In the first part of this lecture, we analyze the noise sensitivity of the Majority function in more detail than last time. In the second part, we discuss the Majority is Stablest theorem, which says that among balanced functions for which each of the individual variables has low influence, the Majority function has the lowest noise sensitivity up to some small error. We also see some applicati...

The standard epistemic justification for inclusiveness in political decision making is the Condorcet Jury Theorem, which states that the probability of a correct decision using majority rule increases in group size (given certain assumptions). Informally, majority rule acts as a mechanism to pool the information contained in the judgements of individual agents. I aim to extend the explanation o...

I introduce a model of corporate voting. I characterize the shareholder majority rule as the unique corporate voting rule that satisfies four axioms: anonymity, neutrality, share monotonicity, and merger, a property that requires consistency in election outcomes following stock-for-stock mergers. JEL classification: D71, D72, K22

A median of a k-tuple =(x1; : : : ; xk) of elements of a 0nite metric space (X; d) is an element x for which ∑k i=1 d(x; xi) is minimum. The function m with domain the set of all k-tuples with k ¿ 0 and de0ned by m( ) = {x: x is a median of } is called the median function on X . Continuing with the program of characterizing m on various metric spaces, this paper presents a characterization of t...

In a general bargaining model with a fixed proposer, we show that the agenda setter’s equilibrium payoff is never lower than her payoff in the static model. There is a cutoff level of voter patience such that below the cutoff, the static equilibrium (possibly along with others) obtains; and above the cutoff, all equilibria are in mixed proposal strategies, and the agenda setter does strictly be...

It is shown that the counting function of n Boolean variables can be implemented with the formulae of size O(n3.06) over the basis of all 2-input Boolean functions and of size O(n4.54) over the standard basis. The same bounds follow for the complexity of any threshold symmetric function of n variables and particularly for the majority function. Any bit of the product of binary numbers of length...

1. The Andre'ev (1987) bound: explicit function with size f = ˜ Ω(n 5 2) 2. The relationship between formula size and circuit depth, and how to accomplish " depth reduction " for a general circuit 3. Monotone formulas for monotone functions (example: monotone formula for the majority function MAJ)

Let X be a linearly ordered set of arbitrary size (finite or infinite). Natural functions on such a set one can define using the linear order include maximum, minimum and median functions. While it is clear what the clone generated by the maximum or the minimum looks like, this is not obvious for median functions. We show that every clone on X contains either no median function or all median fu...

In the context of spatial majority voting games, considerable experimental support exists for the core as a solution hypothesis when it is not empty (Berl et al., 1976; Fiorina and Plott, 1978; Isaacs and Plott, 1978). Specifically, these experiments show that if a simple majority voting game possesses a Core point-a point that cannot be defeated by a majority vote-subjects choose outcomes at o...

A median (antimedian) of a profile of vertices on a graph G is a vertex that minimizes (maximizes) the sum of the distances to the elements in the profile. The median (antimedian) function has as output the set of medians (antimedians) of a profile. It is one of the basic models for the location of a desirable (obnoxious) facility in a network. The median function is well studied. For instance ...