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Person# Stefano Bortolomiol

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Nash equilibrium

In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilib

Externality

In economics, an externality or external cost is an indirect cost or benefit to an uninvolved third party that arises as an effect of another party's (or parties') activity. Externalities can be con

Competition

Competition is a rivalry where two or more parties strive for a common goal which cannot be shared: where one's gain is the other's loss (an example of which is a zero-sum game). Competition can ari

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Michel Bierlaire, Stefano Bortolomiol, Virginie Janine Camille Lurkin

Oligopolistic competition occurs in various transportation markets. In this paper, we introduce a framework to find approximate equilibrium solutions of oligopolistic markets in which demand is modeled at the disaggregate level using discrete choice models, according to random utility theory. Compared with aggregate demand models, the added value of discrete choice models is the possibility to account for more complex and precise representations of individual behaviors. Because of the form of the resulting demand functions, there is no guarantee that an equilibrium solution for the given market exists, nor is it possible to rely on derivative-based methods to find one. Therefore, we propose a model-based algorithmic approach to find approximate equilibria, which is structured as follows. A heuristic reduction of the search space is initially performed. Then, a subgame equilibrium problem is solved using a mixed integer optimization model inspired by the fixed-point iteration algorithm. The optimal solution of the subgame is compared against the best responses of all suppliers over the strategy sets of the original game. Best response strategies are added to the restricted problem until all epsilon-equilibrium conditions are satisfied simultaneously. Numerical experiments show that our methodology can approximate the results of an exact method that finds a pure equilibrium in the case of a multinomial logit model of demand with a single-product offer and homogeneous demand. Furthermore, it succeeds at finding approximate equilibria for two transportation case studies featuring more complex discrete choice models, heterogeneous demand, a multiproduct offer by suppliers, and price differentiation for which no analytical approach exists.

Michel Bierlaire, Stefano Bortolomiol, Virginie Janine Camille Lurkin

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.

2021Many transportation markets are characterized by oligopolistic competition. In these markets customers, suppliers and regulators make decisions that are influenced by the preferences and the decisions of all other agents. In particular, capturing and understanding demand heterogeneity is key for suppliers and regulators to develop optimal strategies and policies. A state-of-the-art approach to model demand at a disaggregate level is discrete choice modeling. This thesis deals with the integration of discrete choice models into optimization and equilibrium problems by means of simulation.First, we analyze a deregulated competitive market. When a disaggregate heterogeneous demand is considered, there is no theoretical guarantee that a market equilibrium solution exists, nor it is possible to rely on derivative-based methods to find one. Therefore, we propose a simulation-based heuristic to find approximate equilibrium solutions. Numerical experiments show that the proposed algorithm can approximate the results of an exact method that finds a pure equilibrium in the case of logit demand with single-product offer and homogeneous customers. Furthermore, the algorithm succeeds at finding approximate equilibria for two transportation case studies featuring more complex discrete choice models, heterogeneous demand, multi-product offer by supplies, and price differentiation, for which no analytical approach exists.Then, the framework is extended to the case of a regulated competitive market. The objective of the regulator is to find optimal price-based policies which affect the behavior of all other agents towards welfare-maximizing outcomes. In transport markets, economic instruments might target specific alternatives, to reduce externalities such as congestion or emissions, or specific segments of the population. A mixed-integer linear optimization model is presented which finds optimal policies subject to supply's profit maximization and demand's utility maximization constraints. This model is included into an adapted version of the simulation-based heuristic framework developed for deregulated competition. Numerical experiments on an intercity travel case study show how the regulator can optimize taxes and subsidies for different objective functions and scenarios.Finally, a deeper analysis is conducted on the choice-based optimization model that represents a fundamental block, and the most computationally expensive one, of the choice-based equilibrium framework. Because of simulation, the problem has a block-diagonal structure that makes it suitable to the use of mathematical decomposition techniques. Specifically, it is shown that a formulation in which all the decision variables of the supplier are discrete is amenable to the use of Benders decomposition. Under this assumption, a Benders decomposition scheme is derived and implemented within a branch-and-cut approach to solve an uncapacitated facility location and pricing problem with disaggregate demand. Numerical experiments that compare this approach with a black-box solver show that the black-box solver is faster at solving small instances, while Benders decomposition is faster on larger instances. The possibility to achieve speed-ups through enhancements available in the literature and to tackle large problems with more complex structures should motivate further investigation of Benders and other decomposition techniques for other classes of choice-based optimization problems.