Tropical convexity and its applications to zero-sum games

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Tropical convexity and its applications to zero-sum games

Minilecture, Part I

INRIA and CMAP, ´Ecole Polytechnique

JGA, Marseille December 16-20, 2013

Works withAkian, Allamigeon, Goubault, Guterman, Katz, Joswig, Meunier, Sergeev, Walsh; highlight: PhD of Benchimol and Qu.

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Max-plus or tropical algebra

In an exotic country, children are taught that:

“a+ b” = max(a,b) “a×b”= a+b So

“2 + 3” =

“2×3” =

“5/2” =

“2³” =

“√

−1” =

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Max-plus or tropical algebra

“2 + 3” = 3

“2×3” =

“5/2” =

“2³” =

“√

−1” =

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Max-plus or tropical algebra

“2 + 3” = 3

“2×3” =

“5/2” =

“2³” =

“√

−1” =

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Max-plus or tropical algebra

“2 + 3” = 3

“2×3” =5

“5/2” =

“2³” =

“√

−1” =

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Max-plus or tropical algebra

“2 + 3” = 3

“2×3” =5

“5/2” =

“2³” =

“√

−1” =

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Max-plus or tropical algebra

“2 + 3” = 3

“2×3” =5

“5/2” =3

“2³” =

“√

−1” =

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Max-plus or tropical algebra

“2 + 3” = 3

“2×3” =5

“5/2” =3

“2³”=

“√

−1” =

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Max-plus or tropical algebra

“2 + 3” = 3

“2×3” =5

“5/2” =3

“2³”=“2×2×2” = 6

“√

−1” =

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Max-plus or tropical algebra

“2 + 3” = 3

“2×3” =5

“5/2” =3

“2³”=“2×2×2” = 6

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Max-plus or tropical algebra

“2 + 3” = 3 “

7 0

−∞ 3 2 1

”=

“2×3” =5

“5/2” =3

“2³”=“2×2×2” = 6

“√

−1” =−0.5

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Max-plus or tropical algebra

“2 + 3” = 3 “

7 0

−∞ 3 2 1

”=

9 4

“2×3” =5

“5/2” =3

3

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The notation a⊕b := max(a,b), a b := a+b, 0:= −∞, 1:= 0 is also used in the tropical/max-plus litterature

Max-plus semiring: R^max = (R∪ {−∞},max,+).

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The sister algebra: min-plus

“a+ b” = min(a,b) “a×b”= a+b

“2 + 3” = 2

“2×3” = 5

Min-plus semiring: R^min.

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Structures called idempotent a +a = a or of characteristic one. Compare with

(p + 1)a = a .

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Exercises (cont)

Find the roots of the max-plus polynomial

“1⁻¹X³ + X² + 2X + 1¹”.

Nota bene: “1^a”= 1×a = a is unambiguous, compare “1” = 0 with “1¹”= 1.

Answer:

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Exercises (cont)

“1⁻¹X³ + X² + 2X + 1¹”.

Answer: max(−1 + 3X,2X,2 +X,1)

= −1 + max(X,−1) + 2 max(X,1.5)

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Exercises (cont)

“1⁻¹X³ + X² + 2X + 1¹”.

Answer:

“1⁻¹X³ +X² + 2X + 1¹ = 1⁻¹(X + 1⁻¹)(X + 1^3/2)²”

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Theorem (Cuninghame-Green & Meijer, 80)

A max-plus polynomial function p = “a_nXⁿ+· · ·+a₀” can be factored uniquely as

p = “a_n(X +α₁). . .(X +α_n)”

= a_n + max(X, α₁) + · · ·+ max(X, α_n) . The α_i are the (tropical) roots.

How to compute (tropical) roots?

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Legendre-Fenchel = tropical Fourier transform

The map which sends coeffs: (i 7→a_i) to the numerical function

X 7→p(X) = max

06i6na_i +i ×X is a special case of Legendre-Fenchel transform

f :Rⁿ →R, f^? : Rⁿ →R∪ {+∞}, f^?(p) = sup

x∈Rⁿ

hp,xi −f(x) .

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Newton polygon

∆(p) = lower concave hull{(i,a_i) | 06 i 6 n}.

Proposition

Let p ∈ R^max[X] be a max-plus polynomial. Roots of p = minus slopes of ∆(p). Multiplicity of root α = length of the interval of ∆(p) of slope −α. [Linear time]

“p = 1⁻¹X³ + 1⁰X² + 1²X + 1¹”. The tropical roots are

−1 (multiplicity 1) and 1.5 (multiplicity 2).

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Exercises (cont.)

Approximate essentially without computation the (usual) roots of

p = 2⁻² + 2²X −2⁵X⁴ + 2X⁶ ∈ C[X]

Answer: −2⁻⁴, 2⁻¹{1,j,j²}, 2²{1,−1},

-0.0625 -0.25-0.433i -0.25+0.433i 0.5 4. - 4.

Check in Scilab:

-0.0624 -0.226-0.434i -0.226+0.434i 0.522 4.00 -4.00

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Exercises (cont.)

p = 2⁻² + 2²X −2⁵X⁴ + 2X⁶ ∈ C[X] Answer: −2⁻⁴, 2⁻¹{1,j,j²}, 2²{1,−1},

-0.0625 -0.25-0.433i -0.25+0.433i 0.5 4. - 4.

Check in Scilab:

-0.0624 -0.226-0.434i -0.226+0.434i 0.522 4.00 -4.00

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Exercises (cont.)

p = 2⁻² + 2²X −2⁵X⁴ + 2X⁶ ∈ C[X] Answer: −2⁻⁴, 2⁻¹{1,j,j²}, 2²{1,−1},

-0.0625 -0.25-0.433i -0.25+0.433i 0.5 4. - 4.

Check in Scilab:

-0.0624 -0.226-0.434i -0.226+0.434i 0.522 4.00 -4.00

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Solution: Associate to P

k a_kX^k ∈ C[X] the max-plus polynomial

X 7→ max

k log₂|a_k|+kX .

p = 2⁻² + 2²X −2⁵X⁴ + 2¹X⁶, tropical roots are −4 (mult. 1), −1 (mult. 3), 2 (mult. 2)

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Theorem (Hadamard, Ostrowski, Poly´a) Let p = P

k a_kX^k with roots ζ_i ∈ C, |ζ₁| > . . . >|ζ_n|, α₁ > . . . >α_n tropical roots of max_k log|a_k|+kX .

1

C_n^k exp(α₁ +· · ·+α_k)

6 |ζ₁· · ·ζ_k| 6 cst_k exp(α₁ +· · ·+α_k)

Corollary

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Hadamard: cst_k 6k + 1 (1891, memoir on Zeta function) Ostrowski: lower bound, cst_k 6 2k + 1 (1940, Graeffe method)

Poly´a cstk < e√

k + 1 (reproduced by Ostrowski).

Proof = variation on Jensen formula

|a₀|R^k

|ζ_n· · ·ζ_n−k+1| 6 exp( 1 2π

Z 2π 0

log|f(Re^iθ)|dθ), ∀R > 0

Akian, SG, Sharify arXiv:1304.2967; more bounds, matrix extension;

decomposition results in Bini, Noferini, Sharify arXiv:1206.3632

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One more exercise

A_ε =





ε 1 ε⁴

0 ε ε⁻²

ε ε² 0



 , Eigenvalues ? →0

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One more exercise

A_ε =





ε 1 ε⁴

0 ε ε⁻²

ε ε² 0



L¹_ε ∼ε⁻¹^/³, L²_ε ∼ jε⁻¹^/³, L³_ε ∼j²ε⁻¹^/³.

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One more exercise

A_ε =





ε 1 ε⁴

0 ε ε⁻²

ε ε² 0



L¹_ε ∼ε⁻¹^/³, L²_ε ∼ jε⁻¹^/³, L³_ε ∼j²ε⁻¹^/³. Answer without computation using tropical algebra, solution later in this lecture

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A partial history of max-plus / tropical algebra

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In the late 80’s in France, the term “alg`ebres exotiques”

was used

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In the late 80’s in France, the term “alg`ebres exotiques”

was used

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The term “exotic” appeared also in the User’s guide of viscosity solutions of Crandall, Ishii, Lions (Bull. AMS, 92)

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The term “exotic” appeared also in the User’s guide of viscosity solutions of Crandall, Ishii, Lions (Bull. AMS, 92)

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The term “tropical” is in the honor of Imre Simon, 1943 - 2009

who lived in Sao Paulo (south tropic).

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These algebras were invented by various schools in the world

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Cuninghame-Green1960-OR (scheduling, optimization) Vorobyev∼65. . .Zimmerman, Butkovic; Optimization

Maslov∼80’-. . .Kolokoltsov, Litvinov, Samborskii, Shpiz. . . Quasi-classic analysis, variations calculus

Simon∼78-. . .Hashiguchi, Leung, Pin, Krob,. . . Automata theory Gondran, Minoux∼77Operations research

Cohen, Quadrat, Viot∼83-. . .Olsder, Baccelli, S.G., Akiandiscrete event systems, optimal control, idempotent probabilities, linear algebra

Nussbaum86-Nonlinear analysis, dynamical systems, also related work in linear algebra,Friedland88,Bapat˜94

Kim, Roush84Incline algebras

Fleming, McEneaney∼00-max-plus approximation of HJB

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Menu: applied tropical geometry, connections between. . .

tropical convexity

dynamic programming / zero-sum games Perron-Frobenius theory

metric geometry

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Some elementary tropical geometry

A tropical line in the plane is the set of (x,y) such that the max in

“ax +by +c” is attained at least twice.

max(x,y,0)

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Some elementary tropical geometry

A tropical line in the plane is the set of (x,y) such that the max in

max(a +x,b+y,c) is attained at least twice.

max(x,y,0)

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Two generic tropical lines meet at a unique point

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By two generic points passes a unique tropical line

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non generic case

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non generic case resolved by perturbation

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Tropical segments:

f

g

[f,g] := {“λf +µg” | λ, µ ∈ R∪ {−∞}, “λ+µ = 1”}.

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Tropical segments:

f

g

[f,g] := {sup(λ+f, µ+g) | λ, µ ∈ R∪ {−∞}, max(λ, µ) = 0}.

(The condition λ, µ > −∞ is automatic.)

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Tropical convex set: f,g ∈ C =⇒ [f,g] ∈ C

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Tropical convex set: f,g ∈ C =⇒ [f,g] ∈ C

Tropical convex cone: ommit “λ+µ = 1”, i.e., replace [f,g] by {sup(λ+f, µ+g) | λ, µ ∈ R∪ {−∞}}

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Homogeneization

A convex set C in Rⁿmax is a cross section of a convex cone ˆC in Rⁿ⁺¹max,

Cˆ := {(λ+u, λ) | u ∈ C, λ ∈ R^max}

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A tropical polytope with four vertices

Structure of the polyhedral complex: Develin, Sturmfels

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The previous drawing was generated byPolymake of Gawrilow and Joswig, in which an extension allows one to handle easily tropical polyhedra. They were drawn withjavaview.

See Joswig arXiv:0809.4694 for more information.

Tropical polyhedra handled by ocaml TPLib,Allamigeon

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Motivation ?

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The tropical point of view arises with log glasses

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Gelfand, Kapranov, and Zelevinsky defined the amoeba of an algebraic variety V ⊂ (C^∗)ⁿ to be the “log-log plot”

A(V) := {(log|z₁|, . . . ,log|z_n|) | (z₁, . . . ,z_n) ∈ V} .

y +x+ 1 = 0

attained twice max(log|x|,log|y|,0)

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y +x+ 1 = 0

|y|6 |x|+ 1, |x| 6|y|+ 1, 1 6 |x|+|y|

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y +x+ 1 = 0

X := log|x|, Y := log|y|

Y 6 log(e^X + 1), X 6 log(e^Y + 1), 0 6log(e^X +e^Y)

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Viro’s log-glasses, related to Maslov’s dequantization a+h b := hlog(e^a/h +e^b/h), h → 0⁺

With h-log glasses, the amoeba of the line retracts to the tropical line as h →0⁺

y +x+ 1 = 0

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Tropical convex sets are deformations of classical convex sets

Briec and Horvath 04

[a,b] := {λa +_p µb, λ, µ >0, λ+_p µ= 1}

a +_p b = (a^p +b^p)¹^/^p

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See Passare & Rullgard, Duke Math. 04

Introduction to amoebas: lecture notes by Alain Yger.

Metric estimates: Avenda˜no, Kogan, Nisse, Rojas, arXiv:1307.3681

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Nonarchimedean valuation point of view

Alternatively [Sturmfels’ point of view], the tropical line

“max(X,Y,0) attained twice” can be seen as the image by the valuation of the line x +y + 1 over the field of complex Puiseux series, C{{t}}, equipped with the valuation vals = − smallest exponent of s.

E.g., val(t^−1/2 −t + 7t^3/2 +. . .) = 1/2

val(z₁ +z₂) 6 max(val(z₁),val(z₂)), with equality when val(z₁) 6= val(z₂).

val(z₁z₂) = val(z₁) + val(z₂)

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tropical hyperplanes (complex version)

Given a ∈ Rⁿmax, a 6≡ −∞,

H := {x ∈ Rⁿmax | “ax = 0”}

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tropical hyperplanes (complex version)

Given a ∈ Rⁿmax, a 6≡ −∞, H := {x ∈ Rⁿmax | max

16i6nai +xi attained twice}

x₂ x₁

x3

max(x₁,x₂,−2 +x₃)

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The rational points of tropical hyperplanes are images of hyperplanes of (C{{t}})ⁿ by the valuation, see:

Theorem (Kapranov) Given p = P

αp_αz^α ∈ C{{t}}[z₁, . . . ,z_n], and Z ∈ Qⁿ,

∃z ∈ (C{{t}})ⁿ, p(z) = 0, Z = valz iff

maxα valp_α+hα,Zi attained twice

Restriction to Q can be avoided by working with Puiseux series with

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real tropical hyperplanes

Given a,b ∈ Rⁿmax, a,b 6≡ −∞, a_i = −∞ or b_i = −∞,

∀i,

H := {x ∈ Rⁿmax | “ax = bx”}

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real tropical hyperplanes

∀i,

H := {x ∈ Rⁿmax | max

16i6na_i +x_i = max

16i6nb_i +x_i}

x₃

x1= max(x2,−2 +x3)

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real tropical hyperplanes

∀i,

16i6na_i +x_i = max

16i6nb_i +x_i}

x₂ x1

x₃

x2= max(x1,−2 +x3)

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real tropical hyperplanes

∀i,

16i6na_i +x_i = max

16i6nb_i +x_i}

x₃

−2 +x3= max(x1,x2)

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Real tropical hyperplanes are images of hyperplanes of R{{t}} by the valuation.

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Given any n points in Rⁿmax in general position, there is a unique (complex) affine tropical hyperplane

passing through them. Richter-Gebert, Sturmfels, Theobalt, 05,

Given any n points in Rⁿmax in general position, there is a unique real affine tropical hyperplane passing through them. Max Plus, 90, see also Akian, SG, Guterman 09

Let these points be given by the columns of a (n+ 1)×n matrix M, in projective coordinates. The vector a such that H = {x | “a ·x = 0”} contains the points is solution

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How are Cramer det defined?

For the complex (RGST 05) version, ignore sign detA = “X

σ

sgnY

i

A_i_σ_(i)”= max

σ

X

i

A_i_σ_(i)

This is an optimal assignment problem

For the real (Max Plus 90) version, the signs of the maximising permutations tells on which side of the equality “ax = bx” the coefficients should be put.

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Proofs

Extensions of tropical semiring, Maxplus 90, Akian, SG, Guterman, 09, 13, Izhakian, Rowen 09, related formalism :

Krasner hyperfield 57 (different axioms but comparable expressivity)

Coherent matching fields Richter-Gebert, Sturmfels, Theobalt, 05, building on Sturmfels and Zelevinsky, The Newton polytope of the product of maximal minors of a (n+ 1)×n matrix is a transportation polytope.

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The symmetrized tropical semiring

Recall that Z = N²/∇, where (a⁰,a⁰⁰)∇(b⁰,b⁰⁰) if a⁰ +b⁰⁰ = a⁰⁰ +b⁰, −(a⁰,a⁰⁰) = (a⁰⁰,a⁰).

Replace N by R^max, S^max = R²max/∼ a ∼ b if a = b or (a∇b and a,b 6 ∇0).

S^max = R^max ∪ R^max∪ R^•max; u = (u,0), u = (0,u), u^• := u u = (u,u) with u ∈ R^max.

S^∨max := R^max∪ R^max: signed elements.

E.g., 2⊕( 3) = 3, but 3 3 = 3^•. Think of u as

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x₁ = max(2 +x₂,7 +x₃) ⇐⇒ x₁ 2x₂ 7x₃∇0 Need to solve aM∇0, where M is of (n+ 1)×n, a ∈ Sⁿ⁺¹max.

Theorem (Transfer theorem)

Any polynomial identity valid in rings is valid in the extensions of semirings.

Akian, SG, Guterman 09, using an idea of Reutenauer and Straubing 86.

Eg P_A(A) = 0 becomes P_A⁺(A) = P_A⁻(A), or P_A(A)∇0, where PA characteristic polynomial.

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Lemma (Elimination)

x∇b, cx∇d , x ∈ S^∨max implies cb∇d .

Cramer theorem is proved by Gaussian elimination

Izhakian introduced the bi-valued tropical semiring, 2⊕2 = 2^•, to remind that max is attained twice. The

“complex” tropical Cramer theorem is proved along the same lines.

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Tropical half-spaces

Given a,b ∈ Rⁿmax, a,b 6≡ −∞,

H := {x ∈ Rⁿmax | “ax 6 bx”}

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Tropical half-spaces

Given a,b ∈ Rⁿmax, a,b 6≡ −∞, H := {x ∈ Rⁿmax | max

16i6nai +xi 6 max

16i6nbi +xi}

x₁ x

x3

max(x₁,x₂,−2 +x₃)

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Tropical half-spaces

16i6nai +xi 6 max

16i6nbi +xi}

x₂ x₁

x3

max(x₁,−2 +x₃)6x₂

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Tropical half-spaces

16i6nai +xi 6 max

16i6nbi +xi}

x₁ x

x3

x₁6max(x₂−2 +x₃)

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Tropical half-spaces

16i6nai +xi 6 max

16i6nbi +xi}

x₂ x₁

x3

max(x₂−2 +x₃)6x₁

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Tropical sesquilinear form and Hilbert’s metric

x/v := max{λ | “λv”6 x}

= min

i (x_i −v_i) if x,v ∈ Rⁿ . δ(x,y) = “(x/y)(y/x)” = min

i (x_i −y_i) + min

j (y_j −x_i) d = −δ is the (additive) Hilbert’s projective metric

d(x,y) =kx −yk , kzk := max z − min z .

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Hilbert’s metric on an open convex set

a

b

¯ a

¯b

b¯ b a

¯ a

d_H(a,b) = log |b −¯a||a−b|¯

|a −¯a|||b −b|¯ . disc: Klein model of the hyperbolic space;

simplex: d_H conjugate to the additive (tropical) Hilbert metric (take exp :Rⁿ → Rⁿ+ and a cross section of Rⁿ+).

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e1 e2

e3

A ball in Hilbert’s metric is classically and tropically convex.

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Projection on a tropical cone

If the tropical convex cone C ⊂Rⁿmax generated by U is stable by arbitrary sups (closed in Scott topology -non-Haussdorf-):

P_C(x) = max{v ∈ C | v 6 x}

= max

u∈U (x/u) +u . Similar to P_C(x) = X

u∈U

hx,uiu

C = Col(A), [P_C(x)]_i = max

k∈[p]min

j∈[n](A_ik−A_jk+x_j), i ∈ [n]

Cuninghame-Green; Gondran, Minoux; Cohen, SG, Quadrat; Ardila; Joswig, Sturmfels, Yu

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Best approximation in Hilbert’s projective metric

Prop.(Cohen, SG, Quadrat, in Bensoussan Festschrift 01) d(x,P_V(x)) = min

y∈V d(x,y) .

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Separation

Goes back to Zimmermann 77, simple geometric construction in Cohen, SG, Quadrat in Ben01, LAA04.

C closed linear cone of R^dmax, or complete semimodule If y 6∈ C, then, the tropical half-space

H := {v | y/v 6 P_C(y)/v} contains C and not y.

Compare with the optimality condition for the projection

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Corollary (Zimmermann; Samborski, Shpiz; Cohen, SG, Quadrat, Singer; Develin, Sturmfels; Joswig. . .)

A tropical convex cone closed (in the Euclidean topology) is the intersection of tropical half-spaces.

R^max is equipped with the topology of the metric (x,y) 7→max_i|e^xⁱ −e^yⁱ| inherited from the Euclidean topology by log-glasses.

The apex −P_C(y) of the algebraic separating half-space H above may have some +∞coordinates, and therefore may not be closed in the Euclidean topology (always Scott closed). The proof needs a perturbation argument, this is where the

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Separation of several convex sets / cyclic projections SG, Sergeev, Fund. i priklad. mat. 07

If V₁ ∩ · · · ∩ V_k = {“0”}, we can find half-spaces H_i such that H_i ⊃ V_i and H₁ ∩ · · · ∩ H_k = {“0”}. The apices of these half-spaces are obtained from an eigenvector u of the cylic projector

“P_V₁· · ·P_V_k(u) =λu”

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The existence of such an eigenvector is obtained by a technique from nonlinear analysis, case of −∞ entries dealt with by perturbation (Collatz-Wielandt theorem), more on this next lecture.

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Theorem (tropical Helly theorem, Briec and Horvath, 04) If a finite collection of cones of R^dmax has a

“zero”intersection, then a subfamily of at most d of them also has a “zero”intersection.

Proved by “dequantization” from classical Helly (passing to the limit).

Alt proof by SG and Sergeev 07 by cyclic projection (at each projection one coordinate decreases)

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Tropical Radon

In SG and Meunier, DCG09, Helly deduced from the tropical Radon’s theorem: a subset of d + 1 vectors in dimension d can be partitioned in two subsets generating cones with a “non-zero”intersection.

Radon theorem follows from tropical Cramer theory (signs provide partition).

x3

x4

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More advanced results of tropical convex geometry, SG and Meunier, DCG09

Barany’s Colorful Caratheodory Theorem

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More advanced results of tropical convex geometry, SG and Meunier, DCG09

Barany’s Colorful Caratheodory Theorem . . . Tropical

r

b

b g

g

C g

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Tropical Tverberg’s theorem, SG and Meunier, DCG09. Let X be a set of (d + 1)(q−1) + 1 points in R^dmax. Then there are q pairwise disjoint subsets X₁,X₂, . . . ,X_q of X whose tropical convex hulls have a common point.

Deduced from the classical one by a limit argument, no direct combinatorial proof known.

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Tropical Tverberg

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Tropical Tverberg

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Tropical Tverberg

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Dutch cheese conjecture

Let q >2, d >1. Sierksma conjectured that for every (d + 1)(q−1) + 1 points in R^d the number of unordered Tverberg partitions is at least ((q−1)!)^d.

SG and Meunier, DCG09: True in the tropical setting!

(development of a “bipartite analogue” of Tverberg’s theorem due to Lindstr¨om, 1970 and Tverberg, 71)

Unfortunately, it is not clear whether this can be transferred to the classical case. In other words, there may be no “Mikhalkin’s correspondence theorem” in the case of inequalities (?)

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Menu of the next lectures

Tropical linear programming, classical linear programming, and mean payoff games Non-linear Perron-Frobenius theory Infinite dimensional tropical convex sets, Metric geometry / boundaries

max-plus approximation, curse of dim reduction in optimal control

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Tropical convexity and its applications to zero-sum games

Minilecture, Part II

[email protected]

INRIA and CMAP, ´Ecole Polytechnique

JGA, Marseille December 16-20, 2013

Works withAkian, Allamigeon, Goubault, Guterman, Katz, Joswig, Meunier, Sergeev, Walsh; highlight: PhD of Benchimol and Qu.

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Today

Equivalence between tropical linear programming and mean payoff games

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Tropical half-spaces

Given a,b ∈ Rⁿmax, a,b 6≡ −∞,

H := {x ∈ Rⁿmax | “ax 6 bx”}

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Tropical half-spaces

16i6nai +xi 6 max

16i6nbi +xi}

x₁ x

x3

max(x₁,x₂,−2 +x₃)

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Tropical half-spaces

16i6nai +xi 6 max

16i6nbi +xi}

x₂ x₁

x3

max(x₁,−2 +x₃)6x₂

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Tropical half-spaces

16i6nai +xi 6 max

16i6nbi +xi}

x₁ x

x3

x₁6max(x₂−2 +x₃)

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Tropical half-spaces

16i6nai +xi 6 max

16i6nbi +xi}

x₂ x₁

x3

max(x₂−2 +x₃)6x₁

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A halfspace can always be written as:

maxi∈I⁻ a_i +x_i 6max

j∈I⁺ b_j +x_j, I⁻∩I⁺ = ∅ . Apex: vi := −max(ai,bi).

If v ∈ Rⁿ, H is the union of sectors of the tropical hyperplane with apex v:

max

16i6nx_i −v_i attained twice

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Tropical polyhedral cones

can be defined as intersections of finitely many half-spaces

x₂ x₁

V x₃

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Tropical polyhedral cones

x2 x₁

V x₃

x2 x₁

x₃

2 +x₁6max(x₂,3 +x₃)

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Tropical polyhedral cones

V

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Feasibility in tropical LP

A tropical polyhedral cone is defined as

C = {x ∈ Rⁿmax | “Ax 6 Bx”}, A,B ∈ R^m×nmax

maxj∈[n] Aij +xj 6max

j∈[n] Bij +xj, ∀i ∈ [m]

A tropical polyhedron is defined as

P = {x ∈ Rⁿmax |“Ax + c 6Bx +d”}

where A,B ∈ R^m×nmax ,c,d ∈ R^mmax. Questions:

is C reduced to “0”? “0” = (−∞, . . . ,−∞)^>

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Example: mean payoff (deterministic) games

G = (V,E) bipartite graph. r_ij price of the arc (i,j) ∈ E.

“Max” and “Min” move a token. The player receives the amount of the arc.

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v_i^k value of MAX, initial state (i,MIN).

v₁^k= min(−2 + 1 +v₁^k−1,−8 + max(−3 +v₁^k−1,−12 +v₂^k−1)) v₂^k= 0 + max(−9 +v₁^k⁻¹,5 +v₂^k−1)

2 1

8

−3

−12 2

1

2

−9 MIN

MAX

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v_i^k value of MAX, initial state (i,MIN).

v₁^k= min(−2 + 1 +v₁^k−1,−8 + max(−3 +v₁^k−1,−12 +v₂^k−1)) v₂^k= 0 + max(−9 +v₁^k⁻¹,5 +v₂^k−1)

2 1

8

−3

−12 0 3 5 2 1

1

2

−9 MIN

MAX

lim_k v^k/k = (−1,5)

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Max and Min flip a coin to decide who makes the move.

Min always pays.

2

3

−1

2

2 1

−1 −8

1 2

3

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Solving the game

v_i^k := value of the k-horizon game starting from node i.

value is defined as the mean reward of Max, assuming both players play optimally

v^k = (v_i^k) ∈ Rⁿ v⁰ = 0

v^k+1 = T(v^k)

where T : Rⁿ → Rⁿ is the Shapley operator

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Solving the game

v^k = (v_i^k) ∈ Rⁿ v⁰ = 0

v^k+1 = T(v^k)

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Solving the game

v^k = (v_i^k) ∈ Rⁿ

v⁰ = 0

v^k+1 = T(v^k)

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Solving the game

v^k = (v_i^k) ∈ Rⁿ v⁰ = 0

v^k+1 = T(v^k)

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Solving the game

v^k = (v_i^k) ∈ Rⁿ v⁰ = 0

v^k+1 = T(v^k)

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2

3

−1

2

2 1

−1 −8

1 2

3

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2

3

−1

2

2 1

−1 −8

1 2

3

v_i^k⁺¹ = 1 2(max

j:i→j(c_ij +v_j^k) + min

j:i→j(c_ij +v_j^k)) .

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Shapley operators

X = C(K), even X = Rⁿ, K = [n]; Shapley operator T, T_i(x) = max

a∈Ai

b∈Bmini,a

r_i^ab + X

16j6n

P_ij^abx_j

, i ∈ [n]

[n] := {1, . . . ,n} set of states

a action of Player I, b action of Player II r_i^ab payment of Player II to Player I

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Shapley operators

a∈Ai

b∈Bmini,a

r_i^ab + X

16j6n

P_ij^abx_j

, i ∈ [n]

T is order preserving, additively homogeneous ⇒ sup-norm nonexpansive:

x 6y =⇒ T(x) 6 T(y)

T(α+ x) = α+T(x), ∀α ∈ R kT(x)−T(y)k6 kx −yk

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Shapley operators

a∈Ai

b∈Bmini,a

r_i^ab + X

16j6n

P_ij^abx_j

, i ∈ [n]

Conversely, any order preserving additively homogeneous operator is a Shapley operator (Kolokoltsov), even with degenerate transition probabilities (deterministic)

Gunawardena, Sparrow; Singer, Rubinov,

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Repeated games

The value of the game in horizon k starting from state i is (T^k(0))_i.

We are interested in the long term payment per time unit χ(T) := lim

k→∞T^k(0)/k

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The mean payoff vector

χ(T) := lim

k→∞T^k(0)/k

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The mean payoff vector

χ(T) := lim

k→∞T^k(0)/k

χi(T) = mean payoff per turn if initial state is i

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The mean payoff vector

χ(T) := lim

k→∞T^k(0)/k

χ_i(T) = mean payoff per turn if initial state is i χ(T) = lim

k→∞T^k(x)/k, ∀x ∈ Rⁿ

for kT^k(x)−T^k(0)k 6kx −0k = kxk

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The mean payoff vector

χ(T) := lim

k→∞T^k(0)/k

χ_i(T) = mean payoff per turn if initial state is i χ(T) = lim

k→∞T^k(x)/k, ∀x ∈ Rⁿ

for kT^k(x)−T^k(0)k 6kx −0k = kxk

Think of xi has a terminal bounty paid by Min to Max if the game ends in state i.

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2

3

−1

2

2 1

−1 −8

1 2

3



 5 0 4





v₁= ¹₂(max(2 +v₁,3 +v₂,−1 +v₃) + min(2 +v₁,3 +v₂,−1 +v₃) v₂= ¹₂(max(−1 +v₁,2 +v₂,−8 +v₃) + min(−1 +v₁,2 +v₂,−8 +v₃)

v3 = ¹(max(2 +v1,1 +v2) + min(2 +v1,1 +v2)

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Optimality certificates

More generally, for u ∈ Rⁿ and λ ∈ R,

T(u) >u =⇒ χ(T) > 0 superfair T(u) 6 u =⇒ χ(T) 60 subfair T(u) = λ+u =⇒ χ(T) = (λ, . . . , λ) . Does χ(T) = lim_k T^k(0)/k exist? Do such certificates exist?

χi = χj is related to ergodicity. May not hold for

deterministic games, what are the certificates then?

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χ(T) = lim_k T^k(0)/k may not exist if the action spaces are infinite (Kohlberg, Neyman). Counter example in dimension 3.

However. Let v_α denote the discounted value v_α = T(αv_α), 0< α < 1 . Theorem (Neyman 04 - book edited with Sorin)

If α 7→ (1−α)v_α has bounded variation as α → 1, then

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Corollary (Neyman 04, Bewley and Kohlberg 76)

If the graph of T is semi-algebraic, then χ(T) does exists.

Then, v_α is a semi-algebraic function of α, it has a Puiseux series expansion, and so (1−α)v_α has BV.

This is the case in particular if the action spaces are finite.

More generally, T definable in an o-minimal model (Bolte, SG, Vigeral 13).

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By subadditivity, the following limits (indep of x ∈ Rⁿ) do exist

lim

k→∞

kT^k(x)−xk_∞

k = inf

k>1

kT^k(x)−xk_∞ k

χ(T) := lim

k→∞

t(T^k(x)−x)

k = inf

k>1

t(T^k(x)−x) k

χ(T) := lim

k→∞

b(T^k(x)−x)

k = sup

k>1

b(T^k(x)−x) k

t(z) := max z , b(z) := min z .

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Think of T as a Perron-Frobenius operator in log-glasses:

F = exp◦T ◦log, Rⁿ+ →Rⁿ+

F extends continuously from intRⁿ+ to Rⁿ+ Burbanks, Nussbaum, Sparrow.

Theorem (non-linear Collatz-Wielandt, Nussbaum, LAA 86) ρ(F) = lim

k→∞kF^k(x)k^1/k, x ∈ IntRⁿ+

= max{µ ∈ R⁺ | F(v) =µv,v ∈ Rⁿ+,v 6= 0}

= max{µ ∈ R⁺ | F(v) > µv,v ∈ Rⁿ+,v 6= 0}

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Stochastic games → tropical convex feasibility

Corollary

Let T be a Shapley operator. Then,

limk maxi[T^k(0)]i/k > 0 iff there is u ∈ (R∪ {−∞})ⁿ, u 6= (−∞, . . . ,−∞)^>, T(u) > u.

Proposition

If T is a Shapley operator, C = {u ∈ Rⁿmax | T(u) >u} is a closed tropical convex cone.

Proof.

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Conversely, any closed tropical convex cone can be written as

C = \

i∈I

H_i

where (H_i)_i∈I is a family of tropical half-spaces.

H_i : “A_ix 6 B_ix”

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Conversely, any closed tropical convex cone can be written as

C = \

i∈I

H_i

where (Hi)_i∈I is a family of tropical half-spaces.

H_i : max

16j6na_ij+x_j 6 max

16k6nb_ik+x_k, a_ij,b_ik ∈ R∪{−∞}

[T(x)]_j = inf

i∈I −a_ij + max

16k6nb_ik +x_k .