Gathering support from rivals: the two rivals case

Two voters must choose between two alternatives. Voters vote in a fixed linear order. If there is not unanimity for any alternative, the procedure is repeated. At every stage, each voter prefers the same alternative to the other, has utilities decreasing with stages, and has an impatience degree representing when it is worth voting for the non-preferred alternative now rather than waiting for the next stage and voting for the preferred alternative. Intuition suggests that the more patient voter will get his preferred alternative. I found that in the unique solution of the sequential voting procedure obtained by backward induction, the first voter get his preferred alternative at the first stage independently from his impatience rate. Keywords: sequential voting, impatience rate, multi-stage voting, unanimit