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We study a natural network creation game, in which each node locally tries to minimize its local diameter or its local average distance to other nodes by swapping one incident edge at a time. The central question is what structure the resulting equilibrium graphs have, in particular, how well they globally minimize diameter. For the local-average-distance version, we prove an upper bound of 2O √(lg n), a lower bound of 3, and a tight bound of exactly 2 for trees, and give evidence of a general polylogarithmic upper bound. For the local-diameter version, we prove a lower bound of Ω(√ n) and a tight upper bound of 3 for trees. The same bounds apply, up to constant factors, to the price of anarchy. Our network creation games are closely related to the previously studied unilateral network creation game. The main difference is that our model has no parameter a for the link creation cost, so our results effectively apply for all values of a without additional effort; furthermore, equilibrium can be checked in polynomial time in our model, unlike in previous models. Our perspective enables simpler proofs that get at the heart of network creation games. [ABSTRACT FROM AUTHOR]

Hat problems have become a popular topic in recreational mathematics. In a typical hat problem, each of n players tries to guess the color of the hat, he or she is wearing by looking at the colors of the hats worn by some of the other players. In this paper we consider several variants of the problem, united by the common theme that the guessing strategies are required to be deterministic and the objective is to maximize the number of correct answers in the worst case. We also summarize what is currently known about the worst-case analysis of deterministic hat guessing problems with a finite number of players. [ABSTRACT FROM AUTHOR]

Hat problems have become a popular topic in recreational mathematics. In a typical hat problem, each of n players tries to guess the color of the hat he or she is wearing by looking at the colors of the hats worn by some of the other players. In this paper we consider several variants of the problem, united by the common theme that the guessing strategies are required to be deterministic and the objective is to maximize the number of correct answers in the worst case. We also summarize what is currently known about the worst-case analysis of deterministic hat guessing problems with a finite number of players. [ABSTRACT FROM AUTHOR]