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Decision Tree Game VideoGame Trees - Artificial Intelligence in Games
Two-person games can also be represented as and-or trees. For the first player to win a game, there must exist a winning move for all moves of the second player.
This is represented in the and-or tree by using disjunction to represent the first player's alternative moves and using conjunction to represent all of the second player's moves.
With a complete game tree, it is possible to "solve" the game — that is to say, find a sequence of moves that either the first or second player can follow that will guarantee the best possible outcome for that player usually a win or a tie.
The algorithm which is generally called backward induction or retrograde analysis can be described recursively as follows.
It is usually possible to solve a game in this technical sense of "solve" using only a subset of the game tree, since in many games a move need not be analyzed if there is another move that is better for the same player for example alpha-beta pruning can be used in many deterministic games.
Any subtree that can be used to solve the game is known as a decision tree , and the sizes of decision trees of various shapes are used as measures of game complexity.
Randomized algorithms can be used in solving game trees. There are two main advantages in this type of implementation: speed and practicality.
Moreover, it is practical because randomized algorithms are capable of "foiling an enemy", meaning an opponent cannot beat the system of game trees by knowing the algorithm used to solve the game tree because the order of solving is random.
The following is an implementation of randomized game tree solution algorithm: . The algorithm makes use of the idea of " short-circuiting ": if the root node is considered an " OR " operator, then once one True is found, the root is classified as True ; conversely, if the root node is considered an " AND " operator then once one False is found, the root is classified as False.
From Wikipedia, the free encyclopedia. For game tree as it is used in game theory not combinatorial game theory , see Extensive-form game.
Artificial Intelligence. Categories : Combinatorial game theory Trees graph theory. In this case, cooperation implies adhering to the agreement, while defection means secretly reneging on the agreement and retaining the nuclear arsenal.
The best outcome for either nation, unfortunately, is to renege on the agreement and retain the nuclear option while the other nation eliminates its arsenal since this will give the former a tremendous hidden advantage over the latter if war ever breaks out between the two.
The second-best option is for both to defect or not cooperate since this retains their status as nuclear powers.
The most common application of the Cournot model is in describing a duopoly or two main producers in a market. For example, assume companies A and B produce an identical product and can produce high or low quantities.
If they both cooperate and agree to produce at low levels, then limited supply will translate into a high price for the product on the market and substantial profits for both companies.
On the other hand, if they defect and produce at high levels, the market will be swamped and result in a low price for the product and consequently lower profits for both.
But if one cooperates i. The payoff matrix for companies A and B is shown figures represent profit in millions of dollars.
In coordination, players earn higher payoffs when they select the same course of action. As an example, consider two technology giants who are deciding between introducing a radical new technology in memory chips that could earn them hundreds of millions in profits, or a revised version of an older technology that would earn them much less.
The payoff matrix is shown below figures represent profit in millions of dollars. In this case, it makes sense for both companies to work together rather than on their own.
This is an extensive-form game in which two players alternately get a chance to take the larger share of a slowly increasing money stash.
The centipede game is sequential since the players make their moves one after another rather than simultaneously; each player also knows the strategies chosen by the players who played before them.
The game concludes as soon as a player takes the stash, with that player getting the larger portion and the other player getting the smaller portion.
But if B passes, A now gets to decide whether to take or pass, and so on. If both write down the same value, the airline will reimburse each of them that amount.
The process of backward induction, for example, can help explain how two companies engaged in a cutthroat competition can steadily ratchet product prices lower in a bid to gain market share , which may result in them incurring increasingly greater losses in the process.
It essentially involves a couple trying to coordinate their evening out. Where should they go? Cell d is the payoff if both make it to the ball game he enjoys it more than she does.
Cell c represents the dissatisfaction if both go not only to the wrong location but also to the event they enjoy least—the woman to the ball game and the man to the play.
The dictator game is closely related to the ultimatum game, in which Player A is given a set amount of money, part of which has to be given to Player B, who can accept or reject the amount given.
The dictator and ultimatum games hold important lessons for issues such as charitable giving and philanthropy. However, if A engages in price cutting war but B does not, A would have a higher payoff of 4 since it may be able to capture substantial market share, and this higher volume would offset lower product prices.
The worst possible outcome is realized if nobody volunteers. For example, consider a company where accounting fraud is rampant but top management is unaware of it.
Game theory can be used very effectively as a tool for decision-making whether in an economical, business or personal setting.
For related reading, see: Game Theory: Beyond the Basics. Business Essentials. Behavioral Economics. Wealth Management. Investing Essentials.