According to the Neumann-Morgenstern theorem, in the classic two-players strategic games /2/ there exists at least one equilibrium point in the mixed strategies domain.

Under assumption that the elements u_ij are fuzzy, the theorem is false. The asymetric equilibrium point, from viewpoint of the I player only, can be established by the so called LPI-sation procedure /6/:

If the player I can state a fuzzy LPI-forecast for the distribution of the moves of the player II, player I can consider Table 1 as a simple LPI-decision and thereby the MaxEmin equilibrium point can be calculated, but only from the viewpoint of the player I.

The determination of a symmetric stability point for players i and II is possible only by changing the rules of the game in Table 1.

Instead of Table 1 we will consider a two stage decision tree, shown below. It is a so called simple decision tree with two-stage moves: first move the player I, then player II. Given are the fuzzy outputs for player I and II at the end points of the tree, the minimal normalized preference indices ai, bi can be determined /2/. Then the LPI-rollback procedure will lead to the MaxEmin-stability strategies for players I and II.

An extension of the tree into finite LPI-decision trees leads to the following theorem which is true for a non-cooperative game /6/:

In any finite extension of a simple fuzzy LPI-decision tree, there exist at least one pair of multi-stage stability strategies for players I and II, supposing, the players know the tree with its LPI structure.

The proof follows by applying the LPI-Roll back procedure in the given decision tree.