Lecture: Condorcet s Theorem
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1 Social Networs and Social Choice Lecture Date: August 3, 00 Lecture: Condorcet s Theorem Lecturer: Elchanan Mossel Scribes: J. Neeman, N. Truong, and S. Troxler Condorcet s theorem, the most basic jury theorem in social choice, is named for Marie Jean Antoine Nicolas de Caritat, marquis de Condorcet (7 Setember March 794), nown as Nicolas de Condorcet. This lecture focuses on the original theorem and some generalizations. The original theorem Condorcet s Jury theorem alies to the following hyothetical situation: suose that there is some decision to be made between two alternatives + or. Assume that one of the two decisions is correct, but we do not now which. Further, suose there are n individuals in a oulation, and the oulation as a whole needs to come to a decision. One reasonable method is a majority vote. So, each individual has a vote X i, taing the value either + or in accordance with his or her oinion, and then the grou decision is either + or deending on whether S n n i X i is ositive or negative. Theorem.. (Condorcet s Theorem) [4] If the individual votes X i, i,..., n are indeendent of one another, and each voter maes the correct decision with robability >, then as n, the robability of the grou coming to a correct decision by majority vote tends to. Proof: This is a consequence of the law of large numbers, see [5] Theorem 4.3. Let a / > 0. Since the roblem is fair in + and, we may without loss of generality assume the correct answer is +. Then EX ( a) + ( Sn + a) a > 0, and the wea law of large numbers states that n converges in robability to EX a, where by converging in robability we mean that for any ɛ, ɛ > 0 there is N large enough such that for every n N, P ( Sn n EX < ɛ ) > ɛ. Taing ɛ a, we see that the robability of a correct decision is ( ) ( ) Sn P (S n > 0) P n > 0 S n P n a < a, which is what we needed to show. This theorem is not realistic for many social situations for a number of reasons, including that individual oinions are not liely to be indeendent, but for the moment we ut such concerns aside. Two natural questions to as within the context of indeendence are: We now the theorem holds for any fixed >, but what if we let n deend on n? How small can n be for the theorem to hold? The theorem tells us that the robability of a correct decision tends to one as n, and we will see that for n > shrining slowly enough to it still holds. But what haens for finite n? How large does n need to be for us to now that the robability of a correct decision is high?
2 Lecture: Condorcet s Theorem We made our decision based on whether S n, a function of X (X,..., X n ), was ositive or negative. But what about other aggregation functions? We could, for instance, use f(x) X, the dictator function, or the more comlicated electoral college function in which individual states tae a majority vote, and then we tae a weighted majority vote of the states. The remainder of the lecture will examine these issues. Generalization: n varies with n First, consider the following sequences n : n n. > n > fixed. n + (log n) n + n /3 n + n. We will see in this section that the robability that a majority vote gives the correct decision when the individual voters are correct indeendently with robability n tends to one for all of the sequences above excet the last. But first, note that for large n the sequences above are nested, i.e. the to one /n is eventually larger than any > > which is eventually larger than + (log n), and so on. The following result lets us determine whether the robability of a correct decision tends to one by comarison, instead of having to deal with each case searately: Proosition.. Let Xi n be indeendent, and equal to with robability n and with robability ( n ), i n. Let Yi n be indeendent, and equal to with robability q n and with robability ( q n ), i n. Define Sn X : i Xn i and Sn Y i Y i n. Then whenever n q n, P (Sn X > 0) P (Sn Y > 0). Proof: By the binomial formula and the fact that Sn X is ositive iff a majority of the Xi n are equal to, we have P (X n > 0) ( ) n n( n ) n ( ) n n( n ) n, and P (Y n > 0) n/< n n/< n n/ ( ) n q n( q n ) n ( ) n q n( q n ) n. n/ It is enough to show that each term in the sum at right is smaller for n than for q n, and we can do this by roving that for each n/, t ( t) n is decreasing in t for t > /. Taing logarithms, we obtain log t + (n ) log( t), and taing derivatives we have t n t. Since n and t t, the derivative is negative, which is enough. Using Proosition. for the list we made at the start of this section imlies that there is a cutoff rate of shrinage toward, above which we get a correct decision in the limit and below which we do not. To find the cutoff, we aeal to the central limit theorem, which in this context states that if X, X,... are i.i.d. taing with robability + a and with robability /, then n(s n /n a) has a limiting
3 Lecture: Condorcet s Theorem 3 distribution that is normal with mean and variance equal to those of X, i.e. mean a and variance 4( ). By limiting distribution we mean that for all t, where Z N(0, 4( )). F n (t) P ( n(s n /n a) t) n P (Z t) If n a n + / with a n 0 then we have that the sum of variances of the first n variables is n( + o()). We can aly the Lyaunov central limit theorem (see []. 37), or the central limit theorem for triangular arrays (see [5], Theorem 5.5) to see that the central limit theorem still holds as long as n converges, so we obtain that (S n n j a j)/ n converges to a N(0, ). In articular if a j << j / then n j a j/ n 0 and the robability of voting for the correct alternative is aroaching / and if a j >> j / then n j a j/ n and the robability of voting for the correct alternative is aroaching. In articular, returning to our list, all of the sequences given excet + n satisfy na n, so the jury theorem holds for all the sequences excet that one. We can actually see that the majority will not be correct in the limit when n + n even without the central limit theorem: suose that an individual s oinion is + with robability /, ± with robability n, and with robability n. Then the / case, in which the jury limit theorem clearly fails because the robability of the majority being correct is exactly, corresonds to individuals voting whenever their oinion is + and a vote of whenever it is ± or. On the other hand, voting with individuals being correct with robability n corresonds to a vote of whenever an individual s oinion is + or ±, and a vote of otherwise. But since the exected number of individuals with oinion ± is n n 0, in the limit there is vanishing chance of any individuals having ± oinions, hence taing n + n is no better than random voting as n. 3 Results for finite n To obtain results regarding the robability of a correct outcome for finite n, we can aeal to large deviations theory. One result by Cramér dating to the early twentieth century [3] states that if Y, Y,..., Y n are iid Bernoulli () random variables, i.e. they are with robability and 0 with robability, and Ȳn denotes their average, then for each b > 0, P ( Ȳn > b) < e bn. We framed the jury theorem in terms of votes of ± rather than votes of 0 or, but if we let Y i when X i and Y i 0 when X i, then the sum of the X i s is ositive iff the average Ȳn is larger than. Hence, when the robability of a correct vote from each individual is is + a, then by taing b a in the large devation result we obtain, using that t / imlies t / a a, P (S n > 0) P ( Ȳ n > ) ( P Ȳ n ) a < a > e an. This rovides a good bound on the error for finite n, and since nothing here revents a n from changing with n since the result holds for each fixed +a and n and is not about a limit, this also rovides another way of deriving the result from the revious section, that the jury limit theorem holds for n iff na n.
4 4 Lecture: Condorcet s Theorem 4 Electoral College You will be ased to derive results regarding the actual United States electoral college as an exercise, but for now consider the following idealized version of the electoral college: We have n m individuals, groued into m states with m individuals each. Their oinions are indeendent and correct with robability n. When we vote, each state taes a majority vote from its m members, and then state casts a single vote of ±, in agreement with its majority vote, in the larger election. For fixed it is easy to see that the jury will still vote correctly, because each state by itself votes correctly with robability tending to one, and then averaging the state votes will give a correct result with robability tending to. For variable n, the result is less clear. If a n m, then the same reasoning, combined with revious results, shows that the jury will mae a correct decision in the limit, but the exact cutoff is not as clear. It turns out, however, that na n (a much weaer condition than ma n ) is still enough. This will be art of the homewor. 5 Other Aggregation Functions It is reasonable to suose that in any society, the decisions made by the overall oulation will deend in some deterministic way on the oinions held by the individuals. So, we assume that there is some function f( ) such that, when X {, } n describes the individual votes or oinions on an issue, then the society as a whole maes a decision of + whenever f(x) > 0 and whenever f(x) < 0. We would lie to exlore what the best and worst reasonable choices of f are, i.e. what are the best and worst ossible ways of deciding a grou oinion from individual oinions, under this model and assuming that the goal is to come to a correct decision as often as ossible. First we need to restrict the ossible f to revent unreasonable choices. First, we assume that f is fair, i.e. f( X) f(x) for all X. We do this because, although, for convenience, we have been letting + denote the correct choice and the incorrect choice, in the real world we do not now what is correct and incorrect, so the function f has to be fair in order to now which choice to mae in realistic situations. For instance f +, i.e. always maing the correct choice, is not reasonable in the real world. We also assume that f is monotone: i.e. f(x) f(x ) whenever X i X i for all f. In words, this says that if the only thing that haens is more eole change their oinions to +, then the grou oinion can only change toward +. This ensures that our worst choice will not be something stuid, lie the minority oinion. It turns out that these two simle restrictions are enough to determine the best and worst ossible functions f. The best is the majority vote f(x) i X i, and the worst is the dictator function f(x) X. The roof that the dictator function is worst is a bit involved and will be ostoned until the next lecture, but we can rove easily that the majority vote is best: Proosition 5.. The best fair monotone function f, in the sense that when the true state of the world is ± then sign(f(x)) is ± with maximal robability and with minimal robability, is the majority vote function. Proof: Because we have restricted attention to fair f, it is enough to find the Bayes rocedure [] under the rior that the world s true state is ± with robability one-half each, since by symmetry the robability of a correct decision is the same regardless of the state of the world, and hence the Bayes rocedure is also the best under any articular state.
5 Lecture: Condorcet s Theorem 5 So, let the true state S of the world equal with robability / and with robability /, and let P (X i s S s) > / and P (X i s S s). Then we can use Bayes rule to comute the robabilities P (X S ) P (X S ) P (S X), P (S X). P (X) P (X) The Bayes rocedure is to mae the decision δ sign(p (S X) P (S X)), i.e. to guess that the state of the world is whichever state has higher osterior robability conditional on the votes X. So our goal is to show that δ sign( i X P (S X) i). To do this, note that δ if and only if P (S X) >. We can comute P (S X) P (S X) P (X S ) P (X S ) i (Xi) ( ) i (Xi ) i (Xi ) ( ) i (Xi) ( ) i (Xi) ( ) i (Xi ) ( ) i (Xi) i (Xi ) ( ) i Xi. Since >, the right side above will be greater than if and only if i X i > 0, and hence the Bayes rocedure and the majority vote agree. References [] P. Bicel and K. Docsum. Mathematical Statistics: Basic Ideas and Selected Toics. Pearson Prentice Hall, nd edition, 007. [] P. Billingsley. Probability and Measure. Wiley, nd edition, 986. [3] A. Dembo and O. Zeitouni. Large deviations techniques and alications. Sringer-Verlag, nd edition, 00. [4] D.M. Estlund. Oinion Leaders, Indeendence, and Condorcet s Jury Theorem. Theorey and Decision, 36():3 6, 994. [5] O. Kallenberg. Foundations of Modern Probability. Sringer, nd edition, 00.
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