|
Backwards Induction
Backward induction is the process of determining a sequence of optimal choices by reasoning from the endpoint of a problem or situation back to its beginning using individual events or actions. Backward induction involves examining the final point in a series of decisions and identifying the optimal process or action required to arrive at that point. This process continues backward until the best action for every possible point along the sequence is determined. Backward induction was first utilized in 1875 by Arthur Cayley, who discovered the method while attempting to solve the secretary problem. In dynamic programming, a method of mathematical optimization, backward induction is used for solving the Bellman equation. In the related fields of automated planning and scheduling and automated theorem proving, the method is called backward search or backward chaining. In chess, it is called retrograde analysis. In game theory, a variant of backward induction is used to compute sub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter "M" first and "Y" last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be '' finite'', as in these examples, or '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero-sum
Zero-sum game is a mathematical representation in game theory and economic theory of a situation that involves two competing entities, where the result is an advantage for one side and an equivalent loss for the other. In other words, player one's gain is equivalent to player two's loss, with the result that the net improvement in benefit of the game is zero. If the total gains of the participants are added up, and the total losses are subtracted, they will sum to zero. Thus, cutting a cake, where taking a more significant piece reduces the amount of cake available for others as much as it increases the amount available for that taker, is a zero-sum game if all participants value each unit of cake equally. Other examples of zero-sum games in daily life include games like poker, chess, sport and bridge where one person gains and another person loses, which results in a zero-net benefit for every player. In the markets and financial instruments, futures contracts and options are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Terminator
''The Terminator'' is a 1984 American science fiction action film directed by James Cameron, written by Cameron and Gale Anne Hurd and produced by Hurd. It stars Arnold Schwarzenegger as the Terminator, a cybernetic assassin sent back in time from 2029 to 1984 to assassinate Sarah Connor (Linda Hamilton), whose unborn son will one day save mankind from extinction by Skynet, a hostile artificial intelligence, in a post-apocalyptic future. Kyle Reese ( Michael Biehn) is a soldier sent back in time to protect Sarah. The screenplay is credited to Cameron and Hurd, while co-writer William Wisher Jr. received an "additional dialogue" credit. Cameron devised the premise of the film from a fever dream he experienced during the release of his first film, '' Piranha II: The Spawning'' (1982), in Rome, and developed the concept in collaboration with Wisher. He sold the rights to the project to fellow New World Pictures alumna Hurd on the condition that she would produce the film ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multi-stage Game
In game theory, a multi-stage game is a sequence of several simultaneous games played one after the other. This is a generalization of a repeated game: a repeated game is a special case of a multi-stage game, in which the stage games are identical. Multi-Stage Game with Different Information Sets As an example, consider a two-stage game in which the stage game in ''Figure 1'' is played in each of two periods: The payoff to each player is the simple sum of the payoffs of both games. Players cannot observe the action of the other player within a round; however, at the beginning of Round 2, Player 2 finds out about Player 1's action in Round 1, while Player 1 does not find out about Player 2's action in Round 1. For Player 1, there are 2^3=8 strategies. For Player 2, there are 2^5=32 strategies. The extensive form of this multi-stage game is shown in ''Figure 2'': In this game, the only Nash Equilibrium in each stage is (B, b). (BB, bb) will be the Nash Equilibrium for the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector (mathematics And Physics)
In mathematics and physics, vector is a term that refers to physical quantity, quantities that cannot be expressed by a single number (a scalar (physics), scalar), or to elements of some vector spaces. Historically, vectors were introduced in geometry and physics (typically in mechanics) for quantities that have both a magnitude and a direction, such as displacement (geometry), displacements, forces and velocity. Such quantities are represented by geometric vectors in the same way as distances, masses and time are represented by real numbers. The term ''vector'' is also used, in some contexts, for tuples, which are finite sequences (of numbers or other objects) of a fixed length. Both geometric vectors and tuples can be added and scaled, and these vector operations led to the concept of a vector space, which is a set (mathematics), set equipped with a vector addition and a scalar multiplication that satisfy some axioms generalizing the main properties of operations on the abov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subgame
In game theory, a subgame is any part (a subset) of a game that meets the following criteria (the following terms allude to a game described in extensive form): #It has a single initial node that is the only member of that node's information set (i.e. the initial node is in a singleton information set). #If a node is contained in the subgame then so are all of its successors. #If a node in a particular information set is in the subgame then all members of that information set belong to the subgame. It is a notion used in the solution concept of subgame perfect Nash equilibrium, a refinement of the Nash equilibrium that eliminates non-credible threats. The key feature of a subgame is that it, when seen in isolation, constitutes a game in its own right. When the initial node of a subgame is reached in a larger game, players can concentrate only on that subgame; they can ignore the history of the rest of the game (provided they know what subgame they are playing). This is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subgame Perfect Equilibrium
In game theory, a subgame perfect equilibrium (SPE), or subgame perfect Nash equilibrium (SPNE), is a refinement of the Nash equilibrium concept, specifically designed for dynamic games where players make sequential decisions. A strategy profile is an SPE if it represents a Nash equilibrium in every possible subgame of the original game. Informally, this means that at any point in the game, the players' behavior from that point onward should represent a Nash equilibrium of the continuation game (i.e. of the subgame), no matter what happened before. This ensures that strategies are credible and rational throughout the entire game, eliminating non-credible threats. Every finite extensive game with complete information (all players know the complete state of the game) and perfect recall (each player remembers all their previous actions and knowledge throughout the game) has a subgame perfect equilibrium. A common method for finding SPE in finite games is backward induction, wher ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extensive-form Game
In game theory, an extensive-form game is a specification of a game allowing for the explicit representation of a number of key aspects, like the sequencing of players' possible moves, their choices at every decision point, the (possibly imperfect) information each player has about the other player's moves when they make a decision, and their payoffs for all possible game outcomes. Extensive-form games also allow for the representation of incomplete information in the form of chance events modeled as " moves by nature". Extensive-form representations differ from normal-form in that they provide a more complete description of the game in question, whereas normal-form simply boils down the game into a payoff matrix. Finite extensive-form games Some authors, particularly in introductory textbooks, initially define the extensive-form game as being just a game tree with payoffs (no imperfect or incomplete information), and add the other elements in subsequent chapters as refinements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Game Tree
In the context of combinatorial game theory, a game tree is a graph representing all possible game states within a sequential game that has perfect information. Such games include chess, checkers, Go, and tic-tac-toe. A game tree can be used to measure the complexity of a game, as it represents all the possible ways that the game can pan out. Due to the large game trees of complex games such as chess, algorithms that are designed to play this class of games will use partial game trees, which makes computation feasible on modern computers. Various methods exist to solve game trees. If a complete game tree can be generated, a deterministic algorithm, such as backward induction or retrograde analysis can be used. Randomized algorithms and minmax algorithms such as MCTS can be used in cases where a complete game tree is not feasible. Understanding the game tree To better understand the game tree, it can be thought of as a technique for analyzing adversarial games, whi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Search Theory
In microeconomics, search theory studies buyers or sellers who cannot instantly find a trading partner, and must therefore search for a partner prior to transacting. It involves determining the best approach to use when looking for a specific item or person in a sizable, uncharted environment. The goal of the theory is to determine the best search strategy, one that maximises the chance of finding the target while minimising search-related expenses. Search theory clarifies how buyers and sellers choose when to acknowledge a coordinating offer for a transaction. Search theory also provides an explanation for why frictional unemployment happens as people look for jobs and corporations look for new employees. Search theory has been used primarily to explain labor market inefficiencies, but also for all forms of "buyers" and "sellers", whether products, homes or even spouses/partners. It can be applied. The clearing price will be met quickly as supply and demand react freely. However ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimal Stopping
In mathematics, the theory of optimal stopping or early stopping : (For French translation, secover storyin the July issue of ''Pour la Science'' (2009).) is concerned with the problem of choosing a time to take a particular action, in order to Optimization (mathematics), maximise an expected reward or minimise an expected cost. Optimal stopping problems can be found in areas of statistics, economics, and mathematical finance (related to the pricing of American options). A key example of an optimal stopping problem is the secretary problem. Optimal stopping problems can often be written in the form of a Bellman equation, and are therefore often solved using dynamic programming. Definition Discrete time case Stopping rule problems are associated with two objects: # A sequence of random variables X_1, X_2, \ldots, whose joint distribution is something assumed to be known # A sequence of 'reward' functions (y_i)_ which depend on the observed values of the random variables in 1: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marginal Utility
Marginal utility, in mainstream economics, describes the change in ''utility'' (pleasure or satisfaction resulting from the consumption) of one unit of a good or service. Marginal utility can be positive, negative, or zero. Negative marginal utility implies that every consumed additional unit of a commodity causes more harm than good, leading to a decrease in overall utility. In contrast, positive marginal utility indicates that every additional unit consumed increases overall utility. In the context of cardinal utility, liberal economists postulate a law of diminishing marginal utility. This law states that the first unit of consumption of a good or service yields more satisfaction or utility than the subsequent units, and there is a continuing reduction in satisfaction or utility for greater amounts. As consumption increases, the additional satisfaction or utility gained from each additional unit consumed falls, a concept known as ''diminishing marginal utility.'' This idea is us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
