Download Strong Nash equilibria and mixed strategies

Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Strong Nash Equilibria and Mixed Strategies2
Eleonora Braggion, Nicola Gatti, Roberto Lucchetti, Tuomas Sandholm
2 Under
review on Games and Economic Behavior, currently available on arXiv.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Strong Nash Equilibria and Mixed Strategies2
Eleonora Braggion, Nicola Gatti, Roberto Lucchetti, Tuomas Sandholm
Toulouse, France, November 19, 2015.
2 Under
review on Games and Economic Behavior, currently available on arXiv.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Strong Nash Equilibria and Mixed Strategies2
Eleonora Braggion, Nicola Gatti, Roberto Lucchetti, Tuomas Sandholm
Toulouse, France, November 19, 2015.
2 Under
review on Games and Economic Behavior, currently available on arXiv.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
The noncooperative setting
To start with:
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
The noncooperative setting
To start with:
Definition
A non cooperative game, with two players, is (X , Y , f : X × Y → R, g :
X × Y → R).
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
The noncooperative setting
To start with:
Definition
A non cooperative game, with two players, is (X , Y , f : X × Y → R, g :
X × Y → R).
X , Y nonempty (strategy) sets, f , g utility functions.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
The noncooperative setting
To start with:
Definition
A non cooperative game, with two players, is (X , Y , f : X × Y → R, g :
X × Y → R).
X , Y nonempty (strategy) sets, f , g utility functions.
Finite case
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
The noncooperative setting
To start with:
Definition
A non cooperative game, with two players, is (X , Y , f : X × Y → R, g :
X × Y → R).
X , Y nonempty (strategy) sets, f , g utility functions.
Finite case
(a, b)
(e, f )
Braggion, Gatti, Lucchetti, Sandholm
(c, d)
(g, h)
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
The optimistic example
One example
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
The optimistic example
One example
(2, 2)
(0, 0)
Braggion, Gatti, Lucchetti, Sandholm
(0, 0)
(1, 1)
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
The optimistic example
One example
(2, 2)
(0, 0)
(0, 0)
(1, 1)
(2, 2) is the “ideal” solution
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
The optimistic example
One example
(2, 2)
(0, 0)
(0, 0)
(1, 1)
(2, 2) is the “ideal” solution
Also (1, 1) makes sense (Individually stable)
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
The optimistic example
One example
(2, 2)
(0, 0)
(0, 0)
(1, 1)
(2, 2) is the “ideal” solution
Also (1, 1) makes sense (Individually stable)
One more example
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
The optimistic example
One example
(2, 2)
(0, 0)
(0, 0)
(1, 1)
(2, 2) is the “ideal” solution
Also (1, 1) makes sense (Individually stable)
One more example
(2, 3)
(1, 5)
Braggion, Gatti, Lucchetti, Sandholm
(4, 4)
(6, 1)
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Efficiency and Individual Rationality
In non–cooperative setting:
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Efficiency and Individual Rationality
In non–cooperative setting:
Efficiency–versus–Individual rationality
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Efficiency and Individual Rationality
In non–cooperative setting:
Efficiency–versus–Individual rationality
(10, 10)
(15, 3)
Braggion, Gatti, Lucchetti, Sandholm
(3, 15)
(5, 5)
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Efficiency and Individual Rationality
In non–cooperative setting:
Efficiency–versus–Individual rationality
(10, 10)
(15, 3)
(3, 15)
(5, 5)
This is the not very well known prisoner dilemma!
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Price of stability/anarchy
To measure the gap between efficiency and individual rationality:
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Price of stability/anarchy
To measure the gap between efficiency and individual rationality:
1
Some social welfare function is defined
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Price of stability/anarchy
To measure the gap between efficiency and individual rationality:
1
Some social welfare function is defined
2
It is calculated on its maximum M in the game and on the worst/best
value E among the Nash Equilibria
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Price of stability/anarchy
To measure the gap between efficiency and individual rationality:
1
Some social welfare function is defined
2
It is calculated on its maximum M in the game and on the worst/best
value E among the Nash Equilibria
3
the ratio
E
M
is considered.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Price of stability/anarchy
To measure the gap between efficiency and individual rationality:
1
Some social welfare function is defined
2
It is calculated on its maximum M in the game and on the worst/best
value E among the Nash Equilibria
3
the ratio
E
M
is considered.
In the worst case this is the price of anarchy.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Price of stability/anarchy
To measure the gap between efficiency and individual rationality:
1
Some social welfare function is defined
2
It is calculated on its maximum M in the game and on the worst/best
value E among the Nash Equilibria
3
the ratio
E
M
is considered.
In the worst case this is the price of anarchy.
In the best case this is the price of stability.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Price of stability/anarchy
To measure the gap between efficiency and individual rationality:
1
Some social welfare function is defined
2
It is calculated on its maximum M in the game and on the worst/best
value E among the Nash Equilibria
3
the ratio
E
M
is considered.
In the worst case this is the price of anarchy.
In the best case this is the price of stability.
This can be defined for a single game, but more interesting for classes of
games.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
When the game represents a lucky situation for the players
Perfect situation:
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
When the game represents a lucky situation for the players
Perfect situation:
When the price of anarchy is exactly one!
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
When the game represents a lucky situation for the players
Perfect situation:
When the price of anarchy is exactly one!
Excellent situation:
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
When the game represents a lucky situation for the players
Perfect situation:
When the price of anarchy is exactly one!
Excellent situation:
When the price of stability is exactly one!
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Strong Nash equilibria
The idea of strong Nash equilibrium3
3 For
formal definitions, see later.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Strong Nash equilibria
The idea of strong Nash equilibrium3
A strong Nash equilibrium is a strategy profile stable not only with respect
to
3 For
formal definitions, see later.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Strong Nash equilibria
The idea of strong Nash equilibrium3
A strong Nash equilibrium is a strategy profile stable not only with respect
to
unilateral deviations of every single player
3 For
formal definitions, see later.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Strong Nash equilibria
The idea of strong Nash equilibrium3
A strong Nash equilibrium is a strategy profile stable not only with respect
to
unilateral deviations of every single player
but also with respect to
3 For
formal definitions, see later.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Strong Nash equilibria
The idea of strong Nash equilibrium3
A strong Nash equilibrium is a strategy profile stable not only with respect
to
unilateral deviations of every single player
but also with respect to
unilateral deviations of every subcoalition of players
3 For
formal definitions, see later.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Strong Nash equilibria
The idea of strong Nash equilibrium3
A strong Nash equilibrium is a strategy profile stable not only with respect
to
unilateral deviations of every single player
but also with respect to
unilateral deviations of every subcoalition of players
Existence of a strong Nash equilibrium: price of stability =1
3 For
formal definitions, see later.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Goal of the paper
Existence of strong Nash equilibria is not guaranteed for standard finite
games
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Goal of the paper
Existence of strong Nash equilibria is not guaranteed for standard finite
games
Question
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Goal of the paper
Existence of strong Nash equilibria is not guaranteed for standard finite
games
Question
“How many games” do possess strong Nash
equilibria?
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Goal of the paper
Existence of strong Nash equilibria is not guaranteed for standard finite
games
Question
“How many games” do possess strong Nash
equilibria?
Conjecture
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Goal of the paper
Existence of strong Nash equilibria is not guaranteed for standard finite
games
Question
“How many games” do possess strong Nash
equilibria?
Conjecture
The set of games with strong Nash equilibria is
“small”
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Definition
Strategic form game (N, A, U):
• N = {1, . . . , n} is the set of the players,
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Definition
Strategic form game (N, A, U):
• N = {1, . . . , n} is the set of the players,
• A = {A1 , . . . , An } is the set of aggregate agents’ actions: Ai is the
set of agent i’s actions
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Definition
Strategic form game (N, A, U):
• N = {1, . . . , n} is the set of the players,
• A = {A1 , . . . , An } is the set of aggregate agents’ actions: Ai is the
set of agent i’s actions
• U = {U1 , . . . , Un } is the set of aggregate agents’ utility arrays where
Ui : A → R is agent i’s utility function.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Definition
Strategic form game (N, A, U):
• N = {1, . . . , n} is the set of the players,
• A = {A1 , . . . , An } is the set of aggregate agents’ actions: Ai is the
set of agent i’s actions
• U = {U1 , . . . , Un } is the set of aggregate agents’ utility arrays where
Ui : A → R is agent i’s utility function.
1
mi : number of actions in Ai , aij , j ∈ {1, ..., mi }: a generic action
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Definition
Strategic form game (N, A, U):
• N = {1, . . . , n} is the set of the players,
• A = {A1 , . . . , An } is the set of aggregate agents’ actions: Ai is the
set of agent i’s actions
• U = {U1 , . . . , Un } is the set of aggregate agents’ utility arrays where
Ui : A → R is agent i’s utility function.
1
2
mi : number of actions in Ai , aij , j ∈ {1, ..., mi }: a generic action
U is a n × m1 × m2 × ... × mn n-matrix. An element of Ui denoted by
Ui (i1 , . . . , in ), i1 = aj1 , . . . , in = ajn
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Definition
Strategic form game (N, A, U):
• N = {1, . . . , n} is the set of the players,
• A = {A1 , . . . , An } is the set of aggregate agents’ actions: Ai is the
set of agent i’s actions
• U = {U1 , . . . , Un } is the set of aggregate agents’ utility arrays where
Ui : A → R is agent i’s utility function.
1
2
3
mi : number of actions in Ai , aij , j ∈ {1, ..., mi }: a generic action
U is a n × m1 × m2 × ... × mn n-matrix. An element of Ui denoted by
Ui (i1 , . . . , in ), i1 = aj1 , . . . , in = ajn
∆i is the simplex of the mixed strategies over Ai , xi a mixed strategy
of agent i: (xi1 , ..., ximi ). A strategy profile is x = {x1 , ..., xn }
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Definition
Strategic form game (N, A, U):
• N = {1, . . . , n} is the set of the players,
• A = {A1 , . . . , An } is the set of aggregate agents’ actions: Ai is the
set of agent i’s actions
• U = {U1 , . . . , Un } is the set of aggregate agents’ utility arrays where
Ui : A → R is agent i’s utility function.
1
2
3
4
mi : number of actions in Ai , aij , j ∈ {1, ..., mi }: a generic action
U is a n × m1 × m2 × ... × mn n-matrix. An element of Ui denoted by
Ui (i1 , . . . , in ), i1 = aj1 , . . . , in = ajn
∆i is the simplex of the mixed strategies over Ai , xi a mixed strategy
of agent i: (xi1 , ..., ximi ). A strategy profile is xP= {x1 , ..., xn }
For a strategy profile
Qx , utility of i is vi (x ) = i1 ,...,in Ui (i1 , . . . , in ) ·
xi1 · · · · · xin := xit Ui j6=i xj
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Nash equilibrium
Definition
A strategy profile x is a Nash equilibrium if for every i ∈ N,
vi (x ) ≥ vi (xi , x −i )
Braggion, Gatti, Lucchetti, Sandholm
∀xi ∈ ∆i .
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Nash equilibrium
Definition
A strategy profile x is a Nash equilibrium if for every i ∈ N,
vi (x ) ≥ vi (xi , x −i )
∀xi ∈ ∆i .
A strategy profile is a Nash equilibrium if it is stable w.r.t. deviations of a
single player
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Nash equilibrium
Finding a NE can be expressed as the problem of finding a profile strategy
x such as:
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Nash equilibrium
Finding a NE can be expressed as the problem of finding a profile strategy
x such as:
Ui |Si
Y
xj = vi∗ · 1
∀i ∈ N
(1a)
j6=i
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Nash equilibrium
Finding a NE can be expressed as the problem of finding a profile strategy
x such as:
Ui |Si
Y
xj = vi∗ · 1
∀i ∈ N
(1a)
xj ≤ vi∗ · 1
∀i ∈ N
(1b)
j6=i
Ui |Sic
Y
j6=i
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Nash equilibrium
Finding a NE can be expressed as the problem of finding a profile strategy
x such as:
Ui |Si
Y
xj = vi∗ · 1
∀i ∈ N
(1a)
xj ≤ vi∗ · 1
∀i ∈ N
(1b)
xij ≥
∀i ∈ N, ∀j ∈ {1, .., mi }
(1c)
j6=i
Ui |Sic
Y
j6=i
0
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Nash equilibrium
Finding a NE can be expressed as the problem of finding a profile strategy
x such as:
Ui |Si
Y
xj = vi∗ · 1
∀i ∈ N
(1a)
xj ≤ vi∗ · 1
∀i ∈ N
(1b)
xij ≥
·1=
∀i ∈ N, ∀j ∈ {1, .., mi }
∀i ∈ N
(1c)
(1d)
j6=i
Ui |Sic
Y
j6=i
xi>
0
1
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Nash equilibrium
Finding a NE can be expressed as the problem of finding a profile strategy
x such as:
Ui |Si
Y
xj = vi∗ · 1
∀i ∈ N
(1a)
xj ≤ vi∗ · 1
∀i ∈ N
(1b)
xij ≥
·1=
∀i ∈ N, ∀j ∈ {1, .., mi }
∀i ∈ N
(1c)
(1d)
j6=i
Ui |Sic
Y
j6=i
xi>
0
1
(1a) is called the Indifference principle
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Pareto Efficiency
Let V : ∆ → Rn , V = (v1 , . . . , vn ), let x̄ = (x̄1 , . . . , x̄n ) be a strategy
profile
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Pareto Efficiency
Let V : ∆ → Rn , V = (v1 , . . . , vn ), let x̄ = (x̄1 , . . . , x̄n ) be a strategy
profile
Definition
x̄ is weakly Pareto dominated if there exists a strategy profile x such that
V (x ) 6= V (x̄ )
∧
V (x ) ∈ V (x̄ ) + Rn+ ,
strictly Pareto dominated if there exists a strategy profile x such that
V (x ) ∈ V (x̄ ) + int Rn+ .
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Pareto Efficiency
Let V : ∆ → Rn , V = (v1 , . . . , vn ), let x̄ = (x̄1 , . . . , x̄n ) be a strategy
profile
Definition
x̄ is weakly Pareto dominated if there exists a strategy profile x such that
V (x ) 6= V (x̄ )
∧
V (x ) ∈ V (x̄ ) + Rn+ ,
strictly Pareto dominated if there exists a strategy profile x such that
V (x ) ∈ V (x̄ ) + int Rn+ .
Definition
x̄ is strictly Pareto efficient if there exists no strategy profile x weakly
Pareto dominating x̄ . x̄ is weakly Pareto efficient if there exists no strategy
profile x strictly Pareto dominating x̄ .
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Pareto Efficiency and KKT conditions
Consider the problem (F,G,H):
max F (x ) : G(x ) ≥ 0, H(x ) = 0
where F : Rk → Rl , G : Rn → Rj , H : Rn → Rs , G, H affine and max is
intended in weak Pareto sense. Then:
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Pareto Efficiency and KKT conditions
Consider the problem (F,G,H):
max F (x ) : G(x ) ≥ 0, H(x ) = 0
where F : Rk → Rl , G : Rn → Rj , H : Rn → Rs , G, H affine and max is
intended in weak Pareto sense. Then:
KKT Conditions: Suppose x is (weakly) efficient for the problem
(F,G,H). Then there are vectors λ, µ, ν verifying the following system:
k
X
i=1
λi ∇fi (x ) +
m
X
µj ∇gj (x ) +
j=1
Braggion, Gatti, Lucchetti, Sandholm
m
X
νj ∇hj (x )
=
0
(2a)
(λ, µ) ≥
0
(2b)
>
j=1
µ g(x )
=
0
(2c)
λ
6=
0
(2d)
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
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The two player case
The case with more players
Strong Nash Equilibrium
Definition
x̄ is a strong Nash equilibrium if it is a Nash equilibrium and weakly
Pareto efficient with respect to all subcoalitions of players. x̄ is a super
strong Nash equilibrium if it is a Nash equilibrium and strictly Pareto
efficient with respect to all subcoalitions of players.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Strong Nash Equilibrium
Definition
x̄ is a strong Nash equilibrium if it is a Nash equilibrium and weakly
Pareto efficient with respect to all subcoalitions of players. x̄ is a super
strong Nash equilibrium if it is a Nash equilibrium and strictly Pareto
efficient with respect to all subcoalitions of players.
Necessary conditions (for a strong Nash):
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Strong Nash Equilibrium
Definition
x̄ is a strong Nash equilibrium if it is a Nash equilibrium and weakly
Pareto efficient with respect to all subcoalitions of players. x̄ is a super
strong Nash equilibrium if it is a Nash equilibrium and strictly Pareto
efficient with respect to all subcoalitions of players.
Necessary conditions (for a strong Nash):
• Indifference principle
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Strong Nash Equilibrium
Definition
x̄ is a strong Nash equilibrium if it is a Nash equilibrium and weakly
Pareto efficient with respect to all subcoalitions of players. x̄ is a super
strong Nash equilibrium if it is a Nash equilibrium and strictly Pareto
efficient with respect to all subcoalitions of players.
Necessary conditions (for a strong Nash):
• Indifference principle
• KKT (for every subcoalition)
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Back to the main question
Let G be the space of all games: G ' Rm1 +···+mn , let SN ⊂ G the subset
of games having (at least) a strong Nash equilibrium:
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Back to the main question
Let G be the space of all games: G ' Rm1 +···+mn , let SN ⊂ G the subset
of games having (at least) a strong Nash equilibrium:
Is SN “small” inside G? f.i. in the Baire category sense
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Back to the main question
Let G be the space of all games: G ' Rm1 +···+mn , let SN ⊂ G the subset
of games having (at least) a strong Nash equilibrium:
Is SN “small” inside G? f.i. in the Baire category sense
It is not! Of course
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Back to the main question
Let G be the space of all games: G ' Rm1 +···+mn , let SN ⊂ G the subset
of games having (at least) a strong Nash equilibrium:
Is SN “small” inside G? f.i. in the Baire category sense
It is not! Of course
(1, 1)
(0, 0)
Braggion, Gatti, Lucchetti, Sandholm
(0, 0)
(0, 0)
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Back to the main question
Let G be the space of all games: G ' Rm1 +···+mn , let SN ⊂ G the subset
of games having (at least) a strong Nash equilibrium:
Is SN “small” inside G? f.i. in the Baire category sense
It is not! Of course
(1, 1)
(0, 0)
(0, 0)
(0, 0)
End of the story?
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Back to the main question
Let G be the space of all games: G ' Rm1 +···+mn , let SN ⊂ G the subset
of games having (at least) a strong Nash equilibrium:
Is SN “small” inside G? f.i. in the Baire category sense
It is not! Of course
(1, 1)
(0, 0)
(0, 0)
(0, 0)
End of the story?
Maybe not
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Back to the main question
Let G be the space of all games: G ' Rm1 +···+mn , let SN ⊂ G the subset
of games having (at least) a strong Nash equilibrium:
Is SN “small” inside G? f.i. in the Baire category sense
It is not! Of course
(1, 1)
(0, 0)
(0, 0)
(0, 0)
End of the story?
Maybe not
Example above: pure strategy SNE. What about mixed strategy SNEs?
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
The case of a fully mixed SNE (two players)
Let (x , y ) be a SNE. We assume U1 y = 0, x t U2 = 0. a is vector, of the
right dimension, whose entries are all a’s.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
The case of a fully mixed SNE (two players)
Let (x , y ) be a SNE. We assume U1 y = 0, x t U2 = 0. a is vector, of the
right dimension, whose entries are all a’s.
Proposition
Let x be a fully mixed strong Nash equilibrium. Then, for some λ = (λ1 , λ2 )
and ν = (ν1 , ν2 ):
λ2 x2t U2 − ν1 1 = 0
(3)
λ1 U1t x1
− ν2 1 = 0
(4)
λ>0
(5)
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
The case of a fully mixed SNE (two players)
Let (x , y ) be a SNE. We assume U1 y = 0, x t U2 = 0. a is vector, of the
right dimension, whose entries are all a’s.
Proposition
Let x be a fully mixed strong Nash equilibrium. Then, for some λ = (λ1 , λ2 )
and ν = (ν1 , ν2 ):
λ2 x2t U2 − ν1 1 = 0
(3)
λ1 U1t x1
− ν2 1 = 0
(4)
λ>0
(5)
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
A further step
In the result above the multiplier λ is non null.
Proposition
Let x be a fully mixed super strong Nash equilibrium. Then it satisfies (3)
(4), and also the relations
U1t x1 = 0,
∧
U2t (x2 ) = 0
with both λ1 and λ2 positive.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
A further step
In the result above the multiplier λ is non null.
Proposition
Let x be a fully mixed super strong Nash equilibrium. Then it satisfies (3)
(4), and also the relations
U1t x1 = 0,
∧
U2t (x2 ) = 0
with both λ1 and λ2 positive. Let x be a fully mixed strong Nash equilibrium,
satisfying the system (3) (4). Then either it satisfies the further conditions
U1t x1 = 0,
∧
U2t (x2 ) = 0,
or else all entries of the bimatrix (U1 , U2 ) lie on a either vertical or horizontal
line through the origin.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
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The case with more players
As a consequence a super strong Nash equilibrium profile x verifies the
system:
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
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The two player case
The case with more players
As a consequence a super strong Nash equilibrium profile x verifies the
system:
U1t x1 = 0, U2t x1 = 0
∧
Braggion, Gatti, Lucchetti, Sandholm
U2 (x2 ) = 0, U2 (x2 ) = 0
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
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The two player case
The case with more players
The main result
Theorem
Let x be a fully mixed SSNE. Then all outcomes in the utility bimatrix U
lie on the same straight line, having non-positive slope.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
The main result
Theorem
Let x be a fully mixed SSNE. Then all outcomes in the utility bimatrix U
lie on the same straight line, having non-positive slope.
Since having all outcomes on the same line means that the game is strictly
competitive
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
The main result
Theorem
Let x be a fully mixed SSNE. Then all outcomes in the utility bimatrix U
lie on the same straight line, having non-positive slope.
Since having all outcomes on the same line means that the game is strictly
competitive
A fully mixed NE is a SNE only when this is trivially true
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
A further step
Theorem
Let x be s1 × s2 mixed-strategy SNE. Then in the s1 × s2 restriction of the
bimatrix U where the outcomes are played with positive probability all the
outcomes lie on the same straight line, with non-positive slope.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
A further step
Theorem
Let x be s1 × s2 mixed-strategy SNE. Then in the s1 × s2 restriction of the
bimatrix U where the outcomes are played with positive probability all the
outcomes lie on the same straight line, with non-positive slope.
Obvious consequence
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
A further step
Theorem
Let x be s1 × s2 mixed-strategy SNE. Then in the s1 × s2 restriction of the
bimatrix U where the outcomes are played with positive probability all the
outcomes lie on the same straight line, with non-positive slope.
Obvious consequence
The set of finite, two player games having a strong Nash equilibrium in
mixed strategies is small
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
SNE complexity
Definition
The SNE–EXISTENCE PROBLEM is defined as:
• input: a game instance
• output: YES if an SNE exists, NO otherwise
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
SNE complexity
Definition
The SNE–EXISTENCE PROBLEM is defined as:
• input: a game instance
• output: YES if an SNE exists, NO otherwise
Theorem
The SNE–EXISTENCE PROBLEM is N P–hard even in two–player
symmetric games [Conitzer and Sandholm, 2007].
There is no efficient algorithm for finding an SNE even in two–player
symmetric games
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Smoothed P complexity
Definition
A problem is in Smoothed–P if and only if there is an algorithm that, once
each entry of the input has been independently perturbed with [−σ, +σ]
with a probability measure D onto [−σ, +σ], has polynomial expected
(w.r.t. D) compute time.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Smoothed P complexity
Definition
A problem is in Smoothed–P if and only if there is an algorithm that, once
each entry of the input has been independently perturbed with [−σ, +σ]
with a probability measure D onto [−σ, +σ], has polynomial expected
(w.r.t. D) compute time.
Smoothed–P problems are problems that are, practically, easy
A necessary condition for a problem to be in Smoothed–P is that hard
instances are confined in a null–measure set
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Designing a Smoothed–P algorithm (1)
We use two conditions computable in polynomial time:
Condition 1 := “there is 2x2 sub bimatrix of (U1 , U2t ) in which all the
entries lie on a line”
Condition 2 := “there is 2x1 sub bimatrix of (U1 , U2t ) in which all the
entries lie on a vertical line or there is 1x2 sub bimatrix of
(U1 , U2t ) in which all the entries lie on a horizontal line”
We use two oracles computable in polynomial time:
Nash existence given S̄ given a bimatrix game (U1 , U2 ) and a support S̄,
returns a Nash equilibrium if it exists [Porter, Nudelman,
Shoham, 2004]
Pareto verification given a bimatrix game (U1 , U2 ) and a strategy profile x̄ ,
returns YES if x̄ is Pareto efficient [Gatti, Rocco, Sandholm,
2013]
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
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Our goal
The two player case
The case with more players
Designing a Smoothed–P algorithm (2)
1: for all pure–strategy profiles x do
2:
if x is a Nash equilibrium then
3:
if x is Pareto efficient then
4:
return x
5:
end if
6:
end if
7: end for
8: if Condition 1 holds or Condition 2 holds then
9:
for all support profiles S̄ do
10:
if there is a Nash equilibrium x ∗ with S(x ) = S̄ (in case of multiple equilibria take x ∗
as the equilibrium maximizing the social welfare) then
11:
if x ∗ is Pareto efficient then
12:
return x ∗
13:
end if
14:
end if
15:
end for
16: end if
17: return NonExistence
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
SNE complexity (new)
Theorem
Finding an SNE in bimatrix games is in Smoothed–P.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
SNE complexity (new)
Theorem
Finding an SNE in bimatrix games is in Smoothed–P.
Curiously, with bimatrix games, finding a Nash equilibrium
(PPAD–complete problem) is easier than finding an SNE (FN P–hard
problem), but, in terms of Smoothed complexity, finding an SNE is easier
than finding a Nash equilibrium (it is not in Smoothed–P)
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
An example with more players
Player one chooses a row, Player two a column, and Player three the
matrix to play.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
An example with more players
Player one chooses a row, Player two a column, and Player three the
matrix to play.
(2, 0, 0) (0, 2, 0)
(0, 0, 2) (0, 0, 0)
(0, 0, 0)
,
(0, 2, 0)
(0, 0, 2)
(2, 0, 0)
Easy to see:
• using equal probabilities for all players is a Nash Equilibrium
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
An example with more players
Player one chooses a row, Player two a column, and Player three the
matrix to play.
(2, 0, 0) (0, 2, 0)
(0, 0, 2) (0, 0, 0)
(0, 0, 0)
,
(0, 2, 0)
(0, 0, 2)
(2, 0, 0)
Easy to see:
• using equal probabilities for all players is a Nash Equilibrium
• the outcomes do not lie on a plane
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
An example with more players
Player one chooses a row, Player two a column, and Player three the
matrix to play.
(2, 0, 0) (0, 2, 0)
(0, 0, 2) (0, 0, 0)
(0, 0, 0)
,
(0, 2, 0)
(0, 0, 2)
(2, 0, 0)
Easy to see:
• using equal probabilities for all players is a Nash Equilibrium
• the outcomes do not lie on a plane
• the subgroups of two players have no incentive to deviate.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
An example with more players
Player one chooses a row, Player two a column, and Player three the
matrix to play.
(2, 0, 0) (0, 2, 0)
(0, 0, 2) (0, 0, 0)
(0, 0, 0)
,
(0, 2, 0)
(0, 0, 2)
(2, 0, 0)
Easy to see:
• using equal probabilities for all players is a Nash Equilibrium
• the outcomes do not lie on a plane
• the subgroups of two players have no incentive to deviate.
Less easy to verify that the three players together do not have incentive to
deviate, but believe me
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
An example with more players
Player one chooses a row, Player two a column, and Player three the
matrix to play.
(2, 0, 0) (0, 2, 0)
(0, 0, 2) (0, 0, 0)
(0, 0, 0)
,
(0, 2, 0)
(0, 0, 2)
(2, 0, 0)
Easy to see:
• using equal probabilities for all players is a Nash Equilibrium
• the outcomes do not lie on a plane
• the subgroups of two players have no incentive to deviate.
Less easy to verify that the three players together do not have incentive to
deviate, but believe me Observe: a Pareto dominated outcome is played
here with positive probability.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
And null measure?
Observe, in the two player case the system a SNE must fulfill is linear.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
And null measure?
Observe, in the two player case the system a SNE must fulfill is linear. This is no longer
true for more than two players.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
And null measure?
Observe, in the two player case the system a SNE must fulfill is linear. This is no longer
true for more than two players.
Definition
A subset A of an Euclidean space is called algebraic if it can be described
as a finite number of polynomial equations. It is called semialgebraic
if it can be described as a finite number of polynomial equalities and
inequalities. A multivalued map between Euclidean spaces is called algebraic
(semialgebraic) if its graph is an algebraic (semialgebraic) set.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
And null measure?
Observe, in the two player case the system a SNE must fulfill is linear. This is no longer
true for more than two players.
Definition
A subset A of an Euclidean space is called algebraic if it can be described
as a finite number of polynomial equations. It is called semialgebraic
if it can be described as a finite number of polynomial equalities and
inequalities. A multivalued map between Euclidean spaces is called algebraic
(semialgebraic) if its graph is an algebraic (semialgebraic) set.
Two basic facts on semialgebraic multimaps
• Given an algebraic set A on X × Y its projection on each space X , Y
is semialgebraic
• For any semialgebraic set-valued mapping Φ between two Euclidean
spaces Φ : E ⇒ Y, if dim Φ (x ) ≤ k for every x ∈ E, then dim Φ (E) ≤
dim E + k.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
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The two player case
The case with more players
The final theorem
Theorem
In a m1 × m2 × . . . × mn game Γ := (U1 , U2 , . . . , Un ), a 2–strong Nash
equilibrium where at least one player randomizes over at least two actions
only exists for a null measure set of games.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Idea of the proof
Idea of the proof The three player case, but only for notational convenience and
m3 ≥ max{m1 , m2 }
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Idea of the proof
Idea of the proof The three player case, but only for notational convenience and
m3 ≥ max{m1 , m2 }
Consider the coalitions made by two players, apply the indifference
principle and KKT conditions: the 2-SSNE must satisfy:


U1 x1 x2 = 0
U2 x1 x2 = 0


U3 x1 x2 = 0
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
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The two player case
The case with more players
Proof:continued
3m
Define the map: Φ : 4m1 × 4m2 ⇒ (Mm1 ×m2 ) 3 defined by
Φ (x1 , x2 ) = (A1 , A2 , ..., A3m3 ) : x1t Ai x2 = 0∀i ,
where Ai are the lines of the equations in the system.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Proof:continued
3m
Define the map: Φ : 4m1 × 4m2 ⇒ (Mm1 ×m2 ) 3 defined by
Φ (x1 , x2 ) = (A1 , A2 , ..., A3m3 ) : x1t Ai x2 = 0∀i ,
where Ai are the lines of the equations in the system.
Observe that the graph of Φ is algebraic and so the set of interest is
semialgebraic.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Proof:continued
3m
Define the map: Φ : 4m1 × 4m2 ⇒ (Mm1 ×m2 ) 3 defined by
Φ (x1 , x2 ) = (A1 , A2 , ..., A3m3 ) : x1t Ai x2 = 0∀i ,
where Ai are the lines of the equations in the system.
Observe that the graph of Φ is algebraic and so the set of interest is
semialgebraic.
Make a calculation of the dimension of Φ(4m1 × 4m2 ) and see that it is
less than 3m1 m2 m3 .
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies
Preliminaries
Efficiency versus rationality
Our goal
The two player case
The case with more players
Final remark
• Let Γ := (U1 , U2t ) be a bimatrix game. The problem of finding a strong
Nash equilibrium of Γ is in Smoothed–P
• Antecedent: Inefficiency of Nash equilibria: Dubey, MOR 1986. He
studies the same problem, in the space of the C 2 functions on the
symplexes, and the natural norm. Application to the finite case are
given. He also shows generic finiteness of the set of Nash equilibria.
Braggion, Gatti, Lucchetti, Sandholm
Strong Nash Equilibria and Mixed Equilibrium Strategies