Ordinary games The category PC Open games Examples Cool stuff Compositional game theory Jules Hedges (University of Oxford) SYCO 1, Birmingham 21 September 2018
Ordinary games The category PC Open games Examples Cool stuff A peek at where we’re going X R q X Y R Z A 1 , X A Z , Y f X R R
Ordinary games The category PC Open games Examples Cool stuff Game theory 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
Ordinary games The category PC Open games Examples Cool stuff Example: penalty shootout a , b ∈ { L , R }
Ordinary games The category PC Open games Examples Cool stuff Example: penalty shootout a , b ∈ { L , R } � (+1 , − 1) if a � = b π ( a , b ) = ( − 1 , +1) if a = b
Ordinary games The category PC Open games Examples Cool stuff 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 �
Ordinary games The category PC Open games Examples Cool stuff 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
Ordinary games The category PC Open games Examples Cool stuff Picturing game theory (1945 – 2018)
Ordinary games The category PC Open games Examples Cool stuff Game theory has some issues Well known: equilibrium as behavioural prediction is experimentally falsified (e.g. ultimatum game)
Ordinary games The category PC Open games Examples Cool stuff 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!)
Ordinary games The category PC Open games Examples Cool stuff 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)
Ordinary games The category PC Open games Examples Cool stuff 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 The category PC Open games Examples Cool stuff 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
Ordinary games The category PC Open games Examples Cool stuff The fundamental headache of social science Beliefs have causal effects
Ordinary games The category PC Open games Examples Cool stuff Defining PC PC is a category where: � X � Objects are pairs of sets S � X � Y � � Morphisms λ : → are pairs of functions: S R v λ : X → Y u λ : X × R → S λ is called a lens
Ordinary games The category PC Open games Examples Cool stuff Defining PC PC is a category where: � X � Objects are pairs of sets S � X � Y � � Morphisms λ : → are pairs of functions: S R v λ : X → Y u λ : X × R → S λ is called a lens We draw it like this: X Y λ S R
Ordinary games The category PC Open games Examples Cool stuff Intuition for PC Approximately . . . First part: physical information X and Y are sets of things an agent can observe or choose
Ordinary games The category PC Open games Examples Cool stuff Intuition for PC Approximately . . . First part: physical information X and Y are sets of things an agent can observe or choose Second part: teleological or counterfactual information R and S are sets of things an agent can optimise or have preferences about
Ordinary games The category PC Open games Examples Cool stuff Intuition for PC Approximately . . . First part: physical information X and Y are sets of things an agent can observe or choose Second part: teleological or counterfactual information R and S are sets of things an agent can optimise or have preferences about A typical example: f : X → Y is a function � X � Y � � Promote to λ : → with v λ = f R R u λ : X × R → R is backpropagation of value If we know x and we know the value of f ( x ) then u λ tells us what the value of x was
Ordinary games The category PC Open games Examples Cool stuff Example: a decision process (aka. a Markov decision process without the probability) Take a state space S , actions A , transition function f : S × A → S × R
Ordinary games The category PC Open games Examples Cool stuff Example: a decision process (aka. a Markov decision process without the probability) Take a state space S , actions A , transition function f : S × A → S × R � S � S � � Every policy function σ : S → A determines a lens λ : → R R by v λ ( s ) = f ( s , σ ( s )) 1 u λ ( s , u ) = f ( s , σ ( s )) 2 + β · u 0 < β < 1 is discount factor
Ordinary games The category PC Open games Examples Cool stuff Composing lenses Given � X � � Y � � Z � µ λ − → − → S R Q � X � Z � � we can compose them to µ ◦ λ : → S Q (Important non-obvious fact: this is associative)
Ordinary games The category PC Open games Examples Cool stuff Composing lenses Given � X � � Y � � Z � µ λ − → − → S R Q � X � Z � � we can compose them to µ ◦ λ : → S Q (Important non-obvious fact: this is associative) λ 1 λ 2 � X 1 � Y 1 � X 2 � Y 2 � � � � Given − → and − → we can compose them to S 1 R 1 S 2 R 2 � X 1 × X 2 � � Y 1 × Y 2 � λ 1 ⊗ λ 2 − − − − → S 2 × S 1 R 2 × R 1 PC is a symmetric monoidal category
Ordinary games The category PC Open games Examples Cool stuff Special lenses � 1 � 1 or f ∗ : � X � Y � � � � f : X → Y lifts to f : → → 1 1 Y X X f Y Y f X
Ordinary games The category PC Open games Examples Cool stuff Special lenses � 1 � 1 or f ∗ : � X � Y � � � � f : X → Y lifts to f : → → 1 1 Y X X f Y Y f X � 1 � X � � Special case: Every is a comonoid, every is a monoid 1 X
Ordinary games The category PC Open games Examples Cool stuff Special lenses � 1 � 1 or f ∗ : � X � Y � � � � f : X → Y lifts to f : → → 1 1 Y X X f Y Y f X � 1 � X � � Special case: Every is a comonoid, every is a monoid 1 X � X � 1 � � There is canonical ε X : → (but no η !) X 1 X X
Ordinary games The category PC Open games Examples Cool stuff The counit law Theorem: ε Y ◦ (( f , 1) ⊗ (1 , id Y )) = ε X ◦ (( id X , 1) ⊗ (1 , f )) aka: X f X = Y Y f
Ordinary games The category PC Open games Examples Cool stuff Interesting facts about PC PC is a dialectica category over a 1-valued logic hence, a sound model of linear logic
Ordinary games The category PC Open games Examples Cool stuff Interesting facts about PC PC is a dialectica category over a 1-valued logic hence, a sound model of linear logic � X � �→ X , λ �→ v λ is a fibration S It’s fibrewise opposite of Jacobs’ simple fibration
Ordinary games The category PC Open games Examples Cool stuff Interesting facts about PC PC is a dialectica category over a 1-valued logic hence, a sound model of linear logic � X � �→ X , λ �→ v λ is a fibration S It’s fibrewise opposite of Jacobs’ simple fibration Hot off the press: PC is complete (if underlying cat is complete, cocomplete, cartesian closed, . . . ) Work in progress: game theory using Span ( PC )
Ordinary games The category PC Open games Examples Cool stuff Interesting facts about PC PC is a dialectica category over a 1-valued logic hence, a sound model of linear logic � X � �→ X , λ �→ v λ is a fibration S It’s fibrewise opposite of Jacobs’ simple fibration Hot off the press: PC is complete (if underlying cat is complete, cocomplete, cartesian closed, . . . ) Work in progress: game theory using Span ( PC ) Really hot off the press: PC can be defined over a monoidal category: � A ∈C �� X � � Y �� hom PC ( C ) , = hom C ( X , A ⊗ Y ) × hom C ( A ⊗ R , S ) S R Needed for probabilistic open games etc Universal property: “freely adding counits” Mitchell Riley, Categories of Optics, arXiv
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