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.     

The closedness of the free-disposal hull of a production set

Summary Assume thatL is a topological vector lattice andY is a closed subset ofL ₊ ×R ^N, whereR ^N denotes theN-dimensional Euclidean space. It is shown that the setY–L ₊ ×R ₊ ^N is closed ifY has appropriate monotonicity properties. The result is applicable to the case ofL equal toL with the Mackey topology, (L ,L ¹).     

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.     

On the representability of Cr preference relations by utility functions

Summary We give a complete proof of Mas-Colell's [7] result that a continuous transitive preference relation on an open setV ^l' that isC ^r (in the sense that the indifference relation is aC ^r hypersurface inV xV) is representable by a C^r utility function without critical points.Without implicating them in remaining errors, I would like to acknowledge the helpful comments of Ket Richter and an anonymous referee.     

A law of large numbers for large economies

Summary LetX(i),i[0; 1] be a collection of identically distributed and pairwise uncorrelated random variables with common finite mean and variance ². This paper shows the law of large numbers, i.e. the fact that ₀ ¹ X(i)di=. It does so by interpreting the integral as a Pettis-integral. Studying Riemann sums, the paper first provides a simple proof involving no more than the calculation of variances, and demonstrates, that the measurability problem pointed out by Judd (1985) is avoided by requiring convergence in mean square rather than convergence almost everywhere. We raise the issue of when a random continuum economy is a good abstraction for a large finite economy and give an example in which it is not.I am indebted to Hugo Hopenhayn. Furthermore I would like to thank Dilip Abreu, Glenn Donaldson, Ed Green, Ramon Marimon, Nabil Al-Najjar, Victor Rios-Rull, Timothy van Zandt and the editor for useful comments. The first version of this paper was written in 1987.     

The Future of EU-Russia Energy Relations in the Context of Decarbonisation

In 2013, there was a joint commitment to “long term strategic EU-Russia energy cooperation”.¹1. EU/RF Roadmap, ‘Roadmap EU-Russia Energy Cooperation until 2050‘, European Commission and Russian Government, March 2013, p. 4, available at <https://ec.europa.eu/energy/sites/ener/files/documents/2013_03_eu_russia_roadmap_2050_signed.pdf>.View all notes Whilst centred on oil and gas, it is noted that “the importance of renewables for EU-Russia energy relations should grow too”,²2. Ibid., p. 21.View all notes and that for energy efficiency, “cooperation potential is immense and could… contribute to the objective of a Pan-European energy area”.³3. Ibid., p. 26.View all notes Given this shared objective, this article analyses EU and Russian energy decarbonisation policy objectives and considers the potential for a supplementary trade relationship based on renewable energy flows and decarbonisation-related technology, as well as the implications for existing energy trade. Despite declarative statements of mutual interest, shared objectives and cooperation in decarbonisation policy, there has been very limited cooperation by early 2016. The EU has set ambitious plans to decarbonise its economy and energy sector by 2050. However, in Russia energy policy is dominated by hydrocarbon exports, decarbonisation targets are modest, and there are major problems with their implementation. The drivers of EU and Russian energy policies are evaluated, and the argument advanced is that different understandings of energy security and types of energy governance provide major obstacles to decarbonisation cooperation and trade. However, it is argued that ideas about energy policy and security are contested and subject to change and there exists significant potential for mutual gain and cooperation in the longer term.     

Optimal learning with costly adjustment

Summary We formulate an infinite-horizon Bayesian learning model in which the planner faces a cost from switching actions that does not approach zero as the size of the change vanishes. We recast the model as a dynamic programming problem which will always have a continuous value function and an optimal policy. We show that the planner's beliefs will converge eventually to some stochastic limit belief which, however, is not necessarily a point mass on the truth. The planner's actions will also converge, although not necessarily to an optimal action given the truth. A key implication of adjustment costs is that the planner will change her action only finitely many times. We present a simple example illustrating how adjustment costs can lead the planner to settle in the long run on an action that is far away from the optimal action given the truth and which yields a reward significantly below that of the optimal action.We would like to thank seminar and conference participants at Brown University, the Social Science Research Council Workshop on Soviet and East European Economics in Pittsburgh, the Econometric Society in Philadelphia, and the Society for Economic Dynamics and Control in Montreal. The bulk of this work was done while Mark Feldman was at the University of Illinois, Champaign-Urbana.     

The challenge of akrasia for the theory of rational choice

The phenomenon of akrasia, in which an actor makes a choice she regrets even while choosing it, appears problematic for theories of rational choice, which assume that an agent prefers any chosen course of action. The apparent possibility of akratic action presents a challenge to rational choice theorists, either to demonstrate that it is illusory or to show that akratic action does not violate the axioms of rational choice. The problematic status of akrasia is exhibited most sharply when set against the backdrop of praxeology. Therefore, this paper will explore whether the idea of akratic action can be reconciled with the fundamental principles of praxeology.

p-Best response set and the robustness of equilibria to incomplete information

In this paper, we use p-best response sets—a set-valued extension of p-dominance—in order to provide a new sufficient condition for the robustness of equilibria to incomplete information: if there exists a set S which is a p-best response set with , and there exists a unique correlated equilibrium μ^* whose support is in S then μ^* is a robust Nash equilibrium.     

The Question of “Equilibrium” in Human Action and the Everyday Paradox of Rationality

This paper first specifies how Schütz's analysis of deliberation determines the limits of rationality where individual human action is concerned. This analysis establishes that there is no equilibrium of alternative possibilities before or after deliberation. Next the paper specifies how Schütz's analysis of the typification that makes successful intersubjective action possible leads to the paradox of rationality on the common sense level. Finally, the paper explains how Schütz's analysis of relevance can provide an account for this paradox, and thereby point to an order of human interaction in the absence of equilibrium, all without violating the postulate of subjective interpretation.     

Constitutions for Local Collective Action and Institutional Interdependence

Collective action can take place at a plurality of levels. It has to be based on a constitution which defines the basic rules of interaction. Here, we are concerned with the problem of the constitutional setting of bottom-up formal institutions with a club nature. The pressure to improve the efficiency of services pushes local administrations to co-ordinate to produce public goods. This process has stimulated the birth of different forms of agencies or private companies with a club nature. The aim of this paper is to discuss the effects of institutional interdependence on the efficiency of this kind of collective action. In order to shed some light on this problem, the paper first discusses the problem of the relativity of efficiency to the institutional setting. A framework of analysis is then discussed to identify the main factors affecting collective action. Finally some evidence will be provided by a comparative institutional analysis performed on some case studies concerning local associational forms among communes in north-eastern Italy.     

Exact equivalence of instrumental variable estimators in an error component structural system

The paper establishes an equivalence result in the context of anm-equation error component structural system, whose disturbances have the usual three-component structure, and whose equations feature explanatory variables of the formz _i, z_t andz _it; the latter vary (respectively) only over individuals, only over time, and over both. Under the stochastic specification assumed, it is shown that the alternative instrumental variables (IV) estimators commonly used in the special cases of this system are all equivalent (numerically identical); the result is a generalization of the equivalences established previously for the special cases. In the single equation (m=1) context, the equivalence requires that the IV set contain variables of the formz _i and/orz _t, and further, in numbers determined by the ranks of (respectively) the individuals-mean and time-mean matrices of the instruments. If such an IV set is common to all equations, the equivalence also holds for the system under joint estimation. The result is used to recommend a couple of estimators for use in panel data, on grounds of computational simplicity.This is a revision of the December 1990 draft with the same title, and is a substantial revision of the April 1990 version entitled: Analysis of an error component structural system. This revision has benefited from comments received from a referee and a editor of this journal. I came to know from an anonymous reader that the equivalence criterion developed in my 1990 a article, used here and the two earlier versions, was infact obtained previously in an unpublished paper by Balestra (1988). Balestra's paper, which was made available to me by Badi Baltagi at the time of this revision, and subsequently by Balestra, considers the equivalence of ^b, ^c and one other estimator which differs from our ^a. Errors, if any, are my responsibility.     

Urbanizing economics

Action takes place at a given time and place. As a science of human action, economics is, therefore, just as much about the spaces where real action occurs as it is about real time. The implications of real time for social order is better recognized than the significance of “action space.” The living city is the principal locus of action space and enabler of social change as well as the source of fundamental concepts in economic theory. Just as a loss of density and diversity in cities tends to retard dynamic discovery and development, the turn in economic theory in the mid-20th century toward static equilibrium reflected a move from an urban-based to a plantation-based conception of the economy—from the city to the farm. Some recent developments in network theory, game theory, and geography, however, can be interpreted as a re-urbanization of economics.

Incentives,regulations, and sustainable land use in Costa Rica

Deforestation in Costa Rica has proceeded at a rapid pace. Of the remaining 2,700 km² of virgin forests on privately-owned land, over 300 km² are being deforested each year. Pressure on National Parks, which cover about 27 percent of Costa Rica is likely to increase in the future. Preliminary information indicates that, contrary to our expectations, most of the deforestation at present is not being done by squatters, but driven by profit and asset maximization motives of the timber industry, banana companies, and large cattle ranchers. Setting aside 27 percent of the country's land as parks and reserves was a major policy decision. Aside from the removal of some perverse incentives operating inside park areas, the main issue there is one of sound management, including protection from intruders, strengthening enforcement, and controlled tourism. On the remaining primary and secondary forest areas on privately-owned land outside the parks, a key question is whether public interests connected with external costs of deforestation warrant public intervention. The paper suggests that a differentiated approach to this and other questions is needed, depending on the costs and benefits involved, and it discusses incentives and regulations which influence land use, and makes proposals for reforms.     

An Austrian Theory of the Firm

The modern Theory of the Firm uses the concept of rent and makes implicit assumptions about equilibrium. An Austrian (Market Process) Theory of the Firm should have something to say about each of these. Two strategic perspectives are analyzed, the neoclassical microeconomic perspective (using the Ricardo-Marshall approach to rent) and the Market Process perspective (using the Fetter approach to rent). In a neoclassical world, rents indicate unsolved or unexploited inefficiencies as every hypothetical outcome is viewed against the standard of perfect competition. By contrast, in the Market Process world there is no single ideal standard by which to measure any particular outcome. All action takes place in an open ended universe in which the future is continually being created, in which competition is a discovery process.     

The aggregated excess demand function and other aggregation procedures

Summary Two theorems are given; the first extends the Sonnenschein-Mantel-Debreu theorem characterizing aggregate demand functions from the set ofn2 commodities to all of the 2 ⁿ –(n+1) subsets of two or more commodities. The second theorem concerns spatial voting models for k2 candidates over a space of n2 issues. The theorem characterizes the sincere elecion rankings of thek candidates over all of the 2 ⁿ –1 subsets of one or more issues. Both theorems have the same kind of conclusion; anything can happen. By demonstrating the mathematical reasons for these conclusions and by recalling related, recent results from statistics, voting, and economics, it is argued that this anything can happen conclusion is the type one must anticipate for aggregation procedures; particularly for the processes commonly used in economic models where the procedure is responsive to changes in agents' preferences, changes in data, etc.I am pleased to acknowledge conversations with L. Hurwicz, A. Mas-Colell and C. Simon about this material. In particular, I want to thank J. Jordan for his several helpful suggestions. Also, I benefited from comments made at meetings and seminars where these results were presented; this includes a June, 1990, conference in Stockholm, Sweden. Some stylistic suggestions were made by Arvid and Padon Kalinen. This research was supported, in part, by NSF Grant IRI-8803505.     

Methods for estimating optimal Dickson and Waters modification dividend barrier

In applications of collective risk theory, complete information for the distribution of individual claims amount is often unknown, but reliable estimates of its first few moments may be available. Dickson and Waters [Dickson, D.C.M. and Waters, H.R., (2004) Some optimal dividends problems, Astin Bulletin, 34, 49–74.] pointed out that shareholders should be liable to cover the deficit at ruin. Thus, they considered b the level of the barrier that maximizes the expectation of the difference between the discounted dividends until ruin and the discounted deficit at ruin. For such a situation, this paper develops methods for estimating the Dickson–Waters modification for the optimal dividend barrier b with the expectation of discounted penalty at ruin. In particular, two De Vylder approximations are explained, and the diffusion approximation for the expectation of discounted penalty at ruin is examined. For several claim amount distributions, the approximate values are compared numerically with exact values.     

Hot vs. Cold: Sequential Responses and Preference Stability in Experimental Games

In experiments with two-person sequential games we analyze whether responses to favorable and unfavorable actions depend on the elicitation procedure. In our hot treatment the second player responds to the first player's observed action while in our cold treatment we follow the strategy method and have the second player decide on a contingent action for each and every possible first player move, without first observing this move. Our analysis centers on the degree to which subjects deviate from the maximization of their pecuniary rewards, as a response to others' actions. Our results show no difference in behavior between the two treatments. We also find evidence of the stability of subjects' preferences with respect to their behavior over time and to the consistency of their choices as first and second mover.