[game theory] Election game

The game is: N players ("candidates") make their moves sequentially. The move is a decision to play or not to play, and if a player chooses to play, then he must take a real number from [0;1] (that would be candidate's "political views", 0 might mean "democrat" and 1 - "republican"). Every player knows all moves of previous players, but does not know moves of next players (e.g. the first player knows no moves, the last one - knows all moves). After all moves are made, votes are counted: number of votes for a candidate X equals the length of a segment, where all points are closer to the number proposed by X on his move, than to numbers, proposed by any other candidate. If k players have chosen the same number, than the length of correspondent segment should be divided by k. The candidate, who has the greatest number of "votes" is the winner and gets a prize = $1. All other players, who played the game, but lost, bear costs of$1 (i.e.their prize=-1). The players, who chose not to play, get $0 (no prize, no costs). Again, in case k players have got the same number of votes, the prize will be divided between these players equally: each player-"semifinalist" will get$(1/k). The goal is of course to win maximum amount of money.
The hypothesis:subgame-perfect Nash equilibrium for this game is: the first player choses the point 0.5 , all other players except the last chose not to play, and the last player also choses the point 0.5.
The problem: This statement is true for 3 players. I see no reason it will not be true for any finite number of players, but I have no idea how to prove that, so I will be grateful for any help or interesting suggestions/ideas.
Thanks in advance The game is: N players ("candidates") make their moves sequentially. The move is a decision to play or not to play, and if a player chooses to play, then he must take a real number from [0;1] (that would be candidate's "political views", 0 might mean "democrat" and 1 - "republican"). Every player knows all moves of previous players, but does not know moves of next players (e.g. the first player knows no moves, the last one - knows all moves). After all moves are made, votes are counted: number of votes for a candidate X equals the length of a segment, where all points are closer to the number proposed by X on his move, than to numbers, proposed by any other candidate. If k players have chosen the same number, than the length of correspondent segment should be divided by k. The candidate, who has the greatest number of "votes" is the winner and gets a prize = $1. All other players, who played the game, but lost, bear costs of$1 (i.e.their prize=-1). The players, who chose not to play, get $0 (no prize, no costs). Again, in case k players have got the same number of votes, the prize will be divided between these players equally: each player-"semifinalist" will get$(1/k). The goal is of course to win maximum amount of money.
The hypothesis:subgame-perfect Nash equilibrium for this game is: the first player choses the point 0.5 , all other players except the last chose not to play, and the last player also choses the point 0.5.
The problem: This statement is true for 3 players. I see no reason it will not be true for any finite number of players, but I have no idea how to prove that, so I will be grateful for any help or interesting suggestions/ideas.