1,586 research outputs found
A compositional treatment of iterated open games
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
Compositional game theory
We introduce open games as a compositional foundation of economic game
theory. A compositional approach potentially allows methods of game theory and
theoretical computer science to be applied to large-scale economic models for
which standard economic tools are not practical. An open game represents a game
played relative to an arbitrary environment and to this end we introduce the
concept of coutility, which is the utility generated by an open game and
returned to its environment. Open games are the morphisms of a symmetric
monoidal category and can therefore be composed by categorical composition into
sequential move games and by monoidal products into simultaneous move games.
Open games can be represented by string diagrams which provide an intuitive but
formal visualisation of the information flows. We show that a variety of games
can be faithfully represented as open games in the sense of having the same
Nash equilibria and off-equilibrium best responses.Comment: This version submitted to LiCS 201
Morphisms of open games
We define a notion of morphisms between open games, exploiting a surprising
connection between lenses in computer science and compositional game theory.
This extends the more intuitively obvious definition of globular morphisms as
mappings between strategy profiles that preserve best responses, and hence in
particular preserve Nash equilibria. We construct a symmetric monoidal double
category in which the horizontal 1-cells are open games, vertical 1-morphisms
are lenses, and 2-cells are morphisms of open games. States (morphisms out of
the monoidal unit) in the vertical category give a flexible solution concept
that includes both Nash and subgame perfect equilibria. Products in the
vertical category give an external choice operator that is reminiscent of
products in game semantics, and is useful in practical examples. We illustrate
the above two features with a simple worked example from microeconomics, the
market entry game
Cooperation in Games and Epistemic Readings of Independence-Friendly Sentences
In the literature on logics of imperfect information it is often stated, incorrectly, that the Game-Theoretical Semantics of Independence-Friendly (IF) quantifiers captures the idea that the players of semantical games are forced to make some moves without knowledge of the moves of other players. We survey here the alternative semantics for IF logic that have been suggested in order to enforce this "epistemic reading" of sentences. We introduce some new proposals, and a more general logical language which distinguishes between "independence from actions" and "independence from strategies". New semantics for IF logic can be obtained by choosing embeddings of the set of IF sentences into this larger language. We compare all the semantics proposed and their purported game-theoretical justifications, and disprove a few claims that have been made in the literature.Peer reviewe
A Logic of Knowing How
In this paper, we propose a single-agent modal logic framework for reasoning
about goal-direct "knowing how" based on ideas from linguistics, philosophy,
modal logic and automated planning. We first define a modal language to express
"I know how to guarantee phi given psi" with a semantics not based on standard
epistemic models but labelled transition systems that represent the agent's
knowledge of his own abilities. A sound and complete proof system is given to
capture the valid reasoning patterns about "knowing how" where the most
important axiom suggests its compositional nature.Comment: 14 pages, a 12-page version accepted by LORI
Compositional Game Theory, compositionally
We present a new compositional approach to compositional game theory (CGT) based upon Arrows, a concept originally from functional programming, closely related to Tambara modules, and operators to build new Arrows from old. We model equilibria as a module over an Arrow and define an operator to build a new Arrow from such a module over an existing Arrow. We also model strategies as graded Arrows and define an operator which builds a new Arrow by taking the colimit of a graded Arrow. A final operator builds a graded Arrow from a graded bimodule. We use this compositional approach to CGT to show how known and previously unknown variants of open games can be proven to form symmetric monoidal categories
- …