We study fair mechanisms for the (asymmetric) one-sided allocation problem with m items and n multi-unit demand agents with additive, unit-sum valuations. The symmetric case (m=n), the one-sided matching problem, has been studied extensively for the class of unit demand agents, in particular with respect to the folklore Random Priority mechanism and the Probabilistic Serial mechanism, introduced by Bogomolnaia and Moulin. Under the assumption of unit-sum valuation functions, Christodoulou et al. proved that the price of anarchy is $Θ(\sqrt{n})$ in the one-sided matching problem for both the Random Priority and Probabilistic Serial mechanisms. Whilst both Random Priority and Probabilistic Serial are ordinal mechanisms, these approximation guarantees are the best possible even for the broader class of cardinal mechanisms.
To extend these results to the general setting there are two technical obstacles. One, asymmetry ($m\neq n$) is problematic especially when the number of items is much greater than the number of items. Two, it is necessary to study multi-unit demand agents rather than simply unit demand agents. Our approach is to study a cardinal mechanism variant of Probabilistic Serial, which we call Cardinal Probabilistic Serial. We present structural theorems for this mechanism and use them to obtain bounds on the price of anarchy. Our first main result is an upper bound of $O(\sqrt{n}\cdot \log m)$ on the price of anarchy for the asymmetric one-sided allocation problem with multi-unit demand agents. This upper bound applies to Probabilistic Serial as well and there is a complementary lower bound of $Ω(\sqrt{n})$ for any fair mechanism. Our second main result is that the price of anarchy degrades with the number of items. Specifically, a logarithmic dependence on the number of items is necessary for both mechanisms.
We consider nonatomic network games with one source and one destination. We examine the asymptotic behavior of the price of anarchy as the inflow increases. In accordance with some empirical observations, we show that, under suitable conditions, the price of anarchy is asymptotic to one. We show with some counterexamples that this is not always the case. The counterexamples occur in very simple parallel graphs.
This paper examines the behavior of the price of anarchy as a function of the traffic inflow in nonatomic congestion games with multiple origin-destination (O/D) pairs. Empirical studies in real-world networks show that the price of anarchy is close to 1 in both light and heavy traffic, thus raising the question: can these observations be justified theoretically? We first show that this is not always the case: the price of anarchy may remain a positive distance away from 1 for all values of the traffic inflow, even in simple three-link networks with a single O/D pair and smooth, convex costs. …
We generalize the notions of user equilibrium and system optimum to non-atomic congestion games with stochastic demands. We establish upper bounds on the price of anarchy for three different settings of link cost functions and demand distributions, namely, (a) affine cost functions and general distributions, (b) polynomial cost functions and general positive-valued distributions, and (c) polynomial cost functions and the normal distributions. All the upper bounds are tight in some special cases, including the case of deterministic demands.
The price of anarchy has become a standard measure of the efficiency of equilibria in games. Most of the literature in this area has focused on establishing worst-case bounds for specific classes of games, such as routing games or more general congestion games. Recently, the price of anarchy in routing games has been studied as a function of the traffic demand, providing asymptotic results in light and heavy traffic. The aim of this paper is to study the price of anarchy in nonatomic routing games in the intermediate region of the demand. To achieve this goal, we begin by establishing some smo…
We consider auctions in which greedy algorithms, paired with first-price or critical-price payment rules, are used to resolve multi-parameter combinatorial allocation problems. We study the price of anarchy for social welfare in such auctions. We show for a variety of equilibrium concepts, including Bayes-Nash equilibrium and correlated equilibrium, the resulting price of anarchy bound is close to the approximation factor of the underlying greedy algorithm.
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…
We study the efficiency of sequential multiunit auctions with two-buyers and complete information. For general valuation functions, we show that the price of anarchy is exactly $1/T$ for auctions with $T$ items for sale. For concave valuation functions, we show that the price of anarchy is bounded below by $1-1/e\simeq 0.632$. This bound is asymptotically tight as the number of items sold tends to infinity.
Game theory has emerged as a fruitful paradigm for the design of networked multiagent systems. A fundamental component of this approach is the design of agents' utility functions so that their self-interested maximization results in a desirable collective behavior. In this work we focus on a well-studied class of distributed resource allocation problems where each agent is requested to select a subset of resources with the goal of optimizing a given system-level objective. Our core contribution is the development of a novel framework to tightly characterize the worst case performance of any re…
We introduce a framework for studying the effect of cooperation on the quality of outcomes in utility games. Our framework is a coalitional analog of the smoothness framework of non-cooperative games. Coalitional smoothness implies bounds on the strong price of anarchy, the loss of quality of coalitionally stable outcomes, as well as bounds on coalitional versions of coarse correlated equilibria and sink equilibria, which we define as out-of-equilibrium myopic behavior as determined by a natural coalitional version of best-response dynamics. Our coalitional smoothness framework captures existi…
Among the many functions a Smart City must support, transportation dominates in terms of resource consumption, strain on the environment, and frustration of its citizens. We study transportation networks under two different routing policies, the commonly assumed selfish user-centric routing policy and a socially-optimal system-centric one. We consider a performance metric of efficiency - the Price of Anarchy (PoA) - defined as the ratio of the total travel latency cost under selfish routing over the corresponding quantity under socially-optimal routing. We develop a data-driven approach to est…
We study Nash equilibria and the price of anarchy in the classic model of Network Creation Games introduced by Fabrikant et al. In this model every agent (node) buys links at a prefixed price $α> 0$ in order to get connected to the network formed by all the $n$ agents. In this setting, the reformulated tree conjecture states that for $α> n$, every Nash equilibrium network is a tree. Moreover, Demaine et al. conjectured that the price of anarchy for this model is constant. Since it was shown that the price of anarchy for trees is constant, if the tree conjecture were true, then the price of ana…
We study Nash equilibria and the price of anarchy in the classic model of Network Creation Games introduced by Fabrikant, Luthra, Maneva, Papadimitriou and Shenker in 2003. This is a selfish network creation model where players correspond to nodes in a network and each of them can create links to the other $n-1$ players at a prefixed price $α> 0$. The player's goal is to minimise the sum of her cost buying edges and her cost for using the resulting network. One of the main conjectures for this model states that the price of anarchy, i.e. the relative cost of the lack of coordination, is consta…
We study cost-sharing games in real-time scheduling systems where the activation cost of the server at any given time is a function of its load. We focus on monomial cost functions and consider both the case when the degree is less than one (inducing positive externalities for the jobs) and when it is greater than one (inducing negative externalities for the jobs). For the former case, we provide tight price of anarchy bounds which show that the price of anarchy grows to infinity as a polynomial of the number of jobs in the game. For the latter, we observe that existing results provide constan…
We consider clustering games in which the players are embedded in a network and want to coordinate (or anti-coordinate) their strategy with their neighbors. The goal of a player is to choose a strategy that maximizes her utility given the strategies of her neighbors. Recent studies show that even very basic variants of these games exhibit a large Price of Anarchy: A large inefficiency between the total utility generated in centralized outcomes and equilibrium outcomes in which players selfishly try to maximize their utility. Our main goal is to understand how structural properties of the netwo…
In opinion formation games with directed graphs, a bounded price of anarchy is only known for weighted Eulerian graphs. Thus, we bound the price of anarchy for a more general class of directed graphs with conditions intuitively meaning that each node does not influence the others more than she is influenced, where the bounds depend on such difference (in a ratio). We also show that there exists an example just slightly violating the conditions with an unbounded price of anarchy.
This paper shows that the PoA in non-atomic congestion games is H{ö}lder continuous w.r.t. combined disturbance on cost functions and demands. We then apply this result to the convergence analysis of the PoA.
We study the inefficiency of equilibria for various classes of games when players are (partially) altruistic. We model altruistic behavior by assuming that player i's perceived cost is a convex combination of 1-α_i times his direct cost and α_i times the social cost. Tuning the parameters α_i allows smooth interpolation between purely selfish and purely altruistic behavior. Within this framework, we study altruistic extensions of linear congestion games, fair cost-sharing games and valid utility games. We derive (tight) bounds on the price of anarchy of these games for several solution concept…
We study {\em bottleneck routing games} where the social cost is determined by the worst congestion on any edge in the network. In the literature, bottleneck games assume player utility costs determined by the worst congested edge in their paths. However, the Nash equilibria of such games are inefficient since the price of anarchy can be very high and proportional to the size of the network. In order to obtain smaller price of anarchy we introduce {\em exponential bottleneck games} where the utility costs of the players are exponential functions of their congestions. We find that exponential b…
Consider an unobservable $M|G|1$ queue with preemptive-resume scheduling and two priority classes. Customers are strategic and may join the premium class for a fee. We analyze the resulting equilibrium outcomes, equilibrium stability, and social welfare. We find that for service distributions with coefficient of variation greater than 1, there exists a unique and stable mixed equilibrium at low loads. We also establish a tight bound on the price of anarchy, which is $4/3$.
We study the price of anarchy of mechanisms in the presence of risk-averse agents. Previous work has focused on agents with quasilinear utilities, possibly with a budget. Our model subsumes this as a special case but also captures that agents might be less sensitive to payments than in the risk-neutral model. We show that many positive price-of-anarchy results proved in the smoothness framework continue to hold in the more general risk-averse setting. A sufficient condition is that agents can never end up with negative quasilinear utility after playing an undominated strategy. This is true, e.…
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.
The Generalized Second Price auction (GSP) has been widely used by search engines to sell ad slots. Previous studies have shown that the pure Price Of Anarchy (POA) of GSP is 1.25 when there are two ad slots and 1.259 when three ad slots. For the cases with more than three ad slots, however, only some untight upper bounds of the pure POA were obtained. In this work, we improve previous results in two aspects: (1) We prove that the pure POA for GSP is 1.259 when there are four ad slots, and (2) We show that the pure POA for GSP with more than four ad slots is also 1.259 given the bidders are ra…
We analyze the network congestion game with atomic players, asymmetric strategies, and the maximum latency among all players as social cost. This important social cost function is much less understood than the average latency. We show that the price of anarchy is at most two, when the network is a ring and the link latencies are linear. Our bound is tight. This is the first sharp bound for the maximum latency objective.
We study Nash equilibria and the price of anarchy in the classical model of Network Creation Games introduced by Fabrikant et al. In this model every agent (node) buys links at a prefixed price $α>0$ in order to get connected to the network formed by all the $n$ agents. In this setting, the reformulated tree conjecture states that for $α> n$, every Nash equilibrium network is a tree. Since it was shown that the price of anarchy for trees is constant, if the tree conjecture were true, then the price of anarchy would be constant for $α>n$. Moreover, Demaine et al. conjectured that the price of a…
In this paper, we analyze a transportation game first introduced by Fotakis, Gourvès, and Monnot in 2017, where players want to be transported to a common destination as quickly as possible and, in order to achieve this goal, they have to choose one of the available buses. We introduce a sequential version of this game and provide bounds for the Sequential Price of Stability and the Sequential Price of Anarchy in both metric and non-metric instances, considering three social cost functions: the total traveled distance by all buses, the maximum distance traveled by a bus, and the sum of the dis…
This paper provides a comprehensive convergence analysis of the PoA of both pure and mixed Nash equilibria in atomic congestion games with unsplittable demands.
Decentralized Multi-product Pricing: Diagonal Dominance, Nash Equilibrium, and Price of Anarchy
Department of Information & Decision Sciences, College of Business Administration, University of Illinois Chicago（伊利诺伊大学芝加哥分校商学院信息与决策科学系）；Department of Industrial Engineering & Decision Analytics, The Hong Kong University of Science and Technology（香港科技大学工业工程与决策分析系）；Department of Technology and Operations, Ross School of Business, University of Michigan（密歇根大学罗斯商学院技术与运营系）
One of the main results shown through Roughgarden's notions of smooth games and robust price of anarchy is that, for any sum-bounded utilitarian social function, the worst-case price of anarchy of coarse correlated equilibria coincides with that of pure Nash equilibria in the class of weighted congestion games with non-negative and non-decreasing latency functions and that such a value can always be derived through the, so called, smoothness argument. We significantly extend this result by proving that, for a variety of (even non-sum-bounded) utilitarian and egalitarian social functions and fo…
We study {\em bottleneck congestion games} where the social cost is determined by the worst congestion of any resource. These games directly relate to network routing problems and also job-shop scheduling problems. In typical bottleneck congestion games, the utility costs of the players are determined by the worst congested resources that they use. However, the resulting Nash equilibria are inefficient, since the price of anarchy is proportional on the number of resources which can be high. Here we show that we can get smaller price of anarchy with the bottleneck social cost metric. We introdu…
Game-theoretic models relevant for computer science applications usually feature a large number of players. The goal of this paper is to develop an analytical framework for bounding the price of anarchy in such models. We demonstrate the wide applicability of our framework through instantiations for several well-studied models, including simultaneous single-item auctions, greedy combinatorial auctions, and routing games. In all cases, we identify conditions under which the POA of large games is much better than that of worst-case instances. Our results also give new senses in which simple auct…
We present a new class of vertex cover and set cover games. The price of anarchy bounds match the best known constant factor approximation guarantees for the centralized optimization problems for linear and also for submodular costs -- in contrast to all previously studied covering games, where the price of anarchy cannot be bounded by a constant (e.g. [6, 7, 11, 5, 2]). In particular, we describe a vertex cover game with a price of anarchy of 2. The rules of the games capture the structure of the linear programming relaxations of the underlying optimization problems, and our bounds are establ…
This paper gives a complete analysis of worst-case equilibria for various versions of weighted congestion games with two players and affine cost functions. The results are exact price of anarchy bounds which are parametric in the weights of the two players, and establish exactly how the primitives of the game enter into the quality of equilibria. Interestingly, some of the worst-cases are attained when the players' weights only differ slightly. Our findings also show that sequential play improves the price of anarchy in all cases, however, this effect vanishes with an increasing difference in …
We study the Price of Anarchy of simultaneous first-price auctions for buyers with submodular and subadditive valuations. The current best upper bounds for the Bayesian Price of Anarchy of these auctions are e/(e-1) [Syrgkanis and Tardos 2013] and 2 [Feldman et al. 2013], respectively. We provide matching lower bounds for both cases even for the case of full information and for mixed Nash equilibria via an explicit construction. We present an alternative proof of the upper bound of e/(e-1) for first-price auctions with fractionally subadditive valuations which reveals the worst-case price dist…
We study the price of anarchy of the first-price auction in the autobidding world, where bidders can be either utility maximizers (i.e., traditional bidders) or value maximizers (i.e., autobidders). We show that with autobidders only, the price of anarchy of the first-price auction is $1/2$, and with both kinds of bidders, the price of anarchy degrades to about $0.457$ (the precise number is given by an optimization). These results complement the recent result by Jin and Lu [2022] showing that the price of anarchy of the first-price auction with traditional bidders only is $1 - 1/e^2$. We furt…
We reconsider the well-studied Selfish Routing game with affine latency functions. The Price of Anarchy for this class of games takes maximum value 4/3; this maximum is attained already for a simple network of two parallel links, known as Pigou's network. We improve upon the value 4/3 by means of Coordination Mechanisms. We increase the latency functions of the edges in the network, i.e., if $\ell_e(x)$ is the latency function of an edge $e$, we replace it by $\hat{\ell}_e(x)$ with $\ell_e(x) \le \hat{\ell}_e(x)$ for all $x$. Then an adversary fixes a demand rate as input. The engineered Price…
We study the Price of Anarchy of mechanisms for the well-known problem of one-sided matching, or house allocation, with respect to the social welfare objective. We consider both ordinal mechanisms, where agents submit preference lists over the items, and cardinal mechanisms, where agents may submit numerical values for the items being allocated. We present a general lower bound of $Ω(\sqrt{n})$ on the Price of Anarchy, which applies to all mechanisms. We show that two well-known mechanisms, Probabilistic Serial, and Random Priority, achieve a matching upper bound. We extend our lower bound to …
We study network formation with n players and link cost α> 0. After the network is built, an adversary randomly deletes one link according to a certain probability distribution. Cost for player v incorporates the expected number of players to which v will become disconnected. We show existence of equilibria and a price of stability of 1+o(1) under moderate assumptions on the adversary and n \geq 9. As the main result, we prove bounds on the price of anarchy for two special adversaries: one removes a link chosen uniformly at random, while the other removes a link that causes a maximum number of…
In this paper we consider the price of anarchy (PoA) in multi-commodity flows where the latency or delay function on an edge has a heterogeneous dependency on the flow commodities, i.e. when the delay on each link is dependent on the flow of individual commodities, rather than on the aggregate flow. An application of this study is the performance analysis of a network with differentiated traffic that may arise when traffic is prioritized according to some type classification. This study has implications in the debate on net-neutrality. We provide price of anarchy bounds for networks with $k$ (…
We study assignment games in which jobs select machines, and in which certain pairs of jobs may conflict, which is to say they may incur an additional cost when they are both assigned to the same machine, beyond that associated with the increase in load. Questions regarding such interactions apply beyond allocating jobs to machines: when people in a social network choose to align themselves with a group or party, they typically do so based upon not only the inherent quality of that group, but also who amongst their friends (or enemies) choose that group as well. We show how semi-smoothness, a …
We consider a game-theoretical problem called selfish 2-dimensional bin packing game, a generalization of the 1-dimensional case already treated in the literature. In this game, the items to be packed are rectangles, and the bins are unit squares. The game starts with a set of items arbitrarily packed in bins. The cost of an item is defined as the ratio between its area and the total occupied area of the respective bin. Each item is a selfish player that wants to minimize its cost. A migration of an item to another bin is allowed only when its cost is decreased. We show that this game always c…
Globally operating suppliers face the rising challenge of wholesale pricing under scarce data about retail demand, in contrast to better informed, locally operating retailers. At the same time, as local businesses proliferate, markets congest and retail competition increases. To capture these strategic considerations, we employ the classic Cournot model and extend it to a two-stage supply chain with an upstream supplier who operates under demand uncertainty and multiple downstream retailers who compete over quantity. The supplier's belief about retail demand is modeled via a continuous probabi…
We consider the capacitated selfish replication (CSR) game with binary preferences, over general undirected networks. We first show that such games have an associated ordinary potential function, and hence always admit a pure-strategy Nash equilibrium (NE). Further, when the minimum degree of the network and the number of resources are of the same order, there exists an exact polynomial time algorithm which can find a NE. Following this, we study the price of anarchy of such games, and show that it is bounded above by 3; we further provide some instances for which the price of anarchy is at le…
Measures of allocation optimality differ significantly when distributing standard tradable goods in peaceful times and scarce resources in crises. While realistic markets offer asymptotic efficiency, they may not necessarily guarantee fair allocation desirable when distributing the critical resources. To achieve fairness, mechanisms often rely on a central authority, which may act inefficiently in times of need when swiftness and good organization are crucial. In this work, we study a hybrid trading system called Crisdis, introduced by Jedličková et al., which combines fair allocation of buyin…
Many algorithms that are originally designed without explicitly considering incentive properties are later combined with simple pricing rules and used as mechanisms. The resulting mechanisms are often natural and simple to understand. But how good are these algorithms as mechanisms? Truthful reporting of valuations is typically not a dominant strategy (certainly not with a pay-your-bid, first-price rule, but it is likely not a good strategy even with a critical value, or second-price style rule either). Our goal is to show that a wide class of approximation algorithms yields this way mechanism…
We study a static game played by a finite number of agents, in which agents are assigned independent and identically distributed random types and each agent minimizes its objective function by choosing from a set of admissible actions that depends on its type. The game is anonymous in the sense that the objective function of each agent depends on the actions of other agents only through the empirical distribution of their type-action pairs. We study the asymptotic behavior of Nash equilibria, as the number of agents tends to infinity, first by deriving laws of large numbers characterizes almos…
Pool block withholding attack is performed among mining pools in digital cryptocurrencies, such as Bitcoin. Instead of mining honestly, pools can be incentivized to infiltrate their own miners into other pools. These infiltrators report partial solutions but withhold full solutions, share block rewards but make no contribution to block mining. The block withholding attack among mining pools can be modeled as a non-cooperative game called "the miner's dilemm", which reduces effective mining power in the system and leads to potential systemic instability in the blockchain. However, existing lite…
Incorporating budget constraints into the analysis of auctions has become increasingly important, as they model practical settings more accurately. The social welfare function, which is the standard measure of efficiency in auctions, is inadequate for settings with budgets, since there may be a large disconnect between the value a bidder derives from obtaining an item and what can be liquidated from her. The Liquid Welfare objective function has been suggested as a natural alternative for settings with budgets. Simple auctions, like simultaneous item auctions, are evaluated by their performanc…
