Best response polytopes - solutions

  1. Define a polytope as a convex hull

    Bookwork: https://vknight.org/gt/chapters/06/#Definition-of-a-Polytope-as-a-convex-hull

  2. Define a polytope as an intersection of halfspaces

    Bookwork: https://vknight.org/gt/chapters/06/#Definition-of-a-Polytope-as-an-intersection-of-halfspaces

  3. For the following set of vertices draw the corresponding polytope and obtain their definition as an intersection of halfspaces:

    A. V = $\{(0, 0), (0, 1), (1, 0), (1, 1)\}$

In [1]:
%matplotlib inline

import matplotlib.pyplot as plt
import numpy as np
import scipy.spatial

V = [np.array([0, 0]), np.array([0, 1]), np.array([1, 0]), np.array([1, 1])]
P = scipy.spatial.ConvexHull(V)
scipy.spatial.convex_hull_plot_2d(P);

The definition as an intersection of halfspaces:

$$ \begin{align} - x_1 & \leq 0\\ -x_2 & \leq 0\\ x_1 & \leq 1\\ x_2 & \leq 1 \end{align} $$

B. V = $\{(0, 0), (0, 1/4), (1, 2/3), (2, 1/5)\}$

In [2]:
V = [np.array([0, 0]), np.array([0, 1 / 4]), np.array([1, 2  / 3]), np.array([2, 1 / 5])]
P = scipy.spatial.ConvexHull(V)
scipy.spatial.convex_hull_plot_2d(P);

Let us first find the equations for each boundary:

  • Between $(0, 0)$ and $(0, 1/4)$: $x_1=0$
  • Between $(0, 1/4)$ and $(1, 2/3)$: $x_2=(5/12)x_1+1/4$
  • Between $(1, 2/3)$ and $(2, 1/5)$: $x_2=(-7/15)x_1+17/15$
  • Between $(2, 1/5)$ and $(0, 0)$: $x_2=(1/10)x_1$

The definition as an intersection of halfspaces (obtained by identifying the lines corresponding to each boundary of the polytope):

$$ \begin{align} -x_1 & \leq 0\\ 12x_2 - 5x_1 & \leq 3\\ 15x_2 + 7x_1 & \leq 17\\ x_1 -10x_2 & \leq 0 \end{align} $$

C. V = $\{(0, 0), (0, 1/4), (1, 2/3), (2, 1/5), (1, 0)\}$

In [3]:
V = [np.array([0, 0]), np.array([0, 1 / 4]), np.array([1, 2  / 3]), np.array([2, 1 / 5]), np.array([1, 0])]
P = scipy.spatial.ConvexHull(V)
scipy.spatial.convex_hull_plot_2d(P);

Let us first find the equations for each boundary:

  • Between $(0, 0)$ and $(0, 1/4)$: $x_1=0$
  • Between $(0, 1/4)$ and $(1, 2/3)$: $x_2=(5/12)x_1+1/4$
  • Between $(1, 2/3)$ and $(2, 1/5)$: $x_2=(-7/15)x_1+17/15$
  • Between $(2, 1/5)$ and $(1, 0)$: $x_2=(1/5)x_1-1/5$
  • Between $(1, 0)$ and $(0, 0)$: $x_2=0$

The definition as an intersection of halfspaces (obtained by identifying the lines corresponding to each boundary of the polytope):

$$ \begin{align} -x_1 & \leq 0\\ 12x_2 - 5x_1 & \leq 3\\ 15x_2 + 7x_1 & \leq 17\\ x_1 - 5 x_2 & \leq 1\\ -x_2 & \leq 0 \end{align} $$

4. Define the best response polytopes.

Bookwork: https://vknight.org/gt/chapters/06/#Definition-of-best-response-polytopes

5. For the following games, obtain the best response polytopes, label the vertices and identify all Nash equilibria:

  1. $ A = \begin{pmatrix} 3 & -1\\ 2 & 7\end{pmatrix} \qquad B = \begin{pmatrix} -3 & 1\\ 1 & -6\end{pmatrix} $

    First we need nonegative valued matrices:

$$ A \to A + 2 = \begin{pmatrix} 5 & 1\\ 4 & 9\end{pmatrix} \qquad B \to B + 7 = \begin{pmatrix} 4 & 8\\ 8 & 1\end{pmatrix} $$

The inequalities of $\mathcal{P}$ are then given by:

$$ \begin{align} -x_1 & \leq 0\\ -x_2 & \leq 0\\ 4x_1 + 8x_2 & \leq 1\\ 8x_1 + 1x_2 & \leq 1\ \end{align} $$

which corresponds to:

$$ \begin{align} x_1 & \geq 0\\ x_2 & \geq 0\\ x_2 & \leq 1/8 - (1/2)x_1\\ x_2 & \leq 1 - 8x_1 \end{align} $$

The vertices are given by:

$$V=\{(0, 0), (0, 1/8), (1/8, 0), (7/60, 1/15)\}$$

The labels of our vertices (using the ordering of the inequalities):

  • $(0, 0)$: $\{0, 1\}$
  • $(0, 1/8)$: $\{0, 2\}$
  • $(1/8, 0)$: $\{1, 3\}$
  • $(7/60, 1/15)$: $\{2, 3\}$

    The inequalities of $\mathcal{Q}$ are given by:

    $$ \begin{align} 5x_1 + x_2 & \leq 1\\ 4x_1 + 9x_2 & \leq 1\\ -x_1 & \leq 0\\ -x_2 & \leq 0 \end{align} $$

    which corresponds to:

    $$ \begin{align} x_2 & \leq 1 - 5x_1\\ x_2 & \leq 1/9 - (4/9)x_1\\ x_1 & \geq 0\\ x_2 & \geq 0 \end{align} $$

    The vertices are given by:

    $$V=\{(0, 0), (0, 1/9), (1/5, 0), (8/41, 1/41)\}$$

    The labels of our vertices (using the ordering of the inequalities):

    • $(0, 0)$: $\{2, 3\}$
    • $(0, 1/9)$: $\{1, 2\}$
    • $(1/5, 0)$: $\{0, 3\}$
    • $(8/41, 1/41)$: $\{0, 1\}$

    The only fully labeled vertex pair is given by:

    $$((7/60, 1/15), (8/41, 1/41))$$

    which gives the normalised Nash equilibrium:

    $$((7/11, 4/11), (8/9, 1/9))$$

    Here is some code to verify this numerically:

In [4]:
import nashpy as nash
# Obtaining the row player best response polytope vertices
B = np.array([[4, 8], [8, 1]])
halfspaces = nash.polytope.build_halfspaces(B.transpose())
for v, l in nash.polytope.non_trivial_vertices(halfspaces):
    print(v)
[ 0.125  0.   ]
[ 0.     0.125]
[ 0.11666667  0.06666667]
In [5]:
# Obtaining the column player best response polytope vertices
A = np.array([[5, 1], [4, 9]])
halfspaces = nash.polytope.build_halfspaces(A)
for v, l in nash.polytope.non_trivial_vertices(halfspaces):
    print(v)
[  6.93889390e-18   1.11111111e-01]
[ 0.19512195  0.02439024]
[  2.00000000e-01  -6.93889390e-18]
In [6]:
# Verifying the Nash equilibria:
A = A - 2  # Scaling back down (note this makes no difference)
B = B - 7 
game = nash.Game(A, B)
list(game.vertex_enumeration()), (7/11, 4/11), (8/9, 1/9)
Out[6]:
([(array([ 0.63636364,  0.36363636]), array([ 0.88888889,  0.11111111]))],
 (0.6363636363636364, 0.36363636363636365),
 (0.8888888888888888, 0.1111111111111111))

2. $ A = \begin{pmatrix} 2 & -1\\ 1 & 3\end{pmatrix} \qquad B = \begin{pmatrix} -2 & 2\\ 1 & -2\end{pmatrix} $

First we need nonegative valued matrices:

$$ A \to A + 2 = \begin{pmatrix} 4 & 1\\ 3 & 5\end{pmatrix} \qquad B \to B + 3 = \begin{pmatrix} 1 & 5\\ 4 & 1\end{pmatrix} $$

The inequalities of $\mathcal{P}$ are then given by:

$$ \begin{align} -x_1 & \leq 0\\ -x_2 & \leq 0\\ x_1 + 4x_2 & \leq 1\\ 5x_1 + x_2 & \leq 1\ \end{align} $$

which corresponds to:

$$ \begin{align} x_1 & \geq 0\\ x_2 & \geq 0\\ x_2 & \leq 1/4 - (1/4)x_1\\ x_2 & \leq 1 - 5x_1 \end{align} $$

The vertices are given by:

$$V=\{(0, 0), (0, 1/4), (1/5, 0), (3/19, 4/19)\}$$

The labels of our vertices (using the ordering of the inequalities):

  • $(0, 0)$: $\{0, 1\}$
  • $(0, 1/4)$: $\{0, 2\}$
  • $(1/5, 0)$: $\{1, 3\}$
  • $(3/19, 4/19)$: $\{2, 3\}$

The inequalities of $\mathcal{Q}$ are given by:

$$ \begin{align} 4x_1 + x_2 & \leq 1\\ 3x_1 + 5x_2 & \leq 1\\ -x_1 & \leq 0\\ -x_2 & \leq 0 \end{align} $$

which corresponds to:

$$ \begin{align} x_2 & \leq 1 - 4x_1\\ x_2 & \leq 1/5 - (3/5)x_1\\ x_1 & \geq 0\\ x_2 & \geq 0 \end{align} $$

The vertices are given by:

$$V=\{(0, 0), (0, 1/5), (1/4, 0), (4/17, 1/17)\}$$

The labels of our vertices (using the ordering of the inequalities):

  • $(0, 0)$: $\{2, 3\}$
  • $(0, 1/5)$: $\{1, 2\}$
  • $(1/4, 0)$: $\{0, 3\}$
  • $(4/17, 1/17)$: $\{0, 1\}$

The only fully labeled vertex pair is given by:

$$((3/19, 4/19), (4/17, 1/17))$$

which gives the normalised Nash equilibrium:

$$((3/7, 4/7), (4/5, 1/5))$$

Here is some code to verify this numerically:

In [7]:
# Obtaining the row player best response polytope vertices
B = np.array([[1, 5], [4, 1]])
halfspaces = nash.polytope.build_halfspaces(B.transpose())
for v, l in nash.polytope.non_trivial_vertices(halfspaces):
    print(v)
[ 0.2  0. ]
[ 0.    0.25]
[ 0.15789474  0.21052632]
In [8]:
# Obtaining the column player best response polytope vertices
A = np.array([[4, 1], [3, 5]])
halfspaces = nash.polytope.build_halfspaces(A)
for v, l in nash.polytope.non_trivial_vertices(halfspaces):
    print(v)
[ 0.   0.2]
[ 0.23529412  0.05882353]
[  2.50000000e-01   1.38777878e-17]
In [9]:
# Verifying the Nash equilibria:
A = A - 2  # Scaling back down (note this makes no difference)
B = B - 3 
game = nash.Game(A, B)
list(game.vertex_enumeration()), (3 / 7, 4 / 7), (4 / 5, 1 / 5)
Out[9]:
([(array([ 0.42857143,  0.57142857]), array([ 0.8,  0.2]))],
 (0.42857142857142855, 0.5714285714285714),
 (0.8, 0.2))
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