Computing optimal strategies for a cooperative hat game

We consider a `hat problem' in which each player has a randomly placed stack of black and white hats on their heads, visible to the other player, but not the wearer. Each player must guess a hat position on their head with the goal of both players guessing a white hat. We address the question of finding the optimal strategy, i.e., the one with the highest probability of winning, for this game. We provide an overview of prior work on this question, and describe several strategies that give the best known lower bound on the probability of winning. Upper bounds are also considered here

Unachievability of network coding capacity

The coding capacity of a network is the supremum of ratios, for which there exists a fractional @ A coding solution, where is the source message dimension and is the maximum edge dimension. The coding capacity is referred to as routing capacity in the case when only routing is allowed. A network is said to achieve its capacity if there is some fractional @ A solution for which equals the capacity. The routing capacity is known to be achievable for arbitrary networks. We give an example of a network whose coding capacity (which is I) cannot be achieved by a network code. We do this by constructing two networks, one of which is solvable if and only if the alphabet size is odd, and the other of which is solvable if and only if the alphabet size is a power of P. No linearity assumptions are made