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Abstract. Different kinds of networks, such as transportation, communication, computer, and supply networks, are susceptible to similar kinds of inefficiencies. These arise when congestion externalities make the cost for each user depend on the other users' choice of routes. If each user chooses the least expensive (e.g., the fastest) route from the users' common point of origin to the common destination, the result may be Pareto inefficient in that an alternative choice of routes would reduce the costs for all users. Braess's paradox represents an extreme kind of inefficiency, in which the equilibrium costs may be reduced by raising the cost curves. As this paper shows, this paradox occurs in an (undirected) twoterminal network if and only if it is not series-parallel. More generally, Pareto inefficient equilibria occur in a network if and only if one of three simple networks is embedded in it. JEL Classification: C72, R41.

Abstract. Different kinds of networks, such as transportation, communication, computer, and supply networks, are susceptible to similar kinds of inefficiencies. These arise when congestion externalities make the cost for each user depend on the other users' choice of routes. If each user chooses the least expensive (e.g., the fastest) route from the users' common point of origin to the common destination, the result may be Pareto inefficient in that an alternative choice of routes would reduce the costs for all users. Braess's paradox represents an extreme kind of inefficiency, in which the equilibrium costs may be reduced by raising the cost curves. As this paper shows, this paradox occurs in an (undirected) twoterminal network if and only if it is not series-parallel. More generally, Pareto inefficient equilibria occur in a network if and only if one of three simple networks is embedded in it. JEL Classification: C72, R41.

Equilibrium flow in a physical network with a large number of users (e.g., transportation, communication, and computer networks) need not be unique if the costs of the network elements are not the same for all users. Such differences among users may arise if they are not equally affected by congestion or have different intrinsic preferences. Whether or not, for all assignments of strictly increasing cost functions, each user's equilibrium cost is the same in all Nash equilibria can be determined from the network topology. Specifically, this paper shows that in a two-terminal network, the equilibrium costs are always unique if and only if the network is one of several simple networks or consists of several such networks connected in series. The complementary class of all two-terminal networks with multiple equilibrium costs for some assignment of (user-specific) strictly increasing cost functions is similarly characterized by an embedded network of a particular simple type. 1. Introduction. Different kinds of networks, such as transportation, communication, and computer networks exhibit congestion effects, whereby increased demand for certain network elements (e.g., roads, telecommunication lines, and servers) tends to downgrade their performance or increase the cost of using them. In such networks, the users' decisions (e.g., choice of routes) are interdependent in that their optimal choices (e.g., the fastest routes) depend on what the others do. If they all choose optimally, given the others' choices, then the users' choices constitute a Nash equilibrium. Even if the users are identical in all respects, due to the congestion externalities, their choices at equilibrium may differ. However, if the number of users is very large and each of them has a negligibly small effect on the others, then they have equal equilibrium payoffs or costs. Moreover, the payoffs or costs in different equilibria are the same (Aashtiani and Magnanti [1]). With a heterogeneous population of users (i.e., a multiclass network; Dafermos [5]), this need not be so. As the following example shows, if the users are not identical, and are differently affected by congestion, equilibrium costs may vary not only across users, but also from one Nash equilibrium to another. Example 1.1. A continuum of three classes of users travels on the two-terminal network shown in Figure 2(a) below. Each user has to choose one of the four routes connecting the users' common point of origin o and the common destination d. The cost of each route is the sum of the costs of its edges. For each user class, the cost of each edge e is given by an increasing affine function of the fraction x of the total population with a route that includes e. The fraction of the population in each class and the corresponding cost functions are given in Figure 1, where blank cells indicate prohibitively high costs. Clearly, users in each class effectively have a choice of only two routes from o to d e 1 and e 2 e 5 for Class I users, e 2 e 5 and e 3 for Class II, and e 3 and e 4 e 5 ...

A crowding game is a noncooperative game in which the payo¤ of each player depends only on the player's action and the size of the set of players choosing that particular action: The larger the set, the smaller the payo¤. Finite, n-player crowding games often have multiple equilibria. However, a large crowding game generically has just one equilibrium, and the equilibrium payo¤s in such a game are always unique. Moreover, the sets of equilibria of the m-replicas of a …nite crowding game generically converge to a singleton as m tends to in…nity. This singleton consists of the unique equilibrium of the "limit" large crowding game. This equilibrium generically has the following graph-theoretic property: The bipartite graph, in which each player in the original, …nite crowding game is joined with all best-response actions for (copies of) that player, does not contain cycles.

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This paper presents a model of group formation based on the assumption that individuals prefer to associate with people similar to them. It is shown that, in general, if the number of groups that can be formed is bounded, then a stable Ž partition of the society into groups may not exist. A partition is defined as stable if . none of the individuals would prefer be in a different group than the one he is in. However, if individuals' characteristics are one-dimensional, then a stable partition always exists. We give sufficient conditions for stable partitions to be segregating Žin the sense that, for example, low-characteristic individuals are in one group and . high-characteristic ones are in another and Pareto efficient. In addition, we propose a dynamic model of individual myopic behavior describing the evolution of group formation to an eventual stable, segregating, and Pareto efficient partition. Journal of Economic Literature

Abstract. An open problem is presented regarding the existence of pure strategy Nash equilibrium (PNE) in network congestion games with a finite number of non-identical players, in which the strategy set of each player is the collection of all paths in a given network that link the player's origin and destination vertices, and congestion increases the costs of edges. A network congestion game in which the players differ only in their origin-destination pairs is a potential game, which implies that, regardless of the exact functional form of the cost functions, it has a PNE. A PNE does not necessarily exist if (i) the dependence of the cost of each edge on the number of users is player-as well as edgespecific or (ii) the (possibly, edge-specific) cost is the same for all players but it is a function (not of the number but) of the total weight of the players using the edge, with each player i having a different weight w i . In a parallel two-terminal network, in which the origin and the destination are the only vertices different edges have in common, a PNE always exists even if the players differ in either their cost functions or weights, but not in both. However, for general twoterminal networks this is not so. The problem is to characterize the class of all two-terminal network topologies for which the existence of a PNE is guaranteed even with player-specific costs, and the corresponding class for player-specific weights. Some progress in solving this problem is reported.