# Social Choice: from Independence axiom to Mixture symmetry axiom

Harsanyi's Utilitarian theorems' has been criticized by Diamond Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparison of Utilities and others that, it is the morally unacceptable for social preference to satisfy the Independence axiom because it would imply that ex ante fairness are of no importance. Epstein and Segal QSWF suggest a Quadratic Social Welfare Function to replace Harsanyi's linear one (weighted sum of individual utilities), equivalently, a "Mixture Symmetry axiom" to replace the Independence axiom.

I am trying to use this QSWF & MS axiom to replace the first order social welfare function and Independence axiom in Professor Uzi's another paper Let's agree all dictatorships are equally bad. Professor Uzi's aim of this paper is not about the social welfare's functional form but I would like to see the difference if I made a change as mentioned above. The maximizer(optimal resources allocation)of the weighted sum of individual utilities is invariant to linear transformation $v_i=au_i+b$ in the original paper. From what I can see now, the optimal allocation is not invariant to linear transformation of utilities, but this is yet to be proved.

It turns out the main job is about how to replicate the Lemma 4 (in Appendix B) of Pro Uzi's original paper (dictatorships) using Mixture Symmetry instead of Independence axiom. One difficulty is that the MS axiom is given in the case of two lotteries:

Mixture symmetry(MS): if lotteries $f \sim g$ socially, then $\alpha f+ (1-\alpha)g \sim \alpha g +(1-\alpha)f$ socially, $\alpha \in [0,1]$.

And there is another axiom called Randomization preference(RP)：

$f \sim g$ socially, but $f \ne g$, this is to say, at least individual strictly prefers $f$ to $g$ and another who strictly prefers $g$ to $f$. Then $\frac{1}{2} f+ \frac{1}{2}g$ $\succ$ $g$. Equiprobable randomization is the best.

MS and RP together implies:

$\frac{1}{2} f+ \frac{1}{2}g$ $\succ$ $\alpha g +(1-\alpha)f$ for any other $\alpha \in [0,1]，\alpha \ne \frac{1}{2}$. The closer the randomization to the equiprobable one the better.

However, the two axioms together do not imply the 3-lottery version of MS:

If $f \sim g \sim r$, $\alpha f+ \beta g + (1-\beta - \alpha)r \sim \alpha g + \beta f + (1-\alpha-\beta)r$.

Because let $g=r$ we would have $\alpha f+ (1-\alpha)g \sim \beta f+ (1-\beta)g$, when $\alpha =\frac{1}{2}$ and $\beta =1$ we would have $\frac{1}{2} f+ \frac{1}{2}g \sim f$ which contradicts with the RP. Similar arguments apply to two mixtures whose probability weights are cyclic permutations of one another.

Hence, it seems to me the current MS combined with RP can only give us the equivalence between 2-lottery mixtures, what if I need to randomize more than 2 lotteries, and compare different mixtures? The above 3-lottery version axiom is not right, but it does not say there is no 3 ore more lottery version, it may take other form. So the question is, how should I generalize the MS to higher dimension if possible? Can I decompose a n-lottery mixture into a sequence of 2-lottery mixture?

BTW, I use MS to replace the Independence axiom, which states: if $f \sim g$, then $\alpha f+ (1-\alpha) r \sim \alpha g+ (1-\alpha) r$. Without this Independence axiom, if we have $a \sim b$, $c \sim d$, I am not sure whether we can have $\alpha a+ (1-\alpha) c \sim \alpha b+ (1-\alpha) d$.

While Uzi's original construction uses all extreme/dictatorial allocation to construct all possible general allocations via the Independence axiom, how can I instead use MS and RP to do so?

Anyone studied this paper before? I am not sure how many technique and model details I should post, I can add more if you want to discuss. The main construction and proof is on page 586 of the paper(dictatorships).

• I would definitely like to see more details. Namely, what the precise definitions of the representations and the axioms are. Otherwise, short of reading all papers you mentioned, I can only recommend to read surveys such as van Zandts excellent survey of independence axioms and utility representations in the Handbook of Rational and Social Choice. – HRSE Feb 17 '16 at 8:40
• @HRSE thanks for comments, do you mean the section in his book "Introduction to the Economics of Uncertainty and Information?" – Bob Feb 17 '16 at 10:02
• no, it is Grant + van Zandt "Expected Utility Theory" in the Handbook of rational and social choice by Anand, Pattanaik + Puppe. the article is very introductory, so if you are already familiar with the works of debreu, gorman, wakker, etc. you may know a lot of it already. – HRSE Feb 18 '16 at 5:25
• @HRSE, sorry to add things this late, a little disorganized as the new semester starts. – Bob Mar 1 '16 at 11:33