Neural Network to Solve Concave Games

Recently, game theory has attracted considerable attention due to its extensive applications in economics, political science, and psychology, as well as logic and biology [1–5]. It has been widely recognized as an important tool in many fields. For game theory, the existence and stability of Nash equilibrium point are the most concerned problems. In past decades, these problems have been widely researched. Up to now, many excellent papers and monographs can be found, such as [6–11]. However, most previously established theory results are difficult to be adopted in practical applications, since the existence results only tell us that, for a given game, the Nash equilibrium point exists, but they do not tell us what it is or how we can calculate it. As is well known, the solving problem of Nash equilibrium point is as important as the existence and stability problems. In order to compute the Nash equilibrium point for a given game, all kinds of optimal algorithms and experiments have been derived in [12–15]. Among these methods, computer technique is one of the most popular ones. For a given game, by utilizing a computer programme to simulate the players, Nash equilibrium point can be approximately solved through computer logic calculation. However, when the quantity of players is large, the computation complexity and converge analysis must be considered. Conversely, projection neural network for solving optimization problems has its distinctly superior. This point is elaborated in [16–18].Thefirst is that it has parallel computing ability. The second is that the solution naturally exists. The last is that it can be implemented by circuits easily. One natural question is, for a given game,whetherwe can compute the Nash equilibrium point by neural network. This idea motivates this study. Combined with concave game theory, projection equation theory, variational inequality, Ky Fan inequality, and neural network method, we first established the relationship between neural network and concave games and pointed out that the equilibrium point of the constructed neural network is the Nash equilibrium point of our concerned game. Then, by using the Lyapunov stable theory, we analyzed the stability of the established neural network. Finally, two classic games are presented to illustrate the validity of the main results.


Introduction
Recently, game theory has attracted considerable attention due to its extensive applications in economics, political science, and psychology, as well as logic and biology [1][2][3][4][5].It has been widely recognized as an important tool in many fields.
For game theory, the existence and stability of Nash equilibrium point are the most concerned problems.In past decades, these problems have been widely researched.Up to now, many excellent papers and monographs can be found, such as [6][7][8][9][10][11].However, most previously established theory results are difficult to be adopted in practical applications, since the existence results only tell us that, for a given game, the Nash equilibrium point exists, but they do not tell us what it is or how we can calculate it.
As is well known, the solving problem of Nash equilibrium point is as important as the existence and stability problems.In order to compute the Nash equilibrium point for a given game, all kinds of optimal algorithms and experiments have been derived in [12][13][14][15].Among these methods, computer technique is one of the most popular ones.For a given game, by utilizing a computer programme to simulate the players, Nash equilibrium point can be approximately solved through computer logic calculation.However, when the quantity of players is large, the computation complexity and converge analysis must be considered.
Conversely, projection neural network for solving optimization problems has its distinctly superior.This point is elaborated in [16][17][18].The first is that it has parallel computing ability.The second is that the solution naturally exists.The last is that it can be implemented by circuits easily.One natural question is, for a given game, whether we can compute the Nash equilibrium point by neural network.This idea motivates this study.
Combined with concave game theory, projection equation theory, variational inequality, Ky Fan inequality, and neural network method, we first established the relationship between neural network and concave games and pointed out that the equilibrium point of the constructed neural network is the Nash equilibrium point of our concerned game.Then, by using the Lyapunov stable theory, we analyzed the stability of the established neural network.Finally, two classic games are presented to illustrate the validity of the main results.

𝑁-Person Noncooperative Games
Consider a typical -person noncooperative game as follows: let  = {1, 2, . . ., } be the set of players.For each  ∈ ,   , a metric space, denotes the strategy set and   :  = Π  =1   → R is the payoff function of th player, respectively.
For each  ∈ , denote î =  \ {}.For -person noncooperative games, one of the most important problem is to research whether there exists  * = ( then  is said to be quasiconcave.
then functional  is said to be lower semicontinuous.
Remark 3. Obviously, function  being concave means that it is quasiconcave; functional  being continuous means that it is lower semicontinuous.

Lemma 4 (Ky Fan inequality).
Let  be a nonempty convex compact subset of Hausdorff linear topological space E;  :  ×  → R satisfies the following.
Then, there exists  * ∈  such that, ∀ ∈ , (,  * ) ≤ 0.  [19]; this relationship provides the theoretical basis to solve the Nash equilibrium point for concave game by neural networks.

The Equivalence between Concave Games and Variational Inequalities
In this section, we will point out that any concave game can be transformed into a variational inequality problem equivalently, and we will utilize Ky Fan inequality to prove the existence of the Nash equilibrium point.Remark 9.This proof is similar to the convex situation in [20].In fact, from [20], one can obtain this result directly.
Here, for the readability, we still give out the proof details.
Remark 10.From Theorem 8, one can see that the Nash equilibrium point problem of a concave game is equivalent to a variational inequality problem.In order to solve the Nash equilibrium point of a concave game, we only need to solve the related variational inequality.
Theorem 11. ∀ ∈ , strategy set   ∈ R   is nonempty, closed, and convex, payoff function Namely, the Nash equilibrium point of the concerned concave game exists.
Remark 12. From Theorem 11, one can see that if   and   satisfy assumptions (1) and (2), the solution of the variational inequality in Theorem 11 exists, which means that the Nash equilibrium point of our concerned concave game exists.Combined with Theorems 8 and 11, we can construct neural network model to solve concave game problems.

Neural Network Model Construction.
To proceed, we first introduce an important lemma as follows.
Lemma 13 (see [21]).Let  : R  → R  be continuous function, and Ω is subset of R  ; then  * satisfies ⟨( * ),  −  * ⟩ ≥ 0 for all  ∈ R  if and only if  * is the fixed point of equation  =  Ω ( − ()), where  is arbitrary positive constant and  Ω ( − ()) is projection operator defined by On the basis of Theorems 8 and 11 and Lemma 13, we can construct the following neural network to solve the Nash equilibrium point of our concerned concave game: where () ∈ R  is the state vector,  = Π  =1   , ∇(()) = (∇  1 ()  1 (()),∇  2 ()  2 (()), . ..,∇   ()   (())), and   :  → R is the payoff function of th player.Remark 14.From Lemma 13, one can see that  * is an equilibrium point of system (9) if and only if it is a Nash equilibrium point of our concerned concave games; notice Theorem 11; one can obtain that the equilibrium point of system (9) exists.Thus, if the equilibrium point of system (9) is asymptotically stable, we can solve the Nash equilibrium point through neural network (9), which can be implemented by electric circuit.This means that Nash equilibrium point can be solved by electric circuit.
Theorem 17.Under assumptions ( 1) and ( 2), the state vector of system (9) globally asymptotically converges to the Nash equilibrium point.
Remark 18.For convex optimization problems, projection neural network has similar property.Compared with previous work, our Lyapunov function sufficiently uses the property of Nash equilibrium point; thus, it is more simple and the proof is more concise.The reason is that, in our proof, the value of  in Lemma 13 is set more appropriately.
Remark 19.Similar to the proof of [23], we can obtain the following propositions.
Remark 22. Proposition 20 means that, for any initial value in the strategy set , the solution through this initial value is unique and bounded, and the strategy set  is an invariant set.Proposition 21 means that the strategy set  is attractive.
Remark 23.On the basis of Propositions 20 and 21, we can further show that the equilibrium point of system ( 9) is not only globally asymptotically stable but also approximately exponentially stable.
Theorem 24.If the domain of payoff function  is R  and continuous, under assumptions ( 1) and ( 2), when  > 0 is sufficiently small, the state vector of system (9) approximately exponentially converges to the Nash equilibrium point, and the approximate convergence exponent is 1.
Set the initial value of system (9) as  0 , and by differential theory and Lemma 13, we have If  0 ∈ , from Proposition 20, we have () ∈ .Notice assumptions (1) and (2); one can obtain that ‖∇(()) − ∇( * )‖ is bounded, which can be assumed as .In this case, we have By Gronwall-Bellman inequality, when  is sufficiently small, one can obtain that which means that, when  0 ∈  and  is sufficiently small, the state vector of system ( 9) approximately exponentially converges to the Nash equilibrium point, and the approximate convergence exponent is 1.
If  0 ∉ , from Proposition 21, there exists  > 0 such that, ∀ > , () ∈ .In this case, we have Since  is continuous on R  , under assumptions By Gronwall-Bellman inequality, when  is sufficiently small, one can obtain that      () −  *     ≤  −(− 0 )      0 −  *      +  ( which means that, when  0 ∉  and  is sufficiently small, the state vector of system ( 9) still approximately exponentially converges to the Nash equilibrium point, and the approximate convergence exponent is 1.This completes the proof.
Remark 25.For system (9), when it is used to convex optimization problem, Theorem 24 does not require that ∇ 2  exists, and ∇ 2  > 0 and ‖∇ 2 ‖ local bounded.Thus, the conditions of Theorem 24 are weaker than those derived in literature [24].

Numerical Examples
In order to show the effectiveness of the technique proposed in this paper, we revisit two classic games as follows.
From Figure 2, one can see that the state vector of neural network also approximately exponentially converges to the Nash equilibrium point.
Additionally, if (),   (  ) are given general nonlinear functions such that   ( 1 ,  2 ) satisfies assumptions (1) and (2), in this case, to calculate the Nash equilibrium point directly is difficult.However, by using the technique established in this paper, we still can get one Nash equilibrium point through simulation tool box.
Example 27 (Hotelling competition).Consider the typical Hotelling game with two firms and a continuum of consumers.These consumers are distributed on a linear city of unit length according to a uniform density function.Each consumer is entitled to buy at most one unit of the commodity.Set   ,   ( 1 ,  2 ),  = 1, 2, which are the firm 's pricing strategy and demand function, respectively. is the distance between the location of the consumer and store and  is the transportation cost coefficient.The production cost is assumed to be identical and equal to  per unit for both firms.Thus, the firms' profit functions can be written as A Hotelling equilibrium ( * 1 ,  * 2 ) means that As is well known, if International Journal of Computer Games Technology State vector q(t) q 1 (t) q 2 (t) where () = [ 1 (),   3 and 4. From Figures 3 and 4, one can

Time t
State vector q(t) q 1 (t) q 2 (t) see that the state vector of neural network approximately exponentially converges to the Nash equilibrium point.Similarly, if   ( 1 ,  2 ),  = 1, 2, are given general nonlinear functions such that   ( 1 ,  2 ) satisfies assumptions (1) and (2), in this case, to calculate the Nash equilibrium directly is difficult.However, by using the technique established in this paper, we still can get one Nash equilibrium point through simulation tool box.Remark 28.The numerical simulation examples show that the results established in this paper are both valid from theoretical and practical points of view.These results provide us a new technique to solve the Nash equilibrium point of concave game.This new technique establishes a bridge between the neural network method and game theory and expands the scientific applications area of neural network.
Remark 29.It is worth to be pointed out that, by using the neural network technique derived in this paper to solve the Nash equilibrium point, the initial strategy value can be out of the strategy set; this phenomenon can provide us a new computer algorithm to solve the Nash equilibrium point.This new method is different from the traditional computer logic calculation and simulation method, which requires that every step's strategy values must be in the strategy set.And the new technique can reduce computation complexity significantly.
Remark 30.The results obtained in this paper show that, theoretically, the Nash equilibrium points of all kinds of concave game can be solved by neural network technique.However, when the payoff function does not satisfy assumption (2), for example, when the payoff function is only lower semicontinuous, upper semicontinuous, or quasi-continuous, how to use neural network technique to solve the Nash equilibrium point still needs to be deeply researched.And this is our future research direction.

Conclusion
The analysis results obtained in this paper imply that every concave game satisfying assumptions (1) and (2) can be equivalently transformed into a neural network model.There exists equivalence between the equilibrium point of neural network and the Nash equilibrium point of concave games.And the equilibrium point's convergence is independent whether the initial value is in the strategy set or not.This means that concave games can be implemented by neural network, or even by hardware.Two classic games' simulation results show that the results established in this paper are valid.