Convergence and approximation results for non-cooperative Bayesian games: Learning theorems

Summary LetT denote a continuous time horizon and {G ^t :tT} be a net (generalized sequence) of Bayesian games. We show that: (i) if {x ^t : tT} is a net of Bayesian Nash Equilibrium (BNE) strategies for G^t we can extract a subsequence which converges to a limit full information BNE strategy for a one shot limit full information Bayesian game, (ii) If {x ^t : tT} is a net of approximate or _t-BNE strategies for the game G^t we can still extract a subsequence which converges to the one shot limit full information equilibrium BNE strategy, (iii) Given a limit full information BNE strategy of a one shot limit full information Bayesian game, we can find a net of _t-BNE strategies {x ^t : tT} in {G ^t :tT} which converges to the limit full information BNE strategy of the one shot game.We wish to thank Larry Blume, Mark Feldman, Jim Jordan, Charlie Kahn, Stefan Krasa, Gregory Michalopoulos, Wayne Shafer, Bart Taub, and Anne Villamil for several useful discussions. The financial support of the University of Illinois at Urbana-Champaign Campus Research Board is gratefully acknowledged.     

From “tax state” to “debt state”

The paper is motivated by Joseph A. Schumpeter's The Crisis of the Tax State. It inquires whether the buildup of government debt in peacetimeprosperity is a threat to the stability, existence or creation of viable tax states. The paper begins by setting out Schumpeter's conception of the tax state and the nature of recent political-economic events which have reinvigorated the concept. Next the paper sets out some simple debt dynamics and sketches a debt-induced business cycle arising from heavy reliance on debt finance in peacetimeprosperity. Finally, the paper assesses threats to the tax state in light of recent work on path dependence and positive feedback. An attempt is made to throw some light on whether the plethora of new, and often small, states spawned by the demise of communism can be viable tax states.Essay on Government, the Tax State and Economic Dynamics submitted to the Third Schumpeter Prize Competition.     

Individual Decision Making in a Negative Externality Experiment

The experimental treatments analysed in this paper are simple in that there is a unique Nash equilibrium resulting in each player having a dominant strategy. However, the data show quite clearly that subjects do not always choose this strategy. In fact, when this dominant strategy is not a focal outcome it does not even describe the average decision adequately. It is shown that average individual decisions are best described by a decision error model based on a censored distribution as opposed to the truncated regression model which is typically used in similar studies. Moreover it is shown that in the treatments where the dominant strategy is not focal dynamics are important with average subject decisions initially corresponding to the focal outcome and then adjusting towards the Nash prediction. Overall, 66.7% of subjects are consistent with Payoff Maximization, 27.8% are consistent with an alternate preference maximization and 5.6% are random.     

Vertical and Horizontal Trust Networks in Bureaucracies: Evidence from the Third Reich

In a seminal contribution to the literature on bureaucracy, Breton and Wintrobe (The Logic of Bureaucratic Conduct: An Economic Analysis of Competition, Exchange, and Efficiency in Private and Public Organization. New York, NY: Cambridge University Press, 1982) develop a model wherein subordinates and superiors in a bureaucratic structure trade with each other to advance the objectives of the superiors. The success of such an organizational arrangement (for superiors) is based upon the development of vertical trust networks in a way that facilitates the promise of informal payments by superiors in return for informal services provided by their subordinates. Breton and Wintrobe [Journal of Political Economy 94 (1986) 905] also provide a theoretical application of their model by describing the Nazi bureaucracy as a conglomeration of competing agencies that zealously carried out the Final Solution to the Jewish question. As an extension, this note develops two compelling empirical examples of vertical and horizontal trust networks within the Nazi regime: Einsatzgruppen As (Special Action Detachments) attempt to liquidate all Lithuanian Jews after the German invasion of the U.S.S.R. in 1941 and the 20 July 1944 attempt to assassinate Adolf Hitler.JEL Classification: D23, D73.     

Equilibria and Pareto optima of markets with adverse selection

Summary This paper examines the efficiency properties of competitive equilibrium in an economy with adverse selection. The agents (firms and households) in this economy exchange contracts, which specify all the relevant aspects of their interaction. Markets are assumed to be complete, in the sense that all possible contracts can, in principle, be traded. Since prices are specified as part of the contract, they cannot be used as free parameters to equate supply and demand in the market for the contract. Instead, equilibrium is achieved by adjusting the probability of trade. If the contract space is sufficiently rich, it can be shown that rationing will not be observed in equilibrium. A further refinement of equilibrium is proposed, restricting agents' beliefs about contracts that are not traded in equilibrium. Incentive-efficient and constrained incentive-efficient allocations are defined to be solutions to appropriately specified mechanism design problems. Constrained incentive efficiency is an artificial construction, obtained by adding the constraint that all contracts yield the same rate of return to firms. Using this notion, analogues of the fundamental theorems of welfare economics can be proved: all refined equilibria are constrained incentive-efficient and all constrained incentive-efficient allocations satisfying some additional conditions can be decentralized as refined equilibria. A constrained incentive-efficient equilibrium is typically not incentive-efficient, however. The source of the inefficiency is the equilibrium condition that forces all firms to earn the same rate of return on each contract.Notation ={ ₁,..., _k } set of outcomes - : ₊ generic contract or lottery - A = () ; - A_o A{, where denotes the null contract or no trade - S={1,...,¦S¦} set of seller types - L(s) number of type-s sellers - M number of buyers - u: × S seller's utility function, which can be extended toA× S by puttingu(, s) ; - v. × S buyer's utility function, which can be extended toA × S by puttingv(, s) ; - f:A ₀ ×S ₊ allocation of sellers - g:A ₀ ×S ₊ allocation of buyers - A ₊ sellers' trading function - :A ×S ₊ buyers' trading function This paper has had a long gestation period, during which I have been influenced by helpful conversations with many persons, by their work, or both. Among those who deserve special mention are Martin Hellwig, Roger Myerson, Edward Prescott, Robert Townsend and Yves Younés. Earlier versions were presented to the NBER/CEME Conference on Decentralization at the University of Toronto and the NBER Conference on General Equilibrium at Brown University. I would like to thank John Geanakoplos, Walter Heller, Andreu Mas Colell, Michael Peters, Michel Poitevin, Lloyd Shapley, John Wooders, Nicholas Yannelis and an anonymous referee for their helpful comments and especially Robert Rosenthal for his careful reading of two drafts. The financial support of the National Science Foundation under Grant No. 912202 is gratefully acknowledged.     

Anonymous sequential games: Existence and characterization of equilibria

Summary In this paper we consider Anonymous Sequential Games with Aggregate Uncertainty. We prove existence of equilibrium when there is a general state space representing aggregate uncertainty. When the economy is stationary and the underlying process governing aggregate uncertainty Markov, we provide Markov representations of the equilibria.Table of notation Agents' characteristics space ( ) - A Action space of each agent (aA) - Y Y = x A - Aggregate distribution on agents' characteristics - (X) Space of probability measures onX - C(X) Space of continuous functions onX - _X Family of Borel sets ofX - State space of aggregate uncertainty ( ) - x _t=1 aggregate uncertainty for the infinite game - = (₁,₂,...,_t,...) - ^{t t} (₁, ₂,..., _t) - L₁(^t,C ×A),v _t Normed space of measurable functions from ^t toC( x A) - 8o(^t,( x A)) Space of measurable functions from ^tto( x A) - X^t X^t= x _s=1 ^t X - _X ^t Borel field onX ^t - v Distribution on - v_t Marginal distribution of v on ^t - v(^t)((¦^t)) Conditional distribution on given ^t - v^t(^s)(v_t(¦^s)) Conditional distribution on ^t given ^s (wheres) - _t Periodt distributional strategy - Distributional strategy for all periods =(₁,₂,...,_t,...) - _t Transition process for agents' types - ( _t,^t,y)(P _t+1(, _t , ^t ,y)) Transition function associated with _t - u _t Utility function - V _t (, a, , ^t) Value function for each collection (, a, , ^t ) - W _t (, , ^t ) Value function given optimal action a - C() Consistency correspondence. Distributions consistent with and characteristics transition functions - B() Best response correspondence (which also satisfy consistency) - E Set of equilibrium distributional strategies - x _t=1 ( ^t , (x A)) - S Expanded state space for Markov construction - (, a, ) Value function for Markov construction - P( _t ^* , _t y)(P(, _t ^* , _t , _y )) Invariant characteristics transition function for Markov game We wish to acknowledge very helpful conversations with C. d'Aspremont, B. Lipman, A. McLennan and J-F. Mertens. The financial support of the SSHRCC and the ARC at Queen's University is gratefully acknowledged. This paper was begun while the first author visited CORE. The financial support of CORE and the excellent research environment is gratefully acknowledged. The usual disclaimer applies.     

Klassische Nationalökonomie gestern und heute

Ohne ZusammenfassungDieser Artikel ist dieungekürzte Fassung des Beitrages Klassische Nationalökonomie des Verfassers zum Staatslexikon (Band IV).     

Measuring the severity of unemployment

A social-welfare (illfare) function framework is applied to compare two demographic groups as to the severity of their unemployment experience. This is based on the assumption that for each individual the disutility of unemployment is an increasing and convex function of spell length. The very concept of spell length and its distribution, however, is not unambiguous. In contrast to previous literature which focuses exclusively on the interrupted spell length in a stock of unemployed, we stress the usefulness of the concept of complete spell length in a cohort of unemployed. We establish an equivalence relationship between second-degree dominance in the cohort and first-degree dominance in the stock. For specific illfare functions the disutilityU(x) when applied to the cohort and the disutilityU(x) when applied to the stock will produce the same value of aggregate welfare (illfare).     

The joint exploitation of a productive asset: a game-theoretic approach

Summary In the present paper we explore the set of equilibria in a game-theoretic model in which players can jointly exploit a productive asset. As in repeated games, we find that under certain circumstances there may be efficient as well as inefficient equilibria. In the model we study, efficient trigger-strategy equilibria may exist from some starting states (stocks of assets) but not others. More precisely, there is a stock level, sayy, such that an efficient trigger-strategy equilibrium exists from starting stocks greater than or equal toy, but not from those strictly less thany. (This statement is meant to include the cases in whichy is zero or infinite.) Under some circumstances, there may exist a new kind of equilibrium, which we call aswitching equilibrium. We show that, in our model, whenever y is positive (and finite), there is an open intervalI with upper endpoint y such that, from any starting stock inI there is an equilibrium of the dynamic game with the following structure: the players follow an inefficient but growing path until the stock reaches the levely, and then follow an (efficient) trigger strategy after that. The use of a continuous-time model enables us to conveniently decouple the delay of information from the time interval between decisions.We thank the C. V. Starr Center at New York University for research support. The views expressed here are those of the authors, and not necessarily those of AT&T Bell Laboratories.     

Biological Adaptation in Human Societies: a ‘Basic Needs’ Approach

The groundzero premise (so to speak) of the biological sciences is that survival and reproduction is the basic, continuing, inescapable problem for all living organisms; life is at bottom a survival enterprise. It follows that survival is the paradigmatic problem for human societies as well; it is a prerequisite for any other, more exalted objectives. Although the term adaptation is also familiar to social scientists, until recently it has been used only selectively, and often very imprecisely. Here a more rigorous and systematic approach to the concept of adaptation is proposed in terms of basic needs. The concept of basic human needs has a venerable history – tracing back at least to Plato and Aristotle. Yet the development of a formal theory of basic needs has lagged far behind. The reason is that the concept of objective, measurable needs is inconsistent with the theoretical assumptions that have dominated economic and social theory for most of this century, namely, valuerelativism and cultural determinism. Nevertheless, there have been a number of efforts over the past 30 years to develop more universalistic criteria for basic needs, both for use in monitoring social wellbeing (social indicators) and for public policy formulation. Here I will advance a strictly biological approach to perationalizing the concept of basic needs. It is argued that much of our economic and social life (and the motivations behind our revealed preferences and subjective utility assessments), not to mention the actions of modern governments, are either directly or indirectly related to the meeting of our basic survival needs. Furthermore, these needs can be specified to a first approximation and supported empirically to varying degrees, with the obvious caveat that there are major individual and contextual variations in their application. Equally important, complex human societies generate an array of instrumental needs which, as the term implies, serve as intermediaries between our primary needs and the specific economic, cultural and political contexts within which these needs must be satisfied. An explicit framework of Survival Indicators, including a profile of Personal Fitness and an aggregate index of Population Fitness, is briefly elucidated. Finally, it is suggested that a basic needs paradigm could provide an analytical tool (a biologic) for examining more closely the relationship between our social, economic and political behaviors and institutions and their survival consequences, as well as providing a predictive tool of some value.     

Information processing for a finite resource allocation mechanism

Summary deal with an organization havingm individual agents (the agents) and one central agent or agency (the coordinator), and the problem of allocating a fixed supply ofn commodities among the agents. The coordinator wishes to distribute the commodities in such a way as to maximize an objective function whose arguments are functions known (implicitly) to the individual agents, but not to the coordinator. The more information the coordinator obtains about the individual agents' functions, the better the allocation decision he or she can make; however, gaining such information is costly, so that the coordinator must balance the potential gain against the cost of acquiring additional information. We find a satisficing solution to this problem which is computable and which terminates in a finite number of steps.We would like to thank Leo Hurwicz, Ken Matheny, Shailendra Mehta, Bill Novshek, Roy Radner, Stan Reiter, Vernon Smith, Mark Walker, and, especially, John Ledyard and Tom Marschak, for helpful comments and suggestions. We must, of course, however reluctantly, accept responsibility for any remaining defects.     

The “Dorfman Steiner” rule

Conclusion When this research was started, it was guessed that the Dorfman-Steiner rule would lose its relevance in an intertemporal setting. This belief has turned out to be false: along the optimal paths ofp (t) ands (t), and must be equal. The only difference with the Dorfman-Steiner result is that they will be different from unity.The author is chargé de cours at the Faculté Universitaire Catholique de Mons (Belgium). He has greatyl benefited from comments by M. Beuthe and J. J. Lambin.     

On economic disequilibrium and free lunch

There is a sharp disagreement between mainstream economists and advocates of energy efficiency as regards the potential for free lunches or no regrets policies to cut greenhouse gas emissions. From an economics perspective, the critical question is whether the economic system is — or is not — close to a Pareto-optimum equilibrium state. If so, it follows that most technological systems now in place are optimum, or nearly so, from an economic perspective. If not, there may be many sub-optimal technologies in place, with corresponding opportunities for very high returns on appropriate investments. This paper presents some of the evidence supporting the latter thesis.     

Fully endogenous mechanism selection on finite outcome sets

Summary This paper defines a choice process over social outcomes in which agents choose the institutional rules ormechanisms themselves without outside interference. Truly endogenizing the mechanism selection process in this way, however, involves facing an infinite regress problem in which outcomes are chosen by games which are themselves chosen by games, ad infinitum. This paper allows the possibility of such an infinite regress which we callfully endogenous mechanism selection.We introduce the notion ofFree Choice which restricts the class of mechanisms in the regress to those which prevent agents from being locked in to an equilibrium outcome by the actions of others. Under this condition, the infinite regress is shown to get truncated with the number of selection iterations endogenously determined. It turns out that the outcomes resulting from a Free Choice-constrained regress are (Weakly) Pareto optimal; in particular, these outcomes solve a weighted Rawlsian Maxmin criterion. We also show that these outcomes are invariant to the equilibrium concept used to evaluate games in the regress.This paper is based on the author's dissertation from the University of Minnesota (November, 1989).I am very grateful for the guidance, advice, and encouragement from my advisor, Marcel K. Richter, and for the many helpful suggestions from David Levine. I have also benefited from conversations with Nabil Al-Najjar, Gerhard Glomm, Leonid Hurwicz, James Jordan, Ramon Marimon, Andrew McClennan, Ariel Rubinstein, and William Thomson.     

Convention, Social Order, and the Two Coordinations

The word coordination has two meanings, and thesemeanings are often conflated. One meaning, associated with ThomasSchelling, is seen in situations like choosing whether to driveon the left or the right; the drivers must coordinate to eachother's behavior. The other meaning, associated with FriedrichHayek, means that a concatenation of activities is arranged soas to produce good results. Along with the Schelling sense ofcoordination comes the notion of convention, such as drivingon the right. Some conventions are consciously designed; othersemerge without design (or are emergent). Along with the Hayeksense of coordination comes the notion of social order. Somesocial orders, such as the skeleton of activities within thefirm or within the hypothetical socialist economy, are consciouslyplanned. Other social orders, such as the catallaxy of the freesociety, function without central planning (or are spontaneous).Distinguishing between the two coordinations (and, in parallelfashion, between convention and social order) clarifies thinkingand resolves some confusions that have arisen in discussionsof coordination and spontaneous order. The key distinctionsare discussed in the context of the thought of, on the one hand,Menger, Schelling, David Lewis, and the recent path-dependencetheorists, and, on the other hand, Smith, Hayek, Polanyi, Coase,and the modern Austrian economists. The paper concludes witha typology that encompasses the several distinctions.     

Sector-specific capital, “Bang-bang” investment,and the Filippov solution

Bang-bang investment in a two-sector growth model with immobile capital is rational and leads to a unique and globally stable long-run equilibrium along a sliding trajectory. This steady state coincides with the stationary equilibrium in the traditional model with non-sector-specific capital.This article was written while the authors were visiting scholars at Cornell University. We gratefully acknowledge financial assistance from the Erasmus University Trust Fund and the Netherlands Scientific Organization. We would like to thank, without implicating, two anonymous referees, Martijn Herrmann, Jean-Marie Viaene, Claus Weddepohl, and the participants of seminars at the University of Maryland, the University of Montreal, and Erasmus University Rotterdam for perceptive remarks and useful comments. Jeroen Hinloopen and Rien Wagenvoort provided able graphical assistance. The views expressed in this article are those of the authors and not necessarily those of the World Bank.     

The measurement of entry rates: Recondiseration and resolution

This paper extends the recent analysis of Audretsch and Fritsch on entry rates, and suggests a resolution to the possible policy confusion which they discuss. British data also show very different results for the determinants of entry depending on whether the ecological or labour market entry rate is used. It is shown that, in addition to the static, size-distribution effect discussed by Audretsch and Fritsch, there are dynamic factors at work which may lead to this result. However, there need be no conflict in policy signals arising from this, because the labour market approach can be shown to be a superior measure of entry rates where spatial analysis is undertaken.     

A New Variant of the Winner's Curse in a Coasian Contracting Game

We examine behavior in a Coasian contracting game with incomplete information. Experimental subjects propose contracts, while automaton property right holders or robot players with uncertain preferences respond to those proposals. The most common pattern of proposals observed in these games results in too many agreements and, in some games, payoffs that are stochastically dominated by those resulting from rational proposals (which imply fewer agreements). In this sense, we observe a winner's curse similar to that observed in bidding games under incomplete information, such as the common value auction (Kagel, J.H. and Levin, D. (1986) American Economic Review. 76, 894–920) and the takeover game (Samuelson, W. and Bazerman, M.H. (1985) In Research in Experimental Economics, Vol. 3. JAI Press, Greenwich, pp. 105–137; Ball, S.B., Bazerman, M.H., and Carroll, J.S. (1990) Organizational Behavior and Human Decision Processes. 48, 1–22; Holt, C. and Sherman, R. (1994) American Economic Review. 84, 642–652). While the naïve model of behavior nicely predicts the winner's curse in those previous bidding games, it does not do so here. Instead, an alternative model we call the guarantor model explains the anomalous behavior best. Hence, we suggest this is a new variant of the winner's curse.     

“Buy Domestic” Campaigns and Optimal Tariffs

This paper attempts to analyze the strategic use of optimal tariffs and to examine the effects of national bias on the optimal trade policy and social welfare in a two-country, two-good, price competition model derived from Neven et al. (1991). The major findings are as follows. (1) If all consumers prefer the domestic good, then buy domestic campaigns will decrease the prohibitive tariff rate and increase local welfare. (2) If at least some consumers prefer the foreign good, but not to a great extent, then buy domestic campaigns will not change the optimal tariff rate, but may improve local welfare. (3) When all consumers greatly prefer the foreign good, then promotion of buy domestic decreases the optimal tariff rate, but it cannot improve social welfare. With this framework, we also prove that buy domestic campaigns serve as a substitute for tariffs with respect to a strategic trade policy.     

Measuring the trade-off between economic growth and a clean environment

This article surveys various aspects of the measurement of environmental quality from the view point of national accounting and welfare economics. It focuses on the question whether GNP or NNP should be corrected for environmental change (green or eco-GNP) or whether physical accounts provide sufficient information for an assessment of the trade-off mentioned above. We conclude that valuation of (services from) environmental capital cannot be avoided for such assessment, but can only be made using a model based approach. Statistical agencies should continue to collect data on environmental quality and to value changes in environmental capital in the context of national resource accounting. However, official statisticians should refrain from correcting GNP or NNP for environmental change, as this correction implicitly contains a political judgment and cannot be based on mere technical knowledge.