ..m} and N, = (m + 1, . . ..I?} where 15 m < n. Since f satisfies PC’, it is clear that for any given issue, the outcome under f is determined only by the preferences of individuals in &, i.e. there exists function f’ : .@“Ix ~!lr-+ Y such that for all ((P,, . . . , P,,,),A) E Yfl x .+,f’[(P,, . . . , P,,,), A] E A and for all SAE.P” if pi= P; for all ~EAJ~,thenfl((PE, . . . , PJ, A] =f(S, A). It is clear that f’ can be considered to be a ‘GDF’ for the smaller ‘society’ Nr, and that since f satisfies LIIA, LM, PC’, LAV and LRI’, f’ satisfies the properties of LIIA, LM, PC’, LAV and LRI with reference to the smaller society A$. Hence Theorem 1 can be applied to f’ to show that there exists an issue A, an individual !E &, (PI, . . . , P,,,) E F1, and PE .4 such that under f’, p is admissible for i given the issue A and the sincere preference .&, and for all admissible Given the (P ,, . . . , P,,) E Y”, if Pi=Pi, then f[(P,,...,P,,,),A]Pi[I(P,,...,P,,,),A]. relation between f’ and f, the required conclusion now follows immediately. The following result follows as an immediate corollary of Theorem 2. Corollary of Theorem 2. Let f E G. Then for some issue A, and some sincere
situation 3, there does not exist arty admissible situakiorl which is a Nash equilibrium and yields the same outcom4v as S. In this section we have mainly concentrated on the notion of admissibility of voting stragegies so as to mark out those Nash equilibria which are obviously implausible; the main impact of Theorem 2 is to show that for every GDF belonging to G, there exists a sincere situation s and “In issue A such that all Nash equilibria which yield the outcome f(gA) must be in this implausible category. It is, however, possible to pursue the notion of admissib;lity to analyse the likely outcomes, independently of the notion of Nash equilibrium. Successive applications of the notion of domination among voting strategie.5 leads one to Farquharson’s (1969) concept of sophisticated voting which we discu.;s in the next section.
4. Outcomes of sophisticated voting under GDFs belonging to F Continuing the assumption, that no co-operation mtrong individuals is possible, in
w
P. K. Paltanaik, M. Sengupta / Admissibk Nash equilibria
n we discussthe problem of ‘sophisticated voting’ * in the context of GDFs g to F. We first define the concept of sophisticated voting strategies. Next hat under every GDF belonging to1F, the outcome of sophisticated voting always determinate for issues with not more than three alternatives, and that for every such GDF in F and for every issue with three distinct elements, one can find a sincere situation such that the determinate outcome of sophisticated voting will be different from the outcome of the sincere situation. Let _f be the given GDF and let s be any given sincere situation. Then for every issue Ca, and for every k N, let Sy($ A) = .Jo and for all positive integers r, let fulfilling the following property: A) be theset of all kS:-’
There does not exist P’E S:‘* ‘(8,A ) such that for all S&E S[- ‘(s,A) x S; ‘(3,A ) x . . . x SL ‘(4 A), if P, = 9 and P; = p’, then f(s’, A)&j(.s, A), and for seine i-?;oriant ~$8 S; ‘(s,A) x 0.0xS;-‘($A), [(Pi=P)&(P;*=P’)&(f(S’,A)Pif(SgA))j. kor all positive integers r, S’($A) =:S#,A) x 5. x Sk@,A). Let S be the smallest E integer such that for all & JV, S;(& A) = S{’ ‘(g,A). Then for all i E Iv, ) is called the set of sophisticuted voting strategies of individual i. We say that the outcome of sophisticated voting is determinate for the issue A iff there exists A suchthat for all SE S@, A), f(s, A) =x. For the motivation of these definitions, the rea kr may refer to Farquharson (1969). l
TICllrorrtrr, 3. Let the GDF f belong to F. (3.1). Fora// SE .fn and for every issue A such that /A 1I 3 the outcome of sophisti-
mted voting is determinate. (3.2). For every issue A such that ;AI= 3, there exists a sincere situation s E .Yn such that f&A) is not an outcome of sophisticated voting, i.e. for all SE Si($,A),
f&4#fWO.
~2, the proof is obvious, given LIL4 and LM. We have only to case where IAl = 3. Let A = (x,y, z}. Since f E F and IAl = 3, there exists an alternative g(A) in A such that for all SE .9”, this alternative emerges as the outcome in the absenceof a P-maximal alternative in A. Without loss of generality assumethat g(A) =x. Let S be any sincere situation. Given LIIA, any individual i can be considered to have six alternative voting strategies numbered as in section 2. Partition N into x is &best}, N2={.ieNjx is Pi-worst}, and N3={idVlx is Pimedium}. Clearly, by Lemma 3, for all SE S’($ A), .Yis Pi-best for all i EN, ; x is Pi-worst
for all id Nl; and Pi = P, for all in NJ. Then given LIIA, one of the fdlowii:g possibilities: must hold: (i) for all s&‘&4), (xP_v&~f%); (ii) for all SE S’@,A), (yfxdk zPx); (iii) for all SE S’(S,A), (yPx& xPz); (iv) for all s E S ‘(s,A ), (zPx & spy). (i) Suppose (i) holds. Then by PC’, for all s E S’($ A), f(s, A j =x. Hence JC is clearly the determinate outcome of sophisticated voting. (ii) Suppose (ii) holds. Then for all SE S’($A), f(s,A)~ (y,2}. If either for all SE S’WQ, .f@,A) =y, or for all SES’(S,A), &A) =z, then the outcome of sophisniaated voting is determinate. Suppose for some s’ E S’(S,.A), f(s’, A ) = y and for some s” E S’(&A ), f(s”, A ) = 7, Then given that for all s E S ‘(3,.4 ), f(s, A ) E ( _Y,,z } and given LIIA, LM and LRI, it is easy to check that for all id (N, U AL) if _vP,z, then yP;z for all P; E $($A) and if ZPiy, thera ZPiy for all P, E Sf(s, A ). Since for all i E N’, x is Pi-best for all Pi E S,!(&A); for all iE Nz, x is Pi-worst for a.11Pi E S,‘(S,4 ); and for all i E NJ, ?#@,A) = {ai) 9 it follows from what we have said above that, for all ie N, $@,A) = I&}. Hence the outcome of sophisticated voting is determinate. (iii) Suppose (iii) holds. Then either for all SE S’(S,A), yPz; or for all SE S’(S, A), zPy; or [for some SES~(S,A), yPz and for some s%S’($A), zP’y). If for all SE S’($A), yPz then given that yPx for all SE Y&A), by PC we have Y =Jjs,A ) for all SE S’(&4). Outcome of sophisticated voting is clearly determinate in this case. Similarly, the outcome of sophisticated voting will be determinate if we have zPy for all s E S’(s, A). Now consider the case where yPz for some s t: S’(S, A) and zP ‘y for some s’ E S ‘(3,A ). We know that (yp~ & XPZ) for all s E S ‘(s,A ). Hence for all SE S ‘(3,14), if y&z, then f&4) =y, and if tpy, thenJ’I$A) =g(A) -xx’. (Clearly, there does not exist SE Sl($A) such thatf($A) = z.) Hence, for all ieArl, zP,y for all P, E S&J); and for all ie IQ, yPiZ for all P, E &,A). Hence for all i E Nl, S,‘(S,A ) = {(2)) and for all i e Nz, S&, A) = {(3)). As we have seen already, for all i E Nj, S,‘(S,A I= (P,) . Hence the outcome of sophisticated voting is determinate. The proof for case (iv) is exactly similar to that for case (iii). (3.2). Let A be any issue with exactly three elements. Let A = {.l;y, z). Without loss of generality assume that g(A)=x (where g(A) is specified as in section 2). Proceeding as in Theorem 1 we can show the following: there exists SE 9” and ic R such that for some &% 9, p is admissible for i [i.e. PE S,‘&4)] and for all SE +“. if s is admissible [i.e. SES’(S,A)], and if Pi=1’, then f(s,A)P,f($A). Throughout the rest of this proof we consider this specific sincere situation S. Note that as in the proof of Theorem (3. I), here also we have four !ogical possibilities: (i) for all sG’@,A), xPy & XPZ; (ii) for all s E S’@, A), yPx & ZPX; (iii) for all s E S ‘(3,A ), yRx & -4-P z; (iv) for all s E S ‘(3,A ), zp.U& xPy.
P. K. Pattanaik, M. Senguptu / Admissible Nash equilibria
$2
If (i) holds then f[S’@,A)] = (x}; if (ii) holds, then f[Sr(&4)] c (y,z); if (iii) holds, then given that g(A)=x, f[Sl($A)) c (x,y}; and lastly, if (iv) holds, then given that &4)=x, f[S’(&A)] G (x,z). Thus in all cases f [SI(.$A)] has no more thun two distinct elements. If f(&4) does not belong tof[Sr(S, A)], then the required conclusion follows directly (since Si(s,A) E Sr@,A)). Suppose f&A) Ef [Sl(s,A)J. Letf(S,A)=a and f [S’($A)1= {a,b} where a&A. By what we have said earlier, for some &N and some PE 9, kS#,A) and for all SE Sr@,A), if Pi= p, then
b==f(s,A)PJ(s,A)=a. Since _f[S1(s,A)]={a,6) it follows that for all F’E ,f, I% Sf(8.A) only if for all s E S,!(g,~1) such that Pi = p’, f(s, A )= 6.Hence it is clear that for all s E S&A), f(s, A) = 6. Hence for all s E S?(&A), f(s, A) = b # f(s, A). in all easesf($ A) is not an outcome of sophisticated voting. 3. The con under GDFs belongingto W In the previous sections we assumed that no co-operation among individuals was aibte. Given this assumption, it was natural to expect that no individual would adopt a dominated preference relation as his voting strategy. If this assumption Red, then the requirement of admissibility loses its appeal. However, the hope of atUning the outcome of the sincere situation through some ‘equilibrium’ does not become much brighter if one is interested in GDFs belonging to H, and if the equilibrium concept is that of a situation belonging to the core. This is what we show in our Theorem 4. Firsi i5.eintroduce the definition of the core.
5. Let f be the given GDF. For any given issue A and any sincere situation 5, the core - to be indicated by Co(J $A) - is defined to be the set of situations s fulfilling the fokiowing property: there does not exist coalition L and preferencerelations pi E 9 (r”E L) such that for all S’E 9” if Pi’= pi for all i E L, then f(s’,A]&fi(s,A) for all ie L. #kfkilior
Tk~mn 4. Let f E H. Then for some issue A and some sincere situation $, there doersnot exist amySE CocI;&A) such that f(s, A) =f(S, A)q Proof. Let A = {qy, t} be the issue. Let L, be the smallest decisive coalition defined as in the proof of Theorem 1. Without loss of generality assume that x&y (where DI. is defined as in the proof of Theorem 1). Let k L. Construct S such that for all iE (L - {k)), for al1 in (N- L),
XFiyFiz; yPi&iX, *OkXP&J.
we have xpy (since x&y), J&T (by LAV) and @x (by the fact that L is the smallest decisive coalition). There are three possibilities: (i) f(S, A) = 2; (ii)
ckriy
A) =x; or (iii) f($A) =y. ’ %otc thatfdcm
two wcescarily satisfy PC’ or LRI’.
(i) Suppose f(S, A) = E. Then assume S= S. Now consider any s E .y)flsuch that J(s, A ) = Z. Clearly -sECOW;S,A) since for all S’E Y such that _yP;#s for all k(N{k)), we have f(s’,A)=y (by LAV and PC), and for all &(N- (k)), pii,:. (ii) Suppose f(S, A) =x. Then assume S= S. Now consider any s E . e such that f(s,A)=x. We sha II show that -s E COW;&A). Let S’E P be such that for all in ((N- t)U (k}), WxP’y. Since L is the smallest decisive set, given LIIA and LM, we have -xP’z & -yP’z. Hence (zP’x& zP’y). Therefore by PC, f(s’, A) = z. Since for all in ((IV- t) U (k}), z&x, it follows that --s E COW;S, A). (iii) Suppose f(S; A) =y. Given LIIA and LM, there are two possibilities: either (i) xD,z, or (ii) z&- L~~. (iii.1) Suppose xD,z. Then let S=S. Let s be any situation such that .f(s, ,4) -= f($A)=y. For all S’E .+” such that yP;zP/y for all k L, we have .YP’z (since xDLz) and xP’y (since x&y) and hence f(s’, A) = x. Given that X&V for all i E I_, and given that f(s’, A) -x for ail S’E .+” such that xP;zP;_vfor all ic L, it is clear that -s E COW;S,A). (iii.2) Suppose zDN_ I_x Let Lo be the smallest subset of (N-L) such that zDLox. (It is easy to check that such Lo exists and ILo1> 1.) Clearly L E (IV- L”.) Let &Lo. Construct SE .+‘Isuch that for all i fz(N- LO), for all ie (LO- {t}),
XPiyPiZ;
yp,zJsiX; &x&y.
Since L E (N- Lo) and since xDLy, we have ,s.-&. By LAV, yp:. Lastly, by the assumption that zDLox, we have 3x. Thus (x& &_I?&& $x). Hence given PC, f($A)=f&4)=y. Now consider any& .+SNsuchthat for all iE[(,K-LO)U(~)], xP$P$ Since L c (N- Lo) and since xDi_y, clearly we have sP’y. Since L” is the sma&st subset of (N- L) such that Z&OX, -zDLo_ III~.. Hence given LIIA and LM, it is clear that xD(+ Lo,uIip. Hence xP’z. Since (.~P’y& xP’z), by PC, f(s’,A)=x. Thus it is clear that for all SE Y such tkt for all k [(N- L”)U (r)], xP;yP$, we have f(s’, A)&f($ A) for all k [(N- L”)U it}]. Hence it is clear that there does not exist s E Codf, 3,A) such that f(s, A) = f(g A ). Thus in all cases there exists SE .F such that for no s E Codf, S,..-I) ate haie f(s, A) = f(S, A). This completes the proof.
6. Concluding remarks The main conclusions of this paper are essentially negative in their impact. The) show that irrespective of whether the decision making process is visualized as a noncooperative or as a co-operative game, under most group decision procedures based on pairwise comparisons it will be impossible to ensure that there will be always some plausible equilibrium yielding the same outcome as the profile of sincere individual preferences. In the non-cooperative contest, the negative conclusion was
P. K. Pattanaik, M. Sengupta / R dmis.rible Nash equilibria
34
atdished using the notion of admissible Nash equilibria while in the co-operative context we used the nMion of the core. In the non-cooperative framework it was also shown that the negative conclusion remains intact even when we give up the notian of Nash equilibrium altogether and switch to the concept of sophisticated vatin@. ‘Ilwsin general, it appears that within the class of group decision procedures on pain&e comparisons, the search for an exactly consistent decision procedure (in the Peleg-Sen sense)is not likely to succeed if exact consistency is definedin terms of certain plausible notions of equilibrium.
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