-
Game Theory and its Applications toNetworks -
Part II: General Games
Corinne Touati
Master ENS Lyon, Fall 2010
-
Game in Extensive Form
An n-person extensive form game is a finite tree structure with:
I A vertex indicating the starting point of the game,
I A pay-off function assigning a n-vector to each terminalvertex of the tree,
I A partition of the nodes of the tree into n + 1 sets (with N i
the set of player i),
I A probability distribution, defined at each vertex of S0, amongthe immediate followers of this vertex,
I A subpartition of each player set N i into information sets ηijsuch that all nodes of a information set has the same numberof children and that no node follows another node of the sameinformation set.
Corinne Touati (INRIA) Strict Competition Games in Extensive Form 2 / 17
-
Game in Extensive Form
An n-person extensive form game is a finite tree structure with:
I A vertex indicating the starting point of the game,
I A pay-off function assigning a n-vector to each terminalvertex of the tree,
I A partition of the nodes of the tree into n + 1 sets (with N i
the set of player i),
I A probability distribution, defined at each vertex of S0, amongthe immediate followers of this vertex,
I A subpartition of each player set N i into information sets ηijsuch that all nodes of a information set has the same numberof children and that no node follows another node of the sameinformation set.
Corinne Touati (INRIA) Strict Competition Games in Extensive Form 2 / 17
Equilibrium
Every finite n-person game with perfectinformation has an equilibrium n-tuple ofstrategies
-
Game in Extensive Form
An n-person extensive form game is a finite tree structure with:
I A vertex indicating the starting point of the game,
I A pay-off function assigning a n-vector to each terminalvertex of the tree,
I A partition of the nodes of the tree into n + 1 sets (with N i
the set of player i),
I A probability distribution, defined at each vertex of S0, amongthe immediate followers of this vertex,
I A subpartition of each player set N i into information sets ηijsuch that all nodes of a information set has the same numberof children and that no node follows another node of the sameinformation set.
Corinne Touati (INRIA) Strict Competition Games in Extensive Form 2 / 17
Be careful
A B
Player 2
RL
Player 1
(2, 1)(0, 0)
(1, 2)
L R
A (0, 0) (2, 1)
B (1, 2) (1, 2)
There are 2 Nash equilibria! ((A,R) and (B,L))BUT, only (A,R) is a subgame perfect equilibrium
-
Surprising Examples
Corinne Touati (INRIA) Strict Competition Games in Extensive Form 3 / 17
Forward Induction
ConcertBook
1, 30, 0
0, 03, 12, 2
Strategic form:B S
Book (2, 2) (2, 2)B (3, 1) (0, 0)S (0, 0) (1, 3)
Chain-Store Game: short-termversus long term
F
CS k
C
OutIn
k
CS
(5, 1)
(2, 2)(0, 0)
Centripede Game
SSSSS
C C C C C C
S(6, 5)
(3, 1) (4, 6)(5, 3)(0, 2)(1, 0) (2, 4)
-
Surprising Examples
Corinne Touati (INRIA) Strict Competition Games in Extensive Form 3 / 17
Forward Induction
Book Concert
1, 30, 0
0, 03, 12, 2
Strategic form:B S
Book (2, 2) (2, 2)B (3, 1) (0, 0)S (0, 0) (1, 3)
Chain-Store Game: short-termversus long term
F
CS k
C
OutIn
k
CS
(5, 1)
(2, 2)(0, 0)
Centripede Game
SSSSS
C C C C C C
S(6, 5)
(3, 1) (4, 6)(5, 3)(0, 2)(1, 0) (2, 4)
-
Stackelberg Equilibrium
Definition.
A Stackelberg game is a two-player extensive game with perfectinformation in which a ”leader” chooses an action from a set A1and a ”follower”, informed of the leader’s choice, chooses anaction from a set A2. The Stackelberg equilibria are solutions ofthe problem:
max(a1,a2)∈A1×A2
u1(a1, a2) s.t. a2 ∈ argmaxa′2∈A2u2(a1, a′2)
Example:Level 1
Level 2
a b c
y n y n y n
(0, 0) (0, 0) (0, 2) (0, 0)(1, 1)(2, 0)
Corinne Touati (INRIA) Strict Competition Games in Extensive Form 4 / 17
-
Game in Normal Form
Definition: Nash Equilibrium.
In a Nash equilibrium, no player has incentive to unilaterallymodify his strategy.
strategy utility
s∗ is a Nash equilibrium iff:
∀p,∀ sp , up(s∗1, . . . , s∗p , . . . s∗n) ≥ up (s∗1, . . . , sp , . . . , s∗n)
In a compact form:∀p,∀sp, up(s∗−p, s∗p) ≥ up(s∗−p, sp)
Corinne Touati (INRIA) Strict Competition Games in Normal Form 5 / 17
-
Nash Equilibrium: Examples
Find the Nash equilibria of these games (with pure strategies)
The prisoner dilemma
collaborate deny
collaborate (1, 1) (3, 0)deny (0, 3) (2, 2)
⇒ not efficient
Battle of the sexes
Paul / Claire Opera Foot
Opera (2, 1) (0, 0)Foot (0, 0) (1, 2)
⇒ not unique
Rock-Scisor-Paper
1/2 P R S
P (0, 0) (1,−1)(−1, 1)R (−1, 1) (0, 0) (1,−1)S (1,−1)(−1, 1) (0, 0)
⇒ No equilibrium
Corinne Touati (INRIA) Strict Competition Games in Normal Form 6 / 17
-
Nash Equilibrium: Examples
Find the Nash equilibria of these games (with pure strategies)
The prisoner dilemma
collaborate deny
collaborate (1, 1) (3, 0)deny (0, 3) (2, 2)⇒ not efficient
Battle of the sexes
Paul / Claire Opera Foot
Opera (2, 1) (0, 0)Foot (0, 0) (1, 2)
⇒ not unique
Rock-Scisor-Paper
1/2 P R S
P (0, 0) (1,−1)(−1, 1)R (−1, 1) (0, 0) (1,−1)S (1,−1)(−1, 1) (0, 0)
⇒ No equilibrium
Corinne Touati (INRIA) Strict Competition Games in Normal Form 6 / 17
-
Nash Equilibrium: Examples
Find the Nash equilibria of these games (with pure strategies)
The prisoner dilemma
collaborate deny
collaborate (1, 1) (3, 0)deny (0, 3) (2, 2)⇒ not efficient
Battle of the sexes
Paul / Claire Opera Foot
Opera (2, 1) (0, 0)Foot (0, 0) (1, 2)⇒ not unique
Rock-Scisor-Paper
1/2 P R S
P (0, 0) (1,−1)(−1, 1)R (−1, 1) (0, 0) (1,−1)S (1,−1)(−1, 1) (0, 0)
⇒ No equilibrium
Corinne Touati (INRIA) Strict Competition Games in Normal Form 6 / 17
-
Nash Equilibrium: Examples
Find the Nash equilibria of these games (with pure strategies)
The prisoner dilemma
collaborate deny
collaborate (1, 1) (3, 0)deny (0, 3) (2, 2)⇒ not efficient
Battle of the sexes
Paul / Claire Opera Foot
Opera (2, 1) (0, 0)Foot (0, 0) (1, 2)⇒ not unique
Rock-Scisor-Paper
1/2 P R S
P (0, 0) (1,−1)(−1, 1)R (−1, 1) (0, 0) (1,−1)S (1,−1)(−1, 1) (0, 0)
⇒ No equilibrium
Corinne Touati (INRIA) Strict Competition Games in Normal Form 6 / 17
-
Nash Equilibrium: Examples
Find the Nash equilibria of these games (with pure strategies)
The prisoner dilemma
collaborate deny
collaborate (1, 1) (3, 0)deny (0, 3) (2, 2)⇒ not efficient
Battle of the sexes
Paul / Claire Opera Foot
Opera (2, 1) (0, 0)Foot (0, 0) (1, 2)⇒ not unique
Rock-Scisor-Paper
1/2 P R S
P (0, 0) (1,−1)(−1, 1)R (−1, 1) (0, 0) (1,−1)S (1,−1)(−1, 1) (0, 0)⇒ No equilibrium
Corinne Touati (INRIA) Strict Competition Games in Normal Form 6 / 17
-
Mixed Nash Equilibria
Corinne Touati (INRIA) Strict Competition Games in Normal Form 7 / 17
Definition: Mixed Strategy Nash Equilibria.
A mixed strategy for player i is a probability distribution over the set ofpure strategies of player i.An equilibrium in mixed strategies is a strategy profile σ∗ of mixedstrategies such that: ∀p,∀σi, up(σ∗−p, σ∗p) ≥ up(σ∗−p, σp).
Theorem 1.
Any finite n-person noncooperative game has at least one equilibriumn-tuple of mixed strategies.
Proof.
Kakutani fixed point theorem: Let S be a non-empty, compact andconvex subset of a Euclidean space. Let f : S → P(S) (the power set ofS) with a closed graph and such that ∀x ∈ S, f(x) is non-empty andconvex. Then f has a fixed point.
Apply Kakutani to f :
{[0, 1]N → P([0, 1]N )σ 7→ ⊗i∈{1,N}Bi(σi)
with Bi(σi)
-
Mixed Nash Equilibria
Corinne Touati (INRIA) Strict Competition Games in Normal Form 7 / 17
Definition: Mixed Strategy Nash Equilibria.
A mixed strategy for player i is a probability distribution over the set ofpure strategies of player i.An equilibrium in mixed strategies is a strategy profile σ∗ of mixedstrategies such that: ∀p,∀σi, up(σ∗−p, σ∗p) ≥ up(σ∗−p, σp).
Theorem 1.
Any finite n-person noncooperative game has at least one equilibriumn-tuple of mixed strategies.
Proof.
Kakutani fixed point theorem: Let S be a non-empty, compact andconvex subset of a Euclidean space. Let f : S → P(S) (the power set ofS) with a closed graph and such that ∀x ∈ S, f(x) is non-empty andconvex. Then f has a fixed point.
Apply Kakutani to f :
{[0, 1]N → P([0, 1]N )σ 7→ ⊗i∈{1,N}Bi(σi)
with Bi(σi)
Consequence:
I The players mixed strategies are independant randomizations.
I In a finite game, up(σ) =∑a
∏p′
σp′(ap′)
ui(a).I Function ui is multilinear
I In a finite game, σ∗ is a Nash equilibrium iff ∀ai in the supportof σi, ai is a best response to σ
∗−i
-
Mixed Nash Equilibria: Examples
Find the Nash equilibria of these games (with mixed strategies)
The prisoner dilemma
collaborate deny
collaborate (1, 1) (3, 0)deny (0, 3) (2, 2)
⇒ No strictly mixed equilibria
Battle of the sexes
Paul / Claire Opera Foot
Opera (2, 1) (0, 0)Foot (0, 0) (1, 2)
σ1 = (2/3, 1/3), σ2 = (1/3, 2/3)
Rock-Scisor-Paper
1/2 P R S
P (0, 0) (1,−1)(−1, 1)R (−1, 1) (0, 0) (1,−1)S (1,−1)(−1, 1) (0, 0)
σ1 = σ2 = (1/3, 1/3, 1/3)
Corinne Touati (INRIA) Strict Competition Games in Normal Form 8 / 17
-
Mixed Nash Equilibria: Examples
Find the Nash equilibria of these games (with mixed strategies)
The prisoner dilemma
collaborate deny
collaborate (1, 1) (3, 0)deny (0, 3) (2, 2)
⇒ No strictly mixed equilibria
Battle of the sexes
Paul / Claire Opera Foot
Opera (2, 1) (0, 0)Foot (0, 0) (1, 2)
σ1 = (2/3, 1/3), σ2 = (1/3, 2/3)
Rock-Scisor-Paper
1/2 P R S
P (0, 0) (1,−1)(−1, 1)R (−1, 1) (0, 0) (1,−1)S (1,−1)(−1, 1) (0, 0)
σ1 = σ2 = (1/3, 1/3, 1/3)
Corinne Touati (INRIA) Strict Competition Games in Normal Form 8 / 17
-
Mixed Nash Equilibria: Examples
Find the Nash equilibria of these games (with mixed strategies)
The prisoner dilemma
collaborate deny
collaborate (1, 1) (3, 0)deny (0, 3) (2, 2)
⇒ No strictly mixed equilibria
Battle of the sexes
Paul / Claire Opera Foot
Opera (2, 1) (0, 0)Foot (0, 0) (1, 2)
σ1 = (2/3, 1/3), σ2 = (1/3, 2/3)
Rock-Scisor-Paper
1/2 P R S
P (0, 0) (1,−1)(−1, 1)R (−1, 1) (0, 0) (1,−1)S (1,−1)(−1, 1) (0, 0)
σ1 = σ2 = (1/3, 1/3, 1/3)
Corinne Touati (INRIA) Strict Competition Games in Normal Form 8 / 17
-
Mixed Nash Equilibria: Examples
Find the Nash equilibria of these games (with mixed strategies)
The prisoner dilemma
collaborate deny
collaborate (1, 1) (3, 0)deny (0, 3) (2, 2)
⇒ No strictly mixed equilibria
Battle of the sexes
Paul / Claire Opera Foot
Opera (2, 1) (0, 0)Foot (0, 0) (1, 2)
σ1 = (2/3, 1/3), σ2 = (1/3, 2/3)
Rock-Scisor-Paper
1/2 P R S
P (0, 0) (1,−1)(−1, 1)R (−1, 1) (0, 0) (1,−1)S (1,−1)(−1, 1) (0, 0)
σ1 = σ2 = (1/3, 1/3, 1/3)
Corinne Touati (INRIA) Strict Competition Games in Normal Form 8 / 17
-
Mixed Nash Equilibria: Examples
Find the Nash equilibria of these games (with mixed strategies)
The prisoner dilemma
collaborate deny
collaborate (1, 1) (3, 0)deny (0, 3) (2, 2)
⇒ No strictly mixed equilibria
Battle of the sexes
Paul / Claire Opera Foot
Opera (2, 1) (0, 0)Foot (0, 0) (1, 2)
σ1 = (2/3, 1/3), σ2 = (1/3, 2/3)
Rock-Scisor-Paper
1/2 P R S
P (0, 0) (1,−1)(−1, 1)R (−1, 1) (0, 0) (1,−1)S (1,−1)(−1, 1) (0, 0)
σ1 = σ2 = (1/3, 1/3, 1/3)Corinne Touati (INRIA) Strict Competition Games in Normal Form 8 / 17
-
Nash Equilibrium with Chance Move
Corinne Touati (INRIA) Strict Competition Chance Moves 9 / 17
Nature take decision w1 or w2 with probability 1/2.(w1) a b
a (0, 0) (6,−3)b (−3, 6) (5, 5)
(w2) a b
a (−20,−20) (−7,−16)b (−16,−7) (−5,−5)
Nash equilibrium if:
I aucun des joueurs ne connâıtl’état:
EN: (b, b), utilité :(0, 0)
I les deux joueurs sont informés:
EN: ((a, a)|w1), ((b, b)|w2),utilité: (−2.5,−2.5)
I Seul le joueur 1 est au courant:
EN: ((a, a)|w1) , ((b, a)|w2),utilité (−8,−3, 5)
Nature
(0, 0) (−5,−5)(−16, 7)(−7,−16)(−20,−20)(5, 5)(−3, 6)(6,−3)
⇒Information can bedetrimental
-
Nash Equilibrium with Chance Move
Corinne Touati (INRIA) Strict Competition Chance Moves 9 / 17
Nature take decision w1 or w2 with probability 1/2.(w1) a b
a (0, 0) (6,−3)b (−3, 6) (5, 5)
(w2) a b
a (−20,−20) (−7,−16)b (−16,−7) (−5,−5)
Nash equilibrium if:
I aucun des joueurs ne connâıtl’état:EN: (b, b), utilité :(0, 0)
I les deux joueurs sont informés:
EN: ((a, a)|w1), ((b, b)|w2),utilité: (−2.5,−2.5)
I Seul le joueur 1 est au courant:
EN: ((a, a)|w1) , ((b, a)|w2),utilité (−8,−3, 5)
Nature
(0, 0) (−5,−5)(−16, 7)(−7,−16)(−20,−20)(5, 5)(−3, 6)(6,−3)
⇒Information can bedetrimental
-
Nash Equilibrium with Chance Move
Corinne Touati (INRIA) Strict Competition Chance Moves 9 / 17
Nature take decision w1 or w2 with probability 1/2.(w1) a b
a (0, 0) (6,−3)b (−3, 6) (5, 5)
(w2) a b
a (−20,−20) (−7,−16)b (−16,−7) (−5,−5)
Nash equilibrium if:
I aucun des joueurs ne connâıtl’état:EN: (b, b), utilité :(0, 0)
I les deux joueurs sont informés:
EN: ((a, a)|w1), ((b, b)|w2),utilité: (−2.5,−2.5)
I Seul le joueur 1 est au courant:
EN: ((a, a)|w1) , ((b, a)|w2),utilité (−8,−3, 5)
Nature
(−5,−5)(0, 0) (6,−3) (−3, 6) (5, 5) (−20,−20)(−7,−16)(−16, 7)
⇒Information can bedetrimental
-
Nash Equilibrium with Chance Move
Corinne Touati (INRIA) Strict Competition Chance Moves 9 / 17
Nature take decision w1 or w2 with probability 1/2.(w1) a b
a (0, 0) (6,−3)b (−3, 6) (5, 5)
(w2) a b
a (−20,−20) (−7,−16)b (−16,−7) (−5,−5)
Nash equilibrium if:
I aucun des joueurs ne connâıtl’état:EN: (b, b), utilité :(0, 0)
I les deux joueurs sont informés:EN: ((a, a)|w1), ((b, b)|w2),utilité: (−2.5,−2.5)
I Seul le joueur 1 est au courant:
EN: ((a, a)|w1) , ((b, a)|w2),utilité (−8,−3, 5)
Nature
(−5,−5)(0, 0) (6,−3) (−3, 6) (5, 5) (−20,−20)(−7,−16)(−16, 7)
⇒Information can bedetrimental
-
Nash Equilibrium with Chance Move
Corinne Touati (INRIA) Strict Competition Chance Moves 9 / 17
Nature take decision w1 or w2 with probability 1/2.(w1) a b
a (0, 0) (6,−3)b (−3, 6) (5, 5)
(w2) a b
a (−20,−20) (−7,−16)b (−16,−7) (−5,−5)
Nash equilibrium if:
I aucun des joueurs ne connâıtl’état:EN: (b, b), utilité :(0, 0)
I les deux joueurs sont informés:EN: ((a, a)|w1), ((b, b)|w2),utilité: (−2.5,−2.5)
I Seul le joueur 1 est au courant:
EN: ((a, a)|w1) , ((b, a)|w2),utilité (−8,−3, 5)
Nature
(−5,−5)(0, 0) (6,−3) (−3, 6) (5, 5) (−20,−20)(−7,−16)(−16, 7)
⇒Information can bedetrimental
-
Nash Equilibrium with Chance Move
Corinne Touati (INRIA) Strict Competition Chance Moves 9 / 17
Nature take decision w1 or w2 with probability 1/2.(w1) a b
a (0, 0) (6,−3)b (−3, 6) (5, 5)
(w2) a b
a (−20,−20) (−7,−16)b (−16,−7) (−5,−5)
Nash equilibrium if:
I aucun des joueurs ne connâıtl’état:EN: (b, b), utilité :(0, 0)
I les deux joueurs sont informés:EN: ((a, a)|w1), ((b, b)|w2),utilité: (−2.5,−2.5)
I Seul le joueur 1 est au courant:EN: ((a, a)|w1) , ((b, a)|w2),utilité (−8,−3, 5)
Nature
(−5,−5)(0, 0) (6,−3) (−3, 6) (5, 5) (−20,−20)(−7,−16)(−16, 7)
⇒Information can bedetrimental
-
Nash Equilibrium with Chance Move
Corinne Touati (INRIA) Strict Competition Chance Moves 9 / 17
Nature take decision w1 or w2 with probability 1/2.(w1) a b
a (0, 0) (6,−3)b (−3, 6) (5, 5)
(w2) a b
a (−20,−20) (−7,−16)b (−16,−7) (−5,−5)
Definition: Bayesian Games.
A Bayesian game consists of:
I (N,A,U), the sets of players, actions and associated utilities
I Ω a set states of natureI For each player:
I A probability measure pi on Ω: a priori belief about the state of nature.I A set of signals TiI A function τi : Ω→ Ti (partial observation)
Then, the posterior belief of player i is
0 if ω 6∈ τ−1i (ti)pi(ω)
pi(τ−1i (ti))
else.
-
Correlated Equilibria
Definition: Correlated Equilibria.
A correlated equilibrium for the bimatrix game A is a correlatedstrategy P s.t.
∀i, ∀k,∑j
aijpij ≥∑j
akjpij
∀i, ∀`,∑i
bijpij ≥∑i
bi`pij
Proposition:
Let (aij , bij){1..N},{1..M} a game. Consider a zero-sum matrixgame C with entries:
c(i,j),(h,k) =
{aij − akj if i = h ∈M,k ∈Mbij − bik if j = h ∈ N, k ∈ N
Then, an optimal mixed strategy for player I of this game is acorrelated equilibrium for the original game.
Corinne Touati (INRIA) Strict Competition Correlated Equilibria 10 / 17
-
Correlated Equilibria
Definition: Correlated Equilibria.
A correlated equilibrium for the bimatrix game A is a correlatedstrategy P s.t.
∀i, ∀k,∑j
aijpij ≥∑j
akjpij
∀i, ∀`,∑i
bijpij ≥∑i
bi`pij
Proposition:
Let (aij , bij){1..N},{1..M} a game. Consider a zero-sum matrixgame C with entries:
c(i,j),(h,k) =
{aij − akj if i = h ∈M,k ∈Mbij − bik if j = h ∈ N, k ∈ N
Then, an optimal mixed strategy for player I of this game is acorrelated equilibrium for the original game.
Corinne Touati (INRIA) Strict Competition Correlated Equilibria 10 / 17
Example
Game
((−10,−10) (5, 0)
(0, 5) (−1,−1)
).
Two pure Nash equilibria (1, 0), (0, 1) and (0, 1), (1, 0).One mixed equilibria (1/16, 5/16), (1/16, 5/16) with an expectedpayoff −5/8.
The correlated equilibria are or the form P =
(p11 p12p21 p22
). with
p11 + p12 + p21 + p22 = 1, max(2.5p11, 0.4p22) ≥ min(p12, p21).
For instance,
(0 0.50.5 0
)gives an expected payoff 2.5.
-
Correlated Equilibria
Definition: Correlated Equilibria.
A correlated equilibrium for the bimatrix game A is a correlatedstrategy P s.t.
∀i, ∀k,∑j
aijpij ≥∑j
akjpij
∀i, ∀`,∑i
bijpij ≥∑i
bi`pij
Proposition:
Let (aij , bij){1..N},{1..M} a game. Consider a zero-sum matrixgame C with entries:
c(i,j),(h,k) =
{aij − akj if i = h ∈M,k ∈Mbij − bik if j = h ∈ N, k ∈ N
Then, an optimal mixed strategy for player I of this game is acorrelated equilibrium for the original game.Corinne Touati (INRIA) Strict Competition Correlated Equilibria 10 / 17
-
Correlated Equilibria
Definition: Correlated Equilibria.
A correlated equilibrium for the bimatrix game A is a correlatedstrategy P s.t.
∀i, ∀k,∑j
aijpij ≥∑j
akjpij
∀i, ∀`,∑i
bijpij ≥∑i
bi`pij
Proposition:
Let (aij , bij){1..N},{1..M} a game. Consider a zero-sum matrixgame C with entries:
c(i,j),(h,k) =
{aij − akj if i = h ∈M,k ∈Mbij − bik if j = h ∈ N, k ∈ N
Then, an optimal mixed strategy for player I of this game is acorrelated equilibrium for the original game.Corinne Touati (INRIA) Strict Competition Correlated Equilibria 10 / 17
Example
Game
((−10,−10) (5, 0)
(0, 5) (−1,−1)
).
Two pure Nash equilibria (1, 0), (0, 1) and (0, 1), (1, 0).One mixed equilibria (1/16, 5/16), (1/16, 5/16) with an expectedpayoff −5/8.
The correlated equilibria are or the form P =
(p11 p12p21 p22
). with
p11 + p12 + p21 + p22 = 1, max(2.5p11, 0.4p22) ≥ min(p12, p21).
For instance,
(0 0.50.5 0
)gives an expected payoff 2.5.
Matrix C =
−10 0 0 −104 0 10 00 10 0 40 −4 −4 0
-
Fictitious Plays Fail
Example
1/2 P R S
P (0, 0) (1,−1)(−1, 1)R (−1, 1) (0, 0) (1,−1)S (1,−1)(−1, 1) (0, 0)
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 0.2 0.4 0.6 0.8 0
Corinne Touati (INRIA) Strict Competition Computing Nash Equilibria 11 / 17
-
Is learning Nash Equilibria simple?
⇒ A natural approach is using Best Response dynamics
This does not always converge to a Nash equilibrium:(17, 8) (4, 9) (3, 4)
(10, 10) (6, 5) (2, 4)
(8, 0) (1, 2) (5,5)
Known to converge in some class of games (for instance, potentialgames)
Corinne Touati (INRIA) Strict Competition Computing Nash Equilibria 12 / 17
-
Is learning Nash Equilibria simple?
⇒ A natural approach is using Best Response dynamics
This does not always converge to a Nash equilibrium:(17, 8) (4, 9) (3, 4)
(10, 10) (6, 5) (2, 4)
(8, 0) (1, 2) (5,5)
Known to converge in some class of games (for instance, potentialgames)
Corinne Touati (INRIA) Strict Competition Computing Nash Equilibria 12 / 17
-
Is learning Nash Equilibria simple?
⇒ A natural approach is using Best Response dynamics
This does not always converge to a Nash equilibrium:(17, 8) (4, 9) (3, 4)
(10, 10) (6, 5) (2, 4)
(8, 0) (1, 2) (5,5)
Known to converge in some class of games (for instance, potentialgames)
Corinne Touati (INRIA) Strict Competition Computing Nash Equilibria 12 / 17
-
Is learning Nash Equilibria simple?
⇒ A natural approach is using Best Response dynamics
This does not always converge to a Nash equilibrium:(17, 8) (4, 9) (3, 4)
(10, 10) (6, 5) (2, 4)
(8, 0) (1, 2) (5,5)
Known to converge in some class of games (for instance, potentialgames)
Corinne Touati (INRIA) Strict Competition Computing Nash Equilibria 12 / 17
-
The Flow Control Problem
Imagine a system with:
I n individual users aiming at optimizing their throughput xnI A routing matrix A giving the set of paths followed by each
connection: Ai,j =
{1 if connection iuses link j0 otherwise
I Capacity constraints on each link C`
I What is the Nash equilibrium of the game? What protocoldoes it corresponds to?
I How can we implement fairness in a distributed way?
Corinne Touati (INRIA) Strict CompetitionThe General Flow Control
Problem 13 / 17
-
The Flow Control Problem:The Non Cooperative Game
Example: A simple network with 3 links
(namSimple.mpeg)
n0→2 = 2, n1→2 = 3, n2→3 = 4Throughput of flow i:
λi.capa
λ1 + λ2
Corinne Touati (INRIA) Strict CompetitionThe General Flow Control
Problem 14 / 17
namSimple.mpegMedia File (video/mpeg)
-
The Flow Control Problem:The Non Cooperative Game
Linksequations:
λ′2 =λ2C
λ2 + λ′1λ′′2 =
λ′2C
λ3 + λ′2= C−λ′3
Throughputreceived byflow 2:
λ′′2 = C−λ3C
λ3 +C
1+λ′1/λ2
=C2
C + λ3(1 + λ′1/λ′2)
1
4
3
2
λ3λ′′1
λ2
λ′1λ1
λ′′2
λ′′2
λ′3
λ′′1
λ′2
Corinne Touati (INRIA) Strict CompetitionThe General Flow Control
Problem 15 / 17
-
The Flow Control Problem:The Non Cooperative Game
(namUDP.mpeg)
Ring topology network, Nidentical links with capacity CSource i uses links i and i+ 1(mod N), hypothesis C >> λ
Exit throughput of flow i:
λ′′ =C2
C + λ(1 + λ′/λ′), and
λ′
λ=
1
2
(√1 +
4C
λ− 1
)∼ Cλ
Then λ′′ ∼ C2
λ
Corinne Touati (INRIA) Strict CompetitionThe General Flow Control
Problem 16 / 17
namCercleUDP.mpegMedia File (video/mpeg)
-
The Flow Control Problem:The Non Cooperative Game
This is network collapse:I The network is fullI Little or no useful information is going through (here
λ′′ ∼ C2
λ→λ→∞ 0)
Observed in 1984 (cf RFC 896) with TCP flows: the protocoldetects a loss, so it retransmits the packet, hence increasing itsincoming throughput...
Since then a flow control mechanism has been impremented inTCP ,
Why hadn’t we observe this kind of phenomena before withtelephony?Corinne Touati (INRIA) Strict Competition
The General Flow ControlProblem 17 / 17
Games in Extensive FormGames in Normal FormChance MovesCorrelated EquilibriaComputing Nash EquilibriaThe General Flow Control Problem