## Can every ranking pattern be obtained?

I address the question of constructing stochastic models that realize an assigned majority graph or more in general an assigned ranking pattern. As regards to the majority graph, I aim to show that this goal can be achieved in terms of hitting times concerning with some one-dependent Markov chains. More precisely, I consider a generic oriented graph without self-loops and 2-cycles, G =([n], E). For any such oriented graph, I can construct a Markov chain and n identically distributed hitting times T_1, ...,T_n, such that the probability of the event {T_i > T_j}, for any i,j = 1,...,n, is larger than $1/2$ if and only if (i,j) is an arc of G, i.e. this stochastic model has the assigned graph as its majority graph. I apply this result to construct a class of games, a generalization of the Penney games, which manifest a paradoxical and unexpected behavior, see [1]. The definition of a ranking pattern can be seen as a multivariate version of a graph. I will show how, for any given ranking pattern, one can explicitly construct stochastic models, in the class of Load-sharing models, which leads to the assigned ranking pattern, see [2]. Such a result has in particular the following application. Consider a set of m candidates and, for any subset A of the m candidates, now let us fix a ranking r_A in an election where only the candidates A are eligible. I wonder whether it is possible to find a population of voters whose preferences, expressed according to the Condorcet’s proposal, give rise to that family of assigned rankings for all the subsets A. The answer to the question is positive, that is, for any ranking pattern one can construct a population of voters that verifies simultaneously the assigned rankings, [3]. References [1] E. De Santis, Ranking graphs through hitting times of Markov chains. Random Structures & Algorithms, 59, (2021), 189–203. [2]E. De Santis, F. Spizzichino, Construction of aggregation paradoxes through load-sharing models. Adv. Appl. Prob., 55, (2023), 223–244. [3]E. De Santis, F. Spizzichino, Construction of voting situations concordant with ranking patterns. Decis. Econ. Finance, 46, (2023), 129–156.

Area: CS53 - Connections between stochastic models and game and voting theory (Emilio De Santis)

Keywords: Penney games, Majority graph, Ranking patterns

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