Building a strategic-form game#
Gambit has full support for constructing and manipulating arbitrary N-player strategic (also known as normal form) games.
For extensive games, Gambit automatically computes the corresponding reduced strategic game. To view the reduced strategic game corresponding to an extensive game, select or click the strategic game table icon on the toolbar.
The strategic games computed by Gambit as the reduced strategic game of an extensive game cannot be modified directly. Instead, edit the original extensive game; Gambit automatically recomputes the strategic game after any changes to the extensive game.
Strategies in a reduced strategic game are assigned numeric labels for identification. These labels are assigned via a deterministic algorithm for constructing the reduced strategic game from an extensive game. Hovering the mouse pointer over a strategy’s row or column label briefly displays a small window listing the action selected at each information set where the reduced strategy specifies an action.
Strategic games may also be input directly. To create a new strategic game, select , or click the new strategic game icon on the toolbar.
Adding players#
Players are added, removed, and reordered on the Players page of the Game properties dialog (); see General concepts. A newly added player has one strategy, by default labeled with the number 1.
Editing strategies#
A player’s strategies can be added, removed, reordered, and renamed by clicking any cell in the strategic game table where one of that player’s strategy labels appears. This opens an Edit strategies dialog for that player, titled with the player’s own label, listing a row for each of the player’s strategies and showing its label.
Adding, removing, and reordering strategies#
Clicking +, below the list of strategies, adds a new strategy with an automatically generated numeric label, which can be relabeled like any other strategy. Each strategy’s row has a ✕ button to remove it, and ↑/↓ buttons to move it earlier or later among the player’s other strategies.
Removing a strategy does not immediately discard the payoffs at the contingencies that use it. 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 strategy, and the payoffs at every contingency involving it, exactly as they were. Nothing is actually deleted until OK is clicked; clicking Cancel leaves the player’s strategies entirely unchanged. At least one strategy must remain: the last remaining strategy cannot be removed.
Renaming versus reordering#
Typing a new label into a strategy’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 strategy, wherever it ends up in the list. The payoffs at every contingency involving it travel with it: renaming a strategy never changes any payoff.
Moving a row up or down changes the strategy’s position among the player’s other strategies, permuting the payoff table to match. Each strategy keeps its own payoffs as it moves: reordering a strategy never changes what it is called.
Because these are separate operations, two strategies can have their labels swapped in place – for instance, relabeling Cooperate to Defect and Defect to Cooperate at the same time, since all the relabeling for a player happens simultaneously when OK is clicked – without touching the payoffs at any contingency. This is different from using the ↑/↓ buttons to swap the positions of Cooperate and Defect, which exchanges the payoffs at the contingencies where each is chosen, while leaving each strategy’s own label attached to it.
To help keep track of which is which while editing, a strategy 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 strategy is shown in bold green text.
Strategy labels must be nonempty, and unique among the strategies currently kept for the player (a removed strategy’s label is not considered). Any field that currently violates one of these rules is highlighted, and a description of the problem is shown below the list of strategies; the OK button is disabled until all fields are valid.
Editing payoffs#
Payoffs for each player are specified individually for each contingency, or collection of strategies, in the game. To edit any payoff in the table, click that cell in the table and edit the payoff. Pressing the Escape key (Esc) cancels any editing of the payoff and restores the previous value.
To speed entry of many payoffs, as is typical when creating a new game, accepting a payoff entry via the Tab key automatically moves the edit control to the next cell to the right. If the payoff is the last payoff listed in a row of the table, the edit control wraps around to the first payoff in the next row; if the payoff is in the last row, the edit control wraps around to the first payoff in the first row. So a strategic game payoff table can be quickly entered by clicking on the first payoff in the upper-left cell of the table, inputting the payoff for the first (row) player, pressing the Tab key, inputting the payoff for the second (column) player, pressing the Tab key, and so forth, until all the payoff entries in the table have been filled.
Merging and splitting outcomes#
Which contingencies share an outcome can itself be changed from the table. Click and drag, or shift-click, across several payoff cells to select them; the selection automatically expands to cover every payoff column of any contingency it touches, since an outcome is shared at the level of a whole contingency, not a single payoff cell. Right-clicking the selection – or, with nothing selected, right-clicking a single cell – opens a menu with three commands:
Set outcome… (labelled Merge into one outcome… when more than one contingency is selected) opens a dialog to give a label and a payoff for each player, then attaches the resulting outcome to every selected contingency. Selecting several contingencies this way merges them into one shared outcome; selecting a single contingency that is currently part of a larger shared outcome splits it off into its own outcome, leaving the rest of the group unchanged. If the selection already matches one outcome’s full membership exactly, this instead edits that outcome’s own label and payoffs in place, rather than creating a new one. The dialog starts from a label and payoffs to edit – the shared outcome’s own, if editing one in place, or the first selected contingency’s current payoffs with a freshly generated label otherwise – and disables OK until the label is nonempty and unique among the game’s outcomes and every payoff is a valid number.
Edit outcome… is the direct route to that same in-place case: it is enabled whenever the selection matches one existing outcome’s full membership exactly (a single contingency is enough, if that contingency’s outcome has no other members), and opens the same label-and-payoffs dialog pre-filled with that outcome’s own values, changing them in place without touching which contingencies share it. Editing a payoff here is no different from typing directly into a payoff cell – it is offered as a second way to reach the same result, alongside also being able to change the label, matching how outcomes are edited in an extensive game.
Remove outcome clears every selected contingency back to having no outcome. It is disabled if none of the selection currently has one.
Clicking any cell afterward clears the selection.