A Theory of Farsightedness in Committee Games

We study the committee decision making process using game theory. A committee here refers to any group of people who have to select one option from a given set of alternatives under a specified rule. Shenoy (1980) introduced two solution concepts, namely, the one-core and a version of bargaining set for committee games. Shortcomings of these solutions concepts are raised and discussed in this paper.These shortcomings are resolved by introducing two new solutions concepts: the farsighted one-core and the bargaining set revised, inspired by an idea of farsightedness initially defined by Rubinstein (1980). It is shown that the farsighted one-core is always non-empty and is better than the one-core. In a well-specified sense, the bargaining set revised is also better than the bargaining set as defined by Shenoy (1980) and it is always non-empty for simple committee games with linear preferences. Other attractive properties are also proved.


Introduction
Our game model is the one considered by Shenoy [1], committee game that generalizes the voting model introduced by von Neumann and Morgenstern [2] under the name of simple game.A committee game consists in any finite group  of persons who have to pick one option from the finite given set of outcomes  through a voting rule V by which the committee arrives at a decision.The rule V is designed such that the decision of the committee will consist of a unique outcome.Any player is allowed to suggest any alternative for consideration by the committee and players get their payoffs only when the committee has made a decision.In such a social choice context, the question generally asked is how a player should behave or should vote when solicited to join a coalition in order to decide over a status quo.Another relevant issue is to determine what could be a suitable choice of a given player  if he is given the opportunity to introduce a motion.
The core is a solution concept in which any player is recommended to vote for  against  whenever he strictly prefers  to  (i.e.,   () >   ()) (  is the player utility function; instead of considering utilities vectors one could consider that each member of the committee has a preference relation which is a weak order on the set of all outcomes, thus yielding a preference profile) if  is opposed to .Furthermore, a committee member should propose an outcome  if it is the (or one of his) best element in the core.An outcome  belongs to the core if it is undominated, that is, there does not exist another outcome , a coalition  powerful on  and all members of which are strictly better off at  than at .This behavioral pattern of the core has been criticized by Shenoy [1] who argued that a player who is making a proposal does not cooperate in any effort to dominate the proposal.In other words such a player cares about undominated outcomes via coalitions not containing him and picks only maximal ones.This yields the definition of the one-core.Unfortunately, the one-core might be empty even if players' preferences are strict.Furthermore, we show through a simple example of 3-person committee game that players are not farsighted while making proposals under one-core behavioral pattern.We resolve this lack of farsightedness by introducing another solution concept, the farsighted one-core for committee games.It is shown that this new solution concept is better than the one-core.Moreover, when preferences are strict orders, the farsighted one-core of any simple committee game is always nonempty.
Another contribution of Shenoy [1] is the introduction of a bargaining set for committee games.The concept of bargaining set was first introduced by Aumann and Maschler [3] in the context of games with side payments.They defined several kinds of bargaining sets.These sets were generalized for games without side payments and studied by Peleg [4], Billera [5,6], D' Aspremont [7], and Asscher [8].Since then, several other modifications of the bargaining set have been studied in different contexts.The bargaining set introduced by Shenoy [1] is relevant for committee games.Like other bargaining sets, it is based on objections and counter-objections.A proposal is said to be -stable if every objection has a counter-objection.Let M be the set of all -stable proposals.A -stable proposal (, ) is said to be maximal if  is at least as good as  for all  such that (, ) ∈ M. The bargaining set M is the set of all maximal -stable proposals.The Shenoy bargaining set can be empty.We revise this solution concept by introducing the revised Shenoy bargaining set which is proved to be better than the latter.Moreover, when indifference is not allowed in individual rankings, the farsighted bargaining set of any simple committee game introduced herein is nonempty.
The rest of the paper is organized as follows.Section 2 is devoted to the model and preliminaries.In Section 3, we define the farsighted one-core and we prove that it is better than the one-core.Moreover, we prove that the farsighted one-core of any committee game is always nonempty, provided that individual preferences are linear orders.A comparison of the farsighted one-core with other solution concepts is conducted therein.In Section 4, we revise Shenoy bargaining set by reconsidering the definition of objection and counter-objection.It is shown that the new bargaining set improves on the latter defined by Shenoy for committee games.Conclusion, which is Section 5, ends the paper.

The Setting and Preliminaries
2.1.The Model.Throughout the paper the set of players, that is, the committee, is denoted by  = {1, 2, . . ., }; the finite set of candidates or outcomes is .It is assumed that  has at least three elements.Nonempty subsets of  are called coalitions and the set of all coalitions of  is denoted by 2  ; || stands for the cardinality of any set .The preference relation of any player on  is a weak order (reflexive and transitive relation).If ⪰  denotes the preference of player  and  and  are two outcomes, ⪰   means that, according to ,  is at least as good as .≻   means that  strictly prefers  to  and ∼   means that  is indifferent between  and .A profile  is a collection of individual preferences,  = (  ) ∈ .
The rules by which the committee members arrive at a decision are called the characteristic function which is a mapping V : 2  → P(), where P() designates the set of subsets of .For any coalition , V() denotes the subset of outcomes that coalition  can realize if the decision is unanimous in .This means that, at any time, an outcome  becomes the final outcome of the game, whenever a coalition  such that  ∈ V() asks for the adoption of .It is assumed that V satisfies the following conditions: Condition (C1) is the well-known monotonicity condition; (C2) means that the whole committee members can enforce any alternative.Condition (C3) ensures that the committee decision consists of at most one outcome.The tuple Γ = (, , V, ) is called an (ordinal) -person committee game. could also be replaced with a utility vector  = (  ), where   :  → R denotes the real-valued ordinal utility function of player .Here, utility is assumed to be nontransferable and interpersonal comparison of utilities has no meaning.
The committee aims at choosing one option from the set  of outcomes.The members of the committee are considered to be situated in one room.As in Shenoy [1], we are primarily concerned with small committees that arrive at a decision after lengthy deliberations.In this respect the model considered here differs fundamentally from the theory of elections where the decision makers (the players) are numerous and spread out extensively.Let us remark that the committee game model fits very well into the more general model of social environments.A social environment is described by a tuple (, , ( →  ) ∈2  , (⪯) ∈ ), where  is the set of players,  the set of outcomes, and { →  } are effectiveness relations defined on .After the move of  to  another coalition  might move to  and so on.Social environments have been considered in many works in the literature including Chwe [9], Xue [10,11], Suziki and Muto [12], Béal et al [13], and Kenfack and Tchantcho [14].In a committee game, if  ∈ V() then  can enforce .Note on the other hand that committee games generalize the model of simple games.In this respect, a committee game Γ = (, , V, ) is said to be simple if ∀ ∈ 2  , V() = 0, or V() = .If V() = 0 then  is a losing coalition and if V() = ,  is a winning coalition.In a simple committee game, a coalition is a minimal winning coalition if it is a winning coalition and if every proper subset of  is a losing coalition.In such a game, the set of winning coalitions is denoted by W while the set of minimal winning is denoted by W  .Player  is said to be a dummy player if  ∉ ⋃ ∈W  .
It is assumed that a particular outcome say  0 is the initial status quo. 0 will be the decision of the committee if it cannot agree on any other outcome or if it specifically picks  0 to be the final decision.There is no agenda (a linear order on  specifying in which order candidates are confronted) and any member  of the committee is allowed to suggest any alternative  at any time for consideration by the committee in the form of a proposal (, ).The game ends at an option  * such that there is no credible contestation.Such an option is said to be stable.

Recall of Dominance Relations.
We give below the definition of 1-dominance that is suitable for committee games.This is a transposition for committee games of the dominance introduced by Rubinstein [15] for social decision systems.According to 1-dominance, a player  should participate in the elimination of motion  for  only if any subsequent deviation from  to  by another coalition  does not worsen the utility of  relative to .Definition 1.Let Γ = (, , V, (⪰  ) ∈ ) be a committee game, ,  ∈ , and  a coalition.
According to the 1-dominance, a coalition will refrain from blocking an alternative, say , by voting for , if  may be blocked later on by another coalition voting for, say,  against , if it turns out that  is worse than  for some of its members.It is obvious that with respect to behavioral purposes, the 1-dominance improves the (classical) dominance recalled below.
and for all  ∈ , ≻  ; (2)  dominates  denoted by  dom  if there exists  ∈ 2  such that  dom  .
It follows from this definition that the rational behavior underlying the core prescribes that a player should vote for an alternative  against another alternative  whenever he prefers  to .
As a comparison between these two dominances, one could verify that for any ,  ∈ , if  1-dom  then  dominates  but the converse is not true.In the next section, we will use the 1-dominance to build our first solution concept: the farsighted one-core.

The Farsighted One-Core of a Committee Game
We introduce this section with the definition of the farsighted one-core, based on the behavioral considerations captured by the 1-dominance relation.Let us denote by  = {(, ) :  ∈ ,  ∈ } the set of all proposals, Ŝ = {(, ) ∈  :  is not 1-dominated via a coalition  ⊆  \ {}},   = {(, ) ∈ Ŝ : ⪰   for all (, ) ∈ Ŝ }.Ŝ represents the set of proposals made by  that are not 1-dominated assuming player 's noncooperation in any effort to 1-dominate his proposal and   represents the maximal proposals in the set Ŝ .Definition 3. Let Γ = (, , V, (⪰  ) ∈ ) be a committee game.The farsighted one-core of Γ denoted by F(Γ) is defined by Intuitively, the farsighted one-core consists of all (maximal) proposals which are not 1-dominated assuming that the player who makes the proposal does not cooperate in any effort to 1-dominate the proposal.For obvious reasons, assuming all proposals in Ŝ to be equally stable, player  picks only the maximal ones.
Before giving some properties of the farsighted one-core, we shall prove through a simple example that the farsighted one-core overcomes a myopic shortcoming observed in the Shenoy one-core.Before that, let us recall the definition of the one-core as introduced by Shenoy [1].
The core was initially studied explicitly by Gillies [16] and Shapley [17] for transferable utilities games.It is defined as follows.
With respect to the core behavioral pattern, a member  of the committee to whom it is given the opportunity to make a proposal should propose the candidate of the core which guarantees a maximal satisfaction.The main shortcoming of the core is its existence, many interesting committee games have empty cores.
The one-core is a solution concept introduced by Shenoy [1] that results from a small modification in the definition of the core.The modification is motivated by behavioral considerations.Indeed, being in the core means not being dominated.The modification provided by Shenoy [1] is that it is better for such a player to propose, instead of that core candidate, a (maximal) proposal which is not dominated assuming that the player himself does not cooperate in any effort to dominate the proposal.Formally, for each  ∈ , define the following.Ĉ = {(, ) ∈  : is not dominated via any  ⊆  \ {}}, the set of proposals made by  that are undominated assuming player 's noncooperation in any effort to dominate his proposal, and   = {(, ) ∈ Ĉ : ⪰   for all (, ) ∈ Ĉ }, the set of maximal (best) proposals in Ĉ .The one-core consists of all (maximal) proposals which are undominated assuming that the player who makes the proposal does not cooperate in any effort to dominate the proposal.
(i) If  is proposed (either by 2 or by 3), then both  and  will be put to vote with the players voting for one of the two motions.It is not in the interest of 2 to vote for .Indeed, if  becomes the new status quo 1 and 3 will certainly enforce the adoption of  as the final outcome.But  is the worst candidate for 2. Thus even if  is opposed to ,  will not win.
(ii) If  is proposed (by 3), then it is obvious that it will be eliminated since the only coalition able to enforce  is 13 and player 1, who made the initial proposal , will not vote for ; hence  will not be defeated!(iii) For the same reason, if  is proposed it will be defeated.
Finally, as we can see, if 1 proposes , then  will be elected.This shows that if 1 is given the possibility to propose it will be better for him to propose  instead of , as recommended by the one-core.
Thanks to Example 6 above, it is obvious that neither the farsighted one-core includes the one-core nor the one-core includes the farsighted one-core.
The Condorcet solution was first defined by Condorcet [18] and rediscovered independently by Dodgson [19].It is defined as follows.
Definition 8. Let Γ be a committee game.A Condorcet solution of Γ is any  ∈  that dominates every other outcome in ; that is, for all  ∈  \ {},  dom .
We show in the following result that if the Condorcet solution exists, then, with respect to the farsighted one-core, every player should propose it.Proposition 9. Let Γ be a committee game such that the Condorcet solution  exists.Then the farsighted one-core is given by F(Γ) = {(1, ), (2, ), . . ., (, )}.
An important solution concept for social environments in general and committee games in particular is the Chwe [9] largest consistent set.It describes situations where deviating coalitions anticipate the ultimate consequences of their initial move.
(3) The largest consistent set LCS(Γ) is the unique consistent set that includes any other consistent set.
In order to compare the farsighted one-core and the largest consistent set, let us consider the following example due to Chwe [9, page 321].
With this Example 11, it is clear that neither the farsighted one-core includes the largest consistent set nor the largest consistent set includes the farsighted one-core.However, when preferences are linear orders, the result below highlights one advantage of the farsighted one-core over the largest consistent set.Proposition 12. Let Γ = (, , V, (≻  ) ∈ ) where || is odd and V the majority rule.Then, for all  ∈ , for all  ∈ , (, ) ∈ F(Γ) ⇒ ⪰   for all  ∈ LCS(Γ).
Proof.It is known that when preferences are linear orders, the set of not 1-dominated outcomes includes the largest consistent set (see Chwe [9, Proposition 7, page 321]).Moreover, for each , the set Ŝ includes the set of not 1-dominated outcomes.That is, LCS(Γ) ⊂ Ŝ (Γ) for all player  and the result follows.
Nonemptiness of the Farsighted One-Core.It is well known that the Condorcet solution has a very strong stability requirement.As a consequence of this, it does not always exist.
As said above, the core of a committee game might be empty.If the core is nonempty, then   is nonempty (implying that the one-core is nonempty) for each player .Furthermore, any player  prefers  to  whenever  ∈   and  ∈ C 1 (Γ).Shenoy [1] illustrated clearly the advantage of the onecore over the core through a simple example and proved that the one-core is nonempty for any -person finite simple committee game when  ≤ 4.
Although the behavioral pattern is improved in the onecore over the core, the problem of existence is still unsolved.Shenoy [1] provides in Example 4.2, page 393, the following 5-player committee game with empty one-core.
and the payoff utility functions are shown in Table 1.
Thus, the farsighted one-core is nonempty.We prove next that when indifference is not allowed in individual rankings or utilities vectors are componentwise different the farsighted one-core of every simple committee game is nonempty.But before that, we need the following result that proves the transitivity of the 1-dominance.Proposition 14.The relation 1-dom is transitive over the set of simple committee games when individual preferences are linear orders.More precisely, for all , ,  ∈ , for all ,  ∈ 2  , [ 1-dom     1-dom  ] ⇒  1-dom  .
Proof.Consider three distinct alternatives , ,  and two winning coalitions  and  such that  1-dom   and  1dom  .To show that  1-dom   it suffices to prove that  dom   and for all  ∈ , ( dom  ⇒ ⪰   for all  ∈ ).
Since  1-dom   then  dom  .In addition  1-dom   and  dom  imply ⪰   for all  ∈ .That is, ≻   for all  ∈  since preferences are linear orders and this means that  dom  , for the committee game, is simple.Next, let  ∈  such that  dom .Since  1-dom   then, for each  ∈ , ⪰  .That is,  dom  .Moreover,  1-dom  .Thus, for each  ∈ , ⪰  ; that is, for all  ∈ , ≻  , or  dom   since preferences are linear orders.Now we prove the main result of this section that deals with the nonemptiness of the farsighted one-core of any simple committee game in which preferences are linear orders and the nonemptiness of the farsighted one-core of any 3-player committee game.Proposition 15.For any simple committee game Γ where preferences are linear orders, F(Γ) ̸ = 0.
For general committee games (that need not be simple) we prove that if the number of players is three, then the farsighted one-core is nonempty.

The Bargaining Set Revised
Shenoy [1] defines a bargaining set that is relevant to the context of a committee game and is presented as an extension of the one-core.It is based, as all the well-known references on bargaining set in the context of games with and without side payments (Aumann and Maschler [3], Peleg [4], Billera [5,6], D' Aspremont [7], Asscher [8], and others), on objections and counter-objections but is quite different from all of them.An objection against proposal (, ) is a triple (, , ) such that  ∈  ∈ 2  ,  ∈ ,  ∉ , and  dom  .The rationale behind the dominance is that a coalition  such that  ∈ V() and members of which prefer  to  should vote for  against  whenever they are given the possibility to.But as argued before, this is a myopic behavior.Our purpose is to explore the consequence of replacing the dominance with the more foresight dominance, the 1-dominance in the definition of objection on one hand and on the definition of counterobjection on the other hand.Definition 17.Let Γ = (, , V, (⪰  ) ∈ ) be a committee game.
(1) An -objection against a proposal (, ) is a triple (, , ) such that (2) An -counter-objection against objection (, , ) to (, ) is a triple (, , ), where (3) A proposal is said to be -stable if every objection has an -counter-objection. Let  M be the set of all -stable proposals.An -stable proposal (, ) is said to be maximal if ⪰   for all  such that (, ) ∈  M.
(4) The farsighted bargaining set FM is the set of all maximal -stable proposals.
An objection against a proposal (, ) is a triple (, , ) such that  ∈  ∈ 2  , ∈\{} ,  ∉ ,  dom  . ( A counter-objection against objection (, , ) to (, ) is a triple (, , ) such that  ∈  ∈ 2  , ∈\{,}, If (, , ) is an objection against a proposal (, ), player  expects the players in coalition  to vote for  which would result in  winning against .The counter-objection which is a reply by player  is made either by player  or by a player who stands to lose if the objection is carried out.If a counterobjection does exist, there is a strong motivation for player  to withdraw his objection.On the other hand, if there is no counter-objection, then player  has a justified objection and player  cannot expect to get his proposal accepted by the committee. 1-dom  ,  1-dom   and by the transitivity of 1-dom, we have  1-dom  .Since (, , ) is an -counter-objection, there exists  ∈  :  dom  and ≻  .Since  1-dom  ,  dom , and  ∈ , it follows that ⪰   which contradicts ≻  .Therefore there is no objection to any proposal (, ) ∈ FM(Γ), meaning that FM(Γ) ⊂ F(Γ).
The following result shows the advantage of the farsighted bargaining set over the farsighted one-core.
Nonemptiness of the Farsighted Bargaining Set.Shenoy [1] proved that if Γ is a finite committee game with a nonempty core, then (i) the Shenoy bargaining set M(Γ) is nonempty; (ii) for any player , there exists an option  such that (, ) ∈ M(Γ); (iii) any player  is better off at any  than at  where (, ) ∈ M(Γ) and  ∈ (Γ).
The above three items still hold with the one-core.
In general committee games, the nonemptiness of the Shenoy bargaining set ensures the nonemptiness of the farsighted bargaining set as we can observe in the result below.
If the committee game is simple and preferences are linear orders, it follows from Propositions 15 and 24 that the farsighted bargaining set is nonemptiness.
For general (not necessarily simple) committee games, we have the following proposition stating the nonemptiness of the farsighted bargaining set (as well as the Shenoy bargaining set) of any 3-player committee game.

Conclusion
We came back on the four different solution concepts studied in relation to committee games by Shenoy [1] with emphasis on the one-core and the bargaining set.After showing that the one-core is subject to a lack of foresight, we introduce another solution concept, the farsighted one-core which is better than the one-core in terms of behavioral standard, which is always nonempty when individual preferences are linear orders, a property not shared neither by the one-core nor by the Shenoy bargaining set.This latter set has also been revised and we introduce a new bargaining set which is also better than the latter for committee games and is always nonempty when restricted to simple committee games in which preferences are strict orders.
We prove furthermore that any 3-player committee game has a nonempty farsighted one-core and we did not succeed in giving a characterization of general committee games inducing a nonempty farsighted one-core.Although the emptiness of the farsighted bargaining set of a simple committee game is guaranteed, the general study of the nonemptiness of this Shenoy revised solution concept is still an open problem.