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Calculating the nth voter utility under the infinite sequential voting with externalities model

We study a model of sequential voting with two alternatives, in which each voter has its own private preference, and some utility externalities. In this model, the voter prefers voting for the chosen winner over voting for his own private preference. Under this model, the voters act strategically as they’re interested in maximizing their utility which depends on 3 factors: (i) the voter’s preferred alternative, (ii) the voter’s vote, and (iii) the outcome of the voting. In this model the voting goes on sequentially until the gap in votes between the two alternatives is at least some large value . Alon-et-al show in [1] that the set of strategies in which every player plays the strategy form a subgame perfect equilibrium. In this work, under the assumption that each player plays the strategy, we offer a way to calculate the expected utility of each voter.