GamesandOptimization,November2016 Ga¨etanFOURNIER LocationGamesonNetworks

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Location Games on Networks

Ga¨etan FOURNIER

IAST, TSE

Games and Optimization, November 2016

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Uniform distribution General distributions Probabilistic Hotelling game

The model and examples Existence result Efficiency result

Hypothesis on buyers

1 Non-strategic continuum of buyers, distributed on a network generated by a metric graph.

2 They buy a given quantity of a good whose price is fixed: they shop to the closest location.

e₁

e₂

e₃

e₄

e₅

e₆ e₇

e₈

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Hypothesis on sellers (= players)

1 A fixed number of strategic sellers simultaneously choose their locations.

2 They want to sell as much as possible.

e₁

e2

e3

e₄ e₅

e6

e7

e8

1 3

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Hypothesis on sellers (= players)

1 A fixed number of strategic sellers simultaneously choose their locations.

2 They want to sell as much as possible.

e₁ e2

e3

e₄ e₅

e6

e7

e8

1 3

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The unit interval [Eaton-Lipsey,1975]

1 For n=2, there exists a pure Nash equilibrium.

2 For n=3, there is no pure Nash equilibrium.

3 For n≥4,there exists a pure Nash equilibrium.

The starS_k(r)

1 For n≤k, there exists a pure Nash equilibrium.

2 For n∈ [k+1,3k−2], there is no pure Nash equilibrium.

3 For n≥3k−1, there exists a pure Nash equilibrium.

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Results with uniform density (Fournier-Scarsini[2015])

ä Existence of pure Nash equilibrium for any graph when the number of player is large enough.

Theorem: Fournier-Scarsini [2015]

On any finite graph, Hotelling games admit a pure Nash equilibrium, provided the number of players is larger than

N∶=3 card(E) + ∑

e∈E

⌈ 5λ(e)

λ^⋆ ⌉ whereλ^⋆=min

E λ( the length of the shortest edge).

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Extensive analysis (also with small number of players): not easy.

Counterexample [D. Palvolgyi]

2 For n∈ {3,4}, there is no pure Nash equilibrium.

3 For n∈ {5,6}, there exists a pure Nash equilibrium.

4 For n∈ {7, . . . ,16}, there is no pure Nash equilibrium.

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ä Existence of pure Nash equilibrium for any graph when the number of player is large enough.

ä Efficiency of these equilibria: not for the players (sellers) but for the consumers.

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Efficiency of the equilibria

Traveling distances of consumers, in equilibrium and in social optimum.

Equilibrium social cost: ? Optimum social cost: ?

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Social costs in equilibrium and in social optimum.

Equilibrium social cost: ¹₈ Optimum social cost: ₁₆¹

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For x ∈Sⁿ, thesocial costσ(x) is given by:

σ(x) ∶= ∫

S min

i∈{1,...,n}d(xi,y)dy The price of anarchy is given by:

PoA(n) ∶= max_x_∈E_n_(H)σ(x) min_x_∈Sⁿσ(x)

,

The price of stability is given by:

PoA(n) ∶=

min_x_∈E_n_(H)σ(x) min_x_∈Sⁿσ(x) , whereE_n(H)is the set of equilibrium withn players.

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On the unit interval, we have:

PoA(n) =

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

2 ifn is even, 2( n

n+1) ifn>3 is odd.

Forn≥4

PoS(n) = n n−2

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ä Efficiency of these equilibria: not for the players (sellers) but for the consumers.

Theorem

Suppose that the gameH(n,G) has an equilibrium. Then PoA(n) Ð→

n→+∞2 PoS(n) Ð→

n→+∞1

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The model Counter-example The result

General distributions of consumers [Fournier 2015]

For a subset A ⊂ G, the quantity of consumers lo- cated inAis:

∫Ag(x)dL(x) where L is the Lebesgue measure, and g >0.

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Counterexamples with general distribution of consumers

Counterexamples

(1) For any>0, the function g ∶ [0,1] →R⁺: g ∶=x ↦1+x

is arbitrary close to1but the game H(n,[0,1],g)doesn’t admit any Nash equilibrium in pure strategies forn>2.

(2) In fact, when the number of player is small, the class of distributions such that the corresponding Hotelling games admit an exact pure Nash equilibrium is small.

Suppose that x = (x₁, . . . ,x_n) is an equilibrium (x₁≤ ⋅ ⋅ ⋅ ≤x_n) in the game H(n,[0,1],f). We first claim that all players are coupled, i.e. thatx1=x2<x3=x4< ⋅ ⋅ ⋅ <xn−1 =xn.

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→ Suppose that 3 players (or more) share the same location xk =xk+1=xk+2

∣ ∣

x_k−1 x_k x_k₊₃

p_k(x₁, . . . ,x_n) = 1 3∫

∪

(1+x)dx Either:

∫ (1+x)dx>p_k(x₁, . . . ,x_n) or:

∫ (1+x)dx>p_k(x₁, . . . ,x_n)

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→ Suppose now that there exists a locationx_k ∈ [0,1]with a single player k. His payoff is equal to a right trapezoid’s area:

∣ ∣

1+^x^k−1₂^+x^k 1+^x^k^+x₂^k+1

x_k−1

xk−1+xk

2

x_k

xk+xk+1

2

x_k+1

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All players are coupled.

0 x₁

x₂

x₃

x₄ x_n−2

x_n−1 x_n

x_n+1 x_n+2 x_n+3 A₁ A₂ A₃ A₄ . . .

A_n−1A_n+1 . . .

p₁(x) =p₂(x) = ^A¹^+A₂ ² =A₁=A₂

◻ A1>A2⇒ player 1 has a profitable deviation: x1−δ.

◻ A₂>A₁⇒ player 1 has a profitable deviation: x₁+δ.

Same: p3(x) =p4(x) =A3=A4

A₂=A₃ ☇

◻ A2<A3⇒ player 2 has a profitable deviation: x3−δ.

◻ A₂>A₃⇒ player 3 has a profitable deviation: x₂+δ.

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Counterexample 2

There exists a pure Nash equilibrium on the unit interval with 4 players and with densityg if and only if g satisfiesQ¹

2

=

Q1 4

+Q3 4

2

0 x ^x+y

2 y 1

A B C D

A=B=C =D⇒x =Q¹

4, y=Q³

4, x+y 2 =Q¹

2

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Definition

A profile of actionsx∶= (x1, . . . ,xn)is a pure additive -equilibrium ofH(n,G,g) if and only if for alli ∈ {1, . . . ,n}and all y∈G, we have:

p_i(x₁, . . . ,x_i₋₁,y,x_i+1, . . . ,x_n) −p_i(x) ≤

Definition

A profile of actionsx ∶= (x1, . . . ,xn) is a pure multiplicative -equilibrium of H(n,G,g) if and only if for alli ∈ {1, . . . ,n} and for ally ∈G we have:

p_i(x₁, . . . ,x_i−1,y,x_i+1, . . . ,x_n) ≤ (1+)p_i(x)

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Asymptotic existence of-equilibrium.

Suppose that:

Ê g isK-Lipschitz

Ë There exist m andM such that for allx, 0<m≤g(x) ≤M Then:

∀>0, ∃N() ∈N, ∀n≥N(),

there exists an −pure equilibrium in the game with n players and density distribution g.

N() ∼ 1

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Sketch of the proof

1/ We approximate the density functiong by a step function ˆg(1), where₁ is a parameter playing a role in the length of the steps.

Becauseg isK-Lipschitz, the step function ˆg(₁) is such that

∥g−gˆ(1)∥

∞≤when 1 is small enough.

2/ We prove that there exists an (exact) equilibrium in pure strategies in the gameH(n,G,gˆ(1)), when the number of players n is larger than a lower-boundN(₁). This lower bound increases when₁ goes to zero.

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3/ If₁ is small enough, the equilibrium constructed in the previous step is a multiplicative-equilibrium in the game H(n,S,g). We obtain therefore a lower boundN()on the number of players n that guarantees the existence of a pure multiplicative

-equilibrium in H(n,G,g).

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The model Results

Probabilistic Hotelling game [Fournier 2016]

Motivation: find a continuous version of the Hotelling game.

Ê n players chose a location in [0,1].

Ë The probability that consumer t∈ [0,1] shops to locationxk

is equal to:

f(∣x_k−t∣)

∑ⁿ_i=1f(∣xi −t∣)

for a given positive and decreasing function f. Ì The payoff of a player k is equal to:

π_k(x₁, . . . ,x_n) = ∫

1 0

f(∣x_k−t∣)

∑ⁿ_i=1f(∣xi−t∣) g(t)dt

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Theorem

Suppose thatf is C², symmetric, strictly positive, decreasing and concave. Then there exists a symmetric equilibrium in pure strategies. This equilibrium is(x, . . . ,x), where x satisfies:

∫

1 0

f^′(x−t)g(t)

f(x−t) dt =0 (1)

i.e:

[ f^′

f ∗g] (x) =0 (2)

Remark:x doesn’t depend on the number of players.

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The model Results

è Simple cases: ifg =1, then x= ¹

2

è In general: either compute ^f_f^′ ∗g, or because π_k(x)is strictly concave inx_k, the best response dynamics is well defined.

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Open questions:

(1) Can we approximate the standard Hotelling game with a sequence of continuous games? We can’t with concave functionf.

(2) Impact of price competition?

(3) Voting models

(4) What if consumers do not always buy?

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The model Results

Theorem

Suppose that consumers are distributed on the real line according toN (0, σ) and that they buy iif v−p−f(d) ≥0⇔d ≤δ. This game always has at least one equilibrium in pure strategies:

(1) _σ^δ <

√

ln 2 2 ,

NE = {(t_A,t_A+2δ) ∣ t_A∈ [α, β] ∈ [−2δ,0]}

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√

ln 2 2 < ^δ

σ <

√

2 ln 2 then

NE = {(−t_A,t_A) ∣ t^A =σ

√

2 ln 2−δ∈ [0, δ]}

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√

2 ln 2< ^δ

σ then

NE = {(0,0)}

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Thank you

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The model Results

Stochastic dominance / Majorization

For a vectorz = (z₁, . . . ,z_n), we denotez_[1]≥ ⋅ ⋅ ⋅ ≥z_[n] its decreasing rearrangement.

Definition

Letx,y ∈ [0,1]ⁿ be such

n

∑

i=1

xi =

n

∑

i=1

yi

if, for allk ∈ {1, . . . ,n}

k

∑

i=1

x_[i_]≤

k

∑

i=1

y_[i]. then we say thatx is majorizedbyy (x ≺y).

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Definition

A functionφ∶Rⁿ→R is saidSchur-convex ifx≺yimplies φ(x) ≤φ(y).

Proposition

Ifψ∶R→Ris a convex function, φ(x₁, . . . ,x_n) =

n

∑

i=1

ψ(x_i), thenφis Schur-convex.

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The model Results

Suppose thatf is C², strictly concave, and bounded below by >0.

Then the functionx_k ↦π_k(x_k,x^−k) is concave and (jointly) continuous.

It follows from theNash-Glicksberg theorem that the game admits an equilibrium in pure strategies.

Moreover, due to the symmetry of the problem, there exists at least one symmetric pure Nash equilibrium.