Posted: February 7th, 2015

Topic in Economic Analysis

Paper, Order, or Assignment Requirements

 
This is a 2000 words coursework and i did type the question and requirement in the Topic question doc.
Topics in Economic Analysis 2015 Week 3 Seminar Solutions 2. Schools in a local area share a common admissions policy, which entails prioritising applicants using characteristics such as whether they have an older sibling already at the school, and distance lived from the school. Suppose there are three schools (A, B and C), each with one remaining place to be filled, and that there are three children (1, 2 and 3) applying. The admissions policy described leads the schools to prioritise these children as shown in Table A below. The children themselves (or perhaps their parents) have the preference ordering, over schools, shown in Table B. Table A: School ‘Preferences’ over Children Table B: Applicant ‘Preferences’ over Schools School 1st Pref. 2nd Pref. 3rd Pref. Child 1st Pref. 2nd Pref. 3rd Pref. A 3 2 1 1 A B C B 1 3 2 2 B A C C 2 1 3 3 B C A (i) Use the Gale-Shapley ‘deferred acceptance’ procedure, treating applicants as the ‘proposers’ to find a ‘stable’ allocation of children to schools. Iteration A B C 1 1 2 3 2 1 2 3 3 2 1 3 4 2 1 3 In the table, rejected proposals are struck-through above and these proposers move to their next highest preference in the next iteration (or propose to no-one if they have exhausted all options). In the first iteration of the procedure, each child ‘proposes’ to her favourite school. School B receives two proposals but only has one place, so rejects Child 2 who then proposes to School A, etc … . The procedure ends when we arrive at an iteration in which no one is rejected. This produces the stable allocation {{1,B}, {2,A}, {3,C}}. (ii) Repeat this procedure, this time treating the schools as ‘proposers’. Iteration 1 2 3 1 B C A Notice that this time, because each School has a different first preference, the algorithm stops after the first iteration. This produces the stable allocation {{1,B}, {2,C}, {3,A}}. (iii) What can we say about the ‘optimality’ properties of the two allocations derived above? We know (from Gale-Shapley) that the first allocation is ‘child-optimal’. This means that all children weakly prefer this allocation to any other stable allocation. The first procedure matches Child 2 with School A and Child 3 with School C, while the second procedure matches Child 2 with School C and Child 3 with School A; and from the perspective both children, the first of these matches is preferable. On the other hand, the second allocation is ‘school-optimal’: all schools weakly ‘prefer’ the second allocation to any other stable allocation. (But the notion of ‘preference’ here is peculiar, because it just arises from the admissions policy.) (iv) Are either (or both) of the allocations above ‘Pareto efficient’ for the children? The second allocation – {{1,B}, {2,C}, {3,A}} – is not even ‘weakly Pareto efficient’ for the children, because there is another allocation – {{1,A}, {2,B}, {3,C}} – that all children strictly prefer. The first allocation – {{1,B}, {2,A}, {3,C}} – is (by ‘Theorem 5’ in the lecture) ‘weakly Pareto efficient’ for the children, but it is not ‘strongly Pareto efficient’ (Pareto efficiency in the usual sense) because there is another allocation – {{1,A}, {2,B}, {3,C}} – that all children weakly prefer and two children strictly prefer.(v) Suppose the ‘top trading cycle’ method is used to revise the allocation derived in part (i) (i.e. we use the deferred acceptance procedure to achieve an ‘initial allocation’, and then the top trading cycle method to facilitate ‘swaps’ between children). What allocation does this lead to? In the first iteration each child ‘points’ to the pair that contains her first-choice school, and we find that there is one ‘cycle’. {1,B} {2,A} {3,C} The children therefore trade within this cycle and (because there is only one other child remaining) the process finishes. This gives us the allocation {{1,A}, {2,B}, {3,C}}. [This is now Pareto efficient (in every sense) among the children, but no longer stable (because Child 3 and School B prefer each other to the School/Child with which they have been matched).] (vi) Which, if any, of all the methods above would you personally recommend to a local education authority? (And why?) The answer to this question is of course entirely subjective, and really quite difficult. There is a strong sense in which the children’s preference should be the ones that ‘matter’; and also we would like the mechanism to be ‘truth revealing’ for the applicants. So these reasons convincingly suggest that the method used in (i) would be better than the method in (ii). But the question arises whether or not we should follow (as in (v)) the Gale-Shapley method with a ‘Top-trading cycle’ procedure in order to achieve an allocation that is Pareto-efficient among the children. Pareto efficiency among the children seems desirable, because the ‘preferences’ of the Schools are created artificially (by the admissions rules). But Pareto efficiency among the children here seems to come into conflict with ‘natural justice’. In (v), Child 2 eventually gets the place at School B which Child 3 covets, even though (by living closer to School B, or having a sibling there), Child 3 has the stronger claim. This would be very difficult for an Education Authority to explain or defend.