A compositional treatment of iterated open games

Abstract

Compositional Game Theory is a new, recently introduced model of economic games based upon the computer science idea of compositionality. In it, complex and irregular games can be built up from smaller and simpler games, and the equilibria of these complex games can be defined recursively from the equilibria of their simpler subgames. This paper extends the model by providing a final coalgebra semantics for infinite games. In the course of this, we introduce a new operator on games to model the economic concept of subgame perfection.

Morphisms of Open Games

“…It is expected that all reasonable definitions of general morphisms between open games will agree on the globular morphisms. However it is demonstrated by example, both in this paper and in [7], that the more general morphisms are necessary. The morphisms we define in this paper are very different to those in [7], but indeed agree on the globular morphisms.…”

“…Having done the work of representing open games in terms of lenses, this definition is almost automatic, but with one crucial twist: the lenses s(α) and t(α) go in the opposite direction to α. If they were covariant then the resulting definition would be similar to that of [7] and have many features in common, including the elegant representation of repeated games. However, we will demonstrate in this paper that the alternative definition has several attractive features.…”

“…A direct solution is given in [7] by defining a 'subgame perfection operator' that modifies an open game's solution concept from Nash to subgame perfect equilibrium. From a practical perspective their solution is similar to the one in this paper, offering flexibility between Nash and subgame perfect equilibrium.…”

“…An alternative definition of morphisms between open games is given in [7], in order to characterise repeated games as final coalgebras of a functor prepending one additional stage. Contravariant lens morphisms are inferior for this purpose, but on the other hand, the two attractive properties of contravariant lens morphisms identified are not shared by the morphisms of [7]. This suggests that there is no single, canonical notion of morphisms between open games, but at least two with different useful properties.…”

Compositional Game Theory with Mixed Strategies: Probabilistic Open Games Using a Distributive Law

“…The second game G 2 : ({σ E , σ NE }, R × R) −→ (Y ×Y, R × R) arises as a "subgame conditioned" game [4,Def. 5] in order to allow the strategies Σ 2 = Y → Σ 1 to depend on the move made in G 1 .…”