Diagonal payoff security and equilibrium existence in quasi-symmetric discontinuous games - Christian Ewerhart (University of Zurich, Department of Economics, 2022)
Previous evidence shows that better insurance coverage increases medical expenditure. However, formal studies on the effect of spending on health outcomes, and especially mental health, are lacking. To fill this gap, we reanalyze data from the Rand Health Insurance Experiment and estimate a joint non-linear model of spending and mental health. We address the endogeneity of spending in a flexible copula regression model with Bernoulli and Tweedie margins and discuss its implementation in the freely available GJRM R package. Results confirm the importance of accounting for endogeneity: in the joint model, a 1000 spending in mental care is estimated to reduce the probability of low mental health by 1.3 percentage points, but this effect is not statistically significant. Ignoring endogeneity leads to a spurious (upwardly biased) estimate.Payoff security combined with reciprocal upper semicontinuity is sufficient for better-reply security, and consequently for the existence of a pure strategy Nash equilibrium in compact, quasiconcave games by Reny's (1999) theorem. Analogously, diagonal payoff security combined with upper semicontinuity of the diagonal payoff function has been widely understood to be sufficient for diagonal better-reply security, and consequently for the existence of a symmetric pure strategy Nash equilibrium in compact, diagonally quasiconcave, quasi-symmetric games. We show by example that this is incorrect. Specifically, diagonal better-reply security may fail to hold, and a symmetric pure strategy equilibrium may fail to exist, if some player's payoff function lacks lower semicontinuity, with respect to the opponents' symmetric strategy profile, at all strategy profiles reached from a non-equilibrium profile on the diagonal by a unilateral better response of that player. These difficulties disappear, both in the game and in its mixed extension, if the lower bound on a player's payoff in the definition of diagonal payoff security is raised to reflect the higher levels that arbitrary better responses may achieve. We also discuss the relationship between our strengthened condition and diagonal payoff security.
Keywords: C62; C72; 330; discontinuous games; equilibrium existence; quasi-symmetric games; diagonal payoff security