A mixed dual to the Nash equilibrium is defined for
The mathematical analysis of both competition and cooperation falls within the realm of game theory, whose systematic development began with von Neumann and Morgenstern [
In this paper we consider
In Section
The following definitions and notation are used. Let
The strategy profile
The strategy profile
Designating a strategy profile
The DE is the mixed extension of the Berge equilibrium (BE) of [
Some results relating DE and NE are now established. Let the function
If
It is first proved that if
We next use the fact that if
Let
In Section
A strategy profile
Consider
Payoff matrix for
Player 2 | |||
---|---|---|---|
|
| ||
Player 1 |
|
(3, 5) | (2, 8) |
|
(2, 6) | (5, 4) |
In this section sufficient conditions are presented for the existence of a DE for
A correspondence
Let
Let
Let
Topological conditions for the existence of a dual equilibrium are now derived. For each player
A strategy profile
Result
If
We show that
More restrictive sufficient conditions are given as a corollary.
If
Fix
For the DE to be useful as a solution concept, it must be computable. Since NEs can be difficult to obtain [
The strategy profile
For a given
The strategy profile
From (
For a given
The following result establishes that mutual cooperation in the form of a DE may be impossible when the number of players is greater than two. As noted above, a DE always exists when
For every
The result is first proved for
Assume that
If
Next utilizing (
Thus from (
For
Payoff matrix for
Player 3 | ||||
---|---|---|---|---|
Player 1 |
|
|
||
Player 2 | Player 2 | |||
|
|
|
| |
|
(1, 1, 0) | (0, 0, 0) | (0, 0, 1) | (0, 0, 0) |
|
(0, 0, 0) | (0, 0, 1) | (0, 0, 0) | (1, 1, 0) |
Theorem
The Berge equilibrium has been extended to a mixed dual equilibrium for the Nash equilibrium. In this duality, NEs embody selfishness for all players, as opposed to selflessness for DEs. The two concepts are closely related mathematically, however. Relabeling each player in a DE yields an NE for the original payoff matrix, from which it follows that a game has at least as many NEs as DEs. But an NE always exists for any
The author declares that there is no conflict of interests regarding the publication of this paper.