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THE HEX GAME THEOREM AND THE ARROWIMPOSSIBILITY THEOREM: THE CASE OF WEAK ORDERS

Yasuhito Tanaka*Doshisha University

(February 2007; revised October 2007)

ABSTRACT

The Arrow impossibility theorem when individual preferences are weak orders is equivalent to theHEX game theorem. Because Gale showed that the Brouwer fixed point theorem is equivalent to theHEX game theorem, this paper indirectly shows the equivalence of the Brouwer fixed point theoremand the Arrow impossibility theorem. Chichilnisky showed the equivalence of her impossibilitytheorem and the Brouwer fixed point theorem, and Baryshnikov showed that the impossibility theoremby Chichilnisky and the Arrow impossibility theorem are very similar. Thus, Chichilnisky and Bary-shnikov are precedents for the result—linking the Arrow impossibility theorem to a fixed pointtheorem.

1. INTRODUCTION

Gale (1979) showed that the so-called HEX game theorem that any HEXgame has one winner is equivalent to the Brouwer fixed point theorem. In thispaper, we will show that under some assumptions about marking rules ofHEX games, the Arrow impossibility theorem (Arrow, 1963) that there existsno binary social choice rule which satisfies transitivity, Pareto principle,independence of irrelevant alternatives (IIA) and has no dictator when indi-vidual preferences are weak orders is equivalent to the HEX game theorem.

In another paper Tanaka (2007), we showed the equivalence of the HEXgame theorem and the Arrow impossibility theorem in the case whereindividual preferences are strong (or linear) orders, that is, individuals are

* I would like to thank two anonymous referees for their valuable comments which substantiallyimproved the quality of this paper. Of course any remaining errors are mine. This research waspartially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aidin Japan.

Metroeconomica 60:1 (2009) 77–90doi: 10.1111/j.1467-999X.2008.00332.x

© 2008 The AuthorJournal compilation © 2008 Blackwell Publishing Ltd, 9600 Garsington Road, Oxford, OX42DQ, UK and 350 Main St, Malden, MA 02148, USA

never indifferent about any pair of two alternatives. In that paper, we used a6 ¥ 6 HEX game. On the other hand, in this paper we will show the equiva-lence in the case where individual preferences are weak orders, that is, indi-viduals may be indifferent about any pair of two alternatives. And in thispaper we will use a 13 ¥ 13 HEX game. The result of this paper is not directlyderived from the result of the previous paper. The chief differences are asfollows:

In the proof of Theorem 2, we consider the social preference, properly speaking, thesociety’s most preferred alternative at the following profiles of individual prefer-ences about three alternatives x1, x2 and x3.

p p p p p pk k k k k k7 4 8 3 9 4123 321 123 312 132 3, , , , , , , ,− − −( ) = ( ) ( ) = ( ) ( ) = 221

213 321 312 23112 4 10 5 9 5

( )( ) = ( ) ( ) = ( )− − −

,

, , , , , , ,p p p p p pk k k k k k(( ) = ( )( ) = ( ) ( ) = ( )− −

132 231

132 312 231 3219 3 11 4

, ,

, , , , , ,p p p p pk k k k kk kp13 4 123 321, ,−( ) = ( )

(p pk k7 4, − ) denotes a profile such that one individual (denoted by k) prefers x1 to x3,

prefers x2 to x3 and he is indifferent between x1 and x2, and all other individuals(denoted by -k) prefer x3 to x2 to x1. (p pk k

12 4, − ) denotes a profile such that individualk prefers x2 to x1, prefers x2 to x3 and he is indifferent between x1 and x3, and all otherindividuals prefer x3 to x2 to x1, and so on. The social preferences at these profiles aredetermined by the social preferences at other profiles such that the preferences ofall individuals are strong orders, and the conditions of transitivity, Pareto principleand IIA.

Because Gale (1979) showed that the Brouwer fixed point theorem isequivalent to the HEX game theorem, this paper indirectly shows theequivalence of the Brouwer fixed point theorem and the Arrow impossibil-ity theorem. In another paper Tanaka (2006), we directly showed thisequivalence using a model according to Baryshnikov (1993). But in thatpaper we used techniques of algebraic topology (homology theory). Topo-logical approaches to social choice problems have been initiated by Chich-ilnisky (1979, 1980). In her model, a space of alternatives is a subset ofEuclidean space, and individual preferences over this set are represented bynormalized gradient fields. Her main result in Chichilnisky (1980) is animpossibility theorem that there exists no continuous social choice rulewhich satisfies unanimity and anonymity. Unanimity is weaker than Paretoprinciple, and anonymity is stronger than the condition of non-existence ofdictator. In Chichilnisky (1979) she showed the equivalence of her impos-sibility theorem and the Brouwer fixed point theorem in the case wherethere are two individuals and the choice space is a subset of 2-dimensionalEuclidian space. Baryshnikov (1993) presented a topological approach to

78 Yasuhito Tanaka

© 2008 The AuthorJournal compilation © 2008 Blackwell Publishing Ltd

the Arrow impossibility theorem in a discrete framework of social choice,and showed that the impossibility theorem in Chichilnisky (1980) and theArrow impossibility theorem are very similar from the point of view ofalgebraic topology. Thus, Chichilnisky (1979, 1980) and Baryshnikov(1993) are precedents for the result—linking the Arrow impossibilitytheorem to a fixed point theorem. See also Chichilnisky (1993) in which hermain results are summarized.

In the next section, according to Gale (1979) we present an outline of theHEX game. In section 3, we will show that the HEX game theorem impliesthe Arrow impossibility theorem when individual preferences are weakorders. And in section 4, we will show that the Arrow impossibility theoremimplies the HEX game theorem.

2. THE HEX GAME

According to Gale (1979), we present an outline of the HEX game. Figure 1represents a 13 ¥ 13 HEX board. Generally, a HEX game is represented byan n ¥ n HEX board where n is a finite positive integer. The rules of the gameare as follows. Two players (called Mr W and Mr B) move alternately,marking any previously unmarked hexagon or tile with a white (by Mr W) ora black (by Mr B) circle, respectively. The game has been won by Mr W (orMr B) if he has succeeded in marking a connected set of tiles which meets theboundary regions W and W′ (or, B and B′). A set S of tiles is connected if any

Figure 1. HEX game.

HEX Game Theorem and Arrow Impossibility Theorem 79


two members of the set h and h′ can be joined by a path P = (h = h1, h2, . . . ,hm = h′) where hi and hi+1 are adjacent. Figure 2 represents a HEX game whichhas been won by Mr B.

About the HEX game Gale (1979) has shown the following theorem.

Theorem 1 (The HEX game theorem): If every tile of the HEX board ismarked by either a white or a black circle, then there is a path connectingregions W and W�, or a path connecting regions B and B�.

Actually he has shown the theorem that any HEX game can never end ina draw, and there always exists at least one winner. But, from his intuitiveexplanation using the following example of river and dam, it is clear thatthere exists only one winner of any HEX game.

Imagine that B and B′ regions are portions of opposite banks of the riverwhich flow from W region to W′ region, and that Mr B is trying to build adam by putting down stones. He will have succeeded in damming the river ifand only if he has placed his stones in a way which enables him to walk onthem from one bank (B region) to the other (B′ region).

The proof of Theorem 1 and also the above intuitive argument do notdepend on the rule ‘two players move alternately’. Therefore, this theorem isvalid for any marking rule.

Figure 3 is obtained by plotting the center of each hexagon, and connect-ing these centers by lines. Rotating this graph 45° in anticlockwise direction,we obtain Figure 4. It is an equivalent representation of the HEX boarddepicted in Figure 1. W and W′ represent the regions of Mr W, and B and B′

Figure 2. HEX game won by Mr B.

80 Yasuhito Tanaka


represent the regions of Mr B. We call it a square HEX board, and call agame represented by a square HEX board a square HEX game. In Figure 4we depict an example of winning marking by Mr B. It corresponds to themarking pattern in Figure 2. A set of marked vertices which represents oneplayer’s victory is called a winning path.

Figure 3. Conversion to square HEX game.

Figure 4. Square HEX game with a winning path.



3. THE HEX GAME THEOREM IMPLIES THE ARROWIMPOSSIBILITY THEOREM

There are m(�3) alternatives and n(�2) individuals. m and n are finite posi-tive integers. The set of individuals is denoted by N. Denote individual i’spreference by pi. A combination of individual preferences, which is called aprofile, is denoted by p(= (p1, p2, . . . , pn)). The set of profiles is denotedby P n. The alternatives are represented by xi, i = 1, 2, . . . , m. Individualpreferences over the alternatives are weak orders, that is, individuals strictlyprefer one alternative to another, or are indifferent between them. We assumethe free triple property, that is, for each set of three alternatives individualpreferences are never restricted.

We consider a binary social choice rule which determines a social prefer-ence corresponding to each profile. Binary social choice rules must satisfy theconditions of transitivity, Pareto principle and IIA. Transitive binary socialchoice rules are called social welfare functions. The meanings of these condi-tions are as follows:

Transitivity If, according to a social welfare function, the society prefers analternative xi to another alternative xj, and prefers xj to another alternativexk, then the society must prefer xi to xk.

Pareto principle When all individuals prefer xi to xj, the society must preferxi to xj.

Independence of irrelevant alternatives (IIA) The social preference aboutevery pair of two alternatives xi and xj is determined by only individualpreferences about these alternatives. Individual preferences about other alter-natives do not affect the social preference about xi and xj.

The Arrow impossibility theorem states that there exists no social welfarefunction which satisfies Pareto principle and IIA and has no dictator, or inother words, there exists a dictator for any social welfare function satisfyingPareto principle and IIA. A dictator is an individual whose strict preferencealways coincides with the social preference.

According to Sen (1979), we define the following terms.

Almost decisiveness If, when all individuals in a group G prefer an alterna-tive xi to another alternative xj, and the other individuals (individuals in N\G)prefer xj to xi, the society prefers xi to xj, then G is almost decisive for xi

against xj.

Decisiveness If, when all individuals in a group G prefer an alternative xi toanother alternative xj, the society prefers xi to xj regardless of the preferencesof the other individuals, then G is decisive for xi against xj.

82 Yasuhito Tanaka


G may consist of one individual. By Pareto principle, N is almost decisive anddecisive about every pair of alternatives. If for a social welfare function anindividual is decisive about every pair of alternatives, then he is the dictatorof the social welfare function.

Sen (1979) (Lemma 3*a) and Suzumura (2000) (Dictator Lemma) haveshown the following result.

Lemma 1: If one individual is almost decisive for one alternative againstanother alternative, then he is the dictator of the social welfare function.

This lemma holds under the conditions of transitivity, Pareto principle andIIA. The conclusion of this lemma is also valid in the case where not anindividual but a group of individuals are almost decisive for one alternativeagainst another alternative. Thus, the following lemma is derived.

Lemma 2: If a group of individuals G is almost decisive for one alternativeagainst another alternative, then this group is decisive about every pair ofalternatives.

Now we confine us to a subset of profiles P n such that all individualsprefer three alternatives x1, x2 and x3 to all other alternatives. Pareto principleimplies that at all such profiles the society also prefers x1, x2 and x3 to all otheralternatives. We denote individual preferences about x1, x2 and x3 in thissubset of profiles as follows.

p p p p p p

p

1 2 3 4 5 6

7

123 132 312 321 231 213

1

= ( ) = ( ) = ( ) = ( ) = ( ) = ( )

=

, , , , , ,

223 123 132 312 231 2138 9 10 11 12

13

( ) = ( ) = ( ) = ( ) = ( ) = ( )=

, , , , , ,p p p p p

p 1123( )

p1 = (123) represents all preferences such that an individual prefers x1 to x2

to x3 to all other alternatives, p1 123= ( ) represents all preferences such thatan individual prefers x1 to x2 and x3 to all other alternatives and is indifferentbetween x2 and x3, p1 132= ( ) represents all preferences such that anindividual prefers x1 and x3 to x2 to all other alternatives and is indifferentbetween x1 and x3, and so on. p1 123= ( ) represents a preference such that anindividual is indifferent about x1, x2 and x3. Although we confine our argu-ments to such a subset of profiles, Lemma 1 with IIA ensures that an indi-vidual who is almost decisive about a pair of alternatives for this subset ofprofiles is the dictator for all profiles.

From Lemma 2 for the profiles in P n we obtain the following result.



Lemma 3: If two groups G and G�, which are not disjoint, are almost decisiveabout a pair of alternatives, then their intersection G � G� is decisive aboutevery pair of alternatives.

Proof: By Lemma 2 G and G′ are decisive about every pair of alternatives.For three alternatives x1, x2 and x3, we consider the following profile in P n.

(1) Individuals in G\(G � G′) prefer x3 to x1 to x2.(2) Individuals in G′\(G � G′) prefer x2 to x3 to x1.(3) Individuals in G � G′ prefer x1 to x2 to x3.(4) Individuals in N\(G � G′) prefer x3 to x2 to x1.

By the decisiveness of G and G′ and transitivity the society must prefer x1 tox2 to x3. As only individuals in G � G′ prefer x1 to x3 and all other individualsprefer x3 to x1, G � G′ is almost decisive for x1 against x3 under IIA. FromLemma 2, it is decisive about every pair of alternatives. �

Further we confine us to a subset of P n such that all but one individualhave the same preferences, and consider a HEX game between one individual(denoted by individual k) and the set of individuals other than k. Represen-tative profiles are denoted by (p pk

ik

j, − ), i = 1, . . . , 13, j = 1, . . . , 13, wherepk

i is individual k’s preference and p kj− denotes the common preference of

the individuals other than k. We relate these profiles to the vertices of a13 ¥ 13 square HEX board as depicted in figure 5. There are 169 vertices inthis HEX board. It represents a square HEX game. k and k′ representindividual k’s regions, and -k and -k′ represent the regions of the set ofindividuals other than k.

We consider the following marking and winning rules of the square HEXgame.

(1) At a profile represented by a vertex of a square HEX board, if thesociety’s unique most preferred alternative is the same as that of indi-vidual k and different from that of the individuals other than k, or that ofthe individuals other than k is not unique, then this vertex is marked bya white circle; conversely if the society’s unique most preferred alternativeis the same as that of the individuals other than k and different from thatof individual k, or that of individual k is not unique, then this vertex ismarked by a black circle.Hereafter we abbreviate ‘the most preferred alternative’ as MPA.

(2) The vertex which corresponds to any other profile is randomly marked bya white or a black circle.

84 Yasuhito Tanaka


(3) The game has been won by individual k (or the set of individuals otherthan k) if he has (or they have) succeeded in marking a connected set ofvertices which meets the boundary regions k and k′ (or -k and -k′).

As a preliminary result we show.

Lemma 4 (Lemma 1 in Baryshnikov, 1993): If the individual preferences aboutx1, x2 and x3 are strong (linear) orders, that is, the individuals are not indif-ferent about any pair of theses alternatives, then the society’s preference aboutthese alternatives are also strong order.

Proof: Assume that at a profile p individual k prefers x1 to x2 and the otherindividuals prefer x2 to x1 and the society is indifferent between them.Suppose that at a profile p′ individual k prefers x1 to x2 to x3 and the otherindividuals prefer x2 to x3 to x1, and at a profile p″ individual k prefers x1 tox3 to x2 and the other individuals prefer x3 to x2 to x1. By Pareto principle andIIA the society should prefer x1 and x2 to x3 and should be indifferentbetween x1 and x2 at p′, and it should prefer x3 to x1 and x2 and should beindifferent between x1 and x2 at p″. Thus the ranking of x1 and x3 depends onthe position of x2 in the preferences of individuals. It is a contradiction.

We can show other cases by similar procedures. �

Figure 5. HEX game representing profiles.



From this lemma when the individual preferences about x1, x2 and x3 arestrong orders, the society’s preference about these alternatives is also strongorder.

A square HEX game is equivalent to the original HEX game. Therefore,there exists one winner for any marking rule. Now we show the followingresult.

Theorem 2: The HEX game theorem implies the existence of a dictator for anysocial welfare function.

Proof: As diagonal vertices are not connected, the diagonal path

p p p p p pk k k k k k1 1 2 2 13 13, , , , , ,− − −( ) ( ) ( )( )�

can not be a winning path. Consider a profile p pk k1 6 123 213, ,−( ) = ( ) ( )( ). By

Pareto principle x3 is not the society’s MPA. Suppose that at this profile thesociety’s MPA is x2 which is the MPA of the individuals other than k. Then,by Pareto principle and IIA the society’s MPA at the following profiles is x2.

p p p pk k k k1 5 2 6, , ,− −( ) ( )

The fact that the society’s MPA at a profile (p pk k2 6, − ) is x2, with Pareto

principle and IIA, implies that the society’s MPA at the following profilesis x2.

p p p p p pk k k k k k3 5 4 5 4 6, , , , ,− − −( ) ( ) ( )

Similarly consider a profile p pk k3 2 312 132, ,−( ) = ( ) ( )( ). By Pareto principle

x2 is not the society’s MPA. Suppose that at this profile the society’s MPA isx1 which is the MPA of the individuals other than k. Then, by Paretoprinciple and IIA the society’s MPA at the following profiles is x1.

p p p pk k k k3 1 4 2, , ,− −( ) ( )



p p p p p pk k k k k k5 1 6 1 6 2, , , , ,− − −( ) ( ) ( )

Similarly consider a profile p pk k5 4 231 321, ,−( ) = ( ) ( )( ). By Pareto principle x1

is not the society’s MPA. Suppose that at this profile the society’s MPA is x3

86 Yasuhito Tanaka


which is the MPA of the individuals other than k. Then, by Pareto principleand IIA the society’s MPA at the following profiles is x3.

p p p pk k k k5 3 6 4, , ,− −( ) ( )



p p p p p pk k k k k k1 3 2 3 2 4, , , , ,− − −( ) ( ) ( )

The results noted above mean that the society’s preference about any pairof alternatives among x1, x2 and x3 coincides with the common preference ofthe individuals other than k when the individual preferences do not includeindifference. Then, by Pareto principle and IIA the society’s MPA at thefollowing profiles is x3.

p p p p p pk k k k k k7 4 8 3 9 4123 321 123 312 132 3, , , , , , , ,− − −( ) = ( ) ( ) = ( ) ( ) = 221

213 32112 4

( )( ) = ( )−

,

, ,p pk k

And the society’s MPA at the following profiles is x2.

p p p pk k k k10 5 9 5312 231 132 231, , , , ,− −( ) = ( ) ( ) = ( )

Further, these results imply that the society’s MPA at the following profilesis x3.

p p p p p pk k k k k k9 3 11 4 13 4132 312 231 321 123, , , , , , ,− − −( ) = ( ) ( ) = ( ) ( ) = ,, 321( )

The vertices which correspond to all of these profiles are marked by blackcircles. Then, we obtain a marking pattern of a square HEX board asdepicted in Figure 6. The set of individuals other than k is the winner of thisHEX game. Therefore, for individual k to be the winner of a square HEXgame, the society’s MPA must coincide with that of individual k at least atone of three profiles (p pk k

1 6, − ), (p pk k3 2, − ) and (p pk k

5 4, − ). It means that individualk must be almost decisive about at least one pair of alternatives, and then byLemma 1 he is the dictator.

If for all k, (k = 1, 2, . . . , n - 1), individual k is not the winner of anysquare HEX game between individual k and the set of individuals other thank, then each set of individuals excluding one individual is the winner of eachsquare HEX game. By Lemma 3 every non-empty intersection of the sets ofindividuals excluding one individual (among 1 to n - 1) is decisive. Then, the



intersection of N\{1}, N\{2}, . . . , N\{n - 1} is decisive. But (N\{1}) �(N\{2}) � . . . (N\{n - 1}) = {n}. Thus, individual n is the dictator. There-fore, the HEX game theorem implies the existence of a dictator for any socialwelfare function. �

By this theorem the HEX game theorem implies the Arrow impossibilitytheorem.

4. THE ARROW IMPOSSIBILITY THEOREM IMPLIES THE HEXGAME THEOREM

Next we will show that the Arrow impossibility theorem implies the HEXgame theorem under an interpretation of dictator. Similarly to the previoussection, we confine us to a subset of profiles such that all individuals preferthree alternatives x1, x2 and x3 to all other alternatives, and the preferences ofindividuals other than one individual (denoted by k) are the same. And weconsider a square HEX game between individual k and the set of individualsother than k. The dictator of a social welfare function is interpreted as anindividual who can determine the MPA of the society when his unique MPAand that of the other individuals are different, or his MPA is unique and that

Figure 6. Winning path of a square HEX game.

88 Yasuhito Tanaka


of the other individuals is not unique, and in a HEX game he can mark tileswith his color in such cases. We denote a vertex of a square HEX board whichcorresponds to a profile (p pk

ik

j, − ) simply by (p pki

kj, − ).

First, consider the case where individual k is the dictator of a social welfarefunction. Then, the following vertices are marked by white circles.

p p p p p p p p p p pk k k k k k k k k k k2 3 2 4 2 5 3 1 3 2 3, , , , , , , , , ,− − − − −( ) ( ) ( ) ( ) ( ) ,, , ,

, , , , , , ,

p p p

p p p p p p p p

k k k

k k k k k k k

− −

− − − −

( ) ( )( ) ( ) ( )

5 3 6

3 7 3 8 3 9 3kk k k k k k k

k k

p p p p p p

p p

11 3 12 3 13 2 10

2 11

( ) ( ) ( ) ( )( )

− − −

−

, , , , , ,

,

Then, we obtain Figure 7. Clearly individual k is the winner of this HEXgame.

Next, consider the case where the dictator of a social welfare function isincluded in the set of individuals other than k. Then, by symmetric consid-eration the set of individuals other than k is the winner of the HEX game.

Thus, the existence of a dictator for a social welfare function implies theexistence of a winner for a HEX game. Therefore, we obtain

Theorem 3: The Arrow impossibility theorem and the HEX game theorem areequivalent.

Figure 7. HEX game won by individual k.



5. CONCLUDING REMARKS

We have considered the relationship between the HEX game theorem and theArrow impossibility theorem when individual preferences are weak orders,and have shown their equivalence. We are now proceeding research onexpansions of the idea of this paper to other theorems of social choice theorysuch as the Gibbard-Satterthwaite theorem for social choice functions.

REFERENCES

Arrow, K. J. (1963): Social Choice and Individual Values, 2nd edn, Yale University Press, NewHaven, CT.

Baryshnikov, Y. (1993): ‘Unifying impossibility theorems: a topological approach’, Advances inApplied Mathematics, 14, pp. 404–15.

Chichilnisky, G. (1979): ‘On fixed point theorems and social choice paradoxes’, EconomicsLetters, 3, pp. 347–51.

Chichilnisky, G. (1980): ‘Social choice and the topology of spaces of preferences’, Advances inMathematics, 37, pp. 165–76.

Chichilnisky, G. (1993): ‘Intersecting families of sets and the topology of cones in economics’,Bulletin of the American Mathematical Society (N.S.), 29, pp. 189–207.

Gale, D. (1979): ‘The game of HEX and the Brouwer fixed-point theorem’, American Math-ematical Monthly, 86, pp. 818–27.

Sen, A. (1979): Collective Choice and Social Welfare, North-Holland, Amsterdam.Suzumura, K. (2000): ‘Welfare economics beyond welfarist-consequentialism’, Japanese Eco-

nomic Review, 51, pp. 1–32.Tanaka, Y. (2006): ‘On the equivalence of the Arrow impossibility theorem and the Brouwer

fixed point theorem’, Applied Mathematics and Computation, 172, pp. 1303–14.Tanaka, Y. (2007): ‘Equivalence of the HEX game theorem and the Arrow impossibility

theorem’, Applied Mathematics and Computation, 186, pp. 509–15.

Yasuhito TanakaDepartment of Faculty of EconomicsKamigyo-kuKyoto 602-8580JapanE-mail: [email protected]

90 Yasuhito Tanaka


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