Chapter 7 NTU-games
Until now we focused on the TU-games. Recall that ‘TU’ stands for a ’trans- ferable utility’ which was represented by real numbers. The presence of the transferable utility allowed us to compare the values in a simple way.
However, in many situations such a scale does not exist. Think for ex- ample about some barter exchange in which the players trade objects that do not have any price tags attached to them. This leads to a study of NTU- games, where ‘NTU’ stands for a ’non-transferable utility’. These games are defined as follows.
As before we consider a fixed setN ={1,2, . . . , n} of players. Addition- ally we assume a set X of outcomes and for each player i a preference relation (i.e., a transitive and complete relation1) i on X. We denote by
≻i the corresponding strict preference relation, defined by x≻i y iff xi y and noty i x.
Finally, for every coalition S we assume that a set V(S)⊆X of outcomes is given.
Intuitively, V(S) is the set of outcomes that coalitionS can obtain solely by means of its joint action.
To clarify the relation with TU-games we first explain how each TU-game can be viewed as an NTU-game. To this end, given a TU-game (N, v) we take as X the set of all allocations, i.e.,
X :=Rn,
1Recall that a binary relationRistransitiveif for allx, yandz,xRyandyRzimplies xRz, andcomplete if for all xandy eitherxRy or yRxor both.
and put
xi yiff xi ≥yi.
So each player ranks the allocations solely on the basis of his share of it.
Finally, for each coalitionS we define V(S) :={x∈Rn|X
i∈S
xi ≤v(S)}.
This corresponds to the earlier intuition that in a TU-gamev(S) stands for the worth that the coalitionScan achieve on its own. Indeed,V(S), when defined as above, simply lists the potential allocations under this viewpoint.
The notion of a core naturally extends to the NTU-games. Given an NTU-game (N, X, V,(i)i∈N) we define its core as follows:
Core(V) :={x∈V(N)| for no coalitionS and y∈V(S), y≻i x for all i∈S}.
To understand this definition say that an outcomex ∈X is blocked by a coalition S if for some y ∈ V(S) we have y ≻i x for all i ∈ S. Then we can say that an outcome x is in the core if it can be achieved by the grand coalition and is not blocked by any coalition. Note that it is similar in spirit to the concept of DCore introduced in Chapter 3.
An important class of NTU-games is the following one.
Example 16 [Exchange economy game]
As in Example 6 we consider a market with k goods and assume that each player has an initial endowment of these goods represented by a vector ωi ∈ Rk+.
However, now the players do not have to their disposal any valuation function that assigns a value to each bundle of these goods. Instead, we assume only that each player can compare these bundles by means of a preference relation i.
To interpret this situation as an NTU-game we take as the set of outcomes the set of all possible sequences of bundles,
X :={(x1, . . .,xn)|xi ∈Rk+ for i∈N},
i.e. X = (Rk+)n, and extend each preference relation i from the set Rk+ of all bundles to the set X of sequences of bundles, by putting for x,y∈X
xi y iff xi i yi.
This simply means that each player is only interested in his own bundle.
Additionally, we assign to each coalitionS the following set of outcomes:
V(S) :={x∈X |P
i∈Sxi =P
i∈Sωi and xj =ωj for all j ∈N \S}.
SoV(S) consists of the set of outcomes that can be achieved by trading between the members of S. The players outside of S do not participate in any trading and hold on to their initial endowments. In particular,
V(N) ={x∈X |P
i∈Nxi =P
i∈Nωi}.
2 In the remainder of this chapter we shall focus on the exchange economy games. These games capture the intuition of an exchange economy that consists of:
• a set N of players,
• k goods,
• for each player i an initial endowment ωi ∈ Rk+ of these goods and a preference relation i over the bundles of these goods.
The exchange economy games and exchange economies can be better understood by introducing prices. Given k goods, a price vector is an element p ∈ Rk+. Given a price vector p and a bundle z of the goods, the inner product pz, defined byPk
j=1pjzj, is then the total cost of the bundle z.
So if p is the price vector and player’s i initial endowment is ωi, he has to his disposal the amount pωi. For this amount he can afford to buy any bundle yi of goods such that pyi ≤pωi.
The following concept is then crucial in the subsequent analysis. Acom- petitive equilibrium of an exchange economy is a pair (p,x) consisting of a price vector p ∈ Rk+, with some entry strictly positive, and a sequence of n bundles x, each xi for player i, such that
(i) P
i∈Nxi =P
i∈Nωi (this sequence of bundles results from trading), (ii) for all i ∈ N, pxi ≤ pωi (assuming the price vector p, each player i
can afford to pay for the bundle xi, given his initial endowment ωi),
(iii) for alli∈N and allyi ∈Rk+, ifpyi ≤pωi, thenxi i yi (assuming the price vector p, each player i prefers the bundle xi among all bundles he can afford to pay for).
Intuitively, a competitive equilibrium consists of a vector of prices for the considered goods and a sequence of bundles of goods that can arise from trading, such that given these prices each player holds from his point of view an optimal bundle. A deep theorem by K. Arrow and G. Debreu states that under natural conditions a competitive equilibrium exists.
Theorem 23 Consider an exchange economy in which each preference re- lation i is continuous and strictly convex.2 Then a competitive equilibrium
exists. 2
We now prove the following theorem that relates the concept of a com- petitive equilibrium to that of a core.
Theorem 24 If(p,x)is a competitive equilibrium of an exchange economy, then x belongs to the core of the corresponding exchange economy game.
Proof. Suppose x is not in the core of the associated exchange economy game. Then for some coalition S andy∈X we have P
i∈Syi =P
i∈Sωi and yi ≻i xi for alli∈S. The latter implies by condition (iii) of the definition of competitive equilibrium pyi >pωi for all i ∈ S. So pP
i∈Syi > pP
i∈Sωi and hence, since the prices are non-negative, P
i∈Syi >P
i∈Sωi, which is a
contradiction. 2
This yields the following important conclusion.
Corollary 25 If each preference relation i is continuous and strictly con- vex, then the core the exchange economy game is non-empty.
Proof. By Theorems 23 and 24. 2
The ‘superiority’ of the sequence of bundles of a competitive equilibrium is defined using the prices. However, an alternative and more direct description exists and allows us to clarify why such sequences of bundles are preferable.
2Recall that a preference relation is strictly convex if for all x, y, if x y and x6=y, then for everyα∈(0,1),αx+ (1−α)y≻y.
To this end we introduce first a concept that has nothing to do with exchange economies.
Consider a sequence of preferences (1, . . .,n) on a set of outcomes X and the associated sequence (≻1, . . .,≻n) of strict preferences. We call an outcome x∈X Pareto efficient w.r.t. X if no outcome y ∈X exists such that for all i∈ {1, . . ., n},y ≻i x. So a Pareto efficient outcome is optimal in the sense that no outcome exists that is strictly better w.r.t. each preference.
We can now state the desired result.
Theorem 26 If(p,x) is a competitive equilibrium of an exchange economy, given the considered sequence of preferences (1, . . .,n), then x is Pareto efficient w.r.t. V(N), i.e., w.r.t. the set of sequences of bundles of goods that can arise from trading in the exchange economy game.
Proof. The proof is very simple. By Theorem 24 x belongs to the core of the corresponding exchange economy game. In particular, it is not blocked by the grand coalition N. So no sequence yof bundles exists such that
• P
i∈Nyi =P
i∈Nωi (that is, y arises from trading),
• y≻i xfor all i∈N.
In other words, xis Pareto efficient w.r.t. the set V(N). 2
