Stability of Generalized Minimax Regret Equilibria with Scalar Set Payoff ()
1. Introduction
In the game, individuals are faced with uncertain strategic choices, and they assume that the decisions of other players will form a subjective assessment of probability. Renou and Schlag [1] introduced the concept of minimax regret equilibria, allowing players to be uncertain about the rationality and conjecture of their opponents, and assumed that regret will lead individuals to form probabilistic assessments and ultimately make choices that minimize regret. Yang and Pu [2] obtained the existence and generic stability of minimax regret equilibria. Recently, Zhang and Chen et al. [3] studied existence of general n person noncooperative game problems and minimax regret equilibria with set payoff by using Kakutani-Fan-Glicksberg fixed point theorem and a nonlinear scalarization function. For more information, refer to [4] [5] [6] [7].
Under the influence of some uncertain factors, the game model is often disturbed. In 1950, Fort [8] first proposed the concept of essential fixed points when studying the stability of fixed points. Kohlberg and Mertens [9] investigated that every game has at least one equilibrium set with a stable strategy. Wu and Jiang [10] put forward the concept of essential equilibrium of finite noncooperative game, and proved that any finite noncooperative game can be approximated arbitrarily by an essential equilibrium game. In the sense of changing order, Luo [11] studied the existence and general stability of Nash equilibrium points in set-valued games. In recent years, Yu and Peng [12] considered the general stability of Nash equilibria in noncooperative differential games, and proved that equilibrium is the essence of differential games to form dense residual sets by using the theory of set-valued analysis. Liu [13] discussed the existence of Nash equilibrium points of generalized set-valued mapping and the stability of Nash equilibrium point sets of generalized set-valued mapping from the perspective of essential equilibrium points. In real locally convex Hausdorff topological linear spaces, He and Chen et al. [14] obtained the general stability of solutions for set-valued generalized strong vector quasi-equilibrium problems when the constraint set-valued mappings are continuous and the target mappings satisfy the cone-true quasi-convex conditions.
Since there are many unpredictable situations in reality, it is very important to study the stability of set-valued game problems. However, to the best of our knowledge, the study of the stability of set-valued game problems is still little. Motivated and inspired by the minimax regret problem in [3], the rest parts are constructed as follows: In Section 2, the generalized minimax regret equilibria problem and some necessary basic knowledge are given. In Section 3, general stability theorem for generalized minimax regret equilibrium with scalar set payoffs is obtained. And a numerical example is given to illustrate our results.
2. Preliminaries
Let Y, K,
,
are real locally convex Hausdorff topological vector spaces. S is a pointed closed convex cone in Y, and
.
Firstly, we introduce the generalized minimax regret equilibrium model in [3].
Let
be a set of players,
be the pure strategy set of ith player and
be the scalar set payoff function of ith player. For each
, denote
. If
, write
. Let
be subset of ith player beliefs conjectures about the play of his opponents.
be the ith player's ex-post regret function relative
, defined as follows:
A strategy
is said to be a minimax regret equilibria point with scalar set payoffs relative to
, if, for each
,
Definition 2.1. [15] Let
be a set-valued mapping,
(i) A set-valued map P is said to be upper semicontinuous (u.s.c) with nonempty compact valued at
, if for any net
with
, and for any
, there exist
and a subnet
of
, such that
;
(ii) A set-valued map P is said to be lower semicontinuous (l.s.c) at
, if for any net
with
, and for any
, there exist
, such that
;
(iii) A set-valued map P is said to be continuous at
, if P is upper and lower semicontinuous at
.
Definition 2.2. Let
and
be true subsets of the metric space, define the Hausdorff distance between
and
as
Theorem 2.1. [16] Let K and Y be two Hausdorff topological spaces,
is a net in
with the Vietoris topology,
is a net of Y and
is a net of real-valued continuous function defined on
. If
under the Vietoris topology,
and
, where
is a real-valued continuous function defined on
, then
.
Theorem 2.2. [15] Let K and Y be two Hausdorff topological spaces, and Y is compact, if set-valued mapping
is closed, then P is upper semicontinuous.
Cited from the literature [3] Theorem 4.2. For each
,
and
are continuous on
and
repectively. For each
,
is
quasiconcave. Then, there exists
such that for each
,
where
is the ith player’s ex-post regret function. Remember this generalized minimax regret problem as
.
Lemma 2.1. [16] Let
be a set-valued mapping with nonempty values. Suppose that
Then,
3. Main Results
In this section, we investigate generic stability theorem of generalized minimax regret equilibria with scalar set payoff when the scalar set payoff functions are disturbed.
For each
, let
be Banach spaces. suppose
be the space composed of
and satisfy all assumption of Literature [3] Theorem 4.2. For
, we define the set of all generalized minimax regret equilibria by
.
Definition 3.1. Since P be a real set-valued mapping, we define the distance
on
by
where
,
,
and
are Hausdorff distance on
. Obviously,
is a complete metric space.
Lemma 3.1. [2] For each
, m is continuous if and only if the set-valued mapping
is lower semicontinuous.
Theorem 3.1. For each
, let
be nonempty compact convex subset. Assume that
and
are continuous with nonempty compact-valued. Then the set-valued mapping
is upper semicontinuous with compact valued.
Proof Since
is nonempty compact subset of
, by Theorem 2.1, it is sufficient to prove that
is closed. i.e., for each
, for any
with
, any
with
. We will prove
.
For any
, there exists
such that
where
.
Because
is continuous and
. We have
when
. Because
is compact-valued, for
, there exists
such that
.
Since
is continuous, for each
, then
By Theorem 2.1 and Lemma 2.1, we have
Thus, for sufficiently large
, for each
, there exists
such that
Hence,
,
is upper semicontinuous with compact valued. This completes the proof.
Remark The proof of this theorem is similar to that of Lemma 4 in Reference [2]. If
,
, then the proof is the same as Lemma 4 in Reference [2].
Lemma 3.2. [17] Let
be a complete metric space and X be a topological space. Suppose that
is upper semicontinuous and nonempty compact valued, there exists a dense residual
such that
is lower semicontinuous on Q.
Theorem 3.2. There exists a dense residual set
such that m is continuous for
.
Proof By Theorem 3.1, the set-valued mapping
is upper semicontinuous with compact valued. By Lemma 3.2, there exists a dense residual
such that
is lower semicontinuous on Q. By Lemma 3.1,m is continuous for any
. This completes the proof.
The following example illustrates that
.
Example 3.1. Consider the generalized minmax regret equilibria problem
. Let
,
, and
be player's scalar set payoff function,
,
be player’s ex-post regret function,
Then
. For each n, define
as follows:
And define
as follows:
Then
,
. It shows that no point in Q is continuous for
. Similarly, for each n, define
as follows:
And define
as follows:
Then
,
. It shows that no point in Q is continuous for
. Consequently,
, So
.
Theorem 3.3. There exists a dense residual set
such that
is a singleton set for any
.
Funding
This research is supported by Yunnan University of Finance and Economics Graduate Innovation Fund (2022YUFEYC072).
Acknowledgements
Sincere thanks to the members of JAMP for their professional performance, and special thanks to managing editor Hellen XU for a rare attitude of high quality.