A simplicial algorithm approach to Nash equilibria in concave games

In this paper we demonstrate a new method for computing approximate Nash equilibria in n-person games. Strategy spaces are assumed to be represented by simplices, while payoff functions are assumed to be concave. Our procedure relies on a simplicial algorithm that traces paths through the set of strategy profiles using a new variant of Sperner's Lemma for labelled triangulations of simplotopes, which we prove in this paper. Our algorithm uses a labelling derived from the satisficing function of Geanakoplos (2003) and can be used to compute approximate Nash equilibria for payoff functions that are not necessarily linear. Finally, in bimatrix games, we can compare our simplicial algorithm to the combinatorial algorithm proposed by Lemke and Howson (1964).simplicial algorithm, Nash equilibria, strategy labelling

A Polytopal Generalization of Sperner\u27s Lemma

We prove the following conjecture of Atanassov (Studia Sci. Math. Hungar.32 (1996), 71–74). Let T be a triangulation of a d-dimensional polytope P with n vertices v1, v2,…,vn. Label the vertices of T by 1,2,…,n in such a way that a vertex of T belonging to the interior of a face F of P can only be labelled by j if vj is on F. Then there are at least n−d full dimensional simplices of T, each labelled with d+1 different labels. We provide two proofs of this result: a non-constructive proof introducing the notion of a pebble set of a polytope, and a constructive proof using a path-following argument. Our non-constructive proof has interesting relations to minimal simplicial covers of convex polyhedra and their chamber complexes, as in Alekseyevskaya (Discrete Math.157 (1996), 15–37) and Billera et al. (J. Combin. Theory Ser. B57 (1993), 258–268)

Sperner\u27s Lemma Implies Kakutani\u27s Fixed Point Theorem

Kakutani’s fixed point theorem has many applications in economics and game theory. One of its most well known applications is in John Nash’s paper [8], where the theorem is used to prove the existence of an equilibrium strategy in n-person games. Sperner’s lemma, on the other hand, is a combinatorial result concerning the labelling of the vertices of simplices and their triangulations. It is known that Sperner’s lemma is equivalent to a result called Brouwer’s fixed point theorem, of which Kakutani’s theorem is a generalization. A natural question that arises is whether we can prove Kakutani’s fixed point theorem directly using Sperner’s lemma without going through Brouwer’s theorem. The objective of this thesis to understand Kakutani’s theorem, Sperner’s lemma, and how they are related. In particular, I explore ways in which Sperner’s lemma can be used to prove Kakutani’s theorem and related results

O teorema de Poincaré-Miranda

Traballo Fin de Grao en Matemáticas. Curso 2019-2020[GL] O teorema de Poincaré-Miranda establece unha condición suficiente para garantir que unha aplicación continua ƒ: → ℜⁿ ten un cero en , onde ⊂ ℜⁿ é un n-cubo e n ∈ N. O obxectivo principal deste traballo é enunciar e demostrar dito resultado, revisando a proba orixinal de Carlo Miranda e proporcionando dúas probas alternativas. Con este fin, introducimos unha serie de conceptos necesarios de combinatoria e topoloxía, incluíndo demostraci óns do lema de Sperner e do teorema do punto fixo de Brouwer. Tamén amosamos unha condición de unicidade e unha posible extensión do teorema de Poincaré-Miranda a dominios máis xerais que o n-cubo. Finalmente, presentamos unha interpretación gráfica do teorema e unha aplicación práctica ao problema dos 1 + 4 corpos.[EN] The Poincaré-Miranda theorem provides a sufficient condition to guarantee that a continuous function ƒ: → ℜⁿ has a zero in , where ⊂ ℜⁿ is an n-cube and n ∈ N. The main objective of this work is to state and prove the aforementioned result, reviewing Carlo Miranda's original proof and describing two alternative proofs. With that goal in mind, we introduce several necessary concepts of combinatorics and topology, including proofs for the Sperner's lemma and the Brouwer's fixed-point theorem. We also produce a uniqueness condition and a possible extension of the Poincaré-Miranda theorem for domains that are more general than n-cubes. Finally, we present a graphical interpretation of the theorem and a practical application to the (1 + 4)-body problem