Sion's mini-max theorem and Nash equilibrium in a multi-players game with two groups which is zero-sum and symmetric in each group

We consider the relation between Sion's minimax theorem for a continuous function and a Nash equilibrium in a multi-players game with two groups which is zero-sum and symmetric in each group. We will show the following results. 1. The existence of Nash equilibrium which is symmetric in each group implies Sion's minimax theorem with the coincidence of the maximin strategy and the minimax strategy for players in each group. %given the values of the strategic variables. 2. Sion's minimax theorem with the coincidence of the maximin strategy and the minimax strategy for players in each group implies the existence of a Nash equilibrium which is symmetric in each group. Thus, they are equivalent. An example of such a game is a relative profit maximization game in each group under oligopoly with two groups such that firms in each group have the same cost functions and maximize their relative profits in each group, and the demand functions are symmetric for the firms in each group.Comment: 14 page

Equivalence of Cournot and Bertrand equilibria in duopoly under relative profit maximization: A general analysis

Abstract. We study the relationship between Cournot equilibrium and Bertrand equilibrium in duopoly with differentiated goods in which each firm maximizes its relative profit. We show that Cournot equilibrium and Bertrand equilibrium coincide under relative profit maximization even with general demand and cost functions. This result is due to the fact that a game of relative profit maximization in duopoly is a two-person zero-sum game.Keywords. Relative profit maximization, Duopoly, Cournot equilibrium, Bertrand equilibrium.JEL. D43, L13

Synthesis, properties and reactivity of an ortho-phenylene-cyclopentene-bridged tetrapyrrole

Increasing the number of meso-methine carbons of porphyrin has led to a creation of a series of vinylogous expanded porphyrins, while the introduction of an ortho-phenylene-unit as a pyrrole-connecting two-carbon bridge usually leads to prevention of effective macrocyclic conjugation. Cyclopentene can serve as a conjugative bridge to increase the macrocyclic conjugation owing to its cis-geometry. In this work, ortho-phenylene-cyclopentene-bridged tetrapyrrole 5 was prepared on the basis of a coupling strategy. The tetrapyrrole 5 exhibited slightly more conjugative features as compared to ortho-phenylene-bridged tetrapyrrole 4. Oxidation of 5 with [bis(trifluoroacetoxy)iodo]benzene (PIFA) at low temperature afforded a partly fused tetrapyrrolic compound 9 having a spiro-connected pyrrolo[2, 1-a]isoindole moiety

Diazadimethano[8]circulene: Synthesis, structure, properties, and isolation of stable radical cation

Hetero[8]circulenes have emerged as novel heteroatom-doped polycyclic aromatic hydrocarbons whose properties depend on the constituent aromatic units. Herein we report a C-doped variant, diazadimethano[8]circulene 3, in which two diphenylcyclopentadiene units are installed into the core of [8]circulene that may prevent effective conjugation. The structure of 3 has been revealed to have two longer C–C bonds in the central eight-membered ring, while the absorption and emission profiles are quite similar to those of tetraaza[8]circulene. Stable radical cation 3⁺ was easily obtained by facile oxidation of 3. X-ray diffraction analysis of 3⁺ showed a slipped dimer arrangement with negligible intermolecular interaction. Interestingly, the lowest-energy absorption of 3⁺ reaches around 2500 nm, while that of 3 is 447 nm

Relative profit maximization and equivalence of Cournot and Bertrand equilibria in asymmetric duopoly

We study the relation between a Cournot equilibrium and a Bertrand equilibrium in an \emph{asymmetric} duopoly with differentiated goods in which each firm maximizes its relative profit that is the difference between its profit and the profit of the rival firm. Both demand and cost functions are linear but asymmetric, that is, demand functions for the goods are asymmetric and the firms have different marginal cots. We will show that a Cournot equilibrium and a Bertrand equilibrium coincide even in an asymmetric duopoly

Two person zero-sum game with two sets of strategic variables

We consider a two-person zero-sum game with two sets of strategic variables which are related by invertible functions. They are denoted by (sA, sB) and (tA, tB) for players A and B. We will show that the following four patterns of competition are equivalent, that is, they yield the same outcome. 1. Player A and B choose sA and sB (competition by (sA, sB)). 2. Player A and B choose tA and tB (competition by (tA, tB)). 3. Player A and B choose tA and sB (competition by (tA, sB)). 4. Player A and B choose sA and tB (competition by (sA, tB))