Existence of Pure Strategy Nash Equilibrium in Bertrand-Edgeworth Oligopolies

This article is searching for necessary and sufficient conditions which are to be imposed on the demand curve to guarantee the existence of pure strategy Nash equilibrium in a Bertrand-Edgeworth game with capacity constraints.

Invariance under type morphisms: the bayesian Nash equilibrium

Ely and Peski (2006) and Friedenberg and Meier (2010) provide examples when changing the type space behind a game, taking a "bigger" type space, induces changes of Bayesian Nash Equilibria, in other words, the Bayesian Nash Equilibrium is not invariant under type morphisms. In this paper we introduce the notion of strong type morphism. Strong type morphisms are stronger than ordinary and conditional type morphisms (Ely and Peski, 2006), and we show that Bayesian Nash Equilibria are not invariant under strong type morphisms either. We present our results in a very simple, finite setting, and conclude that there is no chance to get reasonable assumptions for Bayesian Nash Equilibria to be invariant under any kind of reasonable type morphisms.

A note on the axiomatization of the Nash equilibrium correspondence

A new axiomatization of the Nash equilibrium correspondence for n-person games based on independence of irrelevant strategies is given. Using a flexible general model, it is proved that the Nash equilibrium correspondence is the only solution to satisfy the axioms of non-emptiness, weak one-person rationality, independence of irrelevant strategies and converse independence of irrelevant strategies on the class of subgames of a fixed finite n-person game which admit at least one Nash equilibrium. It is also shown that these axioms are logically independent.

Generalized correlated equilibrium for two-person games in extensive form with perfect information

A correlation scheme (leading to a special equilibrium called “soft” correlated equilibrium) is applied for two-person finite games in extensive form with perfect information. Randomization by an umpire takes place over the leaves of the game tree. At every decision point players have the choice either to follow the recommendation of the umpire blindly or freely choose any other action except the one suggested. This scheme can lead to Pareto-improved outcomes of other correlated equilibria. Computational issues of maximizing a linear function over the set of soft correlated equilibria are considered and a linear-time algorithm in terms of the number of edges in the game tree is given for a special procedure called “subgame perfect optimization”.

Three essays on game theory and bargaining

A PhD Dissertation, presented as part of the requirements for the Degree of Doctor of Philosophy from the NOVA - School of Business and Economics

The implications of mandatory low-cost fuel provision

This work studies fuel retail firms’ strategic behavior in a two-dimensional product differentiation framework. Following the mandatory provision of “low-cost” fuel we consider that capacity constraints force firms to eliminate of one the previously offered qualities. Firms play a two-stage game choosing fuel qualities from three possibilities (low-cost, medium quality and high quality fuel) and then prices having exogenous opposite locations. In the highest level of consumers’ heterogeneity, a subgame perfect Nash equilibrium exists in which firms both choose minimum quality differentiation. Consumers’ are worse off if no differentiation occurs in medium and high qualities. The effect over prices from the mandatory “low-cost” fuel law is ambiguous.

One-seller assignment markets with multi-unit demands

We consider one-seller assignment markets with multi-unit demands and prove that the associated game is buyers-submodular. Therefore the core is non-empty and it has a lattice structure which contains the allocation where every buyer receives his marginal contribution. We prove that in this kind of market, every pairwise-stable outcome is associated to a competitive equilibrium and viceversa. We study conditions under which the buyers-optimal and the seller-optimal core allocations are competitive equilibrium payoff vectors. Moreover, we characterize the markets for which the core coincidences with the set of competitive equilibria payoff vectors. When agents behave strategically, we introduce a procedure that implements the buyers-optimal core allocation as the unique subgame perfect Nash equilibrium outcome.

On the Effects of Mergers on Equilibrium Outcomes in a Common Property Renewable Asset Oligopoly

This paper examines a dynamic game of exploitation of a common pool of some renewable asset by agents that sell the result of their exploitation on an oligopolistic market. A Markov Perfect Nash Equilibrium of the game is used to analyze the effects of a merger of a subset of the agents. We study the impact of the merger on the equilibrium production strategies, on the steady states, and on the profitability of the merger for its members. We show that there exists an interval of the asset's stock such that any merger is profitable if the stock at the time the merger is formed falls within that interval. That includes mergers that are known to be unprofitable in the corresponding static equilibrium framework.

Mixed oligopoly in education

This paper studies oligopolistic competition in education markets when schools can be private and public and when the quality of education depends on ìpeer groupî e§ects. In the Örst stage of our game schools set their quality and in the second stage they Öx their tuition fees. We examine how the (subgame perfect Nash) equilibrium allocation (qualities, tuition fees and welfare) is a§ected by the presence of public schools and by their relative position in the quality range. When there are no peer group e§ects, e¢ ciency is achieved when (at least) all but one school are public. In particular in the two school case, the impact of a public school is spectacular as we go from a setting of extreme di§erentiation to an e¢ cient allocation. However, in the three school case, a single public school will lower welfare compared to the private equilibrium. We then introduce a peer group e§ect which, for any given school is determined by its student with the highest ability. These PGE do have a signiÖcant impact on the results. The mixed equilibrium is now never e¢ cient. However, welfare continues to be improved if all but one school are public. Overall, the presence of PGE reduces the e§ectiveness of public schools as regulatory tool in an otherwise private education sector.

Bargaining over strategies of non-cooperative games

We propose a bargaining process supergame over the strategies to play in a non-cooperative game. The agreement reached by players at the end of the bargaining process is the strategy profile that they will play in the original non-cooperative game. We analyze the subgame perfect equilibria of this supergame, and its implications on the original game. We discuss existence, uniqueness, and efficiency of the agreement reachable through this bargaining process. We illustrate the consequences of applying such a process to several common two-player non-cooperative games: the Prisoner’s Dilemma, the Hawk-Dove Game, the Trust Game, and the Ultimatum Game. In each of them, the proposed bargaining process gives rise to Pareto-efficient agreements that are typically different from the Nash equilibrium of the original games.

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.

Strategic interactions of urban land developers in the housing market

This thesis studies the supply side of the housing market taking into account the strategic interactions that occur between urban land developers. The thesis starts by reviewing the literature on new housing supply, concluding that there are very few studies where strategic interactions are taken into account. Next, we develop a model with two urban land developers, who rst decide the quality of housing and then compete in prices, considering that the marginal production costs depend on the housing quality. First, we analyze the price competition game and characterize the Nash equilibrium of the price game. Finally, we examine the rst stage of the game and determine numerically the subgame perfect Nash equilibrium (SPNE) of the quality-price game. In the price competition game, our results show that the equilibrium price of an urban land developer is an increasing function of its own quality, while it is a non-monotonic function of the rival s quality. The behavior of the equilibrium pro ts reveals that, in general, urban land developers gain by di¤erentiating their quality. However, the urban land developer located at the Central Business District (CBD), may prefer to have the same quality than the rival when transportation costs are high by exploiting its locational advantage. The analysis of the rst stage of the game also reveals that, in general, the rms best response is to di¤erentiate their quality and that, in most cases, there are two subgame perfect Nash equilibria that involve quality di¤erentiation. However, the results depend on transportation costs and the quality valuation parameter. For small quality valuations, in equilibrium, the market is not fully covered and, if the unit transportation costs are high, only the urban land developers located at the CBD operates. For higher quality valuations, all the consumers are served. Furthermore, the equilibrium qualities and pro ts are increasing with quality valuation parameter. RESUMO: Esta tese estuda a oferta no mercado da habitação, tendo em conta as interações es- tratégicas que ocorrem entre os produtores de habitação. A tese revê a literatura sobre a oferta de habitação, concluindo que existem poucos estudos que tenham tido em conta as interações estratégicas. De seguida, desenvolvemos um modelo com dois produtores de habitação, que primeiro decidem a qualidade da habitação e depois competem em preços, considerando que os custos marginais de produção dependem da qualidade. Primeiro analisamos o jogo em preços e caracterizamos o equilíbrio de Nash. Posteriormente, ex- aminamos o primeiro estágio do jogo e determinamos numericamente o equilíbrio perfeito em todos os subjogos (SPNE) do jogo. No jogo de competição em preços, os resultados mostram que, o preço de equilíbrio, é uma função crescente da qualidade da habitação, sendo uma função não monótona da qualidade do rival. O lucro de equilíbrio revela que, geralmente, os produtores de habitação têm ganhos em diferenciar a qualidade. No entanto, o produtor localizado no Centro (CBD), pode preferir oferecer a mesma qualidade que o rival, caso os custos unitários de transporte sejam elevados, através da sua vantagem de localização. A análise do primeiro estágio do jogo, revela que, geralmente, a melhor resposta de um produtor é a de diferenciar a qualidade. Na maior parte dos casos existem dois SPNE que envolvem essa diferenciação. No entanto, os resultados dependem dos custos unitários de transporte e da valorização da qualidade por parte do consumidor. Para uma reduzida valorização da qualidade, em equilíbrio, o mercado não é totalmente coberto e, se o custo unitário de transporte é elevado, apenas o produtor localizado no CBD opera no mercado. Para uma valorização elevada da qualidade, todos os consumidores são servidos. Além disso, as qualidades e os lucros de equilíbrio são crescentes com a valorização da qualidade.

The stability of the equilibrium outcomes in the admission games induced by stable matching rules

A stable matching rule is used as the outcome function for the Admission game where colleges behave straightforwardly and the students` strategies are given by their preferences over the colleges. We show that the college-optimal stable matching rule implements the set of stable matchings via the Nash equilibrium (NE) concept. For any other stable matching rule the strategic behavior of the students may lead to outcomes that are not stable under the true preferences. We then introduce uncertainty about the matching selected and prove that the natural solution concept is that of NE in the strong sense. A general result shows that the random stable matching rule, as well as any stable matching rule, implements the set of stable matchings via NE in the strong sense. Precise answers are given to the strategic questions raised.

Cournot model with investments to change the market size

We present a new deterministic dynamical model on the market size of Cournot competitions, based on Nash equilibria of R&D investment strategies to increase the size of the market of the firms at every period of the game. We compute the unique Nash equilibrium for the second subgame and the profit functions for both firms. Adding uncertainty to the R&D investment strategies, we get a new stochastic dynamical model and we analyse the importance of the uncertainty to reverse the initial advantage of one firm with respect to the other.

Efficient bidding with externalities

We implement a family of efficient proposals to share benefits generated in environments with externalities. These proposals extend the Shapley value to games with externalities and are parametrized through the method by which the externalities are averaged. We construct two slightly different mechanisms: one for environments with negative externalities and the other for positive externalities. We show that the subgame perfect equilibrium outcomes of these mechanisms coincide with the sharing proposals.