On common values ofϕ(n) andσ(m), II

ON COMMON VALUES OF φ(n) AND σ(m), II

For each positive-integer valued arithmetic function f , let Vf ⊂ N denote the image of f , and put Vf (x) := Vf ∩ [1, x] and Vf (x) := #Vf (x). Recently Ford, Luca, and Pomerance showed that Vφ ∩ Vσ is infinite, where φ denotes Euler’s totient function and σ is the usual sum-of-divisors function. Work of Ford shows that Vφ(x) ≍ Vσ(x) as x → ∞. Here we prove a result complementary to that of Fo...

On common neighborhood graphs II

Let G be a simple graph with vertex set V (G). The common neighborhood graph or congraph of G, denoted by con(G), is a graph with vertex set V (G), in which two vertices are adjacent if and only if they have at least one common neighbor in G. We compute the congraphs of some composite graphs. Using these results, the congraphs of several special graphs are determined.

Mechanism Design with Common Values

vi = θ1 + θ2 Assume for simplicity that the θi are i.i.d. random variables with a continuous density on [0, 1]. How would you bid in a first price auction? How would you bid in a second price auction? Does the second price auction have dominant strategy equilibria? Does the game have other equilibria that are similar to dominant strategy equilibria in some sense? Analysis of the first price auc...

Nonparametri Tests for Common Values

Sequential bargaining with common values∗

We study the alternating-offers bargaining problem of assigning an indivisible and commonly valued object to one of two players who jointly own this object. The players are asymmetrically informed about the object’s value and have veto power over any settlement. There is no depreciation during the bargaining process which involves signalling of private information. We characterise the perfect B...