A Further Extension of the KKMS Theorem

Recently Reny and Wooders ([23]) showed that there is some point in the intersection of sets in Shapley's ([24]) generalization of the Knaster-Kuratowski-Mazurkiwicz Theorem with the property that the collection of all sets containing that point is partnered as well as balanced. In this paper we provide a further extension by showing that the collection of all such sets can be chosen to be strictly balanced, implying the Reny-Wooders result. Our proof is topological, based on the Eilenberg-Montgomery fixed point Theorem. Reny and Wooders ([23]) also show that if the collection of partnered points in the intersection is countable, then at least one of them is minimally partnered. Applying degree theory for correspondences, we show that if this collection is only assumed to be zero dimensional (or if the set of partnered and strictly balanced points is of dimension zero), then there is at least one strictly balanced and minimally partnered point in the intersection. The approach presented in this paper sheds a new geometric-topological light on the Reny-Wooders results.

A law of scarcity for games

The "law of scarcity" is that scarceness is rewarded; recall, for example, the diamonds and water paradox. In this paper, furthering research initiated in Kelso and Crawford (1982, Econometrica 50, 1483-1504) for matching models, we demonstrate a law of scarcity for cores and approximate cores of games.cooperative games, games with side payments (TU games), cyclic monotonicity, law of demand, approximate cores, effective small groups, parameterized collections of games.

Approximate cores of games and economies with clubs

We introduce the framework of parameterized collections of games and provide three nonemptiness of approximate core theorems for arbitrary games with and without sidepayments. The parameters bound (a) the number of approx-imate types of players and the size of the approximation and (b) the size of nearly effective groups of players and their distance from exact effectiveness. The theorems are based on a new notion of partition-balanced pro les and approximately partition-balanced profiles. The results are then applied to a new model of an economy with clubs. In contrast to the extant literature, our approach allows both widespread externalities and uniform results.cooperative games, clubs, local public goods, games without side payments (NTU games), large games, approximate cores, effective small groups, parameterized collections of games.

An explicit bound on epsilon for nonemptiness of Epsilon-cores of games

We consider parameterized collections of games without side payments and determine a bound on epsilon so that all suffciently large games in the collection have non-empty epsilon-cores. Our result makes explicit the relationship between the required size of epsilon for non-emptiness of the epsilon-core, the parameters describing the collection of games, and the size of the total player set. Given the parameters describing the collection, the larger the game, the smaller the epsilon that can be chosen.cooperative games, games without side payments (NTU games), large games, approximate cores, effective small groups, parameterized collections of games.

Epsilon cores of games and economies with limited side payments

We introduce the concept of a parameterized collection of games with limited side payments, ruling out large transfers of utility. Under the assumption that the payoff set of the grand coalition is convex, we show that a game with limited side payments has a nonempty epsilon-core. Our main result is that, when some degree of side-paymentness within nearly-effective small groups is assumed, then all payoffs in the epsilon-core treat similar players similarly. A bound on the distance between epsilon-core payoffs of any two similar players is given in terms of the parameters describing the game. These results add to the literature showing that games with many players and small effective groups have the properties of competitive markets.

Endogenous Network Dynamics

In all social and economic interactions, individuals or coalitions choose not only with whom to interact but how to interact, and over time both the structure (the “with whom”) and the strategy (“the how”) of interactions change. Our objectives here are to model the structure and strategy of interactions prevailing at any point in time as a directed network and to address the following open question in the theory of social and economic network formation: given the rules of network and coalition formation, the preferences of individuals over networks, the strategic behavior of coalitions in forming networks, and the trembles of nature, what network and coalitional dynamics are likely to emerge and persist. Our main contributions are (i) to formulate the problem of network and coalition formation as a dynamic, stochastic game, (ii) to show that this game possesses a stationary correlated equilibrium (in network and coalition formation strategies), (iii) to show that, together with the trembles of nature, this stationary correlated equilibrium determines an equilibrium Markov process of network and coalition formation, and (iv) to show that this endogenous process possesses a finite, nonempty set of ergodic measures, and generates a finite, disjoint collection of nonempty subsets of networks and coalitions, each constituting a basin of attraction. We also extend to the setting of endogenous Markov dynamics the notions of pairwise stability (Jackson-Wolinsky, 1996), strong stability (Jacksonvan den Nouweland, 2005), and Nash stability (Bala-Goyal, 2000), and we show that in order for any network-coalition pair to persist and be stable (pairwise, strong, or Nash) it is necessary and sufficient that the pair reside in one of finitely many basins of attraction. The results we obtain here for endogenous network dynamics and stochastic basins of attraction are the dynamic analogs of our earlier results on endogenous network formation and strategic basins of attraction in static, abstract games of network formation (Page and Wooders, 2008), and build on the seminal contributions of Jackson and Watts (2002), Konishi and Ray (2003), and Dutta, Ghosal, and Ray (2005).Endogenous Network Dynamics, Dynamic Stochastic Games of Network Formation, Equilibrium Markov Process of Network Formation, Basins of Attraction, Harris Decomposition, Ergodic Probability Measures, Dynamic Path Dominance Core, Dynamic Pairwise Stability

Elections and strategic positioning games

We formalize the interplay between expected voting behavior and stragetic positioning behavior of candidates as a common agency problem in which the candidates (i.e. the principals) compete for voters (i.e. agents) via the issues they choose and the positions they take. A political situation is defined as a feasible combination of candidate positions and expected political payoffs to the candidates. Taking this approach, we are led naturally to a particular formalization of the candidates’ positioning game, called a political situation game. Within the context of this game, we define the notion of farsighted stability (introduced in an abstract setting by Chwe (1994)) and apply Chwe’s result to obtain existence of farsightedly stable outcomes. We compute the farsightedly stable sets for several examples of political situations games, with outcomes that conform to real-world observations

Endogenous Network Dynamics

In all social and economic interactions, individuals or coalitions choose not only with whom to interact but how to interact, and over time both the structure (the “with whom”) and the strategy (“the how”) of interactions change. Our objectives here are to model the structure and strategy of interactions prevailing at any point in time as a directed network and to address the following open question in the theory of social and economic network formation: given the rules of network and coalition formation, the preferences of individuals over networks, the strategic behavior of coalitions in forming networks, and the trembles of nature, what network and coalitional dynamics are likely to emergence and persist. Our main contributions are (i) to formulate the problem of network and coalition formation as a dynamic, stochastic game, (ii) to show that this game possesses a stationary correlated equilibrium (in network and coalition formation strategies), (iii) to show that, together with the trembles of nature, this stationary correlated equilibrium determines an equilibrium Markov process of network and coalition formation which respects the rules of network and coalition formation and the preferences of individuals, and (iv) to show that, although uncountably many networks may form, this endogenous process of network and coalition formation possesses a nonempty finite set of ergodic measures and generates a finite, disjoint collection of nonempty subsets of networks and coalitions, each constituting a basin of attraction. Moreover, we extend to the setting of endogenous Markov dynamics the notions of pairwise stability (Jackson-Wolinsky, 1996), strong stability (Jackson-van den Nouweland, 2005), and Nash stability (Bala-Goyal, 2000), and we show that in order for any network-coalition pair to be stable (pairwise, strong, or Nash) it is necessary and sufficient that the pair reside in one of finitely many basins of attraction - and hence reside in the support of an ergodic measure. The results we obtain here for endogenous network dynamics and stochastic basins of attraction are the dynamic analogs of our earlier results on endogenous network formation and strategic basins of attraction in static, abstract games of network formation (Page and Wooders, 2008), and build on the seminal contributions of Jackson and Watts (2002), Konishi and Ray (2003), and Dutta, Ghosal, and Ray (2005).

Club Networks with Multiple Memberships and Noncooperative Stability

Modeling club structures as bipartite directed networks, we formulate the problem of club formation as a noncooperative game of network formation and identify conditions on network formation rules and players’ network payoffs sufficient to guarantee that the game has a potential function. Our sufficient conditions on network formation rules require that each player be choose freely and unilaterally those clubs he joins and also his activities within these clubs (subject to his set of feasible actions). We refer to our conditions on rules as noncooperative free mobility. We also require that players’ payoffs be additively separable in player-specific payoffs and externalities (additive separability) and that payoff externalities — a function of club membership, club activities, and crowding — be identical across players (externality homogeneity). We then show that under these conditions, the noncooperative game of club network formation is a potential game over directed club networks and we discuss the implications of this result.

Strategic Basins of Attraction, the Farsighted Core, and Network Formation Games

We make four main contributions to the theory of network formation. (1) The problem of network formation with farsighted agents can be formulated as an abstract network formation game. (2) In any farsighted network formation game the feasible set of networks contains a unique, finite, disjoint collection of nonempty subsets having the property that each subset forms a strategic basin of attraction. These basins of attraction contain all the networks that are likely to emerge and persist if individuals behave farsightedly in playing the network formation game. (3) A von Neumann Morgenstern stable set of the farsighted network formation game is constructed by selecting one network from each basin of attraction. We refer to any such von Neumann-Morgenstern stable set as a farsighted basis. (4) The core of the farsighted network formation game is constructed by selecting one network from each basin of attraction containing a single network. We call this notion of the core, the farsighted core. We conclude that the farsighted core is nonempty if and only if there exists at least one farsighted basin of attraction containing a single network. To relate our three equilibrium and stability notions (basins of attraction, farsighted basis, and farsighted core) to recent work by Jackson and Wolinsky (1996), we define a notion of pairwise stability similar to the Jackson-Wolinsky notion and we show that the farsighted core is contained in the set of pairwise stable networks. Finally, we introduce, via an example, competitive contracting networks and highlight how the analysis of these networks requires the new features of our network formation model.Basins of attraction, Network formation, Supernetworks, Farsighted core, Nash networks