Em-stable points in extensive games with complete information
by
Ezio Marchi*)
Abstract
This paper deals with the general existence theorem concerning em-stable points
for n-persons extensive games with complete information.
Key words and phrases: Extensive games, Complete information, em-stable points
Introduction
*)
Founder, First Director and Member of the Instituto de Matemática Aplicada UNSL, San Luis, 5700,
Argentina.
e-mail: emarchi@unsl.edu.ar and emarchi@sinectis.com.ar
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Nash in [9], introduced the concept of equilibrium point for a general mixed
extension of n-person games in normal form. He proved the existence of them using two
methods. After that, Marchi in [3], [4] and [5] introduced a great variety of new
concepts for non-cooperative mixed extension games. In particular he has proved the
existence of the em-stable points for any structure function of the game. We would like
to say that an em-stable point is an equilibrium point of a resulting suitable game
obtained from the original one. However, in general they are not equilibrium points of
the original game.
On the other hand, the subject of extensive games with complete information
presented in Burger in [1] and Kühn [2] is well known. Here you find the Zermelo
theorem. This asserts that in extensive games with perfect information, we always have
the existence of an equilibrium point in pure strategies.
In this paper we present the analogous of Zermelo theorem for em-stable points.
For further references we quote some bibliography in which the reader will find
related topics, as for example Thomas [13], van Damme [14], Osborne-Rubinstein [10],
Myerson [8] and Selten [11], [12].
In Marchi [6], [7], we have proved the existence of other concepts which are of
interest.
Formulation
We give an n-person extensive game with perfect or complete information by a
finite rooted tree G = {g} with initial root A. The set of players is N = {1, ..., n}. Here
we do not consider chance a player. The set of nodes in G is partitioned into
G   Gi
iN
The ending points are given by e1 , ..., e r . We do not need them explicitly. For
each player i  N and any g  G i we consider all the edges emanating from g, which
are indexed by σ i (g) . We write σ i  {σ i (g)}g  G i as a complete plan to follow by the
player i  N indicating that if the game reaches the node g he chooses σ i (g) with
g  G i at that node. σ i is a complete plan for player i  N , and it is called strategy. The
set of strategies for player i  N is denoted by  i .
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Consider for each i  N the set of players antagonistic to him by e(i)  N  {i} .
Then given the payoff functions,
A i (σ )
σ  (~
σi , ~
σ e(i) , ~
σ N ({i}{e(i)}) )
for σ  (σ1 , ..., σ n )    X  i an em-stable point is a point ~
iN
such that
Fi (~
σi , ~
σ N ({i}{e(i)}) )  Fi (σ i , ~
σ N  ({i}{e(i)}) )
for each i and σ i  Σ i . Here
Fi (σ i , σ N({i}{e(i)}) )  min A i (σ i , σ e(i) , σ N-({i}{e(i)}) ) ,
e ( i )  e ( i )
where the set  f is X  j for f  N . We have  i x e (i ) x N ({i}{e (i )})   . This was
jf
introduced by Marchi in [3].
We need some more notations. Given G i and σ i (g) with g  G i we write
η (g, σ i (g)) as that node g  G being at the end point of σ i (g) . Given σ i   i with
g  G i , then we write σ ig and σ g as the restriction of σ i and σ in the truncation g .
For any g  G i consider the notation
A i (g) (σ i (g), σ η (g, σi (g)) ) ,
or simply A i (g) (σ η (g, σi (g)) ) or A i (σ η (g, σi (g)) ) if there is no confusion. For the payoff
function in the truncation g .
Consider a game  with tree of lenght 0. This means that all the players no not
do anything. In such a case the strategy set for any player is formed by the element do
not do anything.
Then it trivially holds true, for each player
A i (σ i (A))  A i (A)  Fi (σ i (A))  Fi (σ i , σ Nf(i) )  max Fi (σ i , σ Nf(i) )
σi
where f (i)  e(i)  {i} and A is the node. Therefore there exists an em-stable point in
pure strategies.
Now consider the case when the game has lenght   1. That is to say, there is
only a node A and A  G i 0 .
In this case the payoff functions are given as follows:
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A i (σ i , σ e(i) , σ N f(i) )  A i ( ( A, σ i 0 (A)))
for each player.
Now define the strategy σ i 0 (A) such that
max A i 0 (η (A, σ i 0 (A)))  A i (η (A, σ i 0 (A)))
σi 0
which always exists, then we will show that the point {σ i 0 (A)} and any σ i for i  i 0
which means do not do anything is an em-stable equilibrium point.
Let i be the player i 0 , therefore for such a player we have
Fi 0 (σ i 0 , σ N f(i 0 ) )  max min A i 0 (σ i 0 , σ e(i 0 ) , σ N f(i 0 ) )
σi 0
σ e(i 0 )
 max min A i 0 (σ i 0 (η (A, σ i 0 (A)), σ N f(i 0 ) )
σ i 0 (A) σ e(i)
 A i 0 (η (A, σ i 0 (A)))  Fi 0 (σ i 0 , σ N f(i 0 ) )
since A i 0 is independent of all σ i  Σ i i  i 0 .
Secondly if i  i 0 and i 0  e(i) , we have:
Fi (σ i , σ N f(i) )  max min A i (σ i , σ e(i) , σ N f(i) )  min A i (η (σ i (A)))  max Fi (σ i , σ N f(i) )
σi
σ e(i)
σi 0
σi
since it does not depend on the choice σ i 0 .
If i  i 0 and i 0  e(i) , then
Fi (σ i , σ N f(i) )  A i (η (σ i 0 (A)))  max Fi (σ i , σ i 0 , σ N f(i)  {i 0 } )
σi
because such a payoff function is independent σ N  f(i) {i 0 } .
Thus for any extensive game with lenght 0 or one and complete information
there always exists an em-stable equilibrium point.
Consider the payoff function
A (σ (A), η (A, σ (A))
i0
 i0 i0

F' (σ i , σ N f(i) )   min A i (σ i 0 (A), η (A, σ i 0 (A)))
σ (A)
 i0
A i (σ i 0 (A), η (A, σ i 0 (A)))
i  i0
i  i0
i 0  e(i)
i  i0
i 0  e(i)
it determines all the truncations Γ η (A, σ i 0 (A)) . Since all these truncations have length non
greater than k-1, where k is the lenght of the original game, by induction principle we
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assume each of them possess an em-stable point, namely σ
η (A, σ i 0 (A))
. This in terms of the
payoff functions tells us that
Fik 1 (σ i 0 (A), σ i
η (A, σ i 0 (A))
η (A, σ i (A))
, σ N  f(i) 0
)  ηmax
Fik 1 (σ i 0 (A), σ i
(A, σ (A))
σi
η (A, σ i 0 (A))
i0
for a suitable em-stable equilibrium point in Γ η (A, σ i
0
(A))
η (A, σ i (A))
, σ N  f(i) 0
)
and where the F k ' are defined
recursively in a clear way.
By induction hypothesis we also have
Fik 1 (σ i 0 (A), σ i
η (A, σ i 0 (A))
η (A, σ i (A))
, σ N f(i) 0
)  ηmin
A i (σ i 0 (A), σ i
(A, σ i (A))
σ e(i)
η (A, σ i 0 (A))
0
η (A,σ i 0 (A))
, σ e(i)
η (A,σ i (A))
, σ N f(i) 0
)( 2)
for each i. F k 1 denotes the corresponding in the truncation. Now consider i  i 0 . Then
we define the element or strategy σ i (A) such that
Fik0 1 (σ i 0 (A), σ
(η , σ i 0 (A))
η (A, σ i 0 (A))
, σ N f(i)
)  max Fik0 1 (σ i 0 (A), σ
(η , σ i 0 (A))
σ i 0 (A)
η (A, σ i (A))
, σ N f(i) 0
)
which by finiteness, there always exists. Therefore we will show that the point
(σ i 0 (A), σ
η (A, σ i (A))
) is an em-stable point in the original game  .
Consider for the player i  i 0 , then
Fi 0 (σ i 0 , σ N f(i 0 ) )  Fik0 (σ i 0 (A), σ i 0
( , σ i 0 (A))
 max Fik0 -1 (σ i 0 (A), σ
η (A, σ i 0 (A))
, σ N f(i 0 )
( , σ i 0 (A))
σ i 0 (A)
)
η (A, σ i (A))
, σ N f(i 0 )0
)
and by (2)
 max min Fi0 (σ i0 , σ N f(i 0 ) )
σi 0
σ N  f(i 0 )
Now take a player i  i 0 such that i 0  e(i) . Therefore, we have for him
Fi (σ i , σ N f(i) )  min Fik -1 (σ i 0 (A), σ i
σ i 0 (A)
η (A, σ i 0 (A))
η (A, σ i 0 (A))
, σ N f(i)
)
 max F(σ i , σ N  f(i) )
σi
since i 0  e(i) and the choice of the strategy at the point A is independent on i.
Finally, if i 0  e(i) , and i 0  N  f (i) then we have immediately
Fi (σ i , σ N f(i) )  Fik -1 (σ i 0 (A), σ N  f(i) {i 0 } )
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 max Fik -1 (σ i0 (A), σ i
σi
η (A, σ i (A))
η (A, σ i 0 (A))
σ i 0 (A))
, σ ηN(A,
f(i) {i 0 } )
 max F(σ i , σ N  f(i) )
σi
Thus, we have proved the following result:
Theorem: Any game in extensive form with complete information without chance
player, with any given structure which assigns the sets e(i) , has always an em-stable
equilibrium point.
Comments
At this point we would like to emphasize that it is possible to study the existence
of many other concepts as those presented by Marchi [5], Selten in [12] and Myerson in
[8], etc, and the works of Kühn [2], etc, as well as points where the e(i) sets depend on
the nodes.
References
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[1]
Burger, E.: Introduction to the Theory of Games. Prentice Hall. 1959.
[2]
Kühn, H.: Introduction to the Theory of Games. Princeton Univ. Press. 2002.
[3]
Marchi, E.: Simple stability of the general n-person game. Naval Research
Logistic Quart, Vol 14, Nº2, pp. 163-171. 1967.
[4]
Marchi, E.: E-points of games. Proc. Mat. Acad. of Sciences. USA 57, Nº4, pp.
878-882. 1967. (MR 357702)
[5]
Marchi, E.: Foundations of non cooperatives games. Econometric Research
Program of Princeton University. Research Memorandum Nº97315 pp. (1968).
[6]
Marchi, E.: E-points in Extensive Games with Perfect Information (to appear).
[7]
Marchi, E.: E-saddle points in Extensive Game with Perfect Information.
[8]
Myerson, R.B.: Game Theory, Analysis of Conflict. Harvard University Press,
Cambridge, Mass. 1991.
[9]
Nash, J.: Non Cooperative Games. Annals of Mathematics 54, pp. 289-295. 1951.
[10] Osborne, M.J. and Rubinstein, A.: A Course of Game Theory. The MIT Press,
Cambridge, Mass. 1997.
[11] Selten,
R.:
Spiel
theoretische
Behandlung
eines
Oligopolmodells
mit
Nachfrageträgheit Zeitschrift die gesamte Staatswissen Schaft 121, 301-324 and
667-689. 1965.
[12] Selten, R.: Reexamination of the prefectness concepts for equilibrium in extensive
games. Int. Jour of Game Theory 4, 25-55. 1975.
[13] Thomas, L.C.: Games, Theory and Applications. John Wiley & Sons. 1984.
[14] Van Damme, E.: Stability and Perfection of Nash Equilibria. Springer Verlag,
Berlin. 1987.
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