arXiv subjects
Publications and source records attributed to Somdeb Lahiri.
We provide an axiomatic characterization of lexicographic preferences over the set of all random availability functions using two assumptions. The first assumption is strong monotonicity, which in our framework is equivalent to the strong dominance property in microeconomics. The second assumption is independence of worse alternatives and we show that a weaker version of the same suffices for our purpose.
We consider a sub-class of bi-matrix games which we refer to as two-person (hereafter referred to as two-player) additively-separable sum (TPASS) games, where the sum of the pay-offs of the two players is additively separable. The row player's pay-off at each pair of pure strategies, is the sum of two numbers, the first of which may be dependent on the pure strategy chosen by the column player and the second being independent of the pure strategy chosen by the column player. The column player's pay-off at each pair of pure strategies, is also the sum of two numbers, the first of which may be dependent on the pure strategy chosen by the row player and the second being independent of the pure strategy chosen by the row player. The sum of the inter-dependent components of the pay-offs of the two players is assumed to be zero. We prove the existence of equilibrium for such games and show that the set of equilibria for such games is the projection on the set of strategy pairs of the solutions of a pair of linear programming problems that are dual to each other. This result is a generalization of the corresponding and well-known result for two-person zero-sum games. We also show that a (randomized or mixed) strategy pair is an equilibrium of the game if and only if there exist two other real numbers such that the three together solve a certain linear programming problem. In order to prove this result, we need to appeal to the existence of an equilibrium for the TPASS game. The technology we use to prove our results, consists of the duality theorems and the complementary slackness theorem of linear programming.
We introduce a model of infinite horizon linear dynamic optimization and obtain results concerning existence of solution and satisfaction of the competitive condition and transversality condition being unconditionally sufficient for optimality of a trajectory. We also show that under some mild restrictions the optimal trajectory satisfies the Euler condition and a related transversality condition. The optimal trajectory satisfies the functional equation of dynamic programming. Under an additional convexity assumption for the two-period constraint sets, we show that the optimal value function is concave and continuous. Linearity bites when it comes to the definition of optimal decision rules which can no longer be guaranteed to be single-valued. We show that if all the two-period constraint sets are convex, then the optimal decision rule is an upper semi-continuous correspondence. For linear cake-eating problems, we obtain monotonicity results for the optimal value function and a conditional monotonicity result for optimal decision rules. We also introduce the concept of a two-phase linear cake eating problem and obtain a necessary condition that must be satisfied by all solutions of such problems. We show that for a class of linear dynamic optimization problems, known as interlinked linear dynamic optimization problems, a slightly modified version of the functional equation of dynamic programming is satisfied.
We interpret a fuzzy set as a random availability function and provide sufficient conditions under which a preference relation over the set of all random availability functions can be represented by a utility function.
Our first result is a statement of a somewhat general form of a non-substitution theorem for linear programming problems, along with a very easy proof of the same. Subsequently, we provide an easy proof of theorem 1 in a 1979 paper of Olvi L. Mangasarian, based on a new result in terms of two statements that are each equivalent to a given solution of a linear programming problem being its unique solution. We also provide a simple proof of the result that states that the set of optimal solutions of a bounded linear optimization problem is the set of all convex combinations of its basic optimal solutions and the set of basic optimal solutions are the extreme points of the set of optimal solutions. We do so by appealing to the lemma due to Farkas and the well-known result that states that if a linear optimization problem has an optimal solution, it has at least one basic optimal solution. Both results we appeal to have easy proofs. We do not appeal to any version of the Klein-Milman Theorem or any result in advanced polyhedral combinatorics to obtain our results. As an application of this result, we obtain a simple proof of the Birkhoff-von Neumann Theorem.
We introduce a model of infinite horizon linear dynamic optimization with linear constraints and obtain results concerning feasibility of trajectories and optimal solutions necessarily satisfying conditions that resemble the Euler condition and transversality condition. We prove results about optimal trajectories of strictly alternating problems, eventually conclusive problems, strongly eventually conclusive problems and two-phase problems.
We present a simple proof of a well-known axiomatic characterization of state-salient decision rules, using Weak Dominance Criterion and Global Independence of Irrelevant Alternatives. Subsequently we provide a simple axiomatic characterization of the Strict-Condorcet choice function on the domain of all preference profiles that have a strict-Condorcet winner, assuming that if the first two ranks are occupied by the same two alternatives in all states of nature, then the chosen alternative will be the one from these two that is preferred to the other with probability greater than half-provided such an alternative exists. We also show that this result is not valid if we extend the domain to the set of all preference profiles that have a unique weak-Condorcet winner.
In this paper we assemble some results about the upper-semicontinuity and lower-semicontinuity of the feasible correspondence and the solution correspondence of linear programming problems allowing variability of all parameters of such problems. We also prove continuity properties of optimal value functions, once again allowing all parameters to vary. We discuss sensitivity properties of the optimal value function, keeping the coefficient matrix fixed.
We present a theory of expected utility with state-dependent linear utility functions for monetary returns, that incorporates the possibility of loss-aversion. Our results relate to first order stochastic dominance, mean-preserving spread, increasing-concave linear utility profiles and risk aversion. As an application of the expected utility theory developed here, we analyze the contract that a monopolist would offer in an insurance market that allowed for partial coverage of loss.
We provide an exact algorithm to solve the log-linear continuous (fractional) knapsack problem. The algorithm is based on two lemmas that follow from the application of weak duality theorem and complementary slackness theorem to the linear optimization problem with linear objective function that is associated with any solution of a linear optimization problem with (differentiable) concave objective function.
We discuss bundle auctions within the framework of an integer allocation problem. We show that for multi-unit auctions, of which bundle auctions are a special case, market equilibrium and constrained market equilibrium are equivalent concepts. This equivalence, allows us to obtain a computable necessary and sufficient condition for the existence of constrained market equilibrium for bundle auctions. We use this result to obtain a necessary and sufficient condition for the existence of market equilibrium for multi-unit auctions. After obtaining the induced bundle auction of a nonnegative TU game, we show that the existence of market equilibrium implies the existence of a possibly different market equilibrium as well, which corresponds very naturally to an outcome in the matching core of the TU game. Consequently we show that the matching core of the nonnegative TU game is non-empty if and only if the induced market game has a market equilibrium.
This paper proves the existence of non-empty cores for directed network problems with quotas and for those combinatorial allocation problems which permit only exclusive allocations.
We show, that a simple generalization of the Deferred Acceptance Procedure with firms proposing due to Gale and Shapley(1962), yeild outcomes for a two-sided contract choice problem, which necessarily belong to the core and are Weakly Pareto Optimal for firms. Under additional assumptions: (a) given any two distinct workers, the set of yields acheivable by a firm with the first worker is disjoint from the set of yields acheivable by it with the second, and (b) the contract choice problem is pair-wise efficient, we prove that there is no stable outcome at which a firm can get more than what it gets at the unique outcome of our procedure.
In this paper, we consider the problem of choosing a set of multi-party contracts, where each coalition of agents has a non-empty finite set of contracts to choose from. We call such problems, contract choice problems. We provide conditions under which a contract choice problem has a non-empty set of "stable" outcomes. There are two types of stability concepts we study in this paper: cooperative stability and non-cooperative stability. The cooperative stability concept that we invoke here is the core. The non-cooperative stability concept that we study here is individual stability. The final result of this paper states that every contract choice problem has a non-empty weak bargaining set.