When Can Limited Randomness Be Used in Repeated Games?

The central result of classical game theory states that every finite normal form game has a Nash equilibrium, provided that players are allowed to use randomized (mixed) strategies. However, in practice, humans are known to be bad at generating random-like sequences, and true random bits may be unavailable. Even if the players have access to enough random bits for a single instance of the game their randomness might be insufficient if the game is played many times. In this work, we ask whether randomness is necessary for equilibria to exist in finitely repeated games. We show that for a large class of games containing arbitrary two-player zero-sum games, approximate Nash equilibria of the $n$-stage repeated version of the game exist if and only if both players have $\Omega(n)$ random bits. In contrast, we show that there exists a class of games for which no equilibrium exists in pure strategies, yet the $n$-stage repeated version of the game has an exact Nash equilibrium in which each player uses only a constant number of random bits. When the players are assumed to be computationally bounded, if cryptographic pseudorandom generators (or, equivalently, one-way functions) exist, then the players can base their strategies on "random-like" sequences derived from only a small number of truly random bits. We show that, in contrast, in repeated two-player zero-sum games, if pseudorandom generators \emph{do not} exist, then $\Omega(n)$ random bits remain necessary for equilibria to exist

Approximating n-player behavioural strategy nash equilibria using coevolution

Coevolutionary algorithms are plagued with a set of problems related to intransitivity that make it questionable what the end product of a coevolutionary run can achieve. With the introduction of solution concepts into coevolution, part of the issue was alleviated, however efficiently representing and achieving game theoretic solution concepts is still not a trivial task. In this paper we propose a coevolutionary algorithm that approximates behavioural strategy Nash equilibria in n-player zero sum games, by exploiting the minimax solution concept. In order to support our case we provide a set of experiments in both games of known and unknown equilibria. In the case of known equilibria, we can confirm our algorithm converges to the known solution, while in the case of unknown equilibria we can see a steady progress towards Nash. Copyright 2011 ACM

Multipoint-to-multipoint network communication

We have formulated an exact ILP model for the problem of communicating on a virtual network. While this ILP model was successful in solving small problems, it is not recommended to handle larger instances, due to the fact that the number of variables in the model grows exponentially as the graph size grows. However, this ILP model can provide a benchmark for heuristic algorithms developed for this problem. We have also described a heuristic approach, and explored several variants of the algorithm. We found a solution that seems to perform well with reasonable computation time. The heuristic is able to find solutions that respect the degree constraints, but show a small number of violations of the desired time constraints. Tests on small problems show that heuristic is not always able to find feasible solutions, even though the exact method has shown they exist. It would be interesting in the future to look at whether insights gained by looking at exact solutions can be used to improve the heuristic

Playing Smart - Artificial Intelligence in Computer Games

Abstract: With this document we will present an overview of artificial intelligence in general and artificial intelligence in the context of its use in modern computer games in particular. To this end we will firstly provide an introduction to the terminology of artificial intelligence, followed by a brief history of this field of computer science and finally we will discuss the impact which this science has had on the development of computer games. This will be further illustrated by a number of case studies, looking at how artificially intelligent behaviour has been achieved in selected games