The authors study the following model. Let \(G = (E,F,u_1,u_2)\) be a two-person non-zero-sum game with finite players' strategy spaces \(E\) and \(F\), and with players' payoff functions \(u_1\) and \(u_2\), respectively. Let \(G^{0}\) be an {outside option game}, that is the following modification of \(G\): First Player 1 decides between two of his new strategies, ``out'' and ``in''. If he chooses ``out'' the game ends with a payoff vector \((x,y)\) to the players. If Player 1 chooses ``in'' the players play the game \(G\). It is assumed that there exists a {forward induction equilibrium} \(s^*\), which is Player 1's preferred Nash equilibrium in \(G\) satisfying \(u_1(s^*) > x > u_1(s)\) for any other equilibrium \(s\) in \(G\). The paper discusses three properties of the sets of Nash equilibria in outside option game \(G^{0}\). The first result is an example of the game \(G^{0}\) with zero index showing that the outside option component \(M\) for this game (the set of Nash equilibria \((s_1,s_2)\) in \(G^{0}\) with \(s_1(``\text{Out}")>0\)) can be essential. Next, it is shown for the same game that the set \(M\) do not have to be hyperstable. The last result says generally that if the outside option component is an equilibrium evolutionary stable set then the game \(G\) is nongeneric.