A Practical Stochastic Dominance Based Solution to Public Policy Choice When Confronted with a Set of Mutually Exclusive Non-combinable Policy Prospects Gordon Anderson (University of Toronto) Leo Teng Wah (St Francis Xavier University)

Introduction. Stochastic Dominance Comparisons (SDC), have been a cornerstone in the advance of choice theory in many situations. In terms of wellbeing in spite of a well-developed theories for tax policy and EO applications SDC’s are seldom employed by policy makers. As an incomplete ordering, SDC’s often fail to yield an unambiguous choice and they don’t convey a sense of the relative merit of alternatives. (i.e. SDC’s provide no comparative statics or measures of “by how much”). The finance literature long recognized this (Rothschild and Stiglitz (1970, 1971) responded with a concept of Central Dominance characterizing “greater central riskiness" (Gollier 1996) yielding comparative statics for risk and portfolio mix. This could be of use in public policy choice except that alternative competing policies are frequently not divisible or “combinable” in the way that scaling or rebalancing portfolios are (there is no continuity property to exploit).

The Proposal. Here indices are proposed for measuring the extent to which one policy is “better“ than another within the context of a specific dominance class reflecting the particular imperative confronting the policymaker. Conceptually the approach is borrowed from the Stats literature on the choice of the “most powerful test” from a collection of alternatives. There the basis is the eyeballed proximity of each test's power function to the envelope of all power functions of comparable tests reflecting the maximal power obtainable by combining the best bits of each test. Here alternative policy options are considered in the context of a dominance class determined by the policymaker's imperative. The stochastically dominant envelope of policy consequences at the given order of dominance is constructed and measures (indices) of the proximity to this envelope for each policy option can be calculated. Indices and the tests for differences between them are provided.

Dominance Relations, the Basic Idea

The Dominance Rules and the Policy Makers Imperative. From [1] note the dominating distribution is the preferred distribution reflecting the desire for greater E(U(x)), however from [2] increasing orders of dominance attach increasing weight to lower values of x leading to the interpretation that successively higher orders of dominance reflect higher orders of concern for the “poor” permitting interpretation of various forms of dominance as follows: U i=1, (dU(x)/dx > 0) yields a 1 st order SDC, referred to as Utilitarian societal preference it is really an expression of preference for more of x without reference to the spread of x. U i=2, (dU(x)/dx > 0, d 2 U(x)/dx 2 < 0) yields a 2 nd order SDC, referred to as Daltonian societal preferences, expresses preference for more x with weak preference for reduced spread. U i=3, (dU(x)/dx > 0, d 2 U(x)/dx 2 0), yields a 3 rd order SDC, expresses preference for more with weak preference for reduced spread especially at the low end of the distribution.. U ∞ or infinite order SDC, referred to as Rawlsian societal preferences since it attaches infinite weight to the poorest individuals. => The policy maker contemplates the particular order of SDC (choice of i) reflecting her imperative underlying the policy choice.

The Basic Index

The 1 st order index: the area between the 2 curves.

The 2 nd Order index: The Area Between the 2 curves.

The Index in the absence of a first order dominance relation. Suppose the policymakers imperative is utilitarian, but there is no first order dominant policy in the collection. Consider the lower frontier or envelope of all distributions G(x), H(x), J(x),…,K(x) given by LE(x) = min x (G(x), H(x), J(x),…,K(x)). Effectively LE(x) selects the “best policy” at each point x to produce the best policy over the range of x if all policies could be combined. Obviously LE(x) dominates all distributions G(x), H(x), J(x),…,K(x) at the first order and would be the preferred distribution if it existed. Proximity to such a distribution would be of interest in evaluating each of the available distributions at the first order imperative, hence, ignoring normalization, we are led to contemplate:- min M(x) ∫(M(x)-LE(x))dx for M(x) = G(x), H(x), J(x),…,K(x). A sort of “Second Best” solution, it does have an IIA property in that the statistic can be shown to be independent of any policy that is strictly dominated by LE(x).

The Index at higher order imperatives. If the policymakers imperative is represented by the j’th degree dominance criterion we contemplate the lower frontier or envelope of all possible j’th order integrals of the candidate distributions G j (x), H j (x), J j (x),…,K j (x) given by LE j (x) = min(G j (x), H j (x), J j (x),…,K j (x)). Note LE(x) stochastically dominates all distributions G(x), H(x), J(x),…,K(x) at the j’th order and would thus, if it existed, be the preferred distribution at that order. Proximity to such a distribution would be of interest in evaluating the available distributions, hence she would be led to contemplate: min Mj(x) ∫(M j (x)-LE j (x))dx for M j (x) = G j (x), H j (x), J j (x),…,K j (x). Has an independence of irrelevant alternatives property.

A simple 3 alternative policy example Three alternative non-combinable redistributive policies, A, B and C yielding the same expected post tax income to society. The different policies have different re-distributional effects characterized by different revenue neutral tax policies on the initial distribution which shall be denoted policy A. The effect of policy B is that of a proportionate tax t p (x) = t, where 0 < t < 1 whose aggregate proceeds are distributed equally across the population at a level of M per person. The effect of policy C was equivalent to a progressive tax t pr (x) = t 1 + t 2 F(x) (where F(x) is the cumulative density of f(x), the income size distribution of pretax income x and 0 < t 1 + t 2 F(x) < 1 so that 0 < t 1 < 1 and 0 < t 2 < 1-t 1 ) and again the aggregate per capita proceeds M is distributed equally across the population. The empirical analogues are applied to a random sample of n pre-tax weekly incomes x i, i = 1,…, n (where incomes x are ranked highest 1 to lowest n) drawn from the Canadian Labour Force Survey for January 2012.

Example 2: Equality of Opportunity and Educational Attainment in Germany The first PISA assessments on language literacy administered in Germany in 2000 shocked the nation, the country came well below the average overall for all the countries tested and it did no better in mathematics and science than it did in language. It had a higher correlation between family socio-economic status and student achievement than any other OECD country undermining the long held view that the choice of secondary school is based solely on achievement in elementary school. In short there was overwhelming evidence of a dependence of a child’s educational outcomes on the circumstances they confronted, an evident lack of equality of opportunity. These results prompted a €4 billion plan to reform the schooling system, improve student outcomes and equalize opportunities (OECD (2011)). The reforms included more involvement in Kindergarten in teaching German to immigrant children, more money for special language training to help non-German speaking families, more money for all-day school systems, and more money for teacher training programs. In 2003/2004, national education standards were put in place (and subsequently raised in 2007 and over the following years, states in Germany adjusted their tests to represent PISA test standards. The states came together to harmonize their curriculum, and create improved tests to raise the bar again. The question is how to measure the success of the reforms?

Example 2: Equality of Opportunity, the approach. Recently dominance techniques have been applied to equality of opportunity analyses (Lefranc, Pistolesi, and Trannoy 2008, 2009),, where absence of dominance (usually of 1 st or 2nd order) of outcome distributions for different circumstance classes is seen as indicating equality of opportunity. Using Pisa data for German students in 2003 and 2009 we construct an attainment index (ACL) for students who have completed exams in Language, Math and Science and a similar ACL circumstance index for each student based upon its family type, parental income and education. We established 3 equal sized inheritance classes and fitted normal distributions to outcomes of the children the respective inheritance classes. Note that higher circumstances classes First Order Dominate lower circumstance classes in every case rejecting the notion of equality of opportunity for both time periods. But the question is, is there a sense that things have improved over the period ?

Equality Of Opportunity application: Results (3 circumstance categories) Mean Achievements (Standard Deviations) Poor Circumstance Middle Circumstance Rich Circumstance (0.0300) (0.0141) (0.0377) Wellbeing indices Poor v middle poor v upper upper v middle First Order Second Order Third Order Mean Achievements (Standard Deviations) Poor Circumstance Middle Circumstance Rich Circumstance (0.0204) (0.0110) (0.0244) Wellbeing indices Poor v middle poor v upper upper v middle First Order Second Order Third Order

Example 3. Multidimensional comparisons. Drawing from a convergence-polarization study of African nations versus the rest of the world(Anderson et al. 2012), we compare the GDP per capita and life expectancy of nations over the period Observe from table 6, that the rest of the world distribution first order stochastically dominates Africa in both periods so that the index corresponds to a measure of Africa's wellbeing deficiency vis-a-vis the rest of the world. Since the Rest of the World dominance relation is first order, it will also be Second Order as the table indicates. Note that in both cases the deficiency has increased over the period reflecting the fact that Africa and the Rest of the World have polarized.

Table 6: Between Rest of the World & Africa 2 Dimensional 1st Order Dominance Comparisons by Year (FRest 1 FAfrica implies 1st Order Dominance of Rest over Africa) First order Comparisons 1st Order Minimum Difference st Order Maximum Difference Volume Between the Surfaces e e 4 Second order Comparisons 2nd Order Minimum Difference nd Order Maximum Difference Volume Between the Surfaces e e 5

Takeaways. Applying Stochastic Dominance Techniques in the realm of public policy presents two problems. The technique only offers a partial ordering, and it never yields a “by how much“ number presenting a severe problem when alternative policies are not combinable so that the first best solution is not available. Here measures or indices founded on stochastic dominance principles are proposed which provide the policymaker with a measure of how much better one policy is than another in the context of the particular distributional imperative she confronts. The measure provides a complete ordering at any level of comparison deemed appropriate by the policymaker, and it is shown to possess an Independence of Irrelevant Alternatives property. Three distinct examples on re-distributional policy choice, equality of opportunity comparisons, and international wellbeing comparisons illustrate the use of the statistic in a wide range of circumstances where the lack of a completeness property presents a problem.