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The Nash bargaining game is a two-player noncooperative game where two players attempt to divide a good, say a cake, between them.  Each player requests an amount of the cake. If their requests are compatible, each player receives the amount requested; if not, each player receives nothing. The simplest form of the Nash bargaining game assumes the utility function for each player to be a linear function of the amount of cake they get. (We may assume the utility functions of the players are equal since utility functions are determined only up to a nonnegative multiplicative constant and a constant term.) 
According to traditional game theory, an infinite number of Nash equilibria exist for this game. Given any request, the corresponding strategy of the equilibrium pair simply requests the remainder of the cake.  If the first person did not request the entire cake for herself, we have a strict Nash equilibrium.  If the first player did request the whole cake, the equilibrium is not a strict Nash equilibrium since the second player receives the same amount regardless of what she demands. (If she makes her equilibrium demand of 0, then player 2 receives nothing. However, if player 2 makes any nonzero demand, she will still receive the same amount, namely nothing, because any nonzero demand will push the total sum of demands greater than the amount of cake available.)  If both players act to maximize expected utility, traditional game theory dictates each should demand half. Intuitively, this appears not only as the rational thing to do (`rational' meaning maximizing personal expected utility), but also as the "fair" thing to do.