A catalog of normal form games#
This allows us to construct common games directly:
sage: g = game_theory.normal_form_games.PrisonersDilemma()
sage: g
Prisoners dilemma - Normal Form Game with the following utilities: ...
We can then immediately obtain the Nash equilibrium for this game:
sage: g.obtain_nash()
[[(0, 1), (0, 1)]]
When we test whether the game is actually the one in question, sometimes we will build a dictionary to test it, since the printed representation can be platform-dependent, like so:
sage: d = {(0, 0): [-2, -2], (0, 1): [-5, 0], (1, 0): [0, -5], (1, 1): [-4, -4]}
sage: g == d
True
The docstrings give an interpretation of each game.
More information is available in the following references:
REFERENCES:
AUTHOR:
James Campbell and Vince Knight (06-2014)
- sage.game_theory.catalog_normal_form_games.AntiCoordinationGame(A=3, a=3, B=5, b=1, C=1, c=5, D=0, d=0)#
Return a 2 by 2 AntiCoordination Game.
An anti coordination game is a particular type of game where the pure Nash equilibria is for the players to pick different strategies.
In general these are represented as a normal form game using the following two matrices:
\[ \begin{align}\begin{aligned}\begin{split}A = \begin{pmatrix} A&C\\ B&D\\ \end{pmatrix}\end{split}\\\begin{split}B = \begin{pmatrix} a&c\\ b&d\\ \end{pmatrix}\end{split}\end{aligned}\end{align} \]Where \(A < B, D < C\) and \(a < c, d < b\).
An often used version is the following:
\[ \begin{align}\begin{aligned}\begin{split}A = \begin{pmatrix} 3&1\\ 5&0\\ \end{pmatrix}\end{split}\\\begin{split} B = \begin{pmatrix} 3&5\\ 1&0\\ \end{pmatrix}\end{split}\end{aligned}\end{align} \]This is the default version of the game created by this function:
sage: g = game_theory.normal_form_games.AntiCoordinationGame() sage: g Anti coordination game - Normal Form Game with the following utilities: ... sage: d ={(0, 1): [1, 5], (1, 0): [5, 1], ....: (0, 0): [3, 3], (1, 1): [0, 0]} sage: g == d True
There are two pure Nash equilibria and one mixed:
sage: g.obtain_nash() [[(0, 1), (1, 0)], [(1/3, 2/3), (1/3, 2/3)], [(1, 0), (0, 1)]]
We can also pass different values of the input parameters:
sage: g = game_theory.normal_form_games.AntiCoordinationGame(A=2, a=3, ....: B=4, b=2, C=2, c=8, D=1, d=0) sage: g Anti coordination game - Normal Form Game with the following utilities: ... sage: d ={(0, 1): [2, 8], (1, 0): [4, 2], ....: (0, 0): [2, 3], (1, 1): [1, 0]} sage: g == d True sage: g.obtain_nash() [[(0, 1), (1, 0)], [(2/7, 5/7), (1/3, 2/3)], [(1, 0), (0, 1)]]
Note that an error is returned if the defining inequality is not obeyed \(A > B, D > C\) and \(a > c, d > b\):
sage: g = game_theory.normal_form_games.AntiCoordinationGame(A=8, a=3, ....: B=4, b=2, C=2, c=8, D=1, d=0) Traceback (most recent call last): ... TypeError: the input values for an Anti coordination game must be of the form A < B, D < C, a < c and d < b
- sage.game_theory.catalog_normal_form_games.BattleOfTheSexes()#
Return a Battle of the Sexes game.
Consider two payers: Amy and Bob. Amy prefers to play video games and Bob prefers to watch a movie. They both however want to spend their evening together. This can be modeled as a normal form game using the following two matrices [Web2007]:
\[ \begin{align}\begin{aligned}\begin{split}A = \begin{pmatrix} 3&1\\ 0&2\\ \end{pmatrix}\end{split}\\\begin{split} B = \begin{pmatrix} 2&1\\ 0&3\\ \end{pmatrix}\end{split}\end{aligned}\end{align} \]This is a particular type of Coordination Game. There are three Nash equilibria:
Amy and Bob both play video games;
Amy and Bob both watch a movie;
Amy plays video games 75% of the time and Bob watches a movie 75% of the time.
This can be implemented in Sage using the following:
sage: g = game_theory.normal_form_games.BattleOfTheSexes() sage: g Battle of the sexes - Coordination game - Normal Form Game with the following utilities: ... sage: d = {(0, 1): [1, 1], (1, 0): [0, 0], (0, 0): [3, 2], (1, 1): [2, 3]} sage: g == d True sage: g.obtain_nash() [[(0, 1), (0, 1)], [(3/4, 1/4), (1/4, 3/4)], [(1, 0), (1, 0)]]
- sage.game_theory.catalog_normal_form_games.Chicken(A=0, a=0, B=1, b=-1, C=-1, c=1, D=-10, d=-10)#
Return a Chicken game.
Consider two drivers locked in a fierce battle for pride. They drive towards a cliff and the winner is declared as the last one to swerve. If neither player swerves they will both fall off the cliff.
This can be modeled as a particular type of anti coordination game using the following two matrices:
\[ \begin{align}\begin{aligned}\begin{split}A = \begin{pmatrix} A&C\\ B&D\\ \end{pmatrix}\end{split}\\\begin{split}B = \begin{pmatrix} a&c\\ b&d\\ \end{pmatrix}\end{split}\end{aligned}\end{align} \]Where \(A < B, D < C\) and \(a < c, d < b\) but with the extra condition that \(A > C\) and \(a > b\).
Here are the numeric values used by default [Wat2003]:
\[ \begin{align}\begin{aligned}\begin{split}A = \begin{pmatrix} 0&-1\\ 1&-10\\ \end{pmatrix}\end{split}\\\begin{split} B = \begin{pmatrix} 0&1\\ -1&-10\\ \end{pmatrix}\end{split}\end{aligned}\end{align} \]There are three Nash equilibria:
The second player swerving.
The first player swerving.
Both players swerving with 1 out of 10 times.
This can be implemented in Sage using the following:
sage: g = game_theory.normal_form_games.Chicken() sage: g Chicken - Anti coordination game - Normal Form Game with the following utilities: ... sage: d = {(0, 1): [-1, 1], (1, 0): [1, -1], ....: (0, 0): [0, 0], (1, 1): [-10, -10]} sage: g == d True sage: g.obtain_nash() [[(0, 1), (1, 0)], [(9/10, 1/10), (9/10, 1/10)], [(1, 0), (0, 1)]]
Non default values can be passed:
sage: g = game_theory.normal_form_games.Chicken(A=0, a=0, B=2, ....: b=-1, C=-1, c=2, D=-100, d=-100) sage: g Chicken - Anti coordination game - Normal Form Game with the following utilities: ... sage: d = {(0, 1): [-1, 2], (1, 0): [2, -1], ....: (0, 0): [0, 0], (1, 1): [-100, -100]} sage: g == d True sage: g.obtain_nash() [[(0, 1), (1, 0)], [(99/101, 2/101), (99/101, 2/101)], [(1, 0), (0, 1)]]
Note that an error is returned if the defining inequalities are not obeyed \(B > A > C > D\) and \(c > a > b > d\):
sage: g = game_theory.normal_form_games.Chicken(A=8, a=3, B=4, b=2, ....: C=2, c=8, D=1, d=0) Traceback (most recent call last): ... TypeError: the input values for a game of chicken must be of the form B > A > C > D and c > a > b > d
- sage.game_theory.catalog_normal_form_games.CoordinationGame(A=10, a=5, B=0, b=0, C=0, c=0, D=5, d=10)#
Return a 2 by 2 Coordination Game.
A coordination game is a particular type of game where the pure Nash equilibrium is for the players to pick the same strategies [Web2007].
In general these are represented as a normal form game using the following two matrices:
\[ \begin{align}\begin{aligned}\begin{split}A = \begin{pmatrix} A&C\\ B&D\\ \end{pmatrix}\end{split}\\\begin{split}B = \begin{pmatrix} a&c\\ b&d\\ \end{pmatrix}\end{split}\end{aligned}\end{align} \]Where \(A > B, D > C\) and \(a > c, d > b\).
An often used version is the following:
\[ \begin{align}\begin{aligned}\begin{split}A = \begin{pmatrix} 10&0\\ 0&5\\ \end{pmatrix}\end{split}\\\begin{split} B = \begin{pmatrix} 5&0\\ 0&10\\ \end{pmatrix}\end{split}\end{aligned}\end{align} \]This is the default version of the game created by this function:
sage: g = game_theory.normal_form_games.CoordinationGame() sage: g Coordination game - Normal Form Game with the following utilities: ... sage: d = {(0, 1): [0, 0], (1, 0): [0, 0], ....: (0, 0): [10, 5], (1, 1): [5, 10]} sage: g == d True
There are two pure Nash equilibria and one mixed:
sage: g.obtain_nash() [[(0, 1), (0, 1)], [(2/3, 1/3), (1/3, 2/3)], [(1, 0), (1, 0)]]
We can also pass different values of the input parameters:
sage: g = game_theory.normal_form_games.CoordinationGame(A=9, a=6, ....: B=2, b=1, C=0, c=1, D=4, d=11) sage: g Coordination game - Normal Form Game with the following utilities: ... sage: d ={(0, 1): [0, 1], (1, 0): [2, 1], ....: (0, 0): [9, 6], (1, 1): [4, 11]} sage: g == d True sage: g.obtain_nash() [[(0, 1), (0, 1)], [(2/3, 1/3), (4/11, 7/11)], [(1, 0), (1, 0)]]
Note that an error is returned if the defining inequalities are not obeyed \(A > B, D > C\) and \(a > c, d > b\):
sage: g = game_theory.normal_form_games.CoordinationGame(A=9, a=6, ....: B=0, b=1, C=2, c=10, D=4, d=11) Traceback (most recent call last): ... TypeError: the input values for a Coordination game must be of the form A > B, D > C, a > c and d > b
- sage.game_theory.catalog_normal_form_games.HawkDove(v=2, c=3)#
Return a Hawk Dove game.
Suppose two birds of prey must share a limited resource \(v\). The birds can act like a hawk or a dove.
If a dove meets a hawk, the hawk takes the resources.
If two doves meet they share the resources.
If two hawks meet, one will win (with equal expectation) and take the resources while the other will suffer a cost of \(c\) where \(c>v\).
This can be modeled as a normal form game using the following two matrices [Web2007]:
\[ \begin{align}\begin{aligned}\begin{split}A = \begin{pmatrix} v/2-c&v\\ 0&v/2\\ \end{pmatrix}\end{split}\\\begin{split} B = \begin{pmatrix} v/2-c&0\\ v&v/2\\ \end{pmatrix}\end{split}\end{aligned}\end{align} \]Here are the games with the default values of \(v=2\) and \(c=3\).
\[ \begin{align}\begin{aligned}\begin{split}A = \begin{pmatrix} -2&2\\ 0&1\\ \end{pmatrix}\end{split}\\\begin{split} B = \begin{pmatrix} -2&0\\ 2&1\\ \end{pmatrix}\end{split}\end{aligned}\end{align} \]This is a particular example of an anti coordination game. There are three Nash equilibria:
One bird acts like a Hawk and the other like a Dove.
Both birds mix being a Hawk and a Dove
This can be implemented in Sage using the following:
sage: g = game_theory.normal_form_games.HawkDove() sage: g Hawk-Dove - Anti coordination game - Normal Form Game with the following utilities: ... sage: d ={(0, 1): [2, 0], (1, 0): [0, 2], ....: (0, 0): [-2, -2], (1, 1): [1, 1]} sage: g == d True sage: g.obtain_nash() [[(0, 1), (1, 0)], [(1/3, 2/3), (1/3, 2/3)], [(1, 0), (0, 1)]] sage: g = game_theory.normal_form_games.HawkDove(v=1, c=3) sage: g Hawk-Dove - Anti coordination game - Normal Form Game with the following utilities: ... sage: d ={(0, 1): [1, 0], (1, 0): [0, 1], ....: (0, 0): [-5/2, -5/2], (1, 1): [1/2, 1/2]} sage: g == d True sage: g.obtain_nash() [[(0, 1), (1, 0)], [(1/6, 5/6), (1/6, 5/6)], [(1, 0), (0, 1)]]
Note that an error is returned if the defining inequality is not obeyed \(c < v\):
sage: g = game_theory.normal_form_games.HawkDove(v=5, c=1) Traceback (most recent call last): … TypeError: the input values for a Hawk Dove game must be of the form c > v
- sage.game_theory.catalog_normal_form_games.MatchingPennies()#
Return a Matching Pennies game.
Consider two players who can choose to display a coin either Heads facing up or Tails facing up. If both players show the same face then player 1 wins, if not then player 2 wins.
This can be modeled as a zero sum normal form game with the following matrix [Web2007]:
\[\begin{split}A = \begin{pmatrix} 1&-1\\ -1&1\\ \end{pmatrix}\end{split}\]There is a single Nash equilibria at which both players randomly (with equal probability) pick heads or tails.
This can be implemented in Sage using the following:
sage: g = game_theory.normal_form_games.MatchingPennies() sage: g Matching pennies - Normal Form Game with the following utilities: ... sage: d ={(0, 1): [-1, 1], (1, 0): [-1, 1], ....: (0, 0): [1, -1], (1, 1): [1, -1]} sage: g == d True sage: g.obtain_nash('enumeration') [[(1/2, 1/2), (1/2, 1/2)]]
- sage.game_theory.catalog_normal_form_games.Pigs()#
Return a Pigs game.
Consider two pigs. One dominant pig and one subservient pig. These pigs share a pen. There is a lever in the pen that delivers 6 units of food but if either pig pushes the lever it will take them a little while to get to the food as well as cost them 1 unit of food. If the dominant pig pushes the lever, the subservient pig has some time to eat two thirds of the food before being pushed out of the way. If the subservient pig pushes the lever, the dominant pig will eat all the food. Finally if both pigs go to push the lever the subservient pig will be able to eat a third of the food (and they will also both lose 1 unit of food).
This can be modeled as a normal form game using the following two matrices [McM1992] (we assume that the dominant pig’s utilities are given by \(A\)):
\[ \begin{align}\begin{aligned}\begin{split}A = \begin{pmatrix} 3&1\\ 6&0\\ \end{pmatrix}\end{split}\\\begin{split} B = \begin{pmatrix} 1&4\\ -1&0\\ \end{pmatrix}\end{split}\end{aligned}\end{align} \]There is a single Nash equilibrium at which the dominant pig pushes the lever and the subservient pig does not.
This can be implemented in Sage using the following:
sage: g = game_theory.normal_form_games.Pigs() sage: g Pigs - Normal Form Game with the following utilities: ... sage: d ={(0, 1): [1, 4], (1, 0): [6, -1], ....: (0, 0): [3, 1], (1, 1): [0, 0]} sage: g == d True sage: g.obtain_nash() [[(1, 0), (0, 1)]]
- sage.game_theory.catalog_normal_form_games.PrisonersDilemma(R=-2, P=-4, S=-5, T=0)#
Return a Prisoners dilemma game.
Assume two thieves have been caught by the police and separated for questioning. If both thieves cooperate and do not divulge any information they will each get a short sentence. If one defects he/she is offered a deal while the other thief will get a long sentence. If they both defect they both get a medium length sentence.
This can be modeled as a normal form game using the following two matrices [Web2007]:
\[ \begin{align}\begin{aligned}\begin{split}A = \begin{pmatrix} R&S\\ T&P\\ \end{pmatrix}\end{split}\\\begin{split} B = \begin{pmatrix} R&T\\ S&P\\ \end{pmatrix}\end{split}\end{aligned}\end{align} \]Where \(T > R > P > S\).
\(R\) denotes the reward received for cooperating.
\(S\) denotes the ‘sucker’ utility.
\(P\) denotes the utility for punishing the other player.
\(T\) denotes the temptation payoff.
An often used version [Web2007] is the following:
\[ \begin{align}\begin{aligned}\begin{split}A = \begin{pmatrix} -2&-5\\ 0&-4\\ \end{pmatrix}\end{split}\\\begin{split} B = \begin{pmatrix} -2&0\\ -5&-4\\ \end{pmatrix}\end{split}\end{aligned}\end{align} \]There is a single Nash equilibrium for this at which both thieves defect. This can be implemented in Sage using the following:
sage: g = game_theory.normal_form_games.PrisonersDilemma() sage: g Prisoners dilemma - Normal Form Game with the following utilities: ... sage: d = {(0, 0): [-2, -2], (0, 1): [-5, 0], (1, 0): [0, -5], ....: (1, 1): [-4, -4]} sage: g == d True sage: g.obtain_nash() [[(0, 1), (0, 1)]]
Note that we can pass other values of R, P, S, T:
sage: g = game_theory.normal_form_games.PrisonersDilemma(R=-1, P=-2, S=-3, T=0) sage: g Prisoners dilemma - Normal Form Game with the following utilities:... sage: d = {(0, 1): [-3, 0], (1, 0): [0, -3], ....: (0, 0): [-1, -1], (1, 1): [-2, -2]} sage: g == d True sage: g.obtain_nash() [[(0, 1), (0, 1)]]
If we pass values that fail the defining requirement: \(T > R > P > S\) we get an error message:
sage: g = game_theory.normal_form_games.PrisonersDilemma(R=-1, P=-2, S=0, T=5) Traceback (most recent call last): ... TypeError: the input values for a Prisoners Dilemma must be of the form T > R > P > S
- sage.game_theory.catalog_normal_form_games.RPS()#
Return a Rock-Paper-Scissors game.
Rock-Paper-Scissors is a zero sum game usually played between two players where each player simultaneously forms one of three shapes with an outstretched hand.The game has only three possible outcomes other than a tie: a player who decides to play rock will beat another player who has chosen scissors (“rock crushes scissors”) but will lose to one who has played paper (“paper covers rock”); a play of paper will lose to a play of scissors (“scissors cut paper”). If both players throw the same shape, the game is tied and is usually immediately replayed to break the tie.
This can be modeled as a zero sum normal form game with the following matrix [Web2007]:
\[\begin{split}A = \begin{pmatrix} 0 & -1 & 1\\ 1 & 0 & -1\\ -1 & 1 & 0\\ \end{pmatrix}\end{split}\]This can be implemented in Sage using the following:
sage: g = game_theory.normal_form_games.RPS() sage: g Rock-Paper-Scissors - Normal Form Game with the following utilities: ... sage: d = {(0, 1): [-1, 1], (1, 2): [-1, 1], (0, 0): [0, 0], ....: (2, 1): [1, -1], (1, 1): [0, 0], (2, 0): [-1, 1], ....: (2, 2): [0, 0], (1, 0): [1, -1], (0, 2): [1, -1]} sage: g == d True sage: g.obtain_nash('enumeration') [[(1/3, 1/3, 1/3), (1/3, 1/3, 1/3)]]
- sage.game_theory.catalog_normal_form_games.RPSLS()#
Return a Rock-Paper-Scissors-Lizard-Spock game.
Rock-Paper-Scissors-Lizard-Spock is an extension of Rock-Paper-Scissors. It is a zero sum game usually played between two players where each player simultaneously forms one of three shapes with an outstretched hand. This game became popular after appearing on the television show ‘Big Bang Theory’. The rules for the game can be summarised as follows:
Scissors cuts Paper
Paper covers Rock
Rock crushes Lizard
Lizard poisons Spock
Spock smashes Scissors
Scissors decapitates Lizard
Lizard eats Paper
Paper disproves Spock
Spock vaporizes Rock
(and as it always has) Rock crushes Scissors
This can be modeled as a zero sum normal form game with the following matrix:
\[\begin{split}A = \begin{pmatrix} 0 & -1 & 1 & 1 & -1\\ 1 & 0 & -1 & -1 & 1\\ -1 & 1 & 0 & 1 & -1\\ -1 & 1 & -1 & 0 & 1\\ 1 & -1 & 1 & -1 & 0\\ \end{pmatrix}\end{split}\]This can be implemented in Sage using the following:
sage: g = game_theory.normal_form_games.RPSLS() sage: g Rock-Paper-Scissors-Lizard-Spock - Normal Form Game with the following utilities: ... sage: d = {(1, 3): [-1, 1], (3, 0): [-1, 1], (2, 1): [1, -1], ....: (0, 3): [1, -1], (4, 0): [1, -1], (1, 2): [-1, 1], ....: (3, 3): [0, 0], (4, 4): [0, 0], (2, 2): [0, 0], ....: (4, 1): [-1, 1], (1, 1): [0, 0], (3, 2): [-1, 1], ....: (0, 0): [0, 0], (0, 4): [-1, 1], (1, 4): [1, -1], ....: (2, 3): [1, -1], (4, 2): [1, -1], (1, 0): [1, -1], ....: (0, 1): [-1, 1], (3, 1): [1, -1], (2, 4): [-1, 1], ....: (2, 0): [-1, 1], (4, 3): [-1, 1], (3, 4): [1, -1], ....: (0, 2): [1, -1]} sage: g == d True sage: g.obtain_nash('enumeration') [[(1/5, 1/5, 1/5, 1/5, 1/5), (1/5, 1/5, 1/5, 1/5, 1/5)]]
- sage.game_theory.catalog_normal_form_games.StagHunt()#
Return a Stag Hunt game.
Assume two friends go out on a hunt. Each can individually choose to hunt a stag or hunt a hare. Each player must choose an action without knowing the choice of the other. If an individual hunts a stag, he must have the cooperation of his partner in order to succeed. An individual can get a hare by himself, but a hare is worth less than a stag.
This can be modeled as a normal form game using the following two matrices [Sky2003]:
\[ \begin{align}\begin{aligned}\begin{split}A = \begin{pmatrix} 5&0\\ 4&2\\ \end{pmatrix}\end{split}\\\begin{split} B = \begin{pmatrix} 5&4\\ 0&2\\ \end{pmatrix}\end{split}\end{aligned}\end{align} \]This is a particular type of Coordination Game. There are three Nash equilibria:
Both friends hunting the stag.
Both friends hunting the hare.
Both friends hunting the stag 2/3rds of the time.
This can be implemented in Sage using the following:
sage: g = game_theory.normal_form_games.StagHunt() sage: g Stag hunt - Coordination game - Normal Form Game with the following utilities: ... sage: d = {(0, 1): [0, 4], (1, 0): [4, 0], ....: (0, 0): [5, 5], (1, 1): [2, 2]} sage: g == d True sage: g.obtain_nash() [[(0, 1), (0, 1)], [(2/3, 1/3), (2/3, 1/3)], [(1, 0), (1, 0)]]
- sage.game_theory.catalog_normal_form_games.TravellersDilemma(max_value=10)#
Return a Travellers dilemma game.
An airline loses two suitcases belonging to two different travelers. Both suitcases happen to be identical and contain identical antiques. An airline manager tasked to settle the claims of both travelers explains that the airline is liable for a maximum of 10 per suitcase, and in order to determine an honest appraised value of the antiques the manager separates both travelers so they can’t confer, and asks them to write down the amount of their value at no less than 2 and no larger than 10. He also tells them that if both write down the same number, he will treat that number as the true dollar value of both suitcases and reimburse both travelers that amount.
However, if one writes down a smaller number than the other, this smaller number will be taken as the true dollar value, and both travelers will receive that amount along with a bonus/malus: 2 extra will be paid to the traveler who wrote down the lower value and a 2 deduction will be taken from the person who wrote down the higher amount. The challenge is: what strategy should both travelers follow to decide the value they should write down?
This can be modeled as a normal form game using the following two matrices [Ba1994]:
\[ \begin{align}\begin{aligned}\begin{split}A = \begin{pmatrix} 10 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0\\ 11 & 9 & 6 & 5 & 4 & 3 & 2 & 1 & 0\\ 10 & 10 & 8 & 5 & 4 & 3 & 2 & 1 & 0\\ 9 & 9 & 9 & 7 & 4 & 3 & 2 & 1 & 0\\ 8 & 8 & 8 & 8 & 6 & 3 & 2 & 1 & 0\\ 7 & 7 & 7 & 7 & 7 & 5 & 2 & 1 & 0\\ 6 & 6 & 6 & 6 & 6 & 6 & 4 & 1 & 0\\ 5 & 5 & 5 & 5 & 5 & 5 & 5 & 3 & 0\\ 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 2\\ \end{pmatrix}\end{split}\\\begin{split}B = \begin{pmatrix} 10 & 11 & 10 & 9 & 8 & 7 & 6 & 5 & 4\\ 7 & 9 & 10 & 9 & 8 & 7 & 6 & 5 & 4\\ 6 & 6 & 8 & 9 & 8 & 7 & 6 & 5 & 4\\ 5 & 5 & 5 & 7 & 8 & 7 & 6 & 5 & 4\\ 4 & 4 & 4 & 4 & 6 & 7 & 6 & 5 & 4\\ 3 & 3 & 3 & 3 & 3 & 5 & 6 & 5 & 4\\ 2 & 2 & 2 & 2 & 2 & 2 & 4 & 5 & 4\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 3 & 4\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}\end{split}\end{aligned}\end{align} \]There is a single Nash equilibrium to this game resulting in both players naming the smallest possible value.
This can be implemented in Sage using the following:
sage: g = game_theory.normal_form_games.TravellersDilemma() sage: g Travellers dilemma - Normal Form Game with the following utilities: ... sage: d = {(7, 3): [5, 1], (4, 7): [1, 5], (1, 3): [5, 9], ....: (4, 8): [0, 4], (3, 0): [9, 5], (2, 8): [0, 4], ....: (8, 0): [4, 0], (7, 8): [0, 4], (5, 4): [7, 3], ....: (0, 7): [1, 5], (5, 6): [2, 6], (2, 6): [2, 6], ....: (1, 6): [2, 6], (5, 1): [7, 3], (3, 7): [1, 5], ....: (0, 3): [5, 9], (8, 5): [4, 0], (2, 5): [3, 7], ....: (5, 8): [0, 4], (4, 0): [8, 4], (1, 2): [6, 10], ....: (7, 4): [5, 1], (6, 4): [6, 2], (3, 3): [7, 7], ....: (2, 0): [10, 6], (8, 1): [4, 0], (7, 6): [5, 1], ....: (4, 4): [6, 6], (6, 3): [6, 2], (1, 5): [3, 7], ....: (8, 8): [2, 2], (7, 2): [5, 1], (3, 6): [2, 6], ....: (2, 2): [8, 8], (7, 7): [3, 3], (5, 7): [1, 5], ....: (5, 3): [7, 3], (4, 1): [8, 4], (1, 1): [9, 9], ....: (2, 7): [1, 5], (3, 2): [9, 5], (0, 0): [10, 10], ....: (6, 6): [4, 4], (5, 0): [7, 3], (7, 1): [5, 1], ....: (4, 5): [3, 7], (0, 4): [4, 8], (5, 5): [5, 5], ....: (1, 4): [4, 8], (6, 0): [6, 2], (7, 5): [5, 1], ....: (2, 3): [5, 9], (2, 1): [10, 6], (8, 7): [4, 0], ....: (6, 8): [0, 4], (4, 2): [8, 4], (1, 0): [11, 7], ....: (0, 8): [0, 4], (6, 5): [6, 2], (3, 5): [3, 7], ....: (0, 1): [7, 11], (8, 3): [4, 0], (7, 0): [5, 1], ....: (4, 6): [2, 6], (6, 7): [1, 5], (8, 6): [4, 0], ....: (5, 2): [7, 3], (6, 1): [6, 2], (3, 1): [9, 5], ....: (8, 2): [4, 0], (2, 4): [4, 8], (3, 8): [0, 4], ....: (0, 6): [2, 6], (1, 8): [0, 4], (6, 2): [6, 2], ....: (4, 3): [8, 4], (1, 7): [1, 5], (0, 5): [3, 7], ....: (3, 4): [4, 8], (0, 2): [6, 10], (8, 4): [4, 0]} sage: g == d True sage: g.obtain_nash() # optional - lrslib [[(0, 0, 0, 0, 0, 0, 0, 0, 1), (0, 0, 0, 0, 0, 0, 0, 0, 1)]]
Note that this command can be used to create travellers dilemma for a different maximum value of the luggage. Below is an implementation with a maximum value of 5:
sage: g = game_theory.normal_form_games.TravellersDilemma(5) sage: g Travellers dilemma - Normal Form Game with the following utilities: ... sage: d = {(0, 1): [2, 6], (1, 2): [1, 5], (3, 2): [4, 0], ....: (0, 0): [5, 5], (3, 3): [2, 2], (3, 0): [4, 0], ....: (3, 1): [4, 0], (2, 1): [5, 1], (0, 2): [1, 5], ....: (2, 0): [5, 1], (1, 3): [0, 4], (2, 3): [0, 4], ....: (2, 2): [3, 3], (1, 0): [6, 2], (0, 3): [0, 4], ....: (1, 1): [4, 4]} sage: g == d True sage: g.obtain_nash() [[(0, 0, 0, 1), (0, 0, 0, 1)]]