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A Large Population Partnership Formation Game with Associative Preferences and Continuous Time

100%
Ramsey D. M.
Mathematica Applicanda
|
2018
|
tom 46
|
nr 2
EN
 A model of partnership formation is considered in which there are two classes of player (called male and female). There is a continuum of players and two types of both sexes. These two types can be interpreted, e.g. as two subspecies, and each searcher prefers to pair with an individual of the same type. Players begin searching at time zero and search until they find a mutually acceptable prospective partner or the mating season ends. When a pair is formed, both individuals leave the pool of searchers. Hence, the proportion of players still searching and the distribution of types changes over time. Prospective partners are found at a rate which is non- decreasing the proportion of players still searching. Nash equilibria are derived which satisfy the following optimality criterion: each searcher accepts a prospective partner if and only if the reward from such a partnership is greater or equal to the expected reward obtained from future search. So called ”completely symmetric” versions of this game are considered, where the two types of player are equally frequent. A unique Nash equilibrium exists regardless of the precise rule determining the rate at which prospective partners are found. Two examples are given. 
2
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A secretary problem with missing observations

100%
Ramsey D. M.
Mathematica Applicanda
|
2016
|
tom 44
|
nr 1
EN
Two versions of a best choice problem in which an employer views a sequence of $N$ applicants are considered. The employer can hire at most one applicant. Each applicant is available for interview (and, equivalently, for employment) with some probability $p$.The available applicants are interviewed in the order that they are observed and the availability of the $i$-th applicant is ascertained before the employer can observe the $(i+1)$-th applicant. The employer can rank an available applicant with respect to previously interviewed applicants. The employer has no information on the value of applicants who are unavailable for interview. Applicants appear in a random order. An employer can only offer a position to an applicant directly after the interview. If an available applicant is offered the position, then he will be hired. In the first version of the problem, the goal of the employer is to obtain the best of all the applicants. The form of the optimal strategy is derived. In the second version of the problem, the goal of the employer to obtain the best of the available applicants. It is proposed that the optimal strategy for this second version is of the same form as the form of the optimal strategy for the first version. Examples and the results of numerical calculations are given.
3
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On "Games and Mathematics. Subtle Connections" by David Wells

63%
Ramsey D. M., Szajowski K.
Mathematica Applicanda
|
2013
|
tom 41
|
nr 2
PL
Autor w tej ksiazce przedstawia entuzjastycznie matematyke, a własciwie jej uprawianie. Wyjasnia, dlaczego nauka matematyki jest podobny i co powoduje, ze jest rózna do gry w szachy, hex czy go. Porusza wiele zagadnien matematycznych i okolicznosci ich powstania, interpretujac to jako swoiste gry, w szczególnosci z zakresu geometrii i teorii liczb. Robi to rozrywkowym stylu, choc szczatkowy charakter tych esejów powoduje, ze ksiazka nie jest łatwa w czytaniu. W zamysle autora ksiazka jest skierowany do nauczycieli szkół srednich i uczniów z zamiłowaniem do matematyki. Mimo iz wiele przykładów w niej przedstawionych pojawiaja sie w innych ksiazkach o tej tematyce, to potrafi czesto znalezc nowe podejscie do nich. Jednym z oryginalnych ujec, które szczególnie uatrakcyjnia narracje jest to, ze pokazuje przypadki, w których poczatkowe próby znalezienia rozwiazania były rzeczywiscie złe. W ten sposób podkresla, ze matematyki nie mozna uprawiac w ustalony sposób (mechaniczne), ale wymaga namysłu oraz wprowadzania formalizmu.
EN
There is a number of popular books on mathematics and its connection with other human activities. The presented book ``Games and Mathematics. Subtle Connections''   belongs to this class but the author's perspective is original (see [2] in references for links  to other texts and reviews on the book ) by David Graham WellsThe title of the book   is intriguing and outrageous at the same time. However, as it is possible to find in others' opinion (see Thomas~[1]), no one claims that mathematics is a game or bunch of games but doing mathematics is \emph{like} playing a game. This book is on this topic.  In this book, the author presents an enthusiastic view of mathematics and explains why the study of mathematics is similar to the playing of games such as chess, hex and go, but also why it differs. The book considers a large number of mathematical problems and "games", particularly from the field of geometry and number theory in an entertaining style, although he brushes over some of the concepts and as a result the book is not an "easy read". The book is aimed at high school teachers and pupils with an aptitude for mathematics. Although many of the examples presented appear in other such books, the author often finds a novel approach to them. One thing which was particularly refreshing was that the author pointed out cases where initial attempts to find a solution were actually wrong and hence stressed the fact that mathematics is not a mechanical science, but requires insight as well as formalism. The reader does not need a very formal background in mathematics, a more intuitive approach is favored. On the other hand, the book covers a wide range of material (and historical figures) at a very rapid pace. Hence, some of the concepts presented are left unexplained.   
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