Note on the core and compensation in collective choice

Note on the core and compensation in collective choice

Mathematical Social Sciences2 (1982) 323-327 North-Holland 323 Publishing Company NOTE NOTE ON THE CORE AND COMPENSATION IN COLLECTIVE CHOICE Miki...

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Mathematical Social Sciences2 (1982) 323-327 North-Holland


Publishing Company



of Economics,

Toyanra University, 3190 Cofuku, Toyarna 9.30, Japan

Received 14 May 1981

1. Introduction

In real-world collective decision-making situations, people often form coalitions or pressure groups to enhance a decision power, or to demonstrate the intensity of their preferences. Compensatory payments, bribing or vote purchase, whether they are legal or not, are therefore features commonly observed. The purpose of this note is to incorporate these aspects into a formal collective choice model, and thereby examine how choices and payments will be made in the presence of a possibility of compensation. Specifically, we investigate, under several structures of coalitions, the core of collective choice with money as a medium of compensatory payments. The possibility of compensation is introduced into the model through the willingness to pay function of individuals as provided by Groves (1979). We first derive a necessary and sufficient condition for nonemptiness of the core. Then we show that if there exists an individual who is being bribed, i.e., receiving a positive amount of money, the state with such a transfer of money cannot be in the core under the simple majority rule. This is a condition implicit in Kaneko (1975),, where the simple majority rule was investigated by a model of an n-person game in characteristic function form. We next show, when the structure of coalitions exhibits a monotonicity property, that if a state with someone being bribed is in the core, then we necessarily have the structure such that the bribed individual is the one who is in every winning coalition. The individual with the latter property was called by Nakamura (1979) a vetoer in a model without compensation. Thus, in our model the veto power turns into a corresponding ‘price’. In Section 2, we describe our basic model following Groves (1979). In Section 3, the results and discussions are presented. 0165-4896/82/0000-0000/$02.75

0 1982 North-Holland

M. Nakayama / The core and compensalion

2. Tin willingnessto pay function Let AI= ( 1,2, . . . . n) be the set of individuals, and let Y= (x,y, z, . . . } be a nonempty loctof alternatives. In addition, there is a private good which is freely transferable amongthe individuals. We call it money. Let t = (tl, . . . , 1,))E !Rnbe a vector of transfers, whereti denotes the increment (if Zi>0), or decrement (if ti
(ii) [email protected], i,) is strictlyincreasingin ti, and (iii) for any (y, t)e 2 and Y’E Y there exists t% T such that Ui(ypti) = ui(y’, t,f). With thaw assumptions,the willingnessto pay function Wi: Y x Y x T+ R can be defined fos each i&V by

W,cV’;Y, ti) = 0, where

UicV, ti) = [email protected]‘g ti - vi),

It is t&enclear from monotonicity condition (ii) that U,(Yv 4) 3:U,(..V; t[) if and only if

di- t;z Wi(y’;y,ti).

Thus, the quantity vi= wi(y’;y, ti) is the maximum amount of money which k N wouldbe willingto pay for alternativey’ when the present state is (y, ti). Or, if Vi< 0, - o, is the minimumcompensation individual i would require to accept the move. Note that we alwayshave w&y, ti) = 0, by definition.

3. The core with compensation Let W be a nonempty collection of nonempty subsets of N. Each member S in W is called a winningcoalition. A state (y, t) E Z is then said to be dominated by a state (y’, t’) e 2 if for some SE W we have t; -

t;C &(f;J$


for all k S,


In words, a winningcoalitionS c:m make a new proposal against a given state (y, t) if it is possibleto find another alternative y’ and a redistribution of money within S so that every member in S is made better off. The core relative to W is the set of undomhated states(y, t) EZ and is denoted by C( CV). We NW lcharacterizethe core C[ ZV). ?mposftioa 1. A state (~9,t) E Z belongs to C( W) if and only iffor any SE W and any y’ 5 Y,

M. Nakayatna 1 The core and cotttpensation



Proof. We first prove Lemma 2. Lemma 2. A state (y, t) E Z is dominated by a state (y’, t’) E Z if and only if there exists an SE W such that

& (C-G) C ifS C WifY’;_Y,lij,and


c 1; s 0.



Proof. Suppose (a) and (b) are true. By the definition of wi we have Eli(y,ti)= Ui(_Y’, ti - Ui) for all ic S. From (a) and (b) we have C iEs (r; - Ui)< C ie s r,!lO. Hence we can find iE T such that CiEs 6~0 and i$>(ti - 0,) for all iE S. By monotonicity we have [email protected]‘,6) > UiQ’,ti - Di) = Ui(_Ygti) for all ic S. Hence (_I+t) is dominated by (y’, i) E Z. The converse is clear. Proof of Proposition 1. From Lemma 2 it follows that (y, 1) E Z is undominated by any (y’, t’) E Z if and only if for any SE W, C t;sO



C ti 2 C w,cV’;Y9ti) + C tl.




C ,rS


This is equivalent to C t/SO iSS



ti 2



But this is further equivalent to (1). since (~‘,O)E Z for any _V’EY. This completes the proof. Note that putting y’= y in (1) we have 1 ie s ti 10 for all SE W. Proposition 1 indicates that a state (y, t) E Z will be undominated if and only if every winning coalition is compensated enough to accept (y,t). From this proposition several results concerning the transfer of money at a state in the core can be derived. First, we consider the case in which a state is Pareto efficient. A Pareto efficient state is the one which is undominated under the unanimity rule, i.e., I+‘= (R: ). Corollary 3. (Groves, 1979). A slate (y, r)~ Z is Pareto efficienr v and on& if CicNti=O and Cd,NWidY’;y;ti)50SOrUl,y’c Y. Proof. Immediate by letting W = {IV) in Proposition 1. Under a Pareto efficient state, transfers of money are arbitrary provided they satisfy the other condition.


ibf. Nakayamu / The core and compensation

Next, we consider the case of a simple majority rule. We say W is a simple majority and denote it by Wn if W= (SE 2? 1S 1z[n/2] + 13, where (n/2] is the integer not exceeding n/2. CWOI& 4. A state Q, 1)E Z belongs to C( Wm) if and only if ti = 0 for all i EN and 0~ &S wj(y’;y, ti) for ally% Y and all SE Wm. plraol. Immediatefrom Proposition 1. That ti = 0 for all k N follows from the fact Ntj=Os CieNti-ti for all ie N.

The conditiongiven by Corollary 4 is a much stringent one, so that it would be unlikelysatisfiedin general. It indicates, in particular, that if there exists an individual i who is beingbribed at (y, t), i.e., ti>O, then (y, t) cannot be stable under the simplemajorityrule, It may therefore be interesting to examine if it is aa all logically ible to haveundominatedstates through bribing. To considerthis question, let us assume that W satisfies a monotonicity condition, We say W is monotonic if SE W and S’D S together imply S’E IV. Denote by F &‘y monotonic W. The simple majority Wm is monotonic, but not conversely. P

UO~3. Let a state (y, I) E Z belong to C‘;Wmo),and let tiO> 0 for some iOEN.

men, i&S for all SE Wmo.

t-co, we have N- (io) $ Wmoby Proposibion 1. Hence, we Proof. Since CiEN_(iO) , have Se Wmo for all SCN-- (iO} by monotonicity of Wmo. Ti-;is completes the proof.

PropositionS givesa necessarycondition: if bribing should yield an undominated state at ail, the bribed individual must be the one who is indispensable in forming any*winningcoalition. We may further rewrite this result as follows. 6%&I8fy 6. Suppose that a state (y, t)E Z belongs to C( Wmo) and that 2:i&s Wiff;wt ti) >O for someY’E Y and some SE W*? Then, there exists an ioEN SU& that tiO>O and i”ES for all SE PO. Proof, Since Cies ti >O for some SE Wmoby Proposition 1, we have ti,,> 0 for some

i& N. The rest follows from [email protected]


CoalitionSE Wro is opp&d $0 Q, t) E 2 if C iEs WiQ’;J’#ti) >O for some Y’E Y. Under the simple majority rule the state @, t) cannot be a social choice if some timing coalition is opposed to it. But, under a monotonic structure of W other [email protected] Wm, (y, t)i GUIbe a social choice only if there exists an individual who is in [email protected]~Y winrning coalition ar:d who is in fact being bribed. This result would thus Low often bribing can be observed in real world decision-making situations.

M. Nakayama / The core and compensation


We conclude with the fol1owin.gremark. In the case where no transfer of money is allowed, the individual iO with &,tzS for all SE W is called a vetoer (see, e.g., Nakamura (1979)). For, if such individual io exists, the most preferred alternative by icl necessarily cannot be dominated. When money is introduced as in the present model this will not be the case; that is, the veto power can be bought at some positive price.

References T. Groves, Efficient collective choice when compensation is possible, Rev. Econom. Stud. 46 (2) (1979) 227-241. M. Kaneko, Necessary and sufficient conditions for the existence of a nonempty core of a majority game, Internat. J. Game Theory 4 (1975) 295-219. K. Nakamura, The vetoers in a simple game with ordinal preferences, Internat. J. Game Theory 8 (1979) 55-61.