We study optimal election sequences for repeatedly selecting a (very) small group of leaders among a set of participants (players) with publicly known unique ids. In every time slot, every player has to select exactly one player that it considers to be the current leader, oblivious to the selection of the other players, but with the overarching goal of maximizing a given parameterized global (“social”) payoff function in the limit. We consider a quite generic model, where the local payoff achieved by a given player depends, weighted by some arbitrary but fixed real parameter, on the number of different leaders chosen in a round, the number of players that choose the given player as the leader, and whether the chosen leader has changed w.r.t. the previous round or not. The social payoff can be the maximum, average or minimum local payoff of the players. Possible applications include quite diverse examples such as rotating coordinator-based distributed algorithms and long-haul formation flying of social birds. Depending on the weights and the particular social payoff, optimal sequences can be very different, from simple round-robin where all players chose the same leader alternatingly every time slot to very exotic patterns, where a small group of leaders (at most 2) is elected in every time slot. Moreover, we study the question if and when a single player would not benefit w.r.t. its local payoff when deviating from the given optimal sequence, i.e., when our optimal sequences are Nash equilibria in the restricted strategy space of oblivious strategies. As this is the case for many parameterizations of our model, our results reveal that no punishment is needed to make it rational for the players to optimize the social payoff.

We study the following data dissemination problem: In a set of n nodes, every node has a unique piece of information. The communication of the nodes is organized in discrete synchronous lock-step rounds. In each round every node sends all currently known pieces of information to all other nodes. Which nodes receive this message is determined by the actual communication graph, which may change from round to round. Recently, Charron-Bost, Függer, and Nowak proved an upper bound of O(nlogn) rounds for the case where every communication graph is an arbitrary rooted tree. We present a new formalism, which facilitates a concise proof of this result. Moreover, we establish linear-time data dissemination bounds for certain subclasses of rooted trees. In particular, we prove that only (n−1) rounds are needed if the underlying graph is a directed path. An analogous result for undirected paths is also established. Furthermore, for trees with a fixed root we relate the dissemination time to the sizes of the subtrees of the root.

International audience