In view of the intractability of finding a Nash equilibrium, it is important to understand the limits of approximation in this context. A subexponential approximation scheme is known [LMM03], and no approximation better than 1 4 is possible by any algorithm that examines equilibria involving fewer than log n strategies [Alt94]. We give a simple, linear-time algorithm examining just two strategies per player and resulting in a 1 2-approximate Nash equilibrium in any 2-player game. For the more demanding notion of well-supported approximate equilibrium due to [DGP06] no nontrivial bound is known; we show that the problem can be reduced to the case of win-lose games (games with all utilities 0-1), and that an approximation of 5 6 is possible contingent upon a graph-theoretic conjecture.

A note on approximate nash equilibria

We show that deterministic communication complexity can be superlogarithmic in the partition number of the associated communication matrix. We also obtain near-optimal deterministic lower bounds fo...

Deterministic Communication vs. Partition Number.

Information structure design, also sometimes known as signaling or persuasion, is concerned with understanding the effects of information on the outcomes of strategic interactions (the descriptive question), and in characterizing and computing the information sharing strategies which optimize some design objective (the prescriptive question). Both questions are illuminated through the lens of algorithms and complexity, as evidenced by recent work on the topic in the algorithmic game theory community. This monograph is a biased survey of this work, and paints a picture of the current state of progress and challenges ahead.We divide information structure design into single agent and multiple agent models, and further subdivide the multiple agent case into the public channel and private channel modes of information revelation. In each of these three cases, we describe the most prominent models and applications, survey the associated algorithms and complexity results and their structural implications, and outline directions for future work.

Algorithmic information structure design: a survey

We develop a new method to prove communication lower bounds for composed functions of the form $f\circ g^n$, where $f$ is any boolean function on $n$ inputs and $g$ is a sufficiently “hard” two-party gadget. Our main structure theorem states that each rectangle in the communication matrix of $f \circ g^n$ can be simulated by a nonnegative combination of juntas. This is a new formalization for the intuition that each low-communication randomized protocol can only “query” a few inputs of $f$ as encoded by the gadget $g$. Consequently, we characterize the communication complexity of $f\circ g^n$ in all known one-sided (i.e., not closed under complement) zero-communication models by a corresponding query complexity measure of $f$. These models in turn capture important lower bound techniques such as corruption, smooth rectangle bound, relaxed partition bound, and extended discrepancy. As applications, we resolve several open problems from prior work. We show that $\mathsf{SBP}^{\sf cc}$ (a class characterized...

Rectangles Are Nonnegative Juntas.

For any $n$-bit boolean function $f$, we show that the randomized communication complexity of the composed function $f\circ g^n$, where $g$ is an index gadget, is characterized by the randomized decision tree complexity of $f$. In particular, this means that many query complexity separations involving randomized models (e.g., classical vs. quantum) automatically imply analogous separations in communication complexity.

Query-to-Communication Lifting for BPP

On the communication complexity of approximate Nash equilibria

We study the problem of computing approximate Nash equilibria of bimatrix games, in a setting where players initially know their own payoffs but not the payoffs of the other player. In order for a solution of reasonable quality to be found, some amount of communication needs to take place between the players. We are interested in algorithms where the communication is substantially less than the contents of a payoff matrix, for example logarithmic in the size of the matrix. When the communication is polylogarithmic in the number of strategies n, we show how to obtain epsilon-approximate Nash equilibria for epsilon approximately 0.438, and for well-supported approximate equilibria we obtain epsilon approximately 0.732. For one-way communication we show that epsilon=1/2 is achievable, but no constant improvement over 1/2 is possible, even with unlimited one-way communication. For well-supported equilibria, no value of epsilon less than 1 is achievable with one-way communication. When the players do not communicate at all, epsilon-Nash equilibria can be obtained for epsilon=3/4, and we also give a lower bound of slightly more than 1/2 on the lowest constant epsilon achievable.

On the Communication Complexity of Approximate Nash Equilibria

We study the problem of computing approximate Nash equilibria, in a setting where players initially know their own payoffs but not the payoffs of other players. In order for a solution of reasonable quality to be found, some amount of communication needs to take place between the players. We are interested in algorithms where the communication is substantially less than the contents of a payoff matrix, for example logarithmic in the size of the matrix. At one extreme is the case where the players do not communicate at all; for this case (with 2 players having n×n matrices) e-Nash equilibria can be computed for e=3/4, while there is a lower bound of slightly more than 1/2 on the lowest e achievable. When the communication is polylogarithmic in n, we show how to obtain e=0.438. For one-way communication we show that e=1/2 is the exact answer.

On the communication complexity of approximate nash equilibria

Multi-dimensional Bayesian network classifiers are becoming quite popular for multi-label classification. These models have the advantage of a high expressive power, but may induce a prohibitively high runtime of classification. We argue that the high runtime burden originates from their large treewidth. Thus motivated, we present an algorithm for learning multi-classifiers of small treewidth. Experimental results show that these models have a small runtime of classification, without loosing accuracy compared to unconstrained multi-classifiers.

/pdf/multi-classifiers-of-small-treewidth-511uj4qsjo.pdf

Multi-classifiers of Small Treewidth

Monitoring applications of Bayesian networks require computing a sequence of most probable explanations for the observations from a monitored entity at consecutive time steps. Such applications rapidly become impracticable, especially when computations are performed in real time. In this paper, we argue that a sequence of explanations can often be feasibly computed if consecutive time steps share large numbers of observed features. We show more specifically that we can conclude persistence of an explanation at an early stage of propagation. We present an algorithm that exploits this result to forestall unnecessary re-computation of explanations.

/pdf/the-persistence-of-most-probable-explanations-in-bayesian-fayxbg1p5y.pdf

Arnoud Pastink

Papers

On the communication complexity of approximate Nash equilibria

On the Communication Complexity of Approximate Nash Equilibria

On the communication complexity of approximate nash equilibria

Multi-classifiers of Small Treewidth

The persistence of most probable explanations in Bayesian Networks