When confronted with multiple Nash equilibria, decision makers have to refine their choices. Among all known Nash equilibrium refinements, the
Interest for game theoretic applications has been growing in engineering, management and political sciences. A
The Nash equilibrium concept [
Decision makers, confronted to multiple Nash equilibria, have to refine their choices using other rational concepts in addition to the concept of Nash equilibrium. Game theorists introduced many refinement concepts. Among all known Nash equilibrium refinements, the
In this paper, we intend to set an automatic procedure to verify the perfectness of polymatrix games Nash equilibria. Section
Let us define
Each player
Like any
We denote by NE the set of Nash equilibria. This set is the union of a finite number of polytopes called
A Nash subset
Let
Using Selten’s definition of perfect equilibrium for a strategic form game (see [
Let
In other words, a perfect Nash equilibrium
Let
This second characterization describes a perfect Nash equilibrium
In the following, we reformulate the conditions on polymatrix games perfect equilibria to show that every player’s mixed strategic choice is a best response to any combination of the other players pure strategic choices. In other words, we show that every perfect equilibrium of a polymatrix game is undominated and every undominated equilibrium of a polymatrix game is perfect. While this result is known to always be satisfied for bimatrix games, the second part of it is not true in general for games with more than two players. Nevertheless, it appears from the next development that the particular structure of polymatrix games payoffs allows us to extend the perfectness undominance property to polymatrix games. To reach this result, we first show that
The
In the first part of the proof, we show that if
Therefore,
Hence,
If we sum all the
Since
We conclude that a
As shown in [
Let
For any player
If for some combination
By Definition
Thus, we have
If the weakly dominated strategy
Therefore, if we sum, respectively, on all
Also, we know that the mixed strategy vector
Thus,
We now show that every perfect equilibrium of a polymatrix game is undominated and every undominated equilibrium of a polymatrix game is perfect. Theorem
Let
In the first part of the proof, we show that if
Thus, we can write
Therefore, we obtain
Since
We deduce that if
Let
Since
Hence,
We finally conclude that the Nash equilibrium
This shows that
While this perfectness undominance property is generally not right for
Let
This characterization of equilibrium strategies can be used to verify if a Nash equilibrium
The equilibrium
Let
In other words, there is at least one
Therefore, we have
If all the optimal objective functions are equal to zero, for all
Hence, if all the
We note that the linear programs (
Consider a three-player polymatrix game
Extreme Nash equilibria for Example
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As in Audet et al. [
The preceding game example suggests that the enumeration of the extreme Nash equilibria of a polymatrix game leads to a description of the set of Nash perfect equilibria. However, to the best of our knowledge, there are no published results on the geometric properties of the set of Nash perfect equilibria for polymatrix games. By Proposition
Let
Given that
Let
Then, we can write
Given that
Thus, for any vector
Now let us suppose that
Thus, for every
On one hand, condition (
Hence, with
On the other hand, since
Therefore, with
Inequalities (
Since
It is now made clear that Condition (
A set of perfect Nash equilibria belonging to the same Nash subset is called a
Any maximal Selten subset is a convex polytope.
Following Proposition
Given the Nash maximal subsets identified for Example
Quintas [
The set of perfect Nash equilibrium points of a polymatrix game is a finite union of convex polytopes, possibly disjoint.
Any maximal Selten subset is a convex polytope contained in a maximal Nash subset. Therefore, the set of perfect Nash equilibrium points of a polymatrix game is a finite union of convex polytopes. The maximal Selten subsets are possibly disjoint as in Example
Many applications can be found to illustrate how polymatrix games can be used. In the following, we illustrate our results on a three-player chain store competition game and a three-player inspection management game inspired from Fandel and Trockel [
Chain store competition game.
Using exact arithmetics, the
Extreme Nash equilibria for Application 1.
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Perfect |
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No |
2 |
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Yes |
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If the manager (
The current values of the game parameters are displayed in Table
Three-player management inspection game parameters.
( |
( |
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This game has three extreme Nash equilibria and two maximal Nash subsets
Extreme Nash equilibria for Application 1.
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Perfect |
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Yes |
2 |
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1, 2 | No |
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No |
Our computational experiments on randomly generated polymatrix games with different size and density are presented in Table
Computational results on randomly generated polymatrix games.
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ne | np | % | ne | np | % | ne | np | % | ne | np | % | ne | np | % | |
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53 | 32 | 60 | 29 | 22 | 76 | 15 | 13 | 87 | 16 | 16 | 100 | 10 | 10 | 100 |
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110 | 66 | 60 | 76 | 20 | 26 | 29 | 23 | 79 | 20 | 19 | 95 | 16 | 15 | 94 |
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157 | 48 | 31 | 75 | 38 | 51 | 39 | 37 | 95 | 32 | 27 | 84 | 39 | 30 | 77 |
The column (Size) indicates the number of strategies of each player before elimination of strongly dominated strategies as performed in [
The experimental results show that the percentage of extreme perfect equilibria increases with an increase of the polymatrix games density. Hence, reducing the partial payoff matrices density increases the number of extreme Nash equilibria of the polymatrix game and decreases the probability of generating perfect extreme equilibria. For
Game theoretic applications have been increasingly encountered in engineering, management and political sciences. Decision makers can often be represented by autonomous agents such as hardware (central units) or software (program applications), which are unable to distinguish between a set of Nash equilibria unless a refinement procedure is used. This paper presents a new characterization of perfect Nash equilibria for polymatrix games. This characterization shows that a Nash equilibrium is perfect if and only if it is a best response to any combination of the other players pure strategic plays. While this characterization is generally not right for
Moreover, we show that any perfect Nash equilibrium is a convex combination of extreme perfect Nash equilibria. As an immediate implication of this result, the set of perfect Nash equilibria of a polymatrix game is a finite union of convex polytopes. A linear programming formulation to identify perfect equilibria for polymatrix games is presented. Finally, the results of this paper are used to perform computational experiments on randomly generated polymatrix games with different size and density.
The author declares that there is no conflict of interests regarding the publication of this paper.
This work was partially supported by KFUPM Deanship of research under Grant IN101038. This support is gratefully acknowledged. The author thanks Professors Charles Audet, Bernhard von Stengel, and Issam Louhichi for their constructive and valuable comments on the content of this paper.