In this lecture we'll speak about social choice functions. The most famous impossibility results pertain to social welfare function. And in particular the arrows impossibility theorem. It's widely known, and we have discussed that before. Now it might be thought that the problem lies in the fact that social welfare function require you to specify as an output of the process an entire ordering. And that might be highly constraining. But if you only needed to pick a winner as does a social choice function then you would escape these paradoxes. It turns out that the answer is no, but first we need to sort of redefine that criteria a little bit because the notion of Pareto efficiency and independence of irrelevant alternatives just aren't well defined in the context of a social choice function. But we will see there are closely related notions that are well defined. So let's first define Weak Pareto Efficiency. And we'll say that a social choice function C is weakly Pareto efficient if essentially it never elects a dominated outcome or candidate. So, if there is a candidate O2. Such that there is some other candidate O1. That is always preferred to O2 by every voter. Then the weaker candidate, O2 would never be elected. Seemed like a reasonable criterion and we'll call that weak Pareto efficiency. In place of independent of [INAUDIBLE] alternatives we'll have the notion of monotonicity. And informally as is written here. At the bottom, it says that if you have a winner, if we increase the support for that candidate they would still remain the winner. So formally speaking, we'll say that the subject's main function is monotonic if we take any candidate, O, and [INAUDIBLE] the case for any preference profile, this one over here, if under this preference profile O is selected, and if we look at any other preferred profile, if it has the property that for every agent and every other outcome, O prime. If under the original preference O was preferred to o prime which is under the new preference, so that original winner o never lost support, maybe only gained support. Then under those condition it better be the case that under new preference order in which O only got more support it would still be the social choice, the winner. Again, a reasonable property to require. The last notion of dictatorship similar to what you see in social welfare simply says that C is dictatorial if there is some agent who's top choice is always the social choice. This is a bad news, the Muller-Satterthwaite theory tells us that we can't have all three. So if social choice function is Pareto efficient and monotonic, it must be dictatorial. And so, after all, social choice functions aren't more benign than social welfare functions. And we won't go through the proof that the intuition is that, in order to determine the relative ordering among candidates, we need to sort of probe it everywhere. And if we probe it enough, we'll get the entire social welfare function. And since the social choice function must be defined for all inputs as we vary the inputs, we can find the total social welfare ordering. So in fact, what we can use the social choice function to recover the social welfare fund. That's the intuition behind the proof and why social choice functions are as maybe counter intuitive or complicated as social welfare functions. Now, just to test your intuition again, let's consider an example and let's take plurality which, perhaps, on the face of it might contradict the Muller-Satterthwaite Theorem. Clearly, plurality is Pareto-efficient. In other words, if everybody prefers some candidate to another candidate, that other weaker candidate will never be the choice. And it's non-dictatorial, obviously. So the theorem says that it can not be monotonic. Intuitively, I think it is monotonic, but here is a counter example. So here are these seven agents and their three preference profiles. The three of the agent prefer a to b to c etc. And clearly then a would be the winner under plurality. because three agents, I would vote for a. Now, what happens if we go and we modify this, you modify this preference to this preference. We simply make a situation better. C was preferred to a originally, and now a is preferred to c. And all other preferences relative to a remain the same. So surely, you would say a is doing as well as it would have done originally and monotonicity would say that 2 should be the winner, but clearly in this case four agents are voting for b versus only three for a so b would be the winner now, so plurality is not monotonic.