When I have watched Deal or No Deal (I try not to make a habit of it) I always do little sums in my head to work out if the banker is offering a good deal. Where odds drop below "evens" it's easy to see it's a bad deal, but what would be the correct mathematical way to decide if you're getting a good deal?
There are (at least) two factors that mean that simply calculating the average of the remaining options is not enough to describe how someone should play.
Someone's utility is not a predictable function of the amount of money that they win. For instance my utility from winning $\$$5 is more than 100 times my utility from winning 5 cents. However, my utility from winning $\$$100 million is less than 100 times my utility from winning $\$$1 million.
From what I hear about game-shows in general, if your performance does not make it to air, then you don't get anything. Hence you cannot just accept the first amount offered (if it turns out to be a better choice) and expect to get it, since it won't make an interesting show.
In maths, the expected value is the average of how much you'd win if you used the same strategy from the same position a large (approaching infinity) number of times.
We should first note that expected value alone doesn't decide the best option. We also have to take into account risk. Most people (other than gamblers) would prefer a certain dollar rather than a fifty percent chance of two and fifty percent of nothing. To deal with this, we typically define utility instead. Utility varies between individuals and is determined by their risk profile.
Since the dollar amount are quite large for contestants, it would be logical (in theory) to find someone who can afford to take the risk to insure you. This would allow you to receive the gains from a risker strategy.
For deal or no deal, they encourage you to play by making their offers worse than the expected value at the start of the game. Later, (according to Wikipedia) the offers may even exceed the expected value. Without knowing how exactly the offers are calculated (or forming a model), we can't answer this question accurately, but only make general statements.
If we ignore risk and offers being greater than expected utility, you'd always want to go as far as possible in the game. But, as is, the game is extremely difficult to analyse mathematically.