2. Preliminaries
Assume that U is the universe of agents and is a collection of agents with . Assume that each agent has energy levels at which he/she/it can take, where is positive integer. Let be the energy level set of agent i, where 0 represents not participating and . Let be the product set of the energy level sets for agents of N. is a vector that presents current energy levels for every agent in N, where is the energy level that agent i participates at currently for each . Given and , we define and to be the restriction of to K. Furthermore, let be the amount of elements in K, and be the vector satisfying if , and if . is denoted by if no bewilderment can arise. The zero vector in is denoted by .
Let . if for each ; if and ; if for each . We denote . Let . P is comprehensive if and imply . The comprehensive hull of P is denoted to be . The interior of P is denoted to be , and the boundary of P is denoted to be . if .
Definition 1. A multi-choice NTU game is , (Since is fixed in the paper, could be written rather than .) where is a finite collection of agents and V is a characteristic mapping that assigns to every a subset , such that Situation (1) is standard. Comprehensiveness could be treated as a marked weakening of monotonicity. Situation (2) assures that is a proper subset of . It could be treated as a weakening of boundedness. Denote the family of all multi-choice NTU games by .
A payoff vector of is a vector where could be treated as the per unit payoff that agent i obtains for each , hence is the entire payoff that agent i obtains in . x is efficient (EFC) if . x is singlely rational (SR) if for each and for each .
If payoff vector
x is EFC and SRA, then
x is said to be an imputation in
. Denote the collection of imputations in
to be
. Denote the collection of all feasible payoff vectors in
to be
where as
is the collection of efficient payoff vectors of
. In the following, we define the replicated core on multi-choice NTU games.
Definition 2. The replicated core C of a multi-choice NTU game is defined by A standard NTU game is , where is a finite collection of agents and is a characteristic mapping that assigns to every a subset , such that
is non-empty, comprehensive, and closed.
is bounded for each .
The traditional core
of a standard NTU game
is
Assume that
and
is a set of replicated agents with
where, for each
,
. For any
, we denote
to be that
for each
. For
, we define the corresponding replicated vector
to be that for all
,
and vice versa.
Then, we define the replicated NTU game
as follows. For all
,
The following result presents that the replicated core of a multi-choice NTU game coincides with a “replicated subset” of the traditional core of a replicated NTU game.
Proposition 1. Assume that and is the corresponding replicated NTU game. If , then , where, for all , with for all .
Proof of Proposition 1. The proof is trivial. Hence, we omit it. □
3. Balancedness and Non-Emptiness
Scarf [
7] showed that each balanced standard NTU game has a nonempty traditional core. Here, we offer an analogue result on multi-choice NTU games.
Definition 3. Assume that A is a collection of . A is called balanced if there exists for each such that Definition 4. A standard NTU game is called balanced if it holds thatwhere . Remark 1. Scarf [7] showed that the traditional core is nonempty for each balanced standard NTU game. Subsequently, we provide an analogue balancedness on multi-choice NTU games.
Definition 5. Assume that Θ
is a collection of . Θ
is called m-balanced if there exists for each such that, for all , Definition 6. A multi-choice NTU game is m-balanced if for all balanced collections Θ
, and it holds thatwhere . The following theorem presents that the balancedness of a multi-choice NTU game concurs with the balancedness of the corresponding replicated NTU game.
Theorem 1. Assume that and is the corresponding replicated NTU game. Then, is m-balanced if and only if is balanced.
Proof of Theorem 1. Assume that
is a m-balanced. Let
be a balanced collection of
. That is, there exists
for all
such that for all
,
For
, define the vector
by
for all
. Let
be a collection of
. Taking
for all
, we see that for all
,
Hence, is a balanced collection of . By the replication special nature of and the balancedness of , it is easy to see that is balanced.
Conversely, let be balanced. Let be a m-balanced collection of . For , let be the least common multiple of , where , for all and for all .
For
, define
for all
by for all
,
where
. Note that Condition (8) can be done because that
is the least common multiple.
Let
be a collection of
N. Taking
for all
and for all
, we see that for all
,
Hence, A is a balanced collection of N. Similarly, by the replication special nature of and the balancedness of , is m-balanced. □
A multi-choice NTU extension of the Scarf’s [
7] result is as follows.
Theorem 2. The replicated core is nonempty for each m-balanced multi-choice NTU game.
Proof of Theorem 2. Assume that and is the corresponding replicated NTU game. By Proposition 1, if and only if . Hence, this theorem immediately follows Remark 1 and Theorem 1. □
4. Axiomatic Result
A solution on is a map which assigns a subset to each . One would make use of some more axioms as follows. satisfies efficiency (EFCY) if for each . satisfies single rationality (SRY) if x is SR for each and for each . satisfies standard for one-person game (SOPG) if for each with .
Next, we provide a multi-choice NTU extension of the reduction proposed by Peleg [
8]. Assume that
,
and
x is a payoff vector. The reduced game in regard to
T and
x is
defined as follows. For all
,
Consistency asserts that if x is appointed by for , then the projection of x to S should be appointed by in the reduction for each . Thus, the projection of x to S should be consistent with the expectations among the agents of T as mirrored by its reduction. Converse consistency asserts that if the projection of a single rational vector x to each is consistent with the expectations among the agents of T as mirrored by its reduction, then x itself should be supported for total game.
Consistency (CONY): For each , for each , and for each , and .
Converse consistency (CCONY): For each with and for each , if for each such that , and , then .
Here, we confine our consideration to non-level multi-choice NTU games, defined as those games
such that
Situation (9) states that has no level segments, i.e., segments parallel to a coordinate hyperplane. We adopt it to ensure that, if is efficient, then is efficient in .
Next, we would like to adopt SRY, SOPG, CONY, and CCONY to characterize the replicated core. The proofs that the replicated core satisfies CONY and CCONY are in the following steps:
Lemma 1. Assume that , and . Then, the reduction .
Proof of Lemma 1. It is trivial to verify the condition (9) and Definition 1. □
Lemma 2. Assume that , and . Then, if and only if .
Proof of Lemma 2. Clearly, . If , then there exists such that . Thus, and . Thus, . Similarly, there exists such that if . Thus, . Therefore, and . Thus, . □
Lemma 3. Assume that , and . Then, for each , Proof of Lemma 3. Let
,
and
. Let
. Assume that
Then, there exists
such that
Let . Hence, . This contradicts to the fact that . □
Lemma 4. The replicated core satisfies CONY.
Proof of Lemma 4. Let
. Let
and
. By Lemma 1,
. Since
,
. Thus,
by Lemma 2. It remains to show that
for all
. Let
. Since
by Lemma 3. Thus,
for each
if
. □
Lemma 5. The replicated core satisfies CCONY.
Proof of Lemma 5. Assume that with and . Suppose that and for each with . We will prove that . Since , it remains to prove that for all . Two conditions could be distinguished:
Condition 1: :
Let . Suppose .
Consider
with
. Then
by
. Since
. Thus, .
Consider with . The proof is similar to the above processes by applying the reduction .
Condition 2: :
Let .
If there exists
such that
, consider the reduction
. Then,
by
. Since
. Hence, .
Let
or
for all
. Assume that
and
where
. Consider the reduction
. By
,
. Since
. Thus, . □
Lemma 6. If a solution σ satisfies CONY and SOPG, then it also satisfies EFCY.
Proof of Lemma 6. Assume that there exist and such that . Let . Consider the reduction . By CONY of , . By SOPG of , . By Lemma 2, . Therefore, the desired contradiction could be presented. □
Theorem 3. A solution σ on satisfies CONY, CCONY, SOPG, and SRY if and only if for each .
Proof of Theorem 3. By Lemmas 4 and 5, the replicated core satisfies CONY and CCONY. Absolutely, the replicated core satisfies SOPG and SRY.
To analyze uniqueness, suppose that a solution satisfies SOPG, SRY, CONY, and CCONY. By Lemma 6, satisfies EFCY. Let . The following proof proceeds by induction on . By SOPG of , if . Suppose that if , . The condition :
We first prove that . Assume that . Since satisfies SRY and EFCY, . Clearly, for each with by CONY of . Based on the induction hypothesis, for each with . Thus, by CCONY of the replicated core. The remaining proof could be completed analogously by switching the roles of C and . Thus, . □
The following examples present that each of the properties adopted in Theorem 3 should be logically independent of the others. In order to present the independence of the adopted properties, is necessary.
Example 1. Assume that for each . Absolutely, σ satisfies SRY, CONY, and CCONY, but σ violates SOPG.
Example 2. Here, we define a solution σ on to be Absolutely, σ satisfies SOPG, CONY, and CCONY, but σ violates SRY.
Example 3. Assume that for each . Absolutely, σ satisfies SOPG, SRY, and CCONY, but σ violates CONY.
Example 4. Here, we define a solution σ on to be Absolutely, σ satisfies SOPG, SRY, and CONY, but σ violates CCONY.
5. Another Reduction
In this section, we provide a multi-choice NTU extension of the reduction due to Moulin [
9].
Given a payoff vector of some game assigned by a solution, and given a subgroup of agents, Moulin [
9] introduced the reduction as that in which every coalition in the subgroup could reach payoffs to its agents only if they are harmonious with the original payoffs to “all” the agents outside of the subgroup. A natural multi-choice extension of Moulin’s reduction is as follows. Assume that
,
and
x is a payoff vector. The M-reduced game in regard to
T and
x is
defined by for all
,
By replacing “reduction” with “ M-reduction”, we define the properties of M-consistency (MCONY) and converse M-consistency (CMCONY).
Lemma 7. The replicated core satisfies MCONY.
Proof of Lemma 7. Similar to Lemmas 1–4, the proof could be completed. □
In the following, we adopt an example to present that the replicated core violates CMCONY.
Example 5. Define to be , , , , , , , and for all with . Clearly, . Assume that with , . Clearly, for all and for each . Clearly, and . However, . Thus, . The replicated core violates converse M-consistency.
To finalize this section, we show that there exists an unique solution other than the replicated core satisfying SOPG, SRY, MCONY, and CMCONY on .
Lemma 8. On , there exists an unique solution other than the replicated core satisfying SOPG, SRY, MCONY and CMCONY.
Proof of Lemma 8. Clearly, it is shown that the replicated core satisfies SOPG, SRY, and MCONY. Here, we define
by for each
,
where
. Clearly,
for each
. It is easy to verify that the solution
satisfies SOPG and SRY. To verify that
satisfies MCONY and CMCONY, it could similarly be completed by the proof of Lemmas 1–5. Similar to Lemma 6 and Theorem 3, it is easy to show that
is the only solution satisfying SOPG, SRY, MCONY, and CMCONY on
. □
6. Discussion
Some multi-choice NTU extensions of the core and related results have been introduced, such as Hwang and Li [
10], Liu et al. [
11], Tian et al. [
12], and so on. One should compare our works with the works of Hwang and Li [
10], Liu et al. [
11], and Tian et al. [
12].
The core due to Hwang and Li [
10] was defined by determining a payoff for a given agent adopting at a given energy level. Based on the notion of replicated games, we propose the replicated core by considering an entire payoff for a given agent.
Liu et al. [
11] derived an extended existence result of the payoff-dependent balanced core proposed by Bonnisseau and Iehlé [
13] to multi-choice NTU games which implies a multi-choice generalization of the existence result due to Scarf [
7]. By adopting an extended K-K-M-S theorem, Tian et al. [
12] showed that a socially stable core is non-empty in a multi-choice NTU structured game if the transfer rate rule contains several values of power mappings, and this game is payoff-dependent balanced with reference to the rule. Different from the works of Liu et al. [
11] and Tian et al. [
12], we define the replicated core on multi-choice NTU games by means of “replicated NTU games”. The non-emptiness of the replicated core is analyzed by extending the existence result due to Scarf [
7] and the notion of replication simultaneously. Inspired by Peleg [
8], we characterize the replicated core by means of consistency and its converse.
Multi-choice games permit a finite number of energy levels, while fuzzy games admit a continuous level. In a fuzzy game, every agent
i can adopt level
to participate. Fuzzy games and multi-choice games could be treated as continuous generalization and discrete generalization of standard games, respectively. In the framework of fuzzy NTU games, an extended core was introduced by Hwang [
14]. The extended core due to Hwang [
14] and the replicated core are based on the notions of the core on continuum market games (Hart [
4]) and fuzzy TU games (Aubin [
5]). The major difference is that our works are based on the “multi-choice” NTU games and “replicated NTU games”, but Hwang’s [
14] works are based on the “fuzzy” NTU games. Although the notions of axiomatic results are similar, the techniques of the proofs are still different. Furthermore, the notion of replication can not be applied in Hwang [
14].
By applying the replicated core, we would like to analyze cost distribution problems in which the indivisible commodities might be only available under finite natural numbers (energy levels). The replicated core is introduced to offer per unit payoffs (prices) among commodities in multi-choice NTU situations. Assume that is a collection of commodities (or projects) that could be equipped conjunctly by various departments. Let be the cost of offering the components in conjunctly. The mapping K is called a cost-distributing problem (alternatively, K could be considered as a production mapping that appears to be the outcome for any coalition). Modeled by this conception, a cost distribution problem could be considered as a cooperative game, with K being its characteristic mapping. Another condition is investigated when the outcome could be altered continuously. In this condition, it is supposed that commodities are entirely divisible commodities and then quantities of commodities could be estimated with real numbers. This is an adequate approach for situations such as various rural commodities, petroleum commodities (olive oil, gas), and so on. However, there exist many others types of commodities for which this is not possible, such as buildings, cars, and so on. This class of indivisible commodities are only available under finite natural numbers. This is the type of condition that the replicated core would like to analyze: cost distribution problems in which commodities could be provided at finite different levels.