Question on the paper ``Self-Testing/Correcting for Polynomials and for Approximate Functions''
The definition of $g$ should be interpreted in the following way:


  Any function $g$ for which $g(x)$ is a majority value of $\sum_{i=1}^{d+1} \alpha_i P(x+i\cdot t)$ satisfies what follows.


Since it is later proved that $g = f$, it follows that $g(x)$ is always a strict majority (if it weren't, you would be able to define two different realizations of $g$, and prove that both of them are equal to $f$, contradicting their being different).

If you don't like this interpretation, you could fix a tie-breaking rule and use it to define $g$ conclusively. For example, you could take $g(x)$ to be the smallest majority value of $\sum_{i=1}^{d+1} \alpha_i P(x+i\cdot t)$. The exact tie-breaking rule makes no difference, since the proof shows that it is never actually used.

This kind of definition is common in the literature, and it is best to get used to it. Sometimes the authors explicitly mention that ties can be broken arbitrarily, but otherwise it is just implicitly assumed.