A Differentiable Homotopy to Compute Nash Equilibria of n-Person Games

The literature on the computation of Nash equilibria in n-person games is dominated by simplicial methods. This paper is the first to introduce a globally convergent algorithm that fully exploits the differentiability present in the problem. It presents an everywhere differentiable homotopy to do the computations. The homotopy path can therefore be followed by several numerical techniques. Moreover, instead of computing some Nash equilibrium, the algorithm is constructed in such a way that it computes the Nash equilibrium selected by the tracing procedure of Harsanyi and Selten. As a by-product of our proofs it follows that for a generic game the tracing procedure defines an unique feasible path. The numerical performance of the algorithm is illustrated by means of several examples.microeconomics ;

Equilibrium adjustment of disequilibrium prices

We consider an exchange economy in which price rigidities are present. In the short run the non-numeraire commodities have a exible price level with respect to the numeraire commodity but their relative prices are mutually fixed. In the long run prices are assumed to be completely exible. For a given price level and fixed relative prices, markets can be equilibrated by means of quantity rationing on demand and supply. Keeping markets in equilibrium through rationing, we provide an adjustment process in prices and quantities converging from a trivial equilibrium with complete demand rationing on all non-numeraire markets to a Walrasian equilibrium. Along the path initially all relative prices are kept fixed and the price level is increased. Rationing schemes are adjusted to keep markets in equilibrium. Doing so the process reaches a short run equilibrium with only demand rationing and no rationing on the numeraire and at least one of the other commodities. The process allows for a downward price adjustment of non-rationed non-numeraire commodities and reaches a Walrasian equilibrium in the long run.Equilibrium Theory;market economy;Prices;Disequilibrium Theory;Rationing;economic theory

A Globally Convergent Algorithm to Compute Stationary Equilibria in Stochastic Games

Game theory Stochastic games computation of equilibria linear tracing procedure

Loss of treatment benefit due to low compliance with bisphosphonate therapy

Among 8,822 new female bisphosphonate users, non-compliant bisphosphonate use was associated with a 45% increased risk of osteoporotic fracture compared to compliant use (MPR ≥80%). Classifying compliance into five categories, fracture risk gradually increased with poorer compliance. These results emphasize the importance of treatment compliance in obtaining maximal treatment benefit. Introduction: Bisphosphonates are widely used to treat osteoporosis and reduce fracture risk. Low compliance is frequent and will limit treatment benefit. Methods: New female users of alendronate or risedronate between 1999-2004, aged ≥45 years were identified from PHARMO-RLS, including drug-dispensing and hospitalization data of ≥2 million residents of the Netherlands. Patients were followed until first hospitalisation for an osteoporotic fracture, death, or end of study period. Compliance with bisphosphonates during follow-up was measured over 90-day intervals using Medication Possession Ratio (MPR). The association between compliance and fracture risk was analyzed using time-dependent Cox-regression. Results: The study cohort included 8,822 new female bisphosphonate users, contributing in total 22,484 person-years of follow-up. During follow-up, 176 osteoporotic fractures occurred (excluding the first six months). Non-compliant bisphosphonate use was associated with a 45% increased fracture risk compared to compliant use (MPR ≥80%). Classifying compliance into five categories, fracture risk gradually increased with poorer compliance (p-value <0.05 for trend). A MPR <20% was associated with an 80% increased fracture risk compared to a MPR ≥90%. Conclusions: These results show a statistically significant association between level of compliance with bisphosphonates and level of fracture risk, emphasizing the importance of treatment compliance in obtaining maximal treatment benefit