Building an extensive-form game#
The graphical interface provides a flexible set of operations for constructing and editing general extensive games. These are outlined below.
Creating a new extensive game#
To create a new extensive game, select , or click on the new extensive game icon. The extensive game created is a trivial game with two players, named by default Player 1 and Player 2, with one node, which is both the root and terminal node of the game. In addition, extensive games have a special player labeled Chance, which is used to represent random events not controlled by any of the strategic players in the game.
Adding moves#
Right-clicking any node in the tree offers a per-player submenu for adding a move: Append move for at a terminal node, or Insert move for at a nonterminal one. Either way, the submenu lists every player (each shown with its assigned colour), plus the chance player.
Choosing a player opens a small window anchored at the node, meant to be read as a sentence: Append move for (or Insert move for) the chosen player, with a number of actions. The number of actions defaults to two, and can be changed directly – with the arrows beside it, or by typing – at any point; below it, a label field is shown for each action, defaulting to a placeholder number that can be typed over. Tabbing past the last action’s field adds another action and moves into it, so a move can be built up to more than two actions just by typing and tabbing, without touching the number-of-actions field at all. Pressing Enter in any field accepts the move as currently specified; pressing Escape, or clicking anywhere else, cancels it. If any action’s label is currently invalid – left empty, or duplicated among the move’s other actions – accepting is refused: the window stays open, with the problem described and the offending field or fields highlighted, whether that was attempted with Enter or by clicking away.
Choosing the chance player adds a probability field next to each action’s label. These default to splitting evenly across whatever actions currently exist. Adding an action gives the new one whatever share is still needed to bring the total to one (or zero, if the existing actions already summed to one or more), leaving every other action’s probability exactly as typed – so growing the list never silently overwrites a probability already entered. As with action labels, the move cannot be accepted unless every probability is a nonnegative number and they sum to exactly one.
The two submenu labels reflect different underlying operations. At a terminal node, Append move for attaches the new move directly, since nothing is there yet. At a nonterminal node, Insert move for instead makes the new move the parent of the clicked node: the subtree that used to start there becomes the first action’s subtree, and the new move’s other actions each lead to a fresh terminal node. There is currently no way to choose a different action for the displaced subtree at insert time; to relocate it afterward, use drag-and-drop (see Copying and moving subtrees below).
The player who moves at an existing move can be changed separately, using the Assign this move to submenu, also reached by right-clicking a nonterminal node: this lists the other players to which the move can be reassigned, without changing the move’s actions or the subtrees that follow them. A move cannot currently be changed between the chance player and a personal player this way – nor can it be, at present, using any other method in the interface.
A single action can also be added to, or removed from, an existing move using the Move properties dialog described next.
Editing moves#
The properties of an existing move – the label of its information set or event, the player to which it belongs, and the actions available at it – can be changed by right-clicking on any node belonging to the move and choosing Move properties. This displays the Move properties dialog, which shows a dropdown for the player to which the move belongs (each player again shown with its colour), a text field for the information set’s or event’s label, and a row for each action, showing its label and, for moves belonging to the chance player, its probability.
Adding, removing, and reordering actions#
Clicking +, below the list of actions, adds a new action with an automatically generated numeric label, which can be relabeled like any other action. Each action’s row has a ✕ button to remove it, and ↑/↓ buttons to move it earlier or later among the other actions at the move.
Removing an action does not immediately discard the subtree it leads to. Instead, the row is shown disabled, with its label struck through, and its ✕ button is replaced by a ↺ (restore) button, so that what is about to be destroyed remains visible – and reversible – until the dialog is confirmed. Clicking ↺ returns the action, and its subtree, exactly as they were. Nothing is actually deleted until OK is clicked; clicking Cancel leaves the move entirely unchanged. At least one action must remain at the move: the last remaining action cannot be removed.
Renaming versus reordering#
Typing a new label into an action’s row, and moving that row to a different position with the ↑/↓ buttons, are two distinct operations. They can be combined, but neither substitutes for the other:
Changing the text in a row renames that action, wherever it ends up in the list. The subtree it leads to goes with it: renaming an action never changes what happens after it is taken.
Moving a row up or down changes the order in which actions are offered at the move. Each action keeps its own subtree as it moves: reordering an action never changes what it is called.
Because these are separate operations, two actions can have their labels swapped in place – for instance, relabeling Left to Right and Right to Left at the same time, since all the relabeling at a move happens simultaneously when OK is clicked – without touching the subtrees that follow them. This is different from using the ↑/↓ buttons to swap the positions of Left and Right, which exchanges their subtrees while leaving each action’s own label attached to it.
To help keep track of which is which while editing, an action whose label has been changed from what it started as is shown in italic blue text, and shows the original label in a tooltip on hover; a newly added action is shown in bold green text.
Action labels must be nonempty, and unique among the actions currently kept at the move (a removed action’s label is not considered); the information set’s label, if not left blank, must similarly be unique among the information sets belonging to the same player. Any field that currently violates one of these rules is highlighted, and a description of the problem is shown below the list of actions; the OK button is disabled until all fields are valid. For moves belonging to the chance player, the probabilities of the actions kept at the move must also be nonnegative numbers summing to exactly one.
Copying and moving subtrees#
Many extensive games have structures that appear in multiple parts of the tree. It is often efficient to create the structure once, and then copy it as needed elsewhere.
Gambit provides a convenient idiom for this. Clicking on any nonterminal node and dragging it to another node results in a context-aware popup menu. While dragging, the node under the pointer is outlined in green if dropping there is possible, or in red if not. Depending on the destination node, this menu offers the option of copying the subtree rooted at the original node, moving it entirely, or placing the destination node in the same information set as the source node. Moving a subtree onto one of its own descendants is never offered (though shown, disabled), since the destination would cease to exist partway through the move; copying onto a descendant remains available, since the copy is made before anything is removed.
Removing parts of a game tree#
Two deletion operations are supported on extensive games. To delete the entire subtree rooted at a node, right-click that node and choose Delete subtree.
To delete an individual move from the game, right-click one of the direct children of that node, and choose Delete parent. This operation deletes the parent node, as well as all the children of the parent other than the clicked node. The clicked child node now takes the place of the parent node in the tree.
Managing information sets#
Gambit provides several methods to help manage the information structure in an extensive game.
A node can be placed in the same information set as another using the drag-and-drop idiom described in Copying and moving subtrees: dragging one node onto another and choosing Put node in same information set (or, when dropping on a terminal node, Insert move using same information set) from the popup menu. When the node being dragged belongs to the chance player, these read Put node in same event and Insert move using same event. This is currently the only way to have a newly created move share an information set with an existing one; the per-player submenu for adding a move always creates a fresh information set.
The information set to which a node belongs can also be set by right-clicking the node and choosing Node properties (or by double-clicking the node). This displays the node properties dialog, which besides the information set has a text field for the node’s label and, at the bottom, a dropdown for the node’s outcome, offering the same choices and editing behavior described in Outcomes and payoffs below. If the node is the root of the tree, or the root of a proper subgame, the dialog states this beneath the information set dropdown. The Information set dropdown defaults to the current information set to which the node belongs, and contains a list of all other information sets in the game which are compatible with the node, that is, which have the same number of actions. Additionally, the node can be moved to a new, singleton information set by setting this dropdown to the New information set entry. For a node belonging to the chance player the dropdown is labeled Event, lists the game’s events, and offers New event instead.
When building out a game tree using the drag-and-drop approach to copying portions of the tree, the nodes created in the copy of the subtree remain in the same information set as the corresponding nodes in the original subtree. In many cases, though, these trees differ in the information available to some or all of the players. To help speed the process of adjusting information sets in bulk, Gambit offers a “reveal” operation, which breaks information sets based on the action taken at a particular node. Right-click the node at which the action taken is to be made known subsequently to other players, and choose Reveal. This displays a dialog listing the players in the game. Check the boxes next to the players who observe the outcome of the move at the node, and click OK. The information sets at nodes below the clicked one are adjusted based on the action selected at this node.
Note
The reveal operation only has an effect at the time it is done. In particular, it does not enforce the separation of information sets based on this information during subsequent editing of the game.
Outcomes and payoffs#
Gambit supports the specification of payoffs at any node in a game tree, whether terminal or not. Each node is created with no outcome attached; in this case, the payoff at each node is zero to all players. These are indicated in the game tree by the presence of a (u) in light grey to the right of a node.
To set the payoffs at a node, double-click on the (u) to the right of the node. This creates a new outcome at the node, with payoffs of zero for all players, and pops up an editing panel for that outcome.
For nodes with existing outcomes, clicking on any of the displayed payoffs pops up an editing panel for that outcome.
Outcomes may also be moved, or shared between nodes, using drag-and-drop. Left-clicking and dragging an outcome to another node (highlighted, as with moving and copying subtrees, in green or red as a valid or invalid drop target) pops up a context-aware menu which allows the outcome to be moved to the new node, or for the new node to use the same outcome as the original – so that editing the payoffs at either node updates both.
To remove an outcome from a node, right-click the node, and choose Remove outcome.