  In the classical leader election procedure all players toss coins
independently and those who get tails leave the game, while those who get heads
move to the next round where the procedure is repeated. We investigate a
generalizion of this procedure in which the labels (positions) of the players
who remain in the game are determined using an integer-valued random walk. We
study the asymptotics of some relevant quantities for this model such as: the
positions of the persons who remained after $n$ rounds; the total number of
rounds until all the persons among $1,2,\ldots,M$ leave the game; and the
number of players among $1,2,\ldots,M$ who survived the first $n$ rounds. Our
results lead to some interesting connection with Galton-Watson branching
processes and with the solutions of certain stochastic-fixed point equations
arising in the context of the stability of point processes under thinning. We
describe the set of solutions to these equations and thus provide a
characterization of one-dimensional point processes that are stable with
respect to thinning by integer-valued random walks.
