On the game with perfect information with time-claiming alternatives

The following finite stage game with perfect information is considered. In each vertex of the game tree belonging to the set of personal moves of player the finite number of basic alternatives is fixed and for each given basic alternative a closed time interval is defined. The elements of this time interval are interpreted as time necessary to perform basic alternative in a given vertex. Each basic alternative in the multistage game with Time-claiming alternatives is associated with an infinite number of alternatives, the basic alternative with corresponding time values we shall call bunch of alternatives. As usual the strategy of player is a mapping which corresponds to each vertex from the set of personal moves of the player the range consisting from the index of basic alternative, time necessary to realize this alternative and all time values, which is chosen by the players on the previous stages. If the n-tuple of strategies is chosen by players the trajectory of the game path can be uniquely defined. This path consists from the sequence of basic alternatives and corresponding time parameters chosen by players. Payoff function of player for each trajectory of the game continuously depends upon all time values, which is chosen by the players to perform the basic alternative along the trajectory and it is a uniformly bounded function. However it is proved that payoff function of the player not necessary continuously depends upon his strategy (part of his strategy, time necessary to realize basic alternative). This makes impossible the existence of subgame perfect Nash equilibrium. The example of this case is presented and the existence of subgame perfect $\varepsilon$-Nash equilibrium is proved.