 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 is widely known, and we have discussed that before. Now it might be thought that the problem lies in the fact that social welfare functions 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 our criteria a little bit, because the notions 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 that 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 irrelevant 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 a winner. So formally speaking, we'll say that a social choice function is monotonic if we take any candidate O, and if it's the case for any preference profile, this one over here, if under this preference profile, O is selected, then if we look at any other preference 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 also under the new preference. So that original winner O never lost support, maybe only gained support. Then under those conditions, 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's some agent whose top choice is always the social choice. If this is the bad news, the Mueller-Satisweight theorem tells us that we can't have all three. So if a 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 function. And we won't go through the proof, but the intuition is that in order to determine the relative ordering among candidates, we need to sort of probe it everywhere. If we probe it enough, we'll get the entire social welfare function. And since the function, 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, we can use the social choice function to recover the social welfare function. We'll get the intuition behind the proof and why social choice functions are as maybe counterintuitive or complicated as social welfare functions. Now just to test our intuition again, let's consider an example and let's take plurality, which perhaps on the face of it might contradict the Mueller-Satisweight 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 not dictatorial, obviously. So the theorem says it cannot be monotonic, intuitively I think it is monotonic, but here's a counter example. So here are these seven agents and there are three preference profiles. The three of the agents prefer A to B to C, etc. And clearly then A would be the winner under plurality, because three agents would vote for A. Now what happens if we go and we modify this preference to this preference? We simply make A's 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 it should be the winner, but clearly in this case there are now four agents who are voting for B versus only three for A, so B would be the winner now. So plurality is not monotonic.