Implementation of Reduced Form Mechanisms: A Simple Approach and a New Characterization with Sergiu Hart
We provide a new characterization of implementability of reduced form mechanisms in terms of straightforward second-order stochastic dominance. In addition, we present a simple proof of Matthews' (1984) conjecture, proved by Border (1991), on implementability.
Sequential Equilibria of Multi-Stage Games with Infinite Sets of Types and Actions with Roger Myerson
Guided by several key examples, we formulate a definition of co-sequential equilibrium for multi-stage games with infinite type sets and infinite action sets, and we prove its general existence.
Existence of Optimal Mechanisms in Principal-Agent Problems with Ohad Kadan and Jeroen M. Swinkels
We provide general conditions under which principal-agent problems admit mechanisms that are optimal for the principal. Our result covers as special cases those in which the agent has no private information -- i.e., pure moral hazard -- as well as those in which the agent's only action is a participation decision -- i.e., pure adverse selection. We allow multi-dimensional actions and signals, as well as both financial and non-financial rewards. Beyond measurability, we require no a priori restrictions on the space of mechanisms. Consequently, our optimal mechanisms are optimal among all measurable mechanisms. A key to obtaining our result is to permit randomized mechanisms. We also provide conditions under which randomization is unnecessary
Consider a real-valued function that, on a convex subset of a real vector space, is continuous on line segments and has convex contour sets. Inspired by a compelling intuitive argument due to Aumann (1975), we provide a simple proof that no strictly increasing transformation of such a function can be concave unless every pair of contour sets are parallel, i.e., unless for every pair of contour sets, either their affine hulls are disjoint or one of their affine hulls contains the other.
Strategic Approximations of Discontinuous Games
An infinite game is approximated by restricting the players to finite subsets of their pure strategy spaces. A strategic approximation of an infinite game is a countable subset of pure strategies with the property that limits of all equilibria of all sequences of approximating games whose finite strategy sets eventually include each member of the countable set must be equilibria of the infinite game. We provide conditions under which infinite games admit strategic approximations.
Matching to Share Risk with Pierre-Andre Chiappori
We consider a matching model in which individuals belonging to two populations ("males" and "females") can match to share their exogenous income risk. Within each population, individuals can be ranked by risk aversion in the Arrow-Pratt sense. The model permits non transferable utility, a context in which few general results have previously been derived. We show that in this framework a stable matching always exists and it is negatively assortative: for any two matched couples, the more risk averse male is matched with the less risk averse female. We discuss the implications of these results for the empirical analysis of risk-sharing.
Toward a Strategic
Foundation for Rational Expectations Equilibrium with Motty
Perry
A strategic foundation for rational expectations equilibrium is provided by considering a double auction with n buyers and m sellers with interdependent values and affiliated private information. If there are sufficiently many buyers and sellers, and their bids are restricted to a sufficiently fine discrete set of prices, then, generically, there is an equilibrium in nondecreasing bid functions which is arbitrarily close to the unique fully revealing rational expectations equilibrium of the limit market with unrestricted bids and a continuum of agents. In particular, the large double auction equilibrium is almost efficient and almost fully aggregates the agents’ information.
Supplementary
materialOn The Existence of Pure Strategy Monotone Equilibria in Asymmetric First-Price Auctions with Shmuel Zamir
We establish the existence of pure strategy equilibria in monotone bidding functions in first-price auctions with asymmetric bidders, interdependent values and affiliated one-dimensional signals. By extending a monotonicity result due to Milgrom and Weber (1982), we show that single crossing can fail only when ties occur at winning bids or when bids are individually irrational. We avoid these problems by considering limits of ever finer finite bid sets such that no two bidders have a common serious bid, and by recalling that single crossing is needed only at individually rational bids. Two examples suggest that our results cannot be extended to multidimensional signals or to second-price auctions.
A Short Proof of Harsanyi's Purification Theorem with Srihari Govindan and Arthur J. Robson
A short and more general proof of Harsanyi’s
purification theorem is provided through an application of a powerful,
yet intuitive, result from algebraic topology.
We provide an ascending auction that yields an efficient outcome when there are many identical units for sale and bidders have interdependent values and downward-sloping demand. Our ascending auction both extends and generalizes Ausubel's (1997) and yields the same outcome as Perry and Reny's (2002) generalization of Vickrey's (1961) sealed-bid auction. There are two key features of our auction. Bidders are permitted both to express different demands against different bidders, as well as to increase their demands. The equilibrium strategies are closely related to the familiar "drop out when price equals value" strategy of the English auction.
We provide several generalizations of the main results in Reny (1999) for the existence of both pure and mixed strategy Nash equilibria in discontinuous games. We also provide an example demonstrating that one natural additional generalization is not possible.