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Strong risk invariance in multiobjective programming
Socro-Econ. Ph. Sci. Vol. 20. No. 2, 123-125. Printed in Great Britain.
00384121/86 $3.00 + .Xl 0 1986 Pertwnon Press Ltd.
1986
STRONG RISK INVARIANCE IN MULTIOBJECTIVE PROGRAMMING M. A. GROVE Department
of Economics,
University
of Oregon,
Eugene, Oregon
97403, U.S.A.
(Received 4 March 1985; in revisedform 28 October 1985)
Abstract-A
utility function exhibiting strong risk invariance (SRI) is employed to study a problem in which a decision maker’s objectives, y, are linked to his decision variables, x, by the linear system y = Ax, with matrix A random. An application of the model to macroeconomic planning is considered.
INTRODUCITON The focus of this paper is a multiple objective decision making problem in which the objectives, y, of a decision maker (DM) are related to his decision variables, x, by the linear system y = Ax, where x belongs to the polyhedral convex set 5’ = {xtR;]Bx d h}. Matrices A and B are (q X n) and (m X n) respectively, and h is a vector of known constants. Some years ago, Contini [I] introduced uncertainty into this problem by proposing that the relationship between DM’s objectives and his decision variables be written as y = Ax + e, where e is a vector of random variables. If DM has fixed goals, y*, and if e is normally distributed with mean vector 0 and covariance matrix V, then Contini shows that the problem of choosing x is formally analogous to treating y = ku + e as a set of reduced form regression equations, and proceeding as if one were estimating x by generalized least squares. Although they are certainly interesting, Contini’s results depend crucially on the way in which he introduces uncertainty. There are, however, examples of the problem I have outlined where the assumption that uncertainty enters in the additive fashion, suggested by Contini, is inappropriate. In some cases it is natural to assume that the matrix A is if.rc(frandom. I shall explore the implications of making this assumption in the balance of this paper. A model of the decision problem is developed in the following section. The concluding section contains an application and an illustrative example. THE MODEL Suppose, then, that matrix A in our statement of the problem is random. In particular, assume that the n columns A, of A are q-dimensional random variables with finite mean vectors m, and nonsingular covariance matrices V,,. Then y is a q-dimensional random variable with mean vector M = Zrn,x; and covariance matrix v = BBV,&$,. Since the probability distribution of DM’s objectives y depends on the values of his decision variables x, it seems appropriate to apply the expected utility hypothesis. To do so, assume that DM has preferences in probability distributions of y which can be ordered by the von-Neumann Morgenstem utility function u(v)
on R4. We will assume, as usual, that u is concave increasing with continuous first and second derivatives u,, u,,. Recalling the definition of set S above, an assumption that DM chooses x to maximize expected utility requires that he solve problem P: XCS, max E(u). As it stands, problem P is too general to be of much value since little more than the standard Kuhn-Tucker conditions can be extracted from it to guide DM’s choice of x. Clearly, some specialization of P obtained by imposing additional restrictions on u, or on the joint probability distribution of the vectors A,, or both, are required to obtain practical results. The literature on utility functions with multiple attributes is very large, as just a glance at the survey by Farquhar [2] indicates. Though much of it has limited value from an applied point of view, recent contributions by Willig [3] and Morrison [4] on the concept of strong risk invariance (SRI) do have promise in that direction. Let us take a moment to examine it as a source for specializing problem P. Briefly, DM’s utility function displays SRI if for each i, the measure of absolute risk aversion in objective yi, -r~,,/u,, is invariant along indifference surfaces of u. Roughly speaking, this means that DM’s attitude toward “small risks” in each objective y, is independent of the reference point from which the risk is measured-so long as such reference points lie on the same indifference curve. When the measure of absolute risk aversion is (globally) a positive constant for each i, i.e. when -n,,/u, = c, > 0 for each i (globally), Willig shows that the explicit form of u(y) will involve sums and products of the terms exp(-c,y,). A promising special case of this result arises when the arbitrary scaling constants in Willig’s formulation are chosen so that u(v) = -exp(-c’y), where c > 0 is a vector of constantst This form of u is restrictive to be sure: it implies, for example, that rates of substitution among objectives are constant in all directions. But it does have important advantages. For one, it is a straight forward generalization of the standard Pratt-Arrow measure of constant absolute risk aversion to the case of many objectives-the constant ci can be interpreted as the coefficient of constant absolute risk aversion in
defined
t See in particular
123
Willig’s discussion
on page 625.
124
M. A. GROVE
the objective y,. For another, E(u) can be easily calculated when the A, are joint normal random vectors. We can now propose a specialization of problem P which can be of use in an applied setting. To obtain it, we specialize u along the lines suggested above so that u(y) = -exp(-c’y). If we assume further that vectors A, are jointly normally distributed with mean vectors m, and covariance matrices V,, then E(u) = -exp[-c’M + (f)c’Vc], where M = Bm,x, and V = ZZVi++, as before. Given the form of this expression, maximizing E(u) for xtS is equivalent to maximizing the strictly concave function G(x) = Z(c’mi)xi - (~)ZZ(c’V,,c)xix, for x&. Our proposal, then, is to replace problem P by problem P’: xtS, max @(x) when practical results are needed. AN APPLICATION
The econometric model of the U.S. economy constructed by Klein [5] and studied in a planning context by Theil [6] is small enough to provide an interesting example of the type of problem we have considered. The reduced form of Klein’s model contains six endogenous variables which we will denote by q, to &. These variables, which we shall take as targets for a decision making authority, are consumption, profits, the private wage bill, investment, national income, and a measure of the capital stock of the economy. Following Theil’s examples, we will measure these variables in billions of 1934 dollars. Three endogenous variables, the public wage bill, indirect taxes, and government spending appear in the reduced form. We shall take these as decision variables available to our authority, and label them as xl, x2, and x3, respectively. Estimates of the coefficients of the decision variables in the six equations of the reduced form are given in matrix A below. In order to save space, the numbers in parentheses immediately adjoining the coefficients are the variances of the estimates which I have supplied for purposes of the illustration.
ernment spending) can change by as much as 30 percent, but that it is politically infeasible to change x2 (taxes) by more than 5 percent. To make the problem interesting, we assume in addition that a deficit constraint of the form xi - x2 + x3 5 d is a political condition on the decision making authority’s freedom to choose vector x. If we assume, for simplicity, that the matrices V, defined above are such that V, = (0) when i # j and are diagonal with the values reported in matrix A when i = j, then we need only supply values of the constants of vector c in order to apply P’. Details for the solution of three cases are reported below. The rows in each of the tables labeled 1,2, and 3 are the optimal values of xi, x2, and xj for each of the three assumptions about d which head the columns of the tables. Case 1 is based on the assumption that the authority is equally risk averse in all target variables with ci = 1 all i. Case 2 assumes that c, = C~= 2 with the remaining c, = 0.1. This case might be characterized as a short run “proconsumption” policy since the authority is much more risk averse in consumption (yi) and the private wage bill ( JJ~)than it is in the variables associated with long term economic growth. On the other hand, thevaluesofci=c2=c3= 1,cq=c5=2,andc6=3 which underlie Case 3 characterize it as a long run “progrowth” policy, since now the authority is a good deal more risk averse in the variables associated with growth than it is in those associated with consumption. The pattern of the solutions is about what one would expect given the attitudes of the authority which condition these cases. The compromise between the equally weighted targets in Case 1 is accomplished by reducing taxes by the allowable 5 percent and using the allowable deficit to finance government spending. High levels of government spending and taxes are important features of Case 2 as one might guess. Taxes are raised to off-set the decrease in government spending that would otherwise be necessary when the allowTable I.
L
0.666(0.0 14) 0.224(0.008) -0.162(0.034) -0.052(0.053) 0.614(0.012) -0.053(0.006)
-0.188(0.007) -1.281(0.038) -0.204(0.01 1) -0.296(0.016) - I .484(0.06 1) -0.296(0.023)
0.671(0.019) 1.119(0.046) 0.81 l(O.025) 0.259(0.017)
CASE
1
4
6
10
1
3.376
3.376
L.001
1.930(0.226)
2
6.704
6.704
6.704
0.259(0.007)
3
7.328
9.328
12.703
1
Now let D be the matrix of coefficients of the vector w of predetermined variables and structural disturbances given in Theil’s Table 3.4. Then using the values for w he reports on page 88, I’ = dD’ = (37.614,8.955, 22.897, -5.555, 31.852, 201.545) and we can write the equations of the reduced form as q = t + Ax. If we then define y = q - t, we have a system of the same form as that we have considered above. To develop the example further, we need to specify some reasonable constraints on the decision variables. In his study of the problem, Theil uses initial values sy = 4.823, ss = 7.157, and x: = 10.008 for the decision variables. We shall do the same. adding the assumption that ,x, (the public wage bill) and x3 (gov-
CASE
1
2
4
6
10
3.376
3.376
3.694
2
7.410
6.870
3
S-034
9.494
CASE 4
6
6.704 13.010
3 10
3.703
~.58~
2
6.704
6.704
6.704
3
7.001
7.001
7.001
1
4.584
Strong risk invariance
in multiobjective
able deficit is small, and are decreased by the allowable 5 percent only when the allowable deficit reaches 10. The “hands-off” character of Case 3 is evident from the minimal values for taxes and government spending which solve problem P’ for all three values of the allowable deficit. The public wage bill (x,) is a bit larger than its minimum value when the allowable deficit is 6 and 10, but the deficit constraint is still slack in both these instances. REFERENCES
I. B. Contini, A stochastic approach Op. Res. 36, 576-586
(1968).
to goal programming.
programming
125
P. Farquhar, application.
A survey of multiattribute utility theory and In: Starr and Zeleny (eds.), Multiple Criteria Decision Making. North-Holland, Amsterdam (1977).
R. Willig, Risk-invariance and ordinally additive functions. Econometrica 45, 621-640 (1977).
utility
J. Morrison, The structure of a utility function under strong risk invariance. SIAM J. Appl. Math. 31, 93-98
(1976). L. Klein, Economic Fluctuation in the United States, 1921-41. Wiley, New York (1950). 6. H. Theil. Optimal Decision Rules for Government and Industry. North-Holland, Amsterdam (1964).
00384121/86 $3.00 + .Xl 0 1986 Pertwnon Press Ltd.
1986
STRONG RISK INVARIANCE IN MULTIOBJECTIVE PROGRAMMING M. A. GROVE Department
of Economics,
University
of Oregon,
Eugene, Oregon
97403, U.S.A.
(Received 4 March 1985; in revisedform 28 October 1985)
Abstract-A
utility function exhibiting strong risk invariance (SRI) is employed to study a problem in which a decision maker’s objectives, y, are linked to his decision variables, x, by the linear system y = Ax, with matrix A random. An application of the model to macroeconomic planning is considered.
INTRODUCITON The focus of this paper is a multiple objective decision making problem in which the objectives, y, of a decision maker (DM) are related to his decision variables, x, by the linear system y = Ax, where x belongs to the polyhedral convex set 5’ = {xtR;]Bx d h}. Matrices A and B are (q X n) and (m X n) respectively, and h is a vector of known constants. Some years ago, Contini [I] introduced uncertainty into this problem by proposing that the relationship between DM’s objectives and his decision variables be written as y = Ax + e, where e is a vector of random variables. If DM has fixed goals, y*, and if e is normally distributed with mean vector 0 and covariance matrix V, then Contini shows that the problem of choosing x is formally analogous to treating y = ku + e as a set of reduced form regression equations, and proceeding as if one were estimating x by generalized least squares. Although they are certainly interesting, Contini’s results depend crucially on the way in which he introduces uncertainty. There are, however, examples of the problem I have outlined where the assumption that uncertainty enters in the additive fashion, suggested by Contini, is inappropriate. In some cases it is natural to assume that the matrix A is if.rc(frandom. I shall explore the implications of making this assumption in the balance of this paper. A model of the decision problem is developed in the following section. The concluding section contains an application and an illustrative example. THE MODEL Suppose, then, that matrix A in our statement of the problem is random. In particular, assume that the n columns A, of A are q-dimensional random variables with finite mean vectors m, and nonsingular covariance matrices V,,. Then y is a q-dimensional random variable with mean vector M = Zrn,x; and covariance matrix v = BBV,&$,. Since the probability distribution of DM’s objectives y depends on the values of his decision variables x, it seems appropriate to apply the expected utility hypothesis. To do so, assume that DM has preferences in probability distributions of y which can be ordered by the von-Neumann Morgenstem utility function u(v)
on R4. We will assume, as usual, that u is concave increasing with continuous first and second derivatives u,, u,,. Recalling the definition of set S above, an assumption that DM chooses x to maximize expected utility requires that he solve problem P: XCS, max E(u). As it stands, problem P is too general to be of much value since little more than the standard Kuhn-Tucker conditions can be extracted from it to guide DM’s choice of x. Clearly, some specialization of P obtained by imposing additional restrictions on u, or on the joint probability distribution of the vectors A,, or both, are required to obtain practical results. The literature on utility functions with multiple attributes is very large, as just a glance at the survey by Farquhar [2] indicates. Though much of it has limited value from an applied point of view, recent contributions by Willig [3] and Morrison [4] on the concept of strong risk invariance (SRI) do have promise in that direction. Let us take a moment to examine it as a source for specializing problem P. Briefly, DM’s utility function displays SRI if for each i, the measure of absolute risk aversion in objective yi, -r~,,/u,, is invariant along indifference surfaces of u. Roughly speaking, this means that DM’s attitude toward “small risks” in each objective y, is independent of the reference point from which the risk is measured-so long as such reference points lie on the same indifference curve. When the measure of absolute risk aversion is (globally) a positive constant for each i, i.e. when -n,,/u, = c, > 0 for each i (globally), Willig shows that the explicit form of u(y) will involve sums and products of the terms exp(-c,y,). A promising special case of this result arises when the arbitrary scaling constants in Willig’s formulation are chosen so that u(v) = -exp(-c’y), where c > 0 is a vector of constantst This form of u is restrictive to be sure: it implies, for example, that rates of substitution among objectives are constant in all directions. But it does have important advantages. For one, it is a straight forward generalization of the standard Pratt-Arrow measure of constant absolute risk aversion to the case of many objectives-the constant ci can be interpreted as the coefficient of constant absolute risk aversion in
defined
t See in particular
123
Willig’s discussion
on page 625.
124
M. A. GROVE
the objective y,. For another, E(u) can be easily calculated when the A, are joint normal random vectors. We can now propose a specialization of problem P which can be of use in an applied setting. To obtain it, we specialize u along the lines suggested above so that u(y) = -exp(-c’y). If we assume further that vectors A, are jointly normally distributed with mean vectors m, and covariance matrices V,, then E(u) = -exp[-c’M + (f)c’Vc], where M = Bm,x, and V = ZZVi++, as before. Given the form of this expression, maximizing E(u) for xtS is equivalent to maximizing the strictly concave function G(x) = Z(c’mi)xi - (~)ZZ(c’V,,c)xix, for x&. Our proposal, then, is to replace problem P by problem P’: xtS, max @(x) when practical results are needed. AN APPLICATION
The econometric model of the U.S. economy constructed by Klein [5] and studied in a planning context by Theil [6] is small enough to provide an interesting example of the type of problem we have considered. The reduced form of Klein’s model contains six endogenous variables which we will denote by q, to &. These variables, which we shall take as targets for a decision making authority, are consumption, profits, the private wage bill, investment, national income, and a measure of the capital stock of the economy. Following Theil’s examples, we will measure these variables in billions of 1934 dollars. Three endogenous variables, the public wage bill, indirect taxes, and government spending appear in the reduced form. We shall take these as decision variables available to our authority, and label them as xl, x2, and x3, respectively. Estimates of the coefficients of the decision variables in the six equations of the reduced form are given in matrix A below. In order to save space, the numbers in parentheses immediately adjoining the coefficients are the variances of the estimates which I have supplied for purposes of the illustration.
ernment spending) can change by as much as 30 percent, but that it is politically infeasible to change x2 (taxes) by more than 5 percent. To make the problem interesting, we assume in addition that a deficit constraint of the form xi - x2 + x3 5 d is a political condition on the decision making authority’s freedom to choose vector x. If we assume, for simplicity, that the matrices V, defined above are such that V, = (0) when i # j and are diagonal with the values reported in matrix A when i = j, then we need only supply values of the constants of vector c in order to apply P’. Details for the solution of three cases are reported below. The rows in each of the tables labeled 1,2, and 3 are the optimal values of xi, x2, and xj for each of the three assumptions about d which head the columns of the tables. Case 1 is based on the assumption that the authority is equally risk averse in all target variables with ci = 1 all i. Case 2 assumes that c, = C~= 2 with the remaining c, = 0.1. This case might be characterized as a short run “proconsumption” policy since the authority is much more risk averse in consumption (yi) and the private wage bill ( JJ~)than it is in the variables associated with long term economic growth. On the other hand, thevaluesofci=c2=c3= 1,cq=c5=2,andc6=3 which underlie Case 3 characterize it as a long run “progrowth” policy, since now the authority is a good deal more risk averse in the variables associated with growth than it is in those associated with consumption. The pattern of the solutions is about what one would expect given the attitudes of the authority which condition these cases. The compromise between the equally weighted targets in Case 1 is accomplished by reducing taxes by the allowable 5 percent and using the allowable deficit to finance government spending. High levels of government spending and taxes are important features of Case 2 as one might guess. Taxes are raised to off-set the decrease in government spending that would otherwise be necessary when the allowTable I.
L
0.666(0.0 14) 0.224(0.008) -0.162(0.034) -0.052(0.053) 0.614(0.012) -0.053(0.006)
-0.188(0.007) -1.281(0.038) -0.204(0.01 1) -0.296(0.016) - I .484(0.06 1) -0.296(0.023)
0.671(0.019) 1.119(0.046) 0.81 l(O.025) 0.259(0.017)
CASE
1
4
6
10
1
3.376
3.376
L.001
1.930(0.226)
2
6.704
6.704
6.704
0.259(0.007)
3
7.328
9.328
12.703
1
Now let D be the matrix of coefficients of the vector w of predetermined variables and structural disturbances given in Theil’s Table 3.4. Then using the values for w he reports on page 88, I’ = dD’ = (37.614,8.955, 22.897, -5.555, 31.852, 201.545) and we can write the equations of the reduced form as q = t + Ax. If we then define y = q - t, we have a system of the same form as that we have considered above. To develop the example further, we need to specify some reasonable constraints on the decision variables. In his study of the problem, Theil uses initial values sy = 4.823, ss = 7.157, and x: = 10.008 for the decision variables. We shall do the same. adding the assumption that ,x, (the public wage bill) and x3 (gov-
CASE
1
2
4
6
10
3.376
3.376
3.694
2
7.410
6.870
3
S-034
9.494
CASE 4
6
6.704 13.010
3 10
3.703
~.58~
2
6.704
6.704
6.704
3
7.001
7.001
7.001
1
4.584
Strong risk invariance
in multiobjective
able deficit is small, and are decreased by the allowable 5 percent only when the allowable deficit reaches 10. The “hands-off” character of Case 3 is evident from the minimal values for taxes and government spending which solve problem P’ for all three values of the allowable deficit. The public wage bill (x,) is a bit larger than its minimum value when the allowable deficit is 6 and 10, but the deficit constraint is still slack in both these instances. REFERENCES
I. B. Contini, A stochastic approach Op. Res. 36, 576-586
(1968).
to goal programming.
programming
125
P. Farquhar, application.
A survey of multiattribute utility theory and In: Starr and Zeleny (eds.), Multiple Criteria Decision Making. North-Holland, Amsterdam (1977).
R. Willig, Risk-invariance and ordinally additive functions. Econometrica 45, 621-640 (1977).
utility
J. Morrison, The structure of a utility function under strong risk invariance. SIAM J. Appl. Math. 31, 93-98
(1976). L. Klein, Economic Fluctuation in the United States, 1921-41. Wiley, New York (1950). 6. H. Theil. Optimal Decision Rules for Government and Industry. North-Holland, Amsterdam (1964).