I'm thinking about some game theory problem. Here it is,
Problem: Consider the polynomial equation $x^3+Ax^2+Bx+C=0$. A priori, $A$,$B$ and $C$ are "undecided", yet and two players "Boy" and "Girl" are playing game in following way:
- First, the "Boy" chooses a real number.
- Then, the "Girl" decides the position of that number among $A$,$B$ and $C$.
- They repeat this process three times, deciding all values for $A$,$B$ and $C$.
If the final polynomial has all distinct integer roots, the Boy wins. Otherwise, the Girl wins.
Question is: Does one of Boy or Girl has any "Winning Strategy"? If so, who has one?
It seems to me, obviously the boy has great advantage. Here is my attempted argument: If the boy suggest "$0$" at the very first turn, regardless of the girl's decision we can make the polynomial to have three distinct integer roots. Actually, my argument has "almost" worked. for example,
- If the girl put $0$ at position $A$, then boy should suggest "$-84$" at the second round. Then, in any case, we always have three distinct roots, i.e. $(10,-2,-8)$ or $(-3,-4,7)$.
- If the girl put $0$ at position $C$, then boy should suggest "$-1$" at the second round. Then, in any case, we always have three distinct roots, i.e. $(-1,2,0)$ or $(1,-1,0)$.
- HOWEVER, if the girl put $0$ at position $B$, I couldn't find any good number for second round.
Has my winning strategy some problem? Or has the girl a winning strategy somehow?
Thank you for any help in advance!