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10YAK: 517.958
MSC2010: 93C15, 49N30, 49N35
GUARANTEED RISKS AND PAYOFFS IN A ONE-CRITERION
PROBLEM © V. I. Zhukovskiy, M. V. Boldyrev
Moscow State M.V. Lomonosov University Faculty of Computational Mathematics and Cybernetics Department of Optimal Management Lenin's mountains, 1, build. 52, Moscow, 119991, Russian Federation e-mail: [email protected], [email protected]
Guaranteed Risks and Payoffs in a One-criterion Problem.
Zhukovskiy V. I., Boldyrev M. V.
Abstract. The concept of a weakly guaranteed simultaneously under payoffs and risks solution of a one-criterion problem under uncertainty (OCPU) is proposed. Formalization is based on the concept of a vector saddle point from the theory of multicriteria problems with uncertainty.
Keywords: strategy, uncertainty, criterion, Slater optimum, vector saddle point.
Introduction
A one-criterion problem under uncertainty is defined by the following triplet:
• the set of possible decision maker's strategies; a strategy is a rule according to which each current decision maker's state of awareness is assigned some action that is permissible with the given information; in economic systems, these can be: choice of the price of a product, conclusion of supply contracts, introduction of new technologies, distribution of the wage fund, bonuses, and so on;
• the set of possible uncertainties; an uncertainty is the incompleteness or inaccuracy of information about conditions for the implementation of the chosen strategy; uncertainties arise in economic, mechanical controlled systems, during decision making (see more on this in [1-3] and numerous articles; even according to the French proverb "Entre bouche et cuiller, pour un petit de fait, vient souvent encombrier" — while you carry a spoon in your mouth, an obstacle often arises; by the way, an obstacle is a type of uncertainty; moreover, accounting for uncertainties during modeling real conflicts allows one to get more adequate results, which is confirmed, for example, by a large number of publications (over 1 million search results for Google Scholar query "mathematical modelling under uncertainty");
• the criterion, called the payoff function, whose value (winnings) is determined by the decision maker's chosen strategy and the implemented uncertainty (regardless of the decision maker's actions).
Using these three components (the criterion, the set of strategies, and the set of uncertainties), the decision maker finds their risk. Economic literature emphasizes the possibility of the following requirements for decision making in a one-criterion problem under uncertainty [4, p. 32; 5, p. 21]: optimal combination of values of the criterion (payoff, winnings) and risk value. This requirement arised because in a majority of applied problems, the more "profitable" a strategy is, the higher the degree (magnitude) of risk. What is risk? Well-known Russian optimization theory expert Talgat Sirazetdinov believes that there is currently no rigorous mathematical definition of risk [7, p. 31]. In the monograph [8, c. 15], sixteen possible definitions of risk are considered. Most of them require statistical data on uncertainty. However, such information is often missing (for one reason or another). These are the cases that are considered in the present article.
By risk, we mean the possibility of deviation of the implemented values from the desired ones. Note that this definition echoes "ordinary" microeconomic risks, as described, for example, in [9].
The desire for an optimal combination of the payoff value and the magnitude of risk is manifested in the fact that the decision maker evaluates the expected values of gain and risk and chooses a strategy that allows them to get the greatest possible payoff and, at the same time, the lowest risk. From the point of view of the theory of multicriteria problems under uncertainty, [6] in this case, one should consider two equally valid criteria: the initial criterion (the payoff function) and an auxiliary one (the risk function). In such a problem, the decision maker chooses (from the possible ones) the strategy, in which the criterion itself would assume the maximum value, and the risk would assume the minimum value. At the same time, the decision maker has to take into account the fact any uncertainty (from the possible ones) could arise.
Accounting for uncertainties in decision making is not an easy task. The recently published series of articles [11-13] and the book [14] are devoted to possible approaches. While the so-called strong guarantees (oriented on "the worst"—-minimum with respect to uncertainties—values of each individual criterion) were used to take account of the uncertainties in the article [10], in this paper, we focus on the so-called "vector guarantees" are made a focus on; these take into account both the value of the payoff function and the "negative" risk function ("in the spirit of" vector optimization [6, 13]). Then, as in [10], a transition is made to the two-criterion "problem of guarantees", in which uncertainties are no longer present. For the latter, the entire arsenal of different vector maxima, "dictated"
by the theory of multicriteria problems [13]. The fact is that strong guarantees ultimately reduce the possible values of each criterion, and, after all, the goal of the decision maker in the problem of guarantees is to maximize values of both criteria at the same time.
So, the proposed article differs from [10] in that:
first, when choosing the "good for themselves" strategies, the decision maker focuses not only on their payoff, but also on the risk associated with this payoff, and not only when choosing solutions, but also during formalization of guarantees;
second, while an analog of the maximin from [12] is used in [10] for "combatting uncertainty", here, an analog of the vector saddle point from [11] is used.
Finally, in publications on macroeconomics [2, p. 103; 9, p. 5] all decision makers are divided into three categories: risk-averse, risk-loving, and risk-neutral.
How would each of the three types of decision makers approach an OCPU? When making a decision, the concept of a guaranteed result (maximin) appeals the most to the risk-averse, and the principle of minimax regret appeals the most to the risk-loving. The similar question for risk-neutral remained open. [10] and the present work are devoted to attempts to resolve it. Generally speaking, we refer to the risk-averse as to pessimists (they expect "the worst" for themselves), to the risk-loving as to optimists (oppositely, they expect "the best"), and only the risk-neutral unite these two seemingly opposite trends.
So, in this article we will consider the one-criterion problem under uncertainty (OCPU):
r(1) = (X,Y,f (x,y)) , (1)
where the choice of the strategy x from the set X C R is made by the decision maker. The goal of the decision maker is the choice of x E X, for which the scalar criterion f (x,y) assumes the maximum value. In this case, the decision maker must take into account the effect of interference, errors, and other types of uncertainties y, which are only known to assume a value from the given set Y C Rm, the so-called interval uncertainties.
Presence of uncertainties in (1) leads to appearance of a set of results (payoffs, winnings)
f (x, Y) = {f (x,y) |Vy E Y},
"generated" by x E X. The set f (x,y) can be "narrowed down" using risks. As in [10], we confine ourselves to the Niehans-Savage risks. These risks are estimated by the value of the Niehans-Savage risk function
Rf (x,y)=max f (z,y) - f (x,y^ (2)
proposed in the 1950s by American mathematician Leonard Savage and Swiss mathematician Jiirg Niehans. They independently developed the principle of minimax regret (PMR), according to which the solution of (1) is the pair (xr, Rf) E X x R, defined by the following chain of equations:
Rf = minmax Rf (x,y) = max Rf (xr , y). (3)
J x€X yEY y&
Requirement (3) characterizes the decision maker as an optimist, who strives for "the best" payoff.
Note that pessimists follow (xg, fg) E X x R, where
fg = maxmin f (x,y) = min f (xg ,y), (4)
x€X y€Y y€Y
for they focus on "the worst" payoff.
Next, we propose the notion of a weakly guaranteed simultaneous payoffs and risks of a solution to a OCPU (formalization is based on the m vector saddle point from the theory of multicriteria problems with uncertainty [6]). Sufficient conditions of existence are established, with the help of which an explicit form of the introduced solution is found for a fairly general linear-quadratic version of OCPU with limited uncertainty.
1. Definition
Concept of weakly guaranteed solution of a OCPU. As already mentioned in the introduction, economic literature repeatedly emphasizes the relevance of the following requirement for decision making in a one-criterion problem under uncertainty: the solution should optimally combine payoff and risk. That is, after evaluating the expected payoffs and risks, the decision maker chooses the strategy that allows them to get the greatest possible payoff and, at the same time, the smallest possible risk. In fact, this means a transition from a one-criterion OCPU (1) to a two-criterion
(X, Y, {f (x,y), Rf (x,y)» , (5)
in which the two criteria are the payoff function f (x,y) and the risk function Rf (x,y). In problem (5), the decision maker chooses the strategy x E X for which their payoff (value of f (x,y)) would be maximized and at the same time their risk (value of Rf (x,y)) would be minimized. In this case, the decision maker has to take into account that any uncertainty y E Y could arise.
The following concept meets these requirements; in fact, it unites pessimists and optimists:
Definition 1. A weakly guaranteed under payoffs and risks solution (WGPR) of problem (1) is the triplet (xS, fS, RS) G X x R2, for which there is an uncertainty yS G Y such that fS = f (xS,yS),RS = R/(xS,yS), and
(1) for all strategies x G X, the following system of inequalities is incompatible:
f (x,ys) >fS, R/(x,ys) <RS, (6)
(2) for all uncertainties y G Y, the following system of two inequalities is incompatible:
f (xS,y) <fS, R/ (xS,y) >RS, (7)
where the risk function R/(x,y) is defined in (2).
In this case, xS is called a weakly guaranteeing strategy, and (xS, yS) is the pair that implements the WGPR.
According to the definition, for the construction of a WGPR—the weakly guaranteed under payoffs and risks solution (xS, fS, RS) of problem (1)—it is sufficient to find the pair (xS,yS) from the conditions of incompatibility of two inequalities, both (6) and (7), and then build the numbers fS = f (xS,yS), RS = R/(xS,yS).
Let us turn to some meaningful explanations of this definition.
Reduction of (1) to a two-criterion problem. As before, for analytical construction of a WGPR, we must first consider the multicriteria problem under uncertainty
(X, Y, f (x,y)>
and the risk function (2)
R/(x,y) = maxf (z,y) - f (x,y).
Then we construct the two-criterion problem (5) under uncertainty. From the perspective of the theory of multicriteria problems under uncertainty (MCPU) [6], this is not quite an ordinary task, because in it the decision maker strives to maximize the value of the first criterion f (x,y) and to minimize the value of the second criterion R/(x,y) by an appropriate choice of their strategy. At the same time, the decision maker has to take into account the fact any uncertainty y G Y could arise.
Analog of the saddle point. One way to formalize a solution of an MCPU is the "analog of the saddle point" proposed by the first author in the book [6]. It was based on
the notion of the saddle point in the antagonistic game with the scalar payoff function F (x,y):
(X, Y, F(x,y)>. (8)
In (8), the first player chooses their strategy x E X C Rn in order to maximize the scalar payoff function F(x,y) (defined on the product X x Y); on the contrary, the second player chooses their strategy y E Y in order to minimize F(x,y). One of the possible solutions of the game (8) is the saddle point (xo,yo) E X x Y defined by the following chain of equations:
max F (x, yo) = F (xo, yo) = min F (xo, y). (9)
x€X y€Y
In the case where F(x,y) is a vector (not scalar!) payoff function, during the formalization of the solution, scalar optima (9) should be replaced with vector optima: the scalar maximum in the left equation of (9) should be replaced with a vector maximum, the minimum in the right equation of (9) should be replaced with a vector minimum. The obtained pair (xo,yo) is called in [6] a vector saddle point. In the theory of multicriteria problems, various concepts of vector optima have been proposed (according to Slater, Pareto, Borwaine, Geoffrion, as well as the A-optimum; for more details, see [6]). We will now provide one of them, which we just used in Definition 1 during formalization of a weakly guaranteed under payoffs and risks solution of problem (1). So, we consider the MCPU
r = (X, Y, F(x,y)> , (10)
where the strategies chosen by the decision maker are x E X C Rn, uncertainties y E Y C Rm, the components of the now vector criterion F(x,y) = (Fi(x, y), F2(x, y)) defined on X x Y. In problem (10), the decision maker seeks to simultaneously increase both criteria F1(x,y) and F2(x,y) by choosing their strategy x E X, considering that "uncertainty maximally counteracts this" (minimizes F1(x,y) and F2(x,y)). Problem (10) is assigned two auxiliary two-criterion problems
r(yo) = (X, F(x, yo)> , r(xo) = (Y, F(xo, y)> ,
which we obtain from (10) after fixing the uncertainty y = yo E Y and the strategy x = xo E X, respectively.
The strategy xo E X is Slater maximal (synonyms: weakly effective, weakly Pareto maximal) in problem r(yo), if for all x E X the following system of strict inequalities is incompatible:
Fi(x,yo) <Fi(xo,yo) (i = 1, 2).
The uncertainty yo E Y is Slater minimal for problem r(xo) if the following system of strict inequalities is incompatible:
Fi(xo,y) >Fi(xo,yo)(i = 1, 2) Vy E Y.
The pair (xo,yo) that satisfies both conditions (is both Slater maximal and minimal), is named in [6] Slater saddle point of problem (10).
Slater maximal strategy xo of problem r(yo) has the following property: if the decision maker chooses any strategy x G X, both components of the vector criterion from F(x, yo) cannot simultaneously become greater than the corresponding components of the vector F(xo,yo). Similarly for problem r(xo): if any uncertainty y G Y is implemented, both components Fi(x,yo) (i = 1, 2) of the vector F(xo,y) cannot simultaneously become less than F(xo,yo) (with respect to the corresponding components).
Let us clarify the "conceptual meaning" of the requirement for incompatibility of inequalities (7). In these, the decision maker uses strategy xS (from the WGPR (xS, fS, RS)) and any uncertainty (from the set Y). Inequalities (7) are incompatible if, for any implemented y, either they both or at least one of them is violated.
In the first case, there may be f (xS, y) ^ fS and R/(xS, y) ^ RS. Satisfaction of these two inequalities means that payoff f (xS, y) cannot become less than fS and at the same time the corresponding risk R/(xS, y) cannot become greater than RS. Thus, the number fS is the lower bound on the possible payoff f (xS, y), and RS is the upper bound on the possible risk R/(xS,y) (under same uncertainties). System (7) is incompatible if only the first inequality in (7) is violated (that is, for f(xS, y) ^ fS and great risks R/(xS, y) > RS). This case can be interpreted by "familiar" statements from the financial economics: "excessively large gains are associated with large risks" or "large risks can lead to large gains".
Finally, the "conceptual meaning" of inconsistency of (7) is that the player's use of the strategy xS from the WGPR (xS, fS, RS) and the implementation of any uncertainty y G Y result in that the payoff f (xS, y) cannot be less than the guaranteed payoff fS and, at the same time, the corresponding risk R/(xS, y) cannot become greater than the guaranteed risk RS. That is why a triplet was chosen as a solution to problem (1): the strategy xS and the guarantees on the payoffs fS and on the risks R^: using xS, the decision maker guarantees themselves a payoff not less than fS with a simultaneous risk not greater than RS (no matter what uncertainty y G Y is implemented). Note that incompatibility of inequalities (6) means that it is possible to increase the payoff f (x,yS) and at the same time to reduce the risk R/ (x,yS) while expecting "maximum resistance" from the uncertainty yS. Finding guarantees fS = f (xS,yS) and RS = R/ (xS,yS) can be (as shown below) reduced to building a situation of the Nash equilibrium of a special non-cooperative two-person game (effectively constructed using problem (1)) and is carried out by the decision maker independently from the uncertainties that are actually implemented in problem (1).
Remark 1. We emphasise once again that the system of inequalities (6) is built on the basis of the above-mentioned "analog of the saddle point" and corresponds to "the greatest opposition" to the aspirations of the decision maker on the part of uncertainty (analogous to the game with "nature").
Remark 2. If, instead of the risk function Rf(x,y), the criterion — Rf (x,y) is used, then incompatibility of the system (7) is equivalent to incompatibility of the system of two inequalities
f (xS ,y) <fS, — Rf (xS ,y) < —RS Vy E Y, (11)
and incompatibility of (6) is equivalent to incompatibility of
f (x,ys) >f (xS,ys), —Rf(x,ys) > —Rf(xS,ys) Vx E X. (12)
But incompatibility of (11) for all y E Y means that yS E Y is the Slater minimal uncertainty in (Y, {f (xS, y), —Rf (xS, y)}), and incompatibility of (12) for all x E X means that xS E X is the Slater maximal strategy in the two-criterion problem (X, {f (x,ys), —Rf (x,ys)}>.
Thus, the pair (xS,yS) is the Slater saddle point for a two-criterion problem under uncertainty
(X, Y, {f(x,y), —Rf(x,y)}> (13)
(analog of (10), in which the decision maker seeks to maximize the value of each of the two criteria f (x,y) and —Rf (x,y) by making a suitable choice of strategy x E X .At the same time, the the decision maker has to take into account the fact any uncertainty y E Y could arise.
A possible guaranteed solution is the Slater saddle point (xS,yS), determined by incompatibility of the systems (11) and (12). Thus, finding the WGPR (xS, f (xS,yS), Rf (xS,yS)) is equivalent to constructing the Slater saddle point (13).
Remark 3. Pareto (and not Slater, as in Definition 1) optima could also be used as a basis for definition of the WGPR of problem (1). In this case, it would be necessary to apply a Pareto saddle point in problem (13).
Namely, the pair (xP,yP) E X x Y is called a Pareto saddle point of problem (13) if
(a) xP is the Pareto maximal strategy of problem (X, {f (x,y), —Rf (x,y)}> under y = yP, that is, for all x E X, the following system of inequalities is incompatible:
f(x,yp) ^ f (xP,yp), —Rf(x,yp) ^ —Rf(xP,yp),
where at least one of the inequalities is strict;
(b) yP is the Pareto minimal uncertainty in problem (Y, {/ (xP, y), — Rf (xP, y)}), that is, for all y G Y, the following system of inequalities is incompatible:
/(xP,y) ^ /(xP,yp), — Rf(xP,y) ^ — Rf(xP,yp),
where at least one of the two inequalities is strict.
Obviously, a Pareto saddle point is also a Slater saddle point; the opposite is, generally speaking, not true.
2. Sufficient conditions
Method of constructing a weakly guaranteed solution. Suppose that for problem 1 we found
max / (z,y). (14)
Note that function (14) is continuous on Y if /(x,y) is continuous on X x Y and the set X is a non-empty compact in Then the risk function Rf (x,y) is continuous on X x Y (
as a difference of continuous functions). Statement 1. If there are constants а, в G [0,1] and a pair (x?, ys) G X x Y such that
max ( / (x ys) — а max / (z ys)
V / (15)
= / (x? ,ys) — а max / (z,ys), ( / (x?,y) — в max / (z,y)
V ¿ex у (16)
= / (x? ,ys) — в max / (z,ys), zex
then the weakly guaranteed under payoffs solution of problem 1 takes form of (x?,/ (x? ,ys ),Rf (x? ,ys)).
Proof. Deductio ad absurdum. Suppose there are constant а, в G [0,1] and the corresponding pair (x?,ys) G X x Y such that equations (15) and (16) are valid, but this pair satisfies neither requirement of Definition 1. Consider two cases.
Case 1. (15) is valid, but the first requirement of Definition 1 is not. Then there is a strategy x G X, for which
/(x,ys) > /(x?,ys) = /?, Rf (x,ys) < Rf (x?,ys) = R?,
or, equivalently,
f(x,ys) > f(xs), —Rf(x,ys) > —Rf(xs). (17)
After multiplying the first of these inequalities by the number 1 — a (the constant a G [0,1] participates in (15)), the second by a and summing the left and right parts of (17), we have
(1 — a)f (X ys) — aRf(X ys) > (1 — a)f (xS, ys) — aRf(xS, ys) (18)
where the strict inequality sign follows from the following condition: constants a G [0,1] and 1 — a do not simultaneously vanish. Taking into account the explicit form of the risk function Rf (x,ys) = max f (z,ys) — f (x,y) (for all x G X), we find from (18) the following:
(1 — a)f (X ys) — a ^ rz^x^f (Z ys) — f (x, y^ =
= f (x, ys) — a max f (z ys) > > (1 — a)f (x^, ys) — a ( max f ys) — f (x, y)
y zex
= f (x^ ,ys) — a max f (z,x). This is why the following strict inequality is valid:
f (x,ys) — a max f (z,x) > f (x^,ys) — a max f (z,ys)
which contradicts (15).
Case 2. (16) is valid, but the second requirement of Definition 1 is not. In this case, there is an uncertainty y, for which
f(xs,y) < f(xs,ys), Rf(xs,y) > Rf(xs,ys) Again, multiplying the first of these inequalities by the number 1 — £ G [0,1] from (16), and the second one by —£ and summing them, we find, as in case 1, that
f (x^,y) — £maxf (z,y) < f (x^,ys) — £maxf (z,ys). This inequality contradicts (16).
□
Remark 4. From Statement 1 we have the following method of finding the WGPR (xs, fs, RS) of problem 1:
(a) find the vector function x(y) : Y ^ X, based on the equation
maxf (z,y) = f (x(y),y) Vy G Y;
(b) using the found x(y), build two functions
F«(x,y) = f (x,y) - afWy^y^ Fe (x,y) = f (x,y) - ef Wy^y),
where constants a G [0,1] and в G [0,1];
(c) find constants a*, в * G [0,1] and pair (xs , ys) G X x Y such that the following equations are satisfied:
max Fa* (x,ys) = Fa* (xs ,ys),
S S (19)
mm Fa* (x , y) = Fa* (x , ys);
(d) using this pair (xs, ys), write down the guaranteed under payoffs and risks solution
as
(xs, fs, RS) = (xS, f (xs, ys), Rf (xs, ys)). Reduction to finding a saddle point.
Remark 5. If in the method of finding a weakly guaranteed solution (from the remark 4) one required that the constants a = в G [0,1], then the equations (19) transform into
max Fa* (x,ys) = Fa* (xs,ys) = min Fa* (xs,y). (20)
xEX
Satisfcation of the chain of equations (20) means that the pair (xs,ys) G X x Y is the saddle point of the antagonistic game
(X, Y, Fa* (x,y)>. (21)
In game (21), the first player, by choosing their strategy x G X, seeks to maximize the value of the scalar payoff function Fa* (x,y), and the second, on the contrary, seeks to minimize Fa* (x,y) using their strategy y G Y. The solution of game (21) is the saddle point (xs, ys), defined by chain of equations (20). The algorithm for constructing a weakly guaranteed solution proposed in Remarks 4 and 5 will be applied later in obtaining its explicit form for one sufficiently general class of problems of the form (1).
Remark 6. Since
max f (z,ys)
is not a function of x E X, then the strategy xS E X, found from equation (15), coincides with the strategy xS that satisfies the following equation:
max /(x,ys) = /(x ).
Therefore, Statement 1 can be represented in an equivalent form:
If there is a constant ß G [0,1] and a pair of (xS) G X x Y such that
max f (x,ys) = f (xS ),
min(/(xS,y) - ß max f (x,y)) = f (xS) - ß max f (x,ys),
then the WGPR of problem (1) has the form (xS,/(xS),Ä/(xS)).
Conclusion
The simplest conflict problem under uncertainty was and remains "playing with nature" where one should choose an action (strategy) that optimizes a given criterion (for example, profit). In addition, each action is accompanied by incompleteness or inaccuracy of information (uncertainty) about the results of such an action. At the same time, the risk that accompanies the result achieved is also of interest. A particular kind of uncertainty, for which only the bounds of change are known but no statistical characteristics are available, stands out from such research.
An example of such uncertainties is the problem of diversification between the various currencies of a one-year deposit [19].
Uncertainties, of which only the bounds of change are known, were called in Russia "bad uncertainties" because of unpredictability of their implementations. For assessment of the "actions" of such uncertainties, the Niehans-Savage risk function is used, the value of which for a particular strategy is a measure of risk (the decision maker seeks to reduce the risk), and the best value for the decision maker is characterized by zero risk.
In addition, we recall that economists subdivide decision makers into three groups: risk-averse, risk-loving, and risk-neutral. In Definition 1, we restricted ourselves to risk-neutral decision makers, although it would be of undoubted interest to consider cases in games where different players fall into different categories. We hope to consider these questions in the future when transferring the approach to multicriteria problems under uncertainty.
The decision making process in OCPU Г(1) proceeds as follows. The decision maker chooses and uses their strategy x G X С Rn. Independently of decision maker's actions in Г(1), the uncertainty y is implemented (can be any from the set Y). The payoff function f (x,y) is defined for all pairs (x,y) G X x Y .At the substantive level and before the article [10], the decision maker's task was to choose such a strategy (under the said Г(1) procession rule) so that their gain becomes as large as possible. In this case, the decision maker must take into account the possibility of implementation of any uncertainty y G Y. The latter requirement leads to the need for the decision maker to estimate the set
f(x, Y)= U f(x,y).
Such ambiguity of f (x,y), in turn, necessitates choice of such a function f [x] that would have the guaranteeing property. The most obvious and illustrative guarantee for the decision maker in Г(1) is the so-called [12] strong guarantee implemented by the scalar function
f [x] = min f (x,y) (22)
Indeed, from (22) immediately follows satisfation of the following inequality for all situations x G X:
f [x] ^ f (x,y) Vy G Y.
So, f [x] is a guarantee because for all uncertainties y G Y and all situations x G X, the value of f(x,y) cannot become less than f [x]. As proposed in [10] to build
another guarantee min Rf (x,y) = — Rf [x], where Rf(x,y) is the Niehans-Savage risk y
function (2). Finally, the strongly guaranteeing strategy xP, which is a part of the strongly guaranteeing solution (xP, f [xP],Rf [xP]) is the Pareto maximum in the two-criterion problem of "strong guarantees" Г2 = (X, f [x], — Rf [x]> and the problem is reduced to the problem of constructing the maximizer xP in max(f [x] — Rf [x]) = f [xP] — Rf [xP], hereby
x
implementing the analog of maximin proposed in [12] by the first author. In this approach, "in terms of maximin", the internal minimum is replaced by two minima min f (x,y) and
y€Y
min —Rf(x,y), and the external maximum is replaced by the Pareto maximum in Г2.
y€Y
In [12], existence of (xP, f [xP], Rf [xP]) given compactness of X and Y as well as continuity of f (x,y) is established on X x Y. We emphasize once again that in [12], we limited ourselves only to strong guarantees f [x], —Rf [x]. They are called "strong" because they are the "lowest" possible. One could also use the so-called "vector" guarantees, which was done in this article: the components f [x], —Rf [x] form the vector guarantee for the vector (f(x,y), —Rf(x,y)), if for all y G Y and all x G X, two strict inequalities cannot be
satisfied simultaneously:
f (x,y) < f [xL —R/(x,y) < —R/[x],
in other words, all components of the vector guarantee (f[x], — R/[x]) cannot be simultaneously reduced by choosing y E Y using the vector function (f (x,y), —R/(x,y)). From the perspective of the vector optimization theory, for each situation x E X, the vector (f [x], —R/[x]) is the Slater minimum (weakly effective) in the two-criterion problem r(x) = (Y, {f (x,y), —R/(x,y)}>.
Similarly, using other vector optima (more precisely, according to Pareto, Borwaine, Geoffrion minima, and the conical optimality), one can introduce a whole set of vector guarantees (respectively, according to Pareto, Borwaine, and so on). These guarantees have the following property: their value, first, is not less than the corresponding components of the strong guarantee vector (f[x], —R/[x]), but second may also be large. But we are striving for a possible increase in the payoffs of each player (which is achieved, in particular, by increasing their guarantees!). In this regard, the listed vector guarantees are preferable to the strong ones. However, one should not forget: to make the transition from (X, Y, {f (x,y), —R/(x,y)}> under uncertainty to the problem of guarantees (X, {f [x], —R/[x]}> and then use Remarks 4-6, it is necessary that the new guarantee criteria f [x], — R/ [x] are continuous on X.
Finally, the "uncertainty combatting" method for OCPUs is the embodiment of the "analog of a saddle point" method from [11], where in the definition of the saddle point, the minimum is replaced by the Slater minimum, and the maximum is replaced by the Slater maximum.
Why is then the weakly guaranteed under payoffs and risks solution suggested as a "good" solution of an OCPU?
First, it answers the traditional Russian question: "What is to be done?", when strong guarantees from [10] are "too bad"; in response, it is proposed to follow the triplet (xS, f [xS], R/[xS]), formalized by Definition 1.
Second, this strategy xS "provides" the decision maker with the largest payoffs f (xS, y) not less than f [xS] with the risk of R/ (xS, y) not greater than R/ [xS] in case of implementation of any uncertainty y E Y (that is, xS sets the lower bounds for the payoffs with x = xS and the upper bounds for the risks accompanying this implementation).
Third, the situation xS implements the "greatest" (in the "vector sense")—Slater maximal outcomes and the corresponding "negative" risks; in other words, there is no other strategy x = xS that would increase the guarantee on payoffs and at the same time decrease the guarantee R/ [xS] on risks.
Fourth, an increase in decision maker's guaranteed payoffs (as compared to /[x?]) will inevitably cause an increase in guaranteed risks (again, as compared to Rf [x?]); a decrease in such risks, again, automatically "provokes" a decrease in guaranteed profit.
Fifth, if we demand that X, Y are compacts and /(x,y) is continuous on X x Y, then guarantees /[x] and Rf [x] exist and are continuous on X. Therefore, the question of existence of a solution formalized by Definition 1, "rests" on the question of existence of a saddle point for the antagonistic game of "guarantees" (21); here the possibility of wide application of numerous theorems of existence of a saddle point, including from [18] and successors, already arises.
The authors thank participants of the seminar "Risks in complex control systems" of the Faculty of Computational Mathematics and Cybernetics of Moscow State University for their discussion of the work and comments.
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