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We study the convergence of best-response dynamics in lottery contests. We show that best-response dynamics rapidly converges to the (unique) equilibrium for homogeneous agents but may not converge for non-homogeneous agents, even for two non-homogeneous agents. For $2$ homogeneous agents, we show convergence to an $\epsilon$-approximate equilibrium in $\Theta(\log\log(1/\epsilon))$ steps. For $n \ge 3$ agents, the dynamics is not unique because at each step $n-1 \ge 2$ agents can make non-trivial moves. We consider a model where the agent making the move is randomly selected at each time step. We show convergence to an $\epsilon$-approximate equilibrium in $O(\beta \log(n/(\epsilon\delta)))$ steps with probability $1-\delta$, where $\beta$ is a parameter of the agent selection process, e.g., $\beta = n$ if agents are selected uniformly at random at each time step. Our simulations indicate that this bound is tight., Comment: 29 pages; EC'23

We consider the problem of sharing a set of indivisible goods among agents in a fair manner, namely such that the allocation is envy-free up to any good (EFX). We focus on the problem of computing an EFX allocation in the two-agent case and characterize the computational complexity of the problem for most well-known valuation classes. We present a simple greedy algorithm that solves the problem when the agent valuations are weakly well-layered, a class which contains gross substitutes and budget-additive valuations. For the next largest valuation class we prove a negative result: the problem is PLS-complete for submodular valuations. All of our results also hold for the setting where there are many agents with identical valuations.

Lipschitz games, in which there is a limit $\lambda$ (the Lipschitz value of the game) on how much a player's payoffs may change when some other player deviates, were introduced about 10 years ago by Azrieli and Shmaya. They showed via the probabilistic method that $n$-player Lipschitz games with $m$ strategies per player have pure $\epsilon$-approximate Nash equilibria, for $\epsilon\geq\lambda\sqrt{8n\log(2mn)}$. Here we provide the first hardness result for the corresponding computational problem, showing that even for a simple class of Lipschitz games (Lipschitz polymatrix games), finding pure $\epsilon$-approximate equilibria is PPAD-complete, for suitable pairs of values $(\epsilon(n), \lambda(n))$. Novel features of this result include both the proof of PPAD hardness (in which we apply a population game reduction from unrestricted polymatrix games) and the proof of containment in PPAD (by derandomizing the selection of a pure equilibrium from a mixed one). In fact, our approach implies containment in PPAD for any class of Lipschitz games where payoffs from mixed-strategy profiles can be deterministically computed., Comment: 16 pages, accepted for publication in the 15th International Symposium on Algorithmic Game Theory

We consider the problem of partitioning a line segment into two subsets, so that $n$ finite measures all have the same ratio of values for the subsets. Letting $\alpha\in[0,1]$ denote the desired ratio, this generalises the PPA-complete consensus-halving problem, in which $\alpha=\frac{1}{2}$. Stromquist and Woodall showed that for any $\alpha$, there exists a solution using $2n$ cuts of the segment. They also showed that if $\alpha$ is irrational, that upper bound is almost optimal. In this work, we elaborate the bounds for rational values $\alpha$. For $\alpha = \frac{\ell}{k}$, we show a lower bound of $\frac{k-1}{k} \cdot 2n - O(1)$ cuts; we also obtain almost matching upper bounds for a large subset of rational $\alpha$. On the computational side, we explore its dependence on the number of cuts available. More specifically, 1. when using the minimal number of cuts for each instance is required, the problem is NP-hard for any $\alpha$; 2. for a large subset of rational $\alpha = \frac{\ell}{k}$, when $\frac{k-1}{k} \cdot 2n$ cuts are available, the problem is in PPA-$k$ under Turing reduction; 3. when $2n$ cuts are allowed, the problem belongs to PPA for any $\alpha$; more generally, the problem belong to PPA-$p$ for any prime $p$ if $2(p-1)\cdot \frac{\lceil p/2 \rceil}{\lfloor p/2 \rfloor} \cdot n$ cuts are available., Comment: Accepted to ITCS 2023

his report documents the program and the outcomes of Dagstuhl Seminar 17251 "Game Theory Meets Computational Learning Theory". While there have been many Dagstuhl seminars on various aspects of Algorithmic Game Theory, this was the first one to focus on the emerging field of its intersection with computational learning theory.

We study the consensus-halving problem of dividing an object into two portions, such that each of $n$ agents has equal valuation for the two portions. The $\epsilon$-approximate consensus-halving problem allows each agent to have an $\epsilon$ discrepancy on the values of the portions. We prove that computing $\epsilon$-approximate consensus-halving solution using $n$ cuts is in PPA, and is PPAD-hard, where $\epsilon$ is some positive constant; the problem remains PPAD-hard when we allow a constant number of additional cuts. It is NP-hard to decide whether a solution with $n-1$ cuts exists for the problem. As a corollary of our results, we obtain that the approximate computational version of the Continuous Necklace Splitting Problem is PPAD-hard when the number of portions $t$ is two., Comment: Published in MFCS 2018

There are a number of results saying that for certain "path-following" algorithms that solve PPAD-complete problems, the solution obtained by the algorithm is PSPACE-complete to compute. We conjecture that these results are special cases of a much more general principle, that all such algorithms compute PSPACE-complete solutions. Such a general result might shed new light on the complexity class PPAD. In this paper we present a new PSPACE-completeness result for an interesting challenge instance for this conjecture. Chen and Deng~\cite{CD} showed that it is PPAD-complete to find a trichromatic triangle in a concisely-represented Sperner triangulation. The proof of Sperner's lemma --- that such a solution always exists --- identifies one solution in particular, that is found via a natural "path-following" approach. Here we show that it is PSPACE-complete to compute this specific solution, together with a similar result for the computation of the path-following solution of two-dimensional discrete Brouwer functions., 11 pages, 6 figures

PPAD refers to a class of computational problems for which solutions are guaranteed to exist due to a specific combinatorial principle. The most well-known such problem is that of computing a Nash equilibrium of a game. Other examples include the search for market equilibria, and envy-free allocations in the context of cake-cutting. A problem is said to be complete for PPAD if it belongs to PPAD and can be shown to constitute one of the hardest computational challenges within that class. In this paper, I give a relatively informal overview of the proofs used in the PPAD-completeness results. The focus is on the mixed Nash equilibria guaranteed to exist by Nash's theorem. I also give an overview of some recent work that uses these ideas to show PSPACE-completeness for the computation of specific equilibria found by homotopy methods. I give a brief introduction to related problems of searching for market equilibria., 32 pages, 10 figures, 23rd British Combinatorial Conference

We study the problem of computing approximate Nash equilibria (epsilon-Nash equilibria) in normal form games, where the number of players is a small constant. We consider the approach of looking for solutions with constant support size. It is known from recent work that in the 2-player case, a 1/2-Nash equilibrium can be easily found, but in general one cannot achieve a smaller value of epsilon than 1/2. In this paper we extend those results to the k-player case, and find that epsilon = 1-1/k is feasible, but cannot be improved upon. We show how stronger results for the 2-player case may be used in order to slightly improve upon the epsilon = 1-1/k obtained in the k-player case.

Weighted threshold games are coalitional games in which each player has a weight (intuitively corresponding to its voting power), and a coalition is successful if the sum of its weights exceeds a given threshold. Key questions in coalitional games include finding coalitions that are stable (in the sense that no member of the coalition has any rational incentive to leave it), and finding a division of payoffs to coalition members (an imputation) that is fair. We investigate the computational complexity of such questions for weighted threshold games. We study the \emph{core}, the \emph{least core}, and the \emph{nucleolus}, distinguishing those problems that are polynomial-time computable from those that are NP-hard, and providing pseudopolynomial and approximation algorithms for the NP-hard problems.

This paper addresses the problem of fair equilibrium selection in graphical games. Our approach is based on the data structure called the {\em best response policy}, which was proposed by Kearns et al. \cite{kls} as a way to represent all Nash equilibria of a graphical game. In \cite{egg}, it was shown that the best response policy has polynomial size as long as the underlying graph is a path. In this paper, we show that if the underlying graph is a bounded-degree tree and the best response policy has polynomial size then there is an efficient algorithm which constructs a Nash equilibrium that guarantees certain payoffs to all participants. Another attractive solution concept is a Nash equilibrium that maximizes the social welfare. We show that, while exactly computing the latter is infeasible (we prove that solving this problem may involve algebraic numbers of an arbitrarily high degree), there exists an FPTAS for finding such an equilibrium as long as the best response policy has polynomial size. These two algorithms can be combined to produce Nash equilibria that satisfy various fairness criteria., 25 pages. Short version appears in ACM EC'07

Gaussian processes provide natural non-parametric prior distributions over regression functions. In this paper we consider regression problems where there is noise on the output, and the variance of the noise depends on the inputs. If we assume that the noise is a smooth function of the inputs, then it is natural to model the noise variance using a second Gaussian process, in addition to the Gaussian process governing the noise-free output value. We show that prior uncertainty about the parameters controlling both processes can be handled and that the posterior distribution of the noise rate can be sampled from using Markov chain Monte Carlo methods. Our results on a synthetic data set give a posterior noise variance that well-approximates the true variance.

The Vapnik-Chervonenkis (VC) dimension is a combinatorial measure of a certain class of machine learning problems, which may be used to obtain upper and lower bounds on the number of training examples needed to learn to prescribed levels of accuracy. Most of the known bounds apply to the Probably Approximately Correct (PAC) framework, which is the framework within which we work in this paper. For a learning problem with some known VC dimension, much is known about the order of growth of the sample-size requirement of the problem, as a function of the PAC parameters. The exact value of sample-size requirement is however less well-known, and depends heavily on the particular learning algorithm being used. This is a major obstacle to the practical application of the VC dimension. Hence it is important to know exactly how the sample-size requirement depends on VC dimension, and with that in mind, we describe a general algorithm for learning problems having VC dimension 1. Its sample-size requirement is minimal (as a function of the PAC parameters), and turns out to be the same for all non-trivial learning problems having VC dimension 1. While the method used cannot be naively generalised to higher VC dimension, it suggests that optimal algorithm-dependent bounds may improve substantially on current upper bounds.