Facets of the Fully Mixed Nash Equilibrium Conjecture

In this work, we continue the study of the many facets of the Fully Mixed Nash Equilibrium Conjecture, henceforth abbreviated as the FMNE Conjecture, in selfish routing for the special case of n identical users over two (identical) parallel links. We introduce a new measure of Social Cost, defined to be the expectation of the square of the maximum congestion on a link; we call it Quadratic Maximum Social Cost. A Nash equilibrium is a stable state where no user can improve her (expected) latency by switching her mixed strategy; a worst-case Nash equilibrium is one that maximizes Quadratic Maximum Social Cost. In the fully mixed Nash equilibrium, allmixed strategies achieve full support. Formulated within this framework is yet another facet of the FMNE Conjecture, which states that the fully mixed Nash equilibrium is the worst-case Nash equilibrium. We present an extensive proof of the FMNE Conjecture; the proof employs a mixture of combinatorial arguments and ana-lytical estimations. Some of these analytical estimations are derived through some new bounds on generalized medians of the binomial distribution [22] we obtain, which are of independent interest.

Nash equilibria, gale strings, and perfect matchings

This thesis concerns the problem 2-NASH of finding a Nash equilibrium of a bimatrix game, for the special class of so-called “hard-to-solve” bimatrix games. The term “hardto-solve” relates to the exponential running time of the famous and often used Lemke– Howson algorithm for this class of games. The games are constructed with the help of dual cyclic polytopes, where the algorithm can be expressed combinatorially via labeled bitstrings defined by the “Gale evenness condition” that characterise the vertices of these polytopes. We define the combinatorial problem “Another completely labeled Gale string” whose solutions define the Nash equilibria of any game defined by cyclic polytopes, including the games where the Lemke–Howson algorithm takes exponential time. We show that “Another completely labeled Gale string” is solvable in polynomial time by a reduction to the “Perfect matching” problem in Euler graphs. We adapt the Lemke–Howson algorithm to pivot from one perfect matching to another and show that again for a certain class of graphs this leads to exponential behaviour. Furthermore, we prove that completely labeled Gale strings and perfect matchings in Euler graphs come in pairs and that the Lemke–Howson algorithm connects two strings or matchings of opposite signs. The equivalence between Nash Equilibria of bimatrix games derived from cyclic polytopes, completely labeled Gale strings, and perfect matchings in Euler Graphs implies that counting Nash equilibria is #P-complete. Although one Nash equilibrium can be computed in polynomial time, we have not succeeded in finding an algorithm that computes a Nash equilibrium of opposite sign. However, we solve this problem for certain special cases, for example planar graphs. We illustrate the difficulties concerning a general polynomial-time algorithm for this problem by means of negative results that demonstrate why a number of approaches towards such an algorithm are unlikely to be successful

Three Puzzles on Mathematics, Computation, and Games

In this lecture I will talk about three mathematical puzzles involving mathematics and computation that have preoccupied me over the years. The first puzzle is to understand the amazing success of the simplex algorithm for linear programming. The second puzzle is about errors made when votes are counted during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure

Moral hazard and private monitoring.

1This paper incorporates earlier work by Bhaskar [4] and unpublished notes by van Damme. We are grateful to Tilman Börgers, Dilip Mookherjee, Debraj Ray, an anonymous referee, an associate editor, and numerous seminar audiences for useful comments. The first author thanks the CentER for Economic Research (Tilburg) for its hospitality while some of this research was carried out.

Network Formation and Social Coordination

This paper develops a simple model to examine the interaction between partner choice and individual behavior in games of coordination. An important ingredient of our approach is the way we model partner choice: we suppose that a player can establish ties with other players by unilaterally investing in costly pair-wise links. In this context, individual efforts to balance the costs and benefits of links are shown to lead to a unique equilibrium interaction architecture. The dynamics of network formation, however, has powerful effects on individual behavior: if costs of forming links are below a certain threshold then players coordinate on the risk-dominant action, while if costs are above this threshold then they coordinate on the efficient action. These findings are robust to a variety of modifications in the link formation process. For example, it may be posited that, in order for a link to materialize, the link proposal must be two-sided (i.e. put forward by both agents); or that, in case of a unilateral proposal, the link may be refused by the other party (if, say, the latter's net payoff is negative); or that a pair of agents can play the game even if connected only through indirect links.Networks, Links, Coordination games, Equilibrium selection, Risk dominance, Efficiency

Tropical Positivity and Semialgebraic Sets from Polytopes

This dissertation presents recent contributions in tropical geometry with a view towards positivity, and on certain semialgebraic sets which are constructed from polytopes. Tropical geometry is an emerging field in mathematics, combining elements of algebraic geometry and polyhedral geometry. A key in establishing this bridge is the concept of tropicalization, which is often described as mapping an algebraic variety to its 'combinatorial shadow'. This shadow is a polyhedral complex and thus allows to study the algebraic variety by combinatorial means. Recently, the positive part, i.e. the intersection of the variety with the positive orthant, has enjoyed rising attention. A driving question in recent years is: Can we characterize the tropicalization of the positive part? In this thesis we introduce the novel notion of positive-tropical generators, a concept which may serve as a tool for studying positive parts in tropical geometry in a combinatorial fashion. We initiate the study of these as positive analogues of tropical bases, and extend our theory to the notion of signed-tropical generators for more general signed tropicalizations. Applying this to the tropicalization of determinantal varieties, we develop criteria for characterizing their positive part. Motivated by questions from optimization, we focus on the study of low-rank matrices, in particular matrices of rank 2 and 3. We show that in rank 2 the minors form a set of positive-tropical generators, which fully classifies the positive part. In rank 3 we develop the starship criterion, a geometric criterion which certifies non-positivity. Moreover, in the case of square-matrices of corank 1, we fully classify the signed tropicalization of the determinantal variety, even beyond the positive part. Afterwards, we turn to the study of polytropes, which are those polytopes that are both tropically and classically convex. In the literature they are also established as alcoved polytopes of type A. We describe methods from toric geometry for computing multivariate versions of volume, Ehrhart and h^*-polynomials of lattice polytropes. These algorithms are applied to all polytropes of dimensions 2,3 and 4, yielding a large class of integer polynomials. We give a complete combinatorial description of the coefficients of volume polynomials of 3-dimensional polytropes in terms of regular central subdivisions of the fundamental polytope, which is the root polytope of type A. Finally, we provide a partial characterization of the analogous coefficients in dimension 4. In the second half of the thesis, we shift the focus to study semialgebraic sets by combinatorial means. Intersection bodies are objects arising in geometric tomography and are known not to be semialgebraic in general. We study intersection bodies of polytopes and show that such an intersection body is always a semialgebraic set. Computing the irreducible components of the algebraic boundary, we provide an upper bound for the degree of these components. Furthermore, we give a full classification for the convexity of intersection bodies of polytopes in the plane. Towards the end of this thesis, we move to the study of a problem from game theory, considering the correlated equilibrium polytope $P_G$ of a game G from a combinatorial point of view. We introduce the region of full-dimensionality for this class of polytopes, and prove that it is a semialgebraic set for any game. Through the use of oriented matroid strata, we propose a structured method for classifying the possible combinatorial types of $P_G$, and show that for (2 x n)-games, the algebraic boundary of each stratum is a union of coordinate hyperplanes and binomial hypersurfaces. Finally, we provide a computational proof that there exists a unique combinatorial type of maximal dimension for (2 x 3)-games.:Introduction 1. Background 2. Tropical Positivity and Determinantal Varieties 3. Multivariate Volume, Ehrhart, and h^*-Polynomials of Polytropes 4. Combinatorics of Correlated Equilibri

A geometric-combinatorial approach to index and stability in bimatrix games

This thesis provides a new geometric-combinatorial construction to characterise the Nash equilibria of a non-degenerate bimatrix game and their indices. Considering a non-degenerate m x n bimatrix game, the construction yields an (m — 1)-simplex X^ that is simplicially divided into (m — l)-simplices, reflecting the best reply structure of player II. Each (m — 1)-simplex in the triangulation is divided into best reply regions of player I. This yields a division of XA into regions with labels 1,..., m. In this representation, the Nash equilibria are represented by completely labelled points, and the index is the local orientation of the m regions around completely labelled points. For a missing label of player I, the Lemke-Howson algorithm follows paths in XA that are defined by m — 1 labels of player I. This representation of bimatrix games is shown to be related to Sperner's Lemma in dimension m — 1. In particular, the existence of Nash equilibria in non-degenerate bimatrix games is equivalent to Brouwer's fixed point theorem. The construction yields a new strategic characterisation of the index, conjectured by Hofbauer (2000). It is shown that a Nash equilibrium in a non-degenerate bimatrix game has index +1 if and only if one can add strategies to the game such that the equilibrium is the unique equilibrium of the extended game. The construction can be extended to outside option equilibrium components in bimatrix games. The characterisation for such components is shown to be similar to the well-known Index Lemma. As a consequence, index zero boundary labellings allow triangulations that do not contain a completely labelled simplex. The game theoretic counterpart applies to outside option equilibrium components. It is shown that an outside option equilibrium component is hyperessential if and only if it has non-zero index. This question had been open for some time. It is also shown how equilibrium components of arbitrary index can be constructed by means of outside options in bimatrix games