We develop a theory of an equilibrium-econometric analysis of rental housing markets with indivisibilities. It provides a bridge between a (competitive) market equilibrium theory and a statistical/econometric analysis. The listing service of apartments provides the information to both economic agents and an econometric analyzer: each economic agent uses a small part of the data from the service for his economic behavior, and the analyzer uses them to estimate the market structure. It is argued that the latter may be done by assuming that the economic agents take the standard price-taking behavior. We apply our theory to the data in the rental housing markets in the Tokyo area, and examine the law of diminishing marginal utility for household. It holds strictly with respect to the consumption, less with commuting time-distance, and much less with the sizes of apartments.

Inductive game theory has been developed to explore the origin of beliefs of a person from his accumulated experiences of a game situation. It has been restricted to a person's view of the structure not including another person's thoughts. In this paper, we explore the experiential origin of one's view of the other's beliefs about the game situation, especially about the other's payoffs. We restrict our exploration to a 2-role (strategic) game, which has been recurrently played by two people with occasional role-switching. Each person accumulates experiences of both roles, and these experiences become the source for his transpersonal view about the other. Reciprocity in the sense of role-switching is crucial for deriving his own and the other's beliefs. We also consider how a person can use these views for his behavior revision, and we define an equilibrium called an intrapersonal coordination equilibrium. Based on this, we show that cooperation will emerge as the degree of reciprocity increases.

We study the process, called the IEDI process, of iterated elimination of (strictly) dominated strategies and inessential players for finite strategic games. Such elimination may reduce the size of a game considerably, for example, from a game with a large number of players to one with a few players. We extend two existing results to our context; the preservation of Nash equilibria and order-independence. These give a way of computing the set of Nash equilibria for an initial situation from the endgame. Then, we reverse our perspective to ask the question of what initial situations end up at a given final game. We assess what situations underlie an endgame. We give conditions for the pattern of player sets required for a resulting sequence of the IEDI process to an endgame. We illustrate our development with a few extensions of the Battle of the Sexes.

This is the second symposium of the same title "Logic and Economics" in this journal and has four papers studying the issues expressed in the subtitle. In this introduction, the guest editor first describes a general perspective for this symposium and then explains each paper from this perspective. Finally, a few remarks are given.

We explore the inductively derived views obtained by players with partial temporal (short-term) memories. A player derives his personal view of the objective game situation from his accumulated (long-term) memories, and then uses it for decision making. A salient feature that distinguishes this paper from others on inductive game theory is partiality of a memory function of a player. This creates a multiplicity of possibly derived views. Although this is a difficulty for a player in various senses, it is an essential problem of induction. Faced with multiple possible views, a player may try to resolve this using further experiences. The two-way interaction between behavior and personal views is another distinguishing feature of the present paper.

This paper undertakes a simulation study of a player’s learning about the structure of a game situation. In a simple 1-person example called Mike’s Bike Commuting, we simulate the process by which Mike experiences and accumulates memories about the game structure. It is the basic requirement that to keep an experience as a long-term memory, Mike needs enough repetitions of that experience. By the choice of our simple and casual example, we can discuss relevant time spans for learning. In particular, we argue that the limit case of Mike’s learning as time tends to infinity is of little relevance to the problem of learning. We find also that the concept of “marking” introduced by Kaneko-Kline is important for obtaining sufficient structural knowledge in a reasonable time span. The simulation study shows that Mike’s learning can change drastically with the concept. We also consider Mike learning his preferences from his experiences, where we will meet various new conceptual problems.

This paper provides an analysis of discrimination and prejudices from the perspective of inductive game theory. We extend the festival game, originally given by Kaneko-Matsui, to include new constraints on the observability of ethnic identities and on accessible locations for players. We characterize the Nash equilibrium set, which reveals a different variety of segregation patterns and discriminatory behavior. In order to facilitate the analysis of discrimination and prejudices, we introduce a measure of discrimination, which chooses a representative equilibrium with the smallest degree of discrimination. Using this measure, we discuss various new phenomena, such as discrimination in an ethnic hierarchy; similar ethnicities as discriminated and as discriminating; and mutual discrimination. The introduction of limited observability and accessibility enables us to obtain those results. (C) 2011 Elsevier B.V. All rights reserved.

This paper presents a measure of inference in classical and intuitionistic logics in the Gentzen-style sequent calculus. The definition of the measure takes two steps: First, we measure the width of a
given proof. Then the measure of inference assigns, to a given sequent, the minimum value of the widths of its possible proofs. It counts the indispensable cases for possible proofs of a sequent. This measure expresses the degree of difficulty in proving a given sequent. Although our problem is highly proof-theoretic, we are motivated by some general and specific problems in game theory/economics. In this paper, we will define a certain lower bound function, with which we may often obtain the exact value of the measure for a given sequent. We apply our theory a few game theoretical problems and calculate the exact values of the measure.

The aim of this paper is to present the new theory called "inductive game theory". A paper, published by one of the present authors with A. Matsui, discussed some part of inductive game theory in a specific game. Here, we present a more entire picture of the theory moving from the individual raw experiences, through the inductive derivation of a view, to the implications for future behavior. Our developments generate an experiential foundation for game theory and for Nash equilibrium. (C) 2008 Elsevier B.V. All rights reserved.

We present certain duality results on comparative statics on competitive rent vectors in the rental housing market model. In the model, apartments as indivisible goods are classified into a finite number of categories, and are traded for one composite commodity. Our concern is about certain general properties of the behavior of rents with parameter changes. In particular, the rent changes are intimately related to the boundary income changes of the categories of apartments. Both changes are endogenously determined in equilibrium. We will show that these changes exhibit nice dual structures. We will also apply our model and comparative statics to a rental housing market in the Tokyo metropolitan area. (c) 2005 Elsevier Inc. All rights reserved.

Kaneko-Suzuki developed epistemic logics of shallow depths with multiple players for investigations of game theoretical problems. By shallow depth, we mean that nested occurrences of belief operators of players in formulae are restricted, typically to be of finite depths. by a given epistemic structure. In this paper we develop various methods of surgical operations (cut and paste) of epistemic world models. An example is a bouquet-making. i.e., tying several models into a bouquet. Another example is to engraft a model to some branches of another model. By these methods. we obtain various meta-theorems on semantics and syntax on epistemic logics, To illustrate possible uses of our meta-theorems. we present one game theoretical theorem. which is also a meta-theorem in the sense of logic.

This paper presents a multi-modal epistemic logic GLEF of shallow depths. Logic GLEF is a fragment of multi-modal KDn, and enables us to consider interpersonal inferential epistemic complexities. The subscripts E and F are called epistemic structures, which restrict interpersonal epistemic depths involved in formulae and proofs. We give its Gentzen-type sequent formulation as well as its semantics. Also, we provide some meta-theorems in order to evaluate interpersonal inferential epistemic complexities of the provability of a given sequent. These meta-theorems are applied to some game theoretical problems.

Individual decision making is based on predictions about other players' choices as well as on valuations of reactions to predictions. In this sense, a player has a prediction-decision criterion for decision making. We develop a theory of prediction-decision criteria, which enables us to capture new phenomena on individual decision making in games. The decision making situation is described in the epistemic logic GL(EF) of shallow depths. There, each player considers his and other players' decision making down to some shallow depths. It is a point of our theory to investigate inferential complexities of interpersonal introspections. In particular, we can discuss a minimal epistemic inferential structure for prediction-decision making. We will find parallel structures in decision making and prediction making, which is called an inner parallelisms. The climax of the paper is the consideration of inner parallelisms of prediction-decision making.

In order to capture the concept of common knowledge, various extensions of multi-modal epistemic logics, such as fixed-point ones and infinitary ones, have been proposed. Although we have now a good list of such proposed extensions, the relationships among them are still unclear. The purpose of this paper is to draw a map showing the relationships among them. In the propositional case, these extensions turn out to be all Kripke complete and can be comparable in a meaningful manner. F. Wolter showed that the predicate extension of the Halpern-Moses fixed-point type common knowledge logic is Kripke incomplete. However, if we go further to an infinitary extension, Kripke completeness would be recovered. Thus there is some gap in the predicate case. In drawing the map, we focus on what is happening around the gap in the predicate case. The map enables us to better understand the common knowledge logics as a whole. © 2002 Kluwer Academic Publishers.

From the Ex Ante point of view, an axiomatization of decision making in a game with pure strategies is given, while considering its epistemic aspects in propositional game (epistemic) logic. Our axiomatization consists of four base axioms for predicted final decisions. One of them is an epistemic requirement, which together with the others leads to an infinite regress of the knowledge of these axioms. The resulting outcome of this regress is expressed as the common knowledge of the base axioms. We give meta-theoretical evaluations of the derivation of this infinite regress, and consider its implications in solvable and unsolvable games. For a solvable game, it determines predicted decisions to be the common knowledge of a Nash equilibrium, and for an unsolvable game, it is the common knowledge of a subsolution in Nash's sense. The latter result needs the common knowledge of the additional information of which subsolution would be played. We give also meta-theoretical evaluations of these results. (C) 1999 Elsevier Science B.V. All rights reserved.

We show the faithful embedding of common knowledge logic CKL into game logic GL, that is. CKL is embedded into GL and GL is a conservative extension of the fragment obtained by this embedding. Then many results in GL are available in CKL, and vice versa. For example, an epistemic consideration of Nash equilibrium for a game with pun: strategies in GL is carried over to CKL. Another important application is to obtain a Gentzen-style sequent calculus formulation of CKL and its cut-elimination. The faithful embedding theorem is proved for the KD4-type propositional CKL and GL, but it holds for some variants of them.

This paper proposes a new theory, which we call inductive game theory. In this theory, the individual player does not have a priori knowledge of the structure of the game that he plays repeatedly. Instead, he accumulates experiences induced by occasional random trials in the repeated play. A stationary state is required to be stable against intentional deviations based on the player's experiences, and then it turns out to be a Nash equilibrium. The main part of the paper is the consideration of possible individual views of the society based on individual experiences. This view is defined to be a model of the society which the player builds from his experiences. Coherency with these experiences and a condition called rationalization are required for a model. As concrete objects of the theory, this paper analyzes the phenomena of discrimination and prejudice. The development of the new theory is undertaken by contrasting its observational and behavioral aspects with mental and judgmental aspects. The relationship between discrimination and prejudice will emerge in this dichotomous consideration. © 1999 Blackwell Publishers, Inc.

This paper provides a Genzten style formulation of the game logic framework GLm (0 ≤ m ≤ ω), and proves the cut-elimination theorem for GLm. As its application, we prove the term existence theorem for GLω used in Part I. © 1997 Kluwer Academic Publishers.

This paper provides a logic framework for investigations of game theoretical problems. We adopt an infinitary extension of classical predicate logic as the base logic of the framework. The reason for an infinitary extension is to express the common knowledge concept explicitly. Depending upon the choice of axioms on the knowledge operators, there is a hierarchy of logics. The limit case is an infinitary predicate extension of modal propositional logic KD4, and is of special interest in applications. In Part I, we develop the basic framework, and show some applications: an epistemic axiomatization of Nash equilibrium and formal undecidability on the playability of a game. To show the formal undecidability, we use a term existence theorem, which will be proved in Part II.

We consider two models of n‐person bargaining problems with the endogenous determination of disagreement points. In the first model, which is a direct extension of Nash's variable threat bargaining model, the disagreement point is determined as an equilibrium threat point. In the second model, the disagreement point is given as a Nash equilibrium of the underlying noncooperative game. These models are formulated as extensive games, and axiomatizations of solutions are given for both models. It is argued that for games with more than two players, the first bargaining model does not preserve some important properties valid for two‐person games, e.g., the uniqueness of equilibrium payoff vector. We also show that when the number of players is large, any equilibrium threat point becomes approximately a Nash equilibrium in the underlying noncooperative game, and vice versa. This result suggests that the difference between the two models becomes less significant when the number of players is large.

We prove the nonemptiness of the core of a continuum game without side payments where only small coalitions - ones bounded in absolute size of finite cardinality - are permitted. This result covers assignment games with a continuum of players and includes combinations of several assignment games, such as housing and automobile markets.

When perfect recall is not satisfied, the informational contents of mixed and behavior strategies differ and are more than what the information partition describes. First, we consider two kinds of additional information strategies may carry, and show that such information leads to the perfect recall refinement of a given information partition. This does not, however, imply that the strategies compensate fully for the lack of perfect recall. We give a necessary and sufficient condition on an information partition, called A-loss, for the informational content of mixed strategies to fully compensate for the lack of perfect recall. The informational content of behavior strategies never fully compensates.

We consider an international financial problem called debt overhang, by which we mean a situation where a sovereign country has borrowed money from foreign banks and has been unable to fulfill the scheduled repayments for some period. The problem is formulated as a noncooperative game with n lender banks as players where each decides either to sell its loan exposure to the debtor country at the present price of debt on the secondary market, or to wait and keep its exposure. This game has many pure and mixed strategy Nash equilibria. We show, however, that in any Nash equilibrium, the resulting secondary market price remains almost the same as the present price for a large number of banks. We also obtain the comparative statics result that in a mixed strategy equilibrium, a bank with a smaller loan exposure has a greater tendency to sell than one with a larger loan exposure. We discuss the implications of these results for the functioning of the secondary market and the resolution of debt overhang.

This paper provides a simple game-theoretic account of racial discrimination without assuming individual prejudices as a basic concept. We consider a recurrent situation of a festival game with merrymakers, and give a formulation of a stable convention. We show that stable conventions of one type involve discrimination against one group of players. We look closely into the structure of these stable conventions and consider them in comparison with recent developments of sociological thought on ethnic relations. Journal of Economic Literature Classification Numbers 026, 900 and American Mathematical Society Classification Numbers 90D10, 90D80, 90D99.

This note proves that the two person Nash bargaining theory with polyhedral bargaining regions needs only an ordered field (which always includes the rational number field) as its scalar field. The existence of the Nash bargaining solution is the main part of this result and the axiomatic characterization can be proved in the standard way with slight modifications. We prove the existence by giving a finite algorithm to calculate the Nash solution for a polyhedral bargaining problem, whose speed is of order Bm(m - 1) (m is the number of extreme points and B is determined by the extreme points).

This paper attempts to explain the Nash equilibrium concept from the viewpoint of its one-shot play interpretation. We consider a final decision to be made by each player before the game is actually played. We formalize this game situation in terms of an infinitary first-order predicate logic. Then we give an axiom for final decisions-in the two-person case, the central requirement for this axiom is: for players i and j, if x is a possible final decision for player i, then (1) player i knows that x is his final decision; (2) there is a final decision y for player j; and (3) for any final decision y for j, x is a best response to y and player i knows that y is j's final decision. The entire axiom takes the form of the common knowledge of the above requirement by its very nature. We assume that the complete logical abilities of the players are common knowledge. Then we prove that for solvable games in Nash's sense, x is a final decision for player i if it is common knowledge that x is a Nash strategy. A similar result will be obtained for unsolvable games.

We develop a new model of a continuum economy with coalitions consisting of only finite numbers of agents. The core, called the f-core, is the set of allocations that are stable against improvement by finite coalitions and feasible by trade within finite coalitions. Even with widespread externalities—preferences depend on own consumptions and also on the entire allocation up to the null set—we obtain the result that the f-core coincides with the Walrasian allocations. Without widespread externalities, the f-core, the Aumann core, and the Walrasian allocations all coincide; however, with widespread externalities there is no obvious natural definition of the Aumann core.

When permissible coalitions in finite economies are constrained to be small relative to the player set, the continuum model with finite coalitions and its f-core are the limits of large finite economies and their ε-cores. We show convergence both of game-theoretic structures—relatively small coalitions in the finite economies converge to finite coalitions-and of solutions—ε-cores converge to the f-core. Our convergence is carried out in the context of exchange economies with widespread externalities where the requirement that coalitions be small is critical.

This paper attempts to define a new solution concept for n-person noncooperative games. The idea of it is close to that of the von Neumann-Morgenstern stable set, more precisely, is based on their ‘standards of behavior’ interpretation of the stable set. This new approach enables us to consider new interesting problems of information. Further this approach gives us a plausible interpretation of the Nash equilibrium concept. This paper provides the definition and considers certain general properties of our solution concept. Further we consider our solution concept for zero-sum two-person games, the prisoner's dilemma, the battle of the sexes and several simple games with a continuum of players.

In this paper we develop a new model of a cooperative game with a continuum of players. In our model, only finite coalitions - ones containing only finite numbers of players - are permitted to form. Outcomes of cooperative behavior are attainable by partitions of the players into finite coalitions: this is appropriate in view of our restrictions on coalition formation. Once feasible outcomes are properly defined, the core concept is standard - no permissible coalition can improve upon its outcome. We provide a sufficient condition for the nonemptiness of the core in the case where the players can be divided into a finite number of types. This result is applied to a market game and the nonemptiness of the core of the market game is stated under considerably weak conditions (but with finite types). In addition, it is illustrated that the framework applies to assignment games with a continuum of players.

This paper considers an exchange economy called a generalized assignment market, in which sellers and buyers trade one indivisible commodity possibly with product differentiation for a perfectly divisible commodity. The existence of a competitive equilibrium in this economy is proved using Kakutani's fixed point theorem. This existence theorem is applied to a production economy in which sellers are formulated as producers with convex cost functions. Two examples of housing markets are provided and their competitive equilibria are numerically calculated.

In Part I of this paper we introduced extensive games with a non-atomic continuum of players. It was shown that the Nash plays (outcomes) are invariant of the information patterns on the game, provided that no player's unilateral change in moves can be observed by others. This led to an enormous reduction in the Nash plays of the these games, as exemplified in the anti-folk theorem. Our concern in this sequel is to develop a finite version of these results.

In this paper we explore the relation between information patterns and Nash Equilibria in extensive games. By information we mean what players know about moves made by others, as well as by chance. For the most part we confine ourselves to pure strategies. But in Section 2 behavioral strategies are also examined. It turns out that they can be modeled as pure strategies of an appropriately enlarged game. Our results, applied to the enlarged game, can then be reinterpreted in terms of the behavioral strategies of the original game.

This paper provides a new formulation of the Nash social welfare function for a continuum of individuals. The new formulation removes redundant assumptions of Kaneko's original formulation and in particular shows that the continuity axiom on social orderings is unnecessary. The derivation of the Nash social welfare function is also much directer and shorter than Kaneko's original derivation.

This paper argues that interpersonal comparisons of utility levels are difficult in principle and that interpersonal comparisons of utility differences can also hardly be defined based solely on individual choice behavior.

A model of a rental housing market in which houses are treated as indivisible commodities is presented. A recursive equation that determines a competitive equilibrium is provided, and it is argued that this competitive equilibrium is representative of the set of all competitive equilibria. Using this representative equilibrium, several propositions on comparative statics are considered that have to do with how the competitive rents change when certain parameters of the model change. © 1983.

Initially this paper considers an assignment game without side payments and proves the non- emptiness of the core of it. Next, a market model with invisible goods but without the transferable utility assumption is presented, and the non-emptiness of the core and the existence of a competitive equilibrium of the market model are shown, using the first result. Finally this paper presents a generalization of the market model and also shows the non-emptiness of the core and the existence of a competitive equilibrium using the results in the previous model.

It is argued that although the pathological multiplicity of Nash equilibria of super games stated by the folk theorem can be removed by introducing limited observations into super games with a continuum of players, the consideration of super games in terms of the Nash equilibrium concept involves a more fundamental and conceptual difficulty.

In this paper we consider the Nash social welfare function for a society which is given as an arbitrary finite measure space. We show that the Nash social welfare function for the society is derived from certain rationality criteria.

This paper provides an extension of Nash's bargaining model. The extension permits the domain of a solution function to contain non-convex sets. First, it is shown that the proposed extension is a mathematical generalization of Nash's bargaining problem. Next, some relations between the extension and the Nash social welfare function are developed. The formal equivalence of the two constructs is demonstrated, and the intuitive significance of the results is discussed.

This article considers an economy in which there is one public good financed by means of commodity taxes (lump sum transfers being not available). The first part of the paper is devoted to the study of tax equilibria. Sufficient conditions for the existence of an equilibrium with respect to a given tax system are given. When the tax system is modified, the structure of the corresponding set of tax equilibria is analyzed, and continuity properties of equilibria (with respect to the tax system) are stated. In the second part, attention is focused on the Pareto ranking of tax equilibria. In a given equilibrium, the directions of policy tools changes for a Pareto improvement (if any) are characterized. The "size" of the set of second best Pareto optima in the set of tax equilibria is evaluated.

We consider an oligopolistic market as follows. In the market, one good is traded for money. Each oligopolist is a price setter and has the same linear cost function. Each buyer is a price taker and buys the good from oligopolists setting the lowest price. We formulate this market as a cooperative game, and consider two kinds of solution concepts, the core and a bargaining set of the game. First we show that in the monopolistic market, the core gives the monopoly price, but in the oligopolistic market, the core is empty. Second, we obtain the bargaining set of the oligopolistic market.

An economic model of the allocation process with public goods is presented. We define a concept of equilibrium and prove the existence. Next we present a voting game in which a level of the public goods to be produced is decided. We prove that the core of the voting game and the equilibria exist simultaneously, and that they coincide.

In Kaneko [6] we considered the relationship between the ratio equilibria and the core of the voting game G(N, W,r), in which a fixed ratio is given. In this paper we present a new voting game G(N, W) in which no fixed ratio is given, and consider the relationship between the ratio equilibria and the core of G(N, W). We prove that the core of G(N, W) coincides with the ratio equilibria.

We provide a condition for an individual preference ordering to be represented by a function measured in terms of a commodity, i.e., for the commodity to be transferable utility. We also consider the relationships between conditions of the preference ordering and the utility function.

We consider a generalization of the assignment game of Shapley and Shubik [4]. In the market which we consider, s kinds of indivisible goods are exchanged for money. The market consists of buyers and sellers. Each buyer wants to buy at most one unit of the goods, and each seller may sell more than one unit. First, we show that the set of all competitive imputations is given by the solutions of a certain linear programing problem dual to the optimal problem. Second, we show that the core of the market coincides with the set of all competitive imputations under some condition, and consider the core of the market where s=1 and the condition does not hold.

In this paper we provide necessary and sufficient conditions for transitivity of the social preferences given by two types of social decision processes. We consider transitivity of the dominance relation of a proper simple game and that of the social preference given by the simple majority decision rule.

In this note we consider the core of a game with side payments representing simple majority voting. We provide necessary and sufficient conditions for the game to have a nonempty core.