Search game

A search game is a two-person zero-sum game which takes place in a set called the search space. The searcher can choose any continuous trajectory subject to a maximal velocity constraint. It is always assumed that neither the searcher nor the hider has any knowledge about the movement of the other player until their distance apart is less than or equal to the discovery radius and at this very moment capture occurs. As mathematical models, search games can be applied to areas such as hide-and-seek games that children play or representations of some tactical military situations. The area of search games was introduced in the last chapter of Rufus Isaacs' classic book "Differential Games"^[1] and has been developed further by Shmuel Gal ^[2]^[3] and Steve Alpern.^[3]

What is the best way to search a stationary target in a graph? A natural strategy is to find a minimal closed curve L that covers all the arcs of the graph. (L is called a Chinese postman tour). Then, traverse L with probability 1/2 for each direction. This strategy seems to work well if the graph is Eulerian. In general, this random Chinese postman tour is indeed an optimal search strategy if and only if the graph consists of a set of Eulerian graphs connected in a tree-like structure.^[4] A misleadingly simple example of a graph not in this family consists of two nodes connected by three arcs. The random Chinese postman tour (equivalent to traversing the three arcs in a random order) is not optimal. The optimal way to search these three arcs is surprisingly complicated ^[2] .

The princess and monster game deals with a moving target.

Searching unbounded domains is also interesting. In general, the reasonable framework, as in the case of an online algorithm, is to use a normalized cost function (called the competitive ratio in Computer Science literature). The minimax trajectory for problems of these types is always a geometric sequence (or exponential function for continuous problems). This result yields an easy method to find the minimax trajectory by minimizing over a single parameter (the generator of this sequence) instead of searching over the whole trajectory space. This tool has been used for the linear search problem, i.e., finding a target on the infinite line, which has attracted much attention over several decades and has been analyzed as a search game.^[5] It has also been used to find a minimax trajectory for searching a set of concurrent rays. Optimal searching in the plane is performed by using exponential spirals.^[2]^[3]^[6] Searching a set of concurrent rays was later re-discovered in Computer Science literature as the 'cow-path problem'.^[7]

References

↑ Rufus Isaacs, Differential Games, John Wiley and Sons, (1965),
1 2 S. Gal, Search Games, Academic Press, New York (1980)
1 2 3 S. Alpern and S. Gal, The Theory of Search Games and Rendezvous, Springer (2003).
↑ S. Gal, On the optimality of a simple strategy for searching graphs, Int. J. Game Theory (2000).
↑ A. Beck and D.J. Newman. Yet More on the linear search problem. Israel J. Math. (1970).
↑ M. Chrobak, A princess swimming in the fog looking for a monster cow, ACM Sigact news, 35(2), 74–78 (2004).
↑ MY Kao, JH Reif and SR Tate, Searching in an unknown environment: an optimal randomized algorithm for the cow-path problem, SODA 1993.

Topics in game theory

Definitions	Normal-form game Extensive-form game Escalation of commitment 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 n-player game Large Poisson game Nontransitive game Global games Strictly determined game

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 Fair division Fair 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 Rendezvous 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 David M. Kreps Harold W. Kuhn Eric Maskin Jean-François Mertens Paul Milgrom Oskar Morgenstern Hervé Moulin 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 Amos Tversky 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 Alpha–beta pruning Bertrand paradox Bounded rationality No-win situation Coopetition

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