Price-based regulation of oligopolistic markets under discrete choice models of demand

We propose a framework to find optimal price-based policies to regulate markets characterized by oligopolistic competition and in which consumers make a discrete choice among a finite set of alternatives. The framework accommodates general discrete choice models available in the literature in order to capture heterogeneous consumer behavior. In our work, consumers are utility maximizers and are modeled according to random utility theory. Suppliers are modeled as profit maximizers, according to the traditional microeconomic treatment. Market competition is modeled as a non-cooperative game, for which an approximate equilibrium solution is sought. Finally, the regulator can affect the behavior of all other agents by giving subsidies or imposing taxes to consumers. In transport markets, economic instruments might target specific alternatives, to reduce externalities such as congestion or emissions, or specific segments of the population, to achieve social welfare objectives. In public policy, different agents have different individual or social objectives, possibly conflicting, which must be taken into account within a social welfare function. We present a mixed integer optimization model to find optimal policies subject to supplier profit maximization and consumer utility maximization constraints. Then, we propose a model-based heuristic approach based on the fixed-point iteration algorithm that finds an approximate equilibrium solution for the market. Numerical experiments on an intercity travel case study show how the regulator can optimize its decisions under different scenarios.

Sea-unammned aerial vehicle takeoff characteristics analysis method based on approximate equilibrium hypothesis

In this paper, transient multiphase flow computational fluid dynamics simulations based on volume of fluid model are conducted for a sea-unmanned aerial vehicle. The approximate equilibrium hypothesis is implemented after estimating the acceleration in the vertical direction. The complete configuration model and hull model are employed in simulation to predict the aerodynamic and hydrodynamic forces separately for different demands of aerodynamic and hydrodynamic computational fluid dynamics predictions and computing efficiency. In takeoff characteristics analysis, the computational fluid dynamics simulations are conducted as inputs for piecewise interpolation method. The calculated results show that the sea-unmanned aerial vehicle takeoff characteristics are totally different from a conventional aircraft. The drag-peak at hump speed is the obvious feature of the sea-unmanned aerial vehicle/seaplane. In most cases, if a sea-unmanned aerial vehicle will takeoff successfully as long as it can pass the drag peak. The takeoff distance and time calculated by piecewise interpolation method match the experimental data within 7% deviation. The accuracy is acceptable for conceptual design stage of a sea-unmanned aerial vehicle/seaplane. The results are applicable to consultation in choosing takeoff field or choosing powerplant.

Sampling Individually Fundamental Simplexes as Sets of Players’ Mixed Strategies in Finite Noncooperative Game for Applicable Approximate Nash Equilibrium Situations with Possible Concessions

In finite noncooperative game, a method for finding approximate Nash equilibrium situations is developed. The method is prior-based on sampling fundamental simplexes being the sets of players’ mixed strategies. Whereas the sampling is exercised, the sets of players’ mixed strategies are mapped into finite lattices. Sampling steps are envisaged dissimilar. Thus, each player within every dimension of its simplex selects and controls one’s sampling individually. For preventing approximation low quality, however, sampling steps are restricted. According to the restricted sampling steps, a player acting singly with minimal spacing over its lattice cannot change payoff of any player more than by some predetermined magnitude, being specific for each player. The finite lattice is explicitly built by the represented routine, where the player’s mixed strategies are calculated and arranged. The product of all the players’ finite lattices approximates the product of continuous fundamental simplexes. This re-defines the finite noncooperative game in its finite mixed extension on the finite lattices’ product. In such a finite-mixed-extension-defined game, the set of Nash equilibrium situations may be empty. Therefore, approximate Nash equilibrium situations are defined by the introduced possible payoff concessions. A routine for finding approximate equilibrium situations is represented. Approximate strong Nash equilibria with possible concessions are defined, and a routine for finding them is represented as well. Acceleration of finding approximate equilibria is argued also. Finally, the developed method is discussed to be a basis in stating a universal approach for the finite noncooperative game solution approximation implying unification of the game solvability, applicability, realizability, and adaptability.

Approximate Equilibrium Problems and Fixed Points

We find a common element of the set of fixed points of a map and the set of solutions of an approximate equilibrium problem in a Hilbert space. Then, we show that one of the sequences weakly converges. Also we obtain some theorems about equilibrium problems and fixed points.

A NOTE ON THE CONCEPT OF APPROXIMATE EQUILIBRIUM IN ECONOMIC THEORY

The problem of the computability of Walrasian competitive equilibrium is considered from the point of view of concepts of approximate equilibrium. Neither the market-clearing nor Negishi approaches to the proof of existence of Walrasian competitive equilibrium give rise to adequately robust notions of approximate equilibrium. This explains the non-computability of Walrasian competitive equilibrium. The problem lies in the economic conception of markets, in particular the inconsistent treatment of information underlying the Walrasian definition. When trade takes place at disequilibrium prices decentralized market exchange redistributes income and economic welfare, and its equilibrium is path-dependent. The set of such equilibrium outcomes, however, in contrast to the Walrasian competitive equilibrium, is constructive and computable.

Equivalence between the Existence of an Approximate Equilibrium in a Competitive Economy and Sperner's Lemma: A Constructive Analysis

Brouwer's fixed point theorem cannot be constructively proved, so the existence of an equilibrium in a competitive economy also cannot be constructively proved. On the other hand, Sperner's lemma which is used to prove Brouwer's theorem is constructively proved. Some authors have presented a constructive (or an approximate) version of Brouwer's fixed point theorem using Sperner's lemma. In this paper, I prove the existence of an approximate equilibrium in a competitive economy directly by Sperner's lemma. Also I show that the existence of an approximate equilibrium leads to Sperner's lemma. I follow the Bishop style constructive mathematics according to Bishop and Bridges (1985), Bridges and Richman (1987), and Bridges and Vîţă (2006).