Frank H. Page

Strategic Basins of Attraction, the Path Dominance Core, and Network Formation Games

Given the preferences of players and the rules governing network formation, what networks are likely to emerge and persist? And how do individuals and coalitions evaluate possible consequences of their actions in forming networks? To address these questions we introduce a model of network formation whose primitives consist of a feasible set of networks, player preferences, the rules of network formation, and a dominance relation on feasible networks. The rules of network formation may range from non-cooperative, where players may only act unilaterally, to cooperative, where coalitions of players may act in concert. The dominance relation over feasible networks incorporates not only player preferences and the rules of network formation but also assumptions concerning the degree of farsightedness of players. A specification of the primitives induces an abstract game consisting of (i) a feasible set of networks, and (ii) a path dominance relation defined on the feasible set of networks. Using this induced game we characterize sets of network outcomes that are likely to emerge and persist. Finally, we apply our approach and results to characterize the equilibrium of well known models and their rules of network formation, such as those of Jackson and Wolinsky (1996) and Jackson and van den Nouweland (2005).

A necessary and sufficient condition for the compactness of individually rational and feasible outcomes and the existence of an equilibrium

Abstract In an economy with arbitrary closed and convex consumption sets we show that Pages (Journal of Economic Theory, 1987, 41, 392–404) condition of no unbounded arbitrage is necessary and sufficient for compactness of the set of individually rational and feasible allocations. In strictly reconcilable economies, in which at most one agent may have half-lines in his indifference surfaces, we show that Pages condition is necessary and sufficient for compactness of the set of individually rational utility possibilities, for the existence of an equilibrium, for non-emptiness of the core, and for the existence of a Pareto-optimal allocation. These results allow one agent to be risk-neutral, not permitted by Werners (Econometrica, 1987, 55, 1403–1418) assumption of no half-lines in indifference surfaces. Our proofs that no unbounded arbitrage is necessary and sufficient for the existence of an equilibrium and for a Pareto-optimal allocation both rely on the result that non-emptiness of the core implies no unbounded arbitrage.

Three principles of competitive nonlinear pricing

This paper makes three contributions: (1) A competitive revelation principle for contracting games in which several principals compete for one privately informed agent. Specifically, given any profile of incentive compatible indirect contracting mechanisms, there exists an incentive compatible direct contracting mechanism that, in all circumstances, generates the same contract selection as the profile of indirect mechanisms. (2) A competitive taxation principle. That is, given any incentive compatible direct contracting mechanism, there exists a unique profile of nonlinear pricing schedules that implements the mechanism and the converse. (3) Existence of Nash equilibrium for the mixed extension of the nonlinear pricing game. This is proven using the taxation principle (2 above) and a result due to Reny, Econometrica 1999. To appear as a CERMSEM, Paris 1, Working Paper and also on http://www.warwick.ac.uk/fac/soc/Economics/research/twerps.html.

Arbitrage and asset prices

Abstract We generalize the concept of utility arbitrage introduced in Page (Working Paper 401, Graduate School of Business, Indiana University, 1989) and provide a characterization. We also show that in an asset exchange economy with short selling and heterogeneous investors, an absence of unbounded utility arbitrages is necessary and sufficient for the existence of a general equilibrium. This result generalizes similar results by Hammond ( Journal of Economic Theory , 1983, 31, 170–175) and Page (Working Paper 81/82-2-51, Department of Finance, University of Texas, 1982). Utility arbitrage occurs when investors can purchase utility-increasing portfolios for non-positive costs. In contrast, the standard condition of no arbitrage in finance dictates that there are no opportunities to purchase wealth-increasing portfolios for non-positive costs. In general, the absence of wealth-increasing arbitrage is not equivalent to the absence of unity-increasing arbitrage. To underscore the importance of this fact, we construct an example of a competitive asset exchange economy with complete markets and risk-averse investors in which there is an absence of wealth-increasing arbitrages, but there does not exist a competitive general equilibrium.

Arbitrage, equilibrium, and gains from trade: A counterexample

Abstract We present a counterexample to a theorem due to Chichilnisky (Bulletin of the American Mathematical Society, 1993, 29, 189–207; American Economic Review, 1994, 84, 427–434). Chichilniskys theorem states that her condition of limited arbitrage is necessary and sufficient for the existence of an equilibrium in an economy with unbounded short sales. Our counterexample shows that the condition defined by Chichilnisky is not sufficient for existence of equilibrium. We also discuss difficulties in Chichilnisky (Economic Theory, 1995, 5, 79–107).

Existence of optimal auctions in general environments

Abstract We provide a unified approach to the problem of existence of optimal auctions for a wide variety of auction environments. We accomplish this by first establishing a general existence result for a particular Stackelberg revelation game. By systematically specializing our revelation game to cover various types of auctions, we are able to deduce the existence of optimal Bayesian auction mechanisms for single and multiple unit auctions, as well as for contract auctions with moral hazard and adverse selection. In all cases, we allow for risk aversion and multidimensional, stochastically dependent types.

Optimal deterministic contracting mechanisms for principal-agent problems with moral hazard and adverse selection

Abstract. For the principal-agent problem with moral hazard and adverse selection we establish that within the collection of all measurable, deterministic contracting mechanisms satisfying the individual rationality and incentive compatibility constraints there exists one that is optimal for a risk averse principal contracting with a risk averse agent. In addition to demonstrating existence, one of the main contributions of the paper is to show that, in general, centralized contracting implemented via a contracting mechanism is equivalent to delegated contracting implemented via a contract menu. Thus, contracting can always be delegated to the agent without gain or loss to the principal. Based on this result, the existence of an optimal contracting mechanism for the principal-agent problem is established by showing that there exists an optimal contract menu for the equivalent delegated contracting problem.

Arbitrage and Equilibrium in Economies with Externalities

We introduce consumption externalities into a general equilibrium model with arbitrary consumption sets. To treat the problem of existence of equilibrium, a condition of no unbounded arbitrage, extending the condition of Page (1987) and Page and Wooders (1993, 1996) is defined. It is proven that this condition is sufficient for the existence of an equilibrium and both necessary and sufficient for compactness of the set of rational allocations.