Friendly equilibrium points in extensive games
with complete information
by
Ezio Marchi *) **)
Abstract: In this note we prove an existence theorem regarding friendly equilibrium
points in extensive games with complete information. The friendly
equilibrium points is a refinement of equilibrium points.
*) Director del Instituto de Matemática Aplicada.
Universidad Nacional de San Luis, CONICET,
Ejército de los Andes 950. C.P. 5700
San Luis. Argentina
**) This paper has been partially supported by a grant from the CONICET.
1
Introduction and the Formulation
The important notion of equilibrium points was introduced by Nash (8), for nperson games in normal form and its existence was proved in the same paper. However
the existence of equilibrium points in n-person games with complete information in
extensive games was prove in a different context which might be consulted in the
excellent books by Burger (1), Kühn (2), Myerson (6) and/or Van Damme (11).
In a recent note we have studied the existence points of E-points in n-person
games in extensive form with complete information. The reader with interest might read
(4). There, we have given a sufficient and necessary condition for the existence of Epoints, which generalize equilibrium points.
In another non published paper Marchi (5) introduced the notion of friendly
equilibrium points and proved an existence theorem under general condition. All this
material is provided in normal n-person games. Olivera in (9) has extended the friendly
equilibrium points un the context of Garcia Jurado (3), who consequently extended at
this time the proper and perfect equilibrium points due to Myerson (6) and Selten (10).
In this note we are going to prove a general theorem concerning the existence of
friendly equilibrium points in extensive games with complete information.
Consider n-person extensive game with complete information, given by the set
of players i∈N = {1, …, n} and the “chance” player i 0 . The set of the nodes of the
rooted tree is G . The root is A. G is partitioned in the sets G i , i∈N and G i 0 , then
G = ∪ G i ∪ G i0
i∈N
The end points of the tree are E 1 ,…, E r . We do not need explicitly such points.
For each g ∈ G i 0 , i∈N we express by
it p i 0 (g, i0
(g) all the edges emaning from g . Let
i
(g)) , the corresponding assigned probability.
A complete plan for player i∈N is a strategy, namely a Now a
= ( 1 ,..., n
)∈
=X
i∈N
i
i
= { i (g)}g∈G i ∈
i
.
determines a distribution of probability at the
end points of the tree. Therefore we have if at each end point we have the payoff
function, the expectation function is complete determined and their expectations are
called by abuse of language, payoff functions. They are written as
2
A i ( 1 ,..., where
−1
= ( 1 ,..., i −1
,
i +1
,..., n
n
) = A i ( i , −1
)
) as is usual in non cooperative theory of games.
Now for each player i∈N we consider a string of different players f 1 (i) = i ,
f 2 (i),..., f ri (i) . Then a friendly equilibrium point is a point such that
A i ( ) ≥ A i ( i , −1
)
A f 2 (i) ( ) ≥ A f 2 (i) (
i , −1
)
∀i
∀
i
∈ Σi =
∀i
∀
i
∈
∀i
∀
i
∈
2
i
(
)
i
( )
1
i
( )
A f ri ( ) ≥ A
(i)
where
k i
( )=
{
i
∈
r
f i
(i)
k −1 i
( i , −1
)
( ) : A f k −1 ( ) ≥ A f k −1 = ( i , (i)
(i)
−1
}
ri
ri ≥ k ≥ 2
)
First we are going to give an example that an equilibrium point is not a friendly
equilibrium point.
Consider the simple following tree
 a1 
 
 b1 
l1
a2 
 
b2 
g1
a4 
 
b4 
a3 
 
b3 
l2
l3
m1
g2
l4
m2
A
Where G 2 = {A}, G 1 = {g 1 , g 2 } . The point
= ( 1 , 2
) = {(l1 , l 3 ), m 1 } is an
equilibrium point under the condition of the payoff functions a 1 ≥ a 2 and b 1 ≥ b 3 .
However if a 1 = a 2 and b 2 > b 1 the point
is not friendly equilibrium point with the
friendly structure f 1 (1) = 1, f 2 (1) = 2, f 1 (2) = 2 .
~
In such a case the point ~
= (~
1 , 2 ) = {(l 2 , l 3 ), m 1 } is a friendly equilibrium point: this
is easy to see since it is an equilibrium point and
(~) = {(l1 , l 3 ), (l1 , l 4 ), (l 2 , l 3 ), (l 2 , l 4 )}
2 1
3
Now introducing the notation
emaning from g ∈ G i , and for a
(g)) for the node ending at the edge
i
! "$# (g))
i
(g)
)
( &! "'# % i (g))
the payoff in the truncation game Γg with root g and
&! "$# % (g))
i
i
we write
A i (g) ( i (g), )
is the restriction of
(
in
.
Next we present the result of this paper
Theorem: Any n-person extensive game with complete information and any friend
structure has always a friendly equilibrium point.
Proof: We prove it by induction on the length of the tree. Let
be the length of
*
the tree. If λ = 1 . Then if A ∈ G i 0 there is nothing to prove. If A ∈ G i then it is clear
that choosing a point
A i ( + i ) ≥ A i (+ i )
∀
,
A f 2 ( i ) (σ i ) ≥ A f 2 ( i ) (σ i )
∀
+ i
∈
∀
+ i
∈
i
∈ Σi
-
2
i +
( i)
.
A f ri (i ) (σ i ) ≥ A f ri (i ) (σ i )
-
ri
i
(+ i )
then it is friendly equilibrium point. Such a point is evident that exists.
Now we assume that for λ ≤ λ(Γ) − 1 . The theorem is true and we will prove that
it is true for λ(Γ) . Consider the root A and then all the games
abuse of notation also for i = i 0 . Since
6 7$8
principle it has a friendly equilibrium point,
A i (? ) =
E
Σ
i0
(A)
i0
(A)
≥ F Σ p i 0 (L
i0
:; =4> 9 (A))
i
?
p i0 (?
≥ F Σ p i 0 (L
i0
:<; =4> 9 (A)) )
i
i0
(A)
= A i (M i , M
−i
i0
(
(A)) A (
L
i
L
4
if i ∈ N and by
. Now if i = i 0 we have
i0
i0
(A), ( L
L
i 0 (A), (
∀i ∀L
)
0 1 243 / (A))
i
is ≤ λ (Γ) − 1 , then by induction
(A)) A i ( ?
(A)) A i
5
i
(A),
GH IKJ F
GH IKJ F
∈ Σi
?
A B C4D @
i0
i0
(A))
) i , (L
i0
(A))
) i , (L
(A))
)
GH IKJ F
GH IKJ F
i0
(A))
i0
(A))
) −i
) −i
)
)
The same inequalities appear for the players in the friendly structure. Therefore
it is a friendly equilibrium point.
Now if i ≠ i 0 consider the game having length one, rooted by A and where the
payoffs at the end of
N
i
(A) are
QR S4T P (A))
i
A i (A) (O i (A), O
Then by choosing a
A i (A) ( O i (A), O
A f 2 (A) ( O i (A), O
QR SKT P i (A))
O
i
(i)
for j∈ N
(A) such that
) ≥ A i (A) (O i (A), O
QR S4T P i (A))
)
QR S4T P (A))
i
QR SKT P (A))
i
) ≥ A f 2 (A) (O i (A), O
∀
N
)
∀
O
)
∀
O
)
(i)
i
(A)
(A) ∈
U
i
(A) ∈
U
i
2 O
i
( i (A))
V
A f ri (A) ( O i (A), O
(i)
QR SKT P i (A))
QR SKT P (A))
i
) ≥ A f ri (A) (O i (A), O
(i)
ri
i
( O i (A))
where the Ω ik are referred to this game of lenght one, then we have constructed a
O
∈ Σ . Now we will prove that such a point is a friendly equilibrium point, in the entire
game.
For player i we first have
A i (O i , O
−i
) = A i (A) (O i (A), O
QR S4T P (A))
i
= A i (A)  O

i
(A), ( O
≥ A i (A)  O

i
(A), ( O
= A i (A)  O

i
(A), (O
= A i (W i , W
)
−i
)
QR S4T P (A))
i
QR SKT P (A))
i
Q<R S4T P (A))
i
∀
) i , (O
) i , (O
) i , (O
O
i
QR S4T P (A))
i
) -i 

QR SKT P (A))
i
) -i 

Q<R S4T P (A))
i
) -i 

∈ Σi
for
A f k (O i , O
(i)
−i
) = A f k (A) ( O i (A), O
(i)
QR S4T P i (A))
)
= A f k (A)  O i (A ), ( O
(i)

QR S4T P i (A))
≥ A f k (A)  O i (A), ( O
(i)

QR SKT P i (A))
5
) i , (O
) i , (O
QR S4T P i (A))
) -i 

QR SKT P i (A))
) -i 

= A f k (A)  ] i (A), (]
(i)

= A f k (] i , ]
−i
(i)
where
^
k ]
i
( )=
^
k ]
i
( i (A)) ∨ X V
^
k
i
i (A)
Y<Z [4\ X (A))
i
) i , (]
∀ i∈
)
^
Y<Z [4\ X i (A))
) -i 

k ]
i
( ) ∀ k = 1,..., ri
(η(A, ] i (A))) with V disjoint union. The first
inequality in the previous sequences is due to the fact of choosing the point
the second one is due to the fact that
_
]
YZ [4\ X (A))
i
]
i
(A) and
is a friendly equilibrium point in
YZ [4\ X (A)) .
i
For j ≠ i we have
A j (` j , `
−j
) = A ji (A) ( ` i (A), `
= A j (A)  `

i
(A), ( `
≥ A j (A)  `

i
(A), ( `
= A j (` j , `
We have used the fact that
`
−j
bc dKe a (A))
i
)
bc d4e a i (A))
)
bc d4e a (A))
i
bc dKe a i (A))
∀
) j , (`
) j , (`
f
j
bc d4e a (A))
i
) - j 

b<c d4e a i (A))
) j 

∈ Σj
is an equilibrium point in
_
bc d4e a i (A)) .
The
same hold for their friend structure.
Therefore the theorem is proved.
In this way we have obtained a rather powerful tool for decision making under
competition.
Finally, we would like to say that it is possible to extend the friendly equilibrium
points in extensive game with perfect information, when the friendly structure depends
on the node.
6
Bibliography
(1)
Burger, E.: Introduction to the theory of games. Prentice Hall 1959.
(2)
Kühn, H.: Lecture in Game Theory. Princeton University Press (2002)
(3)
Garcia Jurado, I.: Nuevos equilibrios estables ante tendencias al error en juegos no
cooperativos. Doctoral thesis. Univ. Santiago de Compostela 1989.
(4)
Marchi, E.: E-points in extensive games with complete information (to appear)
(5)
Marchi, E.: Friendly equilibrium points (to appear)
(6)
Myerson, R.B.: Game theory, Analysis of Conflict. Harward University Press
Cambridge Mass (1991)
(7)
Thomas, L.C.: Game Theory and Applications. John Wiley & Sons (1984)
(8)
Nash, J.: Non cooperative games. Annals of Mathematics 54, pp. 289-295 (1951)
(9)
Olivera, E.: Refinamientos de puntos de equilibrio en juegos no cooperativos.
Doctoral thesis. Univ. Nacional de San Luis.
(10) Selten, R.: Reexamination of the perfectness concept for equilibrium points in
extensive games. Int. J. Game Theory 4, pp. 35-55 (1975)
(11) Van Damme, E.: Stability and perfection of Nash equilibria. Springer Verlag
Berlin (1987)
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