Sharp Thresholds for Monotone Non Boolean Functions and Social Choice Theory

Sharp Thresholds for Monotone Non Boolean Functions and Social Choice Theory

Gil Kalai Hebrew University of Jerusalem, Yale University, and Microsoft, Israel. Supported by ISF, and NSF awards    Elchanan Mossel U.C. Berkeley and Weizmann Institute of Science. Supported by DMS 0548249 (CAREER) award, by DOD ONR grant N0014-07-1-05-06, by ISF grant 1300/08 and by a Minerva Grant
Abstract

A key fact in the theory of Boolean functions is that they often undergo sharp thresholds. For example: if the function is monotone and symmetric under a transitive action with and then as . Here denotes the product probability measure on where each coordinate takes the value independently with probability .

The fact that symmetric functions undergo sharp thresholds is important in the study of random graphs and constraint satisfaction problems as well as in social choice.

In this paper we prove sharp thresholds for monotone functions taking values in an arbitrary finite sets. We also provide examples of applications of the results to social choice and to random graph problems.

Among the applications is an analog for Condorcet’s jury theorem and an indeterminacy result for a large class of social choice functions.

1 Introduction

1.1 Sharp thresholds

A key fact in the theory of Boolean functions is that monotone symmetric functions undergo sharp thresholds. This fact has fundamental significance in the study of constraint satisfaction problems, random graph processes and percolation and in social choice.

The results of [4] show that for every there exists a such that for all and all which is monotone and symmetric (see definitions below) if and then . This result implies in particular that “Every monotone graph property has a sharp threshold” as [4] is titled. It also implies that symmetric voting systems aggregate information well, see for example [5] and the examples provided in the current paper.

1.2 Notation and Main Results

Let be a finite set. Let . For , a permutation on elements and we denote by the vector satisfying for all . For we write for the vector satisfying for all . For and we write if and for all such that it holds that . In other words if then for all if then . It is easy to see that defines a partial order on .

We say that is monotone if for all and such that it holds that implies that . We say that is symmetric if there exists a transitive group such that for all and . We say that is fair if for all and all it holds that .

Let denote the simplex of probability measures on and let denote the standard probability measure on . For denote by the measure on . We denote by the expected value according to the measure . For any measure , we write for the minimal probability assigns to any of the atoms in .

In our main result we show that:

Theorem 1.1.

There exists an absolute constant such that if is symmetric and monotone then for any and it holds that

The result above establishes sharp threshold for symmetric functions as it shows that for almost all probability measures takes one specific value with probability at least .

Remark 1.2.

It is interesting to compare the results established here to those of [4]. For our results give a threshold interval of length compared to the results of [4] which give threshold interval of length . In the binary case the later result is tight. It is natural to conjecture that the threshold is always of measure .

1.2.1 Other applications

Various applications of the threshold result to problems involving distributions of edge colored graphs are given in subsection 3.2.

1.3 Social choice background, and applications

We will describe two main applications of our main result to social choice theory. The first application gives an extension of Condorcet’s Jury theorem for monotone choice functions for more than two candidates and large classes of voting rules. The second application is to indeterminacy results for generalized social choice functions.

1.3.1 Aggregation of Information

The law of large numbers implies that in an election between two candidates denoted and , if every voter votes for with probability and for with probability and if these votes are independent, then as the number of voters tends to infinity the probability that will be elected tends to one. This fact is referred to as Condorcet’s Jury Theorem.

This theorem can be interpreted as saying that even if agents receive very poor but independent signals indicating which decision is correct, majority voting will nevertheless result in the correct decision being taken with a high probability if there are enough agents (and each agent votes according to the signal he receives). This phenomenon is referred to as asymptotically complete aggregation of information and it plays an important role in theoretical economics.

More recent results studied aggregation of information for general symmetric fair functions . Recall that such a function is fair if . In this setup the results of [4] imply that for every and every symmetric fair function on voters it holds .

Here we derive a similar result in the case of an election between candidates. Note that the conditions of monotonicity and fairness are both natural in this setup.

  • Monotonicity implies that if in a certain vector of voters a certain candidate is elected and if is identical to except that some of the voters changed their mind as to vote , then the outcome of the vote for should also be .

  • Fairness means that all the candidates are treated equally.

Theorem 1.3.

For every there exists a constant for which the following holds for every . Let and satisfy that

Then for every fair monotone function it holds that

In words - the proposition claims that for any measure on the votes that has a bias towards one of the candidates and any fair monotone voting function it holds that will be elected with high probability. The proof is given in subsection 3.1.

1.3.2 Indeterminacy

Arrow’s impossibility theorem asserts that under certain natural conditions, if there are at least three alternatives then every non-dictatorial social choice gives rise to a non-rational choice function, i.e., there exist profiles such that the social choice is not rational. Arrow’s theorem can be seen in the context of Condorcet’s “paradox” which demonstrates that the majority rule may result in the society preferring A over B , B over C and C over A. Arrow’s theorem shows that such “paradoxes” cannot be avoided with any non-dictatorial voting method. It is the general form of Arrow’s theorem, which can be applied to general schemes for aggregating individual rational choices, that made it so important in economic theory.

McGarvey [7] appears to have been the first to show that for every asymmetric relation on a finite set of candidates there is a strict-preferences (linear orders, no ties) voter profile that has the relation as its strict simple majority relation. This implies that we cannot deduce the society’s choice between two candidates even if we know the society’s choice between every other pair of candidates. We refer to this phenomena as “complete indeterminacy”.

Saari [9] proved that the plurality method gives rise to every choice function for sufficiently large societies. (In fact, he proved more: the ranking on any subset of the alternatives can be prescribed). This implies that knowing the outcome of the plurality choice for several examples, where each example consists of a set of alternatives and the chosen element for , cannot teach us anything about the outcome for a set of alternatives which is not among the examples we have already seen.

In [5] McGarvey’s theorem is extended to sequences of neutral social welfare functions in which the maximum Shapley-Shubik power index tends to 0. The proof relied on threshold properties of Boolean functions. In particular, McGarvey’s theorem extends to neutral social welfare functions which are invariant under a transitive group of permutations of the individuals. We describe a similar extension of Saari’s theorem.

2 Proof of the main theorem

The proof follows the same simple idea used in [4]. That is, we use information on the influences in order to deduce that varies quickly as a function of . We thus derive a generalization of Russo’s formula [6, 8] which expresses derivatives in terms of influences. We then use the fact that functions taking only a bounded number of values must have large influence sums.

Depending on the type of influence sum bounds, one obtain different results. Using the results of [1] it is possible to obtain a bound on the threshold interval length of order .Here we derive the better bound of order using a generalization of the results of Talagrand [10]. This generalization was proven in a course taught by the second author on Fall 2005. A draft of the proof was written in scribe notes of the course by Asaf Nachmias at
\(http://www.stat.berkeley.edu/\sim mossel/teach/206af05/scribes/oct25.pdf\). For completeness we have corrected and completed the proof.

It seems like in order to obtain a threshold bound of it would be needed to derive a tighter influence sum bound tailored to the setup here.

From now on, without loss of generality assume . We will consider functions and say that such a function is -monotone if implies that .

Clearly in order to prove Theorem 1.1, it suffices to prove that there exists such that if is symmetric and -monotone then for all it holds that

(1)

2.1 Influences

We will use the influence

Let .

Let , the ’th influence of according to by

In Corollary 5.7 we will derive the following lower bound on influence sum which is a consequence of a generalization of a result of Talagrand [10]. We restate the corollary here:

Corollary 2.1.

There exists some universal constant such that for any probability space and any function which is symmetric it holds that it holds that

2.2 Russo Type Formula

We denote by the tangent space to . The space is easily identified with the space of all vectors in satisfying . The natural derivative on satisfies .

Lemma 2.2.

Let be a -monotone function and a measure on and suppose that is -monotone. Write where and is a probability measure. Let . Then

Proof.

By linearity the derivative equals . The last expression is when is constant. If is not constant then by monotonicity it holds that and therefore the derivative is as needed. Finally, the inequality follows from the fact that if is constant then the variance is and otherwise . ∎

We now prove a generalization of Russo’s formula.

Lemma 2.3.

Let be a -monotone function and . Write where and is a probability measure. Let . Then:

where denotes conditioning on .

Proof.

Use the chain rule:

and Lemma 2.2. ∎

2.3 Proof of the Main Result

We now prove the main result, i.e., (1). In the proof below the constants will denote different constant at different lines. All of them depend on only.

Proof.

Let denote the set of measures in satisfying the following:

  • for all .

Write and note that

(2)

For each measure we look at the interval , where

Our goal is to show that the interval is short by bounding the derivative of

in the interval. In fact we will bound the length of each of the sub-intervals and . Since is valued, it follows that in the interval it holds that . By Lemma 2.3 and Proposition 2.1 we conclude that:

This implies that:

In particular if and then:

So:

Repeating the same argument for the interval we obtain that

Which together (2) with implies that

Taking we obtain the bound:

as needed.

3 Applications

3.1 A general form of Condorcet’s Jury theorem

We begin with a proof of Theorem 1.3. Since the proof is similar to the previous proof, we only sketch the main steps.

Proof.

We only sketch the proof since it is similar to the proof of Theorem 1.1. Assume first that for all . Let where is a probability measure with . As before write for every . Let be chosen so that . From symmetry and monotonicity it follows that

Moreover, the argument in Theorem 1.1 shows that for in the interval it holds that

which implies that

thus proving the statement of the theorem.

It remains to remove the assumption that for all . The general case where some of the probabilities may satisfy requires one additional step at the cost of taking the constant to be . Instead of the measure we first consider the measure where for and .

Note that if satisfies the conditions of the theorem with the constant then satisfies it with the constant . Moreover for all . Therefore by the first part of the proof

On the other hand, by monotonicity we have

which implies the desired result. ∎

3.2 Graph properties

Consider a process on the edges of the complete graph where each edge is labeled by with probability for independently for different edges. This process defines sets where is the set of edges labeled by . In other words, it defines graphs .

Definition 3.1.

A function from the set of partitions of the edges of to parts into is called a graph property if: For all partitions and all permutations it holds that

where .

The function is called a monotone graph property if

  • For every pair of partitions and

  • and all it holds that

If:

  • and

  • for and

  • ,

Then .

Here are a few examples:

  • Given a graph labeled by , let , where is the minimal index for which is maximal. In other words, is the most popular label, where ties are decided by preferring the smaller index.

  • Given a graph labeled by let where has the largest clique. Again ties are decided by preferring the smaller index.

  • Given a graph labeled by let where has the smallest independent set of vertices. Again ties are decided by preferring the smaller index.

Theorem 1.1 implies the following:

Corollary 3.2.

There exists a constant such that for every monotone graph property it holds that

4 Indeterminacy for voting methods

4.1 The setting

Let denote the family of non-empty subsets of .

Definition 4.1.

Given a set of alternatives, a choice function is a mapping which assigns to each nonempty subset of an element . A choice function is thus a map with the additional property that for all .

A choice function is called rational if there is a linear ordering on the alternatives such that is the maximal element of according to that ordering.

A social choice function is a map

of the form where is a choice function on which depends on the profile of individual rational choice functions for the individuals.

Note that there are two different meanings for the term “social choice functions”. Sometimes a social choice function is referred to as a map which associates to the profile of rational individual choices (or preferences) a single “winner” for the society. A social choice function in this sense easily defines a social choice function in our sense by restricting to a subset of alternatives. We can regard social choice functions as election rules which given a set of candidates and (strict) preference relations of the individuals on the candidates, provides a rule for choosing the winner. We regard as the set of all possible candidates, and as the set of available candidates, and we are interested to understand the society’s choice as a function of .

The axiom of Independence of Irrelevant Alternatives (Arrow’s IIA) asserts that may depend only on the preference relations restricted to the set .

We require the stronger property Independence of Rejected Alternatives (IRA), also referred to as Nash’s IIA.

  • (Independence of Rejected Alternative (IRA)) is a function of

    Therefore we can write .

    We will require a few more conditions:

  • (Pareto) .

4.2 Further assumptions

We will make the following additional assumptions:

  • (Unrestricted domains) The social choice function is defined for arbitrary rational profiles of the individuals.

  • (Neutrality) The social choice is invariant under permutations of the alternatives.

  • (Weak anonymity) The social choice is invariant under a transitive group of permutations of the individuals.

  • (Monotonicity) The function is monotone in the following sense: If and for then changing the choice of the -th individual from to will not change .

4.3 The result

Theorem 4.1.

Let be an arbitrary choice function on where is a set of alternatives and let .

Then there is a probability distribution on the space of orderings of the alternatives and a number such that the following holds for every social choice function which satisfies conditions (IRA) (P) (U’) (N1’), (A2’) and (M’):

If the number of individuals is larger than and if every individual makes the choice randomly and independently according to , then with probability at least for all it holds that satisfies with probability of at least (here is the highest ranked alternative in the order of voter ).

Corollary 4.2.

There exists such that when the number of individuals is larger than , every choice function on , where is a set of alternatives, can be written as for every social choice function satisfies conditions (IRA), (P), (U’) (N1’), (A2’) and (M’) applied to that are determined by rankings.

4.4 The proof of Theorem 4.1

Proof.

Let be an arbitrary choice function and consider a profile with individuals such that the plurality leads to . Such a profile exists by Saari’s theorem.

For an ordering of the alternatives let be the number of appearances of the order and let .

Consider a random profile on individuals where for each individual the probability that -th preference relation is described by is (independently for the individuals).

We will show that for all if is sufficiently large, then for it holds with probability at least that satisfies with probability at least . This implies the required result by taking .

To establish the claim above note that are i.i.d. and that for all it holds that . Furthermore the function as a functions of satisfies neutrality and weak anonymity. Thus from Theorem 1.3 it follows that for large enough the probability that is at least as needed.

Note that the proof actually implies the following in the spirit of a theorem by Dasgupta and Maskin [2].

Corollary 4.3.

There exists with the following property: Let be a set of linear orders on alternatives. If for the plurality rule there is a profile restricted to which leads to a choice function for the society, then when this is the case for every social choice function which satisfies conditions (IRA), (P), (N1’), (A2’) and (M’).

4.5 Taking rejected alternatives into account

We now describe a crucial example suggested by Bezalel Peleg. First, we note how we can base on choices on pairs a choice correspondence: Given an asymmetric binary relation on the set of alternatives let be the set of elements of such that the number of such that is maximal. In other words, when we consider the directed graph described by the relation we choose the vertex of maximal out-degree.

Let denote the class of rational choice functions and denote the class of choice correspondences obtained from binary relations as just described. Consider also the class of choice functions obtained from by choosing a single element in according to some fixed order relation on the alternatives. The number of choice functions in and the number of choice correspondences in is exponential in .

Now describe a social choice function as follows: if a majority of the society prefers to .

In this case the social choice of does not depend solely on the individual choices for but also on the preferences among pairs of elements in .

When the individual choices are rational then the social choice still belongs to the class (or ). In this case the choice from is simply those elements of which are Condorcet winners against the maximal number of other elements in . In this example the social choice for a set is typically large but this apparently be corrected by various methods of “tie breaking”.

In these examples the size of the resulting classes of choice functions is exponential in a quadratic function of . It is much smaller than the number of all choice functions which is double exponential in .

The Borda rule can be analyzed by a similar consideration.For this rule is determined as follows: For each alternative let be the number of individuals who ranked in the th place (among the elements of ). Let . The chosen element by the society is the element of with the minimal weight.

Another way to describe the Borda rule is as follows: First construct a directed graph (with multiple edges) with as the set of vertices by introducing an edge from to for every individual that prefers to . Next, define (as before) as the vertex with maximal outdegree.

It is easy to prove that the number of choice functions that arise in this way is at most exponential in . (The choice function can be recovered from the sign patters of (less than) linear expressions in real variables.

To summarize, the size of classes of choice functions that arise from a social choice function such that may depend on the individual preferences of the elements of is at least exponential in and this bound is sharp.

5 A generalization of Talagrand’s result

In this section we prove the bounds on influence sums for arbitrary probability spaces. For this we first recall the notion of Efron-Stein decomposition [3] then generalize Talagrand’s result [10] and finally derive corollaries for -valued and symmetric functions.

5.1 Efron-Stein decomposition

Consider finite probability spaces , with measures . Let be size of the smallest atom of , and set . Let be a real valued function. Write for the Efron-Stein decomposition of which we now recall.

Definition 5.1.

Let be discrete probability spaces . The Efron-Stein decomposition of is given by

(3)

where the functions satisfy:

  • depends only on .

  • For all and all it holds that:

It is well known that the Efron-Stein decomposition exists and that it is unique [3]. We quickly recall the proof of existence. The function is given by:

which implies

Moreover, for we have and for that is a strict subset of we have:

5.2 Generalization of a Result of Talagrand

We now prove:

Theorem 5.2 (Generalization of Talagrand, 1994).

There exists some universal constant such that for any probability spaces and any function it holds that

where .

The proof is almost identical to Talagrand’s proof using Efron-Stein decomposition instead of Fourier expansion and known bounds on the hyper-contractive constants of finite probability spaces. In particular we’ll use the following result of Wolff [11]:

Theorem 5.3.

For , let:

Then for all it holds that:

for all

.

Remark 5.4.

In particular one may take .

Proof.

Let

By Lagrange’s theorem we may take:

for some . Clearly

are decreasing, and therefore

We now prove Theorem 2.1.

Proof.

For a real function from our space, denote

So

Note that the statement of the theorem follows if we prove that for any function with ,

(4)

The statement of the theorem follows by apply (4) to and summing the inequalities. To prove (4) we use hypercontractivity. The following proposition is proved in the end of this note.

Applying Theorem 5.3 gives that for any integer ,

hence

Fix an integer , and sum the previous inequality for all to get

where the last inequality comes from the fact that the ratio between two consecutive summands in the sum is greater than . We now have

(5)

We now choose optimal . Choose largest such that , hence

Plugging this back into (5) gives

An application of Cauchy-Schwartz gives

hence

which concludes the proof of (4) and so we are done. ∎

5.3 The formula for valued functions

For valued functions, Theorem 2.1 has a very simple formulation.

Lemma 5.5.

Let . Then

Proof.

Let be a function with . Note that

Since

we see that

and therefore taking expected value of we obtain that

We now obtain the following corollaries

Corollary 5.6.

There exists some universal constant such that for any probability spaces , and any function it holds that

In particular if for all then: