The Winning Ways of Concurrent Games

Abstract

Abstract-A bicategory of concurrent games, where nondeterministic strategies are formalized as certain maps of event structures, was introduced recently. This paper studies an extension of concurrent games by winning conditions, specifying players' objectives. The introduction of winning conditions raises the question of whether such games are determined, that is, if one of the players has a winning strategy. This paper gives a positive answer to this question when the games are well-founded and satisfy a structural property, race-freedom, which prevents one player from interfering with the moves available to the other. Uncovering the conditions under which concurrent games with winning conditions are determined opens up the possibility of further applications of concurrent games in areas such as logic and verification, where both winning conditions and determinacy are most needed. A concurrent-game semantics for predicate calculus is provided as an illustration.

Borel Determinacy of Concurrent Games

“…Here we extend the result of [4] by providing a much more general determinacy theorem. We consider concurrent games in which plays may be infinite and where the winning set of configurations form a Borel set.…”

“…This situation may lead to a deadlock since, when using σ 2 τ 2 , Player/Opponent would stay waiting for Opponent/Player to play first-which may never happen. ◻ Determinacy and winning conditions: A game with winning conditions [4] comprises G = (A, W ) where A is an event structure with polarity and W ⊆ C(A) consists of the winning configurations for Player. Define the losing conditions to be L = C(A) ∖ W .…”

“…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. from total to partial orders of moves.…”

“…Games there can be thought of as highlyinteractive, distributed games between Player (thought of as a team of players) and Opponent (a team of opponents). The games model, as first introduced in [10], was extended with winning conditions in [4]. There a determinacy result was given for well-founded games (i.e.…”

“…For instance, the games in [1], [8] only allow deterministic strategies-the reason why they are not determined, cf. [4]; Games in logic and formal verification such as those in [2], [3] are not determined with respect to pure strategies. However, in games where the players are allowed to use mixed (i.e.…”