17,000 research outputs found
Monad Transformers for Backtracking Search
This paper extends Escardo and Oliva's selection monad to the selection monad
transformer, a general monadic framework for expressing backtracking search
algorithms in Haskell. The use of the closely related continuation monad
transformer for similar purposes is also discussed, including an implementation
of a DPLL-like SAT solver with no explicit recursion. Continuing a line of work
exploring connections between selection functions and game theory, we use the
selection monad transformer with the nondeterminism monad to obtain an
intuitive notion of backward induction for a certain class of nondeterministic
games.Comment: In Proceedings MSFP 2014, arXiv:1406.153
The game semantics of game theory
We use a reformulation of compositional game theory to reunite game theory
with game semantics, by viewing an open game as the System and its choice of
contexts as the Environment. Specifically, the system is jointly controlled by
noncooperative players, each independently optimising a real-valued
payoff. The goal of the system is to play a Nash equilibrium, and the goal of
the environment is to prevent it. The key to this is the realisation that
lenses (from functional programming) form a dialectica category, which have an
existing game-semantic interpretation.
In the second half of this paper, we apply these ideas to build a compact
closed category of `computable open games' by replacing the underlying
dialectica category with a wave-style geometry of interaction category,
specifically the Int-construction applied to the cartesian monoidal category of
directed-complete partial orders
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
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
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