Generalized Nash stability of voting situations under the Plurality, Nanson and Borda functions

Abstract

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 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. Show more