We give a combinatorial proof of the result of Kahn, Kalai, and Linial [19], which states that every balanced boolean function on the n-dimensional boolean cube has a variable with influence of at least Ω log n n . The methods of the proof are then used to recover additional isoperimetric results for the cube, with improved constants.We also state some conjectures about optimal constants and discuss their possible implications.
The celebrated Erdős-Hajnal conjecture states that for every n-vertex undirected graph H there exists ε(H) > 0 such that every graph G that does not contain H as an induced subgraph contains a clique or an independent set of size at least n ε(H) . A weaker version of the conjecture states that the polynomial-size clique/independent set phenomenon occurs if one excludes both H and its complement H c . We show that the weaker conjecture holds if H is any path with a pendant edge at its third vertex; thus we give a new infinite family of graphs for which the conjecture holds.
A central theme in social choice theory is that of impossibility theorems, such as Arrow's theorem [Arr63] and the Gibbard-Satterthwaite theorem [Gib73,Sat75], which state that under certain natural constraints, social choice mechanisms are impossible to construct. In recent years, beginning in Kalai [Kal01], much work has been done in finding robust versions of these theorems, showing "approximate" impossibility remains even when most, but not all, of the constraints are satisfied. We study a spectrum of settings between the case where society chooses a single outcome (à-la-Gibbard-Satterthwaite) and the choice of a complete order (as in Arrow's theorem). We use algebraic techniques, specifically representation theory of the symmetric group, and also prove robust versions of the theorems that we state. Our relaxations of the constraints involve relaxing of a version of "independence of irrelevant alternatives", rather than relaxing the demand of a transitive outcome, as is done in most other robustness results.
Abstract. We consider elections under the Plurality rule, where all voters are assumed to act strategically. As there are typically many Nash equilibria for every preference profile, and strong equilibria do not always exist, we analyze the most stable outcomes according to their stability scores (the number of coalitions with an interest to deviate). We show a tight connection between the Maximin score of a candidate and the highest stability score of the outcomes where this candidate wins, and show that under mild conditions the Maximin winner will also be the winner in the most stable outcome under Plurality.
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