The two player case
The case with more players
Strong Nash Equilibria in Finite Games
Braggion, Gatti, Lucchetti
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Strong Nash Equilibria in Finite Games
Braggion, Gatti, Lucchetti
Sestri Levante, September 10, 2014
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Strong Nash Equilibria in Finite Games
Braggion, Gatti, Lucchetti
Sestri Levante, September 10, 2014
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Efficiency and Individual Rationality
In non–cooperative setting:
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Efficiency and Individual Rationality
In non–cooperative setting:
Efficiency–versus–Individual rationality
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
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,L.
(3, 15)
(5, 5)
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Price of stability/anarchy
To measure this gap between efficiency and individual rationality:
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Price of stability/anarchy
To measure this gap between efficiency and individual rationality:
1
Some social welfare function is defined
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Price of stability/anarchy
To measure this 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,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Price of stability/anarchy
To measure this 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,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Price of stability/anarchy
To measure this 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,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Price of stability/anarchy
To measure this 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,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Price of stability/anarchy
To measure this 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,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Some social welfare functions
The most common welfare functions W : S → R, where
1
S denotes the set of the outcomes of the game
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Some social welfare functions
The most common welfare functions W : S → R, where
1
S denotes the set of the outcomes of the game
2
N is the set of the players
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Some social welfare functions
The most common welfare functions W : S → R, where
1
S denotes the set of the outcomes of the game
2
N is the set of the players
3
ui is the utility function of player i.
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Some social welfare functions
The most common welfare functions W : S → R, where
1
S denotes the set of the outcomes of the game
2
N is the set of the players
3
ui is the utility function of player i.
• W (s) =
P
i∈N
ui (s), (utilitarian objective)
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Some social welfare functions
The most common welfare functions W : S → R, where
1
S denotes the set of the outcomes of the game
2
N is the set of the players
3
ui is the utility function of player i.
• W (s) =
P
i∈N
ui (s), (utilitarian objective)
• W (s) = mini∈N ui (s), (fairness or egalitarian objective).
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
When the game represents a lucky situation for the players
Perfect situation:
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Strong Nash equilibria
The idea of strong Nash equilibrium1
1 For
formal definitions, see later
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Strong Nash equilibria
The idea of strong Nash equilibrium1
A strong Nash equilibrium is a strategy profile stable not only with respect
to
1 For
formal definitions, see later
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Strong Nash equilibria
The idea of strong Nash equilibrium1
A strong Nash equilibrium is a strategy profile stable not only with respect
to
unilateral deviations of every single player
1 For
formal definitions, see later
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Strong Nash equilibria
The idea of strong Nash equilibrium1
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
1 For
formal definitions, see later
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Strong Nash equilibria
The idea of strong Nash equilibrium1
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.
1 For
formal definitions, see later
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Strong Nash equilibria
The idea of strong Nash equilibrium1
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.
1 For
formal definitions, see later
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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 posses strong Nash
equilibria?
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
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 posses strong Nash
equilibria?
Conjecture
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
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 posses strong Nash
equilibria?
Conjecture
The set of games with strong Nash equilibria is
“small”.
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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 x = {x1 , ..., xn }
For a strategy profile x , utility of i is
X
Y
vi (x ) =
Ui (i1 , . . . , in ) · xi1 · · · · · xin := xit Ui
xj .
i1 ,...,in
Braggion, Gatti,L.
j6=i
Strong Nash Equilibria in finite Games
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,L.
∀xi ∈ ∆i .
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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
j6=i
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
(1a)
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,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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 ≥
0
∀i ∈ N, ∀j ∈ {1, .., mi }
(1c)
x1> · 1 =
1
∀i ∈ N
(1d)
j6=i
Ui |Sic
Y
j6=i
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
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 ≥
0
∀i ∈ N, ∀j ∈ {1, .., mi }
(1c)
x1> · 1 =
1
∀i ∈ N
(1d)
j6=i
Ui |Sic
Y
j6=i
(1a) is called the Indifference principle.
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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,L.
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 in finite Games
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 superstrong
Nash equilibrium if it is a Nash equilibrium and strictly Pareto efficient
with respect to all subcoalitions of players.
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
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 superstrong
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,L.
Strong Nash Equilibria in finite Games
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 superstrong
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,L.
Strong Nash Equilibria in finite Games
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 superstrong
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,L.
Strong Nash Equilibria in finite Games
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 a(t least) strong Nash equilibrium:
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
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 a(t least) strong Nash equilibrium:
Is SN “small” inside G? f.i. in the Baire category sense
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
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 a(t least) strong Nash equilibrium:
Is SN “small” inside G? f.i. in the Baire category sense
It is not! Of course
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
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 a(t least) 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,L.
(0, 0)
(0, 0)
Strong Nash Equilibria in finite Games
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 a(t least) 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,L.
Strong Nash Equilibria in finite Games
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 a(t least) 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,L.
Strong Nash Equilibria in finite Games
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 a(t least) 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 SNE?
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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 = (µ1 , µ2 ):
λ2 x2t U2 − ν1 1 = 0
λ1 x1t U1− ν2 1
= 0
λ>0
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
(3)
(4)
(5)
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 = (µ1 , µ2 ):
λ2 x2t U2 − ν1 1 = 0
λ1 x1t U1− ν2 1
= 0
λ>0
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
(3)
(4)
(5)
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
U1> x1 = 0,
∧
U2t (x2 ) = 0
with both λ1 and λ2 positive.
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
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
U1> 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
U1> 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,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
As a consequence a superstrong Nash equilibrium profile x verifies the
system:
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
As a consequence a superstrong Nash equilibrium profile x verifies the
system:
U1> x1 = 0, U2> x1 = 0
∧
Braggion, Gatti,L.
U2 (x2 ) = 0, U2 (x2 ) = 0.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
The main result
Theorem
Let x be a fully mixed SNE. Then all outcomes in the utility bimatrix U lie
on the same straight line, having non-positive slope.
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
The main result
Theorem
Let x be a fully mixed SNE. 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,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
The main result
Theorem
Let x be a fully mixed SNE. 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,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Idea of the proof
• All outcomes on the same row/column must lie on the same line
(efficiency)
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Idea of the proof
• All outcomes on the same row/column must lie on the same line
(efficiency)
• From above all lines pass through the Nash equilibrium payoff (0, 0)
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Idea of the proof
• All outcomes on the same row/column must lie on the same line
(efficiency)
• From above all lines pass through the Nash equilibrium payoff (0, 0)
• From above the only non trivial case to consider is when a situation
of this type occurs:
Ū =
U1j
(0, 0)
(0, 0) U2k
Suppose there is t > 0 such that
tU1j + (1 − t)U2k = (2a, 2a)
for some a > 0. Consider the strategy profile x̄ = [(t, 1 − t), ( 21 , 12 )].
Then x̄1t Ūi x̄1 = a > 0 for i = 1, 2, contradicting the fact that x is a
strong Nash equilibrium.
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
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,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
The final theorem
Theorem
In a m1 × m2 × m3 game, a fully mixed SNE only exists for a negligible set
of games.
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
The final theorem
Theorem
In a m1 × m2 × m3 game, a fully mixed SNE only exists for a negligible set
of games.
Idea of the proof (three player case)
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
The final theorem
Theorem
In a m1 × m2 × m3 game, a fully mixed SNE only exists for a negligible set
of games.
Idea of the proof (three player case)
Consider the coalitions made by two players, apply the KKT conditions
and see that the SNE must satisfy:


U1 x1 x2 = 1
U2 x1 x2 = 1


U3 x1 x2 = 1
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
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 ) : x1> Ai x2 = bi ∀i ,
where Ai are the lines of the equations in the system.
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
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 ) : x1> Ai x2 = bi ∀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,L.
Strong Nash Equilibria in finite Games
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 ) : x1> Ai x2 = bi ∀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 an easy calculation of the dimensions and conclude.
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Final remarks
• When looking for SNE one can limit the search to pure equilibria
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Final remarks
• When looking for SNE one can limit the search to pure equilibria
• a weakening of super strong Nash equilibrium is given by the k-SNE:
this allows for deviations of coalitions of size not greater than k: our
proof shows that k = 2 suffices to have negligibility
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games
The two player case
The case with more players
Final remarks
• When looking for SNE one can limit the search to pure equilibria
• a weakening of super strong Nash equilibrium is given by the k-SNE:
this allows for deviations of coalitions of size not greater than k: our
proof shows that k = 2 suffices to have negligibility
• the result on the two player case does not hold in the same way for
strong Nash equilibria: a game need not to be strictly competitive to
have a SNE. But this happens only in trivial cases.
Braggion, Gatti,L.
Strong Nash Equilibria in finite Games