Abstract: We study the partially eponymous model of distributed computation, which simultaneously
generalizes the anonymous and the eponymous models. In this model, processors have
identities, which are neither necessarily all identical (as in the anonymous model) nor
necessarily unique (as in the eponymous model). In a decision problem formalized as a
relation, processors receive inputs and seek to reach outputs respecting the relation. We
focus on the partially eponymous ring, and we shall consider the computation of circularly
symmetric relations on it. We consider sets of rings where all rings in the set have the same
multiset of identity multiplicities.
We distinguish between solvability and computability: in solvability, processors are
required to always reach outputs respecting the relation; in computability, they must
do so whenever this is possible, and must otherwise report impossibility.
We present a topological characterization of solvability for a relation on a set of rings,
which can be expressed as an efficiently checkable, number-theoretic predicate.
We present a universal distributed algorithm for computing a relation on a set of
rings; it runs any distributed algorithm for constructing views, followed by local steps.
We derive, as our main result, a universal upper bound on the message complexity to
compute a relation on a set of rings; this bound demonstrates a graceful degradation
with the Least Minimum Base, a parameter indicating the degree of least possible
eponymity for a set of rings. Thereafter, we identify two cases where a relation can be
computed on a set of rings, with rings of size n, with an efficient number of O .n lg n/
messages.

Abstract: We study the fundamental problem 2NASH of computing a Nash equilibrium (NE) point in bimatrix games. We start by proposing a novel characterization of the NE set, via a bijective map to the solution set of a parameterized quadratic program (NEQP), whose feasible space is the highly structured set of correlated equilibria (CE). This is, to our knowledge, the first characterization of the subset of CE points that are in “1–1” correspondence with the NE set of the game, and contributes to the quite lively discussion on the relation between the spaces of CE and NE points in a bimatrix game (e.g., [15], [26] and [33]).
We proceed with studying a property of bimatrix games, which we call mutually concavity (MC), that assures polynomial-time tractability of 2NASH, due to the convexity of a proper parameterized quadratic program (either NEQP, or a parameterized variant of the Mangasarian & Stone formulation [23]) for a particular value of the parameter. We prove various characterizations of the MC-games, which eventually lead us to the conclusion that this class is equivalent to the class of strategically zero-sum (SZS) games of Moulin & Vial [25]. This gives an alternative explanation of the polynomial-time tractability of 2NASH for these games, not depending on the solvability of zero-sum games. Moreover, the recognition of the MC-property for an arbitrary game is much faster than the recognition SZS-property. This, along with the comparable time-complexity of linear programs and convex quadratic programs, leads us to a much faster algorithm for 2NASH in MC-games.
We conclude our discussion with a comparison of MC-games (or, SZS-games) to kk-rank games, which are known to admit for 2NASH a FPTAS when kk is fixed [18], and a polynomial-time algorithm for k=1k=1 [2]. We finally explore some closeness properties under well-known NE set preserving transformations of bimatrix games.

Abstract: In view of the apparent intractability of constructing Nash Equilibria (NE
in short) in polynomial time, even for bimatrix games, understanding the limitations
of the approximability of the problem is an important challenge.
In this work we study the tractability of a notion of approximate equilibria in bimatrix
games, called well supported approximate Nash Equilibria (SuppNE in short).
Roughly speaking, while the typical notion of approximate NE demands that each
player gets a payoff at least an additive term less than the best possible payoff, in a
SuppNE each player is assumed to adopt with positive probability only approximate
pure best responses to the opponent¢s strategy.
As a first step, we demonstrate the existence of SuppNE with small supports and
at the same time good quality. This is a simple corollary of Alth{\"o}fer¢s Approximation
Lemma, and implies a subexponential time algorithm for constructing SuppNE of
arbitrary (constant) precision.
We then propose algorithms for constructing SuppNE in win lose and normalized
bimatrix games (i.e., whose payoff matrices take values from {0, 1} and [0, 1] respectively).
Our methodology for attacking the problem is based on the solvability of zero sum bimatrix games (via its connection to linear programming) and provides a
0.5-SuppNE for win lose games and a 0.667-SuppNE for normalized games.
To our knowledge, this paper provides the first polynomial time algorithms constructing
{\aa}-SuppNE for normalized or win lose bimatrix games, for any nontrivial
constant 0 ≤{\aa} <1, bounded away from 1.