Game Theory as a Design Tool

T. L. Vincent Dept. of Aerospace

and Mechanical Engineering, University of Arizona, Tucson, Ariz. 85721

Game Theory as a Design Tool Game theory encompasses not only the familiar concept of scalar optimization, but the less known concepts of multicriteria optimization and multiple optimizers as well. This paper examines the role of game theory in the engineering design process. This is done by demonstrating how game theory as a design tool applies beyond scalar optimization to the multicriteria optimization problem. The multicriteria optimization task is examined not only from the perspective of a single designer but from the perspective of team design as well. In the latter case the design problem has been assigned to several designers each responsible for one aspect of the total design.

Introduction

There is a large class of engineering design problems (both static and dynamic) for which the design parameters are required to be constants. This paper will deal only with static parametric systems, that is, systems for which no time history is involved.

Optimal design of parametric systems has, in the past, generally referred to only the concept of a minimum (or maximum). However, there are many other optimization concepts defined for parametric systems if we allow optimal design to include not only scalar optimization, but multicriteria optimization with multiple optimizers as well. Such an extended view brings "optimality in parametric systems" into the realm of game theory [1] along with its numerous optimization concepts. The objective of this paper is to demonstrate the usefulness of some of the game theoretic concepts in engineering design.

Generally the variables in a static system are constrained by a system of algebraic equality and inequality equations of the form

g(y)=o

h(y)>0

(1)

(2)

where y is a vector variable and g and h are vector functions, thatisj>=|>,, . . . ,ym]T, g(y)=[gi(y), . . • ,g„(y)]T, and h(y)=[h\ (y), . . . ,hq{y)\T. All vectors are taken to be column vectors; hence T is used to designate the transpose of the row vectors given. An equivalent formulation is given by

g(x,u)=0 (3)

h(x,u)>0 (4)

where components of the vector y have been identified as belonging to a "state" vector x and a "control" vector u, that is x—[xx, . . . ,x„]T and u = [ult . . . ,us]

T, s = n — m. The former notation is classic. The latter notation is often referred to as control notation. In control notation, the design parameter u is explicitly identified. Loosely speaking, the design parameters may be thought of as those components of y that may be freely chosen when all of the inequality con-

Contributed by the Design Automation Committee and presented at the Design Engineering Technical Conference, Hartford, Ct., September 20-23, 1981, of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS. Manuscript

received at ASME Headquarters June 9,1982.

ditions are strict. The n state variables x are then determined from the n equality constraints g(x,u) =0 . The selection process for state and control variables is not unique in the formulation of scalar- and vector-valued optimization problems and hence either notation is often used when discussing these subjects. However, there are computational advantages with control notation when used with scalar optimization. This advantage is apparent with reduced order methods as has been demonstrated in reference [2]. Game theory on the other hand requires a control type of notation in order to make the formulation of game problems tractable.

The equations given by (1) and (2) or (3) and (4) may be thought of as defining the system under study. In mathematical terms the "system" is a constraint set Ydefined by

Y={yeEmsuch that g(y)

where e is used to designate

= 0and / ! (>)>0)

'an element of" and

(5)

is used to designate "an element o t" ana Em

designates an w-dimensional space of real numbers. Thus Yis the set of all points in the m-dimensional variable space which satisfy all the constraints. For each choice of ye Y one or more costs may be defined by means of a cost function

G(y)=[Gi(y),.--,Gr(y)]. (6)

When /•= 1 we have a scalar-valued cost function. For r> 1 we have either a vector-valued cost criterion or a continuous static game. In order to distinguish between the latter two cases, additional formulation is necessary; in particular, we must establish whether one or more designers are involved in selecting the point yeY and the relation of the various costs to the various players.

Optimality in Engineering Design

Engineering design is a multifacet discipline which is hard to identify in any precise way. However, one aspect of it, optimum design, would appear to be more precisely defined and "well understood" as witnessed by numerous texts e.g., [3-5] and the number of algorithms available for "optimizing." Perhaps however, this is not really the case. The familiar concept of a minimum, the most widely used optimality concept, is applicable to only the most straightforward design problem involving a scalar-valued cost (or design objective). For example, in the design of mechanisms,

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minimizing the maximum error as first proposed by Chebychev [6] is a logical design objective. However, cost, weight, and reliability are also important factors and a designer may wish to take explicit account of these objectives in an optimality analysis. To do this a vector cost must be defined.

It has been recognized for some time that multi-criteria or vector-valued optimization theory should play an important role in engineering design. An early introduction to vector minimization methods in engineering design was given in reference [7], They discuss some of the geometric properties of vector minimization and apply these to problems in the design of journal bearings. Two design objectives are formulated: minimum oil film temperature, and minimum oil flow rate. An interesting discussion to the solution of this problem is given. Unfortunately, some of the suggestions which seem to follow from the geometry as presented are incorrect. These suggestions involve some common misconceptions about vector minimization which will be returned to later.

The use of game theory as part of the engineering design process is not as well known. This aspect of optimality may be used when a multi-objective design problem has been assigned to several designers (or design teams) with each designer being responsible for one or more design objectives. Decentralizing responsibility in such a way is a natural choice in the design of large-scale systems such as an entire aircraft or space vehicle system. With several designers each with his own objectives, the nature of the optimization process can take several paths and the total design may not be optimal in the sense that a single designer theoretically could do better. Game theory applied to this situation provides a method for understanding and perhaps guiding the optimization process. As such it provides an important management tool for use in decentralized design.

Optimality problems may be classified according to the following table.

Fig. 1(a) Scalarization is applicable.

Fig. 1(b) Scalarization is not applicable.

Designers one

one

more than one

Objectives one

more than one

one or more for each designer

Classification scalar-valued optimization (nonlinear programing) vector-valued optimization

continuous static game

Concepts minimum

Pareto-minimum, compromise solution, Pareto-minimum, Nash equilibrium, etc.

As noted in the last column the concept of a minimum, as usually defined, is applicable only for the one designer-one objective case. In what follows we first examine optimality concepts for the one designer case and then examine the case of more than one designer. The latter case is often a game situation.

One Designer

With only one designer selecting design parameters we have either a problem in scalar- or vector-valued optimization. It is useful to examine the basic definitions used in each case. The constraint set Y is given by (5) and the cost function G(y) is given by (6).

Definition of a minimum {scalar cost): A point y*eY is a minimum point if and only if there does not exist a point yeY such that G{y) <G(y*). For a local minimum the definition is altered by replacing yeY by yeBD Y where B is a "ball"

centered at y* and BC\Y represents points common to B and Y.

Note that many points y*eY may satisfy the definition; however, for most problems in engineering design, the minimum points y* are isolated.

For vector-valued costs the following definition is used.

Definition of a Pareto-minimum (vector cost): A point y*eY is a Pareto-minimum point if and only if there does not exist a point yeY (yeBDY for local Pareto-minimum) such that GOO <G(y^).

The notion < means "partially less than," that is, G,(y) <G,0-*)forali;e {1 r] and Gj(y) <Gj(y*) for at least one ye {1, . . . ,r}. The "partially less than" notation succinctly captures the essences of the Pareto-minimal property. If y*eYis Pareto-minimal then for any other yeY at least one cost must increase or all costs remain the same. Under this definition it is usually possible for many points y*eY to have the Pareto-minimum property. Indeed for most problems in engineering design, the Pareto-minimal points are not isolated.

Imposing a regularity assumption on the constraint set Y and using the local version of the two foregoing definitions allows for the development of necessary conditions for determining minimum points and Pareto-minimum points. For sake of comparison, these necessary conditions are summarized in the following theorems [1].

Minimum Theorem: If y*eY is a regular local minimal point for the scalar function G, then there exist multipliers X = [X[ \r]

T and n = [^, . . . ,nq]T such that

dL/dy\y*=0 (7)

166/Vol. 105, JUNE 1983 Transactions of the ASME


y* i

GTT. COS-IT

3' Fig. 2 Comparing the Pareto-minimal set obtained using the Pareto-minimal Theorem with the set obtained by minimizing one cost while holding the other constant

•g(y')=o h(y*)>0

HTh(y*) = 0 /*>0

UyXri =G(y) -\Tg(y) ~tiTh(y).

(8) (9)

(10)

(11)

Pareto-minimum Theorem: If y*eY is a regular local Pareto-minimal point for the vector function G then there exist multipliers X = [Xi, . . . ,\r]

T, /*=[/*!> • • • .M?]7^. ij = [iji, . . . ,r\r]

T such that dL/dy\y*=0 (12)

g(y*) = 0 (13) / i ( / ) > 0 (14)

^r/!(j*) = 0 (15) M&0 (16) »/>0 (17)

where ZO^.X,^) = vTG(y) -\Tg(y) ~ixTh(y).

The necessary conditions of the theorems (i.e., (7)-(ll) or (12)—(17)) are used to yield candidate solutions for a minimum or a Pareto-minimum. The first theorem is also known as the Kuhn-Tucker Theorem or the Nonlinear Programing Theorem.

It follows from these theorems that if we define

G = vTG (18) where ?j>0, then the necessary conditions for Pareto-minimum are the same as for minimizing Q subject to yeY. Hence it is tempting to infer that 9 must be minimized at Pareto-minimal points. Indeed this "scalarization" concept (i.e., minimize 9) is often suggested as a way to solve the multicriteria optimization problem. However, 9 need not be minimized at a Pareto-minimal point (it may even be maximized!) and thus seeking only minimal points of Q will, in general, exclude part of the Pareto-minimal solution set. It is easy to demonstrate this possibility by examining 9 in relation with the set of admissible cost vectors in cost space as illustrated in Fig. 1 for the case when r = 2.

In this figure, the cost vector G has two components Gx and G2 which are used as the coordinate axis of cost space. The value of the cost vector for all ye y lies in some subset of T£]of the cost space as indicated in the figure. The Pareto-minimal points for each of these sets are easily determined by applying the Contact Theorem [1]. A vector G is Pareto-minimal if and only if the closed negative orthant (shown dotted) centered at

a. 6 ,

Fig. 3 An illustration of Schmitendorf's (reference [8]) Theorem

G does not contain any points of T£] except G. The Pareto-minimal points are as indicated by the heavy lines in this figure. In Fig. 1(a) r£] is convex and 9 is minimized for every choice of rj >0 on the Pareto-minimal set. In Fig. 1(6) G is not minimized for every choice of t]!>0 on the Pareto-minimal set.

Another procedure often suggested for obtaining Pareto-minimal points is to minimize one of the cost functions holding the others fixed. For example, in the two-dimensional case minimizing G* (Vi Ji) subject to G20, ,y2) = constant for a range of values of the constant will yield Pareto-optimal solutions. This observation is correct. However, the problem with utilizing this observation is knowing the range of values for the constant. For certain values of the constant the foregoing process yields points which are not Pareto-minimum as illustrated in Fig. 2. Lines of constant cost are drawn for two cost functions Gx(yuy2) and G2{yx,y2). The Pareto-minimal points are obtained directly from the Pareto-minimum Theorem. With no equality or inequality constraints the necessary conditions of this theorem reduce to the requirement that the gradients of each cost function be in opposite directions. Note that minimizing Gi(yi,y2) subject to G20'i ,y2) = constant for an improper range of the constant will yield points for which the gradients are in the same direction. Hence an essential property of the Pareto-minimal solution is overlooked by this procedure.

The idea of minimizing one criterion subject to constraints on the other criteria can be formulated so as to produce a necessary and sufficient condition for Pareto-minimality. The following theorem due to Schmitendorf [8] illustrates that Pareto-minimal points for a vector minimization problem may be found by solving a system of scalar minimization problems. However there may be difficulty in utilizing this result numerically [9].

Pareto-minimum Theorem: A point G*e H) is Pareto-minimal if and only if for each je[i, . . . ,r\

G/<Gy for all GeEl; (19) where^; =(GeEl such that G,-<Gf- (=1, . . . ,r,i^j}. The theorem is illustrated in Fig. 3. The point G with components G, and G2 satisfies Gf <GX o n ^ i and Gf <G2 on L Z . 2 '

Suppose that a designer has utilized a Pareto-minimal Theorem to arrive at the Pareto-minimal set. Clearly the design should be chosen on this set, but exactly which point to choose remains an open question. A particular choice involves favoring one design criterion over another. Yet this information was not known a priori. It is often suggested that

Journal of Mechanisms, Transmissions, and Automation in Design JUNE 1983, Vol. 105/167


Fig. 4(a) set

Maximizing the satisfaction function on the Pareto-minimal

Fig. 4(b) The Pareto-minimal set obtained when the satisfaction function is included as one of the design parameters

one can resolve this problem by maximizing some new "satisfaction function" over the Pareto-minimal set. The suggestion is appealing in that a unique point on the Pareto-minimal set is generally then obtained. This would seem to resolve the issue of which Pareto-minimal solution to choose. However, upon reflection it becomes apparent that the idea really begs the issue. If the satisfaction function is important, then it should be used along with the other design criteria in generating the Pareto-minimal set. Figure 4(a) illustrates the use of a satisfaction function in "resolving" which Pareto-minimal solution to choose, and Figure 4(b) illustrates the effect of including the satisfaction function as one of the design parameters. Introducing the satisfaction function as an additional cost criterion increases the dimension of the Pareto-minimal set. Rather than resolving the issue it further adds to the problem of making the ultimate choice.

One concept which may prove useful in some situations is the compromise solution concept of Salakvadze [10]. The idea behind this concept is to find the "closest" point on the Pareto-minimal set to a "utopia point." The Utopia point is a single unique point in the cost space where every design criterion takes on its minimum value. In general the Utopia point is not feasible (that is, it lies off of H ) as illustrated in Fig. 5. The closest point on the Pareto-minimal set is obtained in this case by drawing a line from the utopia point so that it enters perpendicular to the Pareto-minimal set. Since there is no a priori reason for restricting closeness to the special case of the Euclidian norm, the usual measure of distance, the compromise solution, does not in reality resolve the issue of

•UTOPIA VO\*iT

Fig. 5 Using the utopia point to select a Pareto-minimal design point

which Pareto-minimal solution to choose. However, a designer may find it useful to calculate the Utopia point and examine its location in relation to the Pareto-minimal set before making a final design choice.

This section may be summarized by noting that one designer choosing parameters for vector-valued optimization should restrict his choice to the Pareto-minimal set. This set may be obtained by a number of methods. Two theorems are presented here which may be useful. The particular Pareto-minimal solution to use is unresolved unless one introduces additional information. This can include weighting factors, satisfaction functions, and closeness to the Utopia point. All such additional information tends to be "after the fact" in that any particular Pareto-minimal solution choice boils down to deciding the relative importance of each of the costs.

More Than One Designer

Suppose that a large design project has been broken down into a number of subprojects each with a well-defined design objective and each with one designer in charge of minimizing that objective. Since communication is generally possible between the subgroups and since each designer must consider the possible choices made by the other designers, it is envisioned that there will be many tentative designs which will be passed back and forth between the various designers before some ultimate design is finally obtained. Some aspects of this design process will now be examined by considering two questions. Will the process converge to a design acceptable to all the designers? If so, in what sense will the final design be optimal? For simplicity, the discussion which follows will be restricted to two designers.

Assume there are two design parameters ul and u2. Let the first designer have a cost G, («, ,w2) with the design variable «, and the second designer have a cost G2(u{ ,u2) with the design variable u2. For any choice of u2 by the second designer the first designer would rationally choose u, to minimize Gt. Every such pair defines what is known as a rational reaction set [11] for the first designer. Similarly, a rational reaction set for the second designer is also defined. Two such rational reaction sets are illustrated in Fig. 6. If the two rational reaction sets intersect, the point of intersection defines a game theoretic solution known as a Nash solution point [12]. At the Nash point, each designer is on his own rational reaction set and as such, each will perceive his design choice to be optimal. This is because any unilateral design variable change made from the Nash solution by either designer can not result in a lower value to the design cost for the designer who makes the change.



RATIONAL HeACTWM SETS

C SECOr-ir> OESI&MEX-

U,

Fig. 6 A decentralized design process converging to the Nash point

A.-

U*

Fig. 7 A convergent decentralized design example

In general the determination of the rational reaction sets may be quite difficult. For example, if they are to be determined numerically then a large number of scalar optimization problems would have to be solved in order to define the sets. However, for the design process considered here, the rational reaction sets would generally not be determined. Even though each designer has a rational reaction set, the geometric character of the sets such as illustrated in Fig. 6 would usually not be known.

For the situation depicted in Fig. 6, suppose that designer 2 suggests a tentative design to designer 1. This suggestion would no doubt minimize G2{uuu2) with respect to both «, and u2. In other words, designer 2 would assume the most favorable ux and choose u2 accordingly. Designer 1 would then use this value of u2 in Gx{uuu2) and choose ut to be on his own rational reaction set as shown in Fig. 6. Assume the design is then passed back to designer number 2, he now uses this value of ux in G2(wi >"2) a n d chooses u2 to be on his own rational reaction set. If the process is convergent it should continue until each designer perceives himself to be at an optimal solution. That is, the solution converges to the Nash solution.

Unfortunately the Nash solution is usually not on the Pareto-minimal set as indicated in Fig. 6. This being the case,

3 -

2 -

tyvnortAL neAcnoN SETS

15E^l&NE«.<! 2 —

t>ESI&WE1<( I s c PAH.ETO -MINIMUM

Fig. 8 A decentralized design example which does not converge to the Nash solution

there must exist at least one point on the Pareto-minimal set so that the cost function for each designer is less than his Nash cost. Generally there will be a continuum of such points. However, getting such a point as a design point may be difficult since, in general, this will require each designer to choose a design off of his rational reaction set. There are additional game theoretic methods which may be of help in the analysis of a game situation. However, for brevity, only the Nash and Pareto-minimal concept will be used in what follows.

The situation is best illustrated by means of some examples. Figure 7 illustrates a situation in which the design process mentioned above will result in the Nash solution. The two cost functions are given by

Gy =ulu2-3ui +u\ (20)

G1=u\/2-uiu1 (21)

with designer 1 choosing ux and designer 2 choosing u2 with «! > 0 , M 2 aO. The rational reaction sets for each designer are illustrated with the global minimum of Gt at (3/2,0). A global minimum for G2 does not exist. Suppose that designer 1 makes the first tentative design at his global minimum (3/2,0). Designer 2 using u, =3 /2 would choose u2 =1.5 in order to move the design to his rational reaction set. This design is then passed back to designer 1 who using u2 =1.5 would choose ux = 0.75 in order to move the design to his rational reaction set. By continuing this process the solution converges to the Nash solution. The Nash solution in this case is not Pareto-minimal as indicated. Indeed the cost function for both designers can be reduced by moving to any point on the Pareto-minimal set between the two dotted lines. Each of these lines represents a line of constant Nash cost for one of the designers. Clearly in order to obtain a Pareto-minimal solution a different design process than the one just outlined must be used.

What is perhaps surprising, the decentralized design process as described above need not even converge to the Nash solution. Consider two cost functions given by

Gl=ulu2~l.5ul+u2l/4 (22)

G2 = u\/2~uxu2 (23)

where as before, designer 1 chooses ux and designer 2 chooses u2 with «! >0, u2 >0 . Starting with the tentative design with «, =0,u2 =0.8, then passing this information to designer 1 to adjust Mi, then to designer 2 to adjust u2, etc. results in an iterative process which converges to a limit cycle between the designs (3,0) and (3,3) as illustrated in Fig. 8. How the two

Journal of Mechanisms, Transmissions, and Automation in Design JUNE 1983, Vol. 105/169


designers might ultimately choose a solution in this case is difficult to say but in the absence of any additional information or intervention by a third party, the ultimate design would probably be neither Nash nor Pareto-minimal.

Methods exist for obtaining Nash and Pareto-minimal solutions as well as other game theoretic solutions in a game type of setting [1]. These solutions should be useful for analyzing a design but they will not resolve problems associated with the "play of the game," that is, the particular circumstance under which the designers are allowed to make decisions. In any event, it is of interest to know how a given design philosophy (in this case decentralized design) will be reflected in the solution obtained. With a knowledge of game theory, perhaps a manager can accept with proper understanding, the announcement by the design groups that they have arrived at the "optimal" solution or perhaps that they can not resolve their design conflicts.

In summary, it has been shown that optimal design must be carefully defined and requires special attention when more then one design objective is specified. When more than one designer is involved in a large design project, a project manager may have to appeal to game theoretic methods in order to be assured that the ultimate design will be optimal in some sense.

References

1 Vincent, T. L., and Grantham, W. J., Optimality in Parametric Systems, Wiley, New York, 1981.

2 Gabriele, G. A., and Ragsdell, K. M., "The Reduced Gradient Method: A Reliable Tool for Optimal Design," ASME 3rd Design Automation Conference, Sept. 1975,

3 Wilde, D., Globally Optimal Design, Wiley, New York, 1978. 4 Haug, E. J., Jr., and Arora, J. S., Applied Optimal Design, Wiley, New

York, 1979. 5 Gallagher, R. H., and Zienkiewicz, O. C , Optimum Structural Design:

Theory and Applications, Wiley, New York, 1973. 6 Root, R. R., and Ragsdell, K. M., "A Survey of Optimization Methods

Applied to the Design of Mechanisms," ASME Journal of Engineering for Industry, Aug. 1976, pp. 1036-1041.

7 Bartel, D. L., and Marks, R. W., "The Optimum Design of Mechanical Systems With Competing Design Objectives," ASME Journal of Engineering for Industry, Feb. 1974, pp. 171-178.

8 Schmitendorf, W. E., "Cooperative Games and Vector-Valued Criteria Problems," IEEE Trans. Automatic Control, Vol. AC-18, No. 2,1972.

9 Robinson, S. M., "A Characterization of Stability in Linear Programming," Operation Research, Vol. 25, No. 3, 1977.

10 Salukvadze, M. E., Vector-Valued Optimization Problems in Control Theory, Academic Press, New York, 1979.

11 Simaan, M., and Cruz, J. B., Jr., "On the Stackelberg Strategy in Nonzero-Sum Games," Journal of Optimization Theory App., Vol. 11, No. 5, 1973.

12 Nash, J. F., "Non-Cooperative Games," Annals of Mathematics, Vol. 54, No. 2,1951.



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Game Theory as a Design Tool