You and a Professor have agreed to take part in the following "game". Why, I don't know. But there was some clash of ego, and rather than taking it on grades and other nasty things, you guys have decided to face each other in this duel.

At midnight today you will each bring to the Sarovar a cup of coffee spiked with some potent hallucinating agent X. The Professor will drink some coffee from your cup and then some more from her own. You will drink some coffee from the Professor's cup and then some more from your own. You both will sit tight there after this, and see who becomes insane. It is a game of wit, and hence, both of you want to stay sane while watching other go insane. Also, you both have agreed to play this game purely because of your ego. So, your sanity matters, but your ego of overpowering the opponent also matters.

Let me tell you something about the hallucinating agent X.

There is only one type of this agent X available in the IIMK Campus but it comes in different strengths of potency (let's say grade 1, 2, 3, ..., n, where 1 is the weakest grade and n is the highest grade). A small amount of any grade of X is enough for one to go temporarily insane, but it will take a few hours to act. However, there is an antidote to X: a sip of higher grade of X. Taking two doses - one dose followed by a stronger dose - has the same effect as not taking any hallucinating agent at all. However, taking a second dose of equal or weaker potency will not help your sanity at all.

Now, let me tell you something about this Professor.

She teaches Game Theory. Also, She has access to the highest grade of X. (And you don't! You know, faculty privileges...)

So here is the challenge.

Given that you know all about this X thingy, and the fact that the cunning Professor might bring the most potent sample of all, is there a means for you (who is no less intelligent) to survive this cheery game?

Formal questions:

1. Can you model this clash of ego as a game, and find if it has a Nash Equilibrium in pure strategies? Do you win in the Nash Equilibrium?