The Swedish Leader Election Protocol: Analysis and Variations

We analyze in detail a leader election protocol that we call the Swedish leader election protocol. The goal is to select one among n > 0 players, by proceeding through a number of rounds. If there is only one player remaining, the protocol stops and the player is declared the leader. Otherwise, all remaining players flip a biased coin; with probability q the player survives to the next round, with probability p = 1 − q the player loses and plays no further... unless all players lose in a given round, so all them play again. In the classical leader election protocol, any number of null rounds may take place, and with probability 1 some player will ultimately be elected. In the Swedish leader election protocol there is a maximum number τ of consecutive null rounds, and if the threshold is attained the protocol fails without declaring a leader. We analyze several parameters of interest of this protocol as functions of n, q and τ, including the probability of success, the expected number of rounds, the expected number of leftovers (number of players still playing by the time the protocol fails), etc. We also discuss several variations and how to cope with their analysis, e.g., if we bound the total number of null rounds, consecutive or not.