Parent

absorbing state In a Markov chain model, a state that cannot be left once it is entered. Contrast with nonabsorbing state.

conditional event A sequentially dependent event that will occur only if some other event has already taken place. See also time-dependent event.

Constant-sum games: Games in which for every combination of strategies the sum of players' payoff is the same. For example, auction games for risk neutral bidders and a risk-neutral seller are constant-sum games, where a fixed social surplus from exchange is to be divided between the bidders and the bid-taker. More generally, all exchange situations which do neither allow for production nor for destruction of resources are constant-sum games.

Coordination games: Normal form game where the players have the same number of strategies, which can be indexed such that it is always a strict Nash equilibrium for both players to play strategies having the same index.

deterministic model A model in which the results are determined through known relationships among the states and events, and in which a given input will always produce the same output; for example, a model depicting a known chemical reaction. Contrast with stochastic model.

final state The values assumed by the state variables of a system, component, or simulation at the completion of some specified duration of time. Contrast with initial state.

game A physical or mental competition in which the participants, called players, seek to achieve some objective within a given set of rules. See also game theory.

game theory (l) The study of situations involving competing interests, modeled in terms of the strategies, probabilities, actions, gains, and losses of opposing players in a game. See also management game; war game. (2) The study of games to determine the probability of winning, given various strategies.

Game tree: Time structure of possible moves describing an extensive form game. A game tree is a set of nodes some which are linked by edges. A tree is a connected graph with no cycles. The first move of the game is identified with a distinguished node that is called the root of the tree. A play of the game consists of a connected chain of edges starting at the root of the tree and ending, if the game is finite, at a terminal node. The nodes in the tree represent the possible moves in the game. The edges leading away from a node represent the choices or actions available at that move. Each node other than the terminal node is assigned a player's name so that it is known who makes the choice at that move. Each terminal node must be labeled with the consequences for each player if the game ends in the outcome corresponding to that terminal node.

heuristic Pertaining to experimental, especially trial-and-error, methods of problem solving. Note: The resulting solution may not be the most desirable solution to the problem.

initial state The values assumed by the state variables of a system, component, or simulation at the beginning of some specified duration of time. Contrast with final state.

lag variable In a discrete simulation, a variable that is an output of one period and an input for some future period.

Markov chain model A discrete, stochastic model in which the probability that the model is in a given state at a certain time depends only on the value of the immediately preceding state. Also called Markov model. See also semi-Markov model.

Markov process A stochastic process that assumes that in a series of random events, the probability for the occurrence of each event depends only on the immediately preceding outcome. See also semi-Markov process.

Matching pennies: Extremely simplistic, symmetric, two player 2x2 game (which is said to be played by children), in which each player chooses either Head or Tail. If the choices differ, player 1 pays a dollar to player 2; if they are the same, player 2 pays player 1 a dollar. This game does not have an equilibrium in pure strategies, but the unique equilibrium involves each player selecting one of the two actions with equal probability. The game illustrates that interactively optimizing behavior may create the need to take actions randomly, in order not to be predictable by the opponent. For the exact determination of mixed equilibrium strategies, the assumption of expected utility is important. For a real-world situation closely resembling this game, think of penalty shooting in sports: both the goal-keeper and the player who shoots the ball play randomized strategies. They randomize their actions (left or right, upper corner or not) in a way such that the other player cannot improve by either action he takes, given the own probabilities of selecting the actions.

Monte Carlo method In modeling and simulation, any method that employs Monte Carlo simulation to determine estimates for unknown values in a deterministic problem. See also stochastic model.

Normal form vs. extensive form game: In normal (or strategic) form games, the players move (choose their actions) simultaneously. Whenever the strategy spaces of the players are discrete (and finite), the game can be represented compactly as an NxM-game (see below). By contrast, a game in extensive form specifies the complete order of moves (along the direction of time), typically in a game tree (see below), in addition to the complete list of payoffs and the available information at each point in time and under each contingency. As any normal form can be 'inflated' to an extensive form game, concepts of strategic equilibrium in general relate to extensive form games. Whenever the exact timing of actions is irrelevant to the payoffs, however, a game is represented with more parsimony in normal form.

NxM game: A normal form game for two players, where one player has N possible actions and the other one has M possible actions. In such a game, the payoffs pairs to any strategy combination can be neatly arranged in a matrix, and the game is easily analyzable. NxM-games thus provide an easy way to gain an idea of what the structure of a more complex game looks like.

Prisoners' dilemma: Consider the following story. Two suspects in a crime are put into separate cells. If they both confess, each will be sentenced to three years. If only one of them confesses, he will be freed and used to witness against the other, who will receive a sentence of ten years. If neither confesses, they will both convicted of a minor offense and spend just a year in prison. This game is easily put in matrix form as a 2x2 game (see above). Once this is done, it is pretty obvious that each prisoner (player) has a dominant strategy to confess. The unique equilibrium of this game thus leads to the (Pareto) inefficient outcome (efficiency). This provides the most famous example that strategic equilibrium typically implies inefficient outcomes, and even can lead to the worst possible outcome (any other outcome is pareto-dominating the equilibrium outcome.) The prisoners' dilemma game illustrates the structure of interaction in an oil cartel, or any oligolistic industry of quantity competition, where each firm has an incentive to 'spoil' the market by unilaterally increasing its own output. The same structure of interaction characterizes the problem of providing public goods (free rider problem), i.e. of voluntarily paying taxes.

queue In queuing theory, a set of zero or more entities waiting to be serviced by a service facility.

queuing model A model consisting of service facilities and entities waiting in queues to be served; for example, a model depicting teller windows and customers at a bank.

queuing theory The study of queues and of the performance of systems that service entities that are organized into queues

random Pertaining to a process or variable whose outcome or value depends on chance or on a process that simulates chance, often with the implication that all possible outcomes or values have an equal probability of occurrence; for example, the outcome of flipping a coin or executing a computer-programmed random number generator.

Repeated game: 'Super'-game where a fixed group of players plays a given game repeatedly, with the outcome of all previous plays observed before the next play begins. Repetition vastly enlargens the set of possible equilibrium outcomes in a game, as it opens possibilities to 'punish' or 'reward' later actions such that certain strategies form an equilibrium which would not form one in the single, unrepeated ('one-shot') game. For example, repeating the prisoners' dilemma game (often enough) gives rise to many equilibria where both prisoners never confess.

simulation A method for implementing a model over time. Also, a technique for testing, analysis, or training in which real-world systems are used or in which real-world and conceptual systems are reproduced by a model. (l) A model that behaves or operates like a given system when provided with a set of controlled inputs. Also called simulation model. See also emulation. (2) The process of developing or using a model as in (1). (3) Implementation of a special kind of model that represents at least some key internal elements of a system and describes how those elements interact over time. Most combat simulations are implemented as computer programs.

state (1) The internal status of a simulation entity (e.g., fuel level, number of rounds remaining, location of craters). State messages are used to start and restart entities or to update entities concerning the dynamic changes in the environment in their area of interest. See also simulation entity. (2) A condition or mode of existence that a system, component, or simulation may be in; for example, the preflight state of an aircraft navigation program or the input state of a given channel. (3) The values assumed at a given instant by the variables that define the characteristics of a system, component, or simulation. Also called system state. See also final state; initial state; steady state.

state transition A change from one state to another in a system, component, or simulation.

steady state A situation in which a model, process, or device exhibits stable behavior independent of time. Also called equilibrium.

stochastic model A model in which the results are determined by using one or more random variables to represent uncertainty about a process or in which a given input will produce an output according to some statistical distribution; for example, a model that estimates the total dollars spent at each of the checkout stations in a supermarket, based on probable number of customers and probable purchase amount of each customer. Also called probabilistic model. See also Markov chain model. Contrast with deterministic model.

tic-tac-toe 

The usual game of tic-tac-toe (also called ticktacktoe) is 3-in-a-row on a 3x3 board. However, a generalized n-in-row on an UxV board can also be considered. For n=1 and 2 the first player can always win. If the board is at least 3x4, the first player can win for 3.

However, for tic-tac-toe which uses a 3x3 board, a draw can always be obtained. If the board is at least 4x30, the first player can win for n=4. For m=5, a draw can always be obtained on a 5x5 board, but the first player can win if the board is at least  15x15.   On an ENDLESSxENDLESS Board, the first player can win for N=1, 2, 3, and 4, but a tie can always be forced for n>7.

transitive A relation R on a set S is transitive provided that for all x,y and z in S such that xRy and yRz, we also have xRz (e.g. x better than y, y better than z, then x better than z).

yoked variable One of two or more variables that are dependent on one another in such a manner that a change in one automatically causes a change in the others.

unit An aggregate of entities

yoked variable One of two or more variables that are dependent on one another in such a manner that a change in one automatically causes a change in the others.