Modified strategies in a competitive best choice problem with random priority

A zero-sum game version of the full-information best choice problem is considered. Two players observe sequentially a stream of iid random variables (objects) from a known continuous distribution appearing according to some renewal process with the object of choosing the largest one. The horizon of observation is a positive random variable independent of objects. The observation of the random variables is imperfect and the players are informed only whether the object is greater than or less than some levels specified by both of them. Each player can choose at most one object. If both want to accept the same object, a random assignment mechanism is used. If some Player accepts an object, the other Player can change his level and continues the game alone. A similar game with discrete time and random number of objects is considered as a dual problem. The normal form of the game is derived. For the Poisson stream and the exponential horizon the value of the game and the form of the equilibrium strategy are obtained. In discrete-time case a game with geometric number of objects is completely solved.