Mathematical Social Sciences 8 (1984) 15-28 North-Holland

MONOTONICITY, STABILITY AND EGALITARIANISM

William THOMSON * Lkpartment of Economics, University of Rochester, Rochester, NY 14627, U.S.A.

Communicated by J.C. Harsanyi Received 2 February 1983 Revised 7 March 1984

A division problem is a subset of the utility space representing the alternatives available to a group of agents. A solution is a method of selecting from each division problem in some class an equitable compromise. The egalitarian solution selects the only undominated ;rlternative with equal coordinates. This solution is the only one to satis’fy the following axioms: apeok Purero-

optimality, symmetry, continuity, monotonicity (if the number of agents inL,eaSes but not the resources at their disposal, all the original agents should help support the newcomers) and h*eak stubiliry (the compromise should appear equitable to every subgroup).

Key words: Division problem; egalitarian solution; monotonicity; stability.

1. Introduction

In the traditional formulation of the problem of fair division, there is a fixed number of agents among whom IS to be divided some list of objects. A division

problem can also be more abstractly defined as a subset of the utility space repre- senting the alternatives open to the agents. Given a class of such division problems,, a division principle or solution defined on that class is a method of selecting from1 each member of the class a unique feasible alternative called the solution outcome and establishing a fair compromise between the agents’ conflicting interests. The axiomatic study of the problem of fair division consists in formulating &Kiorns expressing desirable properties of solutions and in verifying whether the axioms are compatible. It is this approach, initiated by Nash (1933, that we will follow.

As in Thomson (1983a, 1983b, 1983c), we consider here the more general situation that arises when the number of agents among whom the division is to take place may vary, and the purpose of two of the axioms that we formulate is to provide guidance as to how the result of the division should be affected by such changes.

The first three of our axioms are standard: weak Pareto-oprimaiity, stating that

there should be no feasible way to increase the utility of all agents from the solution

*This research was undertaken at the IMSSS Summer Workshop, Stanford University, and was made possible by NSF under grant SOC 77-0600-Al to the IMSSS Stanford, California and grant SES 8006284. I have benzfited from discussions on related topics with T. Lensberg, and from the comments of fwo anonymous referees.

0165-48%/84/$3.00 0 1984, Elsevier Science Publishers B.V. (North-Holland)

outcome; ,~.~~~~r~tu~r,,‘, saying that if a problem is invariant under all permutations of

the agebnts, then its solution outcome should have equal coordinates; and tnorrthuit.y, rgquiring that mall changes in division problems should lead to small changes in

the sslutisn outcomes. In addition to these axioms, which apply to groups of agents of arbitrary but fixed

cardinslity, we imwse two axioms concerning possible variations in the number of wents. The first one, ori@nally introduced by Thomson (1983~) is an axiom of soli$irrity: it states that if the number of agents were to increase, but not the re- sources at their disposal, then all of the original agents should contribute to the supp~re; of the newcomers, We call it manlotsnicity with respect to changes in the numl,w- of agents, or simply mmotorrkit~. The second one is a minor weakening of art iM3n~ of stability, first introduced by Harsanyi (1959) (see also Harsanyi 11963, 1977) and recently studied estcnsitely by Lensberg (1981a, I981b, 1983): stability says that after attributing to a subgroup of agents their respective payoffs a~ specified by the snlrution outcome of a particular division problem, the sub- problem faced by the remaining wents should be solved at a point giving each of rhcm what was originally attributed to him. An interpretation of the’axiom is that my compromise that is judged acceptable to the whole group should appwr w- ccpt%rble ts any subsrsup. We will use a weakening of this condition, NV& stddity,

which sayo that the subproblem bc solved at a point giving each of the remaining agent% tit lwst as ~~fwh us what was originally attributed to him. Since in most cases this condition will be satisfied only if equality holds, t trc two usitms have mmtidly

the same content.

h is proved here that there is one and only one solution satisfying these five axioms. It is the ugdituiu,, sdMwf, which selects far each divisigrlt problem the maximal feasible alternative with equal coordinates.

The paper is organized as follows. Sxtion 2 specifies the class of division problems under study and spells out the w&mm, Section 3 cmtains the charac- terization result, Section 4 variants of tha,t result, md Section 3 a &x&an of related literarure.

There are n agents, n z 3, indexed by tk subscript i. .P is the family af subsets of (1, . . . . n). RQ... are the generic elements of ;19 The cardinality af P in L@ is clenored 1 Pp. Given P in $ R” is the cart&ail product of 1 PI copies of R, indexed by the members of P. A division pr&em far P is a subset of R c representing in van Neumann-Margenstern utility scales t6:c alternatives achievable by the members of P thro:)gh some joint action. We will cionsider division problems A such that

(iI A is convex, compact and there exists x in A with xr 0. ’

t Given two elemenrs x and means .v, >.:‘: for all i in P.

.u of rr29 xz;u means means

This says that it’ a divWn problem is invariant under all pwmutat icons of the

agents, its solution outcame should haye equal Lwrdinates.

This prevents small changes in the data defining the di\ ision pr~trlzm 10 1~3 w widely different solution outcomes.

These three axicmu are standard. By contrast, the nest two are spxifizatly zon- cetned with variations in the number of agents. In order to state them. w mxd to introduce another piece of notation: given P. Q in + with PI; Q. and .-I C y k’. XE A, we define

I8 IV. Thomson / Monotonicity

where xQ \ p is the PrOjeCtiOn of x OntO i? Oip. (We will also write AP for the pro- jection of A onto RP.)

Monofonicity (MON). For all P, S in :9 with PC S, for all A in Cp and D in Zs, if A = Dp, then QA) g Fp(D).

This says that starting from some original group P facing some division problem, the recognition of the claims of additional agents requires sacrifices from all the agents initially present (in a weak sense since it is only required that none of them should gain). This also means that conversely, if some agent were to leave, all of the remaining agents should benefit (again in the weak sense) from the fact that fewer agents have claims on the resources. The axiom is introduced in Thomson (1983a) under the term “monotonicity with respect to changes in the number of agents”. It can be interpreted as an axiom of solidarity: agents lose or gain together as their number varies; they all ‘pitch in’ when someone comes in, and they all benefit when someone leaves. However, the axiom is in principle compatible with a large amount of flexibility in how individual contributions or bonuses are dkter- mined.

Our next axiom is

StabliC#y (STAB). For all P, S in 9 with P c S, for all A in Zp and D in ZQ, if A = ti(D)# where x = F(D), then xp = F(A).

This says that if the restriction of a division problem, whose solution outcome is some point x, to the subset of its alternatives at which each of the agents in some subgroup gets what he was assigned under x, constitutes a well-defined division problem among the remaining agents, this new problem should be solved at a point giving each of them what he was originally assigned.

The condition, first introduced by Harsanyi (1959), in the weaker form obtained by adding the hypothesis that IPI = 2, is used by him to show that if a solution coincides with the Nash solution for two-person problems, and is stable, then it coincides with the Nash solution for all cardinalities. (It is also the weaker form of the condition that we will use in the sufficiency part of our characterization result.)

The axiom can be interpreted in several wsr?ys. Firs.t of all, if one assumes that satisfactory solutions to two-person problems have been found, it may help in solving n-person problems by appealing to two-person theory. Indeed, the two- person case is much better understood th;u_Ji the n-person case and the axiom may be of great value if it permits such extensions. This is mainly Harsanyi’s motivation.

The axiom can also be seen as stating ;.L requirement of stability under partial implementation, as suggested by Lensberg. After some of the agents have received their payoffs, it appears to the remaining agents that they are facing a new problem. The requirement is that this new problem should be solved at a point that agrees with what they were originally supposed to receive. Agents will be less tr,mpted to renege on an agreement if the requirement is met.

W. Thomson i Monotonicity

Agent 2

/ Agent 3

Fig. 1.

The final motivation is best introduced by considering the particular example of distributing the resources available on the planet Earth; the axiom says that the distribution should be acceptable not only for the Earth’s four and a half billion inhabitants seen as a whole; but also for each continent, each country, each city, each family seen seperately. A solution used at the level of the whole society should not yield outcomes that would be in contradiction with what it would recommend for subgroups.

These last two axioms are illustrated in Fig. 1, in which P= { 1,2} and S = ( 1,2,3).

The agents in S face the problem D, solved at x. If agent 3 were to leave, agents 1 and 2 would face the problem A. The axiom of monotonicity says that this problem should be solved at a point that weakly dominates xp, i.e. F(A)zxp. On the other hand, agents 1 and 2 could consider all the alternatives that leave agent 3’s utility fixed at x3 (the two-dimensional area A’). The axiom of stability says that if these alternatives constitute a well-defined division problem, as is the case of A’, then this division problem should be solved at a point that gives both agent 1 and agent 2 exactly what they were getting initially, i.e. F(A ‘) =.xp.

As mentioned in the introduction, we will use the closely related condition of

Weak stability (W. STAB). For all P, S in ,9 with P c S, for all A in Zp and D in Zs, if A = t;(D), where x = F(D), then F(A) 2 xp.

If a solution satisfies weak Pareto-optimality and x = F(D) happens to be Pareto- optimal then weak stability applied to D is equivalent to stability. Since any problem in our admissible domain can be approximated by a sequence of problems satisfying

20 II;: Thomson / Monotonicity

the additional requirement that their weak Pareto-optima1 and Pareto-optimal boundaries coincide, the two requirements of stability and weak stability are ‘almost’ equivalent for any solution satisfying WPO. The motivation for STAB is that the agents in a subgroup would not want to challenge each other. This idea is preserved by the weaker axiom since it guarantees that 111 members of the subgroup would have the same interests.,

We conclude this section by introducing the egalitarian solution. Given P in L@, and A in 2?, E(A) is the maxima1 point of A (according to the partial ordering of vectors in If? ‘) that has equal coordinates.

Other notation

Given P in Oy, ep designates the vector of lRp that has all of its coordinates equal to 1. Given a list of vectors x1, . . . , xm in IRT and subsets A ‘, . . . , An of IR ‘, cch(x’, . . ..x~. A’ , . . . , A “} designates the smallest convex and comprehensive subset of IRT containing x1, . . . ,xm and A ’ , . . . ,A”. Also, given P in ,p and A in Zp, &I designates the weakly Pareto-optima1 boundary of A. Finally, given S in :?, a subset A of IRS and a vector t of IRS, we designate by cyl(A, t) the cylinder with base A and generators parallel to 1.

3. The results

In this section, we present the results. First we have

Proposition 1. E satisfies WPO, SY, CONT, MON and W. STAB.

Proof. That E satisfies WPO, SY and CONT is clear, although it should be empha- sized that if condition (ii) were removed form the list defining the class of admissible problems, E would fail to satisfy WPO and CONT. The straightforward proof that E satisfies MON, given in Thomson (1983b) is omitted here. We limit ourselves to proving that E satisfies W. STAB. So let P, S aiixd D be as in the statement of that axiom. E(D) has equal coordinates. Also, i: is .a strictly positive vector; therefore A = t$(D), where x= E(D) is an element of 25’. The projection xp of x on II?: also has equal coordinates and clearly E(A) z-up , the desired conclusion. Cl

To prove that if a solution satisfies the five axioms it is the Egalitarian solution, we proceed by way of several lemmas.

Lemma 1. If F satisfies WPO, SY, CONX, MUN and W. STAB, then F= E for two-person problems.

Proof, The proof is organized in several steps in which successively larger and larger classes of problems are considered. The argument is the easiest for problems that

W. Thomson / Monotonicity 21

are supported at their egalitarian outcome by a line of slope - 1. This requirement, that the problems are not too skewed (the maximal utilities attainable by the two agents are not too different from one another), allows them to be ‘expanded’ to three-person problems with enough symmetry properties. The case of arbitrary problems is then taken up, the first step permitting expansions to a much wider class of three-person problems.

In Step 1 we construct a three-person problem given as a function of a parameter t: which has to be chosen positive but not too l&rge. The problem obtained for ): = 0 is easier to describe and represent, which we have done to facilitate the under- standing of the proof. To simplify the notation, we omit c: when it is set equal to 0. (For example, we write I for 1’ when & = 0.)

Sfep 1. If the Pareto-optimal boundary of A, (i) contains a non-degenerate segment 0 centered at E(A) and normal to ep and (ii) is supported at its points of inter- section with the axes by straight lines with finite and non-zero slopes, then F(A) =

E(A)- Without loss of generality we assume that P= { 1,2}. We argue by contradiction

and assume that for some A as just described, F(A)#E(A). Let a be the common value of the coordinates of E(A), a=max{x,Ix6} and b=max(xl/xES). Without loss of generality we assume that a! = 1. By condition (i) imposed on A, we have that as2 and &2. By WPO, either F1(A)>l and &(A)< 1, or F,(A)< 1

and &(A) > 1. Without loss of generality we assume the former. (a)Weintroduceagent3,wesetS~{1,2,3},Q~{2,3}andR~{1,3},wechoose

I Agent 2

Agent 3 Fig. 2.

22 W. Thomson / Monotoreicity

Fig. 3.

e positive but small and construct a problem I’ as follows (see Fig. 2; I= 1’ is represented in Fig. 3).

Let yC-(1 -e)es and DE=lR~ ncyl(A,y”-Q). Let a’-(X’ElR~pXEo S.Z. x;- -x1 and x4=x& C-cch{a’,ae,,aeJ}, and

He- II?: ncyl(c,y~ - +I. Finally, we define I” =DEn.HE. Because both 0” and HE are elements of Zs, SO

is Ic. (b) ahI,, is a vertical line emanating from ael . Since t3Hji goes through ael for all

e, varies continuously with E, and A admits at ael of a line of support with a finite slope (assumption (ii)), it follows that for G small enough A E Hi and therefore I;= A. Similarly, aDR is a horizontal line !emanating from ael. Since aDi goes through ael for all e, varies continuously w&h c, and C admits at ael of a line of support with a finite and non-zero slope (th.% is obvious from the construction of C), it follows that for e small enough C E Gj$ and therefore Ii = c’. We also note that cyl(a,ye - ep) and cyl(a’, y” - eR) are portions of planes, symmetrical with respect to the plane defined by ‘x2 = x3’. Tilrey intersect along a segment that we denote f. For e small (it is enough to take E< +), the points of t” are all Pareto- optimal for fe.

(e) Next, we apply MON three times, by ciomparing what agents 1 and 3 get in 1’ to what they get in 1; and what agent 2 gets in IE to what he gets in 1;. Since C is symmetric, we deduce from WPO and SY that F(C) =eR. We introduce q = I -&(A). By hypothesis q >O. We then have:

W. Thomson / Monotonicitv 23

F,(I’)sFdI;)=F,(C)= 1,

F#) z5FdI;) = F,(A) = 1 - q,

4(%5F3(4) = I;;(C) = 1.

(d) For e= 0, these constraints are satisfied only by a vertical segment S with endpoints eR and jJ=es- vet. For e small enough, they are satisfied by all the points of a narrow triangle 6’ with one vertex at eR and the other two close to J (this is the shaded area of Fig. 2).

For e small and z in 6’, #I”)=cch{(l -&)e2, (1 -e’)e3, (1 -&“)eQ}, where E’ and en are small numbers going to 0 with e; $$(I”) belongs to ZQ. Let t”= FW). By W. STAB, we have that flt#E))gz& but since zs is Pareto-optimal for @I’), we have equality and therefore ~$5 1 - q. (Since for any z in Z, z2 s 1 - q.) But by WPO and SY, we have that fl$(I&))=(l -&“)eQ and since E” goes to 0 with C, we obtain a contradiction for e small enough. This proves Step 1.

Step 2. If the Pareto-optimal boundary of A is supported at E(S) by a straight line normal to ef, then F(A) = E(A).

This follows from the fact that any A as described here can be approximated by a sequence of elements of Zp satisfying the extra assumptions listed in the state- ment of Step I. and the requirement on F that it satisfies CONT.

Step 3. Conclusion of the proof of the Lemma. AS in Step 1, we assume without loss of generality that P= { 1,2}. ‘We argue

by contradiction, and suppose that for some A in Zp, F(A) #E&A). Let (r be the common value of the coordinates of E(A), a= max(x, IA-E A}, and b= max{x21xe A). Without loss of generality, we assume that a = 1. Then either as 2 of b s 2. Without loss of generality, we assume the latter, and we distinguish two cases. Suppose first that:

(i) as2. By WPO, either F,(A) > 1 and F,(A) c 1 or Fl (A) c 1 and &(A) > 1. Without loss of generality we assume the former.

We introduce agent 3, we set S={1,2,3}, Q={2,3}, R={l,3} and we con- struct 15; It?: as follows. Let D= cyl(A, e3), B= cch{ 2ez, 2e3), G = cyl@, et ), C=cch(2ael,2ae3}, H=cyl(C,es -a+) and finally I=DnGnH (see Fig. 4). The following statements can easily be seen to be true. Ip= A (this requires an algebraic argument which is given in the Appendix). bIQ is supported at E(B) = ec’ by a straight line normal to e Q; this implies by Step 1 that F(IQ) = eQ. All the points of

the segment 6= [a+, es[ are Pareto-optimal for I and for any 2 in CF, [i(r) iS

supported at z by a straight line normal to eR. Applying MON three times, we obtain:

F,(I)~F~(lp)=F~(A), F~(I&F’(Ip)=F+l), F3tI)sF3(l~)= 1.

These constraints are satisfied only by the points of an area S made up of a horizontal segment with one endpoint F(A) and of a, triangle in at?. (The triangle may be degenerate; if F(A) happened to be more to the southeast on the boundary

I Agent 2

/ 4gont 3 Fig. 4.

of A than where it is represented in Fig. 4,6 would consist of a horizontal segment with end points F&4) and &4) + e3.) No points of 6 belong to CL For any G‘ in & r;(I) intersects CJ at a point z* whose first coordinate is strictly smaller than cl if E&4) > 0 and aA dzes not coincide with [uel, e& (The desired conclusion for those cases will follow from CONT.) Also, z+ is the egalitarian outcome of t&(I), which is supported there by a line normal to eR, as was shown above. By Step 2, we conclude that &i(l)) = z*. By W. STAB, we have that F&J(J)) g q if z = F(r). Since E+Z~ we obtain a contradiction. Thikl pro,>ves Step 3 when as2. We then assume that:

(ii) a> 2. We define D= cyl(A, e3), B cch{2q, 29 1, G cyl(B, 4,) and finally l= D17C; (see Fig. 5).

If I$(&> 1 and F,IA)< 1 the argument is asin (i) if F,(A) also satisfies &(A)<2. This case can be followed on Fig. 5 by setting F(A) =x’. The same inequalities are derived from MON to determine a zone 6 where z=F(I) should be. (in this case 6 is the segment [flA),F(AI)+q].) But c=@(l) is such that max(xili= 8,3,x~ c‘} s xt(c) and therefore by (i), F(c)= E(c). This is then shown to be incompatible with W. STAB. If Ft(A)g2, use is made of CONT to derive the desired contradic- tion: ‘we introduce a continuous function h: C&l) -@ such that for all il, in [0, 11,

XE h(l)) s2, h(O)= A, and h(1) = A’, where A’ is such that E(A’) =ep, max(WEA’)~t. We then have F,(h(O))z& F,(h(l))= 1 by (i) and for no 1 in

PrsloC. We will show that F= E t’or ail .4 \vhose weak Pareto-optimal boundary c\‘- incides with the Pareto-optimal boundary: the desired conclusion will ~MIcN b’\ ‘U-I application of CONT. We suppose, by way of contradiction, that t’or som S in l

with ISI >Z (we neglect the case IS! = 1 B which can be dealt with by a simple nppc;tl to WPO), and for some D in p whose weak Pareto-optimal and Pareto-optimal boundaries coincide, F(D) #E(D). Without loss of generality, we assun~ &t E(D) = es. Since 413) #E(D), it follows from WPO and the definitim of ?’ ahat there exist i and j in S such that F,(D) > 1 and c,(D) <: 1. Let then p= { L,d ) . A = t:(D), where x= F(o). Since ,4 contains a semi-positive vector and D belongs to Es, then A in fact belongs 10 Zp and, by hypothesis, we know t har F(4 has equal coordinates. But this is in olBntradiction with W. STAB which yields that flA)=> Fp(D) and the facts that F,(D) is Pareto-optimal for A and that 6 (0 > ~Jo,. EJ

26 W. Thomson / Monotonicity

Collecting all the previous results, we have the anncunced characterization of the Egalitarian solution.

Tbuwso. F satisfies WPO, SY, CONT, MON and PI’. STAB if and only if F= E.

4. ResovSng the axioms one at a time

In this section we discuss what additionaY solutionr:s would be made possible by removing one axiom at a time from the list appearir1.g in the Theorem.

(i) W&k Pareto-optimaliry. Given a list Q = i{ a’, P E 9) of ’ on-negative numbers such that for all P, S in 9, a ‘1 CT’ if and oraly if P G S, let II” $ be defined as follows: given P in :y and A in Z’, F,(A) = a ‘ep iif this point belongs to S and F(A)= E(A) otherwise. Any such Fa satisfies SY, MON, W. STAB and CONT. The family of these solutions is introduced in Thomsabn (1983~) under the name of truncated egalitatian solutions.

(ii) Symmetry. Given a list @ = { &, i E { 1, l . . , n) } of increasing and continuous functions @, : R + 3 IR + such that #i(O) = 0 and @i( R,) = IR,, we define F# as follows: given P in :‘ib and A in Cp, F@(A) is the uniqrrie point of intersection of the graph GP of the function #‘: Ii?, +lRf’ defined by #I/‘- @i for ail i in P, with the weak Pareto-optimal boundary of -4. Any such Fe satisfies WPO, MON, W. STAB and CONT. The family of these solutiorrs is discussed in Thomson (1983~) under the name of monotone-path solutions.

(iii) Monotonicity. The Nash solution N, where for all P in 9 and for all A in Cp, N(A) is the unique maximizer of the product n,= Xi for x in A, satisfies WPO, SY, W. STAB and CONT (see Lensberg, 1981a). (Note that in fact N satisfies PO and STAB.)

(iv) Stability, The Kalai-Smorodinsky solution K where for all P in 9 and for all A in 2?, K(A) is the unique maximal feasible point of A oi? the segment connecting the origin to the point a(A) where for each i in P, ai = mm{xi!xe A}, satisfies WPO, SY, MON and CONT (see Thorn&n, 1983a).

(v) Continuity. Many examples of solu$ions satisfying WPO, SY, MON, W. STAB but not CONT can be constructed. &, an example, take F to be the lexico- graphic extension of E for two agents and E for any other number of agents.

We conclude this r;ection with a commep1: on the domain on which we chose to work. We could have considered the subdomain of problems whose Pareto-optimal and weak Pareto-optimal boundaries coincide. On this restricted domain, E satisfies STAB, and in fact can be characterized as the only solution to satisfy PO, SY, CONT, MON and STAB. (It even seems that CONT follows from the other four axioms, but we have not been able to formally establish this fact.) Unfortunately, the proof of that result is significantly longer (although conceptually very close to the one we presented here.) This is because it is much more cumbersome to give an analytical descriptioll of an element of the smaller domain, and why we chose to work with a slightly weaker version of the stability condition.

5. Related literature

I+. Thomson / Monotonicity 27

The axiomatic characterizations of the egalitarian solution and of its lexico- graphic variant that have appeared in the literature can be divided into three groups.

First, a number of authors (Hammond, 1976; Strasnick, 1976; d’Aspremont and Gevers, 1977 and others) have obtained characterizations in the abstract social choice framework. In that literature, very few structural assumptions are made about the set of feasible alternatives or about the admissible preferences.

By contrast, restricted domains are u:sed in the characterizations of the second and third groups. The results of Kalai (19’77), Myerson (1977, 1981), Roth (1979) and Imai (1983) are proved on domains similar to that used by Nash (1950) in his classic paper. In these papers, the number of agents is taken to be an arbitrary but fixed number.

Finally, we have the papers by Thomson (1983b) and Lensberg (1981 b) in which the number of agents is allowed to vary. The egalitarian solution is shown by Thomson to be the only one to satisfy WPO, SY, CONT, MON and IIA (indepen- dence of irrelevant alternatives), a condition used by Nash (not under that name) and saying that the solution outcome of a problem obtained from some other problem by eliminating some of its alternatives but not its solution outcome should coincide with that original solution outcome. Lensberg is concerned with the lexico- graphic extension of the egalitarian solution and shows that it is the only solution to satisfy PO, AN (anonymity), a generalization of SY saying that the names of the agents do not matter, STAB and IND. MON (individual monotonicity), a condition first introduced by Kalai and Smorodinsky (1975) and saying that if a division problem is ‘expanded’ in favor of an agent, then this agent should not lose.

The advamtage of the present characterization over the latter two is that it makes no use of what are the more controversial axioms in these lists, namely IIA and IND. MON.

Appendix

Proof that Ip= A (Step 3, part (i)). It is trivial that Dp= A and that GP ~‘4. To see that HP 2 A we note that all points of A are dominated by some point of tither the segment [epx’] where x1 = (a, b- a(b - 1)) (x1 is the point on the line con- necting be! to ep which has its first coordinate equal to a) or the segment [ep ~‘1, where x*=(a- b(a- I)_ 6) (x2 is the point on the line connecting ael to ep which has its second coordinate equal to b). Both of these segments are below the line of equation

x,+a-1 - =l, 2a ax2

which is the intersection of the plane going through 2ae,, 2aez and es with RP.

2s W. Thomson / Monotonicity

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