European tenure games

We study a variant of the tenure game introduced by Spencer (Theoret. Comput. Sci. 131 (1994) 415). In this version, faculty is not fired, but downgraded to the lowest rung instead.For the upper bound we give a potential function argument showing that the value υd of the game starting with d faculty on the first rung satisfies υd ≤ ⌊log2d + log2 log2d + 1.98⌋. We prove a nearly matching lower bound of ⌊log2d + log2 log2d⌋ using a strategy that can be interpreted as an antirandomization of Spencer's original game. For d tending to infinity, these bounds improve to ⌊ log2d + log2 log2d + 1 + o(1)⌋ ≤ υd ⌊log2d + log2 log2d + 1.73 + o(1)⌋. In particular, the set of all d ∈ N such that the value of the game is precisely ⌊log2d + log2 log2d+ 1⌋ has lower density greater than 1/5.