1. Results

The most important equilibrium of non-cooperative games is the Nash equilibrium point; the perfect equilibrium of Selten and proper equilibrium of Myerson are also considered important. Most authors of non-cooperative games study applications of these three equilibria. However, there was no plan to consider friends. In this work we emphasize this last concept that can be applied to problems of sociology, economics, mathematics, etc.

In the future it can be thought that the existence of this friendly equilibria can be applied to the aforementioned balances. We prove the existence.

Here we are going to introduce the general tools of the game. Consider an n-person game in normal form

where the strategy sets are non-empty and finite. The mixed extension is

Now for a player consider a finite sequence of players

which we call the “successor friends” of player . Then from an intuitive point of view it is natural to assume that the player is “rational”: whenever his payoff is the same he is going to act in such way that first he will maximize the payoff of his first successor friend . In the case that the payoff of this successor friend is equal again he will act to maximize the payoff of his second successor friend and so on until the last .

Consider for the sets

Formally we define a friendly equilibrium point if .

Theorem 1: Given any game and is upper semi-continuous for each , there always exists a friendly equilibrium.

Proof: given any point consider the set of points

since the expectations are multi-linear then each is non-empty convex and compact. Moreover, by the continuity of the expected functions and the hypothesis such, are upper semi-continuous. Therefore using Kakutani’s fixed point applied to there exists a fixed point . Such point is a friendly equilibrium (q.e.d.).

As a simple observation we would like to point out that in the case that for each our new notion coincides with the fundamental Nash equilibrium points.

It is interesting to remark that the existence of a friendly equilibrium is equivalent to the existence of a solution of the non-linear system

where denotes the discrete support of . The proof of the equivalence is rather easy and is left to the reader.

2. Perturbed Friendly Equilibria

Now we will extend the concept of friendly equilibria to the theory of perturbed equilibria. For this, we will follow the approaches of Selten (1975) and Myerson (1978).

Consider a point such that it satisfies

Theorem 2: A point is a friendly equilibrium point if and only if it fulfills B) with for each .

Proof: We prove it by induction on . For the case for each , this is very well known. Assume that it is valid for , then we will show that it is also valid for .

For this, consider that for each . Suppose that for

and

Let be the point defined as

then

which is impossible since and is identified with .

Inversely, consider a point such that

for each , then only if and

therefore for each .

Then following the ideas of Selten (1975), we define a perfect friendly equilibrium as follows: given , a point is called a -perfect friendly equilibria if

A perfect friendly equilibrium is a limit point of a sequence where is an -perfect friendly rational equilibria.

Similarly following the ideas of Myerson (1978) we define -proper friendly equilibria as

Clearly an -proper friendly equilibrium is an -perfect friendly equilibrium. A point is called a proper friendly equilibria if is a limit point of a sequence where is an -proper friendly equilibrium.

When the ’s corresponding concepts become the perfect and proper ones respectively.

Theorem 3: Every normal form game possesses at least one proper friendly equilibrium if , is upper semi-continuous.

Proof: In order to prove this existence theorem let us observe that for given positive integers it always holds true

This can be proven by induction on , which we skip. Therefore, it is clear that

Now given , define

where is the cardinality of , and let

For define the correspondence from by:

We have that is closed and convex for every and the mapping is upper semi-continuous.

Now we will prove that . Consider

for all and . Clearly

Let the point

Clearly .

Now if then and

(1)

(2)

since and the observation given above.

Now if then all

Then performing the same operation as before instead in, in, it is easy to see that

Let be the n-tuple . Then it satisfies the condition of the Kakutani Fixed Point Theorem (Kakutani, 1941) and the fixed point of is an -proper friendly equilibrium. Now for each we have an -proper friendly equilibrium, thus making , it is always possible to find a converging subsequence since the space is compact. The limit point of such subsequence is a friendly equilibrium point. The proof is complete, then the theorem follows.

As a simple consequence of this result we have

Corollary: Every normal form game has at least one perfect friendly equilibrium.

Finally we would like to say that if the do not belong to and are given externally, we have a external new notion of equilibrium. And we would like to emphasize that the previous material might be generalized accordingly for the continuous case. In this situation the upper semi-continuity condition appears more natural.

ACKNOWLEDGMENTS

I am great full for the collaboration of Miss Gimena Páez in editing and taping this paper.

Note

1. This paper was written in the year 1990 in a project supported by CONICET.