Rendezvous problem

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The rendezvous dilemma can be formulated in this way:

Two young 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 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, of course, 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,^[1] and he formalised the continuous version of the problem in 1995.^[2] This has led to much recent research in rendezvous search.^[3] Even the symmetric rendezvous problem played in n discrete locations (sometimes called the Mozart Cafe Rendezvous Problem)^[4] has turned out to be very difficult to solve, and in 1990 Richard Weber and Eddie Anderson conjectured the optimal strategy.^[5] Only recently has the conjecture been proved for n = 3 by Richard Weber.^[6] This was the first non-trivial symmetric rendezvous search problem to be fully solved.

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.

References

↑ Alpern, Steve (1976), Hide and Seek Games, Seminar, Institut fur Hohere Studien, Wien, 26 July .
↑ Alpern, Steve (1995), "The rendezvous search problem", SIAM Journal on Control and Optimization 33 (3): 673–683, doi:10.1137/S0363012993249195, MR 1327232
↑ Alpern, Steve; Gal, Shmuel (2003), The Theory of Search Games and Rendezvous, International Series in Operations Research & Management Science 55, Boston, MA: Kluwer Academic Publishers, ISBN 0-7923-7468-1, MR 2005053 .
↑ Alpern, Steve (2011), "Rendezvous search games", in Cochran, James J., Wiley Encyclopedia of Operations Research and Management Science, Wiley, doi:10.1002/9780470400531.eorms0720 .
↑ Anderson, E. J.; Weber, R. R. (1990), "The rendezvous problem on discrete locations", Journal of Applied Probability 27 (4): 839–851, MR 1077533 .
↑ Weber, Richard (2012), "Optimal symmetric Rendezvous search on three locations", Mathematics of Operations Research 37 (1): 111–122, doi:10.1287/moor.1110.0528, MR 2891149 .

v t e Topics in game theory

Definitions	Normal-form game Extensive-form game Graphical game Cooperative game Succinct game Information set Hierarchy of beliefs Preference

Equilibrium concepts	Nash equilibrium Subgame perfection Mertens-stable equilibrium Bayesian-Nash Perfect Bayesian Trembling hand Proper equilibrium Epsilon-equilibrium Correlated equilibrium Sequential equilibrium Quasi-perfect equilibrium Evolutionarily stable strategy Risk dominance Core Shapley value Pareto efficiency Quantal response equilibrium Self-confirming equilibrium Strong Nash equilibrium Markov perfect equilibrium

Strategies	Dominant strategies Pure strategy Mixed strategy Tit for tat Grim trigger Collusion Backward induction Forward induction Markov strategy

Classes of games	Symmetric game Perfect information Simultaneous game Sequential game Repeated game Signaling game Cheap talk Zero-sum game Mechanism design Bargaining problem Stochastic game Large Poisson game Nontransitive game Global games

Games	Prisoner's dilemma Traveler's dilemma Coordination game Chicken Centipede game Volunteer's dilemma Dollar auction Battle of the sexes Stag hunt Matching pennies Ultimatum game Rock-paper-scissors Pirate game Dictator game Public goods game Blotto games War of attrition El Farol Bar problem Cake cutting Cournot game Deadlock Diner's dilemma Guess 2/3 of the average Kuhn poker Nash bargaining game Screening game Prisoners and hats puzzle Trust game Princess and monster game Monty Hall problem

Theorems	Minimax theorem Nash's theorem Purification theorem Folk theorem Revelation principle Arrow's impossibility theorem

Key figures	Kenneth Arrow Robert Aumann Kenneth Binmore Samuel Bowles Melvin Dresher Merrill M. Flood Drew Fudenberg Donald B. Gillies John Harsanyi Leonid Hurwicz David K. Levine Daniel Kahneman Harold W. Kuhn Eric Maskin Jean-François Mertens Paul Milgrom Oskar Morgenstern Roger Myerson John Nash John von Neumann Ariel Rubinstein Thomas Schelling Reinhard Selten Herbert Simon Lloyd Shapley John Maynard Smith Jean Tirole Albert W. Tucker William Vickrey Robert B. Wilson Peyton Young

See also	Tragedy of the commons Tyranny of small decisions All-pay auction List of games in game theory Confrontation analysis List of game theorists Combinatorial game theory

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Rendezvous problem

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References