The determinacy of Blackwell games

Abstract

Games of infinite length and perfect information have been studied for many years. There are numerous determinacy results for these games, and there is a wide body of work on consequences of their determinacy.Except for games with very special payoff functions, games of infinite length and imperfect information have been little studied. In 1969, David Blackwell [1] introduced a class of such games and proved a determinacy theorem for a subclass. During the intervening time, there has not been much progress in proving the determinacy of Blackwell's games. Orkin [17] extended Blackwell's result to a slightly wider class. Blackwell [2] found a new proof of his own result. Maitra and Sudderth [9, 10] improved Blackwell's result in a different direction from that of Orkin and also generalized to the case of stochastic games. Recently Vervoort [18] has obtained a substantial improvement. Nevertheless, almost all the basic questions have remained open.In this paper we associate with each Blackwell game a family of perfect information games, and we show that the (mixed strategy) determinacy of the former follows from the (pure strategy) determinacy of the latter. The complexity of the payoff function for the Blackwell game is approximately the same as the complexity of the payoff sets for the perfect information games. In particular, this means that the determinacy of Blackwell games with Borel measurable payoff functions follows from the known determinacy of perfect information games with Borel payoff sets.

Borel Determinacy of Concurrent Games

“…for finite games [13]; open games [5]; Borel games [6]; or Blackwell games [7], to mention a few particularly relevant in computer science. Whereas the determinacy theorem in [4] is a concurrent generalisation of Zermelo's determinacy theorem for finite games, the determinacy theorem in this paper generalises the Borel determinacy theorem for infinite games-from trees to event structures, i.e.…”

“…probabilistic) strategies, e.g. as in [3], are known to be determined up to some real value of accuracy; this follows from the determinacy of Blackwell games [7]-a class of imperfect-information games. Since our games require additional structure in order to model imperfectinformation [11] and indeed probability, the determinacy result in this paper does not apply directly to Blackwell games, and so neither to the concurrent games in [3]-though see [12].…”

Recursive Markov Decision Processes and Recursive Stochastic Games

“…Lastly, given instead a strategy τ ∈ Ψ 2 , let q k,·,τ (u,ex) = max σ∈Ψ1 q k,σ,τ (u,ex) , and let q * ,·,τ (u,ex) = sup σ∈Ψ1 q * ,σ,τ (u,ex) . From very general determinacy results (Martin's "Blackwell determinacy" [Mar98] is one such result which applies to all two-player zero-sum stochastic games with countable state spaces) it follows that the games M A are determined, meaning that sup σ∈Ψ1 inf τ ∈Ψ2 q * ,σ,τ (u,ex) = inf τ ∈Ψ2 sup σ∈Ψ1 q * ,σ,τ (u,ex) . Of course, finite SSGs are even memorylessly determined ([Con92]), meaning that the strategies of either player can be restricted to memoryless strategies which ignore the history prior to the current position, without harming the optimal outcome for that player.…”

On Probabilistic Alternating Simulations

“…In such a game there always exists a unique value in [0, 1], on which both players have strategies to guarantee (or infinitely approach) their best performances, regardless of the strategies played by their opponents. Such a supremum value (or infinum value, as for the antagonist) is called the value of the game [14,9]. In a probabilistic multi-player game, we let a group of players A ⊆ Σ be a single player, and A be the other, and the supremal probability for A to enforce an LTL formula ψ starting from a given state s ∈ S can be uniquely determined, as defined by…”

“…Nevertheless, the quantitative version of determinacy [14] ensures that for all LTL formulas ψ and s ∈ S, we have…”