There are $4$ people that they going to split $50$ gold between them. They got one extra gold that they can pay for punishment. All person makes a proposal that how can share the gold. Of the remaining players in the game, including the bidder If more than half (half is not enough) accepts the bid, the gold will be distributed accordingly and the game will end. If the offer is not accepted, the player who made the offer will be removed from the game and the extra gold that the players have the beginning of to game going to be taken away to be given to the player who has the most gold at the end of the game. the bid will move to the next player and the game will continue with the remaining players. All players trying to get as much gold as possible at the end of the game and considering all the possibilities Since the players perform without any mistakes, who is the player who gets the most gold at the end of the game and how much gold does the player get?
Now, consider the situation with just $2$ players. Now, the first one to play takes all the gold since he already has a vote for himself i.e. $50\%$ of the total vote. Next with three players, the first one may suggest to take $49$ gold. This is because of the players (say $a,b,$and $c$ and let $a$ be the first one to play suggesting $49$ gold for himself), if $c$ doesn't agree with $a$ then the next player $b$ will take the entire gold after removing $a$ if $c$ disagrees (as $a$ didn't get at least $50\%$ vote). So, if $c$ disagrees, he will not get even the $1$ gold that he might have obtained. So, he will vote for $a$. Now with $4$ players, $a,b,c$ and $d$, $a$ may suggest $49$ gold again because, when he fails to obtain at least half of the votes, then this situation will be reduce to $3$ persons and again, one will vote and the other will get nothing. So, to obtain at least $1$ gold, at least one of them will vote.