How to make game theory compositional
Jules Hedges
feat.
Neil Ghani, Paulo Oliva, Evguenia Sprits, Viktor Winschel, Philipp Zahn
Queen Mary University of London
February 9, 2016
Jules Hedges (QMUL)
How to make game theory compositional
February 9, 2016
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A motivating rant
Jules Hedges (QMUL)
How to make game theory compositional
February 9, 2016
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A motivating rant
If you have a technique for solving a problem
Jules Hedges (QMUL)
How to make game theory compositional
February 9, 2016
2/7
A motivating rant
If you have a technique for solving a problem
in a non-compositional theory
Jules Hedges (QMUL)
How to make game theory compositional
February 9, 2016
2/7
A motivating rant
If you have a technique for solving a problem
in a non-compositional theory
you know nothing
about solving the same problem an order of magnitude bigger
Jules Hedges (QMUL)
How to make game theory compositional
February 9, 2016
2/7
A motivating rant
If you have a technique for solving a problem
in a non-compositional theory
you know nothing
about solving the same problem an order of magnitude bigger
(in the worst case)
Jules Hedges (QMUL)
How to make game theory compositional
February 9, 2016
2/7
Pregames, a.k.a. open games
G : X ⊗ S∗ → Y ⊗ R∗
Jules Hedges (QMUL)
How to make game theory compositional
February 9, 2016
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Pregames, a.k.a. open games
G : X ⊗ S∗ → Y ⊗ R∗
X
Y
G
S
Jules Hedges (QMUL)
R
How to make game theory compositional
February 9, 2016
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Pregames, a.k.a. open games
G = (ΣG , PG , CG , BG )
G : X ⊗ S∗ → Y ⊗ R∗
X
Y
G
S
Jules Hedges (QMUL)
R
How to make game theory compositional
February 9, 2016
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Pregames, a.k.a. open games
G = (ΣG , PG , CG , BG )
G : X ⊗ S∗ → Y ⊗ R∗
X
Y
ΣG : Set
PG : ΣG × X → Y
CG : ΣG × X × R → S
G
BG : X × (Y → R)
S
Jules Hedges (QMUL)
R
× ΣG → PΣG
How to make game theory compositional
February 9, 2016
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Sequential composition
G = (ΣG , PG , CG , BG )
H = (ΣH , PH , CH , BH )
Y
X
G
T
Jules Hedges (QMUL)
Z
H
S
How to make game theory compositional
R
February 9, 2016
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Sequential composition
G = (ΣG , PG , CG , BG )
H = (ΣH , PH , CH , BH )
Y
X
G
T
Z
H
R
S
ΣH◦G = ΣG × ΣH
···
BH◦G (h, k, (σ, τ )) = BG (h, kτ ◦ , σ) ×
\
BH (PG (σ 0 , x), k, τ )
σ 0 :ΣG
where kτ ◦ y = CH (τ, y , k(PH (τ, k)))
Jules Hedges (QMUL)
How to make game theory compositional
February 9, 2016
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Decision problem
P
π
1
1
I −→
X ⊗ R∗ −→
I
X
π1
P1
R
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How to make game theory compositional
February 9, 2016
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2-player simultaneous
P ⊗P
π
2
2
I −−1−−→
X ⊗ R∗ ⊗ Y ⊗ R∗ −→
I
X
P1
R
π2
Y
P2
R
Jules Hedges (QMUL)
How to make game theory compositional
February 9, 2016
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2-player sequential
X ⊗σX ,R∗
∆ ⊗R∗
P
1
X
I −→
X ⊗ R∗ −−−
−−→ X ⊗ X ⊗ R∗ −−−−−→ X ⊗ R∗ ⊗ X
X ⊗R∗ ⊗P
π
2
2
−−−−−−→
X ⊗ R∗ ⊗ Y ⊗ R∗ −→
I
X
X
P1
R
π2
Y
X
P2
R
Jules Hedges (QMUL)
How to make game theory compositional
February 9, 2016
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