Corrigendum to: Product Quality and Product Compatibility in Network Industries

Using an appropriate game-theoretic approach, this article develops a non-cooperative two-stage game in a Cournot duopolistic network industry in which firms strategically choose whether to produce compatible goods in the first decision-making stage. Quality differentiation affects the sub-game perfect Nash equilibrium (SPNE): (i) the network effect acts differently between low- and high-quality firms, depending on their compatibility choice; (ii) if the network externality is positive (resp. negative), to produce compatible (resp. incompatible) goods is the unique SPNE; however, this equilibrium configuration leads the high-quality firm to be worse off; (iii) there is room for a side payment from the high- to the low-quality firm to deviate toward incompatibility (resp. compatibility) under positive (resp. negative) network externality. This payment represents a Pareto improvement on the firm side but not from a societal perspective, as consumers would be worse off. The article also pinpoints the social welfare outcomes corresponding to the SPNE.

This article considers a product’s compatibility as a strategic variable in a Cournot duopoly with network consumption externalities. It develops a non-cooperative compatibility decision game (CDG) in which firms choose whether to let products be (in)compatible. With costless compatibility, the unique (Pareto-efficient) sub-game perfect Nash equilibrium (SPNE) of the CDG is universal compatibility. With quasi-fixed compatibility costs, the SPNE depends on whether product compatibility is an endogenous (i.e., a profit-maximising) or exogenous (i.e., given by technical constraints) variable. In the case of endogenous compatibility, the SPNE can vary from one unique to multiple regimes of (in)compatibility, such as the anti-prisoner’s dilemma (deadlock), prisoner’s dilemma, and coordination game. In the case of exogenous compatibility, the SPNE can become an anti-coordination game as well. The article also discusses the welfare outcomes corresponding to the SPNE.

Multi-Agent resource allocation has also become a research hotspot in the field of artificial intelligence in recent years. A lot of related work aims to design a procedural resource allocation system with execution efficiency. This paper Studied a parallel allocation method for allocating indivisible items between agents in which every agent asks for an object among those that re-main in every round, and every reported item will be allocated randomly to an agent reporting it. In order to depicts the entire distribution process, the authors propose a method of constructing a game tree of parallel allocation. Study the corresponding relationship between the total number of branches of the game tree and items. In the last part, prof the parallel allocation mechanism is dynamic game and show that the subgame perfect Nash equilibrium (SPNE) can be found by backward introduction the game trees. Provides a method to compute the unique subgame perfect Nash equilibrium in linear time.

Ogawa (Int Tax Public Financ 20(3):474–484, 2013) discusses the analysis of Kempf and Rota-Graziosi (J Public Econ 94(9–10):768–776, 2010a) by taking into account capital ownership in the government’s objective functions. He establishes that the unique subgame perfect Nash equilibrium (SPNE) of the endogenous timing game corresponds to the simultaneous Nash equilibrium. This result contrasts with Kempf and Rota-Graziosi (J Public Econ 94(9–10):768–776, 2010a) who conclude to the existence of two Stackelberg outcomes at the SPNEs. Highlighting the role of plain complementarity or substitutability, we obtain conditions under which leadership still emerges at the equilibrium of the endogenous timing game when capital ownership is considered. Numerical simulations confirm this finding and display a case where plain properties are not monotone and a well-identified Stackelberg outcome is the SPNE.

Bargaining with durable offers and endogenous timing

We follow the framework in Arya and Mittendorf's 2011 Rand Journal of Economics paper but extend their analysis by investigating supplier(s)' equilibrium choices of disclosure or confidentiality regarding their contract terms with the downstream retailers. In the case of a common supplier, we find that the unique subgame perfect Nash equilibrium (SPNE) is for the supplier to choose disclosure. This private incentive is opposite to social incentive, which calls for the regulator to choose confidentiality. In the case of dedicated suppliers, however, there are multiple SPNE due to coordination issues between the suppliers. The case which maximizes social surplus – disclosure – can be supported as a SPNE, together with the case of confidentiality, which maximizes supplier profits at the cost of everyone else. Copyright © 2014 John Wiley & Sons, Ltd.

The extensive form of a game is informationally richer than the normal form since players gather information that allows them to update their subjective priors as the game progresses. For this reason, the study of rationalizability in extensive form games is more complex than the corresponding study in normal form games. There are two ways to use the added information to eliminate strategies that would not be chosen by a rational agent: backward induction and forward induction. The latter is relatively exotic (although more defensible). Backward induction, by far the most popular technique, employs the iterated elimination of weakly dominated strategies, arriving at the subgame perfect Nash equilibria—the equilibria that remain Nash equilibria in all subgames. An extensive form game is considered generic if it has a unique subgame perfect Nash equilibrium. This chapter develops the tools of modal logic and presents Robert Aumann's famous proof that common knowledge of rationality (CKR) implies backward induction. It concludes that Aumann is perfectly correct, and the real culprit is CKR itself. CKR is in fact self-contradictory when applied to extensive form games.

We investigate how the possibility of subsequently subcontracting production to each other influences rivals' initial competition for a contract or a market as a two-stage game. In its first stage, the two firms engage in price competition to supply a contract or a market. In the second stage, the firms may subcontract production to each other. It is supposed that the firms produce the identical product with the same strictly convex cost function. The incentive for subcontracting comesfrom the strictly convex production costs. A firm is obliged to supply the entire quantity demanded at its quoted price. Our analysis discloses that if the winner of the game's first stage determines the terms of the subcontract in its second stage, there exists a unique, subgame perfect Nash equilibrium (SPNE) in pure strategies in which the firms bid the same price in the first stage and both receive zero profits. On the other hand, if the loser of the game's first stage sets the terms of the subcontract in the second stage, there exists a unique SPNE in pure strategies in which the firms bid the same price in the first stage and both receive positive profits. The presence of the possibility of subcontracting supports a unique SPNE in pure strategies, even though no actual subcontracting may occur. The SPNE price is below the socially-optimal price in the first case and is above it in the second case. We also consider other modes of sharing the gains from subcontracting between the two firms, such as the Nash bargaining solution.

We consider a game of endogenous timing of sequential choice of capacity and quantity with observable delay in a mixed duopoly and a private duopoly. In mixed duopoly, we find that a simultaneous play at the capacity stage or at the quantity stage can never be supported as subgame perfect Nash equilibrium (SPNE); whereas a simultaneous play at each stage turns out to be the unique SPNE in a private duopoly. In mixed duopoly there is multiplicity of equilibria and all SPNEs require sequentiality at the capacity as well as quantity stage.

Endogenous price discrimination with asymmetric firms

This study examines the endogenous vertical structure in which each manufacturer sells its product to its exclusive retailer who sells network goods to consumers (i.e. a duopoly in the upstream market) under Bertrand competition and Cournot competition with network externalities. We show that with strong (weak) network externalities under Bertrand competition, (a) it is a dominant strategy for each manufacturer to integrate (separate) its retailer; (b) with strong network externalities, the manufacturers’ profits, consumers’ surplus and social welfare are higher under vertical integration than under vertical separation. Under Cournot competition, (a) vertical separation is a unique subgame perfect Nash equilibrium; (b) with strong network externalities, the manufacturers’ profits, consumers’ surplus and social welfare are higher under vertical separation than under vertical integration.

Incomplete stable structures in symmetric convex games

This work aims to investigate the effects of co-determination in a game-theoretic setting by considering network externalities in consumption. The received theoretical literature, so far focused only on standard (non-network) industries, showed that co-determination might emerge as the sub-game perfect Nash equilibrium (SPNE) of a non-cooperative Cournot duopoly, but this outcome generates a prisoner’s dilemma. The present research shows that (positive) network externalities may solve the prisoner’s dilemma and let co-determination become a Pareto-efficient SPNE with homogeneous or heterogeneous products. In a network industry, co-determination becomes a Pareto-superior institution for society, as firms, consumers and workers are better off than under profit maximisation. These results hold for exogenous (homogeneous and heterogeneous) co-determination and endogenous co-determination.

This research revisits the pioneering work by Katz and Shapiro (Am Econom Rev 75:424–440, 1985) with network (consumption) externalities in a twofold way: first, it considers Corporate Socially Responsible (CSR), instead of profit-maximising, firms; second, it uses a game-theoretic approach and analyses the commitment decision game in which firms face the binary choice to credibly commit (C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C$$\\end{document}) or not to commit (NC\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$NC$$\\end{document}) themselves to an announced output level in the first decision-making stage. Competition at the market stage occurs à la Cournot. Results show a rich spectrum of sub-game perfect Nash equilibrium (SPNE) outcomes, ranging from the prisoner’s dilemma (self-interest and mutual benefit of output commitment conflict) to the anti-prisoner’s dilemma or deadlock (self-interest and mutual benefit of output commitment do not conflict), passing from the coordination to the anti-coordination game. These outcomes depend on the intensity of the social concern in the firm’s objective and the network size. The article also pinpoints the welfare outcomes corresponding to the SPNE and extends the analysis to a Stackelberg rivalry setting.

Price-setting merchants in a simple trade model