Endogenous timing in a mixed duopoly

This paper addresses the issue of endogenizing the equilibrium solution when a private - domestic or foreign - firm competes in the quantities with a public, welfare maximizing firm. Theoretical literature on mixed oligopolies, in fact, provides results and policy implications that crucially rely on the notion of equilibrium assumed, either sequential or simultaneous. In the framework of the endogenous timing model of Hamilton and Slutsky (1990), we show that simultaneous play never emerges as the equilibrium of mixed duopoly games. We provide sufficient conditions for the emergence of public and/or private leadership equilibria. These results are in sharp contrast with those obtained in private duopoly games in which simultaneous play is the general result. We show that the key difference lies in the fact that the objective of a welfare maximizing firm is generally increasing in the rival's output, while the contrary holds for private firms. We develop a comprehensive analysis of a mixed duopoly considering both the cases of domestic and international competition, and the possible strategic complementarity and substitutability. From a methodological viewpoint we make large use of the basic results of the theory of supermodular games in order to avoid extraneous assumptions such as concavity, existence and uniqueness of the equilibria

Endougenous Timing in a Mixed Duopoly

This paper applies the framework of endogenous timing in games to mixed quantity duopoly, wherein a private – domestic or foreign – firm competes with a public, welfare maximizing firm. We show that simultaneous play never emerges as a subgame-perfect equilibrium of the extended game, in sharp contrast to private duopoly games. We provide sufficient conditions for the emergence of public and/or private leadership equilibrium. In all cases, private profits and social welfare are higher than under the corresponding Cournot equilibrium. From a methodological viewpoint we make extensive use of the basic results from the theory of supermodular games in order to avoid common extraneous assumptions such as concavity, existence and uniqueness of the different equilibria, whenever possible. Some policy implications are drawn, in particular those relating to the merits of privatization.Mixed markets, endogenous timing, Cournot equilibrium, Stackelberg equilibrium, privatization.

Mixed duopolies with advance production

Production to order and production in advance has been compared in many frameworks. In this paper we investigate a mixed production in advance version of the capacity-constrained Bertrand-Edgeworth duopoly game and determine the solution of the respective timing game. We show that a pure-strategy (subgame-perfect) Nash-equilibrium point exists for all possible orderings of moves. It is pointed out that unlike the production-to-order case, the equilibrium of the timing game lies at simultaneous moves. An analysis of the public firm's impact on social welfare is also carried out. All the results are compared to those of the production-to order version of the respective game

The Cournot outcome as the result of price competition

In a homogeneous product duopoly with concave revenue and convex costs we study a two stage game in which, first, firms engage simultaneously in capacity (production) and, after production levels are made public, there is sequential price competition in the second stage. Randomizing the order of play in the price subgame, we can find: (i) that the Cournot outcome can be sustained as a pure strategy subgame perfect Nash equilibrium (SPNE) of the whole game, (ii) a SPNE in which firms produce strictly more than the Cournot outcome.subgame perfect Nash equilibrium ; price ; quantity ; pure strategies ; Cournot

Lead, Follow or Cooperate? Endogenous Timing & Cooperation in Symmetric Duopoly Games.

The aim of this paper is to extend Hamilton and Slutsky's (1990) endogenous timing game by including the possibility for players to cooperate. At an initial stage players are assumed to announce both their purpose to play early or late a given duopoly game as well as their intention to cooperate or not with their rival. The cooperation and timing formation rule is rather simple: when both players agree to cooperate and play with a given timing, they end up playing their actions coordinately and simultaneously. Otherwise, they play as singletons with the timing as prescribed by their own announcement. We check for the existence of a subgame perfect Nash equilibrium (in pure strategies) of such a cooperation-timing duopoly game. Two main results on the emergence of cooperation are provided. If players' actions in the symmetric duopoly game are strategic substitutes and there is no discount, cooperating early is a subgame perfect equilibrium of the extended timing-cooperation game. Conversely, cooperating late (at period two) represents an equilibrium when players' strategies are strategic complements. Other equilibria are also possible. Most importantly, our model shows that, in general, the success of cooperation is a¤ected by the endogenous timing of the game. Moreover, the slope of players' best-replies appears crucial both for the success of cooperation as well as for the players' choice of sequencing their market actions.Endogenous Timing, Cooperation

Learning Convex Partitions and Computing Game-theoretic Equilibria from Best Response Queries

Suppose that an $m$-simplex is partitioned into $n$ convex regions having disjoint interiors and distinct labels, and we may learn the label of any point by querying it. The learning objective is to know, for any point in the simplex, a label that occurs within some distance $\epsilon$ from that point. We present two algorithms for this task: Constant-Dimension Generalised Binary Search (CD-GBS), which for constant $m$ uses $poly(n, \log \left( \frac{1}{\epsilon} \right))$ queries, and Constant-Region Generalised Binary Search (CR-GBS), which uses CD-GBS as a subroutine and for constant $n$ uses $poly(m, \log \left( \frac{1}{\epsilon} \right))$ queries. We show via Kakutani's fixed-point theorem that these algorithms provide bounds on the best-response query complexity of computing approximate well-supported equilibria of bimatrix games in which one of the players has a constant number of pure strategies. We also partially extend our results to games with multiple players, establishing further query complexity bounds for computing approximate well-supported equilibria in this setting.Comment: 38 pages, 7 figures, second version strengthens lower bound in Theorem 6, adds footnotes with additional comments and fixes typo

A Note on the Equivalence of Rationalizability Concepts in Generalized Nice Games

Moulin (1984) describes the class of nice games for which the solution concept of point-rationalizability coincides with iterated elimination of strongly dominated strategies. As a consequence nice games have the desirable property that all rationalizability concepts determine the same strategic solution. However, nice games are characterized by rather strong assumptions. For example, only single-valued best responses are admitted and the individual strategy sets have to be convex and compact subsets of the real line R1. This note shows that equivalence of all rationalizability concepts can be extended to multi-valued best response correspondences. The surprising finding is that equivalence does not hold for individual strategy sets that are compact and convex subsets of Rn with n>1.

Endogenous choice of decision variables

In this paper we allow the firms to choose their prices and quantities simultaneously. Quantities are produced in advance and their common sales price is determined by the market. Firms offer their “residual capacities” at their announced prices and the corresponding demand will be served to order. If all firms have small capacities, we obtain the Bertrand solution; while if at least one firm has a sufficiently large capacity, the Cournot outcome and a model of price leadership could emerge

Ambiguity and Social Interaction

We present a non-technical account of ambiguity in strategic games and show how it may be applied to economics and social sciences. Optimistic and pessimistic responses to ambiguity are formally modelled. We show that pessimism has the effect of increasing (decreasing) equilibrium prices under Cournot (Bertrand) competition. In addition the effects of ambiguity on peace-making are examined. It is shown that ambiguity may select equilibria in coordination games with multiple equilibria. Some comparative statics results are derived for the impact of ambiguity in games with strategic complements