11. Estrategia de mercadeo
11.1 Análisis de competitividad
In this Section we focus on the second notion of agents’ types, named “Type II”, that differs from the one used in the previous section as regards the utility functions.
Under this change in the economic environment, the notion of type–symmetry matters;
therefore, we move our focus from the extreme core Ce(E) to the set Ces(E), formed by all the individually rational, Pareto optimal, type–symmetric allocations and we an-alyze its stability when the economy E is populated by s traders of each type i, with s > 1.
The following set of assumptions will be used in the sequel.
(B.1) (Continuity) For every (i, j) ∈ N, Uij is continuous.
(B.2) (Type–wise strict quasi–concavity) For every (i, j) ∈ N, Uij is strictly quasi–
concave with respect to each xTl, l = 1, . . . , k.
(B.2)0 (Type–wise quasi–concavity) For every (i, j) ∈ N, Uij is quasi–concave with re-spect to each xTl, l = 1, . . . , k.
(B.3) (Boundary equivalence) For every (i, j) ∈ N and for every consumption profile x over N , it holds that:
Uij(x) = Uij(0Z, xN \Z) , where Z = {(s, t) ∈ N : xst has a zero component}.
(B.4) (Strictly positive total endowment) P
ij∈Neij 0.
(B.5) (Glove market) For each type i ∈ {1, . . . , k} , there exists a commodity mi ∈ {1, . . . , l} such that emri = 0 for every r 6= i (where emri denotes the mi-th component of the vector er).
(B.6) (Consumption sets and types) If coalition S ⊆ N contains every type of agents, then Xij(S) = Xij(N ) for every (i, j) ∈ S.
Compared with the set of assumption used in the previous section, a (strict) quasi–
concavity assumption on the interdependent utility functions is now added in the economic environment, along with a condition on the consumption sets.
Assumption (B.6) states that, for each trader, the consumption opportunities are the same in the grand coalition N and in every other coalition containing every type of agents 7. Two remarks about Assumptions (B.2) and (B.2)0 are in order.
Remark 3.3 We first note that Assumption (B.2)0 is weaker than assuming the quasi–
concavity of function Uij. For instance, consider the function f (x, y) = x3 + x + y3+ y which is quasi–concave with respect to both x and y. However, f is not quasi–concave.
Indeed, note that both fx0 > 0 and fy0 > 0; moreover, the determinant of the following matrix M :
Remark 3.4 One may ask which examples of interdependent utility functions can fit with Assumption (B.2) or (B.2)0 when types are also taken into account. Here, we provide an example.
Suppose that every agent of type i, i = 1, . . . , k, has a selfish utility function ui. The interdependent utility that agent (i, j) ∈ Ti receives by the consumption profile x = (xT1, . . . , xTk) is given by a sum of three components: the selfish utility that trader (i, j) obtains from his own bundle xij, an average of the utilities that every other agent (i, t) of his same type gets from his own bundle xit, an average of the utilities received by all other agents of types r 6= i, each type r weighted with a coefficient air that, according to
7With reference to the asymmetric information framework, this assumption holds for all the custom-arily used information sharing rules, that is, the private, the fine and the coarse ones. More in general, it is satisfied whenever the consumption set of trader i as a member of coalition S depends just on the types not on the traders actually included in S.
its sign, measures the envy/altruism of type i towards type r.
It is easy to check that for the function Uij both conditions (1) and (2) defining agents of Type II hold.
Moreover, by assuming that every selfish utility function ul, l ∈ {1, . . . , k}, is (strict) quasi–concave, it can be proved that Uij is (strict) quasi–concave with respect to each block xT1, . . . , xTk of variables provided that the coefficient aij are nonnegative.
Let us first prove that Uij is quasi–concave with respect to xTi.
Let A = (xT1, . . . , xTi, . . . , xTk) and B = (xT1, . . . , ¯xTi, . . . , xTk). We want to show that, that is what we had to prove.
Let us prove now that Uij is quasi–concave with respect to xTl, with l 6= i.
Let C = (xT1, . . . , xTl, . . . , xTk) and D = (xT1, . . . , ¯xTl, . . . , xTk). We want to show that, for every λ ∈ [0, 1]:
Uij(λC + (1 − λ)D) ≥ min{Uij(C), Uij(D)} .
It holds that:
We shall prove now that the set Ces(E) formed by all Pareto optimal, type–symmetric, individually rational allocations satisfies a form of crossed stability; precisely, we will show that Ces(E) is internally stable with respect to the α–dominance relation and externally stable with respect to the α2–dominance relation.
The next two preliminary propositions will be used to prove the external stability of the set Ces(E). It is worth noting that for both of them the notion of dominance involved is not relevant because only the grand coalition N is allowed to block. Moreover, the second result is particularly important because it implies that the set Ces(E) is nonempty.
The proof of the next proposition follows the same line of Proposition 3.1 and is omitted.
Proposition 3.2 (Pareto optimality and dominance) Under Assumption (B.1), every individually rational allocation y which is not Pareto optimal can be α–dominated (equivalently, α1–dominated and α2–dominated) by a Pareto optimal allocation through the grand coalition N .
In the statement of the previous proposition, if we add the assumption that the allo-cation y is type-symmetric, then we can prove that it can be dominated by an alloallo-cation which is not only Pareto optimal, but also type–symmetric, provided that quasi–concavity on the utility functions is also assumed. That is, the following result holds:
Proposition 3.3 (Pareto optimality, type–symmetry and dominance) Under As-sumptions (B.1) and (B.2)0, every individually rational, type–symmetric allocation y which is not Pareto optimal can be α–dominated (equivalently, α1–dominated and α2–dominated) by a type–symmetric, Pareto optimal allocation through the grand coalition N .
Proof. Let y be an allocation which is type–symmetric but not Pareto optimal. Since y is not Pareto optimal, by Proposition 3.2, there exists a Pareto optimal allocation h = (hT1, . . . , hTk) that dominates y through the grand coalition N , that is, for all (i, j) ∈ N :
Uij(h) > Uij(y) .
If h is type–symmetric, the proof ends. Otherwise, we consider the allocation ¯h = (¯hT1, . . . , ¯hTk) obtained from h by averaging bundles over traders of the same type; it
The allocation ¯h is both feasible and type–symmetric.
We want to show that, for every (i, j) ∈ N :
l the corresponding consumption profile for type Ti. Finally, let:
For each l = 1, . . . , s, by condition (2) in the definition of agents’ types, Proposition 2.1 and the type-symmetry of y, it holds that:
Uij(gl) = Uij(hTσ1
By (3) and (4), we obtain that for each (i, j) ∈ N : Uij(¯h) > Uij(y)
If the allocation ¯h is Pareto optimal, the proof is concluded. Otherwise, we consider a positive real number δ and the following set:
B = {x ∈ A : x is type-symmetric and Uij(x) ≥ Uij(¯h) + δ, ∀ (i, j) ∈ N }.
B is non empty; indeed, since ¯h is not Pareto optimal, there exists an allocation x ∈ A such that Uij(x) > Uij(¯h), ∀ (i, j) ∈ N . Since ¯h is type–symmetric, as a consequence of what has just been proved, we can assume with no loss of generality that x is also type-symmetric.
Moreover, the set B is closed and, hence, compact.
Let U be the function defined on B as in Proposition 3.2. By assumption (B.1), U is continuous on B and, hence, it has a maximal element on the set B. Let us denote it by g. It holds that g is type–symmetric and Uij(g) ≥ Uij(¯h) + δ, ∀ (i, j) ∈ N .
Moreover, following the same line of reasoning as in Proposition 3.2, we can prove that g is also Pareto optimal.
Therefore, g is a type–symmetric, Pareto optimal allocation which dominates y; that is:
Uij(g) > Uij(y), for every (i, j) ∈ N and this concludes the proof.
We prove now that the set Ces(E) is internally stable with respect to the α–dominance relation.
Theorem 3.4 (Internal stability) Let the economy E satisfy the Assumptions (B.3),(B.4) and (B.5). Then, the symmetric extreme core Ces(E) is α–internally stable.
Proof. By way of contradiction, let x and y be two allocations in V such that x α y.
That is, there exists a coalition S ⊆ N such that:
a. x is an assignment for S;
b. Uij(xS, zN \S) > Uij(y), for all (i, j) ∈ S and for all allocations z such that zS = xS. We first prove that there exists an agent (i, j) ∈ S such that xij 0. Indeed, if for every (i, j) ∈ S, xij has a zero component, then by Assumption (B.3) we get that for every (i, j) ∈ S:
Uij(xS, zN \S) = Uij(0S, zN \S) .
If we consider, in particular, the initial endowment as a redistribution among traders in the complementary coalition N \ S, we get that for every (i, j) ∈ S:
Uij(xS, zN \S) = Uij(0S, zN \S) = Uij(0S, eN \S) = Uij(0N) .
Moreover, since y is individually rational and Assumptions (B.3) and (B.5) hold, Uij(y) ≥ Uij(e) = Uij(0N) and hence the inequality in condition b. cannot hold.
We can conclude that there exists an agent (i, j) ∈ S such that xij 0 and, therefore P
(i,j)∈Sxij 0.
As a consequence of Assumptions (B.4) and (B.5), we can thus infer that coalition S contains every type of agents.
Consider now xN \S itself as a physically feasible redistribution over the complementary coalition N \ S. By the type–symmetry of x and y, condition b. can be rewritten as:
Uij(x) > Uij(y), for every (i, j) ∈ N , which contradicts the Pareto optimality for the allocation y.
Finally, we show the property of external stability.
Theorem 3.5 (External Stability) Let the economy E satisfy the Assumptions (B.1), (B.2) and (B.6). Then, the set Ces(E) is α2–externally stable.
Proof. Let y be an individually rational allocation that does not belong to Ces(E). We have to find an allocation x ∈ V and a non empty coalition S ⊆ N such that:
a. x is an assignment for S;
b. for all i ∈ S and for an allocation z with zS = xS, it holds that Ui(xS, zN \S) > Ui(y).
We distinguish three cases.
First case. Suppose that y is type–symmetric but not Pareto optimal. Then by Propo-sition 3.3, it is dominated by a type-symmetric, Pareto optimal allocation x through the grand coalition N . Since x ∈ V , the proof ends.
Second case. Suppose that y is Pareto optimal but not type–symmetric. Since y is assumed to be non-symmetric, there exist an agents’ type i ∈ {1, . . . , k} and two agents j, t ∈ Ti such that Uij(y) 6= Uit(y). With no loss of generality, we can assume that i = 1 and U1j(y) ≥ U11(y), for every j ∈ {1, . . . , s}, with a strict inequality for at least one index j.
We can also assume that for every other type i 6= 1, it holds:
Uij(y) ≥ Ui1(y) , ∀(i, j) ∈ Ti.
Consider the type–symmetric allocation ¯y obtained from y and defined by:
¯
y = (¯yT1, . . . , ¯yTk) ,
where ¯yTi is the identical consumption bundle allotted to each trader of type i given by
¯
yTi = 1
|Ti| P
(i,j)∈Tiyij, i ∈ {1, . . . , k}.
We repeat now the same construction used for allocation h in the proof of Proposition 3.3.
That is, denote by σ1 the permutation on each set Ti which reindexes (i, j) in (i, j + 1)
In the same way, as a consequence of strict quasi–concavity (B.2), for every type i 6= 1 we get:
Ui1(¯y) > Ui1(y) (8)
Consider the coalition S = {11, 21, . . . , k1}. It satisfies conditions a. and b. through ¯y.
Indeed, together with conditions (7) and (8), it is enough to note that the allocation ¯y is an assignment for coalition S and is also physically feasible for the counter coalition N \S.
In particular, the consumption set feasibility for coalition S is guaranteed by Assumption (B.6).
If ¯y is Pareto optimal, the proof of the external stability of Ces(E) ends.
If ¯y is not Pareto optimal, then by Proposition 3.3 we can find an allocation g such that it is type-symmetric, Pareto optimal and dominates ¯y over the grand coalition N , that is: a. and b. are satisfied by g, N and ¯y. Since ¯y, S and y satisfy a. and b., we get that also g, S and y satisfy the conditions and and this concludes the proof.
Third case. Suppose that y is neither Pareto optimal nor type-symmetric. Since y is not Pareto optimal, by Proposition 3.2, we can find a Pareto optimal allocation h such that h Pareto dominates y.
If h is type-symmetric, then h ∈ V and the proof ends.
If h is not type-symmetric, by the previous case, we can find a type-symmetric and Pareto optimal allocation g and a coalition S such that g, S and h satisfy condition a. and b.:
Therefore we can infer that g, S and y satisfy a. and b. and, since g ∈ V , this concludes the proof.
The following theorem summarizes all the previous results.
Theorem 3.6 (Existence and Uniqueness) Let the economy E satisfy the Assump-tions (B.1)-(B.6). Then:
1. the symmetric extreme core Ces(E) is α-internally stable and α2-externally stable;
2. whenever a set V of symmetric allocations exists which is α-externally stable and α1-internally stable, then V equals the symmetric extreme core Ces(E);
3. whenever a set V of symmetric allocations exists which is α-externally stable and α2-internally stable, then V equals the symmetric extreme core Ces(E);
4. whenever an α-stable set V of symmetric allocations exists, then V equals the sym-metric extreme core Ces(E);
5. whenever an α1-stable set V of symmetric allocations exists, then V equals the subset of the symmetric extreme core Ces(E) formed by all the allocations which do not α1– dominate each other;
6. whenever an α2-stable set V of symmetric allocations exists, then V equals the subset of the symmetric extreme core Ces(E) formed by all the allocations which do not α2– dominate each other.
Notice that, contrary to the framework of traders of Type I, here the uniqueness of the α-stable set can be guaranteed but not the existence.
We conclude this section by analyzing a framework which is intermediate between the selfish and the interdependent ones. Precisely, we consider agents of Type II and suppose that the utility function of each trader j of type i just depends on the consumption bundles of all other traders of type i. That is, we still have interdependence but just within types not across all traders in the market.
In order to distinguish this model from the one analyzed so far, this sort of utility functions will be called partially interdependent preferences.
We note that, when there is just one trader for each type, this model coincides with one with selfish preferences; moreover, the partially interdependent framework can be considered as a particular case of that analyzed above, whenever the utility functions Uij
just depend on xTi. The reason why it is considered on its own is two–fold and concerns the external stability of the set Ces(E): when proving the external stability, in fact, the strict quasi–concavity can be replaced with strict monotonicity and quasi–concavity; moreover, a mechanism of redistribution among traders can be implemented which is not possible when utility functions are totally interdependent.
Formally, we adopt the notion of types provided by Definition 2.11, with two adjustments:
condition (2) is not considered anymore while condition (1) is replaced as follows: for every i = 1, . . . , k and for every consumption bundle xTi = (xi1, . . . , xij, . . . , xit, . . . , xis), it holds that:
Uij(xi1, . . . , xij, . . . , xit, . . . , xis) = Uit(xi1, . . . , xit, . . . , xij, . . . , xis). (9) In the next remark, we provide some examples of functions for which condition (9) holds true.
Remark 3.5 When there are just two traders for each type, it is easy to find examples of functions for which condition (9) holds. For instance, in the case of just one commodity, we can consider the following two functions for traders of type 1:
U11(x11, x12) = 2x11+ x12 U12(x11, x12) = x11+ 2x12
On the other hand, for a framework with two commodities, distinguished by a superscript, we may consider the following utility functions which also identify traders of the same
type according to condition (9):
U11(x111, x211, x112, x212) = q
x111· x211+ x112+ x212 2 U12(x111, x211, x112, x212) = x111+ x211
2 +
q
x112· x212
Let us consider now a more complex context with four agents of type 1 and just one commodity. If traders are endowed with the following utility functions:
U11(x11, x12, x13, x14) = x12· x13· x14, U12(x11, x12, x13, x14) = x11· x13· x14, U13(x11, x12, x13, x14) = x11· x12· x14, U14(x11, x12, x13, x14) = x11· x12· x13,
condition (9) is satisfied and, provided that their endowment is the same, all four traders can be considered of the same type.
For what concerns the notions of dominance relations given for partially interdepen-dent preferences, two points have to be remarked.
First of all, some form of interdependence across types can still be detected in the blocking mechanism; indeed, all traders in N \ S coordinate together their reaction to the deviation thus influencing, through z, the utility of each agent in S.
Second, we notice that, if the allocation to be blocked is type-symmetric, what really matters in a blocking coalition S is the number of agents for each type not the actual identity of traders. More precisely, agents of the same type are interchangeable within a blocking coalition S provided that the number of traders for each type remains the same.
In fact, given condition (9) on the utility functions, consumption bundles in x and in z can be properly permutated within each type in such a way to guarantee a trader that is not originally included in S a utility level grater than that provided by y. Let consider the following example to illustrate this point.
Example 3.1 Consider three types of traders and three traders for each type, that is:
N = {11, 12, 13; 21, 22, 23; 31, 32, 33} .
Suppose that S = {11, 12; 21} blocks a given type–symmetric allocation y through x.
Focusing just on the first type, we have that:
U11(x11, x12, z13) > U11(yT1), U12(x11, x12, z13) > U12(yT1) .
If we substitute agent (1, 1) for agent (1, 3), for instance, we can allot the bundle x11 to trader (1, 3) and the bundle z13 to trader (1, 1) and we get:
U13(z13, x12, x11) = U11(x11, x12, z13) > U11(yT1) = U13(yT1).
Hence, also coalition S0 = {12, 13; 21} blocks the type-symmetric allocation y through the assignment x0 obtained from x by properly permuting consumption bundles8. In the context of partially interdependent preferences we introduce the following new assumption.
(B.7) (Type–strict monotonicity) For every i ∈ {1, . . . , k}, for every consumption profiles xTi and yTi such that xTi > yTi, it holds that:
Uij(xTi) > Uij(yTi), ∀ (i, j) ∈ Ti.
Moeover, given the form of the utility functions, the type–quasi concavity formulated in Assumption (B.2)0 simply takes now the form of quasi–concavity.
The theorems about the internal stability of the symmetric extreme core Ces(E) are identical, both in their statements and in their proofs, to those provided in the general case. On the contrary, we provide the proof of the external stability of the set Ces(E) because the assumption of strict quasi–concavity is now replaced by quasi–concavity and strict monotonicity and, moreover, an interesting mechanism of redistribution among traders of different types is implemented in order to block an allocation y which is Pareto optimal but not type–symmetric.
Theorem 3.7 (External Stability I) Let the economy E satisfy the Assumptions (B.1), (B.2)0, (B.6) and (B.7). Then, the set Ces(E) is α2– externally stable.
Proof. Let y be an individually rational allocation that does not belong to Ces(E). We have to find an allocation x ∈ Ces(E) and a coalition S such that:
a. x is an assignment for S;
b. Uij(zTi) > Uij(yTi), for every (i, j) ∈ S, for an allocation z such that zS = xS.
8We also remark that such interchangeability of traders of the same type within a blocking coalition does not apply anymore for a model with agents of Type I.
We distinguish three cases.
First case. Suppose that y is type–symmetric but not Pareto optimal. Then it is dominated by a type-symmetric, Pareto optimal allocation x through the grand coalition N . Since x ∈ Ces(E), the proof ends.
Second case. Suppose that y is Pareto optimal but not type–symmetric. Since y is assumed to be non-symmetric, there exist an agents’ type i ∈ {1, . . . , k} and two agents j, t ∈ Ti such that Uij(yTi) 6= Uit(yTi). Without loss of generality, we can assume that i = 1 and U1j(yT1) ≥ U11(yT1), for every j ∈ {1, . . . , s}, with a strict inequality for at least one index j.
Consider the type-symmetric allocation ¯y defined by:
¯
y = (¯yT1, . . . , ¯yTk)
which allots to each trader of type i the same consumption bundle given by ¯yTi = 1
|Ti| P
j∈Tiyij, i ∈ {1, . . . , k}.
By Assumption (B.2)0, it holds that:
U11(¯yT1) > U11(yT1).
By the continuity Assumption (B.1), we can pick some 0 < ε < 1 such that:
U11(ε¯yT1) > U11(yT1) .
Consider the coalition S = {11, 21, . . . , k1} and the allocation ¯g defined as follows:
¯
Note that the allocation g is feasible for both the coalition S and the grand coalition N . We want to show that S can dominate the allocation y through ¯g in the sense of condition a. and b.
Indeed, it clearly holds that:
U11(¯gT1) = U11(ε¯yT1) > U11(yT1) .
Moreover, by Assumptions (B.7) and (B.2)0, for every l ∈ {2, 3, . . . , k}:
Ul1(¯gTl) > Ul1(¯yTl) ≥ Ul1(yTl) .
If ¯g is Pareto optimal, the proof of the external stability of Ces(E) ends.
If ¯g is not Pareto optimal, then by Proposition 3.3 we can find an allocation ˜g such that it is type-symmetric, Pareto optimal and dominates ¯g over the grand coalition N , that is:
˜ g N g¯ We can conclude that
˜ g S y and this concludes the proof.
Third case. Suppose that y is neither Pareto optimal nor type-symmetric. Since y is not Pareto optimal, by Proposition 3.2, we can find a Pareto optimal allocation h such that h Nα1 y.
If h is type-symmetric, then h ∈ Ces(E) and the proof ends.
If h is not type-symmetric, by the previous case, we can find a type-symmetric and Pareto optimal allocation ¯g and a coalition S such that S dominates h via g as in conditions a.
and b. Since h Pareto dominates y, we can infer the conclusion.