Generation of quasi-polynomial functions to count ties

Much research has been undertaken in recent decades with the aim of quantifying the frequency of occurrence of certain types of election outcomes for a given voting rule under fixed assumptions on the distribution. Our interest is in the problem of counting the number of ties in certain type of voting rule. In this paper, we will generate and investigate two distinct methods of counting ties and 3-way ties in the Borda Count Voting Rule on 3-candidate elections. The rest of this paper is organized as follows. In Part I we will introduce the Borda Count Voting rule and Hex-Mean Voting rule, definitions and notations such as central profiles, multiplicity of the center points and so on. We will also state some propositions. Then we will use an elementary method to derive the six functions to count the 3-way ties in six different cases. In Part II we will discuss the feasibility of applying Ehrhart Theory to count both ties and 3-way ties. We will also talk about quasi-polynomials, linear constraints and counting lattice points in a convex polytope. Finally, we will generate the quasi-polynomial by running Latte and MATLAB program