We study the problem of allocating a set of indivisible goods among a set of agents with 2-value additive valuations. In this setting, each good is valued either $1$ or $\frac{p}{q}$, for some fixed co-prime numbers $p,q\in N$ such that $1\leq q < p$, and the value of a bundle is the sum of the values of the contained goods. Our goal is to find an allocation which maximizes the Nash social welfare (NSW), i.e., the geometric mean of the valuations of the agents. In this work, we give a complete characterization of polynomial-time tractability of NSW maximization that solely depends on the values of $q$. We start by providing a rather simple polynomial-time algorithm to find a maximum NSW allocation when the valuation functions are integral, that is, $q=1$. We then exploit more involved techniques to get an algorithm producing a maximum NSW allocation for the half-integral case, that is, $q=2$. Finally, we show that such an improvement cannot be further extended to the case $q=3$; indeed, we prove that it is NP-hard to compute an allocation with maximum NSW whenever $q\geq 3$.
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The simple greedy algorithm to find a maximal independent set of a graph can be viewed as a sequential update of a Boolean network, where the update function at each vertex is the conjunction of all the negated variables in its neighbourhood. In general, the convergence of the so-called kernel network is complex. A word (sequence of vertices) fixes the kernel network if applying the updates sequentially according to that word. We prove that determining whether a word fixes the kernel network is coNP-complete. We also consider the so-called permis, which are permutation words that fix the kernel network. We exhibit large classes of graphs that have a permis, but we also construct many graphs without a permis.
Social distance games have been extensively studied as a coalition formation model where the utilities of agents in each coalition were captured using a utility function u that took into account distances in a given social network. In this paper, we consider a non-normalized score-based definition of social distance games where the utility function u_v depends on a generic scoring vector v, which may be customized to match the specifics of each individual application scenario. As our main technical contribution, we establish the tractability of computing a welfare-maximizing partitioning of the agents into coalitions on tree-like networks, for every score-based function u_v. We provide more efficient algorithms when dealing with specific choices of u_v or simpler networks, and also extend all of these results to computing coalitions that are Nash stable or individually rational. We view these results as a further strong indication of the usefulness of the proposed score-based utility function: even on very simple networks, the problem of computing a welfare-maximizing partitioning into coalitions remains open for the originally considered canonical function u.
Estimating the Convex Hull of the Image of a Set with Smooth Boundary: Error Bounds and Applications
We study the problem of estimating the convex hull of the image $f(X)\subset\mathbb{R}^n$ of a compact set $X\subset\mathbb{R}^m$ with smooth boundary through a smooth function $f:\mathbb{R}^m\to\mathbb{R}^n$. Assuming that $f$ is a submersion, we derive a new bound on the Hausdorff distance between the convex hull of $f(X)$ and the convex hull of the images $f(x_i)$ of $M$ sampled inputs $x_i$ on the boundary of $X$. When applied to the problem of geometric inference from a random sample, our results give tighter and more general error bounds than the state of the art. We present applications to the problems of robust optimization, of reachability analysis of dynamical systems, and of robust trajectory optimization under bounded uncertainty.
Evolutionary games are a developing sub-field of game theory. This branch of game theory is used in the study of the adaptation of large, but finite, populations of agents to changes in the environment. It assumes that each agent has no significant influence on the system. Many scientific areas use the theory of evolutionary games. In particular, it is used in biology, medicine and the modelling of wireless networks. In this paper we study an evolutionary game with two levels of interaction between population agents. At the first level, changes in the population state depend on changes in the environment and on increasing or decreasing the resources available to the agents. At the second level, the populations state changes according to how the agents evaluate the state of the environment. These levels form a hierarchical structure. A change in one parameter of the system, which is responsible for the state of the environment, the population or the opinions of the agents, causes a change in the other elements of the system. The study involves the analysis of a modified evolutionary game taking into account the influence of the environment and the opinions of the agents. It also involves the development of computational methods in MATLAB and two sets of numerical experiments.
It is well known that the Euler method for approximating the solutions of a random ordinary differential equation $\mathrm{d}X_t/\mathrm{d}t = f(t, X_t, Y_t)$ driven by a stochastic process $\{Y_t\}_t$ with $\theta$-H\"older sample paths is estimated to be of strong order $\theta$ with respect to the time step, provided $f=f(t, x, y)$ is sufficiently regular and with suitable bounds. Here, it is proved that, in many typical cases, further conditions on the noise can be exploited so that the strong convergence is actually of order 1, regardless of the H\"older regularity of the sample paths. This applies for instance to additive or multiplicative It\^o process noises (such as Wiener, Ornstein-Uhlenbeck, and geometric Brownian motion processes); to point-process noises (such as Poisson point processes and Hawkes self-exciting processes, which even have jump-type discontinuities); and to transport-type processes with sample paths of bounded variation. The result is based on a novel approach, estimating the global error as an iterated integral over both large and small mesh scales, and switching the order of integration to move the critical regularity to the large scale. The work is complemented with numerical simulations illustrating the strong order 1 convergence in those cases, and with an example with fractional Brownian motion noise with Hurst parameter $0 < H < 1/2$ for which the order of convergence is $H + 1/2$, hence lower than the attained order 1 in the examples above, but still higher than the order $H$ of convergence expected from previous works.
The past decade has witnessed the flourishing of a new profession as media content creators, who rely on revenue streams from online content recommendation platforms. The reward mechanism employed by these platforms creates a competitive environment among creators which affect their production choices and, consequently, content distribution and system welfare. It is thus crucial to design the platform's reward mechanism in order to steer the creators' competition towards a desirable welfare outcome in the long run. This work makes two major contributions in this regard: first, we uncover a fundamental limit about a class of widely adopted mechanisms, coined Merit-based Monotone Mechanisms, by showing that they inevitably lead to a constant fraction loss of the optimal welfare. To circumvent this limitation, we introduce Backward Rewarding Mechanisms (BRMs) and show that the competition game resultant from BRMs possesses a potential game structure. BRMs thus naturally induce strategic creators' collective behaviors towards optimizing the potential function, which can be designed to match any given welfare metric. In addition, the BRM class can be parameterized to allow the platform to directly optimize welfare within the feasible mechanism space even when the welfare metric is not explicitly defined.
Directed acyclic graphs (DAGs) are directed graphs in which there is no path from a vertex to itself. DAGs are an omnipresent data structure in computer science and the problem of counting the DAGs of given number of vertices and to sample them uniformly at random has been solved respectively in the 70's and the 00's. In this paper, we propose to explore a new variation of this model where DAGs are endowed with an independent ordering of the out-edges of each vertex, thus allowing to model a wide range of existing data structures. We provide efficient algorithms for sampling objects of this new class, both with or without control on the number of edges, and obtain an asymptotic equivalent of their number. We also show the applicability of our method by providing an effective algorithm for the random generation of classical labelled DAGs with a prescribed number of vertices and edges, based on a similar approach. This is the first known algorithm for sampling labelled DAGs with full control on the number of edges, and it meets a need in terms of applications, that had already been acknowledged in the literature.
This paper presents a novel Importance Sampling (IS) scheme for estimating distribution tails of performance measures modeled with a rich set of tools such as linear programs, integer linear programs, piecewise linear/quadratic objectives, feature maps specified with deep neural networks, etc. The conventional approach of explicitly identifying efficient changes of measure suffers from feasibility and scalability concerns beyond highly stylized models, due to their need to be tailored intricately to the objective and the underlying probability distribution. This bottleneck is overcome in the proposed scheme with an elementary transformation which is capable of implicitly inducing an effective IS distribution in a variety of models by replicating the concentration properties observed in less rare samples. This novel approach is guided by developing a large deviations principle that brings out the phenomenon of self-similarity of optimal IS distributions. The proposed sampler is the first to attain asymptotically optimal variance reduction across a spectrum of multivariate distributions despite being oblivious to the specifics of the underlying model. Its applicability is illustrated with contextual shortest path and portfolio credit risk models informed by neural networks
This paper concludes five years of AI competitions based on Legends of Code and Magic (LOCM), a small Collectible Card Game (CCG), designed with the goal of supporting research and algorithm development. The game was used in a number of events, including Community Contests on the CodinGame platform, and Strategy Card Game AI Competition at the IEEE Congress on Evolutionary Computation and IEEE Conference on Games. LOCM has been used in a number of publications related to areas such as game tree search algorithms, neural networks, evaluation functions, and CCG deckbuilding. We present the rules of the game, the history of organized competitions, and a listing of the participant and their approaches, as well as some general advice on organizing AI competitions for the research community. Although the COG 2022 edition was announced to be the last one, the game remains available and can be played using an online leaderboard arena.
Bid optimization for online advertising from single advertiser's perspective has been thoroughly investigated in both academic research and industrial practice. However, existing work typically assume competitors do not change their bids, i.e., the wining price is fixed, leading to poor performance of the derived solution. Although a few studies use multi-agent reinforcement learning to set up a cooperative game, they still suffer the following drawbacks: (1) They fail to avoid collusion solutions where all the advertisers involved in an auction collude to bid an extremely low price on purpose. (2) Previous works cannot well handle the underlying complex bidding environment, leading to poor model convergence. This problem could be amplified when handling multiple objectives of advertisers which are practical demands but not considered by previous work. In this paper, we propose a novel multi-objective cooperative bid optimization formulation called Multi-Agent Cooperative bidding Games (MACG). MACG sets up a carefully designed multi-objective optimization framework where different objectives of advertisers are incorporated. A global objective to maximize the overall profit of all advertisements is added in order to encourage better cooperation and also to protect self-bidding advertisers. To avoid collusion, we also introduce an extra platform revenue constraint. We analyze the optimal functional form of the bidding formula theoretically and design a policy network accordingly to generate auction-level bids. Then we design an efficient multi-agent evolutionary strategy for model optimization. Offline experiments and online A/B tests conducted on the Taobao platform indicate both single advertiser's objective and global profit have been significantly improved compared to state-of-art methods.
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