Displaying a strategic game

Gambit displays a strategic game as a table. Each cell represents one strategy profile: one strategy choice for every player. The cell shows the payoffs that result when the players choose the strategies named by that cell’s row and column labels.

For a two-player game, Gambit initially uses the familiar matrix arrangement. One player’s strategies label the rows, and the other player’s strategies label the columns.

Each row-and-column combination identifies one strategy profile. For example, a cell in the row labelled Cooperate and the column labelled Defect represents the outcome in which the row player chooses Cooperate and the column player chooses Defect.

The payoffs in each cell are colour-coded by player. In the standard two-player arrangement, Gambit displays the row player’s payoff first and the column player’s payoff second.

Whenever a profile has been computed and selected, hovering the mouse pointer over any strategy’s row or column label also shows that strategy’s probability under the current profile, the payoff its player would get by playing that strategy alone against the others’ current play, and that player’s overall payoff under the current profile. Comparing this payoff-if-played figure across a player’s strategies is how to check whether the current profile is actually a best response for that player.

Row and column hierarchies

Games with more than two players require more than one strategy label to identify each row or column. Gambit handles this by allowing several players to be assigned to the rows, several players to be assigned to the columns, or both.

When several players are assigned to the same side of the table, their strategy labels form a hierarchy. Each level groups together the strategy combinations belonging to the players below it. This is similar to the hierarchical row and column labels used in a pivot table or a table with a multi-level index.

For example, suppose Alice and Bob are both assigned to the rows. If Alice is above Bob in the row hierarchy, Gambit first groups the rows by Alice’s strategy and then lists Bob’s strategies within each group.

Reversing their order groups the rows first by Bob’s strategy and then by Alice’s. The strategy profiles and payoffs do not change; only their arrangement in the table changes.

The same principle applies to the columns. A player placed at a higher level of the column hierarchy forms the outer grouping, while players at lower levels form groups within it.

This arrangement provides a single table view of games that are often printed as a collection of separate payoff matrices. For example, a three-player game can be displayed with one player on the rows and two players in a column hierarchy, rather than as a separate matrix for each strategy of the third player.

Rearranging the table

To change the table arrangement, right-click any row or column label and choose a player from the menu that appears. If that side of the table already has one or more players assigned to it, the menu offers Place player before and Place player after submenus, each listing every player; if not, it offers a single Use as row player (or Use as column player) submenu instead. Gambit updates the display without changing the game itself.

The available positions depend on the current arrangement. Choosing a player from Place player before makes that player a higher level in the hierarchy than the row or column that was right-clicked; choosing one from Place player after makes it a lower level.

Changing the row and column hierarchies may also change the order in which payoffs appear within each cell. The colour of each payoff continues to identify the player to whom it belongs.

Note

Rearranging the table changes only the presentation of the game. It does not change the players, strategies, outcomes, or payoffs.