Perfect information
Game with perfect information, sometimes called game with perfect information, is a term of mathematical game theory. Thus has a game perfect information if every player at the time of a decision, ie, the previously taken decisions of his teammates as well as the random choices they made earlier, are always known the previous game.
Among the board games, most games have perfect information. Examples are Go, chess, checkers, mill, Nim and Mancala as two-person games or even solitaire and Same Game as a solitaire game. Even games with random effect such as backgammon and Parcheesi, the latter regardless of the number of players have perfect information.
A game without perfect information is also referred to as a game of imperfect information. Examples include card games such as Skat and poker, because there each player knows only his own cards. Even a game with simultaneous moves like scissor, paper, stone is not a game with perfect information, there is the crucial players do not know the place simultaneously decision of the counterparty. To that extent are information booths, which are terms of player asymmetric characteristic for games with imperfect information.
- 2.2.1 One or two
- 2.2.2 Wythoffs game
- 2.2.3 Nim
- 2.2.4 Hex
- 2.3.1 One or two
- 2.3.2 The other games
- 2.4.1 One or two
- 2.4.2 Wythoffs game
- 2.4.3 Nim
- 2.4.4 Hex
Properties
1912 proved Ernst Zermelo that a finite two-person zero-sum game with perfect information and without random effect has a uniquely determined match result, in the sense that it can enforce this result regardless of the playing style of opponents a player has at least while the counterparty is possible to prevent an even higher result. For the Zermelo exemplary looked chess game, this means concretely that chess either
- Like a problem White has a winning strategy allowed or
- Black has such a winning strategy, or
- Both players for each can force at least a draw.
Applied to a return game with colors reversed Zermelo's theorem has the consequence that a player with an error-free game can at least force a break even - but no more, if the counterparty is also playing error-free.
The water formed in both sides of an error-free game result is called the value of the game. His practical determination can be very difficult for a given game, even if theoretically there is always the minimax algorithm, a calculation option. Many games, including lady, mill and four wins, have now been fully resolved and the corresponding strategies are known. For some games such as Hex, only the values of the initial position due to symmetry considerations can be determined, without knowing associated strategies.
Games a special subclass are studied within the so-called combinatorial game theory.
For a finite two-person zero-sum game with perfect information and chance influence Zermelo's theorem applies analogously in terms of the expected value of the profit. That is, a player has a strategy so that the expected value of his winnings reached at least this value, while his opponent, there is a strategy which limits the profit expectations of the first player to a maximum of this value. Thus, for example, each position of backgammon, which must be strictly limited to a finite length, a clearly defined value.
For finite games with perfect information, played with more than two people or which have no zero-sum character, Zermelo's theorem does not apply. That is, for such games can generally be no objectively best strategies found. However, there are following a set of Harold Kuhn from 1950 equilibria in pure strategies.
The property of the perfect information can not be seen from the normal shape of a game in general. Within the extensive form, which represents a game on the basis of a directed graph, the perfect information is characterized by single-element amounts of information.
Random Free finite games with two players without a draw
Zermelo's theorem applies even more to those of these games, in which a draw can not occur. In these finite two-person zero-sum games with perfect information and without random draw influence from his sentence follows the statement:
However, Zermelo's argument is a set-theoretical existence argument and contains no algorithm to categorize the positions of a particular game.
As mentioned above, provides the minimax algorithm, which is expected to graph-theoretic approaches to any position they categorize a value. In the following, however, less complex algorithms and criteria are to be presented.
The categorization of positions can work out with pencil and paper, which corresponds to the graph-theoretical categorization of small tableaux. If one has such a feel, it will be possible from this to develop a numerical categorization.
The Nim game (see below) was investigated in detail in 1935 by Roland Sprague and independently in 1939 by P. Grundy and made the prime example of this kind of games (see set of Sprague- Grundy ). Therefore, it marks a starting point of mathematical game theory.
In this Paper of Grundy the Z- position "W" means ( for winning ) and the C- position "L" ( for losing ). It would have to be to say that this is the view of the train donating player. But it is to the marking of a position and not a train as it brings the Z- and C- manner of speaking directly expressed. In Wythoff the Z- position "cold" and the C- position is called " hot ".
Examples
One or two
This match game is one of the simplest:
Between two players is a bunch of matches. Both players alternately take one or two sticks. Who takes the last piece of wood, wins (see one or two and Bachet'sches game).
You can change the rule, in that that loses has to take the last piece of wood.
Wythoffs game
In the " Game of Wythoff " refer to two players alternately stacking of 2 items, from one of the two stacks, or both, but then the same number of each stack. The game ends when a player takes the last object - which he wins the game.
The game has been analyzed by this Dutch mathematician in 1907 mathematically, but is According to Martin Gardner in China under the name捡 石子jiǎn shízǐ ( " take stones " ) have been played ..
In the game of Rufus Isaacs (quoted from mountains ) highlight the player in the first quadrant of the plane with integer lattice ( in formulas ) alternately a node, the successor node seen from the previous node on the same coordinate ( horizontal or vertical ) or the bisector must be located respectively in the direction of origin. The player who reaches first the origin, has won. The initial position is a draw.
The two rules are equivalent. However, the graphical view is particularly well suited for the representation of the graph-theoretical algorithm.
Nim
When Nim game with matches several rows are available. Two players alternately take matches from one of the rows away. How many they take, does not matter; it must be at least a match and there where there is a train only matches a single row to be taken. The player who makes the last train - that takes away the last matches - wins.
Misère Nim
The player who makes the last train ( the last match takes ) has not won, but lost. This rule is called Misère rule.
The Nim game Marienbad, which was known for the film Last Year at Marienbad by Alain Resnais, is a Misère variant.
Hex
The strategic board game Hex is a not " neutral " (English impartial ) game since the two players, called " Red" and "Blue", can only make trains with stones of their color. As a game of this type, it can not be analyzed with the set of Sprague- Grundy.
For game rules, etc. may be made to the
Referenced. This article is only to be shown that the positions can be categorized graph-theoretical as well as with the other deterministic games.
Graph Theoretical categorization
The positions can be for any (finite) tableau in a directed graph with nodes representing the positions and arcs ( directed edges ) for the possible moves. That is, a single potential train is represented by an arc u v leads from a direct predecessor -position to a direct successor position, in characters u → v. This graph reflects the rule of the game restricted to the tableau exactly resist.
The initially established requirement of finiteness is fulfilled only by the freedom cycles of the graph is therefore a directed acyclic graph ( DAG german for directed acyclic graph ). In the form defined in this way he would therefore called position DAG.
Furthermore, we also assume:
If there is an end node with Misère - profit rule, we add to it a bow to an additional node to which train the winner finally reaps victory.
The statement of the following algorithm is simplified by a strict total ordering ≻ of the positions that is compatible with the sequence of moves. You can, for example as follows can be constructed:
Algorithm:
Through the stepwise approach the total order is up to ensure that we scour the entire position space. Already marked nodes remain unchanged; it is also necessary otherwise no action with them.
Proof of the algorithm, and thus further proof of the statement:
In the graph-theoretic way of speaking we have a directed acyclic graph a core ( engl. kernel, French noyau ) constructed, i.e., a stable subset of nodes, which is dominant at the same time. The core is the amount of the Z- node. He always exists and is unique. The stability is the compulsion to go from a Z- node to a C- node, and the dominance of the chance to go from a C- node to a Z- node.
The given algorithm can - at least theoretically - cleared of any position, whether it is a profit or loss position. However, the size of the position DAG, which is still larger than the position space, make a "practical" implementation impossible.
The graphs below represent the results that can be obtained with the described algorithm. The outgoing of the (green) Z- node C actions are represented by red "rays" that make the steps taken by them to nodes ( red ) C- node.
One or two
Even the trivial lash or two, the optimal strategy is not much math to understand right away, can be combined with the above algorithm treat: The staging area is an interval [0, n] of non- negative integers, the instantaneous number of matches represent.
Wythoffs game
In Wythoffs match the properties of the core come out well. The black outline of the rules of the game in the lower right corner symbolizes a kind of " future light cone ", the corresponding " past light cone ", originating from the Z- node (green). If a C- node (red) drawn as the initial position of attracting players can force the win. A Z - node (green) as the initial position of the catching can only reach a C- node, after which his opponent can force the win.
Nim
The position DAG you make your bed at Nim in an obvious way in the n-dimensional interval [0, r1 ] × [0, r2 ] × ... × [0, rn ] of the positions, where n is the number of rows and ri the number of matches in the i- th row. A game position then corresponds to a node with the current numbers of matches than its coordinates. The possible moves ( arcs) are parallel to exactly one to coordinate with the direction of the origin.
Each of the lexicographic orders is compatible with the sequence of moves Totalordung. There is only one end position, which coincides with the origin.
The graph on the right shows two diagrams in the Z- positions (green nodes) and the C- positions ( red nodes) for the standard (top) and the Poverty of rule ( below) of the Nim game - each for simplicity in a tableau of 3 rows (axes) to 5, 4 and 1 piece of wood. In both diagrams, the number with one match by two superimposed layers, the 0 - and 1- level shown, so that the 0 level effectively the presence corresponding to the absence of the third row and the 1- layer 1 piece of wood in this series.
The origin ( 0,0,0) is respectively located on the bottom left front.
Valid moves are movements in the graph in exactly one of the coordinate axes, that is, either horizontally or vertically or even in the third direction of the 1- level on the 0 - level, each closer toward the origin.
The algorithm starts at the origin, the green at the standard rule is colored red in the Misère rule.
It is striking that differ between the 2 Nim variants, the colors only at the nodes very close to the origin.
The winning strategy in 2 rows for the standard rule is: If you can, go with your opponent equal.
With 3 or more rows, it becomes more complicated, as already show the 1-plane in the graph suggestively. The more general cases, especially those with more than three rows, are more difficult to represent.
Hex
If the diamond-shaped board made of n by n hexagonal fields, then there are extreme positions and. Against 3n ² positions overall The described algorithm is thus feasible only for very small n.
As a manner compatible with the turn order total ordering, the lexicographic order of (f, a1, ..., an ²) serve with f ∈ [ 0, n ²] as the number of free fields in the highest lexicographical priority, i ∈ [ 1, n ²] as an index of a field, and
In the example, the position (11 ² = 121, free, free, free, ... ) to position ( 120, red, free, free, free, ...) and this position ( 119, red, clear, blue, free, free, free, ... may follow ).
In position DAG trains alternate to the red stones with which the blue. The core of the position DAG includes positions with both red as blue on the train. Good for red is always the first train in the middle - but he is not the only good. Bad is the first train in a sharp corner, then you can generate a blue Z- position with its first train in the middle.
Minimax algorithm
In contrast to the position of the DAG above algorithm Minimax algorithm requires a game tree, that is, an expanded version of the tree to the root position DAG.
One or two
Based on the very simple game one or two of the minimax algorithm is explained.
In the figure alongside of the game tree is expanded for a tableau of the game with 6 matches. For further below listed reasons, the two players " maximizers " and be called " minimizers ". The nodes ( positions ) and edges (trains ) are green for the maximizer and red for the minimizer. The nodes contain a number and a sign: the number is the number of matches and the sign of the value of the position with the meaning:
The algorithm starts at the leaves of the tree, all of which are available for an end position of 0 matches. As the player who would be the train, has lost a leaf obtained with maximizers train on the green marker 0 -, 0 another red. A green node receives as play value the maximum of the match values of the successor node or a red node whose minimum; hence the name minimax and also the names of the players. Are the values of the successor node different, represent the best train is drawn intensified. For positions with green - and red it is the the train donating player who has won the victory, or can enforce. ( It is Z- positions. Since the Minimax algorithm always comes to a conclusion, the existence and uniqueness of a core position DAG has already proven its applicability through. )
Compared with the above algorithm, the minimax algorithm is somewhat more complex:
- It allows for arbitrary real game values , even if in the random free finite games with two players without a draw only the two values (: = 1 ) and - (: = -1) are needed.
- In the example of the game tree contains repetitions, eg 8 times the position of 1, 5 times with 2, 3 times with 3 and 2 times with 4 matches.
- The nodes of the position DAG, the nodes in the leftmost column. Its core is formed by the nodes with green - and red .
Both algorithms start at the leaves and work your way up to the root, ie against the direction of the game.
The other games
In the other described in this article matches savings of game tree to position DAG yield very fact that there is an arbitrarily large number of moves that lead to the same position. This must be repeated in the tree, whereas it occurs only once in the DAG. In addition, simplified at the games Nim and Wythoffs game of position DAG by the fact that the trains are with common objective in a simple linear order another.
Numerical Categorization
Under numerical categorization is to be understood that there is a formula that clears for each position based on their numerisierten parameter, whether it is a profit or loss position. Is there a formula, then the game can be described as " highly resolved" in the speech of the article Dissolved games.
One or two
The Z- node with the lash or two are exactly the positions in which the number of matches is divisible by 3; with the associated Misère rule ≡ 1 mod 3
Wythoffs game
In Wythoffs game, the Z nodes lie on the coordinates with n ≥ 0 and ( number of the golden ratio ) and the same again mirrored on the bisector.
By the irrationality of results in an aperiodic irregular distribution.
Nim
International Protective Coatings. the formulas for the Nim -sum of the Nim game and the proof of its optimality, we refer to the main article Nim game. The Z- nodes of the graph correspond to positions with straight Nim - sums. The latter are only at a number ≥ 3 of rows, that is, the composition of several " games " really make a difference. In a 2-dimensional graphic which is, however, difficult to make clear.
Hex
A strategy for arbitrarily large n × n ( symmetric ) Hex is not known.
Conclusion
The graph-theoretical algorithm determines the " play value " of a position, bringing the Games within the meaning of Article Dissolved games can be the game "solved weak" as designated. The graphs are naturally always finite and thus indeed for the " own use" of the player, if necessary, sufficient. The expansion of the position graph is often of exponential complexity. They also provide clues for numerical formulas, which can be very different, however. From the mathematical point of view, the formulas are to settle on higher ranks and also require a specially tailored to their proof.
Obviously it is not interesting to play one of those games where both players know the optimal strategy, because then stand winners and losers from the outset. When Hex game this is known, however effective only for small n. In fact, the winner iff it is clear from the outset if one player plays the optimal strategy and gets a C -position.
Since in the present C- position, the Z- positions represent a tiny minority among the successor positions, has a perfect player to start at a Z position - ie left to his opponent a C -position - has, the greater the chances of winning the larger the initial tableau and the more possibilities of error thus made for the opponent. And after the first miss of whose side he is no longer dissuade her from the road of victory.