Rendezvous problem

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 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,[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] In 2012 the conjecture was proved for n = 3 by Richard Weber.[6] 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.

## Deterministic rendezvous problem

Main article: Deterministic rendezvous problem

The deterministic rendezvous problem is a variant of the rendezvous problem where the players, or robots, must find each other by following a deterministic sequence of instructions. Although each robot follows the same instruction sequence, a unique label assigned to each robot is used for symmetry breaking.[7]

## See also

* Coordination game
* Dining philosophers problem
* Probabilistic algorithm
* Rendezvous hashing
* Search games
* Sleeping barber problem
* Superrationality
* Symmetry breaking
* Focal point, a default meeting place

## References

1. ↑ Alpern, Steve (1976), Hide and Seek Games, Seminar, Institut fur Hohere Studien, Wien, 26 July .
2. ↑ "The rendezvous search problem", SIAM Journal on Control and Optimization 33 (3): 673–683, 1995, doi:10.1137/S0363012993249195
3. ↑ The Theory of Search Games and Rendezvous, International Series in Operations Research & Management Science, 55, Boston, MA: Kluwer Academic Publishers, 2003, ISBN 0-7923-7468-1 .
4. ↑ Cochran, James J., ed. (2011), "Rendezvous search games", Wiley Encyclopedia of Operations Research and Management Science, Wiley, doi:10.1002/9780470400531.eorms0720 .
5. ↑ Anderson, E. J. (1990), "The rendezvous problem on discrete locations", Journal of Applied Probability 27 (4): 839–851, doi:10.2307/3214827, http://www.statslab.cam.ac.uk/~rrw1/abstracts/a90a.html .
6. ↑ "Optimal symmetric Rendezvous search on three locations", Mathematics of Operations Research 37 (1): 111–122, 2012, doi:10.1287/moor.1110.0528, http://www.statslab.cam.ac.uk/~rrw1/research/K3 revised.pdf .
7. ↑ Ta-Shma, Amnon; Zwick, Uri (April 2014). "Deterministic rendezvous, treasure hunts, and strongly universal exploration sequences". ACM Transactions on Algorithms 10 (3): 12. doi:10.1145/2601068.

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Topics in game theory

Definitions|

* Cooperative game
* Determinacy
* Escalation of commitment
* Extensive-form game
* First-player and second-player win
* Game complexity
* Graphical game
* Hierarchy of beliefs
* Information set
* Normal-form game
* Preference
* Sequential game
* Simultaneous game
* Simultaneous action selection
* Solved game
* Succinct game

Equilibrium
concepts|

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

Strategies|

* Dominant strategies
* Pure strategy
* Mixed strategy
* Strategy-stealing argument
* Tit for tat
* Grim trigger
* Collusion
* Backward induction
* Forward induction
* Markov strategy

Classes
of games|

* Symmetric game
* Perfect information
* Repeated game
* Signaling game
* Screening game
* Cheap talk
* Zero-sum game
* Mechanism design
* Bargaining problem
* Stochastic game
* n-player game
* Large Poisson game
* Nontransitive game
* Global game
* Strictly determined game
* Potential game

Games|

* Go
* Chess
* Infinite chess
* Checkers
* Tic-tac-toe
* Prisoner's dilemma
* Optional 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 game
* War of attrition
* El Farol Bar problem
* Fair division
* Fair cake-cutting
* Cournot game
* Deadlock
* Diner's dilemma
* Guess 2/3 of the average
* Kuhn poker
* Nash bargaining game
* Prisoners and hats puzzle
* Trust game
* Princess and Monster game
* Rendezvous problem

Theorems|

* Arrow's impossibility theorem
* Aumann's agreement theorem
* Folk theorem
* Minimax theorem
* Nash's theorem
* Purification theorem
* Revelation principle
* Zermelo's theorem

Key
figures|

* Albert W. Tucker
* Amos Tversky
* Ariel Rubinstein
* Claude Shannon
* Daniel Kahneman
* David K. Levine
* David M. Kreps
* Donald B. Gillies
* Drew Fudenberg
* Eric Maskin
* Harold W. Kuhn
* Herbert Simon
* Hervé Moulin
* Jean Tirole
* Jean-François Mertens
* Jennifer Tour Chayes
* John Harsanyi
* John Maynard Smith
* Antoine Augustin Cournot
* John Nash
* John von Neumann
* Kenneth Arrow
* Kenneth Binmore
* Leonid Hurwicz
* Lloyd Shapley
* Melvin Dresher
* Merrill M. Flood
* Olga Bondareva
* Oskar Morgenstern
* Paul Milgrom
* Peyton Young
* Reinhard Selten
* Robert Axelrod
* Robert Aumann
* Robert B. Wilson
* Roger Myerson
* Samuel Bowles
* Suzanne Scotchmer
* Thomas Schelling
* William Vickrey

See also|

* All-pay auction
* Alpha–beta pruning
* Bertrand paradox
* Bounded rationality
* Combinatorial game theory
* Confrontation analysis
* Coopetition
* First-move advantage in chess
* Game mechanics
* Glossary of game theory
* List of game theorists
* List of games in game theory
* No-win situation
* Solving chess
* Topological game
* Tragedy of the commons
* Tyranny of small decisions

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