Embury, Brian Leonard

### Description

In recently published papers, Gibbard (1973) and
Satterthwaite (1975) have proved that for any issue containing at
least three alternatives and under any voting procedure which is
decisive, resolute, non-imposed and non-dictatorial there will always
exist at least one situation in which at least one individual can
secure an outcome which he prefers by following a strategy which does
not reflect his sincere preferences. In a similar vein, Pattanaik
(1973-76) has shown that under very wide...[Show more] classes of voting procedures
there will always exist at least one issue such that at least one
sincere situation will not constitute an equilibrium or strict
equilibrium and hence there will exist at least one individual or at
least one coalition of individuals who can obtain an outcome which he
(they) prefers by following a strategy which does not reflect his
(their) sincere preferences. In Pattanaik's terms, these results paint a bleak picture
of the possibility of finding a voting procedure that is strategyproof
or strictly straLegy-proof. However, even if we know that a
given voting procedure is not strategy-proof or is not strictly strategy-proof, we do not know the probability of any given sincere
situation being an equilibrium or strict equilibrium. If the probability
of any given sincere situation being a strict equilibrium is relatively
high then we might not be too disturbed by the negative thrust of the
results established by Gibbard, Satterthwaite and Pattanaik. However,
if the probability of any given sincere situation being an equilibrium
is relatively low then we have every reason to be concerned. I will
provide a partial answer to these and to other related questions.
I have adopted the structure developed by Pattanaik in
Strategy and Group Choice (Pattanaik:1978). Within this structure, I
have studied three non-binary voting procedures which I refer to as the
Plurality, Nanson and Borda functions. I define the Plurality and Borda
functions as finite ranking rules and I define the Nanson function as a
method of exhaustive voting based on a finite ranking operator. Since
these functions are not resolute, I have adopted the relatively strong
behavioural assumption of maximin behaviour of individuals. Later in
the analysis, I supplement the Plurality function with a tie-breaking
device and employ a much weaker behavioural assumption. In general, I
have assumed that the sincere and the expressed preferences of all
individuals are strict orderings.
I have proved that a necessary condition for any situation to
be a strict equilibrium under the Plurality, Nanson and Borda functions
La that the choice set defined by each of those functions must be a subset
of the choice set defined by the Majority function for the corresponding
sincere situation; in the case of the Plurality function, the
choice set must agree exactly with the choice set defined by the
Majority function for the corresponding sincere situation. I have established a set of necessary and sufficient conditions
for any given sincere situation to be a strict equilibrium under each of
the Plurality, Nanson and Borda functions. From these results, I have
deduced a set of necessary and sufficient conditions for any given sincere
situation to be an equilibrium under each of these functions. For
the Plurality function only, I prove a number of results concerning the
existence of equilibria which satisfy certain additional demands and I
establish a set of necessary and sufficient conditions for the set of
strict equilibria corresponding to any given sincere situation to be
non-empty.
I have utilized these results in a computer simulation of voting
situations and I have determined the probability of the occurrence of the
above phenomena. In the simulation, I have made the following assumptions
(1) the issue consists of exactly three distinct elements, (2) the sincere
as well as the expressed preferences of all individuals are strict orderings,
(3) the distribution of individual preferences satisfies the equiprobability
assumption and (4) the maximin assumption is satisfied. The
simulation has been executed for numbers of voters ranging from 2 to 50.
I have established that the probability of any given sincere
situation being an equilibrium under each of these functions is relatively
high and increases as the number of voters increases. The probability
of any given sincere situation being a strict equilibrium under the
Plurality and Borda functions is relatively low and decreases as the
number of voters increases; however, under the Nanson function, the
probability is relatively high and appears to increase as the number
of voters increases. The results are presented graphically.

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