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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01//EN">
<HTML
><HEAD
><TITLE
>Normal form games</TITLE
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>Gambit: Software Tools for Game Theory</TH
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><DIV
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><H1
CLASS="SECT1"
><A
NAME="GUI.NFG"
>Normal form games</A
></H1
><P
></P
><DIV
CLASS="SECT2"
><H2
CLASS="SECT2"
><A
NAME="GUI.NFG.TABLE"
>The normal form table</A
></H2
><P
>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. </P
><P
>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.</P
><P
>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.</P
><P
>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}</P
></DIV
><DIV
CLASS="SECT2"
><H2
CLASS="SECT2"
><A
NAME="GUI.NFG.STANDARD"
>Standard Nash equilibrium algorithms</A
></H2
><P
>The standard solutions dialog is accessed via the
<B
CLASS="GUIMENUITEM"
>Standard...</B
> menu item on the
<B
CLASS="GUISUBMENU"
>Equilibrium</B
> submenu of the
<B
CLASS="GUIMENU"
>Tools</B
> menu.
There are six options available for two-player games.
<P
></P
><UL
><LI
><P
>Compute one Nash equilibrium (no refinement). This iteratively eliminates
all weakly dominated stratgies, then uses <TT
CLASS="FUNCTION"
>LpSolve</TT
> if
the game is zero-sum, and <TT
CLASS="FUNCTION"
>LcpSolve</TT
> if the game is
not zero-sum.</P
></LI
><LI
><P
>Compute two Nash equilibria (no refinement). This iteratively eliminates
all strictly dominated strategies, then uses <TT
CLASS="FUNCTION"
>EnumMixed</TT
>,
stopping after two equilibria are found (or it is established that there
is a unique equilibrium).</P
></LI
><LI
><P
>Compute all Nash equilibria (no refinement). This iteratively eliminates
all strictly dominated strategies, then uses <TT
CLASS="FUNCTION"
>EnumMixed</TT
>
to enumerate all equilibria.
<A
NAME="AEN498"
HREF="#FTN.AEN498"
>[1]</A
></P
></LI
><LI
><P
>Compute one (trembling-hand) perfect equilibrium. This
eliminates all weakly dominated strategies, then uses
<TT
CLASS="FUNCTION"
>LcpSolve</TT
>.</P
></LI
><LI
><P
>Compute two (trembling-hand) perfect equilibria. This eliminates all
weakly dominated strategies, then uses <TT
CLASS="FUNCTION"
>EnumMixed</TT
>
stopping after two equilibria are found (or it is established that there
is a unique equilibrium once weakly dominated strategies are eliminated).</P
></LI
><LI
><P
>Compute all (trembling-hand) perfect equilibria. This eliminates all
weakly dominated strategies, then uses <TT
CLASS="FUNCTION"
>EnumMixed</TT
>
to enumerate all equilibria surviving the elimination.</P
></LI
></UL
></P
><P
>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:
<P
></P
><UL
><LI
><P
>Compute one Nash equilibrium (no refinement). This iteratively eliminates
all weakly dominated stratgies, then uses <TT
CLASS="FUNCTION"
>SimpDivSolve</TT
>.
Note that <TT
CLASS="FUNCTION"
>SimpDivSolve</TT
> is not guaranteed to locate
an equilibrium.</P
></LI
><LI
><P
>Compute two Nash equilibria (no refinement). This iteratively eliminates
all strictly dominated strategies, then uses <TT
CLASS="FUNCTION"
>LiapSolve</TT
>,
stopping after two equilibria are found. Note that
<TT
CLASS="FUNCTION"
>LiapSolve</TT
> is not guaranteed to locate an equilibrium.</P
></LI
><LI
><P
>Compute all Nash equilibria (no refinement). This iteratively eliminates
all strictly dominated strategies, then uses <TT
CLASS="FUNCTION"
>PolEnumSolve</TT
>
to enumerate all equilibria. This algorithm is guaranteed to find all
equilibria, but takes prohibitively much time for games beyond a rather
small size.</P
></LI
></UL
></P
></DIV
></DIV
><H3
CLASS="FOOTNOTES"
>Notes</H3
><TABLE
BORDER="0"
CLASS="FOOTNOTES"
WIDTH="100%"
><TR
><TD
ALIGN="LEFT"
VALIGN="TOP"
WIDTH="5%"
><A
NAME="FTN.AEN498"
HREF="gui.nfg.html#AEN498"
>[1]</A
></TD
><TD
ALIGN="LEFT"
VALIGN="TOP"
WIDTH="95%"
><P
>In games with non-singleton components of equilibria,
<TT
CLASS="FUNCTION"
>EnumMixed</TT
> computes the extreme points of the components.
See the documentation for <A
HREF="app.alg.enummixed.html"
>EnumMixedSolve</A
> for details.</P
></TD
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