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Princess And Monster Game
In game theory, a princess and monster game is a pursuit–evasion game played by two players in a region. Formal Definition In his book ''Differential Games'' (1965), Rufus Isaacs defined the game as: This game remained a well-known open problem until it was solved by Shmuel Gal in the late 1970s. His optimal strategy for the princess is to move to a random location in the room and stay still for a time interval which is neither too short nor too long, before going to another (independent) random location and repeating the procedure. The proposed optimal search strategy for the monster is based on subdividing the room into many narrow rectangles, picking a rectangle at random and searching it in some specific way, after some time picking another rectangle randomly and independently, and so on. Princess and monster games can be played on a pre-selected graph. It can be demonstrated that for any finite graph an optimal mixed search strategy exists that results in a finite p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Game Theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has applications in all fields of social science, as well as in logic, systems science and computer science. Originally, it addressed two-person zero-sum games, in which each participant's gains or losses are exactly balanced by those of other participants. In the 21st century, game theory applies to a wide range of behavioral relations; it is now an umbrella term for the science of logical decision making in humans, animals, as well as computers. Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum game and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pursuit–evasion
Pursuit–evasion (variants of which are referred to as cops and robbers and graph searching) is a family of problems in mathematics and computer science in which one group attempts to track down members of another group in an environment. Early work on problems of this type modeled the environment geometrically. In 1976, Torrence Parsons introduced a formulation whereby movement is constrained by a graph. The geometric formulation is sometimes called continuous pursuit–evasion, and the graph formulation discrete pursuit–evasion (also called graph searching). Current research is typically limited to one of these two formulations. Discrete formulation In the discrete formulation of the pursuit–evasion problem, the environment is modeled as a graph. Problem definition There are innumerable possible variants of pursuit–evasion, though they tend to share many elements. A typical, basic example is as follows (cops and robber games): Pursuers and evaders occupy nodes of a graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rufus Isaacs (game Theorist)
Rufus Philip Isaacs (June 11, 1914 – January 18, 1981) was a game theorist especially prominent in the 1950s and 1960s with his work on differential games. Biography Isaacs was born on 11 June 1914 in New York City. He worked for the RAND Corporation from 1948 until winter 1954/1955. His investigation stemmed from classic pursuit–evasion type zero-sum dynamic two-player games such as the Princess and monster game. In 1942, he married Rose Bicov, and they had two daughters. His work in pure mathematics included working with monodiffric functions, fractional-order mappings, graph theory, analytic functions, and number theory. In graph theory he constructed the first two infinite families of snarks. In applied mathematics, he worked with aerodynamics, elasticity, optimization, and differential games, which he is most known for. He received his bachelor's degree from MIT in 1936, and received his MA and PhD from Columbia University in 1942 and 1943 respectively. His first ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shmuel Gal
Shmuel Gal ( he, שמואל גל, born 1940) is a mathematician and professor of statistics at the University of Haifa in Israel. He devised the Gal's accurate tables method for the computer evaluation of elementary functions. With Zvi Yehudai he developed in 1993 a new algorithm for Sorting algorithm, sorting which is used by IBM. Gal has solved the Princess and monster game and made several significant contributions to the area of search games. He has been working on rendezvous problems with his collaborative colleagues Steve Alpern, Vic Baston, and John Howard.S. Gal and J. Howard (2005). Rendezvous-evasion search in two boxes, OPERATIONS RESEARCH. Gal received a Ph.D. in mathematics from the Hebrew University of Jerusalem. His thesis advisor was Aryeh Dvoretzky. References External linksProf. Shmuel Gal - Home page {{DEFAULTSORT:Gal, Shmuel Game theorists Hebrew University of Jerusalem alumni Israeli mathematicians Israeli operations researchers University of Haifa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph (discrete Mathematics)
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a Set (mathematics), set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mixed Strategy
In game theory, a player's strategy is any of the options which they choose in a setting where the outcome depends ''not only'' on their own actions ''but'' on the actions of others. The discipline mainly concerns the action of a player in a game affecting the behavior or actions of other players. Some examples of "games" include chess, bridge, poker, monopoly, diplomacy or battleship. A player's strategy will determine the action which the player will take at any stage of the game. In studying game theory, economists enlist a more rational lens in analyzing decisions rather than the psychological or sociological perspectives taken when analyzing relationships between decisions of two or more parties in different disciplines. The strategy concept is sometimes (wrongly) confused with that of a move. A move is an action taken by a player at some point during the play of a game (e.g., in chess, moving white's Bishop a2 to b3). A strategy on the other hand is a complete algorithm for p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steve Alpern
Professor Steve Alpern is a professor of Operational Research at the University of Warwick, where he recently moved after working for many years at the London School of Economics. His early work was mainly in the area of dynamical systems and ergodic theory, but his more recent research has been concentrated in the fields of search games and rendezvous problem, rendezvous. He informally introduced the rendezvous problem as early as 1976.Steve Alpern (1976). ''Hide and Seek Games''. Seminar, Institut fur Hohere Studien, Wien, 26 July His collaborators include Shmuel Gal, Vic Baston and Robbert Fokkink. External links Author profilein the database Zentralblatt MATH, zbMATH References Academics of the University of Warwick Game theorists Living people Courant Institute of Mathematical Sciences alumni Princeton University alumni Year of birth missing (living people) {{UK-academic-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mikhail Zelikin
Mikhail Il'ich Zelikin (russian: Михаил Ильич Зеликин; born 11 February 1936) is a Russian mathematician, who works on differential equations (in particular, Riccati equations), optimal control theory, differential games (for instance, Princess and monster game), the theory of fields of extremals for multiple integrals, the geometry of Grassmannians. He proposed an explanation of ball lightning based on the hypothesis of plasma superconductivity. Journal of Mathematical Sciences, vol. 151, No. 6, pp. 3473–3496. Biography M. I. Zelikin was born in in 1936. He attended |
Search Games
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" and has been developed further by Shmuel GalS. Gal, ''Search Games'', Academic Press, New York (1980)S. Alpern and S. Gal, The Theory of Search Games and Rendezvous', Springer (2003). and Steve Alpern. The princess and monster game deals with a moving target. Strategy A natural strategy to search fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Games In Game Theory
Game theory studies strategic interaction between individuals in situations called games. Classes of these games have been given names. This is a list of the most commonly studied games Explanation of features Games can have several features, a few of the most common are listed here. *Number of players: Each person who makes a choice in a game or who receives a payoff from the outcome of those choices is a player. *Strategies per player: In a game each player chooses from a set of possible actions, known as pure strategies. If the number is the same for all players, it is listed here. *Number of pure strategy Nash equilibria: A Nash equilibrium is a set of strategies which represents mutual best responses to the other strategies. In other words, if every player is playing their part of a Nash equilibrium, no player has an incentive to unilaterally change their strategy. Considering only situations where players play a single strategy without randomizing (a pure strategy) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pursuit–evasion
Pursuit–evasion (variants of which are referred to as cops and robbers and graph searching) is a family of problems in mathematics and computer science in which one group attempts to track down members of another group in an environment. Early work on problems of this type modeled the environment geometrically. In 1976, Torrence Parsons introduced a formulation whereby movement is constrained by a graph. The geometric formulation is sometimes called continuous pursuit–evasion, and the graph formulation discrete pursuit–evasion (also called graph searching). Current research is typically limited to one of these two formulations. Discrete formulation In the discrete formulation of the pursuit–evasion problem, the environment is modeled as a graph. Problem definition There are innumerable possible variants of pursuit–evasion, though they tend to share many elements. A typical, basic example is as follows (cops and robber games): Pursuers and evaders occupy nodes of a graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
