Individual and Social Choices

Transcription

1 Individual and Social Choices Ram Singh Lecture 17 November 07, 2016 Ram Singh: (DSE) Social Choice November 07, / 14

2 Preferences and Choices I Let X be the set of alternatives R i be the weak preference relation for individual i, defined over X; i = 1,..., n P i be the strict preference relation for individual i R be the set of individual preference relations O be the set of individual preference relations that are orderings; O R. (R 1,..., R n ) R n be a profile of preference relations - one for each individuals. That is, R n = {(R 1,..., R n ) R i R for each i = 1,..., n} R be a weak Social preference relation; R R Ram Singh: (DSE) Social Choice November 07, / 14

3 Preferences and Choices II Choice Set: Take any S X. The choice set generated by the preference relation R defined over the set S is given by C(S, R), where x C(S, R) if and only if ( y S) [xry], i.e., C(S, R) = {x ( y S) [xry]} Let S X. An alternative x is a best elements of S iff ( y S)[xRy] A set C(S, R) is the set of best elements of S iff [x C(S, R)] ( y S)[xRy] Ram Singh: (DSE) Social Choice November 07, / 14

4 Preferences and Choices III Let S X. An alternative x is a Maximual elements of S iff: ( y S)(yPx)] A set M(S, R) is the set of Maximual elements of S iff: For all x S, [x M(S, R)] [ ( y S)(yPx)] Suppose xry and yrx. So, M(S, R) = {x, y}. But C(S, R) =. Therefore, x M(S, R) does not mean that for all y S, xry holds. Ram Singh: (DSE) Social Choice November 07, / 14

5 Preferences and Choices IV Proposition For any give S X and preference relation R, C(S, R) M(S, R). Proposition If S X is finite and and preference relation R is quasi-ordering, then M(S, R) is non-empty. Let S = {x 1,..., x n }. Let a 1 = x 1, { x2, if x a 2 = 2 Px 1 a 1, otherwise. { xj+1, if x a j+1 = j+1 Pa j a j, otherwise. You can verify that a n is a maximal element. Ram Singh: (DSE) Social Choice November 07, / 14

6 Social Choice Rules (SCR) I Assumption Every social preference relation R has strict preference relation P and indifference preference relation I associated with it. P and I are such that: For all x, y X xpy xry and (yrx) xiy xry andy Rx Assumption We assume individual preferences are orderings, i.e., are reflexive, complete and transitive. That is, for all i = 1,.., n, R i O. Ram Singh: (DSE) Social Choice November 07, / 14

7 Social Choice Rules (SCR) II A SCR is a function f : R n R, such that, ( (R 1,..., R n ) R n )[f (R 1,..., R n ) = R R]. A SCR f is decisive iff (R 1,..., R n ) R n, the social preference relation generated by f is complete, i.e., iff (R 1,..., R n ) R n, f (R 1,..., R n ) = R is complete. A SCR is rational if (R 1,..., R n ) R n, the social preference relation generated by f, i.e., f (R 1,..., R n ) = R, is an ordering. Ram Singh: (DSE) Social Choice November 07, / 14

8 Pareto Criterion as SCR I Pareto Criterion: For x, y X, x Ry [( i N)[xR i y]] x Py [x Ry & (y Rx)] xīy [x Ry & y Rx] Proposition Relation R is a quasi-ordering. That is, it is reflexive and transitive. Ram Singh: (DSE) Social Choice November 07, / 14

9 Pareto Criterion as SCR II The Pareto Criterion is the SCR iff xry x Ry, xpy [x Ry & (y Rx)] xiy [x Ry & y Rx] Proposition If Pareto Criterion is used as a SCR, then for any finite S X the set of maximal elements for is non-empty. Ram Singh: (DSE) Social Choice November 07, / 14

10 Pareto Criterion as SCR III The SCR is Pareto inclusive, i.e., satisfies the Pareto Criterion if: For all x, y X Note: ( i N)[xR i y], i.e., x Ry xry x Ry and y Rx xpy If Pareto Criterion is used as the SCR, then the SCR is Pareto inclusive However, a Pareto inclusive SCR can (will) be different from the Pareto Criterion Ram Singh: (DSE) Social Choice November 07, / 14

11 Pareto Criterion as SCR IV Proposition Pareto Criterion is a decisive SCR iff ( x, y X)[( i N)[xP i y] ( j N)[xR j y]] Suppose, i N such that xp i y, and at the same time j N such that yp j x. In that case, we have Therefore, the condition is necessary. x Ry and (y Rx), i.e., Ram Singh: (DSE) Social Choice November 07, / 14

12 SCRs: Desirable Features I Take any D O n. A SCR is called a SWF, if f : D R. That is, ( (R 1,..., R n ) D)[f (R 1,..., R n ) = R O]. Condition U: A SCR f satisfies condition of unrestricted domain, if its domain is O n. That is, f generates a social preference relation for every possible profile of individual preferences. Ram Singh: (DSE) Social Choice November 07, / 14

13 SCRs: Desirable Features II Condition P: A SCR f satisfies condition of Weak Pareto Principle, if ( x, y X)( i N)[xP i y xpy]. Suppose, S = {x, y}. Now, condition I implies the following: ( i N)[xR i y xr i y] C(S, R) = C(S, R ), i.e., ( i N)[xR i y xr i y] (xry iff xr y) and (yrx iff yr x). Ram Singh: (DSE) Social Choice November 07, / 14

14 SCRs: Desirable Features III Condition I: Take any S X, and ANY two profiles of individual orderings, say (R 1,..., R n ) and (R 1,..., R n). Let f (R 1,..., R n ) = R and f (R 1,..., R n) = R. A SCR f satisfies condition of independence of irrelevant alternatives if the following holds: ( x, y S)( i N)[xR i y xr i y] C(S, R) = C(S, R ) Condition D: A SCR f satisfies condition of non-dictatorship, if there is NO individual i N such that ( x, y X)[xP i y xpy]. Ram Singh: (DSE) Social Choice November 07, / 14