Searcharxiv⌕ Search

arXiv subjects

Yasuhito Tanaka

Publications and source records attributed to Yasuhito Tanaka.

17 recordsLinked to original sources
arXiv2020

Involuntary unemployment in overlapping generations model due to instability of the economy

The existence of involuntary unemployment advocated by J. M. Keynes is a very important problem of the modern economic theory. Using a three-generations overlapping generations model, we show that the existence of involuntary unemployment is due to the instability of the economy. Instability of the economy is the instability of the difference equation about the equilibrium price around the full-employment equilibrium, which means that a fall in the nominal wage rate caused by the presence of involuntary unemployment further reduces employment. This instability is due to the negative real balance effect that occurs when consumers' net savings (the difference between savings and pensions) are smaller than their debt multiplied by the marginal propensity to consume from childhood consumption.

econ.GN↗
arXiv2019

Dynamic monopolistic competition with sluggish adjustment of entry and exit

We study a steady state of a free entry oligopoly with differentiated goods, that is, a monopolistic competition, with sluggish adjustment of entry and exit of firms under general demand and cost functions by a differential game approach. Mainly we show that the number of firms at the steady state in the open-loop solution of monopolistic competition is smaller than that at the static equilibrium of monopolistic competition, and that the number of firms at the steady state of the memoryless closed-loop monopolistic competition is larger than that at the steady state of the open-loop monopolistic competition, and may be larger than the number of firms at the static equilibrium.

math.OC↗
arXiv2019

A differential game analysis of R&D in oligopoly with differentiated goods under general demand and cost functions: Bertrand vs. Cournot

We study a dynamic oligopoly with differentiated goods by differential game approach under general demand and cost functions. We show that the steady state value of the R&D investment by each firm is decreasing with respect to the number of firms, and the steady state value of the industry R&D investment is increasing with respect to the number of firms. Also we show that if there is no spillover, whether the R&D investment of each firm given the cost level in the memoryless closed-loop case is larger or smaller than that in the open-loop case depends on whether the strategic variables are strategic substitutes or strategic complements. Further we show that the memoryless closed-loop solution and the feedback solution (by the Hamilton-Jacobi-Bellman equation) are equivalent.

math.OC↗
arXiv2019

Advertising in an oligopoly with differentiated goods under general demand and cost functions: A differential game approach

We present an analysis of advertising activities in a dynamic oligopoly with differentiated goods by differential game approach under general demand and cost functions. Mainly we show the following results. The comparison of the open-loop solution and that of the closed-loop solution depends on whether the outputs of the firms are strategic substitutes or strategic complements, and the memoryless closed-loop solution and the feedback solution are equivalent when there is no spillover effect of advertising activities.

math.OC↗
arXiv2018

On zero-sum game formulation of non zero-sum game

We consider a formulation of a non zero-sum n players game by an n+1 players zero-sum game. We suppose the existence of the n+1-th player in addition to n players in the main game, and virtual subsidies to the n players which is provided by the n+1-th player. Its strategic variable affects only the subsidies, and does not affect choice of strategies by the n players in the main game. His objective function is the opposite of the sum of the payoffs of the n players. We will show 1) The minimax theorem by Sion (Sion(1958)) implies the existence of Nash equilibrium in the n players non zero-sum game. 2) The maximin strategy of each player in {1, 2, \dots, n} with the minimax strategy of the n+1-th player is equivalent to the Nash equilibrium strategy of the n players non zero-sum game. 3) The existence of Nash equilibrium in the n players non zero-sum game implies Sion's minimax theorem for pairs of each of the n players and the n+1-th player.

math.OC↗
arXiv2018

Nash equilibrium in asymmetric multi-players zero-sum game with two strategic variables and only one alien

We consider a partially asymmetric multi-players zero-sum game with two strategic variables. All but one players have the same payoff functions, and one player (Player $n$) does not. Two strategic variables are $t_i$'s and $s_i$'s for each player $i$. Mainly we will show the following results. 1) The equilibrium when all players choose $t_i$'s is equivalent to the equilibrium when all but one players choose $t_i$'s and Player $n$ chooses $s_n$ as their strategic variables. 2) The equilibrium when all players choose $s_i$'s is equivalent to the equilibrium when all but one players choose $s_i$'s and Player $n$ chooses $t_n$ as their strategic variables. The equilibrium when all players choose $t_i$'s and the equilibrium when all players choose $s_i$'s are not equivalent although they are equivalent in a symmetric game in which all players have the same payoff functions.

math.OC↗
arXiv2018

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.

math.OC↗
arXiv2018

Nash equilibrium of partially asymmetric three-players zero-sum game with two strategic variables

We consider a partially asymmetric three-players zero-sum game with two strategic variables. Two players (A and B) have the same payoff functions, and Player C does not. Two strategic variables are $t_i$'s and $s_i$'s for $i=A, B, C$. Mainly we will show the following results. 1. The equilibrium when all players choose $t_i$'s is equivalent to the equilibrium when Players A and B choose $t_i$'s and Player C chooses $s_C$ as their strategic variables. 2. The equilibrium when all players choose $s_i$'s is equivalent to the equilibrium when Players A and B choose $s_i$'s and Player C chooses $t_C$ as their strategic variables. The equilibrium when all players choose $t_i$'s and the equilibrium when all players choose $s_i$'s are not equivalent although they are equivalent in a symmetric game in which all players have the same payoff functions.

econ.GN↗
arXiv2018

Sion's mini-max theorem and Nash equilibrium in a five-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 five-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 for a pair of playes in each group. 2. Sion's minimax theorem for a pair of playes in each group imply 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.

econ.GN↗
arXiv2018

Minimax theorem and Nash equilibrium of symmetric multi-players zero-sum game with two strategic variables

We consider a symmetric multi-players zero-sum game with two strategic variables. There are $n$ players, $n\geq 3$. Each player is denoted by $i$. Two strategic variables are $t_i$ and $s_i$, $i\in \{1, \dots, n\}$. They are related by invertible functions. Using the minimax theorem by \cite{sion} we will show that Nash equilibria in the following states are equivalent. 1. All players choose $t_i,\ i\in \{1, \dots, n\}$, (as their strategic variables). 2. Some players choose $t_i$'s and the other players choose $s_i$'s. 3. All players choose $s_i,\ i\in \{1, \dots, n\}$.

q-fin.MF↗
arXiv2018

On the relation between Sion's minimax theorem and existence of Nash equilibrium in asymmetric multi-players zero-sum game with only one alien

We consider the relation between Sion's minimax theorem for a continuous function and a Nash equilibrium in an asymmetric multi-players zero-sum game in which only one player is different from other players, and the game is symmetric for the other players. Then, 1. The existence of a Nash equilibrium, which is symmetric for players other than one player, implies Sion's minimax theorem for pairs of this player and one of other players with symmetry for the other players. 2. Sion's minimax theorem for pairs of one player and one of other players with symmetry for the other players implies the existence of a Nash equilibrium which is symmetric for the other players. Thus, they are equivalent.

econ.EM↗
arXiv2011

Constructive proof of the existence of equilibrium in competitive economy with sequentially locally non-constant excess demand functions

We present a constructive proof of the existence of an equilibrium in a competitive economy with sequentially locally non-constant excess demand functions. And we will show that the existence of such an equilibrium implies Sperner's lemma. Since the existence of an equilibrium is derived from the existence an approximate fixed point of uniformly continuous functions, which is derived from Sperner's lemma, the existence of an equilibrium in a competitive economy with sequentially locally non-constant excess demand functions is equivalent to Sperner's lemma.

math.LO↗
arXiv2011

Constructive proof of Brouwer's fixed point theorem for sequentially locally non-constant functions

We present a constructive proof of Brouwer's fixed point theorem for uniformly continuous and sequentially locally non-constant functions based on the existence of approximate fixed points. And we will show that Brouwer's fixed point theorem for uniformly continuous and sequentially locally non-constant functions implies Sperner's lemma for a simplex. Since the existence of approximate fixed points is derived from Sperner's lemma, our Brouwer's fixed point theorem is equivalent to Sperner's lemma.

math.LO↗