The aim of this paper is to develop an effective method for solving bimatrix games with payoffs of intuitionistic fuzzy value. Firstly, bimatrix game model with intuitionistic fuzzy payoffs (IFPBiG) was put forward. Secondly, two kinds of nonlinear programming algorithms were discussed with the Nash equilibrium of IFPBiG. Thirdly, Nash equilibrium of the algorithm was proved by the fixed point theory and the algorithm was simplified by linear programming methods. Finally, an example was solved through Matlab; it showed the validity, applicability, and superiority.
Since the 1940s, game theory [
In real game situations, due to a lack of information or imprecision of the available information, players could only estimate the payoff value approximately with some imprecise degree. The fuzzy set [
Matrix games with mathematical programming are a mainstream research direction [
However, fuzzy bimatrix games with intuitionistic fuzzy payoffs (IFPBiG) are less studied, although in the real game problems, IFPBiG is very common. For example, in the research about market share games between real estate enterprises, the expectations of market share were very difficult to accurately estimate under the complicated situation, but fuzzy language could be used to express the satisfaction degree and rejection degree of the market share. For example, under the one of the situation, player I has the payoff value as (0.7 and 0.1) which means that for player I, the satisfaction degree is 0.7, the rejection degree is 0.1, and the hesitation degree is 0.2. It could be made clear by voting model that there are 70% of people voted satisfied, 10% of people voted against, and 20% of people abstained from voting. Due to the incompleteness and uncertainty of the market information, the payoff value of players I and II is not necessarily a zerosum, and this kind of phenomenon was very common.
The focus of this paper is considering the effective method for solving IFPBiG problem, with the Nash equilibrium being proved by the fixed point theory.
This paper is arranged as follows. Section
In this section, some basic definitions and operations of intuitionistic fuzzy sets and game theory were briefly reviewed, which are used in the following sections.
Let
Let
At the outcome
If players I and II choose
IFPBiG may be expressed as
Combining with Definition
(1) Consider
In the same way,
From the properties of Definition
(2) Consider
Let
when
as
So
as
Same as
As
so
There is a point; make
From the above, the mixed Nash equilibrium solution of IFPBiG can be obtained by solving the following programming problems:
The above two nonlinear programming models can be transformed into the following linear programming model by “linear exchange”:
To test our algorithm above, the following experiment was made.
There were two major hydropower enterprises competed for the power supply qualification through bidding. Both sides can take the fact that the bidding prices strategies are “high, flat, and low.” And both sides made up a thinktank to vote about the satisfaction degree and reject degree of each situation. The data of corresponding enterprises
“optimistic coefficients”
The solution and expectation of satisfaction degree and reject degree.







0.5  0.5  (0.41, 0.31, 0.28)  (0.18, 0.57, 0.25)  (0.67, 0.31)  (0.51, 0.42) 
0.1  0.1  (0.11, 0.53, 0.36)  (0.38, 0.34, 0.28)  (0.56, 0.41)  (0.48, 0.43) 
0.9  0.9  (0.21, 0.49, 0.30)  (0.28, 0.42, 0.3)  (0.75, 0.17)  (0.78, 0.20) 
0.1  0.9  (0.22, 0.31, 0.47)  (0.31, 0.43, 0.26)  (0.40, 0.41)  (0.52, 0.33) 
0.9  0.1  (0.61, 0.14, 0.25)  (0.17, 0.35, 0.48)  (0.49, 0.41)  (0.48, 0.45) 
The first two columns were the optimistic coefficient of enterprises
(1) Consider
At this time, enterprise
(2) Consider
At this time, enterprise
(3) Consider
At this time, enterprise
(4) Consider
From Table
Under the complicated decision environment, the model and method of this paper were more simple and practical than other general equilibrium game model because of the “linear exchange.” Meanwhile “optimistic coefficients”
Further study can be focused on more people’s bidding online intuitionistic fuzzy matrix game model, dynamic intuitionistic fuzzy bimatrix game model, more people cooperation game model, and so on.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research was supported by the NSFC of China (Grant no. 61305057).