Bertrand Competition Model Research Articles

A bilevel game-theoretic decision-making framework for strategic retailers in both local and wholesale electricity markets

This paper proposes a bilevel game-theoretic model for multiple strategic retailers participating in both wholesale and local electricity markets while considering customers’ switching behaviors. At the upper level, each retailer maximizes its own profit by making optimal pricing decisions in the retail market and bidding decisions in the day-ahead wholesale (DAW) and local power exchange (LPE) markets. The interaction among multiple strategic retailers is formulated using the Bertrand competition model. For the lower level, there are three optimization problems. First, the welfare maximization problem is formulated for customers to model their switching behaviors among different retailers. Second, a market-clearing problem is formulated for the independent system operator (ISO) in the DAW market. Third, a novel LPE market is developed for retailers to facilitate their power balancing. In addition, the bilevel multi-leader multi-follower Stackelberg game forms an equilibrium problem with equilibrium constraints (EPEC) problem, which is solved by the diagonalization algorithm. Numerical results demonstrate the feasibility and effectiveness of the EPEC model and the importance of modeling customers’ switching behaviors. We corroborate that incentivizing customers’ switching behaviors and increasing the number of retailers facilitates retail competition, which results in reducing strategic retailers’ retail prices and profits. Moreover, the relationship between customers’ switching behaviors and welfare is reflected by a balance between the electricity purchasing cost (i.e., electricity price) and the electricity consumption level.

Implications of Price Matching in Supply Chain Negotiation

Problem definition: Few papers have explored the impact of price matching negotiation (PM), in which a channel matches its price with the resulting wholesale price bargained by another channel, on firms’ performances, consumer welfare, and social welfare, with and without supply chain coordination. Academic/practical relevance: Negotiation has been widely seen in determining both uniform and discriminatory wholesale prices, which affect outcomes of competitive supply chain practices. Methodology: To characterize the PM mechanism, we use game theory and Nash bargaining theory to compare PM with simultaneous negotiation (SN) through a common-seller two-buyer differentiated Bertrand competition model. Results: Our analysis reveals that PM can benefit the seller but hurt all buyers, which is at odds with some fair wholesale pricing clauses intending to protect buyers. Under coordination with side payments, however, all firms can conditionally benefit more from PM than from SN. Despite firms’ gains, PM leads to less consumer utility and social welfare compared with SN, unless the second buyer in PM is considerably less powerful than the first buyer. Coordination further worsens PM’s negative impact on consumer utility and social welfare. Moreover, the existence of a spot market can increase the wholesale price in PM, hurting buyers, consumers, and society. Furthermore, the qualitative results about PM remain robust under an alternative disagreement point for PM, multiple buyers, and other extensions. Managerial implications: This paper delivers insights on when price matching in supply chain wholesale price negotiation can benefit a seller, buyers, consumers, and society in a variety of scenarios. It advocates how managers can use PM to their own advantages and provides rationale to decision makers for policy regulations regarding wholesale pricing.

Price Competition Under Mixed Multinomial Logit Demand: Sufficiency Conditions for Validating the Model

This study is motivated by a fundamental question among researchers: do price data correspond to a pure strategy Nash equilibrium of an estimated econometric model of Bertrand competition? The affirmative answer to this question is essential for validating the estimated model. Despite the extensive empirical applications of Bertrand competition over the last three decades, answering the question remains a daunting task for the researchers. This study takes a step to fill this gap, identifying a set of easy‐to‐verify and easy‐to‐interpret conditions under which, by checking how closely the price data fit the system of first‐order conditions, one can easily obtain an answer to the question.

Price competition and private labels in the U.S. packaged salad market

ABSTRACT We study the price competition in the U.S. packaged salads market by testing two price competition hypotheses: Bertrand competition versus sequential pricing model. The former is standard for oligopoly pricing in the IO literature. The latter provides a formal mechanism for second-mover, price-undercutting behaviour by private-label brands, owned by mainstream supermarkets, that grew their market share dramatically in the second half of the 2000s. Applying Vuong and Gasmi (1991)’s test, we show that observed market prices and market shares fit best to Bertrand competition.

Spectrum Sharing with Vehicular Communication in Cognitive Small-Cell Networks

An increasing number of vehicles make spectrum resources face serious challenges in vehicular cognitive small-cell networks. The means of spectrum sharing can greatly alleviate this pressure. In this paper, we introduce a supermodular game theoretic approach to analyze the problem of spectrum sharing. The small-cell BS (primary service provider, PSP) and the vehicle (secondary service provider, SSP) can share the spectrum, where the PSP can sell idle spectrum resources to the SSP. This is taken as a spectrum trading market, and a Bertrand competition model is considered to depict this phenomenon. Different PSPs compete with each other to maximize their individual profits. The Bertrand competition model can be proved as a supermodular game, and the corresponding Nash equilibrium (NE) solution is provided as the optimal price solution. Hence, an improved genetic simulated annealing algorithm is designed to achieve NE. Simulation results demonstrate that the NE point for the price of the primary service provider exists. The change of the exogenous variable is also analyzed on the equilibrium point.

Incentives on the Lightning Network : A Blockchain-based Payment Network

The Lightning Network is a decentralized payment network built on top of a block-chain, in which intermediary nodes provide a trust-less routing service for end users. We provide an overview of the current state of the network and show that it can be well approximated by a scale free generative model with a fitness parameter, which suggests that nodes behave strategically on the network. Those strategic interactions between nodes can be described by a Bertrand competition model with capacity constraints. We show that there is a unique equilibrium in which a centralized network is never optimal, and the routing fee is strictly greater than the marginal cost. When nodes are heterogeneous in their opportunity cost of capital only, the equilibrium network structure can match the current state of the network.

Consumer Search and the Uncertainty Effect

We consider a model of Bertrand competition where consumers are uncertain about the qualities and prices of firms' products. Consumers can inspect all products at zero cost. A share of consumers is expectation-based loss averse. For these consumers, a purchase plan, which involves buying products of varying quality and price with positive probability, creates scale-dependent dis-utility from gain-loss sensations. Even if their degree of loss aversion is modest, they may refrain from inspecting all products and choose an individual default that is first-order stochastically dominated. Firms' strategic behavior can exacerbate the scope for this uncertainty effect', and sellers of inferior products may earn positive profits despite Bertrand competition. We find suggestive evidence for the predicted association between consumer behavior and loss aversion in new survey data.

Ordoliberal theory and gas market in Germany within the framework of international energy cooperation

German economic policy is a special model of a coordinated market economy, which is based on the tradition of ordoliberalism. This theoretical concept is at the core of the regulation of most economic processes in Germany, and its energy markets are no exception. The article analyses the present state of German gas industry, its transformation in connection with the introduction of the Third Energy Package, the liberalization and development of gas projects. The authors compare the main results of the reforms and the stipulations of the ordoliberal theory. The conclusion is that the German government generally upholds this concept, although sometimes the consequences are of negative nature, manifested in rising prices, added complexity of international natural gas trade, especially with the largest supplier – the Russian Federation. Using the Bertrand competition model, the article justifies the conditions under which the export of liquefied natural gas is beneficial for the Russian Federation.

The Myopic Stable Set for Social Environments

We introduce a new solution concept for models of coalition formation, called the myopic stable set. The myopic stable set is defined for a very general class of social environments and allows for an infinite state space. We show that the myopic stable set exists and is non-empty. Under minor continuity conditions, we also demonstrate uniqueness. Furthermore, the myopic stable set is a superset of the core and of the set of pure strategy Nash equilibria in noncooperative games. Additionally, the myopic stable set generalizes and unifies various results from more specific environments. In particular, the myopic stable set coincides with the coalition structure core in coalition function form games if the coalition structure core is non-empty; with the set of stable matchings in the standard one-to-one matching model; with the set of pairwise stable networks and closed cycles in models of network formation; and with the set of pure strategy Nash equilibria in finite supermodular games, finite potential games, and aggregative games. We illustrate the versatility of our concept by characterizing the myopic stable set in a model of Bertrand competition with asymmetric costs, for which the literature so far has not been able to fully characterize the set of all (mixed) Nash equilibria.

Refurbishing authorization strategy in the secondary market for electrical and electronic products

In the electrical and electronic products secondary market, third party resellers (TPR) engaging in product refurbishing are growing rapidly, which provides direct competition to the original equipment manufacturers' (OEMs) new product sales. Irrespective of prices, consumers have shown a higher preference for refurbished products from the OEMs and OEM-authorized TPRs than non-authorized TPRs. As a result, some OEMs choose to provide refurbishing authorization to TPRs who are required to meet specific standards and pay an authorization fee. Besides the strategy of refurbishing authorization, an OEM could choose strategies such as participating in direct refurbishing work or not at all. The main goal of this paper is to examine under what conditions the refurbishing authorization strategy is optimal for the OEM. The essential tradeoff is whether the indirect benefit from authorizing a TPR, which includes an enlarging market share, can outweigh the direct cannibalization effect of authorized refurbished products on new products sales. To that end, we develop Bertrand competition models between an OEM and a TPR using several strategies. Results show that when the consumers' preference for refurbished products relative to new products is not large enough, the OEM should choose the authorization strategy. The OEM and authorized TPR achieve a win-win outcome under some conditions. We also find that a higher authorization fee is not always a better option for the OEM.

Implications of Price Matching in Supply Chain Negotiation

Problem definition: Few papers have explored the impact of price matching negotiation (PM), in which a channel matches its price with the resulting wholesale price bargained by another channel, on firms' performances, consumer welfare, and social welfare, with and without supply chain coordination. Academic/practical relevance: Negotiation has been widely seen in determining both uniform and discriminatory wholesale prices, which affect outcomes of competitive supply chain practices. Methodology: To characterize the PM mechanism, we utilize game theory and Nash bargaining theory to compare PM with simultaneous negotiation (SN) through a common-seller two-buyer differentiated Bertrand competition model. Results: Our analysis reveals that PM can benefit the seller but hurt all buyers, which is at odds with some fair wholesale pricing clauses intending to protect buyers. Under coordination with side payments, however, all firms can conditionally benefit more from PM than from SN. In spite of firms' gains, PM leads to less consumer utility and social welfare as compared to SN, unless the second buyer in PM is considerably less powerful than the first buyer. Coordination further worsens PM's negative impact on consumer utility and social welfare. Moreover, the existence of a spot market can increase the wholesale price in PM, hurting buyers, consumers, and society. Furthermore, the qualitative results about PM remain robust under an alternative disagreement point for PM, multiple buyers, and other extensions. Managerial implications: This paper delivers insights on when price matching in supply chain wholesale price negotiation can benefit a seller, buyers, consumers, and society in a variety of scenarios. It advocates how managers can utilize PM to their own advantages and provides rationale to decision makers for policy regulations regarding wholesale pricing.

Commodity taxes and welfare under endogenous market conduct

We consider consumption taxes in a model of endogenous Cournot versus Bertrand competition. It is argued that when the choice of unit versus ad valorem taxes affects longer-term decisions beyond the customary price or quantity decisions, the mix of the two taxes co-determines market conduct. This gives ad valorem taxes an anti-competitive effect that harms ad valorem taxes’ efficiency in comparison with unit taxes. We show that a mix of the taxes—or a unit tax alone if we compare one or the other of the taxes—is sometimes welfare superior on account of consumer-price and tax revenue effects. A practical implication of our findings is that pass-through rates are only sometimes useful guides for policy. In fact, we show when the proper response to demand for higher revenue is a higher unit tax rate and a lower ad valorem tax rate.

The Myopic Stable Set for Social Environments

We introduce a new solution concept for models of coalition formation, called the myopic stable set. The myopic stable set is defined for a very general class of social environments and allows for an infinite state space. We show that the myopic stable set exists and is non-empty. Under minor continuity conditions, we also demonstrate uniqueness. Furthermore, the myopic stable set is a super-set of the core and of the set of pure strategy Nash equilibria in non-cooperative games.Additionally, the myopic stable set generalizes and unifies various results from more specific environments. In particular, the myopic stable set coincides with the coalition structure core in coalition function form games if the coalition structure core is non-empty; with the set of stable matchings in the standard one-to-one matching model; with the set of pairwise stable networks and closed cycles in models of network formation; and with the set of pure strategy Nash equilibria in finite super-modular games, finite potential games, and aggregative games. We illustrate the versatility of our concept by characterizing the myopic stable set in a model of Bertrand competition with asymmetric costs, for which the literature so far has not been able to fully characterize the set of all (mixed) Nash equilibria.

Product Differentiation, Leniency Programs and Cartel Stability

Competition authorities usually consider the introduction of leniency programs as a success story. The increased incentives to report on fellow cartelists due to the introduction of leniency is likely to have contributed substantially to the clear increase in the number of detected cartels. Critics argue that leniency programs primarily target weaker cartels, whereas the most profitable cartels stay undetected and may even benefit from a shift in the authority’s resources from active cartel screening to the handling of allegedly less important leniency applications. We show that this perception cannot be upheld in a numerical Bertrand competition model with heterogeneous products. In our approach, more profitable cartels consisting of producers of closer substitutes tend to be less stable. Thus leniency programs do in fact threaten more harmful cartels.

The Myopic Stable Set for Social Environments

We introduce a new solution concept for models of coalition formation, called the myopic stable set. The myopic stable set is defined for a very general class of social environments and allows for an infinite state space. We show that the myopic stable set exists and is non-empty. Under minor continuity conditions, we also demonstrate uniqueness. Furthermore, the myopic stable set is a superset of the core and of the set of pure strategy Nash equilibria in noncooperative games. Additionally, the myopic stable set generalizes and unifies various results from more specific environments. In particular, the myopic stable set coincides with the coalition structure core in coalition function form games if the coalition structure core is non-empty; with the set of stable matchings in the standard one-to-one matching model; with the set of pairwise stable networks and closed cycles in models of network formation; and with the set of pure strategy Nash equilibria in finite supermodular games, finite potential games, and aggregative games. We illustrate the versatility of our concept by characterizing the myopic stable set in a model of Bertrand competition with asymmetric costs, for which the literature so far has not been able to fully characterize the set of all (mixed) Nash equilibria.

Two-way information sharing under supply chain competition

We study manufacturer-retailer bilateral information sharing in two competing supply chains (SCs), in which both the manufacturer and the retailer have partial information on demand. Based on Bertrand competition model and Winkler's consensus model, we develop a finite Bayesian Stackelberg game to analyze the two-way information sharing problem under horizontal supply chain (SC) competition. In line with the literature, we find that sharing demand forecast voluntarily in a SC benefits the manufacturer but hurts the retailer. However, we find whether SCs benefit from information sharing depends on competition intensity and forecast error. As competition is intensive, the expected values of information sharing (EVISs) for the entire SCs are high. Moreover, information sharing in one supply chain can improve the rival supply chain's EVIS under some conditions. Numerical experiments are conducted to get some managerial insights.

Research on Cost Information Sharing and Channel Choice in a Dual-Channel Supply Chain

Many studies examine information sharing in an uncertain demand environment in a supply chain. However there is little literature on cost information sharing in a dual-channel structure consisting of a retail channel and a direct sales channel. Assuming that the retail sale cost and direct sale cost are random variables with a general distribution, the paper investigates the retailer’s choice on cost information sharing in a Bertrand competition model. Based on the equilibrium outcome of information sharing, the manufacturer’s channel choice is discussed in detail. Our paper provides several interesting conclusions. In both single- and dual-channel structures, the retailer has little motivation to share its private cost information which is verified to be valuable for the manufacturer. When the cost correlation between the two channels increases, our analyses show that the manufacturer’s profit improves. However, when channel choice is involved, the value of information could play a different role. The paper finds that a dual-channel structure can benefit the manufacturer only when the cost correlation is sufficiently low. In addition, if the cost correlation is weak, the cost fluctuation will bring out the advantage of a dual-channel structure and adding a new direct channel will help in risk pooling.

Profit maximization algorithms for utility companies in an oligopolistic energy market with dynamic prices and intelligent users

Dynamic energy pricing provides a promising solution for the utility companies to incentivize energy users to perform demand side management in order to minimize their electric bills. Moreover, the emerging decentralized smart grid, which is a likely infrastructure scenario for future electrical power networks, allows energy consumers to select their energy provider from among multiple utility companies in any billing period. This paper thus starts by considering an oligopolistic energy market with multiple non-cooperative (competitive) utility companies, and addresses the problem of determining dynamic energy prices for every utility company in this market based on a modified Bertrand Competition Model of user behaviors. Two methods of dynamic energy pricing are proposed for a utility company to maximize its total profit. The first method finds the greatest lower bound on the total profit that can be achieved by the utility company, whereas the second method finds the best response of a utility company to dynamic pricing policies that the other companies have adopted in previous billing periods. To exploit the advantages of each method while compensating their shortcomings, an adaptive dynamic pricing policy is proposed based on a machine learning technique, which finds a good balance between invocations of the two aforesaid methods. Experimental results show that the adaptive policy results in consistently high profit for the utility company no matter what policies are employed by the other companies.

Effects of Spectrum Holdings on Equilibrium in the Wireless Industry

We propose a model of Bertrand competition in which consumers choose firms based on prices and qualities. Service quality depends on congestion, which is a function of capacity and output. We first present theoretical properties of the model. Next, we calibrate the model to the wireless industry and use it to evaluate the impacts of changes in spectrum allocation on consumer welfare and profits. Simulations of the model show that when one firm acquires more spectrum, consumer welfare at all firms increases due to congestion externality effects. We find that a transfer of spectrum from one firm to another can either raise or lower consumer welfare at the firms not involved in the transaction, again due to externality effects. Where it is possible to compare the results of our model to the wireless industry, they are consistent with the data.

Finite purchasing power and computations of Bertrand–Nash equilibrium prices

This article considers the computation of Bertrand---Nash equilibrium prices when the consumer population has finite purchasing power. The literal KKT conditions for equilibria contain "spurious" solutions that are not equilibria but can be computed by existing software, even with prominent regularization strategies for ill-posed problems. We prove a reformulated complementarity problem based on a fixed-point representation of equilibrium prices improves computational reliability and provide computational evidence of its efficiency on an empirically-relevant problem. Scientific inferences from empirical Bertrand competition models with explicit limits on individual purchasing power will benefit significantly from our proposed methods for computing equilibrium prices. An analysis of floating-point computations also implies that any model will have finite purchasing power when implemented on existing computing machines, and thus the techniques discussed here have general value. We discuss a heuristic to identify, and potentially mitigate, the impact of computationally-imposed finite purchasing power on computations of equilibrium prices in any model.