Fuzzy utility and equilibria - Semantic Scholar

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS–PART B: CYBERNETICS, VOL., NO., 2003

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Fuzzy utility and equilibria Philippe De Wilde, Member, IEEE

Abstract— Index Terms— fuzzy utility, fuzzy Cournot equilibrium, fuzzy abstract economies, qualitative reasoning

I. F UZZY CHOICE AND INTELLIGENT AGENTS

E

LECTRONIC commerce allows buyers and sellers to use increasingly complicated decision procedures. Large amounts of information are available, and computers can process this to advise the economic agent in the choice of an alternative. The information gathered, over the web for example, is often inconsistent. If it is to be used in decision making, the decision making process will have to be able to deal with uncertainty. Humans deal with uncertainty in a natural way, via generalization. Automatic procedures use either probability theory or fuzzy logic. We prefer fuzzy logic because it deals with linguistic variables in a more intuitive way that probability theory. The linguistic variables are classes that have evolved over time, in a certain application domain, to be effective in generalization. Linguistic variables are the method adopted by humans to indicate choice and preference. Many intelligent agents aim to capture the preferences of their human owner. If decisions are to be made by a computer agent in e-commerce, it is essential that the computer agent agrees with the human owner. Should we study psychology before implementing a shop-bot? This would create problems, as a psychological analysis of economical behavior returns results that are difficult to implement in a computer algorithm. Most economists hold that their theory, micro-economics based on game theory, gives an accurate description of human economical behavior. They even have applied the micro-economic paradigm to areas such as social interactions, and irrational behavior in households and firms [1]. For the management of resources, a core economic activity, the micro-economic approach is prevailing. This is what e-commerce is mostly about: buying and selling quantifiable resources. If we can allow the quantities to be fuzzy, e-commerce and e-management of resources will be even more widely applied than it is now. E-commerce and e-management of resources can operate automatically using intelligent software agents. To achieve this, we need to re-formulate micro-economy so that it can deal with fuzzy choice and preferences. The Orlovsky choice function is often used as the basis for fuzzy choice [2], [3]. We will start from an entirely different starting point, immediately taking into account prices of resources that affect the choice among alternatives. Another approach, ranking based on pairwise comparisons is described in [4]. Choice among attributes that have P. De Wilde is with the Intelligent and Interactive Systems Group, Department of Electrical Engineering, Imperial College London, London SW7 2BT, United Kingdom. Email: [email protected]

multiple attributes is reviewed in [5]. The attributes of our alternatives will be the prices of goods in the consumption bundle. This will allow us to have more specific procedures for ranking than in [5], [4]. Once a basic concept, such as the Orlovsky choice function is proposed and adopted, scientists usually start refining and generalizing it. This happened to fuzzy choice functions, just as it happened to Nash equilibrium, expert systems, etc. Much of the current theory about fuzzy choice has become so abstract that it is impossible to implement in an e-commerce agent. The refined theory of choice can certainly be used to model particular user’s decisions very accurately, but this matching of theory and user requires extensive human intervention. If the e-commerce agent has to implement fuzzy choice automatically for a large class of users, we have to turn back, and use a more intuitive theory. Kulshreshtha and Shekar [6] have recently attempted to present an intuitive perspective on fuzzy preference. It becomes clear from this paper that there is an array of possible choice functions, with no clear criteria as to which ones to prefer. There are even some intuitive contradictions. The authors point out the need to conduct experiments to find the most appropriate fuzzy preference relations for real life situations. We will not conduct experiments, but consider the crisp theory of preference closer to the application (resource management), before fuzzifying it. II. T WO WEAK AXIOMS OF FUZZY REVEALED PREFERENCE

We now explain the theory of choice, based on [7], and its fuzzification, based on [8]. In the next section, we will propose a radically different fuzzification. The set of alternatives is called X. A preference relation º: X 2 → {0, 1} assigns the number 0 or 1 to two alternatives x and y, where x º y means that alternative x is at least as good as y, in other words, x weakly dominates y. The strict preference relation Â is defined by x Â y ⇔ x º y but not y º x. On the same set of alternatives X, the standard fuzzy preference relation R(x, y) : X 2 → [0, 1] indicates the degree to which x is at least as good as y, a number between 0 and 1. It is clear that the crisp preference relation º is the limit of the fuzzy preference relation R, where the degree can only take on values 0 or 1. Often, more than one alternative is acceptable. It is not possible to implement this via a function; hence the concept of choice structure was introduced. A choice structure is denoted by (B, C). Here B is a set of nonempty subsets of X. An element B ∈ B is called a budget set. It consists of a number of alternatives. C is a choice rule that assigns a non-empty set of chosen elements C(B) ⊂ B for every budget set B. It represents the choice made by the agent of one or more alternatives


from the set B ⊂ X. These alternatives are acceptable alternatives to the agent. A fuzzy choice rule assigns a non-empty fuzzy set of chosen elements C(B) ⊂ ∼ B for every budget set B. A fuzzy subset is defined in the standard way as C(B) ⊂ ∼ B ⇔ µC(B) (x) ≤ µB (x), ∀x ∈ X,

(1)

where µB indicates the membership function of a set B. This reduces to the crisp notion of subset, when the membership functions can only take on the values 0 or 1. For this reason the fuzzy choice function is a simple extension of the crisp one, and we will denote both by (B, C). A choice structure induces a preference relation called the revealed preference relation º∗ defined as follows: x º∗ y ⇔ ∃B ∈ B|x, y ∈ B and x ∈ C(B),

(2)

where we read x º∗ y as “x is revealed as least as good as y”, meaning that both alternatives have to be in a budget set, with the preferred one also in the choice set. Remark that the other alternative can also be in the choice set. It is again possible to define the strict preference relation x Â∗ y ⇔ x º∗ y but not y º∗ x.

(3)

The revealed fuzzy preference relation is defined as follows (following [8], but with our notation) R∗ (x, y) =

max

{B|x,y∈B}

µC(B) (x).

(4)

This definition reduces to (2) for membership functions that can take on only the values 0 and 1, but it is only one amongst many possibilities. It is even possible to use linguistic variables for the values of R∗ , as in [9]. The linguistic variables have to be defined by membership functions, that can be related to µC(B) . Differences in the literature, and counter-intuitive definitions start when one tries to define fuzzy strict preference relations, the fuzzy equivalent of x Â y. A fuzzy strict preference relation can be defined as P ∗ (x, y) = max[R∗ (x, y) − R∗ (y, x), 0].

(5)

This makes the relation P ∗ (x, y) anti-symmetric, as is required for a strict order. In [8], the proposal for a fuzzy strict preference relation is P ∗ (x, y) =

max

{B|x,y∈B}

[µC(B) (x) − µC(B) (y), 0].

(6)

For this relation P ∗ , we have to find a definition based on fuzzy choice rules. Remark that in (4) and (6), the maximum is taken over all budget sets B containing x and y, but R∗ (x, y) is independent of the degree to which y is in the fuzzy choice set C(B). It is the function P ∗ that will be used in the formulation of one of the most fundamental axioms of in the theory of choice, the weak axiom of revealed preference [10]. The weak axiom of revealed crisp preference states that Axiom 1—Crisp weak axiom: If for some B ∈ B with x, y ∈ B we have x ∈ C(B), then for any B 0 ∈ B with x, y, ∈ B 0 , we must also have x ∈ C(B 0 ).

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This means that, if there are two budget sets both containing two alternatives, and one alternative is chosen in the first set, and the second alternative is chosen in the second set, then the fist alternative must also be chosen in the second set. This is an intuitive way of requiring consistency. In terms of revealed preferences, this means that, within a given choice structure, if x is revealed at least as good as y, then y cannot be revealed to be strictly better than x. The weak axiom of revealed fuzzy preference exists in two forms ([8], our notation) Axiom 2—Fuzzy weak axiom, version 1: For all alternatives x and y, P ∗ (x, y) > 0 implies R∗ (y, x) < 1. Axiom 3—Fuzzy weak axiom, version 2: For all alternatives x and y, and for all 0 < k ≤ 1, P ∗ (x, y) > k implies R∗ (y, x) < 1 − k. These two fuzzy axioms are not intuitive, and are not generalizations of the crisp axiom. We try to remedy this in the next section. III. W EAK PROPERTY FOR REVEALED FUZZY PREFERENCE FOR BUDGET SETS

Assume an agent manages L resources, and has quantities x1 , . . . , xL of them. The resources have prices p1 , . . . , pL . These prices can also be virtual, for example when networked agents manage tasks, or when the resources are intangible. However abstract the resources may be, we feel that for agents in an e-commerce context, they can always be quantified and priced. As we study multi-agent systems with many agents, the price will be determined by the market, not by a single agent. The resource vector is denoted by x, and the price vector by p. The agent has a level of wealth w. This can again be virtual or real, and expresses the buying power or power to recruit other agents, that an agent has. The resources that an agent can afford are usually calculated from the Walrasian [7] budget set {x ∈ X : px ≤ w}. As the decision making by software agents in e-commerce is exclusively governed by such constraints, we feel that the Walrasian budget set is the right concept to fuzzify, not the choice function. We will denote by µB (x) the degree to which an alternative x ∈ X belongs to the budget set. The fuzziness arises because the inequality constraint px ≤ w may only hold to a certain degree. It is the leniency that your banker shows you, or the degree to which a company is willing to break its rules to satisfy customers. This flexibility is necessary to break deadlocks. Humans show it, and economic software agents have to have it as a feature. The function µB depends on the alternatives in a special way. The flexibility has to be a function of the difference w − px, because this is the budget surplus or budget deficit. This difference shows how much capability the agent still has (w > px), or whether it has exceeded its wealth. The degree µB will also depend directly on the wealth, for this allows us to take into account such facts as that a small deficit is irrelevant if the wealth is large, etc. On the other hand, µB should not depend directly on p or x. This is because µB is only concerned with wealth, not with prices per unit, or units of resources, the latter two being of a different dimension from w. Moreover, the prices are


1 0.8 µB(4−3x1−5x2)

set by the market, but the tolerance for px to exceed w depends only on the individual decision maker. Two particularly useful functions for µB are  1, px ≤ w,    1 (w − px + d(w)), d(w) µB (w, w − px) = (7)  w ≤ px ≤ w + d(w),   0, w + d(w) ≤ px,

3

0.6 0.4 0.2

and µB (w, w − px) =

0 2

1 {tanh[δ(w)(w − px)] + 1}, 2 (8)

We will call such membership functions fuzzy budget constraints or resource constraints interchangeably in the sequel. Function (7), illustrated in figure 1 indicates that all alternatives x that are in the crisp Walrasian budget set, are in the fuzzy budget set to a degree one. This degree then decreases linearly to 0 until px exceeds w + d(w). The quantity d(w) indicates the flexibility of the wealth limit w. Function (8) illustrated in figure 2 decreases gradually from near 1 to near 0 as px increases, taking the value 1/2 when px = w. The slope δ(w) indicates how hard the budget constraint w is, with a hard constraint implemented via a large δ(w).

1

µB(4−3x1−5x2)

0.8 0.6 0.4 0.2 0 2 1

1.5 0.8

1

0.6 0.4

x2

0

0.2 0

0.8

1

δ(w) > 0.

0.5

1

1.5

x1

Fig. 1. A piecewise linear fuzzy budget set for two goods in quantities x1 and x2 , and with prices 3 resp. 5. The wealth limit equals 4 and can be exceeded by 1.

Walras’ law, px = w for all x, can be exactly satisfied by (7), when µB = 1, but will never be exactly satisfied for (8), where µB can only approach 1 asymptotically. Any linear transformation of prices p or resource amounts x will preserve the membership functions (7) and (8) in the same form. For the same reasons, any hyperplane in the L-dimensional x-space will cut the surfaces (7) and (8) according to a piecewise linear or a tanh function respectively. The Walrasian demand correspondence x(p, w) is the amount of goods x at prices p that can be consumed given wealth w. Normally this is a point within the budget set, or

0.6 0.4

0.5 x2

0

0.2 0

x1

Fig. 2. A smooth fuzzy budget set for two goods in quantities x1 and x2 , with prices 3 resp. 5. The wealth limit 4 can be exceeded to an arbitrary amount.

on the edge of the budget set if Walras’ law px = w is fulfilled. If the budget set is a fuzzy set µB , we define the demand correspondence as a fuzzy set with µ(x, p, w) = µB (w, w − px).

(9)

Many different amounts of good can be consumed, each to a different degree. If µB (w, w − px) = 0, then x(p, w) cannot be consumed. The Walrasian demand correspondence x(p, w) is homogeneous of degree zero if x(αp, αw) = x(p, w) for any p, w and α > 0. In the fuzzy version, homogeneity of degree zero becomes a property of the membership function µB of the fuzzy budget set. It can easily be seen that (7) and (8) will be homogeneous of degree zero if d(w) respectively δ(w) are constants, this means that the flexibility on the budget constraint is independent of the wealth. The Walrasian demand correspondence is different from a price-wealth situation x(p, w). The price-wealth situation is simply the actual consumption of goods x at prices p and wealth w, all crisp numbers. There is nothing fuzzy about a price-wealth situation. The demand correspondence however is fuzzy, because of the fuzzy budget set. It is the originality of our approach that we only introduce fuzziness via fuzzy budget constraints, and not via fuzzy choice relations. Now that the resource constraints are formulated as a fuzzy budget set µB , we are able to formulate a much more intuitive fuzzy weak axiom. The crisp weak axiom of revealed preference for a Walrasian demand function x(p, w) is Axiom 4—Crisp weak axiom: For any (p, w) and (p0 , w0 ), if px(p0 , w0 ) ≤ w and x(p0 , w0 ) 6= x(p, w), then p0 x(p, w) > w0 . It can be shown [7] that this is equivalent to axiom 1. The fuzzy version can now be obtained in an intuitive way, if the membership function µB is introduced. Property 1—Fuzzy weak property: For any two fuzzy budget sets µB (w, w − px(p, w)) and µB 0 (w0 , w0 − p0 x(p0 , w0 )), and any two price-wealth situations x(p, w) 6= x(p0 , w0 ), the fuzzy weak property holds to a degree µW (x) = min{µB [w, w − px(p0 , w0 )],


1 − µB 0 [w0 , w0 − p0 x(p, w)]}. An illustration is given in figure 3.

(10)

0.8

µW(x1,x2)

0.6 0.4

4

Fuzzy graphs [13] are one of the most intuitive ways to quantify linguistic uncertainty, if the membership functions of the variables that are being related in the graph, are well known. Let the universe of discourse be quantities of the L resources. Instead of the numerical values x1 , . . . , xL , of the previous section, the decision maker uses linguistic variables with membership functions µli , i = 1, . . . , m, l = 1, . . . , L. For example, µ21 could be the membership function of a “small” quantity of resource 2, and µ22 the membership function for a “large” quantity of resource 2. The membership functions are functions of real numbers which we will denote by x1 , . . . , xL . The membership functions of the utility (“small utility”, “large utility” etc.) are denoted by µL+1 (xL+1 ), i = 1, . . . , m. i A fuzzy utility can now be defined as a fuzzy graph µU (x1 , . . . , xL , xL+1 ) = max min(µli (xi )). i

l

(11)

0.2 An example is given in figure 4. As the resource vector is also

−1 0

0 4

2

1 0

−2 2

x2

x1 Fig. 3. Two resources, and two budget sets defined by x1 + x2 = 1 and x1 /2 + 2x2 = 1 (the two obliquely intersecting straight lines in the graph). Expression (8) with δ(w) = 1 is chosen for the membership functions µB of the two fuzzy budget sets. The degree to which the fuzzy weak property holds, µW (x1 , x2 ), is plotted, together with a projection of the contour lines. The ridge in µW coincides with the bisector of x1 + x2 = 1 and x1 /2 + 2x2 = 1.

It can be shown that if the budget sets are crisp, the fuzzy weak property always holds to a degree 1, hence is always true. So instead of a fuzzification of the crisp weak axiom, as axioms 2 and 3 at the end of section II, we have found not an axiom, but a property. This property holds to a certain degree. If the membership function of the budget set is ’flat’, then the ridge in figure 3 will not be very pronounced, indicating indifference in the choice of consumption bundle x(p, w). On the other hand, if µB drops sharply at px = w, the ridge will be clearly defined. The fuzzy budget set and the fuzzy weak property have given us an intuitive approach to fuzzy constraints, without the necessity to define fuzzy preference relations. In the next section we will show how the fuzzy budget set can be combined with utility maximization. IV. F UZZY UTILITY MAXIMIZATION UNDER FUZZY RESOURCE CONSTRAINTS

Once constraints on resources are laid down in a budget set, the next step is to maximize utility, given a budget set. Fuzzy constraints go together with fuzzy objectives. The latter can be modelled by a fuzzy utility function. Fuzzy utility functions have been introduced via fuzzy random variables [11], in a desire to find a treatment compatible with Bayesian statistics. Another approach is to use fuzzy numbers in an ordinary utility function [12].

1

µU µ22

0.8

µ2

µ11

1

0.6

µ12

0.4 0.2 0 3 3

2 2 1 x2

1 0

0

x1

Fig. 4. The fuzzy utility of a single resource (L = 1). A small amount of resource 1 has membership function µ11 , a large amount µ12 , low utility µ21 , high utility µ22 . The x1 variable is the amount of resource, the x2 variable quantifies the utility. In most practical applications there will be multiple resources.

denoted x (and the price vector p), note that µU depends on x and xL+1 , the latter variable indicating the utility. If the utility is subject to budget constraints, the membership function will be the minimum of the utility and budget membership functions, min(µU , µB ). The utility maximization problem consists in choosing the resource allocation or consumption bundle x† that maximizes this minimum. x†

= argmax min(µU , µB ) = argmax min[µU (x, xL+1 ), µB (w, w − px)] = argmax min[max i

min

l∈{1,...,L+1}

µB (w, w − px))].

(µli (xi ), (12)

This fuzzy resource-constrained utility maximization problem is computationally intensive as formulated in (12), because of the succession of maximization and minimizations. Fortunately it is possible to significantly simplify (12). This hinges


5

on the observation that, for arbitrary numbers X, Y, Z, T, U , min[X, max(min(Y, Z), min(T, U ))] = max(min(X, Y, Z), min(X, T, U )).

(13)

The easiest way to prove this is by investigating the 32 different possibilities for ranking the variables X, Y, Z, T, U . It is possible to generalize (13) to min[X, max(min(Yi , Zi ))] = max(min(X, Yi , Zi )). i

i

(14)

Property (14) is important for fuzzy graphs with constraints. In words, it says that when a fuzzy graph is constrained (the minimum with X in (14)), this is equivalent to constraining the fuzzy relations (“rectangles”) that make up the fuzzy graph. We can now use (14) to simplify (12): x†

= argmax max min(µli (xi ), µB (w, w − px)), i

l

= argmax min(µli (xi ), µB (w, w − px)), i

l

(15) Fig. 6. How fuzzy utility is affected by fuzzy budget constraints.

where the last minimum is taken over L + 2 membership funcL+1 tions: µ1i , . . . , µL for the utility, and i for the resources, µi µB for the budget constraint. This is illustrated for L = 1 in figure 5.

1 0.8 0.6 0.4 0.2 0 3 3

2 2 1 x2

1 0

0

x1

Fig. 5. The fuzzy utility of a single resource (L = 1), constrained by the fuzzy budget set of figure 2 and equation (7). The resource vector x† , here a single variable x†1 is that value of x1 where the fuzzy constrained utility is maximal. Compare this with figure 4: because of the minimization procedure, the second peak has disappeared, the remaining peak is lower and has a different slope.

The hyperbolic tangent budget constraint (8) can also be applied for an equivalent price and wealth situation to generate the following fuzzy constrained in figures 6 and 7. We can see that the peaks in the fuzzy graphs that represented the statement ’a large amount of resource x has a high utility’ become significantly smaller when a budget constraint is applied. Because of this, ’a large amount of resource x having a high utility’ belongs to the constrained fuzzy utility set to a lesser degree than it does to the unconstrained fuzzy utility set. This is an intuitive result if we consider that all consumers operate in a market environment in which commodities have associated prices and economic agents have personal wealth levels.

Fig. 7. Another example of fuzzy budget constraints lowering fuzzy utility, with different linguistic variables for the amount of resource.

If we assume that a given consumer makes ’rational’ decisions, we should expect that she will attempt to maximize her utility in some sense. For the unconstrained case with just one resource, when a consumer has infinite wealth, we should expect to see her choose a large amount of this resource in order to derive a high utility. However, using the more realistic model in which the consumer has an associated wealth, we find that selecting a large amount of the resource realizes a high utility that belongs to the constrained fuzzy utility set to a lesser degree. In this way, we can view this degree as the likelihood that a high utility will be achieved. For the constrained case it is then seen, as expected, that choosing a small amount of the resource realizes a greater likelihood of a small utility being achieved. Following this theme, we should expect that as the price of the resource increases or the wealth of the consumer


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decreases, the likelihood of achieving a high utility becomes smaller. The following plots show how the constrained fuzzy utility varies for different values of price, wealth and flexibility of the wealth limit. Gaussian membership functions are used to represent the linguistic terms and the budget constraint is given by the hyperbolic tangent function. The flexibility δ(w) in (8) and figures 8-11 is inversely proportional to the leniency that is available to the consumer, a large value represents a harder constraint with less flexibility.

Fig. 10. The same situation as in figure 8, but with less wealth.

Fig. 8. A fuzzy utility reduced by fuzzy budget constraints.

Fig. 11. The same situation as in figure 9, but for a cheaper resource.

V. F UZZY C OURNOT ADJUSTMENT

Fig. 9. The same price-wealth combination as in figure 8, but for a larger δ.

Fuzzy Cournot adjustment is a model with players whose state is a single number. We will limit ourselves here to the case of two players, as the generalization to multiple players is straightforward. One player observes the state of the other player, and adjusts her own state according to her decision function. The other player then does the same, and so on, until an equilibrium is reached. The adjustment process can also diverge, reach a limit cycle or a chaotic attractor. Cournot adjustment is a dynamical system. Denote the state of player i at time t by xti . The time is discrete, t = 0, 1, . . .. A point on the trajectory of the Cournot adjustment at time t is denoted by (xt1 , xt2 ). The initial state


of the system is (x01 , x02 ). The decision function of player i is denoted by di . It is a function of the state of the other player. If player 1 moves first, the points on the trajectory are

7

50 45

d2(x1) d1(x2)

40

x11 = d1 (x02 ) x21 = x11 x12 = x02 x22 = d2 (x11 )

x31 = d1 (x22 ) x32 = x22

... ...

(16)

35 30 x2

When player 2 moves first, the trajectory is

25

x11 = x01 x21 = d1 (x12 ) 1 0 x2 = d2 (x1 ) x22 = x12

x31 = x21 x32 = d2 (x21 )

... ...

(17)

20 15

This is illustrated in figure 12 10 10

5

d2(x1) d1(x2)

9

0 0

10

20

7

40

50

Fig. 13. A Cournot adjustment with 8 attractors. Player 2 starts, hence the attraction basins are vertical strips. The attractors are indicated by diamonds, and the intervals ]xmin , xmax [ of the attraction basins ]xmin , xmax [×] − 1 1 1 1 ∞, +inf ty[ by segments of black dots approximately at the height (x2 -value) of the attractors. When the segments are at x2 = 0, there are no attractors in that vertical strip.

6 x2

30 x1

8

5 4 3 2 1 0 0

2

4

6

8

10

x1

to the equilibrium in the case of non-linear decision functions. What we have observed is that, for the same distance between the initial point and the attractor, the more moves are needed to reach the attractor, the more likely condition (19) is violated, despite the existence of an attractor. The reader should compare figures 12 and 14 for an example.

Fig. 12. Player 2 starting the Cournot adjustment, using its decision function d2 to adjust its state x2 . Player 1 then plays, etc.

10

It is clear that the trajectory only depends on the state of the player who does not move first. When 1 moves first, for example, the trajectory only depends on x02 . This affects the nature of the attraction basins. When 1 moves first, the attraction basins are horizontal strips. For n players, we find Property 2: The attraction basins of an n-player Cournot adjustment starting with player f are

8

n Y

]xmin , xmax [, i i

7

x2

6 5 4 3

(18)

i=1

where xmin = −∞, xmax = +∞. f f An example is shown in fig 13. The existence of an attractor depends on the slope of the decision functions of the players. When the decision functions are linear, with slope d d1 /dx2 resp. d d2 /dx1 , the Cournot adjustment converges when [14] d d1 d d2 < 1. dx2 dx1

d2(x1) d1(x2)

9

(19)

A condition that holds locally will also hold globally for linear decision functions. There is one attractor or none. In the latter case the Cournot adjustment diverges. The condition (19) for the existence of an attractor will still be valid sufficiently close

2 1 0 0

2

4

6

8

10

x

1

Fig. 14. A Cournot adjustment trajectory, with circles indicating the points where condition (19) is violated, despite the existence of an attractor.

In the Fuzzy Cournot adjustment, the player does not know exactly what is the state of the other player. This incomplete information can arise for two reasons. The player may not be able to observe the state of the other player very well, or she may observe the state perfectly but only make decisions based on a coarse discretization of what she observes. We will represent the incomplete information by a linguistic variable. The


different values this variable can take on are characterized by membership functions. We will assume that these membership functions do not overlap, although the case of overlapping membership functions can easily be dealt with, at the expense of a more complicated notation. Every players’ decision function, perfectly know to herself, now takes on discrete values, dependent on the value of the linguistic variable of the state of the other player. The value of the decision function stays the domain of the membership functions. This can be seen in figure 15. 10

x2

Not all intersecting segments are attractors. In Figure 16 the attractors are indicated by a diamond. The whole cartesian product of the two segments intersecting at the diamond is an attractor. In the crisp case, figure 16 would show an alter50 45 40 35 30 x2

d2(x1) d1(x2)

9

8

25

8

20

7

15

6

10

5

5

4

0 0

d2(x1) d1(x2) 10

20

30

40

50

x1

3 2

Fig. 16. A fuzzy Cournot adjustment. The whole cartesian product of the two segments intersecting at the diamond is an attractor.

1 0 0

2

4

6

8

10

x1

Fig. 15. A fuzzy Cournot adjustment. The decision functions d1 (x2 ) and d2 (x1 ) are step functions. The trajectory is the continuous black line. The trajectory leads to the intersection of two segments, and the attractor is the cartesian product of these two segments.

If we denote by u(xi1 , xi+1 1 ) the function that is 1 in the inter[, and 0 elsewhere, and similarly for values of x2 , val [xi1 , xi+1 1 the decision functions of the players can be denoted by d1 (x2 ) =

n2 X i=1

di1 u(xi2 , xi+1 2 ),

d2 (x1 ) =

n1 X

di2 u(xi1 , xi+1 1 ),

nation of stable attractors (diamonds) and unstable points (segment intersections without diamonds). In the fuzzy case, the same graph shows a different dynamical behaviour. As can be seen in figure 15, after an initial move, the trajectory follows the segments of the piecewise linear decision functions. When it terminates in an attractor, its graph stops at the intersection of two segments, but the state of the system can continue to vary in the rectangle formed by the cartesian product of the two intersecting segments. The pair of linguistic variables, however, does not change anymore. When there are p players, their decision functions are p − 1dimensional stepfunctions. For player q, the decision function is

i=1

(20) where n1 resp. n2 are the number of distinct values for the linguistic variables of the state of player 1 resp. 2, and di1 and di2 are numbers. For example, if player 1 only bases her decisions on whether the state of player 2 is large, medium, or small, then n2 = 3, and her decision function d1 (x2 ) will be a step function consisting of three steps. The attractors are not points anymore, but rectangles. If, for a i+1 i i certain i and j, the lines dj2 u(xj1 , xj+1 1 ) and d1 u(x2 , x2 ) interj j+1 i sect, then all points in the rectangle [x1 , x1 [×[x2 , xi+1 2 [ can belong to an attractor. If for example numbers in [xj1 , xj+1 1 [ are denoted by “large”, and numbers in [xi2 , xi+1 [ as “small”, then 2 (“large”,“small”) is a fuzzy equilibrium of the fuzzy Cournot adjustment. This means that player 1 keeps her state at “large”, and player 2 keeps her state at “small”, and this is consistent with the decision functions of both players. This is illustrated in figure 15, where the black line of the trajectory leads to the intersection of two segments, and the attractor is the cartesian product of these two segments.

dq

=

n1 X i1 =1

nq−1

···

X

nq+1

X

iq−1 =1 iq+1 =1

dqi1 ...iq−1 iq+1 ...ip

···

np X ip =1

·

iq−1 iq−1 +1 u([xi11 , xi11 +1 [× · · · × [xq−1 , xq−1 [× iq+1 iq+1 +1 ip ip +1 [xq+1 , xq+1 [× · · · × [xp , xp [),

(21)

where nq is the number of linguistic variables that are needed by the other players to describe the state of player q, the dq are the weights that are now assigned not to segments, as in the case of two players, but to p − 1-dimensional hypercubes. A fuzzy equilibrium in this p player game is a p-dimensional hypercube Qp iq iq +1 [ where a trajectory ends. q=1 [xq , xq VI. E QUILIBRIA IN A BSTRACT F UZZY E CONOMIES Fuzzy equilibria have been defined in the context of abstract fuzzy economies, also called generalized fuzzy games [15]. We repeat their definition here, but slightly simplified and with an


adapted notation, so that a comparison with our definition (21) will be possible. Player q has a set of alternatives Xq . There are p players. Denote by · the closure of a crisp set, and by ·α the alpha-cut of a fuzzy Q set. Every player has a preference correspondence Pq : Xq → F(Xq ), which is a fuzzy set F(Xq ) defined on Xq . This preference correspondence associated a fuzzy set on the players’ alternatives for all crisp states of the other players. Every player also has a strategy corresponQ dence Aq : Xq →QF(Xq ). Every player also has a preference function pi : Xq →]0, 1], which associates a number for every state of the other players, and a strategy function Q ai : Xq →]0, 1]. A fuzzy equilibrium of an abstract fuzzy economy is a state x∗ = (x∗1 , . . . , x∗p ) of all players such that x∗ ∈ (Aq x∗ )aq (x∗ ) (Aq x∗ )aq (x∗ ) ∩ (Pq x∗ )pq (x∗ ) = ∅

for every player q, (22) for every player q. (23)

Is this the same as the fuzzy equilibrium we have defined for fuzzy Cournot adjustment, a p-dimensional hypercube Qp iq iq +1 [ where a trajectory ends? The condition (22) q=1 [xq , xq means that, as the game is played, i.e. the strategy correspondence is iterated, and the equilibrium is a result of this iteration. This is the reason that the existence of such equilibria can de derived from fixed-point theorems. We do have iteration in our fuzzy Cournot adjustment, and the intersection of the segments lies in the closure of the attraction basin. Where our fuzzy equilibrium definition differs from (22) is that we model uncertainty about the states of the other players, and (22), using a strategy correspondence, models uncertainty in the decision of the player. Our model can deal both with a player being uncertain about the state of other players, and with a player who is indifferent about some states of the other players, and expresses her indifference by describing the state of the other players with linguistic variables. An abstract fuzzy economy only deals with the latter. Another major difference is of course that our fuzzy equilibria are regions in state space, where the abstract economy equilibria are points in state space. We feel that when a consensus arises among players with fuzzy preferences, the ensuing equilibrium should be fuzzy, not crisp. Note that the alpha-cut to the level aq (x∗ ) in (22) merely serves to indicate that the membership function of the fuzzy set Aq has to exceed a certain threshold in the strategy correspondence. Condition (23) implies that in the equilibrium, the strategy correspondence cannot be further iterated while improving the preferences. In our framework, this would mean a fuzzy Cournot adjustment, while maximizing the utility. Condition (23) can also be modified to include extra constraints, [15]. VII. C ONCLUSION R EFERENCES [1] Gary S. Becker, The Economic Approach to Human Behavior, University of Chicago Press, Chicago, 1976. [2] Kunal Sengupta, “Fuzzy preference and Orlovsky choice procedure,” Fuzzy Sets and Systems, vol. 93, pp. 231–234, 1998.

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[3] Denis Bouyssou, “Acyclic fuzzy preferences and the Orlovsky choice function: a note,” Fuzzy Sets and Systems, vol. 89, pp. 107–111, 1997. [4] Marc Roubens, “Choice procedures in fuzzy multicriteria decision analysis based on pairwise comparisons,” Fuzzy Sets and Systems, vol. 84, pp. 135–142, 1996. [5] Rita Almeida Ribeiro, “Fuzzy multiple attribute decision making: a review and new preference elicitation techniques,” Fuzzy Sets and Systems, vol. 78, pp. 155–181, 1996. [6] Pankaj Kulshreshtha and B. Shekar, “Interrelationships among fuzzy preference-based choice functions and significance of rationality conditions: a taxonomic and intuitive perspective,” Fuzzy Sets and Systems, vol. 109, pp. 429–445, 2000. [7] Andreu Mas-Colell, Michael D. Whinston, and Jerry R. Green, Microeconomic Theory, Oxford University Press, Oxford, 1995. [8] Asis Banerjee, “Fuzzy choice functions, revealed preference and rationality,” Fuzzy Sets and Systems, vol. 70, pp. 31–43, 1995. [9] Marimin, Motohide Umano, Itsuo Hatono, and Hiroyuki Tamura, “Linguistic labels for expressing fuzzy preference relations in fuzzy group decision making,” IEEE Transactions on Systems, Man, and Cybernetics - Part B: Cybernetics, vol. 28, pp. 205–218, 1998. [10] Amartya K. Sen, “Choice functions and revealed preference,” Review of Economic Studies, vol. 38, pp. 307–317, 1971. [11] Maria Angeles Gil and Pramod Jain, “Comparison of experiments in statistical decision problems with fuzzy utilities,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 22, pp. 662–670, 1992. [12] Chie-Bein Chen and Cerry M. Klein, “A simple approach to ranking a group of aggregated fuzzy utilities,” IEEE Transactions on Systems, Man, and Cybernetics - Part B: Cybernetics, vol. 27, pp. 26–35, 1997. [13] Lotfi A. Zadeh, “Fuzzy logic, neural networks, and soft computing,” Communications of the ACM, vol. 37, pp. 77–83, 1994. [14] Eugene Isaakson and Herbert Bishop Keller, Analysis of Numerical Methods, John Wiley & Sons, New York, 1966. [15] Shih sen Chang and Kok keong Tan, “Equilibria and maximal elements of abstract fuzzy economies and qualitative fuzzy games,” Fuzzy Sets and Systems, vol. 125, pp. 389–399, 2002.