A network experiment in continuous time: The influence of link costs

In recent work on non-cooperative network formation star-shaped networks play an important role. In a particular theoretical model of Bala and Goyal (2000) center-sponsored stars are the only strict Nash networks. In testing this theoretical model, Falk and Kosfeld (2003) do not find experimental evidence that players select the center-sponsored star. Based on a slight modification of Bala and Goyal’s model, we design a network formation experiment in which, depending on link costs, periphery-sponsored stars and the empty network are the only strict Nash networks. We observe that almost all groups not only reach a strict Nash network once but also switch the center player in periphery-sponsored stars several times. The main innovation in our experiment is to use a continuous time framework which we believe to be a more realistic setting to study behavior in network formation situations and which makes coordination on stars much easier than simultaneous strategy adaptation in discrete time. Electronic Supplementary Material Supplementary material is available in the online version of this article at http://dx.doi.org/10.1007/s10683-006-9125-1. JEL Classification C72 · C92 · D81     

Existence of Nash networks in one-way flow models

This paper addresses the existence of Nash equilibria in one-way flow 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 plays a crucial role for the existence of Nash networks. The paper also provides conditions for the existence of Nash networks in models where costs and values of links are heterogeneous. We would like to thank an anonymous referee for useful suggestions. We would also like to thank Hans Haller and participants at the AEA Meetings 2007 for helpful comments. Sudipta Sarangi acknowledges the support of NSF grant HSD-0527315 and the hospitality of CREUSET, Jean Monnet University.     

Group identity in a network formation game with cost sharing

This paper introduces the choice of group identity, which is comprised of characteristics and commitments to these characteristics, in a network formation model where links costs are shared. Players want to link to the largest group given that linking costs for players of the same (different) characteristic are decreasing (increasing) in their commitments. The equilibrium concepts used are Nash equilibrium as well as one that looks for stability allowing for bilateral negotiations. Conditions are shown under which the endogenous choice of characteristics and commitments allow for multiple groups with segregated or connected networks. When group identity is partly endogenous, such that characteristics are fixed and players only choose commitment, it is shown that the equilibrium population commitment profile dictates whether the resulting network is segregated or connected. When group identity is fully endogenous, it is shown that multiple groups and segregated networks are possible equilibrium outcomes but such outcomes are not stable unless the group size additionally affects the costs of link formation.     

One-way flow networks: the role of heterogeneity

I study a one-way flow connections model in which players are heterogeneous with respect to values and the costs of establishing a link. I show that values and costs heterogeneity are equally important in determining the level of connectedness and the architecture of equilibrium networks. I also show that when asymmetries are independent of the potential partner there are distributions of costs and values for which centrality is a distincitive feature of equilibrium networks. This sharply contrasts with the homogeneous case.I thank an anonymous referee for useful comments. The paper has benefited from discussion with Gabriella Conti, Jose Luis Moraga-Gonzalez, Fernando Vega-Redondo and Sanjeev Goyal.     

N‐FIRM OLIGOPOLY WITH GENERAL ISO‐ELASTIC DEMAND

In this note oligopoly with iso‐elastic demand is analysed. Unlike previous studies we consider general iso‐elastic demand rather than the case of unit elasticity. An n‐firm Nash‐Cournot equilibrium for the case of heterogeneous constant marginal costs is derived. The main result is a closed‐form solution that shows the dependency of the equilibrium on the elasticity of demand and the share of industry costs. The result has applications to a wide range of areas in oligopoly theory by allowing comparisons across markets with different elasticities of demand.     

A geometric approach to the price of anarchy in nonatomic congestion games

We present a short, geometric proof for the price-of-anarchy results that have recently been established in a series of papers on selfish routing in multicommodity flow networks and on nonatomic congestion games. This novel proof also facilitates two new types of theoretical results: On the one hand, we give pseudo-approximation results that depend on the class of allowable cost functions. On the other hand, we derive stronger bounds on the inefficiency of equilibria for situations in which the equilibrium costs are within reasonable limits of the fixed costs. These tighter bounds help to explain empirical observations in vehicular traffic networks. Our analysis holds in the more general context of nonatomic congestion games, which provide the framework in which we describe this work.     

The structure of information networks

We develop a strategic model of information acquisition in networks where agents pay for all the pieces of information they acquire, including those through indirect links. The cost of information depends on the distance it traverses in the network. We consider two possibilities in this context: (1) costs increasing with distance, and (2) costs decreasing with distance. The paper also examines situations where it is more expensive to acquire information of higher value. We show that there is almost no divergence between the efficient and Nash equilibrium information architectures. We then study the effect of decay in networks where information through longer paths is cheaper. Finally, we also examine a model with costly link formation that combines both types of cost related assumptionsWe are grateful to Hans Haller, Rob Gilles, Susanne Maria Schmidt, Sumit Joshi, Georg Erber, Beth Allen, Mark Machina, Bob Martin, Johanna Francis, Cheryl Long, Raja Kali, Kaz Miyagiwa, Micheal Kosfeld, Bibhudutta Panda and two anonymous referees for helpful suggestions. The paper has further benefited from the comments of participants at Game Theory and Applications Mumbai 2003, Royal Economic Society Meetings 2003, SED 2004 and GAMES 2004. Sudipta Sarangi acknowledges the hospitality of DIW Berlin where a part of this research was carried out. Rajgopal Kannan acknowledges the support of NSF grants IIS-0329738 and IIS-0312632     

Wage bargaining and turnover costs with heterogenous labor and perfect history screening

We study effects of mobility costs in a model of (Nash) wage bargaining between workers and firms, with instantaneous matching, heterogeneous workers, identical firms and free firm entry, and where firms can screen workers perfectly according to their previous work history but not their actual productivity. We derive the employment level and the minimum worker quality standard, in the market solution, and in the efficient solution established by a social planner. When workers have positive bargaining power, there is always some inefficient unemployment among desired workers in the market solution. The lowest hiring standard chosen by firms is higher than the planner's standard when firing costs are high relative to hiring costs, but may be lower in the opposite case. We show that any higher established hiring standard corresponds to a market equilibrium. The model explains a tendency for a high initial unemployment rate to remain high, particularly for low-skilled workers.     

Strategic Network Disruption and Defense

We study a game between a network designer, who uses costly links to connect nodes in a network, and a network disruptor who tries to disrupt the resulting network as much as possible by deleting either nodes or links. For low linking costs networks with all nodes in symmetric positions are a best response of the designer under both link deletion and node deletion. For high linking costs the designer builds a star network under link deletion, but for node deletion excludes some nodes from the network to build a smaller but stronger network. For intermediate linking costs the designer again builds a symmetric network under node deletion but a star‐like network with weak spots under link deletion.     

Options for Electricity Transmission Regulation in Australia

The pricing of access to electricity transmission networks in Australia is currently under review. Several options have been proposed including those based on nodal pricing and the assignment of transmission rights contracts. As most of the marginal costs of transmission are recovered through wholesale electricity prices we focus on the key issue of regulation and investment incentives. We find that current options are unlikely to be adequate in terms of encouraging socially optimal levels and timing of new transmission investment. As an alternative, we propose a regulatory scheme, based on a related idea by Sappington and Sibley that can overcome this problem. Our scheme can potentially generate first best results and is readily applicable given the current institutional structure of electricity markets in Australia.     

Stackelberg thresholds in network routing games or the value of altruism

It is well known that the Nash equilibrium in network routing games can have strictly higher cost than the optimum cost. In Stackelberg routing games, where a fraction of flow is centrally-controlled, a natural problem is to route the centrally-controlled flow such that the overall cost of the resulting equilibrium is minimized.We consider the scenario where the network administrator wants to know the minimum amount of centrally-controlled flow such that the cost of the resulting equilibrium solution is strictly less than the cost of the Nash equilibrium. We call this threshold the Stackelberg threshold and prove that for networks of parallel links with linear latency functions, it is equal to the minimum of the Nash flows on links carrying more optimum flow than Nash flow.Our approach also provides a simpler proof of characterization of the minimum fraction that must be centrally controlled to induce the optimum solution.     

Principles of network development and evolution: an experimental study

This paper reports on an experimental investigation of the evolution of networks and the individual decision-making processes that guide it. Inasmuch as there is no history of experimental work on network formation, part of the paper is devoted to the formulation of problems that can be examined experimentally. The results are that networks, composed of decentralized decision makers, are capable of overcoming complex coordination and learning problems and converge to stationary configurations. While stationarity is frequently observed, such an achievement is not guaranteed, and when it does not occur, significant and persistent inefficiencies can result. The models of equilibration based on the principle of Nash equilibrium are more reliable than models based on the alternative principles of efficiency seeking or focalness of the network configuration. However, individual decision making within networks is not in accordance with the simple decision rule of Nash best response. Instead, we observe complicated strategies that appear to trade short-term profits in order to signal to and teach other agents the strategies required for long-term profit maximization.     

A NOTE ON MARKET COVERAGE IN VERTICAL DIFFERENTIATION MODELS WITH FIXED COSTS

With fixed costs of quality improvement, we find that a covered market outcome with an interior solution in the price stage is not a Nash equilibrium. When the degree of consumer heterogeneity is high (low) enough, an uncovered market outcome (a covered market outcome with a corner solution in the price stage) is the only Nash equilibrium. When the degree of consumer heterogeneity is moderate, both of the two market outcomes are Nash equilibria, but an uncovered market outcome yields higher social welfare than a covered market outcome with a corner solution in the price stage.     

Spillovers from Foreign Exporters

We develop a general equilibrium model of international trade with heterogeneous firms that accounts for productivity spillovers transmitted by foreign exporters. Everything else equal, stronger spillovers increase welfare. We embed the model framework into a trade policy scenario where countries strategically set inter‐country variable trade costs for the trading partner. In the strategic Nash‐equilibrium policy, governments trade‐off welfare gains from protectionism and those that are due to spillovers from foreign exporters. The equilibrium degree of protectionism is decreasing in the strength of the spillover. Policy coordination induces welfare gains but these gains can be hump‐shaped in the spillover strength.     

A note on control costs and logit rules for strategic games

We consider best response dynamics with endogenous noise based on a finite game in strategic form. A player can reduce the noise level by expending an extra effort and incurring some disutility or control costs. We specify control costs that result in logit adjustment rules. The stochastically stable states of the dynamic process are partial Nash configurations, that is, states where at least one player plays a best response against the others. If the game has a potential, then the stochastically stable states coincide with the Nash equilibria on which the potential is maximized. RID="*" ID="*" Instructive comments of a referee are gratefully acknowledged. Correspondence to:H. Haller     

Knowledge flows and the geography of networks: A strategic model of small world formation

This paper aims to demonstrate that the strategic approach to link formation can generate networks that share some of the main structural properties of most real social networks. For this purpose, we introduce a spatialized variation of the Connections model [Jackson, M.O., Wolinsky, A., 1996. A strategic model of social and economic networks. Journal of Economic Theory 71, 44–74] to describe the strategic formation of links by agents who balance the benefits of forming links resulting from imperfect knowledge flows against their costs, which increase with geographic distance. We show, for intermediate levels of knowledge transferability, clustering occurs in geographical space and a few agents sustain distant connections. Such networks exhibit the small world property (high clustering and short average relational distances). When the costs of link formation are normally distributed across agents, asymmetric degree distributions are also obtained.     

Volunteering a public service: an experimental investigation

In some public goods environments it may be advantageous for heterogeneous groups to be coordinated by a single individual. This “volunteer” will bear private costs for acting as the leader while enabling each member of the group to achieve maximum potential gains. This environment is modeled as a War of Attrition game in which everyone can wait for someone else to volunteer. Since these games generally have multiple Nash equilibria but a unique subgame-perfect equilibrium, we tested experimentally the predictive power of the subgame-perfection criterion. Our data contradict that subjects saw the subgame-perfect strategy combination as the obvious way to play the game. An alternative behavioral hypothesis—that subjects were unable to predict accurately how their opponents would play and tried to maximize their expected payoff—is proposed. This hypothesis fits the observed data generally well.     

Properties and applications of dual reduction

The dual reduction process, introduced by Myerson, allows a finite game to be reduced to a smaller-dimensional game such that any correlated equilibrium of the reduced game is an equilibrium of the original game. We study the properties and applications of this process. It is shown that generic two-player normal form games have a unique full dual reduction (a known refinement of dual reduction) and all strategies that have probability zero in all correlated equilibria are eliminated in all full dual reductions. Among other applications, we give a linear programming proof of the fact that a unique correlated equilibrium is a Nash equilibrium, and improve on a result due to Nau, Gomez-Canovas and Hansen on the geometry of Nash equilibria and correlated equilibria.     

Generic determinacy of Nash equilibrium in network-formation games

This paper proves the generic determinacy of Nash equilibrium in network-formation games: for a generic assignment of utilities to networks, the set of probability distributions on networks induced by Nash equilibria is finite.     

Overbidding and overspreading in rent-seeking experiments: Cost structure and prize allocation rules

We study experimentally the effects of cost structure and prize allocation rules on the performance of rent-seeking contests. Most previous studies use a lottery prize rule and linear cost, and find both overbidding relative to the Nash equilibrium prediction and significant variation of efforts, which we term ‘overspreading.’ We investigate the effects of allocating the prize by a lottery versus sharing it proportionally, and of convex versus linear costs of effort, while holding fixed the Nash equilibrium prediction for effort. We find the share rule results in average effort closer to the Nash prediction, and lower variation of effort. Combining the share rule with a convex cost function further enhances these results. We can explain a significant amount of non-equilibrium behavior by features of the experimental design. These results contribute towards design guidelines for contests based on behavioral principles that take into account implementation features of a contest.