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r""" Normal form games with N players.
This module implements a class for normal form games (strategic form games) [NN2007]_. At present the following algorithms are implemented to compute equilibria of these games:
* ``'enumeration'`` - An implementation of the support enumeration algorithm built in Sage.
* ``'LCP'`` - An interface with the 'gambit' solver's implementation of the Lemke-Howson algorithm.
* ``'lp'`` - A built-in Sage implementation (with a gambit alternative) of a zero-sum game solver using linear programming. See :class:`MixedIntegerLinearProgram` for more on MILP solvers in Sage.
* ``'lrs'`` - A solver interfacing with the 'lrslib' library.
The architecture for the class is based on the gambit architecture to ensure an easy transition between gambit and Sage. At present the algorithms for the computation of equilibria only solve 2 player games.
A very simple and well known example of normal form game is referred to as the 'Battle of the Sexes' in which two players Amy and Bob are modeled. 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 using the following two matrices:
.. MATH::
A = \begin{pmatrix} 3&1\\ 0&2\\ \end{pmatrix}
B = \begin{pmatrix} 2&1\\ 0&3\\ \end{pmatrix}
Matrix `A` represents the utilities of Amy and matrix `B` represents the utility of Bob. The choices of Amy correspond to the rows of the matrices:
* The first row corresponds to video games.
* The second row corresponds to movies.
Similarly Bob's choices are represented by the columns:
* The first column corresponds to video games.
* The second column corresponds to movies.
Thus, if both Amy and Bob choose to play video games: Amy receives a utility of 3 and Bob a utility of 2. If Amy is indeed going to stick with video games Bob has no incentive to deviate (and vice versa).
This situation repeats itself if both Amy and Bob choose to watch a movie: neither has an incentive to deviate.
This loosely described situation is referred to as a Nash Equilibrium. We can use Sage to find them, and more importantly, see if there is any other situation where Amy and Bob have no reason to change their choice of action:
Here is how we create the game in Sage::
sage: A = matrix([[3, 1], [0, 2]]) sage: B = matrix([[2, 1], [0, 3]]) sage: battle_of_the_sexes = NormalFormGame([A, B]) sage: battle_of_the_sexes Normal Form Game with the following utilities: {(0, 1): [1, 1], (1, 0): [0, 0], (0, 0): [3, 2], (1, 1): [2, 3]}
To obtain the Nash equilibria we run the ``obtain_nash()`` method. In the first few examples, we will use the 'support enumeration' algorithm. A discussion about the different algorithms will be given later::
sage: battle_of_the_sexes.obtain_nash(algorithm='enumeration') [[(0, 1), (0, 1)], [(3/4, 1/4), (1/4, 3/4)], [(1, 0), (1, 0)]]
If we look a bit closer at our output we see that a list of three pairs of tuples have been returned. Each of these correspond to a Nash Equilibrium, represented as a probability distribution over the available strategies:
* `[(1, 0), (1, 0)]` corresponds to the first player only playing their first strategy and the second player also only playing their first strategy. In other words Amy and Bob both play video games.
* `[(0, 1), (0, 1)]` corresponds to the first player only playing their second strategy and the second player also only playing their second strategy. In other words Amy and Bob both watch movies.
* `[(3/4, 1/4), (1/4, 3/4)]` corresponds to players `mixing` their strategies. Amy plays video games 75% of the time and Bob watches movies 75% of the time. At this equilibrium point Amy and Bob will only ever do the same activity `3/8` of the time.
We can use Sage to compute the expected utility for any mixed strategy pair `(\sigma_1, \sigma_2)`. The payoff to player 1 is given by the vector/matrix multiplication:
.. MATH::
\sigma_1 A \sigma_2
The payoff to player 2 is given by:
.. MATH::
\sigma_1 B \sigma_2
To compute this in Sage we have::
sage: for ne in battle_of_the_sexes.obtain_nash(algorithm='enumeration'): ....: print("Utility for {}: ".format(ne)) ....: print("{} {}".format(vector(ne[0]) * A * vector(ne[1]), vector(ne[0]) * B * vector(ne[1]))) Utility for [(0, 1), (0, 1)]: 2 3 Utility for [(3/4, 1/4), (1/4, 3/4)]: 3/2 3/2 Utility for [(1, 0), (1, 0)]: 3 2
Allowing players to play mixed strategies ensures that there will always be a Nash Equilibrium for a normal form game. This result is called Nash's Theorem ([Nas1950]_).
Let us consider the game called 'matching pennies' where two players each present a coin with either HEADS or TAILS showing. If the coins show the same side then player 1 wins, otherwise player 2 wins:
.. MATH::
A = \begin{pmatrix} 1&-1\\ -1&1\\ \end{pmatrix}
B = \begin{pmatrix} -1&1\\ 1&-1\\ \end{pmatrix}
It should be relatively straightforward to observe, that there is no situation, where both players always do the same thing, and have no incentive to deviate.
We can plot the utility of player 1 when player 2 is playing a mixed strategy `\sigma_2 = (y, 1-y)` (so that the utility to player 1 for playing strategy number `i` is given by the matrix/vector multiplication `(Ay)_i`, ie element in position `i` of the matrix/vector multiplication `Ay`) ::
sage: y = var('y') sage: A = matrix([[1, -1], [-1, 1]]) sage: p = plot((A * vector([y, 1 - y]))[0], y, 0, 1, color='blue', legend_label='$u_1(r_1, (y, 1-y))$', axes_labels=['$y$', '']) sage: p += plot((A * vector([y, 1 - y]))[1], y, 0, 1, color='red', legend_label='$u_1(r_2, (y, 1-y))$'); p Graphics object consisting of 2 graphics primitives
We see that the only point at which player 1 is indifferent amongst the available strategies is when `y = 1/2`.
If we compute the Nash equilibria we see that this corresponds to a point at which both players are indifferent::
sage: A = matrix([[1, -1], [-1, 1]]) sage: B = matrix([[-1, 1], [1, -1]]) sage: matching_pennies = NormalFormGame([A, B]) sage: matching_pennies.obtain_nash(algorithm='enumeration') [[(1/2, 1/2), (1/2, 1/2)]]
The utilities to both players at this Nash equilibrium is easily computed::
sage: [vector([1/2, 1/2]) * M * vector([1/2, 1/2]) ....: for M in matching_pennies.payoff_matrices()] [0, 0]
Note that the above uses the ``payoff_matrices`` method which returns the payoff matrices for a 2 player game::
sage: matching_pennies.payoff_matrices() ( [ 1 -1] [-1 1] [-1 1], [ 1 -1] )
One can also input a single matrix and then a zero sum game is constructed. Here is an instance of `Rock-Paper-Scissors-Lizard-Spock <http://www.samkass.com/theories/RPSSL.html>`_::
sage: A = matrix([[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]]) sage: g = NormalFormGame([A]) sage: g.obtain_nash(algorithm='enumeration') [[(1/5, 1/5, 1/5, 1/5, 1/5), (1/5, 1/5, 1/5, 1/5, 1/5)]]
We can also study games where players aim to minimize their utility. Here is the Prisoner's Dilemma (where players are aiming to reduce time spent in prison)::
sage: A = matrix([[2, 5], [0, 4]]) sage: B = matrix([[2, 0], [5, 4]]) sage: prisoners_dilemma = NormalFormGame([A, B]) sage: prisoners_dilemma.obtain_nash(algorithm='enumeration', maximization=False) [[(0, 1), (0, 1)]]
When obtaining Nash equilibrium the following algorithms are currently available:
* ``'lp'``: A solver for constant sum 2 player games using linear programming. This constructs a :mod:`MixedIntegerLinearProgram <sage.numerical.MILP>` using the solver which was passed in with ``solver`` to solve the linear programming representation of the game. See :class:`MixedIntegerLinearProgram` for more on MILP solvers in Sage.
* ``'lrs'``: Reverse search vertex enumeration for 2 player games. This algorithm uses the optional 'lrslib' package. To install it, type ``sage -i lrslib`` in the shell. For more information, see [Av2000]_.
* ``'LCP'``: Linear complementarity program algorithm for 2 player games. This algorithm uses the open source game theory package: `Gambit <http://gambit.sourceforge.net/>`_ [Gambit]_. At present this is the only gambit algorithm available in sage but further development will hope to implement more algorithms (in particular for games with more than 2 players). To install it, type ``sage -i gambit`` in the shell.
* ``'enumeration'``: Support enumeration for 2 player games. This algorithm is hard coded in Sage and checks through all potential supports of a strategy. Supports of a given size with a conditionally dominated strategy are ignored. Note: this is not the preferred algorithm. The algorithm implemented is a combination of a basic algorithm described in [NN2007]_ and a pruning component described in [SLB2008]_.
Below we show how the these algorithms are called::
sage: matching_pennies.obtain_nash(algorithm='lrs') # optional - lrslib [[(1/2, 1/2), (1/2, 1/2)]] sage: matching_pennies.obtain_nash(algorithm='LCP') # optional - gambit [[(0.5, 0.5), (0.5, 0.5)]] sage: matching_pennies.obtain_nash(algorithm='lp', solver='PPL') [[(1/2, 1/2), (1/2, 1/2)]] sage: matching_pennies.obtain_nash(algorithm='lp', solver='gambit') # optional - gambit [[(0.5, 0.5), (0.5, 0.5)]] sage: matching_pennies.obtain_nash(algorithm='enumeration') [[(1/2, 1/2), (1/2, 1/2)]]
Note that if no algorithm argument is passed then the default will be selected according to the following order (if the corresponding package is installed):
1. ``'lp'`` (if the game is constant-sum; uses the solver chosen by Sage) 2. ``'lrs'`` (requires 'lrslib') 3. ``'enumeration'``
Here is a game being constructed using gambit syntax (note that a ``NormalFormGame`` object acts like a dictionary with pure strategy tuples as keys and payoffs as their values)::
sage: f = NormalFormGame() sage: f.add_player(2) # Adding first player with 2 strategies sage: f.add_player(2) # Adding second player with 2 strategies sage: f[0,0][0] = 1 sage: f[0,0][1] = 3 sage: f[0,1][0] = 2 sage: f[0,1][1] = 3 sage: f[1,0][0] = 3 sage: f[1,0][1] = 1 sage: f[1,1][0] = 4 sage: f[1,1][1] = 4 sage: f Normal Form Game with the following utilities: {(0, 1): [2, 3], (1, 0): [3, 1], (0, 0): [1, 3], (1, 1): [4, 4]}
Once this game is constructed we can view the payoff matrices and solve the game::
sage: f.payoff_matrices() ( [1 2] [3 3] [3 4], [1 4] ) sage: f.obtain_nash(algorithm='enumeration') [[(0, 1), (0, 1)]]
We can add an extra strategy to the first player::
sage: f.add_strategy(0) sage: f Normal Form Game with the following utilities: {(0, 1): [2, 3], (0, 0): [1, 3], (2, 1): [False, False], (2, 0): [False, False], (1, 0): [3, 1], (1, 1): [4, 4]}
If we do this and try and obtain the Nash equilibrium or view the payoff matrices(without specifying the utilities), an error is returned::
sage: f.obtain_nash() Traceback (most recent call last): ... ValueError: utilities have not been populated sage: f.payoff_matrices() Traceback (most recent call last): ... ValueError: utilities have not been populated
Here we populate the missing utilities::
sage: f[2, 1] = [5, 3] sage: f[2, 0] = [2, 1] sage: f.payoff_matrices() ( [1 2] [3 3] [3 4] [1 4] [2 5], [1 3] ) sage: f.obtain_nash() [[(0, 0, 1), (0, 1)]]
We can use the same syntax as above to create games with more than 2 players::
sage: threegame = NormalFormGame() sage: threegame.add_player(2) # Adding first player with 2 strategies sage: threegame.add_player(2) # Adding second player with 2 strategies sage: threegame.add_player(2) # Adding third player with 2 strategies sage: threegame[0, 0, 0][0] = 3 sage: threegame[0, 0, 0][1] = 1 sage: threegame[0, 0, 0][2] = 4 sage: threegame[0, 0, 1][0] = 1 sage: threegame[0, 0, 1][1] = 5 sage: threegame[0, 0, 1][2] = 9 sage: threegame[0, 1, 0][0] = 2 sage: threegame[0, 1, 0][1] = 6 sage: threegame[0, 1, 0][2] = 5 sage: threegame[0, 1, 1][0] = 3 sage: threegame[0, 1, 1][1] = 5 sage: threegame[0, 1, 1][2] = 8 sage: threegame[1, 0, 0][0] = 9 sage: threegame[1, 0, 0][1] = 7 sage: threegame[1, 0, 0][2] = 9 sage: threegame[1, 0, 1][0] = 3 sage: threegame[1, 0, 1][1] = 2 sage: threegame[1, 0, 1][2] = 3 sage: threegame[1, 1, 0][0] = 8 sage: threegame[1, 1, 0][1] = 4 sage: threegame[1, 1, 0][2] = 6 sage: threegame[1, 1, 1][0] = 2 sage: threegame[1, 1, 1][1] = 6 sage: threegame[1, 1, 1][2] = 4 sage: threegame Normal Form Game with the following utilities: {(0, 1, 1): [3, 5, 8], (1, 1, 0): [8, 4, 6], (1, 0, 0): [9, 7, 9], (0, 0, 1): [1, 5, 9], (1, 0, 1): [3, 2, 3], (0, 0, 0): [3, 1, 4], (0, 1, 0): [2, 6, 5], (1, 1, 1): [2, 6, 4]}
The above requires a lot of input that could be simplified if there is another data structure with our utilities and/or a structure to the utilities. The following example creates a game with a relatively strange utility function::
sage: def utility(strategy_triplet, player): ....: return sum(strategy_triplet) * player sage: threegame = NormalFormGame() sage: threegame.add_player(2) # Adding first player with 2 strategies sage: threegame.add_player(2) # Adding second player with 2 strategies sage: threegame.add_player(2) # Adding third player with 2 strategies sage: for i, j, k in [(i, j, k) for i in [0,1] for j in [0,1] for k in [0,1]]: ....: for p in range(3): ....: threegame[i, j, k][p] = utility([i, j, k], p) sage: threegame Normal Form Game with the following utilities: {(0, 1, 1): [0, 2, 4], (1, 1, 0): [0, 2, 4], (1, 0, 0): [0, 1, 2], (0, 0, 1): [0, 1, 2], (1, 0, 1): [0, 2, 4], (0, 0, 0): [0, 0, 0], (0, 1, 0): [0, 1, 2], (1, 1, 1): [0, 3, 6]}
At present no algorithm has been implemented in Sage for games with more than 2 players::
sage: threegame.obtain_nash() Traceback (most recent call last): ... NotImplementedError: Nash equilibrium for games with more than 2 players have not been implemented yet. Please see the gambit website (http://gambit.sourceforge.net/) that has a variety of available algorithms
There are however a variety of such algorithms available in gambit, further compatibility between Sage and gambit is actively being developed: https://github.com/tturocy/gambit/tree/sage_integration.
It can be shown that linear scaling of the payoff matrices conserves the equilibrium values::
sage: A = matrix([[2, 1], [1, 2.5]]) sage: B = matrix([[-1, 3], [2, 1]]) sage: g = NormalFormGame([A, B]) sage: g.obtain_nash(algorithm='enumeration') [[(1/5, 4/5), (3/5, 2/5)]] sage: g.obtain_nash(algorithm='lrs') # optional - lrslib [[(1/5, 4/5), (3/5, 2/5)]] sage: A = 2 * A sage: g = NormalFormGame([A, B]) sage: g.obtain_nash(algorithm='LCP') # optional - gambit [[(0.2, 0.8), (0.6, 0.4)]]
It is also possible to generate a Normal form game from a gambit Game::
sage: from gambit import Game # optional - gambit sage: gambitgame= Game.new_table([2, 2]) # optional - gambit sage: gambitgame[int(0), int(0)][int(0)] = int(8) # optional - gambit sage: gambitgame[int(0), int(0)][int(1)] = int(8) # optional - gambit sage: gambitgame[int(0), int(1)][int(0)] = int(2) # optional - gambit sage: gambitgame[int(0), int(1)][int(1)] = int(10) # optional - gambit sage: gambitgame[int(1), int(0)][int(0)] = int(10) # optional - gambit sage: gambitgame[int(1), int(0)][int(1)] = int(2) # optional - gambit sage: gambitgame[int(1), int(1)][int(0)] = int(5) # optional - gambit sage: gambitgame[int(1), int(1)][int(1)] = int(5) # optional - gambit sage: g = NormalFormGame(gambitgame) # optional - gambit sage: g # optional - gambit Normal Form Game with the following utilities: {(0, 1): [2.0, 10.0], (1, 0): [10.0, 2.0], (0, 0): [8.0, 8.0], (1, 1): [5.0, 5.0]}
For more information on using Gambit in Sage see: :mod:`Using Gambit in Sage<sage.game_theory.gambit_docs>`. This includes how to access Gambit directly using the version of iPython shipped with Sage and an explanation as to why the ``int`` calls are needed to handle the Sage preparser.
Here is a slightly longer game that would take too long to solve with ``'enumeration'``. Consider the following:
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?
In the following we create the game (with a max value of 10) and solve it::
sage: K = 10 # Modifying this value lets us play with games of any size sage: A = matrix([[min(i,j) + 2 * sign(j-i) for j in range(K, 1, -1)] ....: for i in range(K, 1, -1)]) sage: B = matrix([[min(i,j) + 2 * sign(i-j) for j in range(K, 1, -1)] ....: for i in range(K, 1, -1)]) sage: g = NormalFormGame([A, B]) sage: g.obtain_nash(algorithm='lrs') # optional - lrslib [[(0, 0, 0, 0, 0, 0, 0, 0, 1), (0, 0, 0, 0, 0, 0, 0, 0, 1)]] sage: g.obtain_nash(algorithm='LCP') # optional - gambit [[(0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0), (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0)]]
The output is a pair of vectors (as before) showing the Nash equilibrium. In particular it here shows that out of the 10 possible strategies both players should choose the last. Recall that the above considers a reduced version of the game where individuals can claim integer values from 10 to 2. The equilibrium strategy is thus for both players to state that the value of their suitcase is 2.
Several standard Normal Form Games have also been implemented. For more information on how to access these, see: :mod:`Game Theory Catalog<sage.game_theory.catalog>`. Included is information on the situation each Game models. For example::
sage: g = game_theory.normal_form_games.PrisonersDilemma() sage: g Prisoners dilemma - Normal Form Game with the following utilities: ... sage: d = {(0, 1): [-5, 0], (1, 0): [0, -5], ....: (0, 0): [-2, -2], (1, 1): [-4, -4]} sage: g == d True sage: g.obtain_nash() [[(0, 1), (0, 1)]]
We can easily obtain the best response for a player to a given strategy. In this example we obtain the best responses for Player 1, when Player 2 uses two different strategies::
sage: A = matrix([[3, 0], [0, 3], [1.5, 1.5]]) sage: B = matrix([[4, 3], [2, 6], [3, 1]]) sage: g = NormalFormGame([A, B]) sage: g.best_responses((1/2, 1/2), player=0) [0, 1, 2] sage: g.best_responses((3/4, 1/4), player=0) [0]
Here we do the same for player 2::
sage: g.best_responses((4/5, 1/5, 0), player=1) [0, 1]
We see that for the game `Rock-Paper-Scissors-Lizard-Spock <http://www.samkass.com/theories/RPSSL.html>`_ any pure strategy has two best responses::
sage: g = game_theory.normal_form_games.RPSLS() sage: A, B = g.payoff_matrices() sage: A, B ( [ 0 -1 1 1 -1] [ 0 1 -1 -1 1] [ 1 0 -1 -1 1] [-1 0 1 1 -1] [-1 1 0 1 -1] [ 1 -1 0 -1 1] [-1 1 -1 0 1] [ 1 -1 1 0 -1] [ 1 -1 1 -1 0], [-1 1 -1 1 0] ) sage: g.best_responses((1, 0, 0, 0, 0), player=0) [1, 4] sage: g.best_responses((0, 1, 0, 0, 0), player=0) [2, 3] sage: g.best_responses((0, 0, 1, 0, 0), player=0) [0, 4] sage: g.best_responses((0, 0, 0, 1, 0), player=0) [0, 2] sage: g.best_responses((0, 0, 0, 0, 1), player=0) [1, 3] sage: g.best_responses((1, 0, 0, 0, 0), player=1) [1, 4] sage: g.best_responses((0, 1, 0, 0, 0), player=1) [2, 3] sage: g.best_responses((0, 0, 1, 0, 0), player=1) [0, 4] sage: g.best_responses((0, 0, 0, 1, 0), player=1) [0, 2] sage: g.best_responses((0, 0, 0, 0, 1), player=1) [1, 3]
Note that degenerate games can cause problems for most algorithms. The following example in fact has an infinite quantity of equilibria which is evidenced by the various algorithms returning different solutions::
sage: A = matrix([[3,3],[2,5],[0,6]]) sage: B = matrix([[3,3],[2,6],[3,1]]) sage: degenerate_game = NormalFormGame([A,B]) sage: degenerate_game.obtain_nash(algorithm='lrs') # optional - lrslib [[(0, 1/3, 2/3), (1/3, 2/3)], [(1, 0, 0), (1/2, 3)], [(1, 0, 0), (1, 3)]] sage: degenerate_game.obtain_nash(algorithm='LCP') # optional - gambit [[(0.0, 0.3333333333, 0.6666666667), (0.3333333333, 0.6666666667)], [(1.0, -0.0, 0.0), (0.6666666667, 0.3333333333)], [(1.0, 0.0, 0.0), (1.0, 0.0)]] sage: degenerate_game.obtain_nash(algorithm='enumeration') [[(0, 1/3, 2/3), (1/3, 2/3)], [(1, 0, 0), (1, 0)]]
We can check the cause of this by using ``is_degenerate()``::
sage: degenerate_game.is_degenerate() True
Note the 'negative' `-0.0` output by gambit. This is due to the numerical nature of the algorithm used.
Here is an example with the trivial game where all payoffs are 0::
sage: g = NormalFormGame() sage: g.add_player(3) # Adding first player with 3 strategies sage: g.add_player(3) # Adding second player with 3 strategies sage: for key in g: ....: g[key] = [0, 0] sage: g.payoff_matrices() ( [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0], [0 0 0] ) sage: g.obtain_nash(algorithm='enumeration') [[(0, 0, 1), (0, 0, 1)], [(0, 0, 1), (0, 1, 0)], [(0, 0, 1), (1, 0, 0)], [(0, 1, 0), (0, 0, 1)], [(0, 1, 0), (0, 1, 0)], [(0, 1, 0), (1, 0, 0)], [(1, 0, 0), (0, 0, 1)], [(1, 0, 0), (0, 1, 0)], [(1, 0, 0), (1, 0, 0)]]
A good description of degenerate games can be found in [NN2007]_.
REFERENCES:
- [Nas1950]_
- [NN2007]_
- [Av2000]_
- [Gambit]_
- [SLB2008]_
AUTHOR:
- James Campbell and Vince Knight (06-2014): Original version - Tobenna P. Igwe: Constant-sum game solvers
"""
#***************************************************************************** # Copyright (C) 2014 James Campbell james.campbell@tanti.org.uk # # This program is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation, either version 3 of the License, or # (at your option) any later version. # http://www.gnu.org/licenses/ #*****************************************************************************
from gambit.nash import ExternalLPSolver, ExternalLCPSolver
r""" An object representing a Normal Form Game. Primarily used to compute the Nash Equilibria.
INPUT:
- ``generator`` -- can be a list of 2 matrices, a single matrix or left blank
"""
r""" Initializes a Normal Form game and checks the inputs.
EXAMPLES:
Can have games with more than 2 players::
sage: threegame = NormalFormGame() sage: threegame.add_player(2) # Adding first player with 2 strategies sage: threegame.add_player(2) # Adding second player with 2 strategies sage: threegame.add_player(2) # Adding third player with 2 strategies sage: threegame[0, 0, 0][0] = 3 sage: threegame[0, 0, 0][1] = 1 sage: threegame[0, 0, 0][2] = 4 sage: threegame[0, 0, 1][0] = 1 sage: threegame[0, 0, 1][1] = 5 sage: threegame[0, 0, 1][2] = 9 sage: threegame[0, 1, 0][0] = 2 sage: threegame[0, 1, 0][1] = 6 sage: threegame[0, 1, 0][2] = 5 sage: threegame[0, 1, 1][0] = 3 sage: threegame[0, 1, 1][1] = 5 sage: threegame[0, 1, 1][2] = 8 sage: threegame[1, 0, 0][0] = 9 sage: threegame[1, 0, 0][1] = 7 sage: threegame[1, 0, 0][2] = 9 sage: threegame[1, 0, 1][0] = 3 sage: threegame[1, 0, 1][1] = 2 sage: threegame[1, 0, 1][2] = 3 sage: threegame[1, 1, 0][0] = 8 sage: threegame[1, 1, 0][1] = 4 sage: threegame[1, 1, 0][2] = 6 sage: threegame[1, 1, 1][0] = 2 sage: threegame[1, 1, 1][1] = 6 sage: threegame[1, 1, 1][2] = 4 sage: threegame.obtain_nash() Traceback (most recent call last): ... NotImplementedError: Nash equilibrium for games with more than 2 players have not been implemented yet. Please see the gambit website (http://gambit.sourceforge.net/) that has a variety of available algorithms
Can initialise a game from a gambit game object::
sage: from gambit import Game # optional - gambit sage: gambitgame= Game.new_table([2, 2]) # optional - gambit sage: gambitgame[int(0), int(0)][int(0)] = int(5) # optional - gambit sage: gambitgame[int(0), int(0)][int(1)] = int(8) # optional - gambit sage: gambitgame[int(0), int(1)][int(0)] = int(2) # optional - gambit sage: gambitgame[int(0), int(1)][int(1)] = int(11) # optional - gambit sage: gambitgame[int(1), int(0)][int(0)] = int(10) # optional - gambit sage: gambitgame[int(1), int(0)][int(1)] = int(7) # optional - gambit sage: gambitgame[int(1), int(1)][int(0)] = int(5) # optional - gambit sage: gambitgame[int(1), int(1)][int(1)] = int(5) # optional - gambit sage: g = NormalFormGame(gambitgame) # optional - gambit sage: g # optional - gambit Normal Form Game with the following utilities: {(0, 1): [2.0, 11.0], (1, 0): [10.0, 7.0], (0, 0): [5.0, 8.0], (1, 1): [5.0, 5.0]}
TESTS:
Raise error if matrices aren't the same size::
sage: p1 = matrix([[1, 2], [3, 4]]) sage: p2 = matrix([[3, 3], [1, 4], [6, 6]]) sage: error = NormalFormGame([p1, p2]) Traceback (most recent call last): ... ValueError: matrices must be the same size
Note that when initializing, a single argument must be passed::
sage: p1 = matrix([[1, 2], [3, 4]]) sage: p2 = matrix([[3, 3], [1, 4], [6, 6]]) sage: error = NormalFormGame(p1, p2) Traceback (most recent call last): ... TypeError: __init__() takes at most 2 arguments (3 given)
When initiating, argument passed must be a list or nothing::
sage: error = NormalFormGame({4:6, 6:9}) Traceback (most recent call last): ... TypeError: Generator function must be a list, gambit game or nothing
When passing nothing, the utilities then need to be entered manually::
sage: game = NormalFormGame() sage: game Normal Form Game with the following utilities: {}
"""
game = generator self._gambit_game(game)
r""" This method is one of a collection that aims to make a game instance behave like a dictionary which can be used if a game is to be generated without using a matrix.
Here we set up deleting an element of the utilities dictionary::
sage: A = matrix([[2, 5], [0, 4]]) sage: B = matrix([[2, 0], [5, 4]]) sage: prisoners_dilemma = NormalFormGame([A, B]) sage: prisoners_dilemma Normal Form Game with the following utilities: {(0, 1): [5, 0], (1, 0): [0, 5], (0, 0): [2, 2], (1, 1): [4, 4]} sage: del(prisoners_dilemma[(0,1)]) sage: prisoners_dilemma Normal Form Game with the following utilities: {(1, 0): [0, 5], (0, 0): [2, 2], (1, 1): [4, 4]} """
r""" This method is one of a collection that aims to make a game instance behave like a dictionary which can be used if a game is to be generated without using a matrix.
Here we allow for querying a key::
sage: A = matrix([[2, 5], [0, 4]]) sage: B = matrix([[2, 0], [5, 4]]) sage: prisoners_dilemma = NormalFormGame([A, B]) sage: prisoners_dilemma[(0, 1)] [5, 0] sage: del(prisoners_dilemma[(0,1)]) sage: prisoners_dilemma[(0, 1)] Traceback (most recent call last): ... KeyError: (0, 1) """
r""" This method is one of a collection that aims to make a game instance behave like a dictionary which can be used if a game is to be generated without using a matrix.
Here we allow for iteration over the game to correspond to iteration over keys of the utility dictionary::
sage: A = matrix([[2, 5], [0, 4]]) sage: B = matrix([[2, 0], [5, 4]]) sage: prisoners_dilemma = NormalFormGame([A, B]) sage: for key in prisoners_dilemma: ....: print("The strategy pair {} gives utilities {}".format(key, prisoners_dilemma[key])) The strategy pair (0, 1) gives utilities [5, 0] The strategy pair (1, 0) gives utilities [0, 5] The strategy pair (0, 0) gives utilities [2, 2] The strategy pair (1, 1) gives utilities [4, 4] """
r""" This method is one of a collection that aims to make a game instance behave like a dictionary which can be used if a game is to be generated without using a matrix.
Here we set up setting the value of a key::
sage: A = matrix([[2, 5], [0, 4]]) sage: B = matrix([[2, 0], [5, 4]]) sage: prisoners_dilemma = NormalFormGame([A, B]) sage: del(prisoners_dilemma[(0,1)]) sage: prisoners_dilemma[(0,1)] = [5,6] sage: prisoners_dilemma.payoff_matrices() ( [2 5] [2 6] [0 4], [5 4] )
We can use the dictionary-like interface to overwrite a strategy profile::
sage: prisoners_dilemma[(0,1)] = [-3,-30] sage: prisoners_dilemma.payoff_matrices() ( [ 2 -3] [ 2 -30] [ 0 4], [ 5 4] ) """
r""" Return the length of the game to be the length of the utilities.
EXAMPLES::
sage: A = matrix([[2, 5], [0, 4]]) sage: B = matrix([[2, 0], [5, 4]]) sage: prisoners_dilemma = NormalFormGame([A, B]) sage: len(prisoners_dilemma) 4 """
r""" Return the strategy_profiles of the game.
EXAMPLES:
Basic description of the game shown when calling the game instance::
sage: p1 = matrix([[1, 2], [3, 4]]) sage: p2 = matrix([[3, 3], [1, 4]]) sage: g = NormalFormGame([p1, p2]) sage: g Normal Form Game with the following utilities: {(0, 1): [2, 3], (1, 0): [3, 1], (0, 0): [1, 3], (1, 1): [4, 4]} """
r""" Return the LaTeX code representing the ``NormalFormGame``.
EXAMPLES:
LaTeX method shows the two payoff matrices for a two player game::
sage: A = matrix([[-1, -2], [-12, 2]]) sage: B = matrix([[1, 0], [1, -1]]) sage: g = NormalFormGame([A, B]) sage: latex(g) \left(\left(\begin{array}{rr} -1 & -2 \\ -12 & 2 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 1 & -1 \end{array}\right)\right)
LaTeX method shows nothing interesting for games with more players::
sage: g = NormalFormGame() sage: g.add_player(2) # Adding first player with 2 strategies sage: g.add_player(2) # Adding second player with 2 strategies sage: g.add_player(2) # Creating a game with three players sage: latex(g) \text{\texttt{Normal{ }Form{ }Game{ }...[False,{ }False,{ }False]{\char`\}}}} """
r""" Populate ``self.utilities`` with the values from 2 matrices.
EXAMPLES:
A small example game::
sage: A = matrix([[1, 0], [-2, 3]]) sage: B = matrix([[3, 2], [-1, 0]]) sage: two_game = NormalFormGame() sage: two_game._two_matrix_game([A, B]) """ matrices[1][strategy_profile]]
r""" Creates a ``NormalFormGame`` object from a Gambit game.
TESTS::
sage: from gambit import Game # optional - gambit sage: testgame = Game.new_table([2, 2]) # optional - gambit sage: testgame[int(0), int(0)][int(0)] = int(8) # optional - gambit sage: testgame[int(0), int(0)][int(1)] = int(8) # optional - gambit sage: testgame[int(0), int(1)][int(0)] = int(2) # optional - gambit sage: testgame[int(0), int(1)][int(1)] = int(10) # optional - gambit sage: testgame[int(1), int(0)][int(0)] = int(10) # optional - gambit sage: testgame[int(1), int(0)][int(1)] = int(2) # optional - gambit sage: testgame[int(1), int(1)][int(0)] = int(5) # optional - gambit sage: testgame[int(1), int(1)][int(1)] = int(5) # optional - gambit sage: g = NormalFormGame() # optional - gambit sage: g._gambit_game(testgame) # optional - gambit sage: g # optional - gambit Normal Form Game with the following utilities: {(0, 1): [2.0, 10.0], (1, 0): [10.0, 2.0], (0, 0): [8.0, 8.0], (1, 1): [5.0, 5.0]} """ self.players = [] self.utilities = {} for player in game.players: num_strategies = len(player.strategies) self.add_player(num_strategies) for strategy_profile in self.utilities: utility_vector = [float(game[strategy_profile][i]) for i in range(len(self.players))] self.utilities[strategy_profile] = utility_vector
r""" Creates a Gambit game from a ``NormalFormGame`` object
INPUT:
- ``as_integer`` -- boolean; whether the gambit representation should have the payoffs represented as integers or decimals
- ``maximization`` -- boolean; whether a player is trying to maximize their utility or minimize it
TESTS::
sage: from gambit import Game # optional - gambit sage: A = matrix([[2, 1], [1, 2.5]]) sage: g = NormalFormGame([A]) sage: gg = g._gambit_() # optional - gambit sage: gg # optional - gambit NFG 1 R "" { "1" "2" } <BLANKLINE> { { "1" "2" } { "1" "2" } } "" <BLANKLINE> { { "" 2, -2 } { "" 1, -1 } { "" 1, -1 } { "" 2.5, -2.5 } } 1 2 3 4 <BLANKLINE>
sage: gg = g._gambit_(as_integer=True) # optional - gambit sage: gg # optional - gambit NFG 1 R "" { "1" "2" } <BLANKLINE> { { "1" "2" } { "1" "2" } } "" <BLANKLINE> { { "" 2, -2 } { "" 1, -1 } { "" 1, -1 } { "" 2, -2 } } 1 2 3 4 <BLANKLINE>
::
sage: A = matrix([[2, 1], [1, 2.5]]) sage: B = matrix([[3, 2], [5.5, 4]]) sage: g = NormalFormGame([A, B]) sage: gg = g._gambit_() # optional - gambit sage: gg # optional - gambit NFG 1 R "" { "1" "2" } <BLANKLINE> { { "1" "2" } { "1" "2" } } "" <BLANKLINE> { { "" 2, 3 } { "" 1, 5.5 } { "" 1, 2 } { "" 2.5, 4 } } 1 2 3 4 <BLANKLINE>
sage: gg = g._gambit_(as_integer = True) # optional - gambit sage: gg # optional - gambit NFG 1 R "" { "1" "2" } <BLANKLINE> { { "1" "2" } { "1" "2" } } "" <BLANKLINE> { { "" 2, 3 } { "" 1, 5 } { "" 1, 2 } { "" 2, 4 } } 1 2 3 4 <BLANKLINE>
::
sage: threegame = NormalFormGame() # optional - gambit sage: threegame.add_player(2) # optional - gambit sage: threegame.add_player(2) # optional - gambit sage: threegame.add_player(2) # optional - gambit sage: threegame[0, 0, 0][0] = 3 # optional - gambit sage: threegame[0, 0, 0][1] = 1 # optional - gambit sage: threegame[0, 0, 0][2] = 4 # optional - gambit sage: threegame[0, 0, 1][0] = 1 # optional - gambit sage: threegame[0, 0, 1][1] = 5 # optional - gambit sage: threegame[0, 0, 1][2] = 9 # optional - gambit sage: threegame[0, 1, 0][0] = 2 # optional - gambit sage: threegame[0, 1, 0][1] = 6 # optional - gambit sage: threegame[0, 1, 0][2] = 5 # optional - gambit sage: threegame[0, 1, 1][0] = 3 # optional - gambit sage: threegame[0, 1, 1][1] = 5 # optional - gambit sage: threegame[0, 1, 1][2] = 8 # optional - gambit sage: threegame[1, 0, 0][0] = 9 # optional - gambit sage: threegame[1, 0, 0][1] = 7 # optional - gambit sage: threegame[1, 0, 0][2] = 9 # optional - gambit sage: threegame[1, 0, 1][0] = 3 # optional - gambit sage: threegame[1, 0, 1][1] = 2 # optional - gambit sage: threegame[1, 0, 1][2] = 3 # optional - gambit sage: threegame[1, 1, 0][0] = 8 # optional - gambit sage: threegame[1, 1, 0][1] = 4 # optional - gambit sage: threegame[1, 1, 0][2] = 6 # optional - gambit sage: threegame[1, 1, 1][0] = 2 # optional - gambit sage: threegame[1, 1, 1][1] = 6 # optional - gambit sage: threegame[1, 1, 1][2] = 4 # optional - gambit sage: threegame._gambit_(as_integer = True) # optional - gambit NFG 1 R "" { "1" "2" "3" } <BLANKLINE> { { "1" "2" } { "1" "2" } { "1" "2" } } "" <BLANKLINE> { { "" 3, 1, 4 } { "" 9, 7, 9 } { "" 2, 6, 5 } { "" 8, 4, 6 } { "" 1, 5, 9 } { "" 3, 2, 3 } { "" 3, 5, 8 } { "" 2, 6, 4 } } 1 2 3 4 5 6 7 8 <BLANKLINE> """ from decimal import Decimal strategy_sizes = [p.num_strategies for p in self.players] g = Game.new_table(strategy_sizes)
sgn = 1 if not maximization: sgn = -1
players = len(strategy_sizes)
for strategy_profile in self.utilities: for i in range(players): if as_integer: g[strategy_profile][i] = sgn * int(self.utilities[strategy_profile][i]) else: g[strategy_profile][i] = sgn * Decimal(float(self.utilities[strategy_profile][i])) return g
r""" Checks if the game is constant sum.
EXAMPLES::
sage: A = matrix([[2, 1], [1, 2.5]]) sage: g = NormalFormGame([A]) sage: g.is_constant_sum() True sage: g = NormalFormGame([A, A]) sage: g.is_constant_sum() False sage: A = matrix([[1, 1], [1, 1]]) sage: g = NormalFormGame([A, A]) sage: g.is_constant_sum() True sage: A = matrix([[1, 1, 2], [1, 1, -1], [1, -1, 1]]) sage: B = matrix([[2, 2, 1], [2, 2, 4], [2, 4, 2]]) sage: g = NormalFormGame([A, B]) sage: g.is_constant_sum() True sage: A = matrix([[1, 1, 2], [1, 1, -1], [1, -1, 1]]) sage: B = matrix([[2, 2, 1], [2, 2.1, 4], [2, 4, 2]]) sage: g = NormalFormGame([A, B]) sage: g.is_constant_sum() False """ return False
r""" Return 2 matrices representing the payoffs for each player.
EXAMPLES::
sage: p1 = matrix([[1, 2], [3, 4]]) sage: p2 = matrix([[3, 3], [1, 4]]) sage: g = NormalFormGame([p1, p2]) sage: g.payoff_matrices() ( [1 2] [3 3] [3 4], [1 4] )
If we create a game with 3 players we will not be able to obtain payoff matrices::
sage: g = NormalFormGame() sage: g.add_player(2) # adding first player with 2 strategies sage: g.add_player(2) # adding second player with 2 strategies sage: g.add_player(2) # adding third player with 2 strategies sage: g.payoff_matrices() Traceback (most recent call last): ... ValueError: Only available for 2 player games
If we do create a two player game but it is not complete then an error is also raised::
sage: g = NormalFormGame() sage: g.add_player(1) # Adding first player with 1 strategy sage: g.add_player(1) # Adding second player with 1 strategy sage: g.payoff_matrices() Traceback (most recent call last): ... ValueError: utilities have not been populated
The above creates a 2 player game where each player has a single strategy. Here we populate the strategies and can then view the payoff matrices::
sage: g[0, 0] = [1,2] sage: g.payoff_matrices() ([1], [2]) """
r""" Add a player to a NormalFormGame.
INPUT:
- ``num_strategies`` -- the number of strategies the player should have
EXAMPLES::
sage: g = NormalFormGame() sage: g.add_player(2) # Adding first player with 2 strategies sage: g.add_player(1) # Adding second player with 1 strategy sage: g.add_player(1) # Adding third player with 1 strategy sage: g Normal Form Game with the following utilities: {(1, 0, 0): [False, False, False], (0, 0, 0): [False, False, False]} """
r""" Create all the required keys for ``self.utilities``.
This is used when generating players and/or adding strategies.
INPUT:
- ``replacement`` -- Boolean value of whether previously created profiles should be replaced or not
TESTS::
sage: from sage.game_theory.normal_form_game import _Player sage: g = NormalFormGame() sage: g.players.append(_Player(2)) sage: g.players.append(_Player(2)) sage: g Normal Form Game with the following utilities: {}
sage: g._generate_utilities(True) sage: g Normal Form Game with the following utilities: {(0, 1): [False, False], (1, 0): [False, False], (0, 0): [False, False], (1, 1): [False, False]}
sage: g[(0,1)] = [2, 3] sage: g.add_strategy(1) sage: g._generate_utilities(False) sage: g Normal Form Game with the following utilities: {(0, 1): [2, 3], (1, 2): [False, False], (0, 0): [False, False], (0, 2): [False, False], (1, 0): [False, False], (1, 1): [False, False]}
sage: g._generate_utilities(True) sage: g Normal Form Game with the following utilities: {(0, 1): [False, False], (1, 2): [False, False], (0, 0): [False, False], (1, 1): [False, False], (1, 0): [False, False], (0, 2): [False, False]} """
r""" Add a strategy to a player, will not affect already completed strategy profiles.
INPUT:
- ``player`` -- the index of the player
EXAMPLES:
A simple example::
sage: s = matrix([[1, 0], [-2, 3]]) sage: t = matrix([[3, 2], [-1, 0]]) sage: example = NormalFormGame([s, t]) sage: example Normal Form Game with the following utilities: {(0, 1): [0, 2], (1, 0): [-2, -1], (0, 0): [1, 3], (1, 1): [3, 0]} sage: example.add_strategy(0) sage: example Normal Form Game with the following utilities: {(0, 1): [0, 2], (0, 0): [1, 3], (2, 1): [False, False], (2, 0): [False, False], (1, 0): [-2, -1], (1, 1): [3, 0]}
"""
r""" Check if ``utilities`` has been completed and return a boolean.
EXAMPLES:
A simple example::
sage: s = matrix([[1, 0], [-2, 3]]) sage: t = matrix([[3, 2], [-1, 0]]) sage: example = NormalFormGame([s, t]) sage: example.add_strategy(0) sage: example._is_complete() False """
r""" A function to return the Nash equilibrium for the game. Optional arguments can be used to specify the algorithm used. If no algorithm is passed then an attempt is made to use the most appropriate algorithm.
INPUT:
- ``algorithm`` - the following algorithms should be available through this function:
* ``'lrs'`` - This algorithm is only suited for 2 player games. See the lrs web site (http://cgm.cs.mcgill.ca/~avis/C/lrs.html).
* ``'LCP'`` - This algorithm is only suited for 2 player games. See the gambit web site (http://gambit.sourceforge.net/).
* ``'lp'`` - This algorithm is only suited for 2 player constant sum games. Uses MILP solver determined by the ``solver`` argument.
* ``'enumeration'`` - This is a very inefficient algorithm (in essence a brute force approach).
1. For each k in 1...min(size of strategy sets) 2. For each I,J supports of size k 3. Prune: check if supports are dominated 4. Solve indifference conditions and check that have Nash Equilibrium.
Solving the indifference conditions is done by building the corresponding linear system. If `\rho_1, \rho_2` are the supports player 1 and 2 respectively. Then, indifference implies:
.. MATH::
u_1(s_1,\rho_2) = u_1(s_2, \rho_2)
for all `s_1, s_2` in the support of `\rho_1`. This corresponds to:
.. MATH::
\sum_{j\in S(\rho_2)}A_{s_1,j}{\rho_2}_j = \sum_{j\in S(\rho_2)}A_{s_2,j}{\rho_2}_j
for all `s_1, s_2` in the support of `\rho_1` where `A` is the payoff matrix of player 1. Equivalently we can consider consecutive rows of `A` (instead of all pairs of strategies). Thus the corresponding linear system can be written as:
.. MATH::
\left(\sum_{j \in S(\rho_2)}A_{i,j} - A_{i+1,j}\right){\rho_2}_j
for all `1\leq i \leq |S(\rho_1)|` (where `A` has been modified to only contain the rows corresponding to `S(\rho_1)`). We also require all elements of `\rho_2` to sum to 1:
.. MATH::
\sum_{j\in S(\rho_1)}{\rho_2}_j = 1
- ``maximization`` -- (default: ``True``) whether a player is trying to maximize their utility or minimize it:
* When set to ``True`` it is assumed that players aim to maximise their utility.
* When set to ``False`` it is assumed that players aim to minimise their utility.
- ``solver`` -- (optional) see :class:`MixedIntegerLinearProgram` for more information on the MILP solvers in Sage, may also be ``'gambit'`` to use the MILP solver included with the gambit library. Note that ``None`` means to use the default Sage LP solver, normally GLPK.
EXAMPLES:
A game with 1 equilibrium when ``maximization`` is ``True`` and 3 when ``maximization`` is ``False``::
sage: A = matrix([[10, 500, 44], ....: [15, 10, 105], ....: [19, 204, 55], ....: [20, 200, 590]]) sage: B = matrix([[2, 1, 2], ....: [0, 5, 6], ....: [3, 4, 1], ....: [4, 1, 20]]) sage: g=NormalFormGame([A, B]) sage: g.obtain_nash(algorithm='lrs') # optional - lrslib [[(0, 0, 0, 1), (0, 0, 1)]] sage: g.obtain_nash(algorithm='lrs', maximization=False) # optional - lrslib [[(2/3, 1/12, 1/4, 0), (6333/8045, 247/8045, 293/1609)], [(3/4, 0, 1/4, 0), (0, 11/307, 296/307)], [(5/6, 1/6, 0, 0), (98/99, 1/99, 0)]]
This particular game has 3 Nash equilibria::
sage: A = matrix([[3,3], ....: [2,5], ....: [0,6]]) sage: B = matrix([[3,2], ....: [2,6], ....: [3,1]]) sage: g = NormalFormGame([A, B]) sage: g.obtain_nash(algorithm='enumeration') [[(0, 1/3, 2/3), (1/3, 2/3)], [(4/5, 1/5, 0), (2/3, 1/3)], [(1, 0, 0), (1, 0)]]
Here is a slightly larger game::
sage: A = matrix([[160, 205, 44], ....: [175, 180, 45], ....: [201, 204, 50], ....: [120, 207, 49]]) sage: B = matrix([[2, 2, 2], ....: [1, 0, 0], ....: [3, 4, 1], ....: [4, 1, 2]]) sage: g=NormalFormGame([A, B]) sage: g.obtain_nash(algorithm='enumeration') [[(0, 0, 3/4, 1/4), (1/28, 27/28, 0)]] sage: g.obtain_nash(algorithm='lrs') # optional - lrslib [[(0, 0, 3/4, 1/4), (1/28, 27/28, 0)]] sage: g.obtain_nash(algorithm='LCP') # optional - gambit [[(0.0, 0.0, 0.75, 0.25), (0.0357142857, 0.9642857143, 0.0)]]
2 random matrices::
sage: player1 = matrix([[2, 8, -1, 1, 0], ....: [1, 1, 2, 1, 80], ....: [0, 2, 15, 0, -12], ....: [-2, -2, 1, -20, -1], ....: [1, -2, -1, -2, 1]]) sage: player2 = matrix([[0, 8, 4, 2, -1], ....: [6, 14, -5, 1, 0], ....: [0, -2, -1, 8, -1], ....: [1, -1, 3, -3, 2], ....: [8, -4, 1, 1, -17]]) sage: fivegame = NormalFormGame([player1, player2]) sage: fivegame.obtain_nash(algorithm='enumeration') [[(1, 0, 0, 0, 0), (0, 1, 0, 0, 0)]] sage: fivegame.obtain_nash(algorithm='lrs') # optional - lrslib [[(1, 0, 0, 0, 0), (0, 1, 0, 0, 0)]] sage: fivegame.obtain_nash(algorithm='LCP') # optional - gambit [[(1.0, 0.0, 0.0, 0.0, 0.0), (0.0, 1.0, 0.0, 0.0, 0.0)]]
Here are some examples of finding Nash equilibria for constant-sum games::
sage: A = matrix.identity(2) sage: cg = NormalFormGame([A]) sage: cg.obtain_nash(algorithm='lp') [[(0.5, 0.5), (0.5, 0.5)]] sage: cg.obtain_nash(algorithm='lp', solver='Coin') # optional - cbc [[(0.5, 0.5), (0.5, 0.5)]] sage: cg.obtain_nash(algorithm='lp', solver='PPL') [[(1/2, 1/2), (1/2, 1/2)]] sage: cg.obtain_nash(algorithm='lp', solver='gambit') # optional - gambit [[(0.5, 0.5), (0.5, 0.5)]] sage: A = matrix([[2, 1], [1, 3]]) sage: cg = NormalFormGame([A]) sage: ne = cg.obtain_nash(algorithm='lp', solver='glpk') sage: [[[round(el, 6) for el in v] for v in eq] for eq in ne] [[[0.666667, 0.333333], [0.666667, 0.333333]]] sage: ne = cg.obtain_nash(algorithm='lp', solver='Coin') # optional - cbc sage: [[[round(el, 6) for el in v] for v in eq] for eq in ne] # optional - cbc [[[0.666667, 0.333333], [0.666667, 0.333333]]] sage: cg.obtain_nash(algorithm='lp', solver='PPL') [[(2/3, 1/3), (2/3, 1/3)]] sage: ne = cg.obtain_nash(algorithm='lp', solver='gambit') # optional - gambit sage: [[[round(el, 6) for el in v] for v in eq] for eq in ne] # optional - gambit [[[0.666667, 0.333333], [0.666667, 0.333333]]] sage: A = matrix([[1, 2, 1], [1, 1, 2], [2, 1, 1]]) sage: B = matrix([[2, 1, 2], [2, 2, 1], [1, 2, 2]]) sage: cg = NormalFormGame([A, B]) sage: ne = cg.obtain_nash(algorithm='lp', solver='glpk') sage: [[[round(el, 6) for el in v] for v in eq] for eq in ne] [[[0.333333, 0.333333, 0.333333], [0.333333, 0.333333, 0.333333]]] sage: ne = cg.obtain_nash(algorithm='lp', solver='Coin') # optional - cbc sage: [[[round(el, 6) for el in v] for v in eq] for eq in ne] # optional - cbc [[[0.333333, 0.333333, 0.333333], [0.333333, 0.333333, 0.333333]]] sage: cg.obtain_nash(algorithm='lp', solver='PPL') [[(1/3, 1/3, 1/3), (1/3, 1/3, 1/3)]] sage: ne = cg.obtain_nash(algorithm='lp', solver='gambit') # optional - gambit sage: [[[round(el, 6) for el in v] for v in eq] for eq in ne] # optional - gambit [[[0.333333, 0.333333, 0.333333], [0.333333, 0.333333, 0.333333]]] sage: A = matrix([[160, 205, 44], ....: [175, 180, 45], ....: [201, 204, 50], ....: [120, 207, 49]]) sage: cg = NormalFormGame([A]) sage: cg.obtain_nash(algorithm='lp', solver='PPL') [[(0, 0, 1, 0), (0, 0, 1)]]
Running the constant-sum solver on a game which isn't a constant sum game generates a ``ValueError``::
sage: cg = NormalFormGame([A, A]) sage: cg.obtain_nash(algorithm='lp', solver='glpk') Traceback (most recent call last): ... ValueError: Input game needs to be a two player constant sum game
Here is an example of a 3 by 2 game with 3 Nash equilibrium::
sage: A = matrix([[3,3], ....: [2,5], ....: [0,6]]) sage: B = matrix([[3,2], ....: [2,6], ....: [3,1]]) sage: g = NormalFormGame([A, B]) sage: g.obtain_nash(algorithm='enumeration') [[(0, 1/3, 2/3), (1/3, 2/3)], [(4/5, 1/5, 0), (2/3, 1/3)], [(1, 0, 0), (1, 0)]]
Of the algorithms implemented, only ``'lrs'`` and ``'enumeration'`` are guaranteed to find all Nash equilibria in a game. The solver for constant sum games only ever finds one Nash equilibrium. Although it is possible for the ``'LCP'`` solver to find all Nash equilibria in some instances, there are instances where it will not be able to find all Nash equilibria.::
sage: A = matrix(2, 2) sage: gg = NormalFormGame([A]) sage: gg.obtain_nash(algorithm='enumeration') [[(0, 1), (0, 1)], [(0, 1), (1, 0)], [(1, 0), (0, 1)], [(1, 0), (1, 0)]] sage: gg.obtain_nash(algorithm='lrs') # optional - lrs [[(0, 1), (0, 1)], [(0, 1), (1, 0)], [(1, 0), (0, 1)], [(1, 0), (1, 0)]] sage: gg.obtain_nash(algorithm='lp', solver='glpk') [[(1.0, 0.0), (1.0, 0.0)]] sage: gg.obtain_nash(algorithm='LCP') # optional - gambit [[(1.0, 0.0), (1.0, 0.0)]] sage: gg.obtain_nash(algorithm='enumeration', maximization=False) [[(0, 1), (0, 1)], [(0, 1), (1, 0)], [(1, 0), (0, 1)], [(1, 0), (1, 0)]] sage: gg.obtain_nash(algorithm='lrs', maximization=False) # optional - lrs [[(0, 1), (0, 1)], [(0, 1), (1, 0)], [(1, 0), (0, 1)], [(1, 0), (1, 0)]] sage: gg.obtain_nash(algorithm='lp', solver='glpk', maximization=False) [[(1.0, 0.0), (1.0, 0.0)]] sage: gg.obtain_nash(algorithm='LCP', maximization=False) # optional - gambit [[(1.0, 0.0), (1.0, 0.0)]]
Note that outputs for all algorithms are as lists of lists of tuples and the equilibria have been sorted so that all algorithms give a comparable output (although ``'LCP'`` returns floats)::
sage: enumeration_eqs = g.obtain_nash(algorithm='enumeration') sage: [[type(s) for s in eq] for eq in enumeration_eqs] [[<... 'tuple'>, <... 'tuple'>], [<... 'tuple'>, <... 'tuple'>], [<... 'tuple'>, <... 'tuple'>]] sage: lrs_eqs = g.obtain_nash(algorithm='lrs') # optional - lrslib sage: [[type(s) for s in eq] for eq in lrs_eqs] # optional - lrslib [[<... 'tuple'>, <... 'tuple'>], [<... 'tuple'>, <... 'tuple'>], [<... 'tuple'>, <... 'tuple'>]] sage: LCP_eqs = g.obtain_nash(algorithm='LCP') # optional - gambit sage: [[type(s) for s in eq] for eq in LCP_eqs] # optional - gambit [[<... 'tuple'>, <... 'tuple'>], [<... 'tuple'>, <... 'tuple'>], [<... 'tuple'>, <... 'tuple'>]] sage: enumeration_eqs == sorted(enumeration_eqs) True sage: lrs_eqs == sorted(lrs_eqs) # optional - lrslib True sage: LCP_eqs == sorted(LCP_eqs) # optional - gambit True sage: lrs_eqs == enumeration_eqs # optional - lrslib True sage: enumeration_eqs == LCP_eqs # optional - gambit False sage: [[[round(float(p), 6) for p in str] for str in eq] for eq in enumeration_eqs] == [[[round(float(p), 6) for p in str] for str in eq] for eq in LCP_eqs] # optional - gambit True
Also, not specifying a valid solver would lead to an error::
sage: A = matrix.identity(2) sage: g = NormalFormGame([A]) sage: g.obtain_nash(algorithm="invalid") Traceback (most recent call last): ... ValueError: 'algorithm' should be set to 'enumeration', 'LCP', 'lp' or 'lrs' sage: g.obtain_nash(algorithm="lp", solver="invalid") Traceback (most recent call last): ... ValueError: 'solver' should be set to 'GLPK', ..., None (in which case the default one is used), or a callable. """ "than 2 players have not been " "implemented yet. Please see the gambit " "website (http://gambit.sourceforge.net/) that has a variety of " "available algorithms")
algorithm = "lp" algorithm = "lrs" else:
if not is_package_installed('lrslib'): raise PackageNotFoundError("lrslib")
return self._solve_lrs(maximization)
if Game is None: raise PackageNotFoundError("gambit") return self._solve_LCP(maximization)
r""" EXAMPLES:
A simple game::
sage: A = matrix([[1, 2], [3, 4]]) sage: B = matrix([[3, 3], [1, 4]]) sage: C = NormalFormGame([A, B]) sage: C._solve_lrs() # optional - lrslib [[(0, 1), (0, 1)]]
2 random matrices::
sage: p1 = matrix([[-1, 4, 0, 2, 0], ....: [-17, 246, -5, 1, -2], ....: [0, 1, 1, -4, -4], ....: [1, -3, 9, 6, -1], ....: [2, 53, 0, -5, 0]]) sage: p2 = matrix([[0, 1, 1, 3, 1], ....: [3, 9, 44, -1, -1], ....: [1, -4, -1, -3, 1], ....: [1, 0, 0, 0, 0,], ....: [1, -3, 1, 21, -2]]) sage: biggame = NormalFormGame([p1, p2]) sage: biggame._solve_lrs() # optional - lrslib [[(0, 0, 0, 20/21, 1/21), (11/12, 0, 0, 1/12, 0)]]
Another test::
sage: p1 = matrix([[-7, -5, 5], ....: [5, 5, 3], ....: [1, -6, 1]]) sage: p2 = matrix([[-9, 7, 9], ....: [6, -2, -3], ....: [-4, 6, -10]]) sage: biggame = NormalFormGame([p1, p2]) sage: biggame._solve_lrs() # optional - lrslib [[(0, 1, 0), (1, 0, 0)], [(1/3, 2/3, 0), (0, 1/6, 5/6)], [(1/3, 2/3, 0), (1/7, 0, 6/7)], [(1, 0, 0), (0, 0, 1)]] """ from subprocess import PIPE, Popen m1, m2 = self.payoff_matrices() if maximization is False: m1 = - m1 m2 = - m2 game1_str, game2_str = self._Hrepresentation(m1, m2)
g1_name = tmp_filename() with open(g1_name, 'w') as g1_file: g1_file.write(game1_str) g2_name = tmp_filename() with open(g2_name, 'w') as g2_file: g2_file.write(game2_str)
try: process = Popen(['lrsnash', g1_name, g2_name], stdout=PIPE, stderr=PIPE) except OSError: from sage.misc.package import PackageNotFoundError raise PackageNotFoundError("lrslib")
lrs_output = [row for row in process.stdout] process.terminate()
nasheq = Parser(lrs_output).format_lrs() return sorted(nasheq)
r""" Solve a :class:`NormalFormGame` using Gambit's LCP algorithm.
EXAMPLES::
sage: a = matrix([[1, 0], [1, 4]]) sage: b = matrix([[2, 3], [2, 4]]) sage: c = NormalFormGame([a, b]) sage: c._solve_LCP(maximization=True) # optional - gambit [[(0.0, 1.0), (0.0, 1.0)]] """ g = self._gambit_(maximization) output = ExternalLCPSolver().solve(g) nasheq = Parser(output).format_gambit(g) return sorted(nasheq)
r""" Solve a constant sum :class:`NormalFormGame` using Gambit's LP implementation.
EXAMPLES::
sage: A = matrix([[2, 1], [1, 2.5]]) sage: g = NormalFormGame([A]) sage: g._solve_gambit_LP() # optional - gambit [[(0.6, 0.4), (0.6, 0.4)]] sage: A = matrix.identity(2) sage: g = NormalFormGame([A]) sage: g._solve_gambit_LP() # optional - gambit [[(0.5, 0.5), (0.5, 0.5)]] sage: g = NormalFormGame([A,A]) sage: g._solve_gambit_LP() # optional - gambit Traceback (most recent call last): ... RuntimeError: Method only valid for constant-sum games. """ if Game is None: raise NotImplementedError("gambit is not installed") g = self._gambit_(maximization = maximization) output = ExternalLPSolver().solve(g) nasheq = Parser(output).format_gambit(g) return sorted(nasheq)
r""" Solves a constant sum :class:`NormalFormGame` using the specified LP solver.
INPUT:
- ``solver`` -- the solver to be used to solve the LP:
* ``'gambit'`` - his uses the solver included within the gambit library to create and solve the LP
* for further possible values, see :class:`MixedIntegerLinearProgram`
EXAMPLES::
sage: A = matrix.identity(2) sage: g = NormalFormGame([A]) sage: g._solve_LP() [[(0.5, 0.5), (0.5, 0.5)]] sage: g._solve_LP('gambit') # optional - gambit [[(0.5, 0.5), (0.5, 0.5)]] sage: g._solve_LP('Coin') # optional - cbc [[(0.5, 0.5), (0.5, 0.5)]] sage: g._solve_LP('PPL') [[(1/2, 1/2), (1/2, 1/2)]] sage: A = matrix([[2, 1], [1, 3]]) sage: g = NormalFormGame([A]) sage: ne = g._solve_LP() sage: [[[round(el, 6) for el in v] for v in eq] for eq in ne] [[[0.666667, 0.333333], [0.666667, 0.333333]]] sage: ne = g._solve_LP('gambit') # optional - gambit sage: [[[round(el, 6) for el in v] for v in eq] for eq in ne] # optional - gambit [[[0.666667, 0.333333], [0.666667, 0.333333]]] sage: ne = g._solve_LP('Coin') # optional - cbc sage: [[[round(el, 6) for el in v] for v in eq] for eq in ne] # optional - cbc [[[0.666667, 0.333333], [0.666667, 0.333333]]] sage: g._solve_LP('PPL') [[(2/3, 1/3), (2/3, 1/3)]]
An exception is raised if the input game is not constant sum::
sage: A = matrix.identity(2) sage: B = A.transpose() sage: g = NormalFormGame([A, B]) sage: g._solve_LP() Traceback (most recent call last): ... ValueError: Input game needs to be a two player constant sum game """ return self._solve_gambit_LP(maximization)
r""" Obtain the Nash equilibria using support enumeration.
Algorithm implemented here is Algorithm 3.4 of [NN2007]_ with an aspect of pruning from [SLB2008]_.
1. For each k in 1...min(size of strategy sets) 2. For each I,J supports of size k 3. Prune: check if supports are dominated 4. Solve indifference conditions and check that have Nash Equilibrium.
EXAMPLES:
A Game::
sage: A = matrix([[160, 205, 44], ....: [175, 180, 45], ....: [201, 204, 50], ....: [120, 207, 49]]) sage: B = matrix([[2, 2, 2], ....: [1, 0, 0], ....: [3, 4, 1], ....: [4, 1, 2]]) sage: g=NormalFormGame([A, B]) sage: g._solve_enumeration() [[(0, 0, 3/4, 1/4), (1/28, 27/28, 0)]]
A game with 3 equilibria::
sage: A = matrix([[3,3], ....: [2,5], ....: [0,6]]) sage: B = matrix([[3,2], ....: [2,6], ....: [3,1]]) sage: g = NormalFormGame([A, B]) sage: g._solve_enumeration(maximization=False) [[(1, 0, 0), (0, 1)]]
A simple example::
sage: s = matrix([[1, 0], [-2, 3]]) sage: t = matrix([[3, 2], [-1, 0]]) sage: example = NormalFormGame([s, t]) sage: example._solve_enumeration() [[(0, 1), (0, 1)], [(1/2, 1/2), (1/2, 1/2)], [(1, 0), (1, 0)]]
Another::
sage: A = matrix([[0, 1, 7, 1], ....: [2, 1, 3, 1], ....: [3, 1, 3, 5], ....: [6, 4, 2, 7]]) sage: B = matrix([[3, 2, 8, 4], ....: [6, 2, 0, 3], ....: [1, 3, -1, 1], ....: [3, 2, 1, 1]]) sage: C = NormalFormGame([A, B]) sage: C._solve_enumeration() [[(0, 0, 0, 1), (1, 0, 0, 0)], [(2/7, 0, 0, 5/7), (5/11, 0, 6/11, 0)], [(1, 0, 0, 0), (0, 0, 1, 0)]]
Again::
sage: X = matrix([[1, 4, 2], ....: [4, 0, 3], ....: [2, 3, 5]]) sage: Y = matrix([[3, 9, 2], ....: [0, 3, 1], ....: [5, 4, 6]]) sage: Z = NormalFormGame([X, Y]) sage: Z._solve_enumeration() [[(0, 0, 1), (0, 0, 1)], [(2/9, 0, 7/9), (0, 3/4, 1/4)], [(1, 0, 0), (0, 1, 0)]]
TESTS:
Due to the nature of the linear equations solved in this algorithm some negative vectors can be returned. Here is a test that ensures this doesn't happen (the particular payoff matrices chosen give a linear system that would have negative valued vectors as solution)::
sage: a = matrix([[-13, 59], ....: [27, 86]]) sage: b = matrix([[14, 6], ....: [58, -14]]) sage: c = NormalFormGame([a, b]) sage: c._solve_enumeration() [[(0, 1), (1, 0)]]
Testing against an error in `_is_NE`. Note that 1 equilibrium is missing: ``[(2/3, 1/3), (0, 1)]``, however this equilibrium has supports of different sizes. This only occurs in degenerate games and is not supported in the `enumeration` algorithm::
sage: N = NormalFormGame([matrix(2,[0,-1,-2,-1]),matrix(2,[1,0,0,2])]) sage: N._solve_enumeration() [[(0, 1), (0, 1)], [(1, 0), (1, 0)]]
In this instance the `lrs` algorithm is able to find all three equilibria::
sage: N = NormalFormGame([matrix(2,[0,-1,-2,-1]),matrix(2,[1,0,0,2])]) sage: N.obtain_nash(algorithm='lrs') # optional - lrslib [[(0, 1), (0, 1)], [(2/3, 1/3), (0, 1)], [(1, 0), (1, 0)]]
Here is another::
sage: N = NormalFormGame([matrix(2,[7,-8,-4,-8,7,0]),matrix(2,[-9,-1,-8,3,2,3])]) sage: N._solve_enumeration() [[(0, 1), (0, 0, 1)]] """
powerset(range(player.num_strategies))] for player in self.players]
# Check if any supports are dominated for row player # Check if any supports are dominated for col player and self._row_cond_dominance(pair[1], pair[0], M2.transpose())):
r""" Check if any row strategies of a sub matrix defined by a given pair of supports are conditionally dominated. Return ``False`` if a row is conditionally dominated.
TESTS:
A matrix that depending on the support for the column player has a dominated row::
sage: g = NormalFormGame() sage: A = matrix([[1, 1, 5], [2, 2, 0]]) sage: g._row_cond_dominance((0, 1), (0, 1), A) False
or does not have a dominated row::
sage: g._row_cond_dominance((0, 1), (0, 2), A) True """
r""" For support1, retrns the strategy with support: support2 that makes the column player indifferent for the utilities given by M.
This is done by building the corresponding linear system. If `\rho_1, \rho_2` are the supports of player 1 and 2 respectively. Then, indifference for player 1 implies:
.. MATH::
u_1(s_1,\rho_2) = u_1(s_2, \rho_2)
for all `s_1, s_2` in the support of `\rho_1`. This corresponds to:
.. MATH::
\sum_{j\in S(\rho_2)}A_{s_1,j}{\rho_2}_j = \sum_{j\in S(\rho_2)}A_{s_2,j}{\rho_2}_j
for all `s_1, s_2` in the support of `\rho_1` where `A` is the payoff matrix of player 1. Equivalently we can consider consecutive rows of `A` (instead of all pairs of strategies). Thus the corresponding linear system can be written as:
.. MATH::
\left(\sum_{j \in S(\rho_2)}^{A_{i,j} - A_{i+1,j}\right){\rho_2}_j
for all `1\leq i \leq |S(\rho_1)|` (where `A` has been modified to only contain the row corresponding to `S(\rho_1)`). We also require all elements of `\rho_2` to sum to 1:
.. MATH::
\sum_{j\in S(\rho_1)}{\rho_2}_j = 1.
TESTS:
Find the indifference vector for a support pair that has no dominated strategies::
sage: A = matrix([[1, 1, 5], [2, 2, 0]]) sage: g = NormalFormGame([A]) sage: g._solve_indifference((0, 1), (0, 2), A) (1/3, 2/3) sage: g._solve_indifference((0, 2), (0, 1), -A.transpose()) (5/6, 0, 1/6)
When a support pair has a dominated strategy there is no solution to the indifference equation::
sage: g._solve_indifference((0, 1), (0, 1), -A.transpose()) <BLANKLINE>
Particular case of a game with 1 strategy for each for each player::
sage: A = matrix([[10]]) sage: g = NormalFormGame([A]) sage: g._solve_indifference((0,), (0,), -A.transpose()) (1) """
# Build linear system for player 1 # Checking particular case of supports of pure strategies M[strategy1][strategy2]: else: # Coefficients of linear system that ensure indifference # between two consecutive strategies of the support M[strategy1][support2[strategy_pair2]] -\ M[strategy1][support2[strategy_pair2 - 1]] # Coefficients of linear system that ensure the vector is # a probability vector. ie. sum to 1 # Create rhs of linear systems
# Solve both linear systems
r""" For vectors that obey indifference for a given support pair, checks if it corresponds to a Nash equilibria (support is obeyed and no negative values, also that no player has incentive to deviate out of supports).
TESTS::
sage: X = matrix([[1, 4, 2], ....: [4, 0, 3], ....: [2, 3, 5]]) sage: Y = matrix([[3, 9, 2], ....: [0, 3, 1], ....: [5, 4, 6]]) sage: Z = NormalFormGame([X, Y]) sage: Z._is_NE([0, 1/4, 3/4], [3/5, 2/5, 0], (1, 2,), (0, 1,), X, Y) False
sage: Z._is_NE([2/9, 0, 7/9], [0, 3/4, 1/4], (0, 2), (1, 2), X, Y) True
Checking pure strategies are not forgotten::
sage: A = matrix(2, [0, -1, -2, -1]) sage: B = matrix(2, [1, 0, 0, 2]) sage: N = NormalFormGame([A, B]) sage: N._is_NE([1, 0], [1, 0], (0,), (0,), A, B) True sage: N._is_NE([0, 1], [0, 1], (1,), (1,), A, B) True sage: N._is_NE([1, 0], [0, 1], (0,), (1,), A, B) False sage: N._is_NE([0, 1], [1, 0], (1,), (0,), A, B) False
sage: A = matrix(3, [-7, -5, 5, 5, 5, 3, 1, -6, 1]) sage: B = matrix(3, [-9, 7, 9, 6, -2, -3, -4, 6, -10]) sage: N = NormalFormGame([A, B]) sage: N._is_NE([1, 0, 0], [0, 0, 1], (0,), (2,), A, B) True sage: N._is_NE([0, 1, 0], [1, 0, 0], (1,), (0,), A, B) True sage: N._is_NE([0, 1, 0], [0, 1, 0], (1,), (1,), A, B) False sage: N._is_NE([0, 0, 1], [0, 1, 0], (2,), (1,), A, B) False sage: N._is_NE([0, 0, 1], [0, 0, 1], (2,), (2,), A, B) False """ # Check that supports are obeyed all([b[j] > 0 for j in p2_support]) and all([a[i] == 0 for i in range(len(a)) if i not in p1_support]) and all([b[j] == 0 for j in range(len(b)) if j not in p2_support])):
# Check that have pair of best responses
for row in M1.rows()] for col in M2.columns()]
#if p1_payoffs.index(max(p1_payoffs)) not in p1_support: if x == max(p1_payoffs)): if x == max(p2_payoffs)):
r""" Create the H-representation strings required to use lrs nash.
EXAMPLES::
sage: A = matrix([[1, 2], [3, 4]]) sage: B = matrix([[3, 3], [1, 4]]) sage: C = NormalFormGame([A, B]) sage: print(C._Hrepresentation(A, B)[0]) H-representation linearity 1 5 begin 5 4 rational 0 1 0 0 0 0 1 0 0 -3 -1 1 0 -3 -4 1 -1 1 1 0 end <BLANKLINE> sage: print(C._Hrepresentation(A, B)[1]) H-representation linearity 1 5 begin 5 4 rational 0 -1 -2 1 0 -3 -4 1 0 1 0 0 0 0 1 0 -1 1 1 0 end <BLANKLINE>
"""
""" A function to check whether the game is degenerate or not. Will return a boolean.
A two-player game is called nondegenerate if no mixed strategy of support size `k` has more than `k` pure best responses [NN2007]_. In a degenerate game, this definition is violated, for example if there is a pure strategy that has two pure best responses.
The implementation here transforms the search over mixed strategies to a search over supports which is a discrete search. A full explanation of this is given in [CK2015]_. This problem is known to be NP-Hard [Du2009]_. Another possible implementation is via best response polytopes, see :trac:`18958`.
The game Rock-Paper-Scissors is an example of a non-degenerate game,::
sage: g = game_theory.normal_form_games.RPS() sage: g.is_degenerate() False
whereas `Rock-Paper-Scissors-Lizard-Spock <http://www.samkass.com/theories/RPSSL.html>`_ is degenerate because for every pure strategy there are two best responses.::
sage: g = game_theory.normal_form_games.RPSLS() sage: g.is_degenerate() True
EXAMPLES:
Here is an example of a degenerate game given in [DGRB2010]_::
sage: A = matrix([[3, 3], [2, 5], [0, 6]]) sage: B = matrix([[3, 3], [2, 6], [3, 1]]) sage: degenerate_game = NormalFormGame([A,B]) sage: degenerate_game.is_degenerate() True
Here is an example of a degenerate game given in [NN2007]_::
sage: A = matrix([[0, 6], [2, 5], [3, 3]]) sage: B = matrix([[1, 0], [0, 2], [4, 4]]) sage: d_game = NormalFormGame([A, B]) sage: d_game.is_degenerate() True
Here are some other examples of degenerate games::
sage: M = matrix([[2, 1], [1, 1]]) sage: N = matrix([[1, 1], [1, 2]]) sage: game = NormalFormGame([M, N]) sage: game.is_degenerate() True
If more information is required, it may be useful to use ``certificate=True``. This will return a boolean of whether the game is degenerate or not, and if True; a tuple containing the strategy where degeneracy was found and the player it belongs to. ``0`` is the row player and ``1`` is the column player.::
sage: M = matrix([[2, 1], [1, 1]]) sage: N = matrix([[1, 1], [1, 2]]) sage: g = NormalFormGame([M, N]) sage: test, certificate = g.is_degenerate(certificate=True) sage: test, certificate (True, ((1, 0), 0))
Using the output, we see that the opponent has more best responses than the size of the support of the strategy in question ``(1, 0)``. (We specify the player as ``(player + 1) % 2`` to ensure that we have the opponent's index.)::
sage: g.best_responses(certificate[0], (certificate[1] + 1) % 2) [0, 1]
Another example with a mixed strategy causing degeneracy.::
sage: A = matrix([[3, 0], [0, 3], [1.5, 1.5]]) sage: B = matrix([[4, 3], [2, 6], [3, 1]]) sage: g = NormalFormGame([A, B]) sage: test, certificate = g.is_degenerate(certificate=True) sage: test, certificate (True, ((1/2, 1/2), 1))
Again, we see that the opponent has more best responses than the size of the support of the strategy in question ``(1/2, 1/2)``.::
sage: g.best_responses(certificate[0], (certificate[1] + 1) % 2) [0, 1, 2]
Sometimes, the different algorithms for obtaining nash_equilibria don't agree with each other. This can happen when games are degenerate::
sage: a = matrix([[-75, 18, 45, 33], ....: [42, -8, -77, -18], ....: [83, 18, 11, 40], ....: [-10, -38, 76, -9]]) sage: b = matrix([[62, 64, 87, 51], ....: [-41, -27, -69, 52], ....: [-17, 25, -97, -82], ....: [30, 31, -1, 50]]) sage: d_game = NormalFormGame([a, b]) sage: d_game.obtain_nash(algorithm='lrs') # optional - lrslib [[(0, 0, 1, 0), (0, 1, 0, 0)], [(17/29, 0, 0, 12/29), (0, 0, 42/73, 31/73)], [(122/145, 0, 23/145, 0), (0, 1, 0, 0)]] sage: d_game.obtain_nash(algorithm='LCP') # optional - gambit [[(0.5862068966, 0.0, 0.0, 0.4137931034), (0.0, 0.0, 0.5753424658, 0.4246575342)]] sage: d_game.obtain_nash(algorithm='enumeration') [[(0, 0, 1, 0), (0, 1, 0, 0)], [(17/29, 0, 0, 12/29), (0, 0, 42/73, 31/73)]] sage: d_game.is_degenerate() True
TESTS::
sage: g = NormalFormGame() sage: g.add_player(3) # Adding first player with 3 strategies sage: g.add_player(3) # Adding second player with 3 strategies sage: for key in g: ....: g[key] = [0, 0] sage: g.is_degenerate() True
sage: A = matrix([[3, 0], [0, 3], [1.5, 1.5]]) sage: B = matrix([[4, 3], [2, 6], [3, 1]]) sage: g = NormalFormGame([A, B]) sage: g.is_degenerate() True
sage: A = matrix([[1, -1], [-1, 1]]) sage: B = matrix([[-1, 1], [1, -1]]) sage: matching_pennies = NormalFormGame([A, B]) sage: matching_pennies.is_degenerate() False
sage: A = matrix([[2, 5], [0, 4]]) sage: B = matrix([[2, 0], [5, 4]]) sage: prisoners_dilemma = NormalFormGame([A, B]) sage: prisoners_dilemma.is_degenerate() False
sage: g = NormalFormGame() sage: g.add_player(2) sage: g.add_player(2) sage: g.add_player(2) sage: g.is_degenerate() Traceback (most recent call last): ... NotImplementedError: Tests for Degeneracy is not yet implemented for games with more than two players. """ "implemented for games with more than " "two players.")
powerset(range(player.num_strategies))] for player in self.players]
# filter out all supports that are pure or empty for k in potential_supports]
product(*potential_supports) if len(pair[0]) != len(pair[1])]
# Sort so that solve small linear systems first
if certificate: return True, (strat, 0) else: return True else:
return False, () else:
""" For a given strategy for a player and the index of the opponent, computes the payoff for the opponent and returns a list of the indices of the best responses. Only implemented for two player games
INPUT:
- ``strategy`` -- a probability distribution vector
- ``player`` -- the index of the opponent, ``0`` for the row player, ``1`` for the column player.
EXAMPLES::
sage: A = matrix([[3, 0], [0, 3], [1.5, 1.5]]) sage: B = matrix([[4, 3], [2, 6], [3, 1]]) sage: g = NormalFormGame([A, B])
Now we can obtain the best responses for Player 1, when Player 2 uses different strategies::
sage: g.best_responses((1/2, 1/2), player=0) [0, 1, 2] sage: g.best_responses((3/4, 1/4), player=0) [0]
To get the best responses for Player 2 we pass the argument :code:`player=1`
sage: g.best_responses((4/5, 1/5, 0), player=1) [0, 1]
sage: A = matrix([[1, 0], [0, 1], [0, 0]]) sage: B = matrix([[1, 0], [0, 1], [0.7, 0.8]]) sage: g = NormalFormGame([A, B]) sage: g.best_responses((0, 1, 0), player=1) [1]
sage: A = matrix([[3,3],[2,5],[0,6]]) sage: B = matrix([[3,3],[2,6],[3,1]]) sage: degenerate_game = NormalFormGame([A,B]) sage: degenerate_game.best_responses((1, 0, 0), player=1) [0, 1]
sage: A = matrix([[3, 0], [0, 3], [1.5, 1.5]]) sage: B = matrix([[4, 3], [2, 6], [3, 1]]) sage: g = NormalFormGame([A, B]) sage: g.best_responses((1/3, 1/3, 1/3), player=1) [1]
Note that this has only been implemented for 2 player games::
sage: g = NormalFormGame() sage: g.add_player(2) # adding first player with 2 strategies sage: g.add_player(2) # adding second player with 2 strategies sage: g.add_player(2) # adding third player with 2 strategies sage: g.best_responses((1/2, 1/2), player=2) Traceback (most recent call last): ... ValueError: Only available for 2 player games
If the strategy is not of the correct dimension for the given player then an error is returned::
sage: A = matrix([[3, 0], [0, 3], [1.5, 1.5]]) sage: B = matrix([[4, 3], [2, 6], [3, 1]]) sage: g = NormalFormGame([A, B]) sage: g.best_responses((1/2, 1/2), player=1) Traceback (most recent call last): ... ValueError: Strategy is not of correct dimension
sage: g.best_responses((1/3, 1/3, 1/3), player=0) Traceback (most recent call last): ... ValueError: Strategy is not of correct dimension
If the strategy is not a true probability vector then an error is passed:
sage: A = matrix([[3, 0], [0, 3], [1.5, 1.5]]) sage: B = matrix([[4, 3], [2, 6], [3, 1]]) sage: g = NormalFormGame([A, B]) sage: g.best_responses((1/3, 1/2, 0), player=1) Traceback (most recent call last): ... ValueError: Strategy is not a probability distribution vector
sage: A = matrix([[3, 0], [0, 3], [1.5, 1.5]]) sage: B = matrix([[4, 3], [2, 6], [3, 1]]) sage: g = NormalFormGame([A, B]) sage: g.best_responses((3/2, -1/2), player=0) Traceback (most recent call last): ... ValueError: Strategy is not a probability distribution vector
If the player specified is not `0` or `1`, an error is raised::
sage: A = matrix([[3, 0], [0, 3], [1.5, 1.5]]) sage: B = matrix([[4, 3], [2, 6], [3, 1]]) sage: g = NormalFormGame([A, B]) sage: g.best_responses((1/2, 1/2), player='Player1') Traceback (most recent call last): ... ValueError: Player1 is not an index of the oponent, must be 0 or 1 """
""" Checks whether a game is degenerate in pure strategies.
TESTS::
sage: A = matrix([[3,3],[2,5],[0,6]]) sage: B = matrix([[3,3],[2,6],[3,1]]) sage: degenerate_game = NormalFormGame([A,B]) sage: degenerate_game._is_degenerate_pure() True
sage: A = matrix([[1, 0], [0, 1], [0, 0]]) sage: B = matrix([[1, 0], [0, 1], [0.7, 0.8]]) sage: g = NormalFormGame([A, B]) sage: g._is_degenerate_pure() False
sage: A = matrix([[2, 5], [0, 4]]) sage: B = matrix([[2, 0], [5, 4]]) sage: prisoners_dilemma = NormalFormGame([A, B]) sage: prisoners_dilemma._is_degenerate_pure() False
sage: A = matrix([[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]]) sage: g = NormalFormGame([A]) sage: g._is_degenerate_pure() True
Whilst this game is not degenerate in pure strategies, it is actually degenerate, but only in mixed strategies.
sage: A = matrix([[3, 0], [0, 3], [1.5, 1.5]]) sage: B = matrix([[4, 3], [2, 6], [3, 1]]) sage: g = NormalFormGame([A, B]) sage: g._is_degenerate_pure() False """ else:
strat = [0 for k in range(M1.ncols())] strat[j] = 1 return True, (tuple(strat), 1) else:
r""" TESTS::
sage: from sage.game_theory.normal_form_game import _Player sage: p = _Player(5) sage: p.num_strategies 5 """
r""" TESTS::
sage: from sage.game_theory.normal_form_game import _Player sage: p = _Player(5) sage: p.add_strategy() sage: p.num_strategies 6 """ |