Uncertainty and Information in Economics - IMS

Uncertainty and Information in Economics
(9 May - 3 July 2005)

~ Abstracts ~

Computational mechanism design and auctions
David C. Parkes, Harvard University

Computational Mechanism Design (CMD) aims to develop protocols to implement desirable outcomes in a distributed system of self-interested agents. An outcome might define a joint plan of action, or an allocation of resources to agents. CMD brings together economic concepts of game-theoretic equilibrium and incentive-compatibility constraints with the traditional computer science concerns of computational tractability and communication complexity.

This tutorial is designed for a general audience in computer science and economics, and it should have interest to both novices and experts in computational mechanism design and computational game theory. We will begin with direct-revelation and truthful mechanisms, and study the Vickrey-Clarke-Groves mechanism and then more general characterization results. Continuing, we will study iterative mechanisms, including ascending-price auctions, and introduce the general problem of incentive-compatible preference elicitation. In closing, we will define the problem of mechanism design for sequential decision problems and mention some challenges in combining learning with mechanism design.

Iterated strict dominance in general games
Xiao Luo, Institute of Economics, Academia Sinica, Taiwan

Following Milgrom and Roberts [Econometrica 58(1990), 1255-1278], we offer a definition of iterated elimination of strictly dominated strategies (IESDS) for games with (in)…finite players, (non)compact strategy sets, and (dis)continuous payoff functions. IESDS is always a well-defined order independent procedure that can be used to solve out Nash equilibrium in dominance-solvable games. We characterize IESDS by means of a "stability" criterion. We show by an example that IESDS might generate spurious Nash equilibria in the class of Reny's better-reply secure games. We provide sufficient conditions under which IESDS preserves the set of Nash equilibria. JEL Classifi…cation: C70, C72.

Trade with heterogeneous multiple priors
Atsushi Kajii, Kyoto University

This paper presents a general framework to understand the possibility of a purely speculative trade under asymmetric information, where the decision making rule of each trader conforms to the multiple priors model (Gibloa and Schmeidler, 1989): the agents are interested in the minimum of the conditional expected value of trade where the minimum is taken over the set of posteriors. In this framework, we derive a necessary and sufficient condition on the sets of posteriors, thus implicitly on the updating rules adopted by the agents, for non-existence of trade such that it is always common knowledge that every agent expects a positive gain.

Applications of rich measure spaces formed from nonstandard models
Peter A. Loeb, University of Illinois at Urbana-Champaign

We review recent work by several authors that utilize the rich structure of what are called "Loeb measure spaces" in the literature. In particular, we show that there are many results, some used for decades without a rigorous foundation, that are only true for spaces with this rich structure.

A reformulation of the rational expectations equilibrium (REE): Walrasian Bayesian equilibrium
Nicholas Yannelis, University of Illinois at Urbana-Champaign

The REE as introduced by Radner in general doesn’t exist, it fails to be fully Pareto optimal and incentive compatible and also it is not implementable as a perfect Bayesian equilibrium of an extensive form game. The lack of all the above properties is mainly due to the fact the agents are supposed to predict the equilibrium market clearing price (as agent’s expected maximized utility is conditioned on the information that equilibrium prices reveal) which is compatible with the idea that agents know all the primitives in the economy, i.e., random initial endowments, random utility function and private information sets. To get around this ad hoc equilibrium notion, we introduce a new concept called Bayesian Walrasian equilibrium (BWE) which has Bayesian features. In particular, agents try to predict the market clearing prices using a Bayesian updating and evaluate their consumption in terms of Bayesian price estimates which are different for each individual. In this framework agents maximize expected utility based on their own private information, subject to a Bayesian estimated budget constraint. The market clearing it not state wise, it is in expected terms. However, once learning is introduced (i.e. filtration of the information partitions) our BWE becomes fully revealing and leads to a stare wise market clearing. This new BWE exists under the standard assumptions, it is Pareto optimal and it is incentive compatible contrary to the REE.

Uncoupled dynamics and Nash equilibrium
Sergiu Hart, Hebrew University of Jerusalem

We call a dynamical system "uncoupled" if the dynamic for each player does not depend on the payoff functions of the other players. This is a natural informational restriction. We study convergence of uncoupled dynamics to Nash equilibria, and present a number of possibility and impossibility results. (joint work with Andreu Mas-Colell)

See http://www.ma.huji.ac.il/hart/abs/uncoupl-st.html and also http://www.ma.huji.ac.il/hart/abs/uncoupl.html.

Redoing the foundations of decision theory: Decision theory with subjective state spaces
Joseph Y. Halpern, Cornell University

The standard approach in decision theory (going back to Savage) is to place a preference order on acts, where an act is a function from states to outcomes. If the preference order satisfies appropriate postulates, then the decision maker can be viewed as acting as if he has a probability on states and a utility function on outcomes, and is maximizing expected utility. This framework implicitly assumes that the decision maker knows what the states and outcomes are. That isn't reasonable in a complex situation. For example, in trying to decide whether or not to attack Iraq, what are the states and what are the outcomes? We redo Savage viewing acts essentially as syntactic programs. We don't need to assume either states or outcomes. However, among other things, we can get representation theorems in the spirit of Savage's theorems; for Savage, the agent's probability and utility are subjective; for us, in addition to the probability and utility being subjective, so is the state space and the outcome space. I discuss the benefits, both conceptual and pragmatic, of this approach.

This is joint work with Larry Blume and David Easley. No prior knowledge of Savage's work is assumed.

Information and market design
Ruqu Wang, National University of Singapore and Queens University, Canada

The effects of information on market design are explored in a simple setting where firms have private information about their correlated fixed costs and the government aims to maximize its expected revenue conditional on achieving efficient allocations. Government revenues are higher when the costs are less correlated (or are more of a private value). The reduced correlation increases the firms' information rents, but a change in the information structure also changes the expected market structures with positive effects on government revenues. If the government faces the no-deficit constraint, there are situations where efficient allocations are achieved under asymmetric information but not under symmetric information.

Strategic uncertainty and the Hirshleifer effect
Murali Agastya, University of Sydney

The Hirshleifer effect refers to a situation where an increase in the amount of information leads to a reduction in welfare in the Pareto sense, the intuition being that a "helicopter drop" of information destroys hedging opportunities and consequently a decline in ex-ante utility. In many strategic situations, the amount of information that is ultimately common-knowledge, and hence the set of hedging opportunities that are available is *endogenous* -- it depends on the equilibrium the players may coordinate upon. In the context of a stylized model of insider trading, we show that an informationally inefficient (pooling) equilibrium can Paerto dominate to an informationally efficient separating equilibrium.

Efficiently regulated competition in insurance economies with adverse selection
Peter Hammond, Stanford University

In insurance economies with a continuum of agents and adverse selection, it is shown that incentive-constrained Pareto efficient allocations correspond to regulated competitive (or compensated competitive) equilibria in markets with non-linear pricing for options to buy insurance contracts. These options make the incentive constraints self-enforcing. Efficiency is achieved through a "universal service" requirement allowing only new contracts or blocking coalitions that benefit all potential types of each agent. This regulation prevents "cream skimming" intended to exclude high-risk agents. Under suitable assumptions, regulated equilibria are shown to exist and be characterized as "regulated core" allocations.

Cominimum additive operator
Takashi Ui, Yokohama National University

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The communication requirements of social choice rules and supporting budget sets
Ilya Segal, Stanford University

The paper examines the communication requirements of social choice rules when the (sincere) agents privately know their preferences. It shows that for a large class of choice rules, any minimally informative way to verify that a given alternative is in the choice rule is by verifying a "budget equilibrium", i.e., that the alternative is optimal to each agent within a "budget set" given to him. Therefore, any communication mechanism realizing the choice rule must …nd a supporting budget equilibrium. We characterize the class of choice rules that have this property. Furthermore, for any rule from the class, we characterize the minimally informative messages (budget equilibria) verifying it. This characterization is used to identify the amount of communication needed to realize a choice rules, measured with the number of transmitted bits or real variables. Applications include efficiency in convex economies, exact or approximate surplus maximization in combinatorial auctions, the core in indivisible-good economies, and stable many-to-one matchings.

Uncertainty and hyperbolic discounting
Eric Maskin, Institute for Advanced Study, Princeton

We propose an evolutionary explanation for the pattern of intertemporal preference reversals often ascribed to “hyperbolic discounting.” We take the view that preferences— manifested, for example, in urges, cravings, and inclinations— are the outcome of evolutionary forces, and so will induce animals or humans to make survival-maximizing choices in “typical” decision problems. We show that if the typical problem involves payoffs whose realization times are uncertain, then optimal preferences give rise to relatively patient behavior when the time horizon is long but induce a switch to impatience when the horizon grows short.

Such reversals do not entail dynamic inconsistency in typical decision problems; behavior there is optimal. However, if a decision-maker is confronted with a choice for which the realization-time uncertainty falls outside the evolutionary norm, her preferences may well prompt her to behave inconsistently. We argue that, if such a choice problem recurs, her evolutionarily endowed ability to learn will lead her to make self-commitments against these urges.

The comparative statics of constrained optimization problems
John Quah, University of Oxford

The objective of this paper is to develop and apply some new results in the theory of monotone comparative statics. Let f be a real-valued function defined on R^l, and consider the problem of maximizing f(x) when x is constrained to lie in some subset C of R^l. We develop a natural way of ordering constraint sets and identify the conditions on f which guarantee that the solution to the maximization problem increases as the constraint set changes. We apply this to a variety of problems in economic theory.

Game theory
Sudhir Shah, University of Delhi

1) Games in extensive form
2) Information structures in games
3) Application I: Auction design
4) Application II: Repeated games

Safe votes, sincere votes, and strategizing
Rohit Parikh, City University of New York

We examine the basic notion of strategizing in the statement of the Gibbard-Satterthwaite theorem and note that to talk about strategizing presumes that we already know what it means to vote sincerely. For a strategic vote is one which differs from a sincere vote, taking into account how others are voting. Now the notion of a sincere vote is clear enough in most commonly used voting systems (which we identify with the aggregation function Ag) but there is no general notion of sincerity. It follows that there cannot be a general notion of strategizing either. In this paper we define a notion of safe vote and study its properties. Safe votes exist with plurality, approval voting and the Borda count, where they concide with sincere votes. Safe votes do not exist with STV.

The nature of equilibria under noncollusive product design and collusive pricing
Kali Rath, University of Notre Dame

It is well known that in a two stage duopoly model of product choice with quadratic transportation cost, the firms locate at the extreme endpoints of the market. This paper examines this model in an infinite horizon setting where in the initial period the firms choose locations and in subsequent periods choose prices. The firms collude in prices and share the profits on the profit possibility frontier. It is shown that under very general conditions, both the firms locating at the center is an equilibrium. It is not necessarily unique and multiple symmetric equilibria can exist. So, the products are not minimally differentiated and the degree of differentiation can vary. Sufficient conditions for three types of equilibria are given: a unique equilibrium at the center of the market, multiple symmetric equilibria and multiple asymmetric agglomerated equilibria. The first two cases obtain if the firms share profits equally when they are located at the same point and the last case otherwise.

Ex ante efficiency and incentive compatibility
Yeneng Sun, National University of Singapore

We show that when agents become informationally negligible in a large economy with asymmetric information, every ex ante efficient allocation must be incentive compatible. Thus, any ex ante core allocation or ex ante Walrasian equilibrium is incentive compatible. Furthermore, asymptotic analogs for such results can be obtained for large but finite asymmetric information economies. These results are false in fixed finite-agent economies with asymmetric information. This is joint work with Nicholas Yannelis of the University of Illinois at Urbana-Champaign.

The g-core and coalition formation
Parkash Chander, National University of Singapore

This paper reinterprets the g-core (Chander and Tulkens (1995, 1997)) and justifies it as well as its prediction that the efficient coalition structure is stable in terms of the coalition formation theory. We assume that coalitions can freely merge or break apart and are farsighted (that is, it is the final and not the immediate payoffs that matter to the coalitions). We then show that subsequent to a deviation by a coalition, it is ex post optimal for the nonmembers to break apart into singletons, as is assumed in the definition of the g-characteristic function, and the grand coalition is a stable coalition structure. Our analysis not only justifies the g-core in terms of the coalition formation theory but also shows that in order to rule out the stability of the efficient coalition structures one must either place some exogenous restrictions on the process of coalition formation or assume that the coalitions are not farsighted.

Factor taxation and growth under asymmetric information
Yong Wang City, University of Hong Kong

The literature on growth effects of factor taxation has revealed that the desirability of capital income taxation, relative to that of labor income taxation, tends to vary with the modeling assumption made about life-cycle considerations. In particular, by considering overlapping generations models in which government taxes both capital and labor income, Uhlig and Yanagawa (1996) and Caballé (1998) show that increasing the tax rate on capital income leads to higher economic growth. The objective of this paper is to reexamine the growth effects of factor taxation in an overlapping generations economy with asymmetric information. The key feature of our model is that capital accumulation takes place through investment projects mediated by a credit market in which asymmetric information between borrowers and lenders is present. In such an environment, we show that the presence of asymmetric information introduces an additional adverse effect of capital income taxation on growth. In so doing, we present a new argument in favor of using a lower (higher) capital (labor) income taxation to fund public spending. As a corollary, we also show that there is a negative relationship between the optimal tax rate on capital income and the severity of asymmetric information in the credit market.

Stock return autocorrelation is not spurious
Robert Anderson, University of California at Berkeley

We find compelling evidence that stock return autocorrelation is not spurious; partial price adjustment is an important source, and in some cases the main source, of the autocorrelation. Our tests of partial price adjustment are direct, using disjoint time intervals, separated by a trade, to eliminate the nonsynchronous trading effect and minimize bid-ask bounce. We find evidence for partial price adjustment in individual stock returns, portfolio returns, and in an unlikely setting: the incorporation of ETF price information into individual stock prices. We find that partial price adjustment generates a very substantial fraction of the autocorrelation. (joint with Kyong Shik Eom, Sang) Buhm Hahn and Jong-Ho Park.

Implementation with interdependent valuation
Richard McLean, Rutgers University

There is a large literature aimed at characterizing the social choice functions that can be implemented in Bayes Nash equilibria. This literature typically takes agents' information as exogenously given and fixed throughout the analysis. While for some problems this may be appropriate, the assumption is problematic for others. A typical analysis, relying on the revelation principle, maximizes some objective function subject to truthful revelation being a Bayes equilibrium. It is often the case that truthful revelation is not "ex post incentive compatible", that is, for a given agent, there are some vectors of the other agents' types for which the agent may be better off by misreporting his type than truthfully revealing it. Truthful revelation, of course, may still be a Bayes equilibrium, because agents announce their types without knowing other agents types: choices must be made on the basis of their beliefs about other agents' types. The difficulty with assuming that agents' information is exogenous is that when truthful revelation is not ex post incentive compatible, agents have incentives to learn other agents' types. To the extent that an agent can, at some cost, learn something about other agents' types, agents' beliefs when a mechanism is applied must be treated as endogenous. A planner who designs a mechanism for which truthful revelation is ex post incentive compatible can legitimately ignore agents' incentives to engage in espionage to discover other agents' types, and consequently, ex post incentive compatibility is desirable. The Clarke-Groves-Vickrey mechanism (hereafter CGV) for private values environments is a classic example of a mechanism for which truthful revelation is ex post incentive compatible.

There has recently been renewed interest in mechanisms for which truthful revelation is ex post incentive compatible. While ex post incentive compatibility is desirable, nontrivial mechanisms for which truthful revelation is ex post incentive compatible fail to exist for a large set of important problems with interdependent valuations. We introduce in this paper a notion of e-ex post incentive compatibility: a mechanism is e-ex post incentive compatibile if truthful revelation is ex post incentive compatible with probability at least 1-e. If truthful revelation is e-ex post incentive compatible for a mechanism, agents' incentive to collect information about other agents' is bounded by e times the maximal gain from espionage. If espionage is costly, a mechanism designer can be relatively comfortable in taking agents' beliefs as exogenous when e is sufficiently small. We show that the existence of mechanisms for which there are e-incentive compatible equilibria is related to the concept of informational size introduced in McLean and Postlewaite (2001, 2002). When agents have private information, the posterior probability distribution on the set of states of the world Q will vary depending on a given agent's type. Roughly, an agent's informational size corresponds to the maximal expected change in the posterior on Q as his type varies, fixing other agents' types. We show that for any e, there exists d such that if each agent's informational size is less than d, there exists an efficient mechanism for which truthful revelation is an e-incentive compatible equilibrium.

Bankruptcy and firm finance
Anne Villamil, University of Illinois at Urbana-Champaign

This paper analyzes how an enforcement mechanism that resembles a court affects firm finance. The court is described by two parameters that correspond to enforcement costs and the amount of creditor/debtor protection. We provide a theoretical and quantitative characterization of the effect of these enforcement parameters on the contract loan rate, the default probability and welfare. We show that when constraints bind, which give agents an incentive to default and pursue bankruptcy, the enforcement parameters have a sharply non-linear effect on finance and welfare. The results provide guidance on when models which abstract from enforcement provide good approximations and when they do not. The bankruptcy rule corresponds to firm liquidation.

Uncertainty in mechanism design
Chris Shannon, University of California at Berkeley

We consider mechanism design problems with interdependent values in which agents perceive Knightian uncertainty. Uncertainty is formalized using incomplete preferences, as in Bewley (1986). We show that the seller can extract all gains from trade with a direct mechanism in which truth-telling is a Nash equilibrium, in the sense that no buyer has a unilateral incentive to misrepresent his type. In these mechanisms, however, truth-telling and misrepresenting may be incomparable alternatives. We also consider an equilibrium refinement in which truth-telling is optimal, i.e. at least as good as any alternative. In this case the full extraction of all gains from trade is feasible only if there is sufficient disagreement in beliefs across types. If all buyers types have sufficiently similar beliefs, an ex-post incentive compatible mechanism is optimal for the seller.

The theory of coalition formation
Parkash Chander, National University of Singapore

This tutorial will attempt to provide a comprehensive theory of coalition formation using both partition function and non-cooperative game theory approaches. Alternative equilibrium concepts will be discussed and their implications on the coalition structure will be derived. Applications of the theory to global environmental problems and industrial organization will be presented.

Main Reading list:

Barrett, S. (1994), “Self-enforcing international environmental agreements”, Oxford Economic Papers, 46, pp. 878-894.
Bloch, F. (1996), “Sequential formation of coalitions in games with externalities and fixed payoff division”, Games and Economic Behavior, 14, pp. 90-123.
Chander, P. (2005), “The gamma-core and coalition formation”, CORE Discussion Paper No. 2003/46. (electronic version available on request by e-mail to the author).
Chander, P. and H. Tulkens (1997), “The core of an economy with multilateral environmental externalities”, International Journal of Game Theory, 26, pp. 379-406.
Chander, P. and H. Tulkens (1995), “A core-theoretic solution for the design of cooperative agreements on transfrontier pollution”, International Tax and Public Finance, 2, pp. 279-293.
Chwe, M.S.Y. (1994), “Farsighted coalition stability”, Journal of Economic Theory, 54, pp. 299-325.
Ray, D. and R. Vohra (1997), “Equilibrium binding agreements”, Journal of Economic Theory, 73, pp. 30-78.
Yi, S. (1997), “Stable coalition structures with externalities”, Games and Economic Behavior, 20, pp. 201-223.

A non-cooperative theory of quantity-rationing global pollution
Sudhir Shah, University of Delhi

We study an incomplete information, non-cooperative game model of the determination of emissions in a Kyoto-type quantity-rationing setting with heterogeneous nations. We model the emission capping negotiations using the best response dynamic process and provide conditions under which the process has a unique and globally asymptotically stable stationary point. We then analyze the link between the type profile and the stationary point to derive the type-contingent ordering of emission allocations. Finally, we study the investment strategies that nations can use prior to the negotiations in order to manipulate the equilibrium emission caps.

Auctions for dynamic environments: WiFi, last-minute tickets and grid computing
David Parkes, Harvard University

I consider the problem of auction design for dynamic environments, in which agents arrive and depart dynamically and in which goods are inherently temporal. Motivating examples are drawn from the problem of WiFi allocation in coffee houses, last-minute tickets, and scientific grid computing. We provide a chracterization for the design of truthful online auctions, such that it is a domnant strategy equilibrium for bidders to reveal their true value for resources immediately upon arrival into a system. The auctions are online, in the sense that they make allocation decisions without knowledge of the future. In a setting without priors, we provide an e-competitive (for efficiency) truthful auction for a limited-supply unit-demand problem, drawing an analogy with the classic secretary problem. We also present a 2-competitive auction (wrt efficiency) for a setting with a reusable resource, and and describe a randomized online auction that achieves a competitive ratio for revenue of O(log h), where h is the ratio of maximum value to minimum value among the agents. In closing, we discuss approaches that utilize priors as an interesting direction for future work.

Approximate generalizations and applied equilibrium analysis
Felix Kubler, Universitaet Mannheim

In this paper I derive conditions on the fundamentals of general equilibrium models that allow for a generalization of finitely many examples to statements about (infinite) classes of economies and I show how these approximate generalizations can be applied in computational experiments.

If there exist upper bounds on the number of connected components of one-dimensional linear subsets of the set of parameters for which a conjecture is true, one can conclude that it is correct for all parameter values in the class considered, except for a small residual set, once one has verified it for a predetermined finite set of points. I spell out assumptions on economic fundamentals which ensure that these bounds on the number of connected components exist, and that the residual set can be bounded from above.

I argue that utility- and production functions used in applied equilibrium analysis satisfy these conditions. Using the theoretical results, I show how computational experiments can be used to explore qualitative and quantitative implications of economic models. I give examples for actual upper bounds in realistically calibrated economies and discuss both deterministic and random algorithms for generalizing examples in these economies.

Manipulation of endowments and sunspot equilibria
Aditya Goenka, National University of Singapore

We consider the connection between occurrence of manipulation via reallocating endowments by coalitions and sunspot equilibria. The un- certainty about which coalition will form introduces extrinsic uncertainty into the economy. Under certain conditions, manipulation of endowments by coalitions can occur if and only if sunspots matter.

Minimal rationality
Isaac Levi, Columbia University

Principles of minimal rationality ought to prescribe the minimum necessary to insure coherent inquiry into controversial issues. I argue here for principles of minimum rationality that retain the core features of Bayesian rationality (Independence) while abandoning the dogmatic insistence on the requirement that rational agents should evaluate their options according to a weak ordering that can define optimal options. Conflict in values and indeterminacy in probability argue against this. I shall summarize elements of an account of rational choice that allows for indeterminacy in probability and value (or utility) judgment and breaks with the view that binary comparison is fundamental to the evaluation of options.

Asset price volatility with heterogeneous and rational beliefs
Ho-Mou Wu, National Taiwan University

This paper develops a theoretical model with a continuum of speculators trading on differences of opinion to analyzes the effects of margin requirements on asset price volatility. In our model optimistic speculators conduct speculation by expanding their credit with margin trading. We show the existence of a unique general equilibrium with heterogeneous and rational beliefs. We demonstrate that a decrease in margin requirement may increase the current price since it raises the demand of optimistic speculators, and result in pyramiding/depyramiding phenomenon for the future prices. Our results also suggest that a decrease in margin requirement is anticipated, under certain conditions, to increase asset price volatility.

Independent random matching
Yeneng Sun, National University of Singapore

We provide micro-foundations for independent random matching of a large population, as widely used in the economics and genetics literatures. We consider both static and dynamic systems with random mutation, partial matching arising from search, and type changes induced by matching. Under independence assumptions at each randomization step, we show that there is an almost-sure constant cross-sectional distribution of types in a large population, and moreover that the time evolution of the cross-sectional type process is completely determined from a Markov chain with known transition matrices. We also construct a joint agent-probability space, and randomized mutation, partial matching, and match-induced type-changing functions that satisfy the required independence conditions. This is joint work with Darrell Duffie of Stanford University.

Repetitive risk aversion
Parkash Chander, National University of Singapore

This paper introduces and investigates the concept of repetitive risk aversion. The risk aversion of an increasing and concave utility function is repetitive if the fear of ruin, which measures agent’s aversion to risking his entire income, is also increasing and concave. This is shown to be equivalent to the behaviorally meaningful condition that the risk premium is increasing at a non-increasing rate with the size of the bet. We find an additional justification for mixed risk aversion, which is known to be stronger than standard (and thus proper) risk aversion, in terms of this concept. We discuss several economic applications of repetitive risk aversion.

State-space partitioning methods for pricing high-dimensional American-style options
Xing Jin, National University of Singapore

The pricing of American options by simulation-based methods is an important and difficult task primarily due to the feature of early exercise, particularly for high-dimensional derivatives. In this paper, a bundling method based on quasi-Monte Carlo sequences is proposed to price high-dimensional American-style options. The proposed method extends Tilley's bundling algorithm to higher-dimensional situations. By using low-discrepancy points, this approach partitions the state space and forms bundles. A dynamic programming algorithm is then applied to the bundles to estimate the continuation value of an American option. A convergence proof of the algorithm is provided. A variety of examples with up to 15 dimensions are investigated numerically and accurate results are obtained.

Hart effect and equilibrium in incomplete markets
Roberto Raimondo, University of Melbourne

It is well known that the GEI Model can fail to have equilibrium since there are bad spot prices. Using a system of polynomials and matrix algebra I show that the set of "bad" spot prices is typically empty if the information tree, the set of tradable securities satisfies a certain inequality and randomness condition. I also show that this condition is always satisfied if the GEI model is the discretization of a GEI model with continuous distributions. Finally I prove that under this condition the equilibrium always exists and I address the computational aspects of my result.

When and how to dismantle nuclear weapons: more on optimal auction with externalities
Jingfeng Lu, National University of Singapore

This paper derives the optimal auction with externalities between seller and bidders in addition to those among bidders. An auction with the following features is established as the optimal auction for the symmetric setting. (i) Every participant pays a nonnegative entry fee, which equals to the absolute value of the least externality he might suffer or enjoy. (ii) If only bidder i does not show up, the item is assigned to the one (including the seller) generating him the least externality given this least externality is non-positive, otherwise the item is destroyed by the seller. (iii) If all bidders participate, the highest bidder wins if his bid is higher than the reserve price, and he pays the second highest bid or the reserve price whichever is higher. Each losing bidder pays an additional payment (positive or negative) equal to the externality to him/her incurred by the one keeping the item (including the seller). The optimal reserve price is set differently depending on whether the summation of the seller's valuation, the destroying cost of the seller and the total externalities to the bidders when seller keeps the item is positive or negative. (iv) If no bidder bids higher than the reserve price, the seller may keep the item by himself/herself or destroy the item. The necessary and sufficient condition for the seller to destroy the auctioned item (dismantling nuclear weapon) on the equilibrium path is that the above specified summation is negative. (v) All bidders participate at the equilibrium.

On equilibrium in pure strategies in games with many players
Myrna Wooders, Vanderbilt Universities and University of Warwick

Treating games of incomplete information with countable sets of actions and types and finite but large player sets we demonstrate that for every mixed strategy profile there is a pure strategy profile that is ‘ε- equivalent’. Our framework introduces and exploits a distinction between crowding attributes of players (their external effects on others) and their taste attributes (their payoff functions and any other attributes that are not directly relevant to other players). The main assumption is a ‘large game’ property,’ dictating that the actions of relatively small subsets of players cannot have large effects on the payoffs of others Since it is well known that, even allowing mixed strategies, with a countable set of actions a Nash equilibrium may not exist, we provide an existence of equilibrium theorem. The proof of existence relies on a relationship between the ‘better reply security’ property of Reny (1999) and a stronger version of the large game property. Our purification theorem are based on a new mathematical result, of independent interest, applicable to countable strategy spaces.

Pure-strategy equilibrium in large games with infinite types
Ali Khan, Johns Hopkins University

In this expository overview of ongoing work with K. Rath and Y. Sun, I discuss "large" games with a "large" type spaces. Earlier work (see Khan-Sun (PNAS 1996 and JME 1999) for discussion, results and references) done under the rubric of games with private information and with a finite number of agents can then be seen as a consideration of "large" games with finite number of type spaces. Thus, this talk represents a stock-taking in which the analytical difficulties are identified and the signal position of Loeb probability spaces, and the law of large numbers based on such spaces, is highlighted. In particular, I distinguish the question of the existence of a pure-strategy equilibrium in such games from that of the purification of their mixed-strategy equilibria, and examine in this regard the implications of the theory of vector integration of corresondences. I also take this opportunity to consider earlier work of Dvoretsky-Wald-Wolfowitz as well as some ongoing recent work of other authors.

Market games, inequality and an equal treatment property of the core
Myrna Wooders, Vanderbilt Universities and University of Warwick

It is shown that games with many players whose attributes are drawn from a compact set, are approximately market games in the sense that they are representable by economies where all players have concave utility functions. The games include those derived from economies with local public goods, with clubs, and with indivisibilities and nonconvexities. The conditions required on the games are only small group effectiveness and that there be no externalities between groups of players. Small groups are effective if almost all gains to collective activities can be achieved by groups bounded in absolute size. Examples demonstrate that the assumption of per capita boundedness (finiteness of the supremum of average payoff) introduced in prior research of this author is not adequate for our current results. To obtain our results we also provide another Theorem demonstrating that small group effectiveness is equivalent to small group negligibility, the condition dictating that vanishingly small sets of players can have at most negligible effects on aggregate payoffs of large sets of players. Finally, for the case of a finite number of player types, for small ε, an ε-core payoff has the property that most players who have many close substitutes are treated nearly equally — some, but only limited discrimination can occur. An example demonstrates that, simultaneously, players with ‘scarce’ attributes may be treated vastly unequally.