WebOptimization and concave/convex functions Examples: • The functions and are concave • The function is concave on its domain • Any affine function is both concave and convex … WebJan 5, 2024 · Thus, J is a diagonal matrix with negative diagonal elements. This implies that J is negative definite. Henceforth [J+ \(J^T\)] is also negative definite, and according to Theorem (6) in, , the weighted sum of the utility functions \(\psi (q_{s},x)\) is diagonally strictly concave. Thus, the Nash equilibrium point is unique.

Proof that a strictly diagonally dominant matrix is invertible

1. A differentiable function f is (strictly) concave on an interval if and only if its derivative function f ′ is (strictly) monotonically decreasing on that interval, that is, a concave function has a non-increasing (decreasing) slope. 2. Points where concavity changes (between concave and convex) are inflection points. chloe kim 2018 olympics

game theory - Rosen

WebFeb 25, 2010 · (3), where hi is a concave function, and there exists some x˜i ∈ Rm i such that hi (x˜i) > 0. Assume also that the payoff functions (u1, ... , uI) are diagonally … WebWe say that a graph is concave up if the line between two points is above the graph, or alternatively if the first derivative is increasing. (In finance, such a curve is said to be convex.) Similarly, we say that a graph is concave down if the line between two points is below the graph, or alternatively if the first derivative is decreasing. WebSep 29, 2024 · 1. I will answer some of your questions. The entries of a Hessian matrix H of f are second partials H i j = ∂ i ∂ j f and it is a standard result in multivariable calculus that ∂ i ∂ j f = ∂ j ∂ i f provided both second partials are continuous functions. In your case, the entries of the Hessian are constants so are continuous functions. chloe kim and toby miller