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In this paper, we investigate the impact of reward schemes and committee sizes motivated by governance systems over blockchain communities. We introduce a model for elections with a binary outcome space where there is a ground truth (i.e., a "correct" outcome), and where stakeholders can only choose to delegate their voting power to a set of delegation representatives (DReps). Moreover, the effort (cost) invested by each DRep positively influences both (i) her ability to vote correctly and (ii) the total delegation that she attracts, thereby increasing her voting power. This model constitutes the natural counterpart of delegated proof-of-stake (PoS) protocols, where delegated stakes are used to elect the block builders. As a way to motivate the representatives to exert effort, a reward scheme can be used based on the delegation attracted by each DRep. We analyze both the game-theoretic aspects and the optimization counterpart of this model. Our primary focus is on selecting a committee that maximizes the probability of reaching the correct outcome, given a fixed monetary budget allocated for rewarding the delegates. Our findings provide insights into the design of effective reward mechanisms and optimal committee structures (i.e., how many DReps are enough) in these PoS-like governance systems.

In the face of limited block size, miners (e.g., in Bitcoin) prioritize high-bid transactions, forming a large part of their revenue. If the block size were to expand significantly, meeting all transaction demand due to infrastructure or protocol improvements, bids could drop to zero or to a minimum fee, reducing mining incentives and potentially affecting security. To address this, Lavi et al. (2022) introduced a monopolistic pricing mechanism where miners only include transactions paying a minimum price, ensuring some revenue but resulting in an unbounded loss in welfare. Nisan (2023) expands this by modeling bidders as patient, who wait indefinitely long for lower prices, causing price fluctuations even with stable demand. In order to capture users' diminishing interest in having their transactions added to the ledger over time, we consider a more realistic setting with quasi-patient users, where only a fraction $\delta \in [0,1]$ of pending transactions remains in the next round. This richer model encompasses both Lavi et al.'s impatient users ($\delta=0$) and Nisan's patient users ($\delta=1$) as special cases. We demonstrate that Nisan's fluctuating dynamics persist for $\delta$ close to 1, while for $\delta$ close to 0, the dynamics resemble the impatient case. For $\delta \in (0,1)$, we establish new bounds on price dynamics, revealing unexpected effects. Unlike the fully patient case, the bounds of the dynamics for $\delta<1$ depend on the demand curve and undergo a "transition phase". For some $\delta$, the model mirrors the fully patient setting, and for smaller $\delta' < \delta$, it stabilizes at the highest monopolist price, thus collapsing to the impatient case. We provide quantitative bounds and analytical results, showing that the bounds for $\delta=1$ are generally not tight for $\delta<1$, and we give guarantees on the minimum ("admission") price for transactions.

We consider long-term equilibria that arise in the tokenomics design of proof-of-stake (PoS) blockchain systems that comprise of users and validators, both striving to maximize their own utilities. Validators are system maintainers who get rewarded with tokens for performing the work necessary for the system to function properly, while users compete and pay with such tokens for getting a desired portion of the system service. We study how the system service provision and suitable rewards schemes together can lead to equilibria with desirable characteristics (1) viability: the system keeps parties engaged, (2) decentralization: multiple validators are participating, (3) stability: the price path of the underlying token used to transact with the system does not change widely over time, and (4) feasibility: the mechanism is easy to implement as a smart contract, i.e., it does not require fiat reserves on-chain for buy back of tokens or to perform bookkeeping of exponentially growing token holdings. Our analysis enables to put forward a novel generic mechanism for blockchain ``monetary policy'' that we call {\em quantitative rewarding} (QR). We investigate how to implement QR in single-token and two-token proof of stake (PoS) blockchain systems. The latter are systems that utilize one token for the users to pay the transaction fees and a different token for the validators to participate in the PoS protocol and get rewarded. Our approach demonstrates a concrete advantage of the two-token setting in terms of the ability of the QR mechanism to be realized effectively and provide good equilibria. Our analysis also reveals an inherent limitation of the single token setting in terms of implementing an effective blockchain monetary policy - a distinction that is, to the best of our knowledge, highlighted for the first time.

Explainable AI aims to render model behavior understandable by humans, which can be seen as an intermediate step in extracting causal relations from correlative patterns. Due to the high risk of possible fatal decisions in image-based clinical diagnostics, it is necessary to integrate explainable AI into these safety-critical systems. Current explanatory methods typically assign attribution scores to pixel regions in the input image, indicating their importance for a model's decision. However, they fall short when explaining why a visual feature is used. We propose a framework that utilizes interpretable disentangled representations for downstream-task prediction. Through visualizing the disentangled representations, we enable experts to investigate possible causation effects by leveraging their domain knowledge. Additionally, we deploy a multi-path attribution mapping for enriching and validating explanations. We demonstrate the effectiveness of our approach on a synthetic benchmark suite and two medical datasets. We show that the framework not only acts as a catalyst for causal relation extraction but also enhances model robustness by enabling shortcut detection without the need for testing under distribution shifts.

We study the problem of estimating a rank-1 additive deformation of a Gaussian tensor according to the \emph{maximum-likelihood estimator} (MLE). The analysis is carried out in the sparse setting, where the underlying signal has a support that scales sublinearly with the total number of dimensions. We show that for Bernoulli distributed signals, the MLE undergoes an \emph{all-or-nothing} (AoN) phase transition, already established for the minimum mean-square-error estimator (MMSE) in the same problem. The result follows from two main technical points: (i) the connection established between the MLE and the MMSE, using the first and second-moment methods in the constrained signal space, (ii) a recovery regime for the MMSE stricter than the simple error vanishing characterization given in the standard AoN, that is here proved as a general result.

In this work, we consider the problem of recovery a planted $k$-densest sub-hypergraph on $d$-uniform hypergraphs. This fundamental problem appears in different contexts, e.g., community detection, average-case complexity, and neuroscience applications as a structural variant of tensor-PCA problem. We provide tight \emph{information-theoretic} upper and lower bounds for the exact recovery threshold by the maximum-likelihood estimator, as well as \emph{algorithmic} bounds based on approximate message passing algorithms. The problem exhibits a typical statistical-to-computational gap observed in analogous sparse settings that widen with increasing sparsity of the problem. The bounds show that the signal structure impacts the location of the statistical and computational phase transition that the known existing bounds for the tensor-PCA model do not capture. This effect is due to the generic planted signal prior that this latter model addresses.

In this work, we establish a frequency-domain framework for analyzing gradient-based algorithms in linear minimax optimization problems; specifically, our approach is based on the Z-transform, a powerful tool applied in Control Theory and Signal Processing in order to characterize linear discrete-time systems. We employ our framework to obtain the first tight analysis of stability of Optimistic Gradient Descent/Ascent (OGDA), a natural variant of Gradient Descent/Ascent that was shown to exhibit last-iterate convergence in bilinear games by Daskalakis et al. \cite{DBLP:journals/corr/abs-1711-00141}. Importantly, our analysis is considerably simpler and more concise than the existing ones. Moreover, building on the intuition of OGDA, we consider a general family of gradient-based algorithms that augment the memory of the optimization through multiple historical steps. We reduce the convergence -- to a saddle-point -- of the dynamics in bilinear games to the stability of a polynomial, for which efficient algorithmic schemes are well-established. As an immediate corollary, we obtain a broad class of algorithms -- that contains OGDA as a special case -- with a last-iterate convergence guarantee to the space of Nash equilibria of the game.

In this work, we establish near-linear and strong convergence for a natural first-order iterative algorithm that simulates Von Neumann's Alternating Projections method in zero-sum games. First, we provide a precise analysis of Optimistic Gradient Descent/Ascent (OGDA) -- an optimistic variant of Gradient Descent/Ascent -- for \emph{unconstrained} bilinear games, extending and strengthening prior results along several directions. Our characterization is based on a closed-form solution we derive for the dynamics, while our results also reveal several surprising properties. Indeed, our main algorithmic contribution is founded on a geometric feature of OGDA we discovered; namely, the limit points of the dynamics are the orthogonal projection of the initial state to the space of attractors. Motivated by this property, we show that the equilibria for a natural class of \emph{constrained} bilinear games are the intersection of the unconstrained stationary points with the corresponding probability simplexes. Thus, we employ OGDA to implement an Alternating Projections procedure, converging to an $\epsilon$-approximate Nash equilibrium in $\widetilde{\mathcal{O}}(\log^2(1/\epsilon))$ iterations. Our techniques supplement the recent work in pursuing last-iterate guarantees in min-max optimization. Finally, we illustrate an -- in principle -- trivial reduction from any game to the assumed class of instances, without altering the space of equilibria.

The realization that selfish interests need to be accounted for in the design of algorithms has produced many contributions in computer science under the umbrella of algorithmic mechanism design. Novel algorithmic properties and paradigms have been identified and studied. Our work stems from the observation that selfishness is different from rationality; agents will attempt to strategize whenever they perceive it to be convenient according to their imperfect rationality. Recent work has focused on a particular notion of imperfect rationality, namely absence of contingent reasoning skills, and defined obvious strategyproofness (OSP) as a way to deal with the selfishness of these agents. Essentially, this definition states that to care for the incentives of these agents, we need not only pay attention about the relationship between input and output, but also about the way the algorithm is run. However, it is not clear what algorithmic approaches must be used for OSP. In this paper, we show that, for binary allocation problems, OSP is fully captured by a combination of two well-known algorithmic techniques: forward and reverse greedy. We call two-way greedy this algorithmic design paradigm. Our main technical contribution establishes the connection between OSP and two-way greedy. We build upon the recently introduced cycle monotonicity technique for OSP. By means of novel structural properties of cycles and queries of OSP mechanisms, we fully characterize these mechanisms in terms of extremal implementations. These are protocols that ask each agent to consistently separate one extreme of their domain at the current history from the rest. Through the connection with the greedy paradigm, we are able to import a host of approximation bounds to OSP and strengthen the strategic properties of this family of algorithms. Finally, we begin exploring the power of two-way greedy for set systems.

Generative models for networks with communities have been studied extensively for being a fertile ground to establish information-theoretic and computational thresholds. In this paper we propose a new toy model for planted generative models called planted Random Energy Model (REM), inspired by Derrida's REM. For this model we provide the asymptotic behaviour of the probability of error for the maximum likelihood estimator and hence the exact recovery threshold. As an application, we further consider the 2 non-equally sized community Weighted Stochastic Block Model (2-WSBM) on $h$-uniform hypergraphs, that is equivalent to the P-REM on both sides of the spectrum, for high and low edge cardinality $h$. We provide upper and lower bounds for the exact recoverability for any $h$, mapping these problems to the aforementioned P-REM. To the best of our knowledge these are the first consistency results for the 2-WSBM on graphs and on hypergraphs with non-equally sized community.

This work introduces a natural variant of the online machine scheduling problem on unrelated machines, which we refer to as the favorite machine model. In this model, each job has a minimum processing time on a certain set of machines, called favorite machines, and some longer processing times on other machines. This type of costs (processing times) arise quite naturally in many practical problems. In the online version, jobs arrive one by one and must be allocated irrevocably upon each arrival without knowing the future jobs. We consider online algorithms for allocating jobs in order to minimize the makespan. We obtain tight bounds on the competitive ratio of the greedy algorithm and characterize the optimal competitive ratio for the favorite machine model. Our bounds generalize the previous results of the greedy algorithm and the optimal algorithm for the unrelated machines and the identical machines. We also study a further restriction of the model, called the symmetric favorite machine model, where the machines are partitioned equally into two groups and each job has one of the groups as favorite machines. We obtain a 2.675-competitive algorithm for this case, and the best possible algorithm for the two machines case.

Catering to the incentives of people with limited rationality is a challenging research direction that requires novel paradigms to design mechanisms and approximation algorithms. Obviously strategyproof (OSP) mechanisms have recently emerged as the concept of interest to this research agenda. However, the majority of the literature in the area has either highlighted the shortcomings of OSP or focused on the "right" definition rather than on the construction of these mechanisms. We here give the first set of tight results on the approximation guarantee of OSP mechanisms for scheduling related machines and a characterization of optimal OSP mechanisms for set system problems. By extending the well-known cycle monotonicity technique, we are able to concentrate on the algorithmic component of OSP mechanisms and provide some novel paradigms for their design., Comment: Polished throughout to improve readability

We consider the problem of sorting $n$ elements in the case of \emph{persistent} comparison errors. In this model (Braverman and Mossel, SODA'08), each comparison between two elements can be wrong with some fixed (small) probability $p$, and \emph{comparisons cannot be repeated}. Sorting perfectly in this model is impossible, and the objective is to minimize the \emph{dislocation} of each element in the output sequence, that is, the difference between its true rank and its position. Existing lower bounds for this problem show that no algorithm can guarantee, with high probability, \emph{maximum dislocation} and \emph{total dislocation} better than $\Omega(\log n)$ and $\Omega(n)$, respectively, regardless of its running time. In this paper, we present the first \emph{$O(n\log n)$-time} sorting algorithm that guarantees both \emph{$O(\log n)$ maximum dislocation} and \emph{$O(n)$ total dislocation} with high probability. Besides improving over the previous state-of-the art algorithms -- the best known algorithm had running time $\tilde{O}(n^{3/2})$ -- our result indicates that comparison errors do not make the problem computationally more difficult: a sequence with the best possible dislocation can be obtained in $O(n\log n)$ time and, even without comparison errors, $\Omega(n\log n)$ time is necessary to guarantee such dislocation bounds. In order to achieve this optimal result, we solve two sub-problems, and the respective methods have their own merits for further application. One is how to locate a position in which to insert an element in an almost-sorted sequence having $O(\log n)$ maximum dislocation in such a way that the dislocation of the resulting sequence will still be $O(\log n)$. The other is how to simultaneously insert $m$ elements into an almost sorted sequence of $m$ different elements, such that the resulting sequence of $2m$ elements remains almost sorted., Comment: 21 pages, 1 figure

This work gives the first natural non-utilitarian problems for which the trivial $n$ approximation via VCG mechanisms is the best possible. That is, no truthful mechanism can be better than $n$ approximate, where $n$ is the number of agents. The problems are the min-max variant of shortest path and (directed) minimum spanning tree mechanism design problems. In these procurement auctions, agents own the edges of a network, and the corresponding edge costs are private. Instead of the total weight of the subnetwork, in the min-max variant we aim to minimize the maximum agent cost., Comment: to appear in Games and Economic Behavior

We present a sorting algorithm for the case of recurrent random comparison errors. The algorithm essentially achieves simultaneously good properties of previous algorithms for sorting $n$ distinct elements in this model. In particular, it runs in $O(n^2)$ time, the maximum dislocation of the elements in the output is $O(\log n)$, while the total dislocation is $O(n)$. These guarantees are the best possible since we prove that even randomized algorithms cannot achieve $o(\log n)$ maximum dislocation with high probability, or $o(n)$ total dislocation in expectation, regardless of their running time., Comment: 15 pages, ISAAC 2017

We consider the well-studied game-theoretic version of machine scheduling in which jobs correspond to self-interested users and machines correspond to resources. Here each user chooses a machine trying to minimize her own cost, and such selfish behavior typically results in some equilibrium which is not globally optimal: An equilibrium is an allocation where no user can reduce her own cost by moving to another machine, which in general need not minimize the makespan, i.e., the maximum load over the machines. We provide tight bounds on two well-studied notions in algorithmic game theory, namely, the price of anarchy and the strong price of anarchy on machine scheduling setting which lies in between the related and the unrelated machine case. Both notions study the social cost (makespan) of the worst equilibrium compared to the optimum, with the strong price of anarchy restricting to a stronger form of equilibria. Our results extend a prior study comparing the price of anarchy to the strong price of anarchy for two related machines (Epstein, Acta Informatica 2010), thus providing further insights on the relation between these concepts. Our exact bounds give a qualitative and quantitative comparison between the two models. The bounds also show that the setting is indeed easier than the two unrelated machines: In the latter, the strong price of anarchy is $2$, while in ours it is strictly smaller., Comment: Extended abstract to appear in COCOA'17

We study the game-theoretic task of selecting mobile agents to deliver multiple items on a network. An instance is given by $m$ messages (physical objects) which have to be transported between specified source-target pairs in a weighted undirected graph, and $k$ mobile heterogeneous agents, each being able to transport one message at a time. Following a recent model by [B\"artschi et al. 2016], each agent $i$ consumes energy proportional to the distance it travels in the graph, where the different rates of energy consumption are given by weight factors $w_i$. We are interested in optimizing or approximating the total energy consumption over all selected agents. Unlike previous research, we assume the weights to be private values known only to the respective agents. We present three different mechanisms which select, route and pay the agents in a truthful way that guarantees voluntary participation of the agents, while approximating the optimum energy consumption by a constant factor. To this end we analyze a previous structural result and an approximation algorithm given by [B\"artschi et al. 2017]. Finally, we show that for some instances in the case of a single package, the sum of the payments can be bounded in terms of the optimum., Comment: 17 pages. An extended abstract of this paper will be published at the Workshop on Algorithmic Approaches for Transportation Modeling, Optimization, and Systems 2017, ATMOS'17

We consider the classical machine scheduling, where $n$ jobs need to be scheduled on $m$ machines, and where job $j$ scheduled on machine $i$ contributes $p_{i,j}\in \mathbb{R}$ to the load of machine $i$, with the goal of minimizing the makespan, i.e., the maximum load of any machine in the schedule. We study inefficiency of schedules that are obtained when jobs arrive sequentially one by one, and the jobs choose themselves the machine on which they will be scheduled, aiming at being scheduled on a machine with small load. We measure the inefficiency of a schedule as the ratio of the makespan obtained in the worst-case equilibrium schedule, and of the optimum makespan. This ratio is known as the \emph{sequential price of anarchy}. We also introduce two alternative inefficiency measures, which allow for a favorable choice of the order in which the jobs make their decisions. As our first result, we disprove the conjecture of Hassin and Yovel claiming that the sequential price of anarchy for $m=2$ machines is at most 3. We show that the sequential price of anarchy grows at least linearly with the number $n$ of players, i.e., we show that $SPoA = \Omega(n)$. Furthermore, we show that there exists an order of the jobs, resulting in makespan that is at most linearly larger than the optimum makespan. To the end, we show that if an authority can change the order of the jobs adaptively to the decisions made by the jobs so far (but cannot influence the decisions of the jobs), then there exists an adaptive ordering in which the jobs end up in an optimum schedule.

We study very simple sorting algorithms based on a probabilistic comparator model. In our model, errors in comparing two elements are due to (1) the energy or effort put in the comparison and (2) the difference between the compared elements. Such algorithms keep comparing pairs of randomly chosen elements, and they correspond to Markovian processes. The study of these Markov chains reveals an interesting phenomenon. Namely, in several cases, the algorithm which repeatedly compares only adjacent elements is better than the one making arbitrary comparisons: on the long-run, the former algorithm produces sequences that are "better sorted". The analysis of the underlying Markov chain poses new interesting questions as the latter algorithm yields a non-reversible chain and therefore its stationary distribution seems difficult to calculate explicitly.

We consider the problem of delivering $m$ messages between specified source-target pairs in a weighted undirected graph, by $k$ mobile agents initially located at distinct nodes of the graph. Each agent consumes energy proportional to the distance it travels in the graph and we are interested in optimizing the total energy consumption for the team of agents. Unlike previous related work, we consider heterogeneous agents with different rates of energy consumption (weights~$w_i$). To solve the delivery problem, agents face three major challenges: \emph{Collaboration} (how to work together on each message), \emph{Planning} (which route to take) and \emph{Coordination} (how to assign agents to messages). We first show that the delivery problem can be 2-approximated \emph{without} collaborating and that this is best possible, i.e., we show that the \emph{benefit of collaboration} is 2 in general. We also show that the benefit of collaboration for a single message is~$1/\ln 2 \approx 1.44$. Planning turns out to be \NP-hard to approximate even for a single agent, but can be 2-approximated in polynomial time if agents have unit capacities and do not collaborate. We further show that coordination is \NP-hard even for agents with unit capacity, but can be efficiently solved exactly if they have uniform weights. Finally, we give a polynomial-time $(4\max\tfrac{w_i}{w_j})$-approximation for message delivery with unit capacities.

We study an independent best-response dynamics on network games in which the nodes (players) decide to revise their strategies independently with some probability. We provide several bounds on the convergence time to an equilibrium as a function of this probability, the degree of the network, and the potential of the underlying games. These dynamics are somewhat more suitable for distributed environments than the classical better- and best-response dynamics where players revise their strategies "sequentially", i.e., no two players revise their strategies simultaneously.

In the classical framework of local distributed network computing, it is generally assumed that the entities executing distributed algorithms are altruistic. However, in various scenarios, the value of the output produced by an entity may have a tremendous impact on its future. This is for instance the case of tasks such as computing maximal independent sets (MIS) in networks. Indeed, a node belonging to the MIS may be later asked more than to a node not in the MIS, e.g., because MIS in networks are often used as backbones to collect, transfer, and broadcast information, which is costly. In this paper, we revisit typical local distributed network computing tasks in the framework of algorithmic game theory. Specifically, we focus on the construction of solutions for locally checkable labeling (LCL) tasks, which form a large class of distributed tasks, including MIS, coloring, maximal matching, etc., and which have been studied for more than 20 years in distributed computing. Given an LCL task, the nodes are collectively aiming at computing a solution, but, at the individual level, every node plays rationally and selfishly with the objective of optimizing its own profit. Surprisingly, the classical frameworks for game theory are not fully appropriate for the purpose of our study. Moreover, we show that classical notions like Nash equilibria may yield algorithms requiring an arbitrarily large number of rounds to converge. Nevertheless, by extending and generalizing core results from game theory, we establish the existence of a so-called trembling-hand perfect equilibria, a subset of Nash equilibria that is well suited to LCL construction tasks. The main outcome of the paper is therefore that, for essentially all distributed tasks whose solutions are locally checkable, there exist construction algorithms which are robust to selfishness.

Counting the number of permutations of a given total displacement is equivalent to counting weighted Motzkin paths of a given area (Guay-Paquet and Petersen, 2014). The former combinatorial problem is still open. In this work, we show that this connection allows to construct efficient algorithms for counting and for sampling such permutations. These algorithms provide a tool to better understand the original combinatorial problem. A by-product of our approach is a different way of counting based on certain building sequences for Motzkin paths, which may be of independent interest., Comment: 19 pages. An extended abstract of this paper will be published at the 27th International Workshop on Combinatorial Algorithms 2016, IWOCA'16

Schummer (Journal of Economic Theory 2000) introduced the concept of bribeproof mechanism which, in a context where monetary transfer between agents is possible, requires that manipulations through bribes are ruled out. Unfortunately, in many domains, the only bribeproof mechanisms are the trivial ones which return a fixed outcome. This work presents one of the few constructions of non-trivial bribeproof mechanisms for these quasi-linear environments. Though the suggested construction applies to rather restricted domains, the results obtained are tight: For several natural problems, the method yields the only possible bribeproof mechanism and no such mechanism is possible on more general domains., Comment: Extended abstract accepted to SAGT 2016. This ArXiv version corrects typos in the proofs of Theorem 7 and Claims 28-29 of prior ArXiv version

Logit-response dynamics (Alos-Ferrer and Netzer, Games and Economic Behavior 2010) are a rich and natural class of noisy best-response dynamics. In this work we revise the price of anarchy and the price of stability by considering the quality of long-run equilibria in these dynamics. Our results show that prior studies on simpler dynamics of this type can strongly depend on a synchronous schedule of the players' moves. In particular, a small noise by itself is not enough to improve the quality of equilibria as soon as other very natural schedules are used.

In this paper we study the inefficiency ratio of stable equilibria in load balancing games introduced by Asadpour and Saberi [3]. We prove tighter lower and upper bounds of 7/6 and 4/3, respectively. This improves over the best known bounds in problem (19/18 and 3/2, respectively). Equivalently, the results apply to the question of how well the optimum for the $L_2$ -norm can approximate the $L_{\infty}$-norm (makespan) in identical machines scheduling.

We present the first general bounds on the mixing time of the Markov chain associated to the logit dynamics for wide classes of strategic games. The logit dynamics with inverse noise beta describes the behavior of a complex system whose individual components act selfishly and keep responding according to some partial ("noisy") knowledge of the system, where the capacity of the agent to know the system and compute her best move is measured by the inverse of the parameter beta. In particular, we prove nearly tight bounds for potential games and games with dominant strategies. Our results show that, for potential games, the mixing time is upper and lower bounded by an exponential in the inverse of the noise and in the maximum potential difference. Instead, for games with dominant strategies, the mixing time cannot grow arbitrarily with the inverse of the noise. Finally, we refine our analysis for a subclass of potential games called graphical coordination games, a class of games that have been previously studied in Physics and, more recently, in Computer Science in the context of diffusion of new technologies. We give evidence that the mixing time of the logit dynamics for these games strongly depends on the structure of the underlying graph. We prove that the mixing time of the logit dynamics for these games can be upper bounded by a function that is exponential in the cutwidth of the underlying graph and in the inverse of noise. Moreover, we consider two specific and popular network topologies, the clique and the ring. For games played on a clique we prove an almost matching lower bound on the mixing time of the logit dynamics that is exponential in the inverse of the noise and in the maximum potential difference, while for games played on a ring we prove that the time of convergence of the logit dynamics to its stationary distribution is significantly shorter.

Best-response mechanisms (Nisan, Schapira, Valiant, Zohar, 2011) provide a unifying framework for studying various distributed protocols in which the participants are instructed to repeatedly best respond to each others' strategies. Two fundamental features of these mechanisms are convergence and incentive compatibility. This work investigates convergence and incentive compatibility conditions of such mechanisms when players are not guaranteed to always best respond but they rather play an imperfect best-response strategy. That is, at every time step every player deviates from the prescribed best-response strategy according to some probability parameter. The results explain to what extent convergence and incentive compatibility depend on the assumption that players never make mistakes, and how robust such protocols are to "noise" or "mistakes"., Comment: In the conference version of this work, we claimed that in a modified version of PageRank games, there exists a subgame which is a potential game and thus our results can be used to obtain a good approximation of the logit dynamics for these games. Unfortunately, this claim was wrong and the logit dynamics for this subgame is in general not easy to analyze

Logit choice dynamics are a family of randomized best response dynamics based on the logit choice function [McFadden, 1974], used for modeling players with limited rationality and knowledge. In this paper we study the all-logit dynamics, where at each time step all players concurrently update their strategies according to the logit choice function. In the well studied one-logit dynamics [Blume, 1993] instead at each step only one randomly chosen player is allowed to update. We study properties of the all-logit dynamics in the context of local interaction games, a class of games that has been used to model complex social phenomena and physical systems. In a local interaction game, players are the vertices of a social graph whose edges are two-player potential games. Each player picks one strategy to be played for all the games she is involved in and the payoff of the player is the sum of the payoffs from each of the games. We prove that local interaction games characterize the class of games for which the all-logit dynamics is reversible. We then compare the stationary behavior of one-logit and all-logit dynamics. Specifically, we look at the expected value of a notable class of observables, that we call decomposable observables. We prove that the difference between the expected values of the observables at stationarity for the two dynamics depends only on the rationality level beta and on the distance of the social graph from a bipartite graph. In particular, if the social graph is bipartite then decomposable observables have the same expected value. Finally, we show that the mixing time of the all-logit dynamics has the same twofold behavior that has been highlighted in the case of the one-logit: for some games it exponentially depends on the rationality level beta, whereas for other games it can be upper bounded by a function independent from beta.