Refinements of Equilibria for Population Games Based on Bounded Rationality of Agents

We consider stability of equilibria for population games against slight perturbation on the social state space. We provide a necessary and sufficient condition for the existence of Nash equilibria for perturbed population games, which is very important and interesting. ,en, refinements of equilibria for population games are introduced. Equivalent characterizations of perfect equilibrium are given. At last, it is shown that each population game admits at least one perfect (proper, weakly proper, and robust) equilibrium.


Introduction and Preliminaries
Nash [1] proposed the fundamental notion of Nash equilibria for noncooperative games and obtained the Nobel Prize in 1994 due to his contribution to game theory. Much early research in evolutionary game theory has focused on the models generated by a symmetric normal form game. In fact, in Nash's unpublished Ph.D. dissertation [2], he has already provided a "mass action" interpretation for his equilibrium concept, which is close to the perspective of evolutionary game theory. It is a pity that his interpretation seemed to have gone unnoticed for 45 years. Until 1995, many researchers [3][4][5] reexplored Nash's dissertation and realized the appeal of his interpretation for constructing evolutionary population models. Since then the research of population game theory has proceeded like a raging fire. In fact, early in the 1970s, Maynard and Price [6] introduced the ESS concept as a way of understanding animal conflict. In 1982, Maynard [7] described the interaction of populations consisting of individuals as "playing the field." Weibull [8] provided an overview of the developments that have taken place in evolutionary game theory. e monograph of Sandholm [9] which presents a rigorous and unified description of population game theory was published in 2010.
In a rather long period, many game theorists pay their attention to the definition of rationality of individuals. Consequently, various refinements of equilibria were proposed. See [10][11][12][13]. It is worth mentioning that [13] was the first paper of refinements in view of bounded rationality. Actually, early in 1962, Wu and Jiang [14] proposed the notion of essential equilibria for n-person noncooperative games and proved that every game can be closely approximated arbitrarily by an essential game.
It is assumed that the agents are always completely rational in [9], which means they are omniscient and omnipotent.
is assumption however is too restrictive in reality, especially in population games involving numerous agents. It is reasonable to assume that agents are bounded rational since everyone may make mistakes. Consequently, some agents will sometimes use nonoptimal strategies. Under the bounded rational assumption, we propose some refinements of equilibria for population games, such as perfect equilibria, proper equilibria, weakly proper equilibria, and robust equilibria. Furthermore, we prove the existence of these equilibria and reveal the relations among them.
In this section, we introduce some elementary concepts and notations for population games. In Section 2, perturbed population games are proposed and the concept of perfect equilibrium is defined. We proved in Lemma 4 that equilibria for population games with continuous payoff functions are equivalent to solutions of variational inequalities.
is conclusion is meaningful and important since the algorithms for solving variational inequalities are abundant. In Section 3, we define proper equilibria and weakly proper equilibria for population games. e existence of that two kinds of equilibria are then proved. In Section 4, robust equilibria for population games are introduced. Although its idea is the same as weakly proper equilibria, our robust equilibria may be better to describe the bounded rationality of agents. Now some definitions and notations will be given for population games. Let P � 1, 2, . . . , p be a society consisting of p ≥ 1 populations of agents. e set of strategies for agents in population p ∈ P is denoted S p � 1, 2, . . . , n p { }, and m � p∈P n p is the total number of strategies in all populations. X p � x p ∈ R n p + : n p i�1 x p i � 1 denotes the population states for population p ∈ P, where each component x p i of x p represents the share of agents in population p who choose the pure strategy i in S p . For each p ∈ P, e society states, denoted X � p∈P X p , describe the behavior in all populations.
A payoff function F: X ⟶ R m is a continuous map that assigns each social state a vector of payoffs, one for each strategy in each population. For each p ∈ P, F p i : X ⟶ R denotes the payoff function for strategy i ∈ S p , while F p : X ⟶ R n p denotes the payoff functions for all strategies in S p . A population game is denoted by G � (P, X, F). e definition of a Nash equilibrium for population games is as follows.
Definition 1 (Definition of [9], p. 24). Let G � (P, X, F) be a population game. A social state x ∈ X is a Nash equilibrium of G if it satisfies: e notion of best response is now given, and an equivalent characterization of the equilibria for population games will be easily obtained with it.
Definition 2 (Definition of [9], p. 23). Let G � (P, X, F) be a population game. e best response correspondence for population p ∈ P is given by the following equation: (2) We say that y ∈ X is a best response to x ∈ X if y p is a best response to x for all p ∈ P, that is, Evidently, a social state x ∈ X is a Nash equilibrium if and only if it is a best response to itself, i.e., x ∈ B(x). erefore, we can prove the existence of Nash equilibria for population games by Kakutani fixed point theorem.
An important conclusion we will use is stated as follows.
Lemma 2 (Corollary of [15], p. 148). Let X ⊂ R m be nonempty, compact, and convex, and let F: X ⟶ R m be continuous. en, there exists x ∈ X such that the following variational inequality holds:

Perfect Equilibria
As stated in Introduction, we assume that agents are bounded rational; thus, a small number of agents may make mistakes and use nonoptimal strategies. Mathematically, this idea is modelled via perturbed population games, in which every strategy in each population may be chosen by agents.
Definition 3. Let G � (P, X, F) be a population game, and for each population, p ∈ P and η p � (η e perturbed population state set of population p ∈ P is given by the following equation: Furthermore, let η � (η 1 , η 2 , . . . , η p ) and X(η) � p∈P X p (η p ). en, G(η) � (P, X(η), F) is the perturbed population game for the population game G with perturbation η. e Nash equilibrium concept for perturbed population games is analogous to Definition 1.
Notice that the social states set for a perturbed population game X(η) is a subset of X of the original population game, so one may query the existence of an equilibrium for perturbed population games.
if and only if x is a Nash equilibrium of G(η).
Proof. Suppose x ∈ X(η) satisfies (6). If x is not a Nash equilibrium of G(η), then 2 Mathematical Problems in Engineering Let the pure strategy j ∈ S p 0 be such that F and , in which each component is the same as the corresponding component in x except the i th 0 and the j th components of population p 0 . Obviously, y ∈ X(η). en, we have is contradicts with (6).
It is obvious that For each p ∈ P and i ∈ S p , we set λ For each p ∈ P and y p ∈ X p (η p ), since i∈S p y p i � 1 and y p i ≥ η p i for all i ∈ S p , we have erefore, for each p ∈ P and y p ∈ X p (η p ), en, we have □ e proof is complete. Lemma 4 implies that the solution of the variational inequality (6) which is defined on the nonempty, compact, and convex set X(η) must be a Nash equilibrium of the perturbed population game G(η). Moreover, the variational inequality (6) possesses at least one solution on the perturbed social state space X(η) according to Lemma 2. en, we have the following theorem.

Theorem 1. Every perturbed population game admits at least one Nash equilibrium.
With the concepts having been given above, we now give the formal definition of a perfect equilibrium.
n�1 and x (n) ∞ n�1 satisfying the following: Proof. Suppose that the perfect equilibrium x of G is not a Nash equilibrium of G. en, is a Nash equilibrium of the perturbed population game G(η (n) ) for each n. In view of the continuity of F and the continuity of the maximum operator, it is clear that as n ⟶ ∞.
Consequently, for sufficiently large n, according to (16) and (17) and the property of limits, we have Because x (n) is a Nash equilibrium of the perturbed population game G(η (n) ), we have that, for sufficiently large n, is contradicts with our assumption x p 0 i 0 > 0. Hence, a perfect equilibrium of G must be a Nash equilibrium of G. e proof is complete.

□
Since the perfectness concept only requires Nash equilibrium to be robust with respect to some low-share perturbation, existence is easy to establish.

Theorem 2. Every population game G possesses at least one perfect equilibrium.
Proof. For any perturbed population game sequence G(η (n) ) ∞ n�1 for the population game G with η (n) ⟶ 0 as n ⟶ ∞, according to eorem 1, G(η (n) ) possesses at least one Nash equilibrium x (n) for each n.
Since x (n) ∞ n�1 is a sequence in the compact set X, we can assume without loss of generality that x (n) ∞ n�1 converges to some x ∈ X. en, x is a perfect equilibrium of G. e proof is complete. Next, we will give another description for perfect equilibria which is similar to Myerson's definition in [11]. Definition 6. Let G � (P, X, F) be a population game. A social state x ∈ X is completely mixed if each component of x is positive, that is, x p i > 0, for each i ∈ S p and p ∈ P.
Definition 7. Let G � (P, X, F) be a population game and let ϵ > 0. A completely mixed social state x ∈ X is an ϵ-perfect Next theorem shows that the above characterization of perfect equilibria is equivalent. Theorem 3. Let G � (P, X, F) be a population game. e following assertions are equivalent: such that x(ϵ n ) ⟶ x and ϵ n ⟶ 0 as n ⟶ ∞, where x(ϵ n ) is an ϵ n -perfect equilibrium of G for all n (3) ere exists a sequence ϵ n ∞ n�1 and a completely mixed social state sequence x(ϵ n ) ∞ n�1 , such that x(ϵ n ) ⟶ x and ϵ n ⟶ 0 as n ⟶ ∞, with the property that x is a best response to x(ϵ n ) for all n.
Proof (1) ⇒(2): since x is a perfect equilibrium of G, there exists a sequence η (n) ∞ n�1 such that lim n⟶∞ η (n) � 0 and lim n⟶∞ x (n) � x with x (n) a Nash equilibrium of the perturbed population game G(η (n) ) for all n. en, we have Let ϵ n � max p,i η (n)p i , then x (n) is an ϵ n -perfect equilibrium of G for all n.
(2)⇒(3): we show that if x p i > 0, then F p i (x(ϵ n )) � max k∈S p F p k (x(ϵ n )) for sufficiently large n.Since lim n⟶∞ x(ϵ n ) � x and lim n⟶∞ ϵ n � 0, we have x p i (ϵ n ) > ϵ n for sufficiently large n. Since x(ϵ n ) is an ϵ n -perfect equilibrium of G for each n, this implies F p i (x(ϵ n )) � max k∈S p F p k (x(ϵ n )) for sufficiently large n.
(3)⇒(1): since lim n⟶∞ ϵ n � 0, without loss of generality for every p ∈ P, there exists j ∈ S p such that x p j (ϵ n ) > ϵ n for all n. For each p ∈ P, define η (n)p , n � 1, 2, . . ., as Obviously, for all n and p ∈ P, η (n)p satisfies η (n)p i > 0 for all i ∈ S p and i∈S p η (n)p i < 1. As ϵ n converges to 0 as n ⟶ ∞, those η (n)p i which are equal to ϵ n converge to 0. Since x(ϵ n ) converges to x as n ⟶ ∞, for p ∈ P and i ∉ C(x p ), we have lim n⟶∞ x p i (ϵ n ) � 0. erefore, η (n) converges to 0 as n ⟶ ∞.
en, we obtain a sequence of perturbed population games G(η (n) ) ∞ n�1 . In the following, it will be shown that x(ϵ n ) is a Nash equilibrium of the perturbed population game G(η (n) ) for all n. For p ∈ P and i ∈ S p , if F p i (x(ϵ n )) < max k∈S p F p k (x(ϵ n )), then x p i � 0, since x is a best response to x(ϵ n ) for all n. en, i ∉ C(x p ), and according to the definition of η . erefore, x(ϵ n ) is a Nash equilibrium of the perturbed population game G(η (n) ) for all n. e proof is complete.
We conclude this section with the following example, which shows that the concept of perfectness really refines the Nash equilibria for population games.

Proper Equilibria and Weakly Proper Equilibria
In 1978, Myerson [11] proposed the notion of proper equilibria for n-person finite noncooperative games when he was a visiting scholar in Bielefeld University. is notion requires more rationality for players and seems more reasonable than perfect equilibria. Similar to Myerson's idea, we generalize this concept to population games considering that each agent will try her best to make herself better and better. Furthermore, even if the difference of payoffs of two strategies are sufficiently small, properness also requires the share of the worse one to be sufficiently small, which is too demanding. Agents often have no incentive to change if their choice is not too bad comparing another strategy. Considering this unpleasant requirement of proper equilibria, we introduce the notion of weakly proper equilibria. We first define the ϵ-proper equilibrium which is stronger than the ϵ-perfect equilibrium. (22) e basic idea underlying the ϵ-proper equilibrium is that every agent in each population will do her best to achieve better strategies although she cannot reach the predictive Nash equilibrium strategy because of the inadequacy of ability. Consequently, the share of a better strategy should be larger than a worse one. Using the concept of ϵ-proper equilibrium, we now introduce the definition of a proper equilibrium.
Definition 9. Let G � (P, X, F) be a population game. A social state x ∈ X is a proper equilibrium of G if there exist some sequences ϵ n ∞ n�1 and x(ϵ n ) ∞ n�1 satisfying the following: (1) lim n⟶∞ ϵ n � 0 (2) x(ϵ n ) is a ϵ n -proper equilibrium of the population game G for each n (3) lim n⟶∞ x(ϵ n ) � x Remark 1. Every proper equilibrium is perfect since each ϵ-proper equilibrium is ϵ-perfect.
en, every proper equilibrium must be a Nash Equilibrium. e existence of proper equilibria is also easy to establish as long as we can prove the existence of ϵ-proper equilibrium for every sufficiently small ϵ.

Theorem 4. Every population game G possesses at least one proper equilibrium.
Proof. Let ϵ ∈ (0, 1) and for each sufficiently small ϵ. For any p ∈ P, define vector η p by η p i (ϵ |S p | /|S p |) for each i ∈ S p , where | · | indicates the cardinality. en, we have a perturbation of population game G. For each p ∈ P, we define a set-valued map Φ p : We first show that, for any x ∈ X(η), Φ p (x) ≠ ∅. For each x ∈ X(η) and i ∈ S p , we let en, it is obvious that y p ∈ Φ p (x). We next proceed to the proof that Φ p (x) is a closed and convex set for any x ∈ X(η). For any y p , z p ∈ Φ p (x) and for all i, j ∈ S p . Passing to the limit, we have y p i ≤ ϵy p j , which means that y p ∈ Φ p (x), namely, Φ(x) is a closed set for each x ∈ X(η).
Finally, we prove that the correspondence Φ p is upper semicontinuous. It suffices to show that the graph of Φ p is closed. Suppose that x (n) ⟶ x, y (n)p ∈ Φ p (x (n) ), and y (n)p ⟶ y p as n ⟶ ∞, we show that for sufficiently large n since x (n) ⟶ x and payoff function F is continuous. Furthermore, we have y (n)p i ≤ ϵy (n)p j because y (n)p ∈ Φ p (x (n) ). Since y (n)p ⟶ y p , this implies y p i ≤ ϵy p j for sufficiently large n. en, we have y p ∈ Φ p (x). erefore, the correspondence Φ is upper semicontinuous.
For every x ∈ X(η), define Φ(x) � p∈P Φ p (x). It is clear that Φ satisfies all conditions of Lemma 3. So there exists some x ∈ X(η) such that x ∈ Φ(x). Obviously, this x is an ϵ-proper equilibrium.
Consequently, from the proof above, for any sequence ϵ n converging to 0, we have a corresponding sequence x (n) in which every social state x (n) ∈ X is an ϵ n -proper equilibrium. Furthermore, since X is compact, there exists a convergent subsequence of x (n) and the limit x is clearly a proper equilibrium. e proof is complete. □ e following example shows that a perfect equilibrium for population game need not be proper which means the set of proper equilibria may be a proper subset of the set of perfect equilibria.
Definition 10. Let G � (P, X, F) be a population game and let ϵ > 0. A completely mixed social state x ∈ X is an ϵ-weakly proper equilibrium against some x ∈ X of G if it satisfies: Definition 11. Let G � (P, X, F) be a population game. A social state x ∈ X is a weakly proper equilibrium of G if there Mathematical Problems in Engineering 5 exist some sequences ϵ n ∞ n�1 and x(ϵ n ) ∞ n�1 satisfying the following: (1) lim n⟶∞ ϵ n � 0 (2) x(ϵ n ) is a ϵ n -weakly proper equilibrium against x of the population game G for each n (3) x(ϵ n ) is an ϵ n -perfect equilibrium of G for each n (4) lim n⟶∞ x(ϵ n ) � x Remark 2. It is clear from eorem 3 that every weakly proper equilibrium is perfect since condition (3) in Definition 11 holds.
Next, we give an example which indicates that a weakly proper equilibrium needs not to be proper.
We can easily have the existence theorem of weakly proper equilibrium since every proper equilibrium is weakly proper.
Theorem 5. Every population game G possesses at least one weakly proper equilibrium.

Conclusion
is paper has generalized the notions of perfect equilibria, proper equilibria, and weakly proper equilibria to models of population games. And we have proposed the so-called robust equilibrium for population games. Lemma 4 is an interesting and important result for perturbed population games. We have shown the existences of the above equilibria without the assumption of the convexity of the payoff. Furthermore, we have proved that our robust equilibrium is equivalent to the weakly robust equilibrium. From Example 1, we have seen the perfectness really refines the Nash equilibria for population games, and from Example 2, the properness really refines perfect equilibria. Example 3 shows the set of proper equilibria may be a proper subset of weakly proper equilibria. From Example 5, we have seen the robustness concept is a strict refinement of perfectness concept. For the future work, we will consider the dynamical stability of the mentioned refinement concepts.

Data Availability
No data were used to support this study.

Disclosure
Part of this work was done at Tilburg University while the author was a visiting scholar of Econometrics and Operations Research.