Gambit: Software Tools for Game Theory | ||
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In the normal form representation, the game is viewed as a 2-dimensional window into an N-dimensional matrix. Each 2D window shows a table of payoffs for each player as a function of the strategies for two of the players, holding the strategies of all of the other players (if there are any) fixed. Note that this organization requires each game to have at least two players. Each cell contains the payoff vector with one entry per player for the strategy profile determined by this cell. The profile itself is set by using a combination of settings for the row and column players plus the strategy settings for the rest of the players.
The payoff vector can be edited by double-clicking on the cell and entering new values in the dialog. To enter an entire game matrix, an accelerator key may be useful. Assuming the default accelerators are used, pressing the {\tt TAB} key will move the cursor to the next cell of the normal form, and call up the payoff edit dialog. So an entire matrix can be entered by first double clicking on the first (upper left) cell, editing the entries (using tab to advance to successive entries), hitting return, then successively pressing tab, editing the entries and pressing return to enter the remaining cells of the matrix.
The row and column player choice boxes determine which players' strategies get displayed on the horizontal and vertical axis of the matrix. You can select a new player by clicking on the arrow at the right of the box, and selecting a new player from the list of players in the game. The dimensions of the matrix are determined by the number of strategies for the row and column players. Note that it is meaningless to select the same player for both the row and the column. If the game has more than two players, a warning will be issued and no action will be taken. In the case of a two player game, the row and the column players will be switched.
The array of choice boxes labeled "Profile" at all times reflects the strategies picked by each player to achieve the payoffs shown in the highlighted cell. The nth choice box can contain a value from 1 to the total number of strategies the nth player has. When one of the choice boxes is changed, one of two things can happen: \begin{enumerate} \item If the choice box number is not equal to either the row or the column player, the entire matrix will be updated with new values to reflect the new 2D view into the matrix. \item If the choice box number was either the row or the column player, the highlighted cell will move to reflect the new strategy. \end{enumerate}
The standard solutions dialog is accessed via the Standard... menu item on the Equilibrium submenu of the Tools menu. There are six options available for two-player games.
Compute one Nash equilibrium (no refinement). This iteratively eliminates all weakly dominated stratgies, then uses LpSolve if the game is zero-sum, and LcpSolve if the game is not zero-sum.
Compute two Nash equilibria (no refinement). This iteratively eliminates all strictly dominated strategies, then uses EnumMixed, stopping after two equilibria are found (or it is established that there is a unique equilibrium).
Compute all Nash equilibria (no refinement). This iteratively eliminates all strictly dominated strategies, then uses EnumMixed to enumerate all equilibria. [1]
Compute one (trembling-hand) perfect equilibrium. This eliminates all weakly dominated strategies, then uses LcpSolve.
Compute two (trembling-hand) perfect equilibria. This eliminates all weakly dominated strategies, then uses EnumMixed stopping after two equilibria are found (or it is established that there is a unique equilibrium once weakly dominated strategies are eliminated).
Compute all (trembling-hand) perfect equilibria. This eliminates all weakly dominated strategies, then uses EnumMixed to enumerate all equilibria surviving the elimination.
For games with more than two players, Gambit does not currently provide any algorithms which are guaranteed to find perfect equilibria in normal form games. There are only three options available for these games:
Compute one Nash equilibrium (no refinement). This iteratively eliminates all weakly dominated stratgies, then uses SimpDivSolve. Note that SimpDivSolve is not guaranteed to locate an equilibrium.
Compute two Nash equilibria (no refinement). This iteratively eliminates all strictly dominated strategies, then uses LiapSolve, stopping after two equilibria are found. Note that LiapSolve is not guaranteed to locate an equilibrium.
Compute all Nash equilibria (no refinement). This iteratively eliminates all strictly dominated strategies, then uses PolEnumSolve to enumerate all equilibria. This algorithm is guaranteed to find all equilibria, but takes prohibitively much time for games beyond a rather small size.
[1] | In games with non-singleton components of equilibria, EnumMixed computes the extreme points of the components. See the documentation for EnumMixedSolve for details. |
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