Applications to Mathematical Economics

Chapter 9

Applications to Mathematical Economics Contents 9.1 A Debreu-Gale-Nikaïdo Theorem 261 9.2 Application to Walras Equilibrium Model 263 9.3 Nash Equilibria Within n-Pole Economy 267

9.4 Pareto Optimality for n-Person Games Note

270 275

Learning never exhausts the mind. Leonardo da Vinci (1452–1519)

Chapter points • This chapter is dedicated to show some application of the equilibrium theory to mathematical economics. • Two models are considered: the Walrasian equilibrium model and the n-pole economy. • The chapter also discusses Pareto optima for n-person games, in the case when the players are permitted to exchange information and to collaborate.

9.1 A DEBREU-GALE-NIKAÏDO THEOREM The Debreu-Gale-Nikaïdo theorem, which can be proven by Ky Fan’s minimax inequality (Theorem 3.3) is a potential tool to prove the existence of a market equilibrium price. Specifically, we will use it to prove the existence of an equilibrium price within the Walras equilibrium model (see Section 9.2 below). To state the Debreu-Gale-Nikaïdo theorem we first need the following definition. Let K be a compact topological space, Y be a real normed space and ϕ : K ⇒ Y a given set-valued map. As before, we denote by Y ∗ the topological dual space of Y . The norm of Y ∗ will be denoted by  · ∗ . For any fixed p ∈ Y ∗ , define the support function related to ϕ by σ (ϕ(x), p) := sup p, y,

for all x ∈ K.

(9.1)

y∈ϕ(x) Equilibrium Problems and Applications. https://doi.org/10.1016/B978-0-12-811029-4.00017-1 Copyright © 2019 Elsevier Inc. All rights reserved.

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262 Equilibrium Problems and Applications

Definition 9.1. The set-valued map ϕ : K ⇒ Y is called upper hemicontinuous at x0 ∈ K if for all p ∈ Y ∗ the support function x → σ (ϕ(x), p) is upper semicontinuous at x0 . The map ϕ is said to be upper hemicontinuous, if it is upper hemicontinuous at every x0 ∈ K. The next result provides a sufficient condition for upper hemicontinuity. If x0 ∈ K we denote by V(x0 ) the collection of all neighborhoods of x0 and by B the open unit ball in Y centered at the origin. Lemma 9.1. If the mapping ϕ is upper semicontinuous (as a set-valued mapping), then it is also upper hemicontinuous. Proof. Let x0 ∈ K be fixed. By the hypothesis ϕ is upper semicontinuous at x0 , that is (see Chapter 1), for each open set V ⊂ Y such that ϕ(x0 ) ⊂ V , there exists N (x0 ) ∈ V(x0 ) with ϕ(x) ⊂ V

for all x ∈ N (x0 ).

(9.2)

For every  > 0 let V := ϕ(x0 ) + B ⊂ Y , which is an open set. By (9.2) we conclude that ∃ N (x0 ) ∈ V(x0 ) : ϕ(x) ⊂ ϕ(x0 ) + B

for all x ∈ N (x0 ).

(9.3)

Fix arbitrary  > 0 and p ∈ Y ∗ . Then by (9.3) σ (ϕ(x), p) ≤ σ (ϕ(x0 ), p) + σ (B, p)

for all x ∈ N (x0 ),

that is, σ (ϕ(x), p) ≤ σ (ϕ(x0 ), p) + p∗

for all x ∈ N (x0 ).

Hence the mapping x → σ (ϕ(x), p) is upper semicontinuous at x0 . Since x0 was arbitrary, it follows that x → σ (ϕ(x), p) is upper semicontinuous. In what follows we consider the unit simplex   n  xi = 1 . M n := x ∈ Rn+ : i=1

Theorem 9.1. (Debreu-Gale-Nikaïdo) Let C : M n ⇒ Rn be a set-valued map with nonempty compact values. Suppose that (i) C is upper hemicontinuous; (ii) ∀ x ∈ M n : C(x) − Rn+ is a convex and closed set; (iii) ∀ x ∈ M n : σ (C(x), x) ≥ 0, ¯ ∩ Rn+ = ∅. Then there exists x¯ ∈ M n such that C(x)

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Proof. Consider the function φ : M n × M n → R, φ(x, y) = σ (C(x), y). It is easy to check that φ satisfies the assumptions of Ky Fan’s minimax inequality theorem (Theorem 3.3). Indeed, 1. φ(x, x) ≥ 0, ∀ x ∈ M n , by assumption (iii); 2. φ(·, y) : M n → R is upper semicontinuous by assumption (i); 3. φ(x, ·) : M n → R is convex for all x ∈ M n , since y → σ (C(x), y) is convex. Since M n is convex and compact, we may apply Theorem 3.3, and such there exists x¯ ∈ M n : φ(x, ¯ y) ≥ 0, ∀ y ∈ M n , or, equivalently σ (C(x), ¯ y) ≥ 0, ∀ y ∈ M n .

(9.4)

Taking into account the definition of σ , it can be seen immediately that the latter implies σ (C(x), ¯ y) ≥ 0, ∀ y ∈ Rn+ . We show that this condition is equivalent to σ (C(x) ¯ − Rn+ , y) ≥ 0, ∀ y ∈ Rn .

(9.5)

Indeed, let S := C(x) ¯ − Rn+ . Since σ (−Rn+ , y) = 0 for y ∈ Rn+ and n σ (−R+ , y) = +∞ for y ∈ / Rn+ , we obtain that σ (S, y) = σ (C(x), ¯ y) + σ (−Rn+ , y) ≥ 0, ∀ y ∈ Rn . Now S is a closed convex set by (ii). Suppose that 0 ∈ / S. Then by the separation theorem {0} and S can be strongly separated, that is, ∃ y ∈ Rn : supz, y < inf z, y = 0, z∈S

z∈{0}

n , hence C(x)∩R n =∅. ¯ which contradicts (9.5). In conclusion 0∈S=C(x)−R ¯ + + The proof is now complete.

9.2 APPLICATION TO WALRAS EQUILIBRIUM MODEL Consider an economy with l types of elementary commodities each with a unit of measurement. An elementary commodity is described besides its physical properties also by other characteristics such as its location and/or the date when it will be available and, in case of uncertainty, the event which will take place,

264 Equilibrium Problems and Applications

etc. A set of commodities is a vector x ∈ Rl which describes the quantity xj of each elementary commodity for j = 1, ..., l. Suppose that there are n consumers and the set M ⊂ Rl represents the available commodities. Next, we present the so-called classical Walrasian model. The consumption set for each of the n consumers will be denoted by Li ⊂ Rl . This is interpreted as the set of commodities which the ith consumer needs. If x ∈ Li , then the j th component xj represents the consumer’s demand for the elementary commodity j if xj ≥ 0 and |xj | represents the supply of this elementary commodity if xj < 0. It is natural to ask whether the consumers can share an available commodity. In order to examine this situation, the concept of allocation is needed. An allol n cation is an nelement x ∈ (R ) consisting on n commodities xi ∈ Li such that the value i=1 xi is available. The set of all allocations is then defined as  K := x ∈

n  i=1

Li :

n 

 xi ∈ M .

i=1

Assuming that the set K of allocations is nonempty, our aim is to describe a mechanism which allows each consumer to choose its own allocation. The mechanism does not require each consumer to know the set M of available commodities and the choices of the other consumers, but only require each consumer to know his own particular environment and to have access to common information about the state of the economy. This common information will take the form of a price (or price system). The price is regarded as a linear func∗ tional p ∈ Rl which associates to each commodity x ∈ Rl the value p, x ∈ R expressed in monetary units. Suppose that the elementary commodity i is represented by the unit vector ej = (0, ..., 0, 1, 0, ..., 0) of the canonical basis of Rl , the components p, ej  of the price p represent the price of the commodity j . We denote the price simplex by ⎧ ⎫ l ⎨ ⎬  ∗ pj = 1 . M l := p ∈ Rl+ : ⎩ ⎭ j =1

In the case of the Walrasian mechanism, each consumer is described by the set-valued map Di : M l × R ⇒ Li which associates a subset of consumptions ∗ Di (p, r) ⊂ Li to each price system p ∈ Rl and each income r ∈ R. The support function σM of the set M of available commodities, already discussed in Section 9.1 and given by σM (p) := sup p, y, y∈M

(9.6)

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is regarded as the collective-income function which is the maximum value of the commodities available for each price p. An essential assumption is given by ∃ ri : M l → R, i = 1, . . . , n :

n 

ri (p) = σM (p),

(9.7)

i=1

which means that the collective income is shared between the n consumers. The income of each consumer is ri (P ) and he is thus led to choose a consumption xi ∈ Di(p, ri (P )). This choice is decentralized; it depends only on the price p and is independent of the choice of other consumers. The following feasibility arises: is there any price p¯ such that the problem  ¯ ri (p)) ¯ is available (belongs sum of the consumptions ni=1 x¯i ∈ ni=1 Di (p, ¯ ri (p)) ¯ form an allocation? to M) or such that the consumptions x¯i ∈ Di (p, The next definition is of special importance for our purposes. Definition 9.2. A price p¯ ∈ M l is called a Walrasian equilibrium price if it is a solution of the inclusion 0∈

n 

Di (p, ¯ ri (p)) ¯ − M.

(9.8)

i=1

The set-valued map E : M l ⇒ Rl given by E(p) :=

n 

Di (p, ri (p)) − M

(9.9)

i=1

is called the excess-demand correspondence. It is then obvious that the Walrasian equilibrium prices are the zeros of the excess-demand correspondence. The demand correspondences should satisfy the collective Walras law expressed as follows:

 n n   l ∀ p ∈ M , ∀ xi ∈ Di (p, ri ) : p, xi ≤ ri (p), (9.10) i=1

i=1

that is, the consumers cannot spend more than their total income. A stronger, decentralized law, called Walras law is given below. Every correspondence Di satisfies the condition ∀ p ∈ M l , ∀ x ∈ Di (p, r) : p, x ≤ r.

(9.11)

Now we are in the position to prove the main result of this section: the existence of a Walrasian equilibrium. The next theorem will be proved via the Debreu-Gale-Nikaïdo theorem.

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Theorem 9.2. (see Theorem 10.1 in [20]) Suppose that the following assumptions are satisfied: (i) the set M is convex and may be written as M = M0 − Rl+ , where M0 is compact; (ii) the set-valued maps Di : M l × R ⇒ Li are upper hemicontinuous with convex, compact values and satisfy the collective Walras law; (iii) the income functions ri are continuous. Then there exists a Walrasian equilibrium. Proof. Consider the set-valued map C : M l ⇒ Rl given by C(p) := M0 −

n 

Di (p, ri (p)).

(9.12)

i=1

We shall apply for this C the Debreu-Gale-Nikaïdo theorem (Theorem 9.1). To this aim let us verify that all assumptions of Theorem 9.1 are satisfied. It is clear that C has nonempty and compact values, since the sets M0 and Di (p, ri (p)) are compact by (i). It is also clear that C is upper hemicontinuous. By assumptions (i) and (ii), for all p ∈ M l the set C(p) − Rl+ is convex and closed. Also, for every p ∈ M l we have that n 

ri (p) = σ (M, p) = σ (M0 − Rl+ , p)

i=1

= σ (M0 , p) + σ (−Rl+ , p) = σ (M0 , p).

(9.13)

By the collective Walras law (9.10) −

n 

ri (p) ≤

i=1

n  −p, xi , ∀ xi ∈ Di (p, ri (p)). i=1

Passing to supremum over xi ∈ Di (p, ri (p)) we obtain −

n  i=1

ri (p) ≤

sup

n 

xi ∈Di (p,ri (p)) i=1

−p, xi .

Thus σ (C(p), p) = σ (M0 , p) +

sup

n  −p, xi 

xi ∈Di (p,ri (p)) i=1

(9.14)

Applications to Mathematical Economics Chapter | 9

(9.14)

≥ σ (M0 , p) −

n 

267

ri (p)

i=1 (9.13)

= σ (M0 , p) − σ (M0 , p) = 0.

Consequently we may apply the Debreu-Gale-Nikaïdo (Theorem 9.1) and such ∃ p¯ ∈ M l : C(p) ¯ ∩ Rl+ = ∅, or, in other words 0 ∈ C(p) ¯ − Rl+ = M0 − =M −

n 

Di (p, ¯ ri (p)) ¯ − Rl+

i=1 n 

Di (p, ¯ ri (p)) ¯

i=1

= −E(p). ¯ Hence 0 ∈ −E(p), ¯ and therefore, p¯ is a Walrasian equilibrium price. An important particular case is obtained when M := w − Rl+ is the set of commodities less than the available commodity w ∈ Rl+ . Corollary 9.1. Suppose that assumptions (ii) and (iii) of Theorem 9.2 hold and w=

n 

wi

i=1

is allocated to the n consumers. Then there exists a Walrasian equilibrium price p¯ and consumptions   x¯i ∈ Di (p, ¯ p, ¯ wi ) such that ni=1 x¯i ≤ ni=1 wi .

9.3 NASH EQUILIBRIA WITHIN n-POLE ECONOMY In this section we illustrate the concept of a Nash equilibrium point defined in Chapter 2, Subsection 2.2.4 with one of the easiest models in economy, the so-called n-pole economy. In this economy we have n producers with n ≥ 2 which compete on the market, that is, they produce and sell the same product. We consider the producers as players in an n-person game where the strategy sets are given by Si = R+ (i = 1, 2, ..., n). The strategy s ∈ R+ stands for the production of s units of the product. It is reasonable to assume that the price of the product on the market is determined by demand. More precisely, we assume

268 Equilibrium Problems and Applications

that there is an affine relationship of the form:  p(s1 , . . . , sn ) = α − β

n 

 si

(9.15)

i=1

where α, β are positive constants. It is easy to see that once the total production increases, the price decreases. Assume that the individual cost functions for each producer are given by ci (si ) = γi si + δi ,

(9.16)

where γi , δi are fixed costs for i = 1, . . . , n. By these data we can easily calculate the gain of each producer (player), which is fi (s1 , . . . , sn ) = p(s1 , . . . , sn )si − ci (si ), i = 1, . . . , n.

(9.17)

Using relations (9.15) and (9.16) the formula (9.17) can be evaluated as  fi (s1 , . . . , sn ) = α − β 

n 

 si si − γi si − δi

i=1

 n α − γi  = βsi si − δi − β i=1   n  = βsi ui − si − δi , i = 1, . . . , n,

(9.18)

i=1 i where ui := α−γ β .  Let U = U1 × . . . × Un ⊆ ni=1 Si the common feasible strategy set, where Ui = [0, ui ], i = 1, . . . , n. In what follows, for the sake of simplicity, we will suppose that

⎧ δi = 0, i = 1, . . . , n ⎪ ⎪ ⎪ ⎨ γi = γ > 0, i = 1, . . . , n ⎪ β =1 ⎪ ⎪ ⎩ u1 = u2 = . . . = un = u. Thus, the profit function (9.18) for s ∈ U is given by  fi (s) = si u −

n  i=1

 si , i = 1, . . . , n.

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By summing up these relations with respect to i, we can easily determine the maximal total profit, since n 

fi (s) = u

i=1

attains its maximum at be

u2 4 .

n 

 si −

i=1

n

i=1 si

n 

2 si

i=1

= u2 , and such, the maximal total profit will

Let us introduce the decision functions gi : U1 × . . . × Ui−1 × Ui+1 × . . . × Un → Ui , i = 1, . . . , n,

(9.19)

and examine the first player’s optimal decision function (rule). His decision g1 (s2 , . . . , sn ) depends on the other players decisions and must be taken in such a way that f1 (g1 (s2 , . . . , sn ), s2 , . . . , sn ) = max f1 (s). s1 ∈[0,u]

Thus  f1 (g1 (s2 , . . . , sn ), s2 , . . . , sn ) = max f1 (s) = max s1 u − s1 ∈[0,u]

s1 ∈[0,u]

= max (s1 u − s12 − s1 s1 ∈[0,u]

n 

n 

 si

i=1

si ).

i=2

Since in s1 we have apolynomial function of degree 2, the maximum of f1 is   attained at s1 = u − ni=2 si /2, therefore g1 (s2 , . . . , sn ) =

u−

n

i=2 si

2

.

In a similar way we can determine the optimal decisions of the other players.  u − ni=1 si i=k gk (s1 , . . . , sk−1 , sk+1 , . . . , sn ) = , k = 1, . . . , n. 2 In what follows, let us calculate the Nash equilibrium point of this game. It can be seen that s¯ = (¯s1 , . . . , s¯n ) is a Nash equilibrium point if and only if ⎧ g (¯s , . . . , s¯n ) = s¯1 ⎪ ⎪ ⎨ 1 2 .. . ⎪ ⎪ ⎩ gn (¯s1 , . . . , s¯n−1 ) = s¯n .

270 Equilibrium Problems and Applications

By using the decision functions the system above can be written as ⎧ ⎪ ⎪s¯1 = ⎪ ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ . ⎪ ⎪ ⎪ ⎨ s¯j = ⎪ ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎩ s¯n =

 u− ni=2 si 2  u− ni=1 si i=j 2  u− n−1 i=1 si 2

⎧ 2¯s1 + s¯2 + . . . + s¯n = u ⎪ ⎪ ⎪ ⎪ ⎪ . ⎪ . ⎪ ⎪ ⎨. ⇔ s¯1 + . . . 2¯sj + . . . + s¯n = u ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎩ s¯1 + s¯2 + . . . + 2¯sn = u.

This system has a unique solution, namely   u u ,..., , s¯ = (¯s1 , . . . , s¯n ) = n+1 n+1 which is the Nash equilibrium point of the game. This means that each player u has the same optimal strategy n+1 , and the same optimal gain   nu u2 u fi (¯s ) = u− = , i = 1, . . . , n. n+1 n+1 (n + 1)2 It is worth mentioning that in this case the sum of the gains

nu2 (n+1)2

is less

u2 4 .

than the maximal total gain This means that in case the players cooperate u and everyone agrees to choose the same 2n strategy, then fi

u u u  u  u2 u2 ,..., = u− = > , 2n 2n 2n 2 4n (n + 1)2

as n ≥ 2 and in this way the total profit is maximal, since equals

u2 4 .

9.4 PARETO OPTIMALITY FOR n-PERSON GAMES Consider two players Alex and Bob and assume that they choose their strategies using their loss functions fA and fB from A × B to R. We say that a pair (x, ¯ y) ¯ ∈ A × B is a noncooperative equilibrium if and only if fA (x, ¯ y) ¯ = inf fA (x, y) ¯ x∈A

and fB (x, ¯ y) ¯ = inf fB (x, ¯ y). y∈B

It follows that a noncooperative equilibrium is an alternative in which any player optimizes his own criterion, assuming that his partner’s choice is fixed. This corresponds to a situation with individual stability.

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If we assume that the players communicate, exchange information, and cooperate then the notion of noncooperative equilibrium does not provide the only reasonable scheme for solution of a game in strategic form. More precisely, it is possible to find a strategy pair (x, y) ∈ A × B such that fA (x, y) < fA (x, ¯ y) ¯ and

fB (x, y) < fB (x, ¯ y). ¯

In such a case, the players Alex and Bob have losses strictly less than in the case of noncooperative equilibrium (x, ¯ y). ¯ Thus, there is a lack of collective stability, in the sense that the two players can each find better strategies for themselves. The corresponding notion of Pareto optimum is associated to an allocation for which there are no possible alternative allocations whose realization would cause every player to gain. Definition 9.3. We say that a strategy pair (x∗ , y∗ ) ∈ A × B is Pareto1 optimal (or Pareto-efficient) if there are no other strategy pairs (x, y) ∈ A × B such that fA (x, y) < fA (x∗ , y∗ ) and fB (x, y) < fB (x∗ , y∗ ). As in the cases of two-person games, the decision rules in the case of n players are determined by loss functions. More precisely, the behavior of the kth player (1 ≤ k ≤ n) is defined by a loss function f k : E → R, which evaluates the loss f k (x) inflicted on the kth player by each multistrategy x. Accordingly, we define the multiloss function f : E → Rn by f (x) := (f 1 (x), . . . , f n (x))

for all x ∈ E.

Definition 9.4. We say that a multistrategy x ∈ E is Pareto optimal if there are no other multistrategies x ∈ E such that f i (x) < f i (x) for all i = 1, ..., n.

(9.20)

In the case of two-person games we have observed that there may be a number of Pareto optima. Thus, a natural problem that arises is to choose these optima. Let us attribute a weight λk ≥ 0 to the kth player. If the player accept this weighting, they may agree to collaborate and to minimize the weighted function fλ (x) :=

n 

λi f i (x)

(9.21)

i=1

over the set E. If the vector λ = (λ1 , . . . , λn ) is not zero, we observe that any multistrategy x ∈ E which minimizes fλ (x) is a Pareto minimum. Indeed, arguing by contradiction, we find x satisfying inequalities (9.20). Multiplying these relations by λi ≥ 0 and summing them, we obtain fλ (x) < fλ (x), which is a contradiction.

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If n players could be made to agree on a weighting λ, we would no longer have a game problem proper, but a simple optimization problem. However, it is interesting to know the conditions under which Pareto optimum may be obtained by minimizing a function fλ associated with a weighting λ which is borne in some way by this Pareto optimum. This question has a positive answer in the following property, by applying convexity arguments. Proposition 9.1. Suppose that the strategy sets E i are convex and that the loss functions f i : E → R are convex. Then any Pareto optimum x may be associated with a nonzero weight λ ∈ Rn such that x minimizes the function fλ over E. Proof. We first observe that f (E) + int(Rn+ ) is a convex set. It follows that an element x ∈ E is a Pareto minimum if and only if f (x) ∈ f (E) + int(Rn+ ). By the separation theorem for convex sets, we deduce that there exists λ ∈ Rn , λ = 0, such that λ, f (x) =

inf

x∈E u∈int(Rn+ )

(λ, f (x) + λ, u) .

It follows that λ is positive and that x minimizes the function x → fλ (x) = λ, f (x) over E. We point out that a Pareto minimum also minimizes other functions. For example, we introduce the virtual minimum α, which is defined by its components α i := inf f i (x). x∈E

We say that the game is bounded below if α i > −∞ for all i = 1, . . . , n. In this case, we take β i < α i for all i and set β := (β 1 , ..., β n ) ∈ Rn . Proposition 9.2. Suppose that the game is bounded below. Then an element x ∈ E is a Pareto minimum if and only if there exists λ ∈ int(Rn+ ) such that x minimizes the function gλ defined by gλ (x) := max

i=i,...,n

over E.

1 i (f (x) − β i ) λi

(9.22)

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Proof. (a) Assume that x ∈ E minimizes gλ on E and is not a Pareto minimum. Then there exists x ∈ E satisfying inequalities (9.20). Subtracting β i , and multiplying by λ1i and taking the maximum of the two terms, we obtain the contradiction gλ (x) < gλ (x). (b) Let x be a Pareto minimum. We take λi = f i (x) − β i > 0 such that gλ (x) = 1. Let x ∈ E be such that gλ (x) < gλ (x). Then  max

i=1,...,n

f i (x) − β i f i (x) − β i

 < 1,

which implies inequalities (9.20). We can also define conservative strategies for the players. We set ˆ

f i (x i ) := sup f i (x i , x i ). x iˆ ∈E iˆ

Definition 9.5. We say that x i ∈ E i is a conservative strategy for the ith player if ˆ

f i (x i ) = inf sup f i (x i , x i ) x i ∈E i

x iˆ ∈E iˆ



and we say the number vi defined by ˆ



vi := inf sup f i (x i , x i ) x i ∈E i

ˆ x i∈E

iˆ

is the conservative value of the game. 

We point out that the conservative value vi may be used as a threat, by refusing to accept any multistrategy x such that 

f i (x) > vi

since by playing a conservative strategy x i the loss f i (x i , x i ) is strictly less than f i (x). Suppose that 

vi > α i

for all i = 1, ..., n.

This assumption says that the conservative value is strictly greater than the virtual minimum.

274 Equilibrium Problems and Applications

Consider the function g0 : E → R defined by g0 (x) = max

f i (x) − α i

i=1,...,n



vi − α i

.



Taking β i = α i and λi = vi − α i , Proposition 9.2 implies that if x0 ∈ E minimizes g0 on E, then x0 is a Pareto minimum. If d := minx∈E g0 (x), it follows that x0 minimizes g0 on E if and only if 

f i (x0i ) ≤ (1 − d)vi + dα i

for all i, ..., n.

This property suggests that such choices of Pareto optima should be viewed as best compromise solutions. Other methods of selection by optimization involve minimizing functions   f n (x) − α n f 1 (x) − α 1 , ..., (9.23) x → s   v1 − α 1 vn − α n on E, where the function s satisfies the following increasing property if a i > bi for all i, then s(a) > s(b). We observe that x ∈ E that minimizes (9.23) is a Pareto minimum. We also note that the function (9.23) remains invariant whenever the loss functions f i are replaced by functions ai f i + bi , where ai > 0. We say that by replacing the functions f i by the functions g i g i (x) =

f i (x) − α i 

vi − α i

,

then we have normalized the same game. For the normalized game the virtual minimum is zero and the conservative value is 1. In a general setting (not only concerning n-person games), a state of affairs is Pareto-optimal if there is no alternative state that would make some people better off without making anyone worse off. Alternatively, a state of affairs x is said to be Pareto-inefficient (or suboptimal) if there is some state of affairs y such that no one strictly prefers x to y and at least one person strictly prefers y to x. We conclude that the concept of Pareto-optimality assumes that anyone would prefer an option that is cheaper, more efficient, or more reliable or that otherwise comparatively improves one’s condition. In his most influential work Manuale d’economia politica (1906), Pareto further developed his theory of pure economics and his analysis of ophelimity

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(power to give satisfaction). He laid the foundation of modern welfare economics with his concept of Pareto optimum, stating that “the optimum allocation of the resources of a society is not attained so long as it is possible to make at least one individual better off in his own estimation while keeping others as well off as before in their own estimation”.

NOTE 1. Vilfredo Pareto (1848–1923), Italian engineer and economist, who used the concept in his studies of economic efficiency and income distribution. The concept has been applied in academic fields such as economics, engineering, and the life sciences. He shared with Léon Walras the conviction of the applicability of mathematics to the social sciences.