We introduce new stable, natural merge sort algorithms, called 2-merge sort and α-merge sort. We prove upper and lower bounds for several merge sort algorithms, including Timsort, Shiver’s sort, α-stack sorts, and our new 2-merge and α-merge sorts. The upper and lower bounds have the forms c · n logm and c · n logn for inputs of length n comprising m runs. For Timsort, we prove a lower bound of...

An example put forward by Dawkins to describe the evolution of strategies in the conflict of the sexes over parental investment is discussed by means of a simple dynamic system. It is shown that the equilibrium of the strategies is not evolutionarily stable, but equal to the time average of the endlessly oscillating strategies. The theory of games and the notion of evolutionarily stable strateg...

In evolutionary game theory, evolutionarily stable states are characterised by the folk theorem because exact solutions to the replicator equation are difficult to obtain. It is generally assumed that the folk theorem, which is the fundamental theory for non-cooperative games, defines all Nash equilibria in infinitely repeated games. Here, we prove that Nash equilibria that are not characterise...

The concept of evolutionarily stable strategies (ESS) has been central to applications of game theory in evolutionary biology, and it has also had an influence on the modern development of game theory. A regular ESS is an important refinement the ESS concept. Although there is a substantial literature on computing evolutionarily stable strategies, the precise computational complexity of determi...

Abstract Natural evolutionary strategies (NES) are a family of gradient-free black-box optimization algorithms. This study illustrates their use for the randomly initialized parameterized quantum circuits (PQCs) in region vanishing gradients. We show that using NES gradient estimator exponential decrease variance can be alleviated. implement two specific approaches, and separable NES, parameter...

Since the workshop held at BIRS in 2006 on the same topic, evolutionary game theory has continued to expand in the directions identified there as well as in several new directions. There seems to be a tendency to reach out from the founding concept of an evolutionarily stable strategy, towards related concepts such as continuously stable strategies and stochastically stable strategies, towards ...