**Sankhya: The Indian Journal of Statistics**

1996, Volume 58, Series A, Pt. 2, pp. 186--193

**ON CONTINUOUS-TIME TWO PERSON FULL-INFORMATION BEST CHOICE PROBLEM WITH IMPERFECT OBSERVATION**

By

ZDZISLAW POROSINSKI and KRZYSZTOF SZAJOWSKI, *Technical University of Wroclaw*

SUMMARY. A zero-sum game version of the continuous-time full-information best choice problem is considered. Two players observe sequentially a stream of i.i.d. random variables 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 observations. The observations of the random variable are imperfect and the players are informed only whether it is greater than or less than some levels specified by both of them. The normal form of the game is derived. For the Poisson stream and the exponential horizon of the value of the game and the form of the optimal strategy are obtained. It is worth to emphasize the difference from of solution of the game for the various relation of the intensity of the Poisson stream and the parameter of the exponential horizon.

*AMS (1991) subject classification.* 60G40, 90D60.

*Key words and phrases.* the best choice problem, stopping time, optimal stopping, zero-sum game, mixed strategy, Poisson process, random horizon.