ISSN 0005-1179, Automation and Remote Control, 2017, Vol. 78, No. 8, pp. 1512–1522.
Pleiades Publishing, Ltd., 2017.
Original Russian Text
E.N. Konovalchikova, 2015, published in Upravlenie Bol’shimi Sistemami, 2015, No. 54, pp. 114–133.
LARGE SCALE SYSTEMS CONTROL
A Model of Best Choice under Incomplete Information
E. N. Konovalchikova
Transbaikal State University, Chita, Russia
Received February 4, 2015
Abstract—This paper suggests two approaches to the construction of a two-player game of
best choice under incomplete information with the choice priority of one player and the equal
weights of both players. We consider a sequence of independent identically distributed random
), i =1...,n, which represent the quality of incoming objects. The ﬁrst compo-
nent is announced to the players and the second component is hidden. Each player chooses an
object based on the information available. The winner is the player whose object has a greater
sum of the quality components than the opponent’s object. We derive the optimal threshold
strategies and compare them for both approaches.
This paper is dedicated to a game-theoretic model of best choice under incomplete information.
The ﬁrst solution of the best-choice problem also known as the secretary problem dates back to
the paper  published in 1961. Some researchers use the term the best object choice problem. In
the minimax setting, the best-choice problem was originally studied in .
The game-theoretic models of best choice were considered in [8, 9, 11, 15, 16]. One may identify
two approaches to modeling of best-choice games. According to the ﬁrst approach, players observe
the same sequence of objects x
, striving to choose (a) one or several objects with highest
quality or (b) an object having higher quality than the ones chosen by the opponents. The second
approach proceeds from the assumption that each player has an individual sequence of observations.
The goal remains the same—it is necessary to choose a better object than the opponents do. For
Best-choice problems are also characterized by available information on the quality of observed
objects. The classical secretary problem involves objects with linear ordering but the quality itself
is not deﬁned. At the same time, models with complete information treat the quality of objects as a
random variable obeying a known probability distribution [10, 13]. Moreover, best-choice problems
may have partial information (a known probability with unknown parameters ) or incomplete
information (an unknown exact value of a random variable [7, 17]). In the above statements,
objects (or secretaries) always agree with the decision of a receiving party. Alternative statements
are also possible. As an example, we mention the secretary problem with the potential refusal of
a candidate, see . In addition, candidates can also choose among existing oﬀers. Such settings
are called mutual choice problems [5, 12, 14].
The present paper considers a two-player game where each player seeks to choose a better object
than the opponent does. Note that the information about an object is incomplete, i.e., each player
takes the risk of accepting not the best candidate. We compare the resulting solutions for two
scenarios, namely, when one of the players has choice priority and when both players have equal