Reasoning about knowledge of unawareness

Awareness has been shown to be a useful addition to standard epistemic logic. However, standard propositional logics for knowledge and awareness cannot express the fact that an agent knows that there are facts of which he is unaware without there being an explicit fact that the agent knows he is unaware of. We extend Fagin and Halpern's logic of general awareness to a logic that allows quantification over variables, so that there is a formula in the language that says “an agent explicitly knows that there exists a fact of which he is unaware.” Moreover, that formula can be true without the agent explicitly knowing that he is unaware of any particular formula. We provide a sound and complete axiomatization of the logic. Finally, we show that the validity problem for the logic is recursively enumerable, but not decidable.     

Dynamics of inductive inference in a unified framework

We present a model of inductive inference that includes, as special cases, Bayesian reasoning, case-based reasoning, and rule-based reasoning. This unified framework allows us to examine how the various modes of inductive inference can be combined and how their relative weights change endogenously. For example, we establish conditions under which an agent who does not know the structure of the data generating process will decrease, over the course of her reasoning, the weight of credence put on Bayesian vs. non-Bayesian reasoning. We illustrate circumstances under which probabilistic models are used until an unexpected outcome occurs, whereupon the agent resorts to more basic reasoning techniques, such as case-based and rule-based reasoning, until enough data are gathered to formulate a new probabilistic model.     

The geometry of inductive reasoning in games

Summary.  This paper contributes to the recent focus on dynamics in noncooperative games when players use inductive learning. The most well-known inductive learning rule, Brown’s fictitious play, is known to converge for games, yet many examples exist where fictitious play reasoning fails to converge to a Nash equilibrium. Building on ideas from chaotic dynamics, this paper develops a geometric conceptualization of instability in games, allowing for a reinterpretation of existing results and suggesting avenues for new results. Received: October 27, 1995 revised version May 2, 1996     

Cognitive ability and strategic sophistication

In three experiments we examine the extent to which strategic sophistication (i.e., inductive reasoning, iterative dominance and level-k thinking) is determined by broader cognitive skills. In the first experiment we replicate previous results showing strong associations between cognitive ability and sophistication in a game of iterative dominance and show that similar results arise in a game requiring induction. In the second two experiments we extend the literature in new directions. In Experiment 2 we modify the games to better capture participantsʼ ability to reason inductively and predict the sophistication of others and, again, find strong associations between cognitive ability, measured using a common IQ test, and sophistication. In Experiment 3 we examine more closely the causal nature of the relationship between cognitive ability and sophistication. We use a standard tool from cognitive psychology to randomly shock the cognitive ability of participants and show that this significantly affects game performance.     

Bounded Rationality in Economics: A Game Theorist's View

In this paper, we argue that it is essential to incorporate bounded rationality into game theory. Game theory has been applied to aspects of economics such as industrial organizati on on the basis of the naive interpret ation of game theory, which requires players to be ideally rational in an extremely unrealistic way. We stress the importance of establishing the perceptive interpretation of game theory by taking boundedly rational players' inductive reasoning processes into account. We explain my recent work, Matsushima (1997), which shows that the subjective games perceived by players in the long run are entirely different from the true objective game, and are trivial games in the sense that there exists a strictly dominant and subjectively Pareto-efficient strategy profile among the set of pure strategy profiles.
JEL Classification Numbers: C70, C90, D43, D80     

Substantive Rationality and Backward Induction

S. Modica and A. Rustichini (1994, Theory and Decision37, 107–124) provided a logic for reasoning about knowledge where agents may be unaware of certain propositions. However, their original approach had the unpleasant property that nontrivial unawareness was incompatible with partitional information structures. More recently, Modica and Rustichini (1999, Games Econ. Behav.27, 265–298) have provided an approach that allows for nontrivial unawareness in partitional information structures. Here it is shown that their approach can be viewed as a special case of a general approach to unawareness considered by R. Fagin and J. Y. Halpern (1988, Artificial Intelligence34, 39–76). Journal of Economic Literature Classification Numbers: D80, D83.     

Cyclical growth in a Goodwin–Kalecki–Marx model

This paper presents a disequilibrium macrodynamic model that incorporates certain elements from Goodwin (the dynamics of the rate of employment and income distribution), Kalecki (an investment function independent of savings, and mark-up pricing in oligopolistic goods markets), and Marx (the reserve-army and reserve-army-creation effects). The model has a system of differential equations for the rate of utilization, profit share, and rate of employment. We show that there exist limit cycles that depend on the sizes of the reserve-army effect and reserve-army-creation effect. This implies that there exists a situation in which the economy experiences endogenous and perpetual growth cycles. Moreover, we show that if the stable long-run equilibrium corresponds to the profit-led growth regime, an increase in the bargaining power of workers increases the rate of unemployment; conversely, if the equilibrium corresponds to the wage-led growth-regime, an increase in the bargaining power of workers decreases the rate of unemployment.     

Team reasoning without a hive mind

The theory of team reasoning has been developed to resolve a long-lasting niggle in orthodox game theory. Despite its intuitive appeal, the theory has received little attention from mainstream game theorists and economists to date. We believe that this is so because of two theoretic issues, which the theory of team reasoning itself needs to resolve. One of these presents a worry that the theory achieves its explanatory and predictive success by abandoning ontological individualism — a fundamental precept in mainstream economics, including game theory. Here we argue that the theory of team reasoning is compatible with ontological individualism. We show that the core principles of the theory — those that give rise to the above worry — are in fact implicitly assumed in some branches of orthodox game theory itself. We also argue against the methodological approach that construes team reasoning as involving a transformation of the interacting players’ payoffs in modelled games.     

Strongly stable networks

We analyze the formation of networks among individuals. In particular, we examine the existence of networks that are stable against changes in links by any coalition of individuals. We show that to investigate the existence of such strongly stable networks one can restrict focus on a component-wise egalitarian allocation of value. We show that when such strongly stable networks exist they coincide with the set of efficient networks (those maximizing the total productive value). We show that the existence of strongly stable networks is equivalent to core existence in a derived cooperative game and use that result to characterize the class of value functions for which there exist strongly stable networks via a “top convexity” condition on the value function on networks. We also consider a variation on strong stability where players can make side payments, and examine situations where value functions may be non-anonymous—depending on player labels.     

A two-stage core with applications to asset market and differential information economies

Summary. We introduce a new core concept, called the two-stage core, which is appropriate for economies with sequential trade. We prove a general existence theorem and present two applications of the two-stage core: (i) In asset markets economies where we extend our existence proof to the case of consumption sets with no lower bound, in order to capture the case of arbitrary short sales of assets. Further, we show that the two-stage core is non empty in the Hart (1975) example where a rational expectations equilibrium fails to exist. (ii) In differential information economies where we provide sufficient conditions for the incentive compatibility of trades. Namely, that no coalition of agents can misreport the true state and provide improvements to all its members, even by redistributing the benefits from misreporting. Received: December 20, 1995; revised version: December 6, 1996     

Hotelling's competition with general customer distributions

Hotelling and his critics assumed uniformly distributed demand. In this note the solution is characterized for any distribution F and some propositions follow. For any F an equilibrium does not exist if the duopolists are located too closely together, but another distribution can be always found for which an equilibrium does exist. When we observe duopolies in equilibrium with closer and closer proximity we can expect the density in the area of contention to go up and prices and profits to go down. Hotelling's known suggestion that any duopolist has an advantage to move his location towards the other, and thus creating a tendency for the differentiation to diminish, holds whenever an equilibrium exists.     

Green consumers and environmental policy

The present paper examines how improvements in consumers' environmental awareness influence the choice between output and emission taxes, within a framework of imperfect competition and endogenous choice of abatement level. We first show that in the absence of policy intervention, there exists a level of environmental awareness beyond which welfare is decreasing as market imperfections become more prominent relative to environmental concerns. We also confirm that both output and emission taxes are welfare superior to the free-market case. What is surprising, however, is that the welfare performance of an optimally chosen emissions tax is monotonically decreasing in consumers' environmental sensitivity, while the opposite is true for an output tax up to a certain level. At low levels of consumers' environmental awareness an emissions tax is welfare superior, but eventually, there is a level of environmental awareness beyond which an output-tax welfare dominates an emissions tax. Therefore, an emissions tax is better suited to societies that have not yet developed high levels of environmental awareness, while societies characterized by high levels of environmental awareness should prefer an output tax.     

Multi-Dimensional Product Differentiation

We analyze product differentiation in a multi-dimensional model with non-uniform consumer distribution. The level of product differentiation is measured by both unit transport costs and firms’ locations. Our analysis concerns both measures. First, fixing firms’ locations, we show that equilibrium prices can increase or decrease with unit transport costs. The overall result depends on the interplay of a shifting effect and a rotating effect—the latter exists only in multi-dimensional models. Second, fixing unit transport costs, we find that under non-uniform distribution, there may exist no equilibrium where firms maximize differentiation on one dimension but minimize differentiation on other dimensions. Instead, there may exist an equilibrium where firms choose intermediate locations, contrary to common findings in existing studies which assume uniform distribution.     

Overstrained citizens? The number of ballot propositions and the quality of the decision process in direct democracy

We study how the number of ballot propositions affects the quality of decision making in direct democracy, as reflected in citizens’ knowledge, voting behavior, and attitudes toward democracy. Using three comprehensive data sets from Switzerland with over 3,500 propositions, we exploit variation in the number of federal and cantonal propositions. Voters know the most about the content of federal propositions when they are exclusively presented and less with a high number of concurrent cantonal propositions on the ballot. Across other outcomes we find no consistent indications that – for the observed variation in the exposure to popular votes – a high number of propositions impedes the quality of decision making in Swiss federal direct democracy. In the medium to longer term, more federal propositions on the ballot rather relate to higher perceived political influence and satisfaction with democracy.     

Substitutes and stability for matching with contracts

We consider the matching with contracts framework of Hatfield and Milgrom [20], and we introduce new concepts of bilateral and unilateral substitutes. We show that the bilateral substitutes condition is a sufficient condition for the existence of a stable allocation in this framework. However, the set of stable allocations does not form a lattice under this condition, and there does not necessarily exist a doctor-optimal stable allocation. Under a slightly stronger condition, unilateral substitutes, the set of stable allocations still does not necessarily form a lattice with respect to doctors' preferences, but there does exist a doctor-optimal stable allocation, and other key results such as incentive compatibility and the rural hospitals theorem are recovered.     

Endogenous market incompleteness without market frictions: Dynamic suboptimality of competitive equilibrium in multiperiod overlapping generations economies

In this paper, we show that within the set of stochastic three-period-lived OLG economies with productive assets (such as land), markets are necessarily sequentially incomplete, and agents in the model do not share risk optimally. We start by characterizing perfect risk-sharing and find that it requires state-dependent consumption claims which depend only on the exogenous shock realizations. We show then that the recursive competitive equilibrium of any overlapping generations economy with weakly more than three generations is not strongly stationary. This then allows us to show directly that there are short-run Pareto improvements possible in terms of risk-sharing and hence, that the recursive competitive equilibrium is not Pareto optimal. We then show that a financial reform which eliminates the equity asset and replaces it with zero net supply insurance contracts (Arrow securities) will implement to Pareto optimal stochastic steady-state known to exist in the model. Finally, we also show via numerical simulations that a system of government taxes and transfers can lead to a Pareto improvement over the competitive equilibrium in the model.