Hatem Smaoui

An example of probability computations under the IAC assumption: The stability of scoring rules

A society facing a choice problem has also to choose the voting rule itself from a set of different possible voting rules. A voting rule is self-selective if it chooses itself when it is also used in choosing the voting rule. A set of voting rules is said to be stable if it contains at least one self-selective voting rule at each profile of preferences on voting rules. We consider in this paper a society which makes a choice from a set of three alternatives {a,b,c} and a set of the three well-known scoring voting rules {Borda, Plurality, Antiplurality}. We will derive an a priori probability for the stability of this triplet of voting rules, under the Impartial Anonymous Culture assumption (IAC). In order to solve this problem, we need to specify Ehrhart polynomials, which count the number of integer points inside a (convex) polytope. We discuss briefly a recent algorithmic solution to this method before applying it. We also discuss the impact of different behavioral assumptions for the voters (consequentialist or nonconsequentialist) on the probability of stability for the triplet {Borda, Plurality, Antiplurality}.

The Condorcet Efficiency of Voting Rules with Mutually Coherent Voter Preferences: A Borda Compromise

The Condorcet Efficiency of a voting rule is defined as the conditional probability that the voting rule elects the Pairwise Majority Rule Winner (PMRW), given that a PMRW exists. Five simple voting rules are considered in this paper: Plurality Rule, Negative Plurality Rule, Borda Rule, Plurality Elimination Rule and Negative Plurality Elimination Rule. In order to study the impact that the presence of degrees of group mutual coherence in voting situations will have on the probability of selecting the PMRW for each of these rules, we develop representations for their Condorcet Efficiency as a function of the proximity of voters preferences on candidates to being perfectly singlepeaked, perfectly single-troughed or perfectly polarized. The results we obtain lead us to appeal for a Borda Compromise.

Monotonicity paradoxes in three-candidate elections using scoring elimination rules

Scoring elimination rules (SER), that give points to candidates according to their rank in voters’ preference orders and eliminate the candidate(s) with the lowest number of points, constitute an important class of voting rules. This class of rules, that includes some famous voting methods such as Plurality Runoff or Coombs Rule, suffers from a severe pathology known as monotonicity paradox or monotonicity failure, that is, getting more points from voters can make a candidate a loser and getting fewer points can make a candidate a winner. In this paper, we study three-candidate elections and we identify, under various conditions, which SER minimizes the probability that a monotonicity paradox occurs. We also analyze some strategic aspects of these monotonicity failures. The probability model on which our results are based is the impartial anonymous culture condition, often used in this kind of study.