Let's play a little game

Level pending

Players A and B are playing a game consisting of \(n\) turns. There are two buttons placed in front of both A and B, labelled with 0 and 1 respectively. At one turn they both have to choose one of the buttons in front of them and press it simultaneously. After that we calculate the sum of the values of the labels on the two pressed buttons and call it \(S\). Suppose that if \(S=0\), B has to give $1 to A, if \(S=1\), A has to hand $2 to B, and finally if \(S=2\), B has to give $\(a\) to A with \(a>0\). After \(n\) turns have been played, we name a winner based on who has more money. As \(n\) goes to infinity, determine the maximum value of \(x\) for which B always has a winning strategy if \(a<x\).

Details and assumptions: A and B can not see their opponent's choice before revealing his. They can not communicate with each other during the game either. Assume that we allow the players to have a negative amount of money, for instance in the form of loan and debt.

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