Keywords

1 Introduction

A new model of “consensus” and a definition of “consensus building” are proposed in this work within a framework of voting committees (Peleg 1984; Yamazaki et al. 2000). A committee (Peleg 1984) expresses a group decision-making situation. A negotiation process among decision makers (DMs) in the situation is expressed as a sequence of decision makers’ permissible ranges, and “consensus” is defined as a permissible range of decision makers with a “stable alternative (Yamazaki et al. 2000).” Then, “consensus building” is defined as a sequence of decision makers’ permissible ranges from “status quo” to “consensus.”

“Consensus” and “consensus building” are mentioned in the literature as

Consensus building is a process of seeking unanimous agreement. It involves a good-faith effort to meet the interests of all stakeholders. Consensus has been reached when everyone agrees they can live with whatever is proposed after every effort has been made to meet the interests of all stakeholding parties (page 6 in Susskind (1999)).

and

A group reaches consensus on a decision when every member can agree to support that decision. Each person may not think it’s the very best decision, but he or she can buy into it and actively support its implementation. No one in the group feels that his or her fundamental interests have been compromised. Consensus is not “almost everybody.” It’s unanimous support for a decision, in the same way that a jury returns a unanimous verdict (page 58 in Straus (2002)).

Since both Susskind (1999) and Straus (2002) deal with the words “agree,” “unanimous,” and “interests,” a model of “consensus” and a definition of “consensus building” should involve these words as keywords. Also, the phrase “live with (Susskind 1999),” which has the meaning “to accept something unpleasant (Oxford Advanced Learner’s Dictionary 2000),” is almost the same meaning as the phrase “buy into (Straus 2002)” accompanied with the sentence “[e]ach person may not think it’s the very best decision (Straus 2002).” A DM, therefore, should be modeled as an agent who may agree to a decision which is not the best for him/her.

With respect to the difference between “consensus” and “agree,” moreover, referring to the sentences “[c]onsensus has been reached when everyone agrees (Susskind 1999),” “A group reaches consensus (Straus 2002),” and “every member can agree (Straus 2002),” the author uses “agree” for describing an individual’s state and “consensus” for expressing a group’s state. More specifically, in this work, a group is said to reach “consensus” on a decision, if and only if every DM in the group “agrees” to the decision.

In the next section, a framework of voting committees with DMs’ permissible ranges is provided on the basis of the framework in Yamazaki et al. (2000). In Sect. 11.3, mathematical definitions of consensus and consensus building are newly proposed, and Sect. 11.4 verifies a relationship between consensus building in a committee and the core of the voting committee. Section 11.5 treats a strategic aspect of consensus building and investigates a relationship between consensus and Nash equilibrium (Nash 1950, 1951). Section 11.6 verifies the existence of consensus in a committee. The last section is devoted to conclusions.

2 Preliminaries (Peleg 1984; Yamazaki et al. 2000)

On the basis of the frameworks in Peleg (1984) and Yamazaki et al. (2000), this section gives a framework of voting committees with DMs’ permissible ranges. Mathematical definitions of simple games, properness, unanimity, committees, cores of committees, and efficiency are provided in Sect. 11.2.1. Then, in Sect. 11.2.2, definitions of permissible ranges, stable coalitions, and stable alternatives are provided, and two propositions verified in Yamazaki et al. (2000) are introduced.

2.1 Committees and Core (Peleg 1984)

A simple game specifies the set of all DMs in a group decision-making situation and the decision-making rule adopted in the situation.

Definition 1 (Simple Games)

A simple game is a pair \((N, {\mathbb {W}})\) of a set N of all DMs and a set \({\mathbb {W}}\) of all winning coalitions, where (i) \({\emptyset }\notin {\mathbb {W}}\) and \(N\in {\mathbb {W}}\) and (ii) if S ⊆ T ⊆ N and \(S\in {\mathbb {W}}\), then \(T\in {\mathbb {W}}\). □

A winning coalition is assumed to have enough power to make the coalition’s opinion be the final decision of the group, if every DM in the coalition agrees to the opinion.

Under the properness of a simple game, no two disjoint winning coalitions can be formed at the same time.

Definition 2 (Properness of Simple Games)

A simple game \((N, {\mathbb {W}})\) is said to be proper if and only if for all S ⊂ N, \(S\in {\mathbb {W}}\) implies \(N{\backslash } S\notin {\mathbb {W}}\). □

A unanimous decision rule, on which this work concentrates, is expressed by a unanimous simple game.

Definition 3 (Unanimous Simple Games)

A simple game \((N, {\mathbb {W}})\) is said to be unanimous if and only if \({\mathbb {W}}=\{N\}\). □

Evidently, a unanimous simple game is proper.

A group decision-making situation is represented by a committee.

Definition 4 (Committees)

A committee C is a 4-tuple \({(N, {\mathbb {W}}, A, (\succsim _i)_{i\in N})}\) of a set N of all DMs, a set \({\mathbb {W}}\) of all winning coalitions, a set A of all alternatives, and a list (≿i)i ∈ N of preferences ≿i on A of DM i for each i ∈ N, where \((N, {\mathbb {W}})\) is a simple game, 2 ≤|N| < ∞, and 2 ≤|A| < ∞. For any i ∈ N, the preference ≿i on A of DM i is an element of the set L(A) of all linear orderings on A. □

A relation \(\succsim \) on A is said to be a linear ordering on A if and only if \(\succsim \) is complete, transitive, and anti-symmetric; that is, (i) for x and y in A, \(x \succsim y\) or \(y \succsim x\) (complete), (ii) for x, y, and z in A, if \(x \succsim y\) and \(y \succsim z\), then \(x \succsim z\) (transitive), and (iii) for x and y in A, if \(x \succsim y\) and \(y \succsim x\), then x = y (anti-symmetric). Therefore, L(A) is the set of all complete, transitive, and anti-symmetric relations on A.

When we see a linear ordering \(\succsim \) on A as a DM’s preference, for x and y in A, \(x \succsim y\) means that x is equally or more preferred to y. x ≻ y is defined as \(x \succsim y\) and \(\lnot (y \succsim x)\), where ¬ denotes “not.” If x ≠ y, then \(x \succsim y\) implies x ≻ y, because \(\succsim \) is a linear ordering (in particular, an anti-symmetric relation). For \(\succsim \in L(A)\), moreover, \({\mathit {max}} \succsim \) denotes the most preferred alternative in A in terms of \(\succsim \), that is, \({\mathit {max}} \succsim =a\) if and only if for all x ∈ A, \(a \succsim x\). \({\mathit {max}} \succsim \) is uniquely determined, because \(\succsim \) is a linear ordering. Furthermore, for \(\succsim \in L(A)\), \(\succsim =[x,y,z]\) denotes that x is more preferred to y and y is more preferred to z (and hence x is more preferred to z by the transitivity of r), that is, x ≻ y and y ≻ z (and hence x ≻ z), with respect to \(\succsim \).

Definition 5 (Cores of Committees)

Consider a committee \(C=(N, {\mathbb {W}}, A, (\succsim _i)_{i\in N})\), and the relation Dom on A, which is defined as, for all a and b in A, aDomb if and only if there exists \(S\in {\mathbb {W}}\) such that a ≿i b for all i ∈ S. For all a and b in A, moreover, aDom ∕ b means “not aDomb.” The core of C, denoted by Core(C), is defined as the set {a ∈ A|∀b ∈ A∖{a}, bDom ∕ a}. □

As shown in Appendix, in the case that the simple game \((N, {\mathbb {W}})\) of the committee \(C={(N, {\mathbb {W}}, A, (\succsim _i)_{i\in N})}\) is unanimous, the following three propositions are mutually equivalent: (i) x ∈Core(C), (ii) x is Pareto efficient, and (iii) x is strongly Pareto efficient, where Pareto efficiency and strong Pareto efficiency are defined as follows:

Definition 6 (Pareto Efficiency (p. 7 in Osborne and Rubinstein (1994)))

Consider a committee \(C={(N, {\mathbb {W}}, A, (\succsim _i)_{i\in N})}\). x is said to be Pareto efficient if and only if no b ∈ A satisfies that b ≻i x for all i ∈ N. More, x is said to be strongly Pareto efficient if and only if no b ∈ A satisfies that b ≿i x for all i ∈ N and b ≻i x for some i ∈ N.

2.2 Committees with Permissible Ranges (Yamazaki et al. 2000)

Permissible ranges of DMs allow us to treat the flexibility of the DMs and make it possible to model agents who may agree to a decision which is not the best for them.

Definition 7 (Committees with Permissible Ranges)

A committee C with permissible range P, denoted by C(P), is a pair of a committee \(C={(N, {\mathbb {W}}, A, (\succsim _i)_{i\in N})}\) and a list P = (P i)i ∈ N of permissible ranges P i of DM i for each i ∈ N, where ≿i ∈ P i ⊆ L(A) for each i ∈ N. It is assumed that for all i ∈ N and all x and y in A, if x ≿i y and there exists \(\succsim \in P_i\) such that \({\mathit {max}} \succsim =y\), then there exists \(\succsim '\in P_i\) such that \({\mathit {max}} \succsim '=x\). The set of all permissible ranges P i of DM i is denoted by \({\mathbb {P}}_i\). □

For i ∈ N and a ∈ A, DM i is said to have a as one of his/her permissible alternatives if and only if there exists \(\succsim \in P_i\) such that \({\mathit {max}} \succsim =a\). The set of all DM i’s permissible alternatives is denoted by maxP i, that is, \({\mathit {max}} P_i=\{a\in A | \exists \succsim \in P_i, {\mathit {max}} \succsim =a\}\). The assumption in Definition 7 can be expressed as follows: if x ≿i y and y ∈maxP i, then x ∈maxP i, which can be regarded as a kind of monotonicity. This assumption reflects the idea that each DM considers his/her interests even when he/she agrees to an alternative which is not the best for him/her.

Let S a be the set \(\{i\in S | \exists \succsim \in P_i, {\mathit {max}} \succsim =a\}\), that is, S a denotes the set of all DMs who are members of coalition S and have a as one of their permissible alternatives. Moreover, \({{\mathbb {W}}_{C(P)}}=\{S\in W | \exists a\in A, S_a\in {\mathbb {W}}\}\), that is, \({{\mathbb {W}}_{C(P)}}\) denotes the set of all winning coalitions S such that S a forms a winning coalition for some a ∈ A. In other words, \({{\mathbb {W}}_{C(P)}}\) is the set of all winning coalitions which have possibility of cooperation to make their permissible alternatives be chosen as the final decision of the group. An alternative must be permissible for all members in a winning coalition in \({{\mathbb {W}}_{C(P)}}\), in order to be the final decision of the group, and such alternatives constitute the set A C(P), that is, \({A_{C(P)}}=\{a\in A | \exists S\in {\mathbb {W}}, S_a\in {\mathbb {W}}\}\), or equivalently, \({A_{C(P)}}=\{a\in A | N_a\in {\mathbb {W}}\}\).

There may exist a winning coalition \(S\in {{\mathbb {W}}_{C(P)}}\) such that all DMs in S have an alternative a ∈ A as their common permissible alternative, and for each DM i ∈ S, the alternative a is the best for him/her among the alternatives in A C(P). Such a coalition is quite stable in the group decision situation, because each DM in S has no incentives to deviate from the coalition, and there is no need for the DMs in S to invite other DMs into S in order to obtain bigger power. This type of winning coalitions is said to be stable, in this work.

Definition 8 (Stable Coalitions)

Consider a committee C with permissible range P, that is, C(P), where \(C={(N, {\mathbb {W}}, A, (\succsim _i)_{i\in N})}\), and \({{\mathbb {W}}_{C(P)}}\). A winning coalition \(S\in {{\mathbb {W}}_{C(P)}}\) is said to be stable if and only if there exists a ∈ A such that (i) S a = S, and (ii) for all i ∈ S and all b ∈ A∖{a}, if b ≿i a, then b∉A C(P). The set of all stable coalitions in C(P) is denoted by \({\overline {{{\mathbb {W}}_{C(P)}}}}\). □

An alternative that has possibility to be selected as the final choice by some stable coalitions is called a stable alternative, and the set of all stable alternatives, that is, the set

$$\displaystyle \begin{aligned} \{a\in A\ | \exists S\in{{\mathbb{W}}_{C(P)}}, S_a=S \land (\forall i\in S, \forall b\in A{\backslash}\{a\}, b \succsim_i a \to b\notin{A_{C(P)}})\}, \end{aligned}$$

is denoted by \({\overline {{A_{C(P)}}}}\).

The next proposition validates that the number of stable alternatives in a committee with a proper simple game is at most one.

Proposition 1 (Coincidence of Final Choice (Yamazaki et al. 2000))

Consider a committee C with permissible range P, that is, C(P), where \(C={(N, {\mathbb {W}}, A, (\succsim _i)_{i\in N})}\) , and assume that the simple game \((N, {\mathbb {W}})\) is proper. Then, the number of elements in \({\overline {{A_{C(P)}}}}\) is one, at most. □

Consider a committee \(C={(N, {\mathbb {W}}, A, (\succsim _i)_{i\in N})}\) and an alternative x ∈ A. For i ∈ N, \(P^x_i\) denotes the set \(\{\succsim \in L(A)\ |\ ({\mathit {max}} \succsim ) \succsim _i x\}\), which expresses that the DM i’s permissible alternatives are those that he/she equally or more prefers to x in terms of DM i’s preference ≿i. P x denotes the list \((P^x_i)_{i\in N}\) of \(P^x_i\) for each i ∈ N, and C(P x) is a committee with permissible range P x. In this case, in particular, all DMs have x as one of their permissible alternatives.

The next proposition gives a characterization of the stable alternatives in a committee C with permissible range P x with respect to the core Core(C) of C.

Proposition 2 (Yamazaki et al. 2000)

Consider a committee \(C=(N, {\mathbb {W}}, A, (\succsim _i)_{i\in N})\) , and assume that the simple game \((N, {\mathbb {W}})\) is proper. For an alternative x ∈ A, it is satisfied that \({\overline {A_{C(P^x)}}}=\{x\}\) if and only if x ∈Core(C). □

3 Consensus and Consensus Building

This section proposes mathematical definitions of consensus and consensus building.

A negotiation process in a group decision situation is expressed by a sequence of DMs’ permissible ranges in a committee. It is assumed in this work that the negotiation process starts from the state, called the status quo, in which each DM agrees only to his/her best alternative. In the process, however, each DM may change his/her permissible range and may come to agree to an alternative which is not the best for him/her.

Definition 9 (Negotiation Processes in Committees)

Consider a committee \(C={(N, {\mathbb {W}}, A, (\succsim _i)_{i\in N})}\). A negotiation process in C is a sequence (P t)t ∈ T of DMs’ permissible ranges \(P^t=(P^t_i)_{i\in N}\) at time t for each t ∈ T, where T = {0, 1, 2, …}. \(P^0=(P^0_i)_{i\in N}=(\{\succsim _i\})_{i\in N}\) is called status quo. □

Consensus is defined as a state with a stable alternative in a negotiation process, and a sequence of DMs’ permissible ranges from the status quo to the consensus is called consensus building.

Definition 10 (Consensus and Consensus Building)

Consider a committee \(C={(N, {\mathbb {W}}, A, (\succsim _i)_{i\in N})}\). A negotiation process (P t)t ∈ T in C is said to reach consensus at t ∗∈ T on x ∈ A if and only if either (i) t ∗ = 0 and \({\overline {A_{C(P^0)}}}=\{x\}\) or (ii) t ∗ > 0, \({\overline {{A_{C(P^t)}}}}={\emptyset }\) for all t such that 0 ≤ t < t ∗, and \({\overline {{A_{C(P^{t^*})}}}}=\{x\}\). In either cases, the sequence \((P^0, P^1,\ldots , P^{t^*})\) is called the consensus building on x in C, and x is said to be consensus through the sequence \((P^0, P^1,\ldots , P^{t^*})\) in C. □

The next example shows that the consensus may change depending on the consensus building process.

Example 1

Consider a committee \(C={(N, {\mathbb {W}}, A, (\succsim _i)_{i\in N})}\) such that N = {1, 2, 3}; \({\mathbb {W}}=\{\{1, 2, 3\}\}\); A = a, b, c; ≿1 = [a, b, c]; ≿2 = [b, c, a]; ≿3 = [c, b, a]. Note that the simple game \((N, {\mathbb {W}})\) is unanimous. Consider, moreover, the following permission ranges of DMs:

$$\displaystyle \begin{aligned} P_{11}=\{[a,b,c]\}&; P_{12}=\{[a,b,c],[b,a,c]\}; P_{13}=\{[a,b,c],[b,c,a],[c,a,b]\}; \\ P_{21}=\{[b,c,a]\}&; P_{22}=\{[b,c,a],[c,b,a]\}; P_{23}=\{[b,c,a],[c,a,b],[a,b,c]\}; \\ P_{31}=\{[c,b,a]\}&; P_{32}=\{[c,b,a],[b,c,a]\}; P_{33}=\{[c,b,a],[b,a,c],[a,c,b]\}. \\ \end{aligned}$$

Then, (i) P 0 = (P 11, P 21, P 31), P 1 = (P 12, P 21, P 31), P 2 = (P 12, P 21, P 32) is a consensus building on b ∈ A in C; in fact, \({\overline {A_{C(P^0)}}}={\overline {A_{C(P^1)}}}={\emptyset }\) and \({\overline {A_{C(P^2)}}}=\{b\}\); (ii) P 0 = (P 11, P 21, P 31), P 1 = (P 12, P 21, P 31), P 2 = (P 13, P 21, P 31), P 3 = (P 13, P 22, P 31) is a consensus building on c ∈ A in C; and (iii) P 0 = (P 11, P 21, P 31), P 1 = (P 11, P 21, P 32), P 2 = (P 11, P 21, P 33), P 3 = (P 11, P 22, P 33), P 4 = (P 11, P 23, P 33) is a consensus building on a ∈ A in C.

4 Consensus and Core

The following proposition gives a relationship between consensus building in a committee and the core of the committee.

Proposition 3

Consider a committee \(C={(N, {\mathbb {W}}, A, (\succsim _i)_{i\in N})}\) . Assume that the simple game \((N, {\mathbb {W}})\) is unanimous. Then, for an alternative x ∈ A, there exists a negotiation process (P t)t ∈ T in C which reaches consensus at t ∗∈ T on x ∈ A for some t ∗∈ T if and only if x ∈Core(C). □

Proof

Consider a committee \(C={(N, {\mathbb {W}}, A, (\succsim _i)_{i\in N})}\), where the simple game \((N, {\mathbb {W}})\) is unanimous, and an alternative x ∈ A.

First, assume that there exists a negotiation process (P t)t ∈ T in C which reaches consensus at t ∗∈ T on x ∈ A for some t ∗∈ T. Then, we immediately have from the definition of consensus (Definition 10) that \({\overline {{A_{C(P^{t^*})}}}}=\{x\}\). Since the simple game \((N, {\mathbb {W}})\) is unanimous, that is, \({\mathbb {W}}=\{N\}\), it is implied that N x = N, that is,

$$\displaystyle \begin{aligned} \mbox{for all }i\in N, x\in{\mathit{max}} P_i, \end{aligned} $$
(11.1)

and

$$\displaystyle \begin{aligned} \mbox{for all }i\in N\mbox{ and all }b\in A{\backslash}\{x\}\mbox{, if }b \succsim_i x\mbox{, then }b\notin{A_{C(P)}} \end{aligned} $$
(11.2)

(see Definition 8).

If x∉Core(C), then the unanimity of the simple game \((N, {\mathbb {W}})\) implies that

$$\displaystyle \begin{aligned} \mbox{there exists }b\in A{\backslash}\{x\}\mbox{ such that for all }i\in N, b \succsim_i x. \end{aligned} $$
(11.3)

The alternative b in (11.3) satisfies that for all i ∈ N, b ∈maxP i, which implies b ∈ A C(P), by (11.1) and the assumption on DMs’ permissible ranges (see Definition 7). Equation (11.3) and b ∈ A C(P) imply the existence of b ∈ A∖{x} such that for all i ∈ N, b ≿i x, and b ∈ A C(P), which contradicts (11.2).

Thus, if there exists a negotiation process (P t)t ∈ T in C which reaches consensus at t ∗∈ T on x ∈ A for some t ∗∈ T, then we have that x ∈Core(C).

Second, assume that x ∈Core(C). If \({\overline {A_{C(P^0)}}}=\{x\}\), where \(P^0=(P^0_i)_{i\in N}=(\{\succsim _i\})_{i\in N}\), then the negotiation process (P t)t ∈ T reaches consensus on x ∈ A at t ∗ = 0 (see Definition 10). That is, the sequence (P 0) is the consensus building on x.

If \({\overline {A_{C(P^0)}}}\neq \{x\}\), then consider the sequence (P 0, P 1), where \(P^0=(P^0_i)_{i\in N}=(\{\succsim _i\})_{i\in N}\) and \(P^1=(P^1_i)_{i\in N}=(P^x_i)_{i\in N}\). Then, we have, by Proposition 2 and the assumption that x ∈Core(C), that \({\overline {A_{C(P^1)}}}={\overline {A_{C(P^x)}}}=\{x\}\), which implies that the negotiation process (P t)t ∈ T reaches consensus on x ∈ A at t ∗ = 1.

Thus, if x ∈Core(C), then there exists a negotiation process (P t)t ∈ T in C which reaches consensus at t ∗∈ T on x ∈ A for some t ∗∈ T. ■

This proposition implies that for an alternative in a committee, being a consensus through some sequences is equivalent to be an element of the core of the committee.

5 Consensus and Nash Equilibrium

Consider a committee \(C={(N, {\mathbb {W}}, A, (\succsim _i)_{i\in N})}\). Then, we can define a game \(G_C={(N, ({\mathbb {P}}_i)_{i\in N}, (\succsim ^{\prime }_i)_{i\in N})}\) in normal form by defining \(\succsim ^{\prime }_i\) for each i ∈ N based on C as follows: for P = (P i)i ∈ N and \(P'=(P^{\prime }_i)_{i\in N}\) in \({\mathbb {P}}=\prod _{i\in N}{\mathbb {P}}_i\), \(P \succsim ^{\prime }_i P'\), if and only if either

  • \({\overline {{A_{C(P)}}}}=\{a\}\), \({\overline {{A_{C(P')}}}}=\{b\}\), and a ≿i b,

  • \({\overline {{A_{C(P)}}}}={\overline {{A_{C(P')}}}}={\emptyset }\),

  • \({\overline {{A_{C(P)}}}}=\{a\}\), \({\overline {{A_{C(P')}}}}={\emptyset }\), and a ∈maxP i, or

  • \({\overline {{A_{C(P)}}}}={\emptyset }\), \({\overline {{A_{C(P')}}}}=\{a\}\), and \(a\notin {\mathit {max}} P^{\prime }_i\).

Among these four conditions, fourth one cannot hold for a committee \(C={(N, {\mathbb {W}}, A, (\succsim _i)_{i\in N})}\) such that the simple game \((N, {\mathbb {W}})\) is unanimous, that is, \({\mathbb {W}}=\{N\}\), because we always have \(a\in {\mathit {max}} P^{\prime }_i\) if \({\overline {{A_{C(P')}}}}=\{a\}\). Moreover, if the simple game \((N, {\mathbb {W}})\) in the committee \(C={(N, {\mathbb {W}}, A, (\succsim _i)_{i\in N})}\) is unanimous, then we have the next lemma, which is used in the proof of Proposition 4.

Lemma 1

Consider P = (P i)i ∈ N = (P i, P −i) and \(P'=(P^{\prime }_i)_{i\in N}=(P^{\prime }_i, P^{\prime }_{-i})\) in \(\prod _{i\in N}{\mathbb {P}}_i\) , and assume that \(P_{-i}=P^{\prime }_{-i}\) . If \({\mathit {max}} P^{\prime }_i\subseteq {\mathit {max}} P_i\) , then \({A_{C(P')}}\subseteq {A_{C(P)}}\). □

Proof

For x ∈ A, consider the sets N x and \(N^{\prime }_x\), which are defined as {i ∈ N|x ∈maxP i} and \(\{i\in N | x\in {\mathit {max}} P^{\prime }_i\}\), respectively. Since it is assumed that \(P_{-i}=P^{\prime }_{-i}\), we have \(N^{\prime }_x\subseteq N_x\) from \({\mathit {max}} P^{\prime }_i\subseteq {\mathit {max}} P_i\). Therefore, it is satisfied that if \(N^{\prime }_x=N\), then N x = N, which implies by the unanimity of the simple game \((N, {\mathbb {W}})\) that \({A_{C(P')}}=\{a\in A | N^{\prime }_a=N\}\subseteq \{a\in A | N_a=N\}={A_{C(P)}}\). ■

The next proposition shows that if a sequence \((P^0, P^1,\ldots , P^{t^*})\) is consensus building on some alternative x ∈ A in a committee \(C={(N, {\mathbb {W}}, A, (\succsim _i)_{i\in N})}\), then \(P^{t^*}=(P^{t^*}_i)_{i\in N}\in \prod _{i\in N}{\mathbb {P}}_i\) is a Nash equilibrium in the game \(G_C={(N, ({\mathbb {P}}_i)_{i\in N}, (\succsim ^{\prime }_i)_{i\in N})}\), where \(P=(P_i)_{i\in N}\in \prod _{i\in N}{\mathbb {P}}_i\) is said to be a Nash equilibrium (Nash 1950, 1951) in G, if and only if \((P_i, P_{-i}) \succsim ^{\prime }_i (P^{\prime }_i, P_{-i})\) for all i ∈ N and all \(P^{\prime }_i\in {\mathbb {P}}_i\), where \(P_{-i}=(P_1, P_2,\ldots , P_{i-1}, P_{i+1},\ldots , P_n)\in \prod _{j\neq i}{\mathbb {P}}_j\).

Proposition 4

Consider a committee \(C={(N, {\mathbb {W}}, A, (\succsim _i)_{i\in N})}\) and the game \(G_C={(N, ({\mathbb {P}}_i)_{i\in N}, (\succsim ^{\prime }_i)_{i\in N})}\) . Assume that the simple game \((N, {\mathbb {W}})\) is unanimous. Then, for \(P=(P_i)_{i\in N}\in \prod _{i\in N}{\mathbb {P}}_i\) , if \({\overline {{A_{C(P)}}}}=\{x\}\) for some x ∈ A in C, then P is Nash equilibrium in G C. □

Proof

It suffices to verify \(P\succsim ^{\prime }_i P'\) for i ∈ N and \(P'=(P^{\prime }_i)_{i\in N}=(P^{\prime }_i, P^{\prime }_{-i})\in \prod _{i\in N}{\mathbb {P}}_i\) such that \(P_{-i}=P^{\prime }_{-i}\). Consider the following three cases: (a) \({\mathit {max}} P^{\prime }_i\subseteq {\mathit {max}} P_i\) and \(x\in {\mathit {max}} P^{\prime }_i\), (b) \({\mathit {max}} P^{\prime }_i\subseteq {\mathit {max}} P_i\) and \(x\notin {\mathit {max}} P^{\prime }_i\), and (c) \({\mathit {max}} P^{\prime }_i\supseteq {\mathit {max}} P_i\).

  1. (a)

    Cases \({\mathit {max}} P^{\prime }_i\subseteq {\mathit {max}} P_i\) and \(x\in {\mathit {max}} P^{\prime }_i\):

    First, \(x\in {\mathit {max}} P^{\prime }_i\) and \(P_{-i}=P^{\prime }_{-i}\) implies that

    $$\displaystyle \begin{aligned} \mbox{if }N_x=N\mbox{ then }N^{\prime}_x=N, \end{aligned} $$
    (11.4)

    where for x ∈ A, N x and \(N^{\prime }_x\) are defined as the sets {j ∈ N|x ∈maxP j} and \(\{j\in N | x\in {\mathit {max}} P^{\prime }_i\}\), respectively.

    From \({\mathit {max}} P^{\prime }_i\subseteq {\mathit {max}} P_i\) and Lemma 1, we have \({A_{C(P')}}\subseteq {A_{C(P)}}\), which implies that

    $$\displaystyle \begin{aligned} \mbox{if }b\notin{A_{C(P)}}\mbox{ then }b\notin{A_{C(P')}}. \end{aligned} $$
    (11.5)

    Then, from the unanimity of the simple game \((N, {\mathbb {W}})\),

    $$\displaystyle \begin{aligned} {\overline{{A_{C(P)}}}}=\{x\} &\Rightarrow N_x=N\ \mbox{and}\ \forall i\in N, \forall b\in A{\backslash}\{x\}, (b \succsim_i x \to b\notin{A_{C(P)}}) \\ &\Rightarrow N^{\prime}_x=N\ \mbox{and}\ \forall i\in N, \forall b\in A{\backslash}\{x\}, (b \succsim_i x \to b\notin{A_{C(P')}})\\ &\qquad \mbox{(by (11.4) and (11.5))} \\ &\Rightarrow {\overline{{A_{C(P')}}}}=\{x\}\quad \mbox{(by Proposition 1)} \end{aligned}$$

    Thus, in this case, \(P\succsim ^{\prime }_i P'\) holds by the definition of \(\succsim ^{\prime }_i\).

  2. (b)

    Cases \({\mathit {max}} P^{\prime }_i\subseteq {\mathit {max}} P_i\) and \(x\notin {\mathit {max}} P^{\prime }_i\):

    Since \(x\notin {\mathit {max}} P^{\prime }_i\) implies \(i\notin N^{\prime }_x\), we have \(x\notin {\overline {{A_{C(P')}}}}\).

    Assume that \({\overline {{A_{C(P')}}}}=\{y\}\) for some y ∈ A such that y ≠ x. Then, we have to have \(N^{\prime }_y=N\), where \(N^{\prime }_y\) is defined as the set \(\{j\in N | y\in {\mathit {max}} P^{\prime }_j\}\). \(N^{\prime }_y=N\) implies \(y\in {A_{C(P')}}\), and y ∈ A C(P) follows by \({\mathit {max}} P^{\prime }_i\subseteq {\mathit {max}} P_i\) and Lemma 1.

    \(N^{\prime }_y=N\) implies \(y\in {\mathit {max}} P^{\prime }_i\), too. Then, we have y ≿i x under the completeness of ≿i. In fact, if we do not have y ≿i x, then we need to have x ≿i y by the completeness of ≿i. By the assumption on DMs’ permissible ranges (see Definition 7), x ≿i y and \(y\in {\mathit {max}} P^{\prime }_i\) imply \(x\in {\mathit {max}} P^{\prime }_i\), which contradicts the assumption that \(x\notin {\mathit {max}} P^{\prime }_i\).

    We see that y ∈ A satisfies that y ≿i x and y ∈ A C(P), which contradict \({\overline {{A_{C(P)}}}}=\{x\}\) and y ≠ x.

    Therefore, \({\overline {{A_{C(P')}}}}=\{y\}\) for some y ∈ A such that y ≠ x cannot be satisfied, and then, we have \({\overline {{A_{C(P')}}}}={\emptyset }\).

    Thus, in this case, \(P\succsim ^{\prime }_i P'\) holds by the definition of \(\succsim ^{\prime }_i\).

  3. (c)

    Cases \({\mathit {max}} P^{\prime }_i\supseteq {\mathit {max}} P_i\):

    It suffices to show that \({\overline {{A_{C(P')}}}}=\{y\}\) for some y ∈ A such that y ≠ x cannot be satisfied, because this implies from Proposition 1 that either \({\overline {{A_{C(P')}}}}={\emptyset }\) or \({\overline {{A_{C(P')}}}}=\{x\}\), and thus, we have \(P\succsim ^{\prime }_i P'\).

    Assume that \({\overline {{A_{C(P')}}}}=\{y\}\) for some y ∈ A such that y ≠ x. If it is satisfied that x ≿i y and \(x\in {A_{C(P')}}\), then it contradicts \({\overline {{A_{C(P')}}}}=\{y\}\) by the definition of \({\overline {{A_{C(P')}}}}\).

    If we do not have x ≿i y, then we need to have y ≿i x by the completeness of ≿i. The assumption \({\overline {{A_{C(P)}}}}=\{x\}\) means that x ∈ A C(P), which implies x ∈maxP i. By the assumption on DMs’ permissible ranges (see Definition 7), y ≿i x and x ∈maxP i imply y ∈maxP i.

    \({\overline {{A_{C(P')}}}}=\{y\}\) implies \(y\in {A_{C(P')}}\), which means \(N^{\prime }_y=N\), where \(N^{\prime }_y\) is defined as the set \(\{j\in N | y\in {\mathit {max}} P^{\prime }_j\}\). Since \(P_{-i}=P^{\prime }_{-i}\), we have that y ∈ A C(P) from y ∈maxP i.

    y ≿i x and y ∈ A C(P) contradict \(x\in {\overline {{A_{C(P)}}}}\), and thus, we have x ≿i y.

    From the assumption of \({\mathit {max}} P^{\prime }_i\supseteq {\mathit {max}} P_i\) and Lemma 1, we have \({A_{C(P')}}\supseteq {A_{C(P)}}\), which implies \(x\in {A_{C(P')}}\), because x ∈ A C(P) follows \({\overline {{A_{C(P)}}}}=\{x\}\).

    Consequently, we have both x ≿i y and \(x\in {A_{C(P')}}\).

■

By this proposition, we see that a stable alternative, and consequently, consensus (see Definition 10), in a committee is actually stable when we see the committee as a strategic decision situation.

6 Existence of Consensus

This section deals with the existence of consensus.

Consider a committee \(C={(N, {\mathbb {W}}, A, (\succsim _i)_{i\in N})}\). Then, we can define max(≿i)i ∈ N as a set {x ∈ A |∃i ∈ N, x = max ≿i} of alternatives. Then, we have the next proposition.

Proposition 5

Consider a committee \(C={(N, {\mathbb {W}}, A, (\succsim _i)_{i\in N})}\) , and assume that the simple game \((N, {\mathbb {W}})\) of the committee \(C={(N, {\mathbb {W}}, A, (\succsim _i)_{i\in N})}\) is unanimous. Then, we have that ∅ ≠ max(≿i)i ∈ N ⊆Core(C). □

Proof

From the argument in the proof of Proposition 6 in Appendix, it is satisfied, in the setting of this work, that Core(C) = {x ∈ A|∀b ∈ A∖{x}, ∃i ∈ N, x ≻i b}.

If x ∈max(≿i)i ∈ N, then for some i ∈ N, \(x=\max \succsim _i\), that is, ∃i ∈ N, ∀b ∈ A∖{x}, x ≻i b, which logically implies that ∀b ∈ A∖{x}, ∃i ∈ N, x ≻i b. Thus, we have max(≿i)i ∈ N ⊆Core(C).

We have max(≿i)i ∈ N is non-empty, because max ≿i is uniquely determined for each i ∈ N from the assumption that ≿i is a linear ordering for each i ∈ N. ■

Proposition 5 together with Proposition 3 implies that in a committee \(C={(N, {\mathbb {W}}, A, (\succsim _i)_{i\in N})}\) with a unanimous simple game \((N, {\mathbb {W}})\), there always exists a negotiation process (P t)t ∈ T in C which reaches consensus at t ∗ on x for some t ∗∈ T and some x ∈ A.

The next example shows that max(≿i)i ∈ N = Core(C) is not always true.

Example 2

Consider a committee \(C={(N, {\mathbb {W}}, A, (\succsim _i)_{i\in N})}\) such that N = {1, 2, 3}; \({\mathbb {W}}=\{\{1, 2, 3\}\}\); A = {a, b, c, d}; ≿1 = [b, a, c, d], ≿2 = [c, a, d, b], ≿3 = [d, a, b, c]. Then, we see that max(≿i)i ∈ N = {b, c, d} and Core(C) = {a, b, c, d}. In fact, a ∈ A is not dominated by any one of the other alternatives. □

7 Conclusions

This work proposed a new model of consensus and a definition of consensus building on the basis of the frameworks in Peleg (1984) and Yamazaki et al. (2000) (Sect. 11.3) and verified some relationships between consensus and core (Proposition 3), between consensus and Nash equilibrium (Proposition 4), and the existence of consensus (Proposition 5). More, Proposition 6 in Appendix indirectly showed a relationship between consensus and efficiency.

Through these propositions, we obtained the following insights on consensus and consensus building:

  • For an alternative in committee, being a consensus through some sequences is equivalent to be an element of the core of the committee (Proposition 3).

  • Consensus is stable in the sense that it constitutes a Nash equilibrium in the game in normal form, which describes the strategic aspect of the committee (Proposition 4).

  • There always exists a negotiation process which reaches consensus (Propositions 3 and 5).

  • Consensus is efficient (Propositions 3 and 6).

This work treated stability of consensus as a state in a group decision situation in Sect. 11.5. Instead, we should investigate stability of consensus building as a negotiation process in future research opportunities. This requires modelling a group decision situation as a game in extensive form game (Eichberger 1993; Osborne and Rubinstein 1994) or a graph model within the framework of the Graph Model for Conflict Resolution (Fang et al. 1993; Yasui and Inohara 2007). In order to generalize the existence result in Proposition 5, we need to think of Nakamura’s theorem (Nakamura 1979) on the relationship between the non-emptiness of cores of committees and the cardinality of the set of all alternatives. Moreover, strategic information exchange should be involved in the model, and the models by Gibbard (1973) and Satterthwaite (1975) and that by Inohara (2002) may be useful.