An evolutionary game is introduced which considers game-theoretic strategies in the context of non-linear population matrix models. This game considers the states and actions of the organisms of the evolving population, and a notion of dynamic equilibrium between strategies is described. The game’s formalism is expounded and a proof about equilibrium is given; specifically that any stable equilibrium can be described by proportions of pure strategies; particularly when population matrices are not defective.

We show how some concentration inequalities for sampling without replacement can be used for bounding future samples. This process can be extended to bound the sum of future samples from multiple populations, and we analyse an illustrative sample allocation problem.

Upper confidence bound multi-armed bandit algorithms (UCB) typically rely on concentration in- equalities (such as Hoeffding’s inequality) for the creation of the upper confidence bound. Intu- itively, the tighter the bound is, the more likely the respective arm is or isn’t judged appropriately for selection. Hence we derive and utilise an optimal inequality. Usually the sample mean (and sometimes the sample variance) of previous rewards are the information which are used in the bounds which drive the algorithm, but intuitively the more infor- mation that taken from the previous rewards, the tighter the bound could be. Hence our inequality explicitly considers the values of each and every past reward into the upper bound expression which drives the method. We show how this UCB method fits into the broader scope of other information theoretic UCB algorithms, but unlike them is free from assumptions about the distribution of the data, We conclude by reporting some already established regret information, and give some numerical simulations to demonstrate the method’s effectiveness.

We derive a concentration inequality for the uncertainty in stratied random sampling. Minimising this inequality leads to an iterated online method for choosing samples from the strata. The inequality is versatile and considers a range of factors including: the data ranges, weights, sizes of the strata, as well as the number of samples taken, the estimated sample variances and whether strata are sampled with or without replacement. We evaluate the improvement this method reliably offers against other methods over sets of synthetic data, and also in approximating the Shapley value of cooperative games. The method is seen to be competitive with the performance of perfect Neyman sampling, even without prior information on strata variances. We supply a multidimensional extension of our inequality and discuss some future applications.