### DIMACS Workshop on Computational Issues in Game Theory and Mechanism Design

#### October 31 - November 2, 2001 DIMACS Center, Rutgers University, Piscataway

Organizers:
Vijay Vazirani, Georgia Tech, vazirani@cc.gatech.edu
Noam Nisan, Hebrew University, Jerusalem, Israel, noam@cs.huji.ac.il
Presented under the auspices of Next Generation Networks Technologies and Applications and Social Science Methods and Information Technology.

#### Abstracts:

1.

Daniel Lehmann, Hebrew University
Title: Combinatorial Auctions with Decreasing Marginal Utilities (joint work
with Benny Lehmann and Noam Nisan)

Abstract:
In most of microeconomic theory, consumers are assumed to exhibit decreasing
marginal utilities. We consider combinatorial auctions among such buyers.
The valuations of such buyers are placed within a hierarchy of valuations
that exhibit no complementarities, a hierarchy that includes also OR and XOR
combinations of singleton valuations, and valuations satisfying the gross
substitutes property. While we show that the allocation problem among
valuations with decreasing marginal utilities is NP-hard, we present an
efficient greedy $2$-approximation algorithm for this case. No such
approximation algorithm exists in a setting allowing for complementarities.
Some results about strategic aspects of combinatorial auctions among players
with decreasing marginal utilities are also presented. Some algorithmic
results in the more general case of players
with limited complementarities are also presented.

2. Correlated Q-Learning Amy Greenwald and Keith Hall Bowling [2000] puts forth two desiderata for multiagent learning algorithms in general-sum Markov games: rationality and convergence. This talk introduces correlated-Q learning, a natural generalization of existing algorithms that satisfies these criteria. Nash-Q [Hu and Wellman 1998] satisfies rationality, but in general it does not converge. FF-Q [Littman 1994 and 2001] satisfies convergence, but in general it is not rational. Correlated-Q satisfies rationality by construction. Moreover, we demonstrate (empirical) convergence on the standard testbed of general-sum Markov games, including those for which deterministic equilibrium policies do not exist.
3. Suresh Mutuswami, Essex University Subscription Mechanisms for Network Formation We analyze a model of network formation where the costs of link formation are publicly known but individual benefits are not known to the social planner. The objective is to design a simple mechanism ensuring efficiency, budget balance and equity. We propose two mechanisms towards this end; the first ensures efficiency and budget balance but not equity. The second mechanism corrects the asymmetry in payoffs through a two-stage variant of the first mechanism. We also discuss an extension of the basic model to cover the case of directed graphs and give conditions under which the proposed mechanisms are immune to coalitional deviations.
4. Design of Competitive Mechanisms Andrew V. Goldberg, Research Fellow, InterTrust STAR Lab ABSTRACT The development of Internet brings up mechanism design problems that involve many agents, both human and computerized. In this context, it is often desirable to have truthful (or strategyproof) mechanisms with an objective function that is suited for the application. Classical mechanisms maximize either the total welfare of the system or divide the operating cost among the participants. In some cases, however, it is desirable to achieve different objectives. In particular, in application with a single seller and one or more buyers, it may be desirable to maximize seller's revenue. This requires mechanisms which are different from those provided by the classical game theory. Auctions are natural mechanisms with a seller and buyers. In the spirit of on-line mechanisms, one can compare revenue of a truthful auction with that of a nontruthful pricing mechanism. A competitive auction is a truthful auction that for any input has an expected revenue that is within a constant factor of the highest possible for such an auction. In this talk we give a high-level overview of competitive auctions, including motivation and definitions, positive and negative results, generalizations, and open problems. Joint work with Amoz Fiat, Jason Hartline, Anna Karlin, and Andrew Wright.
5. Sven de Vries, Munchen Linear Programming and Ascending Auctions (joined work with Sushil Bikhchandani, James Schummer, and Rakesh V. Vohra) The Vickrey sealed bid auction occupies a central place in auction theory because of its efficiency and incentive properties. Implementing the auction requires the auctioneer to solve $n+1$ optimization problems, where $n$ is the number of bidders. In this talk we survey various environments (some old and some new) where the payments bidders make under the Vickrey auction correspond to dual variables in certain linear programs. Thus, in these environments, at most two optimization problems must be solved to determine the Vickrey outcome. Furthermore, primal-dual algorithms for some of these linear programs suggest ascending auctions that implement the Vickrey outcome.
6. Rahul Sami Yale University Department of Computer Science TITLE: Approximation and Collusion in Multicast Cost Sharing We investigate multicast cost sharing from both computational and economic points of view. Recent work in economics [Moulin and Shenker,1997] leads naturally to the consideration of two mechanisms: Marginal Cost (MC), which is efficient and strategyproof, and Shapley value (SH), which is budget-balanced and group-strategyproof and, among all mechanisms with these two properties, minimizes the worst-case efficiency loss. Subsequent work in computer science [Feigenbaum, Papadimitriou and Shenker,2000] shows that the MC mechanism can be computed by a simple, distributed algorithm that uses only two messages per link of the multicast tree but that computing the SH mechanism seems, in the worst case, to require a number of messages that is quadratic in the size of the multicast tree. We extend these results in two directions. First, we give a group-strategyproof mechanism that exhibits a tradeoff between the other properties of the Shapley value: It can be computed by a distributed algorithm that is more communication-efficient than the natural algorithm (exponentially more so in the worst case), but it might fail to achieve exact budget balance or exact minimum efficiency loss (albeit by a bounded amount). Second, we completely characterize the groups that can strategize successfully against the MC mechanism. Joint work with Joan Feigenbaum, Arvind Krishnamurthy, and Scott Shenker
7. On the complexity of auctions Rudolf Muller Maastricht University Many publications on combinatorial auctions are motivated by two claims: the winner determination problem is NP-complete, therefore one should develop fast exact or approximation algorithms, and, progressive auctions are preferable because they do not require buyers to report their full type. However, if we consider the plain case of a single-item auction, a sealed bid auction is polynomial, while the English auction is exponential in the input. Also, if buyers would report their full type in the VCG mechanism for the multi-object case, the winner determination problem would become tractable. It is therefore necessary to develop more precise models of the complexity of auctions. Such models have to consider carefully the encoding length of the input, which is the information seller and buyers have when the auction starts, as well as the cost of communication during the auction. Questions to be answered include in which cases we may expect a polynomial auction mechanism at all, and in which not? Furthermore, how efficient can progressive auctions be, if the plain English auction is exponential? We present ongoing research on such questions.
8. Fairness Measures in Optimization Jon Kleinberg Cornell University In a typical discrete optimization problem, the quality of the solution -- the value of the objective function -- is a single figure of merit that must be maximized or minimized. But if we consider settings in which resources must be allocated or provisioned for a set of individuals, there is really a separate objective function associated with each individual. (How much bandwidth was I allocated? How far am I from the nearest open facility? When will my job complete?) The quality' of the solution in such a case is most naturally expressed as a vector with a coordinate specifying the quality of the outcome for each individual. Once we consider optimization problems of this flavor, it is far from clear how to define the optimal' solution. And without a clear notion of an optimum, it is harder still to quantify the notion of an approximate' solution. We discuss how one can formulate notions of optimization and approximation in these settings using fairness measures,' which intuitively seek solutions that are favorable for all individuals. Several of these measures originated in the study of bandwidth allocation, where they can be usefully applied; we will consider how they can also be applied in other domains, including scheduling and facility location problems. The talk is based on joint work with Amit Kumar, Yuval Rabani, and Eva Tardos.
9. A Generic Analysis of Selfish Routing Eric Friedman, Cornell University We show that for most transmission rates the losses due to selfish routing in networks are bounded by $O(\log(C))$, where $C$ is a measure sensitivity of the network to changes in transmission rates for arbitrary latency functions. This contrasts with the losses for the worst rate which can be $O(C)$. These results also hold for the restricted case of typical queuing functions (such as M/M/1) and for systems operating under congestion control algorithms (such as TCP).
10. A Cryptographic Solution to a Game Theoretic Problem Yevgeniy Dodis, Shai Halevi and Tal Rabin In this work we use cryptography to solve a game-theoretic problem which arises naturally in the area of two party strategic games. The standard game-theoretic solution concept for such games is that of an equilibrium. It is known that for many games the expected equilibrium payoffs can be much higher when a trusted third party (a mediator'') assists the players in choosing their moves (correlated equilibria), than when each player has to choose its move on its own (Nash equilibria). It is natural to ask whether there exists a mechanism that eliminates the need for the mediator yet allows the players to maintain the high payoffs offered by mediator-assisted strategies. We answer this question affirmatively provided the players are computationally bounded and can have free communication (so-called cheap talk'') prior to playing the game.
11. Title: The Effect of False-name Bids on Auction Protocols Speaker: Makoto Yokoo Abstract: In this talk, we examine the effect of a new type of frauds called false-name bids, which can be a serious problem in Internet auctions. False-name bids are bids submitted by a single agent under multiple fictitious names (e.g., multiple e-mail addresses). From the game-theoretic/mechanism design viewpoint, the strategy space of each agent/player is expanded if agents can submit false-name bids. Thus, a protocol that is dominant-strategy incentive compatible (i.e., honesty is the best policy) without the possibility of false-name bids might no longer has a dominant strategy equilibrium with the possibility of false-name bids. For example, the Generalized Vickrey Auction protocol (GVA) satisfies dominant-strategy incentive compatibility, Pareto efficiency, and individual rationality without the possibility of false-name bids. On the other hand, with the possibility of false-name bids, the GVA fails to satisfy the dominant-strategy incentive compatibility. Furthermore, we prove that it is theoretically impossible for any combinatorial auction protocol to simultaneously satisfy these three properties. We develop a new combinatorial auction protocol and a double auction protocol that satisfy dominant-strategy incentive compatibility and individual rationality, and can achieve relatively good (but not Pareto efficient) social surplus even if the agents can submit false-name bids.
12. Title: On approximating optimal auctions Author: Amir Ronen, Stanford Computational problems that arise in game theoretical environments are very different from "usual" algorithmic problems. In this talk I will focus on a fundamental problem in economics called optimal auction design: A seller wishes to sell an item to a group of self-interested agents. Each agent $i$ has a {\em privately} known valuation $v^i$ for the object. Given a distribution on these valuations, our goal is to design an auction (i.e. a protocol) that maximizes the seller's expected revenue. We present a simple generic protocol that {\bf guarantees} at least half of the optimal revenue. When the designer can only sample the distribution, we show a tight bound of $\ln h$ on the approximation ratio where $h$ denotes that ratio between the highest and the lowest possible valuations. Finally we suggest a generalization of our auction and argue that it will generate a revenue which is close to optimal for reasonable distributions. In particular we show this under an independence assumption. Several open problems are presented as well.
13. Minimal Preference Elicitation: An Equilibrium Approach David C. Parkes Division of Engineering and Applied Sciences, Harvard University The core role of price-based mechanisms, such as iBundle, in reducing the complexity of the agent valuation problem in combinatorial auctions is demonstrated with a simple observation. A large class of dynamic preference elicitation methods for efficient allocation must collect enough information from agents to compute a set of competitive equilibrium prices, because this information is implied in the primal solution. As such, price-based mechanisms can usefully guide preference elicitation, and can be combined with methods to leverage structured preference representations. Connections with Vickrey payments also lead to a coherent integration of incentive considerations into dynamic mechanism design.
14. Graphical Models for Game Theory Michael Kearns Syntek Capital In order to scale up to the analysis of large multi-agent systems, classical game theory must be augmented with compact representations of multi-player matrix games, and these representations should permit efficient computation of equilbria and related concepts. In this work, we introduce a graph-theoretic representation of one-stage, multi-player games that may yield an exponential reduction in space in many natural cases. Our main results are efficient algorithms for computing Nash equilibria in graphical games that are trees or are sparse. Joint work with Michael Littman and Satinder Singh.
16. TITLE: Two Results On Competitive Auctions SPEAKER: Jason Hartline ABSTRACT: We consider the problem of selling identical items via a truthful (or strategy-proof) auction, with the goal of maximizing the seller's revenue. We aim to design "competitive" auctions, auctions which achieve a revenue that is within a constant factor of the revenue of the optimal (non-truthful) fixed-price auction. In this talk, we present two recent results for competitive auctions. First, we prove that no deterministic truthful auction is competitive. We then present a randomized auction, based on the Shapley Value mechanism, that is 4-competitive. This auction has the property that it can be used as a building block for revenue-maximizing mechanisms for other applications. Joint work with Andrew Goldberg and Andrew Wright.
17. Eva Tardos, Cornell University How Bad is Selfish Routing? We consider the problem of routing traffic in a congested network. We are given a network, a rate of traffic between each pair of nodes, and a latency function for each edge specifying the time needed to traverse the edge given its congestion. We consider a "selfish" routing in which each network user routes its traffic on the minimum-latency path available to it, ignoring the latency of all other users. We will compare this "selfish" routing to a social optimum, where the objective is to route traffic such that the sum of all travel times---the total latency---is minimized. In general the "selfish" assignment of traffic to paths will not minimize the total latency, i.e., the lack of central regulation degrades network performance. In this talk we will quantify the degradation of network performance due to unregulated traffic. The talk is based on joint work with Tim Roughgarden.
18. Tim Roughgarden, Cornell University Title: Designing Networks for Selfish Users is Hard Abstract: We consider a directed network in which every edge possesses a latency function specifying the time needed to traverse the edge given its congestion. Selfish, noncooperative agents constitute the network traffic and wish to travel from a source $s$ to a sink $t$ as quickly as possible. Since the route chosen by one network user affects the congestion (and hence the latency) experienced by others, we model the problem as a noncooperative game. Assuming each agent controls only a negligible portion of the overall traffic, Nash equilibria in this noncooperative game correspond to $s$-$t$ flows in which all flow paths have equal latency. A natural measure for the performance of a network used by selfish agents is the common latency experienced by each user in a Nash equilibrium. It is a counterintuitive but well-known fact that removing edges from a network may {\em improve} its performance; the most famous example of this phenomenon is the so-called {\em Braess's Paradox}. This fact motivates the following network design problem: given such a network, which edges should be removed to obtain the best possible flow at Nash equilibrium? Equivalently, given a large network of candidate edges to be built, which subnetwork will exhibit the best performance when used selfishly? We give optimal inapproximability results and approximation algorithms for several network design problems of this type. For example, we prove that for networks with $n$ nodes and continuous, nondecreasing latency functions, there is no approximation algorithm for this problem with approximation ratio less than $n/2$ (unless $P=NP$). We also prove this hardness result to be best possible by exhibiting an $n/2$-approximation algorithm. For networks in which the latency of each edge is a linear function of the congestion, we prove that there is no $(\frac{4}{3}-\eps)$-approximation algorithm for the problem (for any $\eps > 0$, unless $P=NP$); the existence of a $\frac{4}{3}$-approximation algorithm follows easily from existing work, proving this hardness result sharp. Moreover, we prove that an optimal approximation algorithm for these problems is what we call the {\em trivial algorithm}: given a network of candidate edges, build the entire network. Roughly, this result implies that the presence of harmful extra edges in a network (a phenomenon that can lead to extremely poor performance in large networks with general latency functions) is impossible to detect efficiently.
19. Bidding Agents with Complex Valuation Problems in Auctions Tuomas Sandholm Associate Professor Carnegie Mellon University Computer Science Department This talk analyzes automated bidding agents that do not know their valuations for the items or bundles of items that are being auctioned a priori, but rather have to compute the valuations. This is the case, for example, in transportation exchanges where a carrier company's cost of handling a set of deliveries is the cost of her vehicle routing solution including those tasks minus the cost of her routing solution without them. In practice, the routing problem instances are too large to solve optimally. They can be approximated using anytime algorithms. This spawns a host of questions related to bounded rationality: how should an agent allocate its limited or costly computation? An agent needs to decide on which bundles to allocate its computation. The agent may also want to allocate computation on the other agents' valuation problems in order to be able to strategically tailor her bids. Furthermore, the other bidders' deliberation strategies affect what the agent's best deliberation strategy is, and vice versa. We first introduce a normative, performance profile based deliberation control method for anytime algorithms. We combine this with a game-theoretic solution concept to treat each agent's deliberation actions (on which agent-bundle pair to invest the next computation step) and bids uniformly as part of the agent's strategy. We show under which auction mechanisms and which models of bounded rationality (limited or costly computation), different types of strategic computation (computing on others' valuation problems) occur. To our knowledge, this work is the first to treat deliberation actions strategically. This is joint work with Kate Larson. The talk covers the following two papers. (Other papers on related topics are available at http://www.cs.cmu.edu/~sandholm ). Larson, K. and Sandholm, T. 2001. Costly Valuation Computation in Auctions: Deliberation Equilibrium. Theoretical Aspects of Reasoning about Knowledge (TARK), pp. 169-182, Siena, Italy, July. http://www.cs.cmu.edu/~sandholm/tark01.pdf Larson, K. and Sandholm, T. 2001. Computationally Limited Agents in Auctions. International Conference on Autonomous Agents, Workshop on Agent-based Approaches to B2B, pp. 27-34, Montreal, Canada, May 28th. http://www.cs.cmu.edu/~sandholm/computationally_limited.agents01ws.pdf
20. TITLE: Open Problems in Competitive Auction Design Anna R. Karlin University of Washington ABSTRACT: Consider the problem of selling identical items via a truthful (or strategyproof) auction. The classic Vickrey auction, when faced with the task of selling k identical items, sells them to the k highest bidders at the (k+1)-st highest price bid. Suppose, however, that the auctioneer is willing to sell fewer than k items in order to increase his revenue. For example, if the top two bids are sufficiently high, his revenue might be greatly increased by selling only one of the items, say to the highest bidder at the second highest price. This motivates the problem of designing truthful auctions that attempt to maximize revenue to the auctioneer. Specifically, we are interested in designing "competitive" auctions, auctions which achieve a revenue that is within a constant factor of the revenue of the optimal (non-truthful) fixed-price auction. We present a number of intriguing open problems and conjectures in the area of competitive auction design, as well as limited progress towards resolving these problems. Joint work with Amos Fiat, Jason Hartline and Andrew Goldberg.
21. Cooperative Facility Location Games Michel X. Goemans MIT The location of facilities in order to provide service for customers is a well-studied problem in the operations research literature. In the basic model, there is a predefined cost for opening a facility and also for connecting a customer to a facility, the goal being to minimize the total cost. Often, both in the case of public facilities (such as libraries, municipal swimming pools, fire stations, ...) and private facilities (such as distribution centers, switching stations, ...), we may want to find a fair' allocation of the total cost to the customers --- this is known as the cost allocation problem. A central question in cooperative game theory is whether the total cost can be allocated to the customers such that no coalition of customers has any incentive to build their own facility, i.e. whether the core is non-empty. We establish strong connections between fair cost allocations and linear programming relaxations for several variants of the facility location problem. In particular, we show that a fair cost allocation exists if and only if there is no integrality gap for a corresponding linear programming relaxation. We use this insight in order to give proofs for the existence of fair cost allocations for several classes of instances based on a subtle variant of randomized rounding. We also establish that it is NP-complete to decide whether a fair cost allocation exists and whether a given allocation is fair.
This is based on joint work with Martin Skutella. 22. The communication complexity of efficient discrete allocations Ilya Segal, Stanford We analyze the communication complexity of mechanisms implementing value-maximizing discrete allocations, combinatorial auctions being the leading example. We study both the continuous model, in which communication is measured with the dimensionality of the message space, and the discrete model, in which communication is measured with the number of exchanged bits. Our basic technique applies to both cases and yields a lower bound on the amount of communication. This bound is exponential in the number of objects when the class of agents' possible preferences is sufficiently rich, in particular when it includes all submodular preferences. In contrast, when agents' preferences satisfy the gross substitute property, efficiency can be achieved with polynomial communication (using the Walrasian equilibrium with single-item prices or Ausubel's deterministic mechanism). Lower bounds are also obtained for the implementation of approximately efficient allocations. Joint work with Noam Nisan.
23. Kamal Jain, Microsoft Research Equitable, Group Strategyproof Cost Allocations via Primal-Dual-Type Algorithms We build on the well-studied egalitarian cost-sharing method of Dutta and Ray [1987] as follows. Using a primal-dual-type algorithm, we show how to construct, efficiently, a class of cost-sharing methods for a submodular cost function, parametrized by $n$ functions, one for each user. Submodular functions capture the notion of economies of scale and such cost funcitons arise naturally in practice, e.g., in multicast routing. All methods in our class are cross-monotone and therefore lead to group strategyproof mechanisms (i.e., the dominant strategy for each user is to report her true utility, even in the presence of collusions). Our class contains cost-sharing methods satisfying a wide range of fairness criteria, hence the name equitable''. For instance, if the $n$ functions are identical, we get the Dutta-Ray egalitarian method, which attempts to equalize the amounts charged from the users. If they are probability distributions from which users pick their utilities, we get a new method that attempts to equalize the probabilities of users getting sevice. (This is a joint work with Vijay Vazirani.)
24. Bhaskar Dutta Cost Monotonicity, Consistency and Minimum Cost Spanning Tree Games We propose a new cost allocation rule for minimum cost spanning tree games. The new rule is a core selection and also satisfies cost monotonicity. We also give characterization theorems for the new rule as well as the much-studied Bird allocation. We show that the principal difference between these two rules is in terms of their consistency properties. Mixing Streaming and Flow-Controlled traffic in Networks: Distributed Control and Incentives Peter B. Key Microsoft Research, Cambridge In the current Internet, TCP is the dominant flow-control scheme, appropriate for so-called elastic applications that can tolerate wide variations in throughput. Conversely, streaming media typically can only operate at one rate, or at one of a small number of allowable rates. We show how the use of distributed control signals based on resource-prices allows the conflicting demands of the two types of traffic to be traded-off, and specific user-strategies constructed for streaming traffic. The signals can be encoded by a simple packet-marking scheme, for example by suitable use of the ECN field in IP. At present file transfers are categorised as elastic traffic, and use flow-control protocols; we discuss how under certain assumptions this may not be an optimal strategy. Specifically, we discuss the implications for Web mice and Web elephants.
25. Title: An Evolutionary Games Analysis of Bidding Strategies in a Scheduling Auction Authors: Jeffrey K. MacKie-Mason and Michael P. Wellman Abstract: We have embarked on a project to design and study the performance of market-based mechanisms for decentralized scheduling problems. Such problems are hard due to naturally arising nonconvexities (from complementarities and integer constraints). We call the problem "decentralized" when self-interested agents possess private information relevant to a good solution; respecting private information places an additional constraint on the design of good mechanisms. In this paper we address a long-standing problem in the literature on mechanism design for any complex problem: the evaluation of mechanism performance when optimal agent behavior is unknown. More often than not it is impossible to derive Bayesian-Nash strategies for agents participating in complex mechanisms (e.g., the FCC spectrum auctions). Yet, absent empirical data, analysis of mechanism performance requires assumptions about how agents will interact with that mechanism. An evolutionary approach to performance analysis has received increasing attention as a way to understand which strategies are "fit": stable and successful against other strategies when strategic encounters are frequently repeated. We take an evolutionary approach to isolate strategies or sets of strategies that are relatively (mutually) fit, and then evaluate the performance of market mechanisms for scheduling against these strategies. To implement this we form a population of agents playing strategies drawn from a parameterized space, and then use selection and updating rules to modify the composition of the population and the characteristics of the strategies the agents play.
26. Elias Koutsoupias, UCLA Selfish resource allocation. We consider selfish resource allocation and in particular selfish routing over a network consisting of parallel links. We assume a collection of users, each employing a mixed strategy ---a probability distribution over the links--- to control the routing of its own traffic. The expected latency through a link is assumed to be equal to the expectation of the traffic assigned to the link; all users sharing a link incur the same expected latency cost, taken to be the link's expected latency. In a Nash equilibrium, each user selfishly routes its traffic on those links that minimize its expected latency cost, given the network congestion caused by the other users. We study how much the performance of the network deteriorates due to the selfish nature of its users. More specifically, we study the coordination ratio, defined as the ratio of the maximum (over all links) expected latency in the worst possible Nash equlibrium, over the least possible maximum latency had global regulation been available.
27. Ilan Kremer Assistant Professor of Finance Graduate School of Business Stanford University 518 Memorial Way, CA 94305-5015 Phone 650-736-0288 Fax 650-725-6152 Divisible Good Auctions - the role of allocation rules Uniform price auctions are used in many financial and non-financial markets. Beginning with Wilson (1979), the theoretical literature has argued that these auctions are subject to possible collusion among bidders. In this paper we show that these results are due to the way the asset is being divided rather than the choice of a uniform price format. We focus on allocation rules that specify the way the asset is divided in cases of excess demand, and show that these rules have a dramatic effect on the set of equilibrium prices. In particular, we show that the allocation rule used in virtually all uniform price auctions has a negative effect on equilibrium prices. We also question a common assumption regarding agents' strategy sets. We show that restricting agents to use only continuous strategies may distort the set of equilibria.
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