Command-line tools

Gambit provides command-line interfaces for each method for computing Nash equilibria. These are suitable for scripting or calling from other programs. This chapter describes the use of these programs. For a general overview of methods for computing equilibria, see the survey of [McKMcL96].

The graphical interface also provides a frontend for calling these programs and evaluating their output. Direct use of the command-line programs is intended for advanced users and applications.

These programs take an extensive or strategic game file, which can be specified on the command line or piped via standard input, and output a list of equilibria computed. The default output format is to present equilibria computed as a list of comma-separated probabilities, preceded by the tag NE. For mixed strategy profiles, the probabilities are sorted lexicographically by player, then by strategy. For behavior strategy profiles, the probabilites are sorted by player, then information set, then action number, where the information sets for a player are sorted by the order in which they are encountered in a depth-first traversal of the game tree. Many programs take an option -D, which, if specified, instead prints a more verbose, human-friendly description of each strategy profile computed.

Many of the programs optionally output additional information about the operation of the algorithm. These outputs have other, program-specific tags, described in the individual program documentation.

gambit-enumpure: Enumerate pure-strategy equilibria of a game

gambit-enumpure reads a game on standard input and searches for pure-strategy Nash equilibria.

Changed in version 14.0.2: The effect of the -S switch is now purely cosmetic, determining how the equilibria computed are represented in the output. Previously, -S computed using the strategic game; if this was not specified for an extensive game, the agent form equilibria were returned.

-S

Report equilibria in reduced strategic form strategies, even if the game is an extensive game. By default, if passed an extensive game, the output will be in behavior strategies. Specifying this switch does not imply any change in operation internally, as pure-strategy equilibria are defined in terms of reduced strategic form strategies.

-D

New in version 14.0.2.

The default output format for computed equilibria is a comma-separated list of strategy or action probabilities, suitable for postprocessing by automated tools. Specifying -D instead causes the program to output greater detail on each equilbrium profile computed.

-A

New in version 14.0.2.

Report agent form equilibria, that is, equilibria which consider only deviations at one information set. Only has an effect for extensive games, as strategic games have only one information set per player.

-P

By default, the program computes all pure-strategy Nash equilibria in an extensive game. This switch instructs the program to find only pure-strategy Nash equilibria which are subgame perfect. (This has no effect for strategic games, since there are no proper subgames of a strategic game.)

-h

Prints a help message listing the available options.

-q

Suppresses printing of the banner at program launch.

Computing the pure-strategy equilibria of extensive game e02.efg, the example in Figure 2 of Selten (International Journal of Game Theory, 1975):

$ gambit-enumpure e02.efg
Search for Nash equilibria in pure strategies
Gambit version 15.1.0, Copyright (C) 1994-2014, The Gambit Project
This is free software, distributed under the GNU GPL

NE,1,0,0,0,1,0

With the -S switch, the set of equilibria returned is the same, except expressed in strategic game strategies rather than behavior strategies:

$ gambit-enumpure -S e02.efg
Search for Nash equilibria in pure strategies
Gambit version 15.1.0, Copyright (C) 1994-2014, The Gambit Project
This is free software, distributed under the GNU GPL

NE,1,0,0,1,0

The -A switch considers only behavior strategy profiles where there is no way for a player to improve his payoff by changing action at only one information set; therefore the set of solutions is larger:

$ gambit-enumpure -A e02.efg
Search for Nash equilibria in pure strategies
Gambit version 15.1.0, Copyright (C) 1994-2014, The Gambit Project
This is free software, distributed under the GNU GPL

NE,1,0,1,0,1,0
NE,1,0,1,0,0,1
NE,1,0,0,1,1,0

gambit-enumpoly: Compute equilibria of a game using polynomial systems of equations

gambit-enumpoly reads a game on standard input and computes Nash equilibria by solving systems of polynomial equations and inequalities.

This program searches for all Nash equilibria in a strategic game using a support enumeration approach. This approach computes all the supports which could, in principle, be the support of a Nash equilibrium, and then searches for a totally mixed equilibrium on that support by solving a system of polynomial equalities and inequalities formed by the Nash equilibrium conditions. The ordering of the supports is done in such a way as to maximize use of previously computed information, making it suited to the computation of all Nash equilibria.

When the verbose switch -v is used, the program outputs each support as it is considered. The supports are presented as a comma-separated list of binary strings, where each entry represents one player. The digit 1 represents a strategy which is present in the support, and the digit 0 represents a strategy which is not present. Each candidate support is printed with the label “candidate,”.

Note that the subroutine to compute a solution to the system of polynomial equations and inequalities will fail in degenerate cases. When the verbose switch -v is used, these supports are identified on standard output with the label “singular,”. It is possible that there exist equilibria, often a connected component of equilibria, on these singular supports.

-d

Express all output using decimal representations with the specified number of digits.

-h

Prints a help message listing the available options.

-H

By default, the program uses an enumeration method designed to visit as few supports as possible in searching for all equilibria. With this switch, the program uses a heuristic search method based on Porter, Nudelman, and Shoham [PNS04], which is designed to minimize the time until the first equilibrium is found. This switch only has an effect when solving strategic games.

-S

By default, the program uses behavior strategies for extensive games; this switch instructs the program to use reduced strategic game strategies for extensive games. (This has no effect for strategic games, since a strategic game is its own reduced strategic game.)

-q

Suppresses printing of the banner at program launch.

-v

Sets verbose mode. In verbose mode, supports are printed on standard output with the label “candidate” as they are considered, and singular supports are identified with the label “singular.” By default, no information about supports is printed.

Computing equilbria of the extensive game e01.efg, the example in Figure 1 of Selten (International Journal of Game Theory, 1975) sometimes called “Selten’s horse”:

$ gambit-enumpoly e01.efg
Compute Nash equilibria by solving polynomial systems
Gambit version 15.1.0, Copyright (C) 1994-2014, The Gambit Project
Heuristic search implementation Copyright (C) 2006, Litao Wei
This is free software, distributed under the GNU GPL

NE,0.000000,1.000000,0.333333,0.666667,1.000000,0.000000
NE,1.000000,0.000000,1.000000,0.000000,0.250000,0.750000
NE,1.000000,0.000000,1.000000,0.000000,0.000000,0.000000
NE,0.000000,1.000000,0.000000,0.000000,1.000000,0.000000

gambit-enummixed: Enumerate equilibria in a two-player game

gambit-enummixed reads a two-player game on standard input and computes Nash equilibria using extreme point enumeration.

In a two-player strategic game, the set of Nash equilibria can be expressed as the union of convex sets. This program generates all the extreme points of those convex sets. (Mangasarian [Man64]) This is a superset of the points generated by the path-following procedure of Lemke and Howson (see gambit-lcp: Compute equilibria in a two-player game via linear complementarity). It was shown by Shapley [Sha74] that there are equilibria not accessible via the method in gambit-lcp: Compute equilibria in a two-player game via linear complementarity, whereas the output of gambit-enummixed is guaranteed to return all the extreme points.

-d

By default, this program computes using exact rational arithmetic. Since the extreme points computed by this method are guaranteed to be rational when the payoffs in the game are rational, this permits exact computation of the equilibrium set. Computation using rational arithmetic is in general slow, however. For most games, acceptable results can be obtained by computing using the computer’s native floating-point arithmetic. Using this flag enables computation in floating-point, and expresses all output using decimal representations with the specified number of digits.

-D

Since all Nash equilibria involve only strategies which survive iterative elimination of strictly dominated strategies, the program carries out the elimination automatically prior to computation. This is recommended, since it almost always results in superior performance. Specifying -D skips the elimination step and performs the enumeration on the full game.

-c

The program outputs the extreme equilibria as it finds them, prefixed by the tag NE . If this option is specified, once all extreme equilbria are identified, the program computes the convex sets which make up the set of equilibria. The program then additionally outputs each convex set, prefixed by convex-N , where N indexes the set. The set of all equilibria, then, is the union of these convex sets.

-h

Prints a help message listing the available options.

-q

Suppresses printing of the banner at program launch.

-L

Use lrslib by David Avis to carry out the enumeration process. This is an experimental feature that has not been widely tested.

Computing the equilibria, in mixed strategies, of e02.nfg, the reduced strategic form of the example in Figure 2 of Selten (International Journal of Game Theory, 1975):

$ gambit-enummixed e02.nfg
Compute Nash equilibria by enumerating extreme points
Gambit version 15.1.0, Copyright (C) 1994-2014, The Gambit Project
Enumeration code based on lrslib 4.2b,
Copyright (C) 1995-2005 by David Avis (avis@cs.mcgill.ca)
This is free software, distributed under the GNU GPL

NE,1,0,0,1,0
NE,1,0,0,1/2,1/2

In fact, the game e02.nfg has a one-dimensional continuum of equilibria. This fact can be observed by examining the connectedness information using the -c switch:

$ gambit-enummixed -c e02.nfg
Compute Nash equilibria by enumerating extreme points
Gambit version 15.1.0, Copyright (C) 1994-2014, The Gambit Project
Enumeration code based on lrslib 4.2b,
Copyright (C) 1995-2005 by David Avis (avis@cs.mcgill.ca)
This is free software, distributed under the GNU GPL

NE,1,0,0,1,0
NE,1,0,0,1/2,1/2
convex-1,1,0,0,1/2,1/2
convex-1,1,0,0,1,0

gambit-gnm: Compute Nash equilibria in a strategic game using a global Newton method

gambit-gnm reads a game on standard input and computes Nash equilibria using a global Newton method approach developed by Govindan and Wilson [GovWil03]. This program is a wrapper around the Gametracer 0.2 implementation by Ben Blum and Christian Shelton.

-d

Express all output using decimal representations with the specified number of digits.

-h

Prints a help message listing the available options.

-n

Randomly generate the specified number of perturbation vectors.

-q

Suppresses printing of the banner at program launch.

-s

Specifies a file containing a list of starting points for the algorithm. The format of the file is comma-separated values, one mixed strategy profile per line, in the same format used for output of equilibria (excluding the initial NE tag).

-v

Show intermediate output of the algorithm. If this option is not specified, only the equilibria found are reported.

Computing an equilibrium of e02.nfg, the reduced strategic form of the example in Figure 2 of Selten (International Journal of Game Theory, 1975):

$ gambit-gnm e02.nfg
Compute Nash equilibria using a global Newton method
Gametracer version 0.2, Copyright (C) 2002, Ben Blum and Christian Shelton
Gambit version 15.1.0, Copyright (C) 1994-2014, The Gambit Project
This is free software, distributed under the GNU GPL

NE,1,0,2.99905e-12,0.5,0.5

Note

This is an experimental program and has not been extensively tested.

See also

gambit-ipa: Compute Nash equilibria in a strategic game using iterated polymatrix approximation.

gambit-ipa: Compute Nash equilibria in a strategic game using iterated polymatrix approximation

gambit-ipa reads a game on standard input and computes Nash equilibria using an iterated polymatrix approximation approach developed by Govindan and Wilson [GovWil04]. This program is a wrapper around the Gametracer 0.2 implementation by Ben Blum and Christian Shelton.

-d

Express all output using decimal representations with the specified number of digits.

-h

Prints a help message listing the available options.

-q

Suppresses printing of the banner at program launch.

Computing an equilibrium of e02.nfg, the reduced strategic form of the example in Figure 2 of Selten (International Journal of Game Theory, 1975):

$ gambit-ipa e02.nfg
Compute Nash equilibria using iterated polymatrix approximation
Gametracer version 0.2, Copyright (C) 2002, Ben Blum and Christian Shelton
Gambit version 15.1.0, Copyright (C) 1994-2014, The Gambit Project
This is free software, distributed under the GNU GPL

NE,1.000000,0.000000,0.000000,1.000000,0.000000

Note

This is an experimental program and has not been extensively tested.

See also

gambit-gnm: Compute Nash equilibria in a strategic game using a global Newton method.

gambit-lcp: Compute equilibria in a two-player game via linear complementarity

gambit-lcp reads a two-player game on standard input and computes Nash equilibria by finding solutions to a linear complementarity problem. For extensive games, the program uses the sequence form representation of the extensive game, as defined by Koller, Megiddo, and von Stengel [KolMegSte94], and applies the algorithm developed by Lemke. For strategic games, the program using the method of Lemke and Howson [LemHow64]. There exist strategic games for which some equilibria cannot be located by this method; see Shapley [Sha74].

In a two-player strategic game, the set of Nash equilibria can be expressed as the union of convex sets. This program will find extreme points of those convex sets. See gambit-enummixed: Enumerate equilibria in a two-player game for a method which is guaranteed to find all the extreme points for a strategic game.

-d

By default, this program computes using exact rational arithmetic. Since the extreme points computed by this method are guaranteed to be rational when the payoffs in the game are rational, this permits exact computation of the equilibrium set. Computation using rational arithmetic is in general slow, however. For most games, acceptable results can be obtained by computing using the computer’s native floating-point arithmetic. Using this flag enables computation in floating-point, and expresses all output using decimal representations with the specified number of digits.

-S

By default, the program uses behavior strategies for extensive games; this switch instructs the program to use reduced strategic game strategies for extensive games. (This has no effect for strategic games, since a strategic game is its own reduced strategic game.)

-D

New in version 14.0.2.

The default output format for computed equilibria is a comma-separated list of strategy or action probabilities, suitable for postprocessing by automated tools. Specifying -D instead causes the program to output greater detail on each equilbrium profile computed.

-P

By default, the program computes Nash equilibria in an extensive game. This switch instructs the program to find only equilibria which are subgame perfect. (This has no effect for strategic games, since there are no proper subgames of a strategic game.)

-h

Prints a help message listing the available options.

-q

Suppresses printing of the banner at program launch.

Computing an equilibrium of extensive game e02.efg, the example in Figure 2 of Selten (International Journal of Game Theory, 1975):

$ gambit-lcp e02.efg
Compute Nash equilibria by solving a linear complementarity program
Gambit version 15.1.0, Copyright (C) 1994-2014, The Gambit Project
This is free software, distributed under the GNU GPL

NE,1,0,1/2,1/2,1/2,1/2

gambit-lp: Compute equilibria in a two-player constant-sum game via linear programming

gambit-lp reads a two-player constant-sum game on standard input and computes a Nash equilibrium by solving a linear program. The program uses the sequence form formulation of Koller, Megiddo, and von Stengel [KolMegSte94] for extensive games.

While the set of equilibria in a two-player constant-sum strategic game is convex, this method will only identify one of the extreme points of that set.

-d

By default, this program computes using exact rational arithmetic. Since the extreme points computed by this method are guaranteed to be rational when the payoffs in the game are rational, this permits exact computation of an equilibrium. Computation using rational arithmetic is in general slow, however. For most games, acceptable results can be obtained by computing using the computer’s native floating-point arithmetic. Using this flag enables computation in floating-point, and expresses all output using decimal representations with the specified number of digits.

-S

By default, the program uses behavior strategies for extensive games; this switch instructs the program to use reduced strategic game strategies for extensive games. (This has no effect for strategic games, since a strategic game is its own reduced strategic game.)

-D

New in version 14.0.3.

The default output format for computed equilibria is a comma-separated list of strategy or action probabilities, suitable for postprocessing by automated tools. Specifying -D instead causes the program to output greater detail on each equilbrium profile computed.

-P

By default, the program computes Nash equilibria in an extensive game. This switch instructs the program to find only equilibria which are subgame perfect. (This has no effect for strategic games, since there are no proper subgames of a strategic game.)

-h

Prints a help message listing the available options.

-q

Suppresses printing of the banner at program launch.

Computing an equilibrium of the game 2x2const.nfg, a game with two players with two strategies each, with a unique equilibrium in mixed strategies:

$ gambit-lp 2x2const.nfg
Compute Nash equilibria by solving a linear program
Gambit version 15.1.0, Copyright (C) 1994-2014, The Gambit Project
This is free software, distributed under the GNU GPL

NE,1/3,2/3,1/3,2/3

gambit-liap: Compute Nash equilibria using function minimization

gambit-liap reads a game on standard input and computes approximate Nash equilibria using a function minimization approach.

This procedure searches for equilibria by generating random starting points and using conjugate gradient descent to minimize the Lyapunov function of the game. This function is a nonnegative function which is zero exactly at strategy profiles which are Nash equilibria.

Note that this procedure is not globally convergent. That is, it is not guaranteed to find all, or even any, Nash equilibria.

-d

Express all output using decimal representations with the specified number of digits.

-n

Specify the number of starting points to randomly generate.

-h

Prints a help message listing the available options.

-q

Suppresses printing of the banner at program launch.

-s

Specifies a file containing a list of starting points for the algorithm. The format of the file is comma-separated values, one mixed strategy profile per line, in the same format used for output of equilibria (excluding the initial NE tag).

-S

By default, the program uses behavior strategies for extensive games; this switch instructs the program to use reduced strategic game strategies for extensive games. (This has no effect for strategic games, since a strategic game is its own reduced strategic game.)

-v

Sets verbose mode. In verbose mode, initial points, as well as points at which the minimization fails at a constrained local minimum that is not a Nash equilibrium, are all output, in addition to any equilibria found.

Computing an equilibrium in mixed strategies of e02.efg, the example in Figure 2 of Selten (International Journal of Game Theory, 1975):

$ gambit-liap e02.nfg
Compute Nash equilibria by minimizing the Lyapunov function
Gambit version 15.1.0, Copyright (C) 1994-2014, The Gambit Project
This is free software, distributed under the GNU GPL

NE, 0.998701, 0.000229, 0.001070, 0.618833, 0.381167

gambit-simpdiv: Compute equilibria via simplicial subdivision

gambit-simpdiv reads a game on standard input and computes approximations to Nash equilibria using a simplicial subdivision approach.

This program implements the algorithm of van der Laan, Talman, and van Der Heyden [VTH87]. The algorithm proceeds by constructing a triangulated grid over the space of mixed strategy profiles, and uses a path-following method to compute an approximate fixed point. This approximate fixed point can then be used as a starting point on a refinement of the grid. The program continues this process with finer and finer grids until locating a mixed strategy profile at which the maximum regret is small.

The algorithm begins with any mixed strategy profile consisting of rational numbers as probabilities. Without any options, the algorithm begins with the centroid, and computes one Nash equilibrium. To attempt to compute other equilibria that may exist, use the gambit-simpdiv -r or gambit-simpdiv -s options to specify additional starting points for the algorithm.

-g

Sets the granularity of the grid refinement. By default, when the grid is refined, the stepsize is cut in half, which corresponds to specifying -g 2. If this parameter is specified, the grid is refined at each step by a multiple of MULT .

-h

Prints a help message listing the available options.

-n

Randomly generate COUNT starting points. Only applicable if option gambit-simpdiv -r is also specified.

-q

Suppresses printing of the banner at program launch.

-r

Generate random starting points with denominator DENOM. Since this algorithm operates on a grid, by its nature the probabilities it works with are always rational numbers. If this parameter is specified, starting points for the procedure are generated randomly using the uniform distribution over strategy profiles with probabilities having denominator DENOM.

-s

Specifies a file containing a list of starting points for the algorithm. The format of the file is comma-separated values, one mixed strategy profile per line, in the same format used for output of equilibria (excluding the initial NE tag).

-v

Sets verbose mode. In verbose mode, initial points, as well as the approximations computed at each grid refinement, are all output, in addition to the approximate equilibrium profile found.

Computing an equilibrium in mixed strategies of e02.efg, the example in Figure 2 of Selten (International Journal of Game Theory, 1975):

$ gambit-simpdiv e02.nfg
Compute Nash equilibria using simplicial subdivision
Gambit version 15.1.0, Copyright (C) 1994-2014, The Gambit Project
This is free software, distributed under the GNU GPL

NE,1,0,0,1,0

gambit-logit: Compute quantal response equilbria

gambit-logit reads a game on standard input and computes the principal branch of the (logit) quantal response correspondence.

The method is based on the procedure described in Turocy [Tur05] for strategic games and Turocy [Tur10] for extensive games. It uses standard path-following methods (as described in Allgower and Georg’s “Numerical Continuation Methods”) to adaptively trace the principal branch of the correspondence efficiently and securely.

The method used is a predictor-corrector method, which first generates a prediction using the differential equations describing the branch of the correspondence, followed by a corrector step which refines the prediction using Newton’s method for finding a zero of a function. Two parameters control the operation of this tracing. The option -s sets the initial step size for the predictor phase of the tracing. This step size is then dynamically adjusted based on the rate of convergence of Newton’s method in the corrector step. If the convergence is fast, the step size is adjusted upward (accelerated); if it is slow, the step size is decreased (decelerated). The option -a sets the maximum acceleration (or deceleration). As described in Turocy [Tur05], this acceleration helps to efficiently trace the correspondence when it reaches its asymptotic phase for large values of the precision parameter lambda.

-d

Express all output using decimal representations with the specified number of digits. The default is -d 6.

-s

Sets the initial step size for the predictor phase of the tracing procedure. The default value is .03. The step size is specified in terms of the arclength along the branch of the correspondence, and not the size of the step measured in terms of lambda. So, for example, if the step size is currently .03, but the position of the strategy profile on the branch is changing rapidly with lambda, then lambda will change by much less then .03 between points reported by the program.

-a

Sets the maximum acceleration of the step size during the tracing procedure. This is interpreted as a multiplier. The default is 1.1, which means the step size is increased or decreased by no more than ten percent of its current value at every step. A value close to one would keep the step size (almost) constant at every step.

-m

Stop when reaching the specified value of the parameter lambda. By default, the tracing stops when lambda reaches 1,000,000, which is usually suitable for computing a good approximation to a Nash equilibrium. For applications, such as to laboratory experiments, where the behavior of the correspondence for small values of lambda is of interest and the asymptotic behavior is not relevant, setting MAXLAMBDA to a much smaller value may be indicated.

-l

While tracing, compute the logit equilibrium points with parameter LAMBDA accurately.

-S

By default, the program uses behavior strategies for extensive games; this switch instructs the program to use reduced strategic game strategies for extensive games. (This has no effect for strategic games, since a strategic game is its own reduced strategic game.)

-h

Prints a help message listing the available options.

-e

By default, all points computed are output by the program. If this switch is specified, only the approximation to the Nash equilibrium at the end of the branch is output.

Computing the principal branch, in mixed strategies, of e02.nfg, the reduced strategic form of the example in Figure 2 of Selten (International Journal of Game Theory, 1975):

$ gambit-logit e02.nfg
Compute a branch of the logit equilibrium correspondence
Gambit version 15.1.0, Copyright (C) 1994-2014, The Gambit Project
This is free software, distributed under the GNU GPL

0.000000,0.333333,0.333333,0.333333,0.5,0.5
0.022853,0.335873,0.328284,0.335843,0.501962,0.498038
0.047978,0.338668,0.322803,0.33853,0.504249,0.495751
0.075600,0.341747,0.316863,0.34139,0.506915,0.493085
0.105965,0.345145,0.310443,0.344413,0.510023,0.489977
0.139346,0.348902,0.303519,0.347578,0.51364,0.48636

...

735614.794714,1,0,4.40659e-11,0.500016,0.499984
809176.283787,1,0,3.66976e-11,0.500015,0.499985
890093.921767,1,0,3.05596e-11,0.500014,0.499986
979103.323545,1,0,2.54469e-11,0.500012,0.499988
1077013.665501,1,0,2.11883e-11,0.500011,0.499989

gambit-convert: Convert games among various representations

gambit-convert reads a game on standard input in any supported format and converts it to another text representation. Currently, this tool supports outputting the strategic form of the game in one of these formats:

• A standard HTML table.
• A LaTeX fragment in the format of Martin Osborne’s sgame macros (see http://www.economics.utoronto.ca/osborne/latex/index.html).

-O FORMAT

Required. Specifies the output format. Supported options for FORMAT are html or sgame.

-r PLAYER

Specifies the player number to place on the rows of the tables. The default if not specified is to place player 1 on the rows.

-c PLAYER

Specifies the player number to place on the columns of the tables. The default if not specified is to place player 2 on the columns.

-h

Prints a help message listing the available options.

-q

Suppresses printing of the banner at program launch.

Example invocation for HTML output:

$ gambit-convert -O html 2x2.nfg
Convert games among various file formats
Gambit version 15.1.0, Copyright (C) 1994-2014, The Gambit Project
This is free software, distributed under the GNU GPL

<center><h1>Two person 2 x 2 game with unique mixed equilibrium</h1></center>
<table><tr><td></td><td align=center><b>1</b></td><td
align=center><b>2</b></td></tr><tr><td align=center><b>1</b></td><td
align=center>2,0</td><td align=center>0,1</td></tr><tr><td
align=center><b>2</b></td><td align=center>0,1</td><td
align=center>1,0</td></tr></table>

Example invocation for LaTeX output:

$ gambit-convert -O sgame 2x2.nfg
Convert games among various file formats
Gambit version 15.1.0, Copyright (C) 1994-2014, The Gambit Project
This is free software, distributed under the GNU GPL

\begin{game}{2}{2}[Player 1][Player 2]
&1 & 2\\
1 &  $2,0$  &  $0,1$ \\
2 &  $0,1$  &  $1,0$
\end{game}