This article is a survey of Nash’s contributions to algebraic geometry, focusing on the topology of real algebraic sets and on arc spaces of singularities. Nash wrote two papers in algebraic geometry, one at the beginning of his career [Nash52] and one in 1968, after the onset of his long illness; the latter was published only much later [Nash95]. With these papers Nash was ahead of the times; ...

We study the existence and tractability of a notion of approximate equilibria in bimatrix games, called well supported approximate Nash Equilibria (SuppNE in short). We prove existence of ε−SuppNE for any constant ε ∈ (0, 1), with only logarithmic support sizes for both players. Also we propose a polynomial–time construction of SuppNE, both for win lose and for arbitrary (normalized) bimatrix g...

We study how the structure of the interaction graph of a game affects the existence of pure Nash equilibria. In particular, for a fixed interaction graph, we are interested in whether there exist pure Nash equilibria which arise when random utility tables are assigned to the players. We provide conditions for the structure of the graph under which equilibria are likely to exist and complementar...

We focus on the problem of computing approximate Nash equilibria and well-supported approximate Nash equilibria in random bimatrix games, where each player's payoffs are bounded and independent random variables, not necessarily identically distributed, but with common expectations. We show that the completely mixed uniform strategy profile, i.e. the combination of mixed strategies (one per play...

In patients with nonalcoholic steatohepatitis (NASH), the prevalence of cirrhosis is higher among women than men, and hepatocellular carcinoma (HCC) develops mainly in the cirrhotic stage among women. However, the long-term outcomes in female patients with NASH have not been fully elucidated, and age, gender and BMI were not simultaneously adjusted in previous studies on the prognosis of NASH. ...