Editing Rendezvous problem (section)

{{short description|Logical dilemma}}
The '''rendezvous dilemma''' is a logical dilemma, typically formulated in this way:

:Two people have a date in a park they have never been to before. Arriving separately in the park, they are both surprised to discover that it is a huge area and consequently they cannot find one another. In this situation each person has to choose between waiting in a [[Focal point (game theory)|fixed place]] in the hope that the other will find them, or else starting to look for the other in the hope that ''they'' have chosen to wait somewhere.

If they both choose to wait, they will never meet. If they both choose to walk there are chances that they meet and chances that they do not. If one chooses to wait and the other chooses to walk, then there is a theoretical certainty that they will meet eventually; in practice, though, it may take too long for it to be guaranteed. The question posed, then, is: what strategies should they choose to maximize their probability of meeting?

Examples of this class of problems are known as '''rendezvous problems'''.  These problems were first introduced informally by [[Steve Alpern]] in 1976,<ref>{{citation | last = Alpern | first = Steve | author-link = Steve Alpern | year = 1976 | title = Hide and Seek Games | publisher = Seminar, Institut fur Hohere Studien, Wien, 26 July}}.</ref> and he formalised the continuous version of the problem in 1995.<ref>{{citation
 | last = Alpern | first = Steve | author-link = Steve Alpern
 | doi = 10.1137/S0363012993249195
 | issue = 3
 | journal = SIAM Journal on Control and Optimization
 | mr = 1327232
 | pages = 673–683
 | title = The rendezvous search problem
 | volume = 33
 | year = 1995}}</ref> This has led to much recent research in rendezvous search.<ref>{{citation
 | last1 = Alpern | first1 = Steve | author1-link = Steve Alpern
 | last2 = Gal | first2 = Shmuel | author2-link = Shmuel Gal
 | isbn = 0-7923-7468-1
 | location = Boston, MA
 | mr = 2005053
 | publisher = Kluwer Academic Publishers
 | series = International Series in Operations Research & Management Science
 | title = The Theory of Search Games and Rendezvous
 | volume = 55
 | year = 2003}}.</ref> Even the symmetric rendezvous problem played in ''n'' discrete locations (sometimes called the ''Mozart Cafe Rendezvous Problem'')<ref>{{citation
 | last = Alpern | first = Steve | author-link = Steve Alpern
 | editor-last = Cochran | editor-first = James J.
 | contribution = Rendezvous search games
 | doi = 10.1002/9780470400531.eorms0720
 | publisher = Wiley
 | title = Wiley Encyclopedia of Operations Research and Management Science
 | year = 2011}}.</ref> has turned out to be very difficult to solve, and in 1990 [[Richard Weber (mathematician)|Richard Weber]] and Eddie Anderson conjectured the optimal strategy.<ref>{{citation
 | last1 = Anderson | first1 = E. J.
 | last2 = Weber | first2 = R. R. | author2-link = Richard Weber (mathematician)
 | issue = 4
 | journal = Journal of Applied Probability
 | mr = 1077533
 | pages = 839–851
 | title = The rendezvous problem on discrete locations
 | url = http://www.statslab.cam.ac.uk/~rrw1/abstracts/a90a.html
 | volume = 27
 | year = 1990 | doi=10.2307/3214827| jstor = 3214827
 | s2cid = 122587972
 }}.</ref> In 2012 the conjecture was proved for ''n'' = 3 by [[Richard Weber (mathematician)|Richard Weber]].<ref>{{citation
 | last = Weber | first = Richard | author-link = Richard Weber (mathematician)
 | doi = 10.1287/moor.1110.0528
 | issue = 1
 | journal = [[Mathematics of Operations Research]]
 | mr = 2891149
 | pages = 111–122
 | title = Optimal symmetric Rendezvous search on three locations
 | url = http://www.statslab.cam.ac.uk/~rrw1/research/K3%20revised.pdf
 | volume = 37
 | year = 2012}}.</ref> This was the first non-trivial symmetric rendezvous search problem to be fully solved. The corresponding asymmetric rendezvous problem has a simple optimal solution: one player stays put and the other player visits a random permutation of the locations. 

As well as being problems of theoretical interest, rendezvous problems include real-world problems with applications in the fields of [[synchronization]], [[operating system]] design, [[operations research]], and even [[search and rescue]] operations planning.