Intro to game theory Compositionality Open games Introducing open games Jules Hedges Joint work with Neil Ghani Viktor Winschel Philipp Zahn University of Oxford February 24, 2017
Intro to game theory Compositionality Open games Motivation: powerful machinery vs. hard problems
Intro to game theory Compositionality Open games What game theory is Mathematical theory of interacting “rational” agents
Intro to game theory Compositionality Open games What game theory is Mathematical theory of interacting “rational” agents Players make observations and then make choices
Intro to game theory Compositionality Open games What game theory is Mathematical theory of interacting “rational” agents Players make observations and then make choices Group choices determine payoffs
Intro to game theory Compositionality Open games What game theory is Mathematical theory of interacting “rational” agents Players make observations and then make choices Group choices determine payoffs “Local view” of rationality: players act to maximise payoff
Intro to game theory Compositionality Open games What game theory is Mathematical theory of interacting “rational” agents Players make observations and then make choices Group choices determine payoffs “Local view” of rationality: players act to maximise payoff “Global view”: equilibrium strategies
Intro to game theory Compositionality Open games Example: penalty shootout a , b ∈ { L , R }
Intro to game theory Compositionality Open games Example: penalty shootout a , b ∈ { L , R } � (+1 , − 1) if a � = b π ( a , b ) = ( − 1 , +1) if a = b
Intro to game theory Compositionality Open games Example: penalty shootout a , b ∈ { L , R } � (+1 , − 1) if a � = b π ( a , b ) = ( − 1 , +1) if a = b Unique (probabilistic) equilibrium: a = b = 1 2 | L � + 1 2 | R �
Intro to game theory Compositionality Open games Example: penalty shootout a , b ∈ { L , R } � (+1 , − 1) if a � = b π ( a , b ) = ( − 1 , +1) if a = b Unique (probabilistic) equilibrium: a = b = 1 2 | L � + 1 2 | R � Nash’s theorem generalises this situation
Intro to game theory Compositionality Open games Example: the $$$ auction
Intro to game theory Compositionality Open games Game theory has some issues Well known: equilibrium as behavioural prediction is experimentally falsified (e.g. ultimatum game)
Intro to game theory Compositionality Open games Game theory has some issues Well known: equilibrium as behavioural prediction is experimentally falsified (e.g. ultimatum game) Harsanyi type spaces are accurate but underfit (and mathematically hard!)
Intro to game theory Compositionality Open games Game theory has some issues Well known: equilibrium as behavioural prediction is experimentally falsified (e.g. ultimatum game) Harsanyi type spaces are accurate but underfit (and mathematically hard!) There is no accepted operational theory (or “equilibriating process”) (c.f. evolutionary game theory)
Intro to game theory Compositionality Open games Game theory has some issues Well known: equilibrium as behavioural prediction is experimentally falsified (e.g. ultimatum game) Harsanyi type spaces are accurate but underfit (and mathematically hard!) There is no accepted operational theory (or “equilibriating process”) (c.f. evolutionary game theory) Serious computability/complexity issues (algorithmic game theory)
Intro to game theory Compositionality Open games Game theory has some issues Well known: equilibrium as behavioural prediction is experimentally falsified (e.g. ultimatum game) Harsanyi type spaces are accurate but underfit (and mathematically hard!) There is no accepted operational theory (or “equilibriating process”) (c.f. evolutionary game theory) Serious computability/complexity issues (algorithmic game theory) Ordinary games do not compose/scale
Intro to game theory Compositionality Open games The fundamental headache of social science Beliefs have causal effects
Intro to game theory Compositionality Open games What is compositionality? A theory is compositional if:
Intro to game theory Compositionality Open games What is compositionality? A theory is compositional if: Objects of interest are black boxes with interfaces
Intro to game theory Compositionality Open games What is compositionality? A theory is compositional if: Objects of interest are black boxes with interfaces Objects can be composed without knowing how they are defined
Intro to game theory Compositionality Open games What is compositionality? A theory is compositional if: Objects of interest are black boxes with interfaces Objects can be composed without knowing how they are defined Consequence: can easily change one component in a large structure
Intro to game theory Compositionality Open games What is compositionality? A theory is compositional if: Objects of interest are black boxes with interfaces Objects can be composed without knowing how they are defined Consequence: can easily change one component in a large structure All reasoning is by structural induction on composition
Intro to game theory Compositionality Open games Examples of compositional systems Any serious programming language
Intro to game theory Compositionality Open games Examples of compositional systems Any serious programming language Internal reasoning in a logic, algebraic structure, category, etc.
Intro to game theory Compositionality Open games Examples of compositional systems Any serious programming language Internal reasoning in a logic, algebraic structure, category, etc. Most systems designed by human engineers
Intro to game theory Compositionality Open games Examples of compositional systems Any serious programming language Internal reasoning in a logic, algebraic structure, category, etc. Most systems designed by human engineers Organisations made of people (e.g. a company)
Intro to game theory Compositionality Open games Examples of compositional systems Any serious programming language Internal reasoning in a logic, algebraic structure, category, etc. Most systems designed by human engineers Organisations made of people (e.g. a company) Non-examples: Analytic solution of differential equations
Intro to game theory Compositionality Open games Examples of compositional systems Any serious programming language Internal reasoning in a logic, algebraic structure, category, etc. Most systems designed by human engineers Organisations made of people (e.g. a company) Non-examples: Analytic solution of differential equations Natural (biological) systems
Intro to game theory Compositionality Open games Examples of compositional systems Any serious programming language Internal reasoning in a logic, algebraic structure, category, etc. Most systems designed by human engineers Organisations made of people (e.g. a company) Non-examples: Analytic solution of differential equations Natural (biological) systems Economic systems
Intro to game theory Compositionality Open games The compositionality hypothesis If a theory is compositional, then reasoning with it is scalable
Intro to game theory Compositionality Open games The compositionality hypothesis If a theory is compositional, then reasoning with it is scalable Hypothesis Compositionality is the only way to be scalable
Intro to game theory Compositionality Open games Compositionality is delicate Compositionality is not all-or-nothing
Intro to game theory Compositionality Open games Compositionality is delicate Compositionality is not all-or-nothing In programming languages it is harmed by: Goto Mutable global state Inheritance Type classes a la Haskell
Intro to game theory Compositionality Open games Compositionality is delicate Compositionality is not all-or-nothing In programming languages it is harmed by: Goto Mutable global state Inheritance Type classes a la Haskell More generally: Leaky abstractions Emergent behaviour
Intro to game theory Compositionality Open games Compositionality via symmetric monoidal categories (1) Objects (aka. interfaces, types, systems) X X
Intro to game theory Compositionality Open games Compositionality via symmetric monoidal categories (1) Objects (aka. interfaces, types, systems) X X Morphisms (aka. black boxes, processes) f : X → Y Y f X
Intro to game theory Compositionality Open games Compositionality via symmetric monoidal categories (1) Compound object X 1 ⊗ X 2 Objects (aka. interfaces, types, systems) X X 1 X 2 X Morphisms (aka. black boxes, processes) f : X → Y Y f X
Intro to game theory Compositionality Open games Compositionality via symmetric monoidal categories (1) Compound object X 1 ⊗ X 2 Objects (aka. interfaces, types, systems) X X 1 X 2 X Monoidal product (aka. tensor product, simultaneous/spatial Morphisms (aka. black boxes, composition) processes) f : X → Y f 1 ⊗ f 2 : X 1 ⊗ X 2 → Y 1 ⊗ Y 2 Y Y 1 Y 2 f f 1 f 2 X X 1 X 2
Intro to game theory Compositionality Open games Compositionality via symmetric monoidal categories (2) Categorical composition (aka. sequential/temporal composition) g ◦ f : X → Z Z g Y f X
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