Bicategories of Concurrent Games

This paper summarises recent results on bicategories of concurrent games and strategies. Nondeterministic concurrent strategies, those nondeterministic plays of a game left essentially unchanged by composition with copy-cat strategies, have recently been characterized as certain maps of event structures. This leads to a bicategory of general concurrent games in which the maps are nondeterministic concurrent strategies. It is shown how the bicategory can be refined to a bicategory of winning strategies by adjoining winning conditions to games. Assigning “access levels” to moves addresses situations where Player or Opponent have imperfect information as to what has occurred in the game. Finally, a bicategory of deterministic “linear” strategies, a recently discovered model of MALL (multiplicative-additive linear logic), is described. All the bicategories become equivalent to simpler order-enriched categories when restricted to deterministic strategies.