Existence of Nash Networks in One-Way Flow Models (Revised Version of LSU Working Paper 2006-05)

This paper addresses the existence of Nash networks for the one-way flow model of Bala and Goyal (2000) in a number of different settings. First, we provide conditions for he existence of Nash networks in models where costs and values of links are heterogenous and players obtain resources from others only through the directed path between them. We find that costs of establishing links play a vital role in the existence of Nash networks. Next we examine the existence of Nash networks when there are congestion effects in the model. Then, we provide conditions for the existence of Nash networks in a model where a player’s payoff depends on the number of links she has established as well as on the number of links that other players in the population have created. More precisely, we show that convexity and increasing (decreasing) differences allow for the existence of Nash networks.

Partner Heterogeneity Increases the Set of Strict Nash Networks

Galeotti et al. (2006, [2]) show that all minimal networks can be strict Nash in two-way flow models with full parameter heterogeneity while only inward pointing stars and the empty network can be strict Nash in the homogeneous parameter model of Bala and Goyal (2000, [1]). In this note we show that the introduction of partner heterogeneity plays a major role in substantially increasing the set of strict Nash equilibria.

On the Interaction between Heterogeneity and Decay in Directed Networks

In this paper, we examine the role played by heterogeneity in the connection model. In sharp contrast to the homogeneous cases we show that under heterogeneity involving only two degrees of freedom, all networks can be supported as Nash or efficient. Moreover, we show that there does not always exist Nash networks. However, we show that on reducing heterogeneity, both the earlier “anything goes” result and the non existence problem disappear.

Local Spillovers, Convexity and the Strategic Substitutes Property in Networks

We provide existence results in a game with local spillovers where the payoff function satisfies both convexity and the strategic substitutes property. We show that there always exists a stable pairwise network in this game, and provide a condition which ensures the existence of pairwise equilibrium networks. Moreover, our existence proof allows us to characterize a pairwise equilibrium of these networks.

Existence of Nash Networks and Partner Heterogeneity

In this paper, we pursue the work of H. Haller and al. (2005, [10]) and examine the existence of equilibrium networks, called Nash networks, in the noncooperative two-way flow model (Bala and Goyal, 2000, [1]) with partner heterogeneous agents. We show through an example that Nash networks do not always exist in such a context. We then restrict the payoff function, in order to find conditions under which Nash networks always exist. We give two properties : increasing differences and convexity in the first argument of the payoff function, that ensure the existence of Nash networks. It is worth noting that linear payoff functions satisfy the previous properties.Nash networks; two-way flow models; partner heterogeneity

Existence of Nash Networks in One-Way Flow Models

This paper addresses the existence of Nash networks for the one-way flow model of Bala and Goyal (2000) in a number of different settings. First, we provide conditions for he existence of Nash networks in models where costs and values of links are heterogenous and players obtain resources from others only through the directed path between them. We find that costs of establishing links play a vital role in the existence of Nash networks. Next we examine the existence of Nash networks when there are congestion effects in the model. Then, we provide conditions for the existence of Nash networks in a model where a player’s payoff depends on the number of links she has established as well as on the number of links that other players in the population have created. More precisely, we show that convexity and increasing (decreasing) differences allow for the existence of Nash networks.

Existence of Nash Networks in One-Way Flow Models

This paper addresses the existence of Nash equilibria in one-way flow or directed network models in a number of different settings. In these models players form costly links with other players and obtain resources from them through the directed path connecting them. We find that heterogeneity in the costs of establishing links play a crucial role in the existence of Nash networks. We also provide conditions for the existence of Nash networks in models where costs and values of links are heterogeneous.Network Formation, Non-cooperative Games

Heterogeneity and Link Imperfections in Nash Networks

Heterogeneity in Nash networks with two-way flow can arise due to differences in the follow- ing four variables: (i) the value of information held by players, (ii) the rate at which information decays as it traverses the network, (iii) the probability with which a link transmits information, and (iv) the cost of forming a link. Observe that the second and third forms of heterogeneity are also instances of link imperfections. In sharp contrast to the homogeneous cases in this paper we show that for any type of link imperfection, under heterogeneity involving only two degrees of freedom, all networks can be supported as Nash or efficient. To address the question of conflict between stability and efficiency, we then identify conditions under which efficient networks are also Nash. We also find that cost heterogeneity leads to non-existence of Nash networks in models with and without link imperfections. We show that in general there is no relationship between the decay and probabilistic models of network formation. Finally, we show that on reducing heterogeneity the earlier “anything goes” result disappears.

On the Interaction between Heterogeneity and Decay in Two-way Flow Models

In this paper we examine the role played by heterogeneity in the popular “connections model” of Jackson andWolinsky (1996). We prove that under heterogeneity in values or decay involving only two degrees of freedom, all networks can supported as Nash. Moreover, we show that Nash networks may not always exist. In the absence of decay, neither result can be found in a model with value heterogeneity. Finally, we show that on reducing heterogeneity, both the earlier “anything goes” result and the non-existence problem disappear.connections model; decay; two-way flow models

A Note on Existence of Nash Networks in One-way Flow

In this note we provide conditions which ensure the existence of Nash networks in One-way flow models with cost heterogeneity.