Igal Milchtaich

Stability and Segregation in Group Formation

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 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 (in 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.

Topological Conditions for Uniqueness of Equilibrium in Networks

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, forall assignments of strictly increasing cost functions, each users 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 forsome assignment of (user-specific) strictly increasing cost functions is similarly characterized by an embedded network of a particular simple type.

Congestion games with player-specific constants

We consider a special case of weighted congestion games with playerspecific latency functions where each player uses for each particular resource a fixed (non-decreasing) delay function together with a player-specific constant. For each particular resource, the resource-specific delay function and the playerspecific constant (for that resource) are composed by means of a group operation (such as addition or multiplication) into a player-specific latency function. We assume that the underlying group is a totally ordered abelian group. In this way, we obtain the class of weighted congestion games with player-specific constants; we observe that this class is contained in the new intuitive class of dominance weighted congestion games. We obtain the following results: Games on parallel links: - Every unweighted congestion game has a generalized ordinal potential. - There is a weighted congestion game with 3 players on 3 parallel links that does not have the Finite Best-Improvement Property. - There is a particular best-improvement cycle for general weighted congestion games with player-specific latency functions and 3 players whose outlaw implies the existence of a pure Nash equilibrium. This cycle is indeed outlawed for dominance weighted congestion games with 3 players - and hence for weighted congestion games with player-specific constants and 3 players. Network congestion games: For unweighted symmetric network congestion games with player-specific additive constants, it is PLS-complete to find a pure Nash equilibrium. Arbitrary (non-network) congestion games: Every weighted congestion game with linear delay functions and player-specific additive constants has a weighted potential.

Random-Player Games

This paper introduces games of incomplete information in which the number, as well as the identity, of the participating players is determined by chance. The participation of certian players may not be independent of the participation of others, and hence the very fact that a particular player was chosen to play may give that player a clue as to the number and the identity of the other players chosen. However, players have to choose their strategies before the identity of the other players is fully revealed to them and thus, effectively, before they know whether or not they will take part in the game. Pure-strategy, mix-strategy, and correlated equilibria of random-player games are defined. These definitions extend the corresponding definitions for finite games, Bayesian games with consistent beliefs, and Poisson games-all of which can be seen as special cases of random-player games. Sufficient conditions for the existence of pure and mixed-strategy equilibria are given.

Social optimality and cooperation in nonatomic congestion games

Abstract Congestion externalities may result in nonoptimal equilibria. For these to occur, it suffices that facilities differ in their fixed utilities or costs. As this paper shows, the only case in which equilibria are always socially optimal, regardless of the fixed components, in that in which the costs increase logarithmically with the size of the set of users. Therefore, achieving a socially optimal choice of facilities generally requires some form of external intervention or cooperation. For heterogeneous populations (in which the fixed utilities or costs vary across users as well as across facilities), this raises the question of utility or cost sharing. The sharing rule proposed in this paper is the Harsanyi transferable-utility value of the game—which is based on the users’ marginal contributions to the bargaining power of coalitions.

The equilibrium existence problem in finite network congestion games

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 players 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 edge-specific 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 wi. 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 two-terminal 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.

First and Second Best Voting Rules in Committees

A group of people with identical preferences but different abilities in identifying the best alternative (e.g., a jury) takes a vote to decide between two alternatives. The first best voting rule is a weighted voting rule that takes into account the different individual competences, and is therefore not anonymous. Under such a rule, it is rational for group members to vote informatively, i.e., according to their private information. The use of any (non-trivial) anonymous voting rule may provide an incentive for some group members to vote strategically, non-informatively. However, this paper shows that the identity of the best anonymous voting rule does not depend on whether or not they actually choose to do so; a single, second best, rule maximizes utility in both cases.

Weighted congestion games with separable preferences

Players in a congestion game may differ from one another in their intrinsic preferences (e.g., the benefit they get from using a specific resource), their contribution to congestion, or both. In many cases of interest, intrinsic preferences and the negative effect of congestion are (additively or multiplicatively) separable. This paper considers the implications of separability for the existence of pure-strategy Nash equilibrium and the prospects of spontaneous convergence to equilibrium. It is shown that these properties may or may not be guaranteed, depending on the exact nature of player heterogeneity.

Implementability of Correlated and Communication Equilibrium Outcomes in Incomplete Information Games

In a correlated equilibrium, the players’ choice of actions is directed by correlated random messages received from an outside source, or mechanism. These messages allow for more equilibrium outcomes than without any messages (pure-strategy equilibrium) or with statistically independent ones (mixed-strategy equilibrium). In an incomplete information game, the messages may also reflect the types of the players, either because they are affected by extraneous factors that also affect the types (correlated equilibrium) or because the players themselves report their types to the mechanism (communication equilibrium). Mechanisms may be further differentiated by the connections between the messages that the players receive and their own and the other players’ types, by whether the messages are statistically dependent or independent, and by whether they are random or deterministic. Consequently, whereas for complete information games there are only three classes of equilibrium outcomes, with incomplete information the corresponding number is 14 or 15 for correlated equilibria and even larger—15, 16 or 17—for communication equilibria. For both solution concepts, the implication relations between the different kinds of equilibria form a two-dimensional lattice, which is considerably more intricate than the single-dimensional one of the complete information case.

Network topology and equilibrium existence in weighted network congestion games

Every finite game can be represented as a weighted network congestion game on some undirected two-terminal network. The network topology may reflect certain properties of the game. This paper solves the topological equilibrium-existence problem of identifying all networks on which every weighted network congestion game has a pure-strategy equilibrium.