Soc Choice Welfare (1994) 11:283-285

Social Choice Welfare

© Springer-Verlag 1994

Further notes on Craven's conjecture

John Craven

University of Kent at Canterbury, Canterbury, Kent CT2 7NZ, UK

Received: 29 January 1993/ Accepted: 8 February 1994

Abstract. This paper extends the work of Fishburn [4] and Abello and Johnson [1] by increasing lower bounds on the number of possible value-restricted preference orderings, and by considering cases in which the number of alter- natives is not a power of 2.

Craven's conjecture (see [3]) is that if there are n alternatives, then 2" - 1 is the maximum number of distinct linear (strict preference) orderings which simulta- neously satisfy the conditions of Sen and Pattanaik [6] which ensure that majority voting is acyclic. The conjecture is true for n - -3 since the orderings must be value restricted. Kim and Roush [5] had in fact demonstrated earlier than the conjecture is false for n >__ 4 for which the maximum is 9 (Fishburn [4] provides a simple example).

Abello and Johnson [1] provide a formula for a lower bound on the number of orderings for general n: this lower bound is 3 . 2 n - 2 - 4 for n > 4. For n - -5 , the bound is 20; for n = 6 the bound is 44. There is no known example of a consistent set of more than 20 orderings when n = 5, but Chameni-Nembua [2] attributes to an unpublished paper of Monjardet an example of 45 orderings when n = 6. Fishburn [4] gives a bound for the maximum for certain cases where n is a power of 2 which exceeds 3 . 2 n - 2 - 4 .

We refer to a set of orderings which are simultaneously value restricted on each triple as consistent: the maximum size of such a set for n alternatives is f(n). For completeness, we define f ( l ) = l and f ( 2 ) = 2 reflecting the total number of orderings of 1 and 2 alternatives. We define g ( ) as the greatest lower bound given in the previous paragraph and so f ( 3 ) = g (3 )= 4, f ( 4 ) = g (4 )= 9, f ( 5 ) _ > g ( 5 ) = 2 0 , f (6) >_ g (6) = 45, f(n)>_g(n)=3.2"-2-4 for n > 6.

I am grateful to Peter Fishburn for comments on an earlier version, and for drawing my attention to earlier papers on this topic.

284

Table 1. Lower bounds on f (n): previously known [g (n)] and as in (2) [h (n)]

J. Craven

n g(n) h(n) n g(n) h(n)

1 1 2 2 4 4 9 9

20 20 45 45 92 92

188 188

9 10 11 12 13 14 15 16

380 405 764 900

1 532 2 025 3 068 4140 6 140 8 464

12 284 18 225 24 572 40 500 49 148 91 125

Let P (n) be the set of all partitions of { 1,. . . , n } excluding { 1 .. . . . n } itself and the partition into n singletons. Consider Q = { Q1, Q2 .... , Qk } ~ P (n ). Each Qi can be ordered in f ( I Qil ) ways without violating value restriction on any triple in Qi. In addition, we can order the k subsets in each of f ( k ) ways so that if a ~ Qh, b ~ Qi, c ~ Qj then the triple {a, b, c} is value restricted. Finally, if a, b ~ Q~ and c ~ Qj then c is never middle in {a, b, c } however the Q~ are ordered. The partition Q therefore generates a consistent set of

k

~ ° ( Q ) = f ( k ) ' l - I f ( I Q;[) (1) i = 1

orderings of the n alternatives and so f ( n ) >_ max ~0 (Q). P(n)

So for each n we can calculate new lower bounds h (n) from known values as~

h ( n ) = m a x (n) ,maxh(k) ' I -[ h(I Qil) (2) P ( n ) i= 1

Table 1 shows h(n) and g(n) for n_< 16. For n>_8, h(n) > g(n). Define the partition Q of P ( n + m ) as {1 .. . . . n} and m

{ n ÷ 1 },.. . , { n ÷ m }. Then the method gives singletons

f (n ÷rn) >_ ~o (Q) = f (m ÷ 1 ) . f ( n )

and hence the lower bound on the asymptotic growth rate of f ( ) becomes 1

arbitrarily close to max [ f ( m + 1)m]. For the range of m in table 1, the largest m

of these values is 451/5= 2.1411... This method includes Fishburn's computations [4] as a special case in which

all the subsets are the same size and he considers only n for some powers of 2. These computations give f (16) _> 95 = 59 049; we have increased the lower bound for f (16) to 91 125. Further Fishburn's computations give f(256)_> 317o. We have f (256) _> 91 125.4548 = 4551. Since 45 = 3 3.4649..., w e have increased the lower bound on f (256) to 3 176.7....

Further notes on Craven's conjecture 285

References

1. Abello JM, Johnson CR (1984) How large are transitive simple majority domains? SIAM J Algebra Disc Methods 5:603-18

2. Chameni-Nembua C (1989) R6gle majoritaire et distributivit~ dans la permutoedre. Math, Informat Sci Hum 108:5-22

3. Craven J (1992) Social Choice. Cambridge: Cambridge University Press 4. Fishburn P (1992) Notes on Craven's conjecture. Soc Choice Welfare 9:259-62 5. Kim KH, Roush FW (1980) Introduction to mathematical consensus theory. New York:

Marcel Dekker 6. Sen A, Pattanaik PK (1969) Necessary and sufficient conditions for rational choice under

majority decision. J Econ Theory 1:178-202