This article establishes that, in a one-dimensional diffusion world, any contingent claim's delta is bounded by its delta at maturity and, if its payoff is convex, its current value is convex in the underlying's value. A decline in the present value of the exercise price can be associated with a decline in a call's price. Bounds on call prices and deltas are derived for the case when the underlying's volatility is bounded. If the underlying follows a multi-dimensional diffusion (a stochastic volatility world), or a discontinuous or non-Markovian process, call prices can be decreasing, concave function of the underlying's value.

We consider a repeated two-person zero-sum game in which the payoffs in the stage game are given by a 2 % 2 matrix. This is chosen (once) by chance, at the beginning of the game, to be either G1 or G², with probabilities p and 1 - p respectively. The maximizer is informed of the actual payoff matrix chosen but the minimizer is not. Denote by vn(p) the value of the n -times repeated game (with the payoff function defined as the average payoff per stage), and by v%(p) the value of the infinitely repeated game. It is proved that vn(p)=v%(p) + %(p)%(p)/%n + %%1/%n% , where %(p) is on appropriately scaled normal distributiondensity function evaluated at its p-quantile, and the coefficient K(p) is either 0 or the absolute value of a linear function in p.

Starting with the analysis of arms control and disarmament problems in the sixties, inspection games have evolved into a special area of game theory with specific theoretical aspects, and, equally important, practical applications in various fields of human activities where inspection is mandatory. In this contribution, a survey of applications is given first. Then, the general problem of inspection is presented in a game theoretic framework as an extension of a statistical hypothesis presented in a testing problem. Using this framework, two important models are solved: material accountancy and dataverification. A second important aspect of inspection games are limited inspection resources that have to be used strategically. This is presented in the context of sequential inspection games, where many mathematically challenging models have been studies. Finally, the important concept of leadership, where the inspector becomes a leader by announcing and committing himself to his strategy, is shown to apply naturally to inspection games.

Formal Interactive Epistemology deals with the logic of knowledge and belief when there is more than one agent or "player". One is interested not only in each person's knowledge about substantive matters, but also in his knowledge about the others' knowledge. These notes examine two parallel approaches to the subject. The first is the semantic approach, in which knowledge is represented by a space % of states of the world, together with partitions %i of % for each player i; the atom of %i containing a given state % of the world represents the set of those states that i cannot distinguish from %. The second is the syntactic approach, in which knowledge is represented by abstract formulas constructed according to certain syntactic rules. These notes examine the relation between the two approaches, and show that they are in a sense equivalent. In game theory and economics, the semantic approach has heretofore been most prevalent. A question that often arises in this connection is whether, in what sense, and why the space % and the partitions %i can be taken as given and commonly known by the players. An answer to this question is provided by the syntactic approach. Other topics that are taken up include various formalizations of "common knowledge", and the "Agreement Theorem" of J. Cave and M. Bacharach. The notes end with an application of these ideas to the context of probabilistic beliefs.

It has been argued that in the presence of an `Atmosphere Externality' and competitive behavior by households, a uniform commodity tax on an externality - generating good attains the first best. It is demonstrated, however, that if income redistribution is desirable then personalized taxes are required for a second-best optimum. Each of these taxes is the sum of a uniform (across households) tax and a component, positive or negative, which depends on the household's income and demand elasticities. Second-best optimal indirect taxes and rules for investment in externality-reducing measures are also considered.

A survey of bounded rationality models and ideas in Game Theory. Topics covered include: The evolutionary approach to optimization – and specifically to game theory – and its implications for the idea of bounded rationality; evolutionary dynamics; "rule rationality" as opposed to "act rationality"; "trembles" and refinements in general; "crazy" perturbations; failure of common knowledge of rationality; limiting average payoff in infinitelyrepeated games; epsilon equilibria; players modeled as computers, finite state automata, or Turing machines; paradoxes (such as Ellsberg or Allais); laboratory experiments; and finally, an open problem.

The reactive bargaining set (Granot [1994]) is the set of outcomes for which no justified objection exists. Here, in a justified objection the objector first watches how the target tries to act (if he has such an option), and then reacts by making a profit and ruining the target's attempt to maintain his share. In this paper we explore properties of the reactive bargaining set, set up the system of inequalities that defines it, and construct a dynamic system in the sense of Stearn's transfer scheme that leads the players to this set. We also extend the definition of the reactive bargaining set to NTU games in a way that keeps it nonempty. To shed light on its nature and its relative ease of computation, we compute the reactive bargaining set for games that played important role in the game theory literature.

A reply to Margalit and Yaari's paper "Rationality and Comprehension", in which they comment on my papers "Agreeing to disagree" [1] and "Notes on Interactive Epistemology" [2]. Inter alia, we point out that contrary to Margalit and Yaari's claim, in [1] the agents need not condition on the same events; and in [2], the state space is not assumed as analytic knowledge, but is derived. In addition, a simple resolution of the "hangman's paradox" is offered.

Motivated by recent auctions of licenses for the radio-frequency spectrum, we consider situations where multiple objects are auctioned simultaneously by means of a second-price, sealed-bid auction. For some buyers, called global bidders, the value of multiple objects exceeds the sum of objects' values separately. Others, called local bidders, are interested in only one object. In a simple independent private values setting, we (a) characterize an equilibrium that is symmetric among the global bidders; (b) show that the addition of bidders often leads to less aggressive bidding; and (c) compare the revenues obtained from the simultaneous auction to those from its sequential counterpart.

For a class of nonatomic congestion games, two solution concepts, a noncooperative one and a cooperative one, are compared. Each player in the game chooses one of several common facilities. The player's payoff is the difference between the reward and the cost associated with choosing that facility. The rewards are fixed and player-specific. The costs are uniform, but variable: they strictly increase with the measure of the set of players using the facility. The noncooperative solution of the game is the (unique) Nash equilibrium outcome. The cooperative one is the Aumann-Shapley value of the cooperative game that results when players are willing to cooperate in order to minimize the total utility. Using a new result in the theory of values of nonatomic games, we derive a formula for the value. We show that there is exactly one case in which the Nash equilibrium outcome andthe value always coincide: this is the case in which the costs increase logarithmically with the measure of the set of users.

The following generalization of a theorem of Aumann and Shapley is proved: A vector measure game of the form f'$^\circ$%, where % is a nonatomic banach-space measure of bounded variation and f is a weakly continuously differentiable real-valued function defined on the closed convex hull of the range of % such that f(0)=0, is in pNA. If the game is monotonic, then the conclusion holds even if at 0 f is only continuous, and not differentiable. The value of the game is given by the diagonal formula. These results are used for giving a new, relatively short, proof to the result that, under certain conditions, a market game is in pNA.

In a series of experiments, we demonstrate that people are reluctant to exchange lottery tickets. In other words, when given a small incentive to exchange a lottery ticket with which they had just been endowed for a different one, with the same probability of winning the same prize, only about 50% choose to do so. In contrast, when given the same incentive to exchange a pen with which they had just been endowed for another pen just like it, over 90% choose to do so. We discuss – and rule out – a series of possible explanations for this effect, including: distorted subjective probabilities; fear of finding out that you gave up a wining ticket; lack of sufficient incentive (i.e.,transaction cost); general confusion or "paranoia"; etc. We conclude that people will not exchange ex ante identical tokens of the same type unless the two tokens will be identical ex post as well. A lottery ticket with which one has been endowed becomes at once the status quo, or reference point, with respect to which changes are evaluated for possible gains and losses. Since losses loom larger than gains, two lottery tickets which are symmetrical before they pass into one's possession are no longer symmetrical once one of them becomes one's own.

In an epistemic framework due to Aumann we characterize the minimal condition on the rationality of the players that implies backward induction in perfect information games in agent form. This condition requires each player to know that the players are rational at later, but not at previous decision nodes.

We study the war of attrition and the all-pay auction when players' signals are affiliated and symmetrically distributed. We (a) find sufficient conditions for the existence of symmetric monotonic equilibrium biddingstrategies; and (b) examine the performance of these auction forms in terms of the expected revenue accruing to the seller. Under our conditions the war of attrition raises greater expected revenue than all other known sealed did auction forms.

It is a striking fact that different solutions become equivalent in the setting of perfectly competitive economies. We provide an axiomatic approach to this equivalence phenomenon. A solution is viewed as a correspondence which maps each economy to a subset of its individually rational and Pareto-optimal allocations. Four axioms are placed on the correspondence: anonymity, equity, consistency and restricted continuity. It is shown that the axioms categorically determine the Walrasian correspondence. The equivalence of other solutions, such as the core or value allocations, now follows by checkingthat they too satisfy the axioms.

A non-negative function f defined on the class of subsets of a finite set of factors of production describes the production possibilities at each date. The problem of allocating the surplus among the factors is studied in a dynamic learning model. Representatives for the factors (called players) make wage demands naively based on precedent and ignorant of each others' utilities for this good. A global convergence result shows that players learn to reach some (and only a) core allocation in the long run. If players make mistakes however, only a strict subset of the core allocations are likely, i.e., stochastically stable. The main result shows that in the limit, these stable allocations for a particular set of players, converge to the allocation that maximizes the product of all the players' utilities over core allocations.

We formulate precisely and prove the proposition that if common knowledge of rationality obtains in a game of perfect information, then the backward induction outcome is reached.

We analyze the problem of extending a given bilateral principle of justice to a consistent n-creditor bankruptcy rule. Based on the bilateral principle, we build a family of binary relations on the set of creditors in order to make bilateral comparisons between them. We find that the possibility of extending a specific bilateral principle of justice in a consistent way is closely related to the quasi-transitivity of the binary relations mentioned above.

In this paper we study finite-agent exchange economies. We extend the classical model by adding an imports-exports vector, which defines the markets clearing conditions of the economy. Equipped with this new definition, self-consistency properties are naturally defined. We show that the Core correspondence and the Walrasian allocations correspondence are self-consistent. In addition, we present an axiomatic characterization of the Walrasian allocations correspondence for a class of convex and smooth economies. All the axioms presented in the characterization are satisfied by the Core, except for a converse-consistency property, which can be interpreted as a requirement of decentralization.

It is shown that the core of a non-atomic glove-market game which is defined as the minimum of finitely many non-atomic probability measures is a von-Neumann Morgenstern stable set. This result is used to characterize some stable set of large games which have a decreasing returns to scale property. We also study exact non-atomic glove-market games. In particular we show that in a glove-market game which consists of the minimum of finitely many mutually singular non-atomic measures, the core is a von-Neumann Morgenstern stable set if the game is exact. We also discuss the intuitive appeal of the equivalence of the core and stable set. We do this by employing the theory of social situations [5] and highlighting the negotiation processes that underlie these two notions.