All right, first some initial thoughts:
- You don't really need game theory for this, although you could.
Take it to a multi-person level and basically what you have is
- Unless there is some other motivation to
take the job (for example because one is required to do an
internship and it's one of the only ones offered) you would not
compete to 0, but to your reservation wage, below which you would
refuse to work at all. Of course you can always say that you've
normalized the reservation wage to 0. Moreover, the reservation wage is not the same for everyone because we have different alternative opportunities, typically.
- If there is an other reason to take the job, as e.g. with the earlier mentioned internship, there is no reason why competition could not drive the wage below 0, that is people will actually pay to take the job if they want it bad enough.
As for your question in principle nothing happens to demand (in the short-run); only the quantity demanded changes. Suppose before the game is played we have the initial situation like this:
The staircase function is the supply of laborers, the dotted function the demand. In this situation person 1 is willing to work at wage=1, and person 2 at wage=2. In principle in this case the firm would hire person 1.
Now the game is played and person 2, undercuts person 1, who will undercut person 2 in turn etc.. What this does is bring the staircase steps down until they are all aligned at 0 like this:
The supply curve becomes a horizontal line at 0. The quantity demanded for the number of laborers changes from 1 to 2.
In a model involving more people that all have the same reservation wage the supply curve would look like a straight upward-sloping line and competition would induce the supply curve to turn until it is horizontal at the reservation wage.
Why does that not happen in real life? Given that typically everyone has not the same reservation wage because they have different outside options, people do drop out of the competition earlier and the supply curve stays an upward sloping line.