Partial monotonicity of entropy revisited

It is a classical and interesting problem to find a Nash equilibrium of noncooperative games in the strategic form. It is well known that the game always has a mixed-strategy Nash equilibrium, but it does not necessarily have a pure-strategy Nash equilibrium. Takeshita and Kawasaki proved that every noncooperative partially monotone game has a pure-strategy Nash equilibrium, that is, the partial monotonicity is a sufficient condition for a noncooperative game to have a pure-strategy Nash equilibrium. In this paper, we prove the necessary and sufficient condition for a noncooperative [Formula: see text]-person game with [Formula: see text] to be partially monotone. This result is an improvement of Takeshita and Kawasaki’s result.

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ON POSITIVITY OF SEVERAL COMPONENTS OF SOLUTION VECTOR FOR SYSTEMS OF LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

Glasgow Mathematical Journal ◽

10.1017/s0017089509990218 ◽

2009 ◽

Vol 52 (1) ◽

pp. 115-136 ◽

Cited By ~ 10

Author(s):

RAVI P. AGARWAL ◽

ALEXANDER DOMOSHNITSKY

Keyword(s):

Differential Equations ◽

Functional Differential Equations ◽

Sufficient Conditions ◽

Main Idea ◽

Cauchy Matrix ◽

Solution Vector ◽

Green’S Matrix ◽

Functional Differential ◽

Linear Differential ◽

Partial Monotonicity

AbstractIn the classical theorems about lower and upper vector functions for systems of linear differential equations very heavy restrictions on the signs of coefficients are assumed. These restrictions in many cases become necessary if we wish to compare all the components of a solution vector. The formulas of the integral representation of the general solution explain that these theorems claim actually the positivity of all elements of the Green's matrix. In this paper we define a principle of partial monotonicity (comparison of only several components of the solution vector), which assumes only the positivity of elements in a corresponding row of the Green's matrix. The main theorem of the paper claims the equivalence of positivity of all elements in the nth row of the Green's matrices of the initial and two other problems, non-oscillation of the nth component of the solution vector and a corresponding assertion about differential inequality of the de La Vallee Poussin type. Necessary and sufficient conditions of the partial monotonicity are obtained. It is demonstrated that our sufficient tests of positivity of the elements in the nth row of the Cauchy matrix are exact in corresponding cases. The main idea in our approach is a construction of an equation for the nth component of the solution vector. In this sense we can say that an analog of the classical Gauss method for solving systems of functional differential equations is proposed in the paper.

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