Market -Welfare Arrows Impossibility Theorem Suppose the aggregation method(or social welfare function) had three properties: are complete, transitive and reflexive(unlike majority voting 2. If everyone prefers some allocation a to another. b, then a should be socially preferred to b 3. Social preferences between a and b should only depend on the way that agents rank a and b and not on the elative rank of any other irrelevant"allocation Theorem: Arrou's impossibility theorem states that any mechanism for the aggregation of preferences(social elfare function) which satisfies the above three properties is a dictatorship In other words, the social rankings correspond exactly with one individuals rankings -the social welfare function There is no"perfect"way to rank allocations. How can the government choose between allocations? Market- welfare Social Welfare Obviously, they use a less than"perfect "social welfare function. Either one of the properties is not satisfied or they use a dictatorship welfare function. Surely this is bad? Perhaps not Formally, a social welfare function(SWF)maps individual utility functions to a umber social welfare. Consider the Rawlsian and Utilitarian social welfare functions which are given respectively by: (u…,un)=∑ Consider the Rawlsian SWF. It satisfies property 1- since it is simply a utility function and hence complete, transitive and reflexive. It satisfies property 2. since if everyone prefers a to b then the minimum utility individual prefers a to b and hence the SwF ranks a above b It is independent of irrelevant alternatives, c, since all individuals(including the one with the smallest utility)rank and b independently of c. Hence, there must be a dictator. There is. The least well off agent. Is this so bad? The Utilitarian SWF also either breaks one of the three conditions or is a dictatorship. Exercise: Which?Market — Welfare 9 Arrow’s Impossibility Theorem • Suppose the aggregation method (or social welfare function) had three properties: 1. Given a set of complete, transitive and reflexive preferences, aggregation should result in social preferences that are complete, transitive and reflexive (unlike majority voting). 2. If everyone prefers some allocation a to another, b, then a should be socially preferred to b. 3. Social preferences between a and b should only depend on the way that agents rank a and b and not on the relative rank of any other “irrelevant” allocation c. • Theorem: Arrow’s impossibility theorem states that any mechanism for the aggregation of preferences (social welfare function) which satisfies the above three properties is a dictatorship. • In other words, the social rankings correspond exactly with one individual’s rankings — the social welfare function is simply the utility function of a particular agent. • There is no “perfect” way to rank allocations. How can the government choose between allocations? Market — Welfare 10 Social Welfare • Obviously, they use a less than “perfect” social welfare function. Either one of the properties is not satisfied or they use a dictatorship welfare function. Surely this is bad? Perhaps not. • Formally, a social welfare function (SWF) maps individual utility functions to a number — social welfare. • Consider the Rawlsian and Utilitarian social welfare functions which are given respectively by: WR(u1, . . . , un) = min i ui and WU (u1, . . . , un) = Xn i=1 ui • Consider the Rawlsian SWF. It satisfies property 1 — since it is simply a utility function and hence complete, transitive and reflexive. It satisfies property 2, since if everyone prefers a to b then the minimum utility individual prefers a to b and hence the SWF ranks a above b. • It is independent of irrelevant alternatives, c, since all individuals (including the one with the smallest utility) rank a and b independently of c. Hence, there must be a dictator. There is. The least well off agent. Is this so bad? • The Utilitarian SWF also either breaks one of the three conditions or is a dictatorship. Exercise: Which?