Gambit: Software Tools for Game Theory | ||
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Gambit displays extensive form games as a tree, with time flowing from left to right. As the tree is built, Gambit uses an automatic layout algorithm to render the tree. [1] Some options are provided to customize the display of the tree; see Changing how the tree is displayed for details. The current objective of these options is to provide sufficient customizability to be able to use Gambit to create figures suitable for use in classroom materials and working papers; however, it might not be sufficient at this state for users who desire fine control over tree figures.
Extensive form games have one or more players. The players are specified when the game is created (see the section called Creating a new extensive form game); players may also be added on the fly using the Extensive form properties dialog (see the section called Viewing and changing extensive form game properties). [2] Each player is indexed by her player number, which ranges from 1 to the number of players. The player number 0 is reserved for the player called "chance" or "nature". For each player, the nodes at which she chooses are partitioned into information sets. For convenience, Gambit assigns an ordered pair to each information set of the form (player number, information set number). The information set number ranges from 1 to the number of information sets a player has, and information set numbers are only unique within a player; therefore, information set (1,1), which is player 1's information set number 1, is distinct from information set (2,1), which is player 2's information set number 1. The information set ID's are arbitrary, and have no significance in terms of analysis. By default, Gambit displays information set ID's below each node; this feature can be configured by editing the display settings (see Changing how the tree is displayed).
Gambit has a flexible interpretation of the concept of an outcome. An outcome is an object which has associated with it a vector of payoffs, one for each player. A particular outcome may appear at multiple nodes in a tree, a feature used in the poker game example in the section called A simple game of poker in the chapter called Getting Started with the Graphical Interface. Outcomes may even appear at non-terminal nodes within a game; in such cases, they are interpreted as incremental payoffs. Permitting outcomes to appear at non-terminal nodes makes it easier to construct trees where a stage game is repeated, with the supergame payoffs being the sum of the stage game payoffs earned. [3]
Most of the operations involving constructing and editing games require the selection of a node. A node is selected by clicking with the left mouse button on a node, which may be represented by a circle, square, diamond, or horizontal line, depending on the selected style. A selected node is drawn using a thicker line than its non-selected counterparts. Left-clicking the tree window away from any node unselects all nodes. Shortcuts are provided for quickly navigating through a tree; see the section called Navigating extensive form game trees for a complete list.
This dialog is used to add a move to the extensive form game, and is accessed through the {\bf Edit->Node->AddMove} or {\bf Edit->Node->InsertMove} menu items in the extensive form GUI. Select the player to be in control of the move by clicking on a player in the Player choicebox. The player can be either chance or an existing player, or a "New Player." If New Player is chosen, a new player will be created and given a default name "Player \#." If this node is to belong to an existing information set, choose the desired infoset from the Infoset choicebox. If a new information set is to be created, just enter the number of branches desired. If the node created was a CHANCE node, you will be prompted for the probabilities associated with each branch.
The graphical interface offers a number of options which allow the user to customize the display of game trees to his or her own tastes.
The layout of a game tree can be customized using the Layout Parameters dialog, which is accessible by way of the Layout item on the Format menu.
All lengths are in pixels. The Show Infoset Lines choicebox is used to select a mode for display of information sets on the extensive form. None means no lines are drawn connecting members of information sets. Same Level means only members of an information set at the same level of the tree are connected with lines. All Levels means that all nodes are connected, using lagged lines if necessary. (See Figure~\ref{fig_lag_iset} in the extensive form section for an illustration.)
The text labels displayed on nodes and branches can be customized using the Legends dialog (Format menu, Legends item).
There are two positions where information can be displayed for each branch (above and below.) In each of these positions, you can select to display information about the game tree (such as branch numbers or action names), or information about the currently selected solution (such as action probabilities or action values). Similarly, there are three positions at which information can be displayed for a node (above, below, and to the right). In the positions above and below the node, you can select to display information about the game tree (such as infoset name or ID), or information about the currently selected solution (such as realization probabilities, node values, belief probabilities). The position to the right of the node is reserved for information relating to outcomes, and here you can select to display the outcome name or outcome vector.
[1] | That is, there is no mechanism currently for customizing the location of nodes in the tree display. This is an item on the list for future enhancement. |
[2] | Note that on-the-fly player adding is not currently supported in normal form games. |
[3] | A two-stage Bayesian game using outcomes at non-terminal nodes in this fashion can be found in the bayes2a.efg game in the standard Gambit distribution. |
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