27.2. 27.2 Method of equilibrium

In this chapter we assume that decision makers interested in the selection of a mutual decision alternative. Let denote the objective function of the th decision maker, which is also called payoff function in the game theory literature. Depending on the decision makers relationship to each other we can speak about cooperative and non-cooperative games. In the first case the decision makers care about only their own benefits, while in the second case they strive for an agreement when every one of them are better off than in the non-cooperative case. In this chapter we will discuss the non-cooperative case, while the cooperative case will be topic of the next chapter.

Let's denote for and , the set of the decision alternatives into which the th decision maker can move over without the others' support. Evidently .

Definition 27.3 An alternative is equilibrium if for all and ,

This definition can also be formulated that is stable in the sense that none of the decision makers can change the decision alternative from alone to change any objective function value for the better. In the case of non-cooperative games, the equilibrium are the solutions of the game.

For any and decision maker, the set

is called the set of the best answers of the th decision maker to alternative . It is clear that the elements of are those alternatives which the th decision maker can move over from , and which ensure the best objective functions out of all the possible alternatives. According to inequality (27.22) it is also clear that is an equilibrium if and only if for all , , that is is mutual fixed point of the point-to-set maps. Thus, the existence of equilibrium can be traced to the existence of mutual fixed point of point-to-set maps, so the problem can be solved by the usual methods.

It is a very common case when the collective decision is made up by the personal decisions of the certain decision makers. Let denote the set of the th decision maker's alternatives, let be the concrete alternatives, and let be the objective function of the th decision maker. That is the collective decision is . In this case

and the (27.22) definition of equilibrium is modified as follows:

In the game theory literature the equilibrium is also called Nash-equilibrium.

The existence of an equilibrium is not guaranteed in general. To illustrate this let's consider the case, when both decision makers can choose between to alternatives: and . The objective function values are shown in Figure 27.10, where the the first number in the parentheses shows the first, the second number shows the second decision maker's objective function value. If equilibrium exists, it might not be unique, what can be proved by the case of constant objective functions, when every decision alternative is an equilibrium.

Figure 27.10.  Game with no equilibrium.

Game with no equilibrium.


If the sets are finite, the equilibrium can be found easily by the method of reckoning, when we check for all of the decision vectors whether the component can be changed for the better of the objective function. If the answer is yes, is not equilibrium. If none of the components can be changed in such manner, is equilibrium. For the formal algorithm, let's assume that .

Equilibrium-Search

  1  
                  FOR
                
                
               
                  TO
                
                  2    
                  DO
                
               
                  FOR
                
                
               
                  TO
                
                  3          4          
                  DO
                
               
                  FOR
                
                
               
                  TO
                
                  5             
                  DO
                
                  6             
                  FOR
                
                
               
                  TO
                
                  7                
                  DO
                
               
                  FOR
                
                
               
                  TO
                
                  8                   
                  DO
                
               
                  IF
                
                  9                      
                  THEN
                
                and go to 10  10             
                  IF
                   
                 11                
                  THEN
                
                is equilibrium 

The existence of equilibrium is guaranteed by the following theorem.

Theorem 27.4 Assume that for all

(i) is convex, bounded and closed in a final dimensional Euclidean space;

(ii) is continuous on the set ;

(iii) for any fixed , is concave in .

Then there is at least one equilibrium.

Determination of the equilibrium is usually based on the observation that for all , is the solution of the

optimum task. Writing the necessary conditions of the optimal solution (for example the Kuhn-Tucker conditions) we can get an equality-inequality system which solutions include the equilibrium. To illustrate this method let's assume that

where is a finite dimensional vector and is a vector-valued function. In this way (27.25) can be rewritten as follows:

In this case the Kuhn-Tucker necessary conditions are:

where denotes the gradient at , and is a vector which has the same length as . If we formulate the (27.27) conditions for , we get an equality-inequality system which can be solved by computer methods. It is easy to see that (27.27) can also be rewritten to an nonlinear optimization task:

If the optimal objective function value is positive, the (27.27) system doesn't have a solution, and if the optimal objective function value is zero, any optimal solution is also a solution of the (27.27) system, so the equilibrium are among the optimal solutions. We know about the sufficiency of the Kuhn-Tucker conditions that if is concave in with all , the Kuhn-Tucker conditions are also sufficient, thus every solution of (27.27) gives an equilibrium.

The formal algorithm is as follows:

Kuhn–Tucker-Equilibrium

  1  
                  FOR
                
                
               
                  TO
                
                  2    
                  DO
                
                  3          4  the solution of the (27.28) optimum task   5  
                  IF
                
                  6    
                  THEN
                
               
                  RETURN
                "there is no equilibrium"   7    
                  ELSE
                
               
                  RETURN
                () 

Example 27.8 Assume that production plant produce some water purification device sold into households. Let denote the quantity produced by the th production plant, let be the cost function of it, and let be the sale price, which depends on the total quantity to be put on the market. Furthermore, be is the capacity of the th production plant. Thus, the possible decision set is the closed interval, which can be defined by the

conditions, so

The objective function of the th production plant is the profit of that:

Since is two-dimensional, is a two-element vector as well, and the (27.28) optimum task has the following form:

Let's introduce the new variables, and for the sake of notational convenience be , then taking account of the last condition, we get the following problem:

Let's notice that in case of optimum , so the last term of the objective function can be neglected.

Consider the special case of , , , . The (27.32) problem is now simplified as follows:

Using a simple computer program we can get the optimal solution:

which is the equilibrium as well.

Exercises

27.2-1 Let , , , . Formulate the (27.27) conditions and solve them as well.

27.2-2 Formulate and solve the optimum problem (27.28) for the previous exercise.

27.2-3 Let again . , , . Repeat Exercise 27.2-1.

27.2-4 Repeat Exercise 27.2-2 for the problem given in the previous exercise.