@Cardinal hasn't elaborated on his/her correct answer so I'll do so...
To learn the expected value of the whole we calculate the expected value of the parts. Let's start with the optional winner's bet.
Expected winnings = Value of each prize * the probability of each prize.
$\$$50 * 0.6 = $\$$30
$\$$150 * 0.3 = $\$$45
$\$$200 * 0.1 = $\$$20
30+45+20 = $\$$95
The winner's bet has expected winnings of $\$$95 but the gambler always has a choice between the winner's bet and a guaranteed $\$$100. $\$$100 > $\$$95 so a gambler maximizing their expected winnings will never take the winner's bet. Knowing that the winner's bet is a bad idea, we can ignore it when calculating the expected value of the overall game. The game reduces to a 50% chance of winning $\$$100 and a 50% chance of losing $\$$100, resulting in expected winnings of $\$$0.
Therefore, the gambler can either play the game (and always decline the winner's bet) or not play the game at all, and have the same amount of money in expectation.