A century ago the pioneers in game theory John von Neumann (1903—1957) and Oskar Morgenstern (1902—1978) were born. At the age of 23 (December 7, 1926) John von Neumann presented for the Mathematical Society in Göttingen his paper 'Zur Theorie der Gesellschaftspiele', which appeared in 1928 ([1]; for an English translation, see [2]). This paper, inspired by parlour games (cf. [3]), contained al...

For a controlled stochastic differential equation with a Bolza type performance functional, a variational formula for the functional in a given control process direction is derived, by means of backward stochastic differential equations. As applications, some Pontryagin type maximum principles are established for optimal controls of control problems, for saddle points of open-loop two-person ze...

A joint derivation of utility and value for two-person zero-sum games is obtained using a decision theoretic approach. Acts map states to consequences. The latter are lotteries over prizes, and the set of states is a product of two finite sets (m rows and n columns). Preferences over acts are complete, transitive, continuous, monotonic and certainty-independent (Gilboa and Schmeidler (1989», an...

This paper considers uniformly bounded classes of non-zero-sum strategic-form games with large finite or compact action spaces. The central class considered is assumed to be defined via a semi-algebraic condition. We show that for each ɛ>0, the support size required ɛ-equilibrium can taken uniform over entire class. As corollary, value zero-sum games, as function single-variable, well-behaved i...

Counterfactual Regret Minimization (CFR) is the most popular iterative algorithm for solving zero-sum imperfect-information games. Regret-Based Pruning (RBP) is an improvement that allows poorly-performing actions to be temporarily pruned, thus speeding up CFR. We introduce Total RBP, a new form of RBP that reduces the space requirements of CFR as actions are pruned. We prove that in zero-sum g...

Inspired by previous work on information theoretical optimization problems, the basics of an axiomatic theory of certain special twoperson zero-sum games is developed. Among the two players, one – “Observer” – is imagined to have a “mind” , the other – “Nature” – not. Expressing such ideas leads to un-symmetric modeling as the two players are treated quite differently. We demonstrate that the t...

Zero-sum games are two player games played on a matrix M ∈ Matm×n(R). The row player, denoted R, chooses a row i ∈ [m] and the column player C chooses a column j ∈ [n], simultaneously. The payoff to the row player is Mij and the payoff to the column player is −Mij (hence the game is “zero-sum”). We can also consider randomized strategies, where R chooses a probability distribution p on [m], whi...

This paper proposes a novel way to compare classes of strategic games based on their sets pure Nash equilibria. approach is then used relate the zero-sum games, polymatrix, and k-polymatrix games. concludes with conjecture that form an increasing chain classes.

Moving assets through a transportation network is a crucial challenge in hostile environments such as future battlefields where malicious adversaries have strong incentives to attack vulnerable patrols and supply convoys. Intelligent agents must balance network costs with the harm that can be inflicted by adversaries who are in turn acting rationally to maximize harm while trading off against t...

Multilevel games are abstractions of situations where decision makers are distributed in a network environment. In Part I of this paper, the authors present several of the challenging problems that arise in the analysis of multilevel games. In this paper a specific set up is considered where the two games being played are zero-sum games and where the decision makers use the linear reward-inacti...