Nash_Equilibrium.ipynb

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        "# Calculating Nash Equilibrium on Quantum Annealers\n",
        "by the Team of Dark Star Quantum Lab, North Carolina, USA \\\\\n",
        "Olga Okrut  \\\\\n",
        "Keith Cannon \\\\\n",
        "Kareem El-safty \\\\\n",
        "Faisal Shah Khan \\\\\n",
        "Nada Elsokkary \\\\\n",
        "\n",
        "Email: faisal@darkstarquantumlab.com\n",
        "\n"
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        "# Nash Equilibrium As a Quadratic Function \n",
        "## Same number of pure strategies\n",
        "\n",
        "$max: p^{T}(M+N)q -\\alpha-\\beta$\n",
        "\n",
        "subject to: \n",
        "\n",
        "$Mq - \\alpha e <=0$ \\\n",
        "$ N^{T} p - \\beta l <=0 $\\\n",
        "$e^{T} p -1 = 0$\\\n",
        "$L^{T} q -1 = 0$\\\n",
        "$p, q >=0 $\\\n",
        "where $e$ and $l$ are vectors of ones with dimentions $n x 1$ and $m x 1$ respectively. $\\alpha$ and $\\beta$ are scalar values. \n",
        "\n",
        "Reformulating linear inequality constrains to equalities by introducing slack variable, we obtain the following constrains:\n",
        "\n",
        "$Mq - \\alpha e + \\zeta =0$ \\\n",
        "$ N^{T} p - \\beta l + \\eta =0 $\\\n",
        "$e^{T} p -1 = 0$\\\n",
        "$L^{T} q -1 = 0$\\\n",
        "$p, q >=0 $\n",
        "\n",
        "\n",
        "For more details, refer to the paper [*Calculating Nash Equilibrium on Quantum Annealers*.](https://arxiv.org/abs/2112.12583)"
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        "# Battle of the Sexes \n",
        "\n",
        "We shall now consider an example of a two-person nonzero-sum game - the Battle of the Sexes. \n",
        "\n",
        "Problem formulation: A couple, Adam and Beth, decide independently whether\n",
        "to go to a soccer game or to the ballet in the evening. Each person likes to do something together with the other, but the man prefers soccer, and the woman prefers ballet. \n",
        "\n",
        "We shall assume that the total payoff for each player is the sum of the payoffs in terms of satisfaction of being at the preffered place with the partner. For instance, for Adam chossing \"soccer\" (+2 points to satisfaction), Beth's best response would be \"soccer\" as well (+1 point to satisfaction). For Beth choosing \"ballet\" (+2 points), Adam's best responce is \"ballet\" (+1 point). If both decide to visit different places, both of them would be dissatisfied. The table below depicts payoffs for the dicsussed situations. \n",
        "\n",
        "\n",
        "\n"
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        "![Battle of Sexes.png](data:image/png;base64,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)"
      ],
      "metadata": {
        "id": "becWqN5kbFD4"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "We further assume that Adam and Beth can not negotiate preplay. \n",
        "\n",
        "The Battle of the Sexes is the game with more than one Nash  equlibrium that exists in pure and mixed strategies. Because of the current quantum hardware limitations, we shall explore Nash equilibrium at pure strategy only. \n",
        "\n",
        "As we can see from the table, the Nash equlibrium reaches at \"soccer - soccer\" or \"ballet-ballet\" with the total payoff in terms of satisfaction of 3.\n",
        "\n",
        "Let $P$ represent Adam's choise of strategy with the payoff matrix $M$.\n",
        "\n",
        "\\begin{align}\n",
        "        \\ P = \\begin{pmatrix}\n",
        "        \\ p_1  \\\\\n",
        "        \\ p_2\n",
        "        \\end{pmatrix}.\n",
        "    \\end{align}\n",
        "\n",
        "  \\begin{align}\n",
        "        \\ M = \\begin{pmatrix}\n",
        "        \\ 2 & \\ -1 \\\\\n",
        "        \\ -1 & \\ 1 \n",
        "        \\end{pmatrix}.\n",
        "    \\end{align} \\\\\n",
        "And $Q$ represents Beth's choise of startegy with the payoff matrix $N$\n",
        "\\begin{align}\n",
        "        \\ Q = \\begin{pmatrix}\n",
        "        \\ q_1  \\\\\n",
        "        \\ q_2\n",
        "        \\end{pmatrix}.\n",
        "    \\end{align} \n",
        "\n",
        "  \\begin{align}\n",
        "        \\ N = \\begin{pmatrix}\n",
        "        \\ 1 & \\ -1 \\\\\n",
        "        \\ -1 & \\ 2 \n",
        "        \\end{pmatrix}.\n",
        "    \\end{align}\n",
        "\n",
        "  To find equilibrium points $P$ and $Q$ (e.g. the vector of pure strategies) that yeild the optimum payoffs for both players we shall solve:\n",
        "\n",
        "$ max: F = P^T (M + N) Q$ \\\\\n",
        "\n",
        "Taking into account that DWave Quantum annealer minimizes an objective function $F$, we shall consider $-F$.\n",
        "\n",
        "$ min: -F = -1 * \\begin{pmatrix}\n",
        "        \\ p_1 & p_2\\end{pmatrix} * \\begin{align}\n",
        "                                    \\begin{pmatrix}\n",
        "                                  \\ 3 & \\ -2 \\\\\\\\ -2 & \\ 3 \n",
        "                                  \\end{pmatrix}\\end{align} * \\begin{pmatrix} \\ q_1  \\\\\\ q_2\\end{pmatrix} + \\alpha + \\beta$\n",
        "\n",
        "subject to: \n",
        "\n",
        "$ 2q_1-q_2 - \\alpha \\leq 0 $ \\\\\n",
        "\n",
        "$ -q_1 + q_2 - \\alpha \\leq 0 $ \\\\\n",
        "\n",
        "$ p_1- p_2 - \\beta \\leq 0 $ \\\\\n",
        "\n",
        "$-p_1 + 2p_2 - \\beta \\leq  0 $ \\\\\n",
        "\n",
        "$p_1 + p_2 -1 = 0$ \\\\\n",
        "\n",
        "$q_1 + q_2 -1 = 0$ \\\\\n",
        "\n",
        "The condition $p_i,q_i \\geq 0$ is satisfied since $p$ and $q$ are binary: $ (p,q \\in {0,1}) $\n",
        "\n",
        "\n"
      ],
      "metadata": {
        "id": "zjdIw0P3a_le"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "#modules installation\n",
        "!python -m pip install dwave-ocean-sdk\n",
        "!python -m pip install dwave-neal"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "RB_rZUhlHnOL",
        "outputId": "9046ece9-2f87-429f-807c-02214ba73ab7"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Collecting dwave-ocean-sdk\n",
            "  Downloading dwave_ocean_sdk-4.5.0-py3-none-any.whl (8.6 kB)\n",
            "Collecting minorminer==0.2.7.post0\n",
            "  Downloading minorminer-0.2.7.post0-cp37-cp37m-manylinux_2_12_x86_64.manylinux2010_x86_64.whl (5.0 MB)\n",
            "\u001b[K     |████████████████████████████████| 5.0 MB 5.4 MB/s \n",
            "\u001b[?25hCollecting dwave-inspector==0.2.10\n",
            "  Downloading dwave_inspector-0.2.10-py3-none-any.whl (26 kB)\n",
            "Collecting dwave-hybrid==0.6.6\n",
            "  Downloading dwave_hybrid-0.6.6-py3-none-any.whl (69 kB)\n",
            "\u001b[K     |████████████████████████████████| 69 kB 3.3 MB/s \n",
            "\u001b[?25hCollecting penaltymodel-lp==0.1.6\n",
            "  Downloading penaltymodel_lp-0.1.6-py3-none-any.whl (7.3 kB)\n",
            "Collecting dwave-greedy==0.2.3\n",
            "  Downloading dwave_greedy-0.2.3-cp37-cp37m-manylinux_2_17_x86_64.manylinux2014_x86_64.whl (655 kB)\n",
            "\u001b[K     |████████████████████████████████| 655 kB 11.8 MB/s \n",
            "\u001b[?25hCollecting dwave-neal==0.5.9\n",
            "  Downloading dwave_neal-0.5.9-cp37-cp37m-manylinux_2_5_x86_64.manylinux1_x86_64.manylinux_2_12_x86_64.manylinux2010_x86_64.whl (497 kB)\n",
            "\u001b[K     |████████████████████████████████| 497 kB 16.7 MB/s \n",
            "\u001b[?25hCollecting dwave-tabu==0.4.4\n",
            "  Downloading dwave_tabu-0.4.4-cp37-cp37m-manylinux_2_17_x86_64.manylinux2014_x86_64.whl (696 kB)\n",
            "\u001b[K     |████████████████████████████████| 696 kB 40.1 MB/s \n",
            "\u001b[?25hCollecting dwavebinarycsp==0.1.4\n",
            "  Downloading dwavebinarycsp-0.1.4-py3-none-any.whl (35 kB)\n",
            "Collecting dwave-cloud-client==0.9.5\n",
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            "\u001b[K     |████████████████████████████████| 108 kB 41.1 MB/s \n",
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            "  Downloading penaltymodel_mip-0.2.6-py3-none-any.whl (7.8 kB)\n",
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            "\u001b[K     |████████████████████████████████| 13.2 MB 13.7 MB/s \n",
            "\u001b[?25hCollecting dwave-preprocessing==0.3.2\n",
            "  Downloading dwave_preprocessing-0.3.2-cp37-cp37m-manylinux_2_12_x86_64.manylinux2010_x86_64.whl (935 kB)\n",
            "\u001b[K     |████████████████████████████████| 935 kB 42.2 MB/s \n",
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            "  Downloading pyqubo-1.2.0-cp37-cp37m-manylinux_2_17_x86_64.manylinux2014_x86_64.whl (229 kB)\n",
            "\u001b[K     |████████████████████████████████| 229 kB 39.7 MB/s \n",
            "\u001b[?25hCollecting penaltymodel==0.16.6\n",
            "  Downloading penaltymodel-0.16.6-py3-none-any.whl (13 kB)\n",
            "Collecting dwave-qbsolv==0.3.4\n",
            "  Downloading dwave_qbsolv-0.3.4-cp37-cp37m-manylinux_2_17_x86_64.manylinux2014_x86_64.whl (271 kB)\n",
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            "  Downloading plucky-0.4.3-py2.py3-none-any.whl (10 kB)\n",
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            "Collecting fasteners\n",
            "  Downloading fasteners-0.17.3-py3-none-any.whl (18 kB)\n",
            "Collecting rectangle-packer>=2.0.1\n",
            "  Downloading rectangle_packer-2.0.1-cp37-cp37m-manylinux2010_x86_64.whl (246 kB)\n",
            "\u001b[K     |████████████████████████████████| 246 kB 48.1 MB/s \n",
            "\u001b[?25hCollecting scipy\n",
            "  Downloading scipy-1.7.3-cp37-cp37m-manylinux_2_12_x86_64.manylinux2010_x86_64.whl (38.1 MB)\n",
            "\u001b[K     |████████████████████████████████| 38.1 MB 1.2 MB/s \n",
            "\u001b[?25hCollecting ortools<10.0.0,>=6.6.4659\n",
            "  Downloading ortools-9.3.10497-cp37-cp37m-manylinux_2_17_x86_64.manylinux2014_x86_64.whl (15.5 MB)\n",
            "\u001b[K     |████████████████████████████████| 15.5 MB 37.2 MB/s \n",
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            "Collecting Deprecated>=1.2.12\n",
            "  Downloading Deprecated-1.2.13-py2.py3-none-any.whl (9.6 kB)\n",
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            "Requirement already satisfied: typing-extensions>=3.6.4 in /usr/local/lib/python3.7/dist-packages (from importlib-metadata>=1.0->dwave-greedy==0.2.3->dwave-ocean-sdk) (4.2.0)\n",
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            "Requirement already satisfied: absl-py>=0.13 in /usr/local/lib/python3.7/dist-packages (from ortools<10.0.0,>=6.6.4659->penaltymodel-mip==0.2.6->dwave-ocean-sdk) (1.0.0)\n",
            "Collecting protobuf>=3.19.4\n",
            "  Downloading protobuf-3.20.1-cp37-cp37m-manylinux_2_5_x86_64.manylinux1_x86_64.whl (1.0 MB)\n",
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            "Requirement already satisfied: urllib3!=1.25.0,!=1.25.1,<1.26,>=1.21.1 in /usr/local/lib/python3.7/dist-packages (from requests[socks]>=2.18->dwave-cloud-client==0.9.5->dwave-ocean-sdk) (1.24.3)\n",
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            "Requirement already satisfied: PySocks!=1.5.7,>=1.5.6 in /usr/local/lib/python3.7/dist-packages (from requests[socks]>=2.18->dwave-cloud-client==0.9.5->dwave-ocean-sdk) (1.7.1)\n",
            "Installing collected packages: pyparsing, dimod, scipy, rectangle-packer, pydantic, plucky, homebase, fasteners, dwave-networkx, diskcache, protobuf, penaltymodel, minorminer, dwave-tabu, dwave-preprocessing, dwave-cloud-client, penaltymodel-lp, penaltymodel-cache, ortools, dwave-system, dwave-neal, dwave-greedy, Deprecated, pyqubo, penaltymodel-mip, dwavebinarycsp, dwave-qbsolv, dwave-inspector, dwave-hybrid, dwave-ocean-sdk\n",
            "  Attempting uninstall: pyparsing\n",
            "    Found existing installation: pyparsing 3.0.8\n",
            "    Uninstalling pyparsing-3.0.8:\n",
            "      Successfully uninstalled pyparsing-3.0.8\n",
            "  Attempting uninstall: scipy\n",
            "    Found existing installation: scipy 1.4.1\n",
            "    Uninstalling scipy-1.4.1:\n",
            "      Successfully uninstalled scipy-1.4.1\n",
            "  Attempting uninstall: protobuf\n",
            "    Found existing installation: protobuf 3.17.3\n",
            "    Uninstalling protobuf-3.17.3:\n",
            "      Successfully uninstalled protobuf-3.17.3\n",
            "\u001b[31mERROR: pip's dependency resolver does not currently take into account all the packages that are installed. This behaviour is the source of the following dependency conflicts.\n",
            "tensorflow 2.8.0 requires tf-estimator-nightly==2.8.0.dev2021122109, which is not installed.\n",
            "albumentations 0.1.12 requires imgaug<0.2.7,>=0.2.5, but you have imgaug 0.2.9 which is incompatible.\u001b[0m\n",
            "Successfully installed Deprecated-1.2.13 dimod-0.10.17 diskcache-5.4.0 dwave-cloud-client-0.9.5 dwave-greedy-0.2.3 dwave-hybrid-0.6.6 dwave-inspector-0.2.10 dwave-neal-0.5.9 dwave-networkx-0.8.12 dwave-ocean-sdk-4.5.0 dwave-preprocessing-0.3.2 dwave-qbsolv-0.3.4 dwave-system-1.13.1 dwave-tabu-0.4.4 dwavebinarycsp-0.1.4 fasteners-0.17.3 homebase-1.0.1 minorminer-0.2.7.post0 ortools-9.3.10497 penaltymodel-0.16.6 penaltymodel-cache-0.4.5 penaltymodel-lp-0.1.6 penaltymodel-mip-0.2.6 plucky-0.4.3 protobuf-3.20.1 pydantic-1.9.0 pyparsing-2.4.7 pyqubo-1.2.0 rectangle-packer-2.0.1 scipy-1.7.3\n",
            "Requirement already satisfied: dwave-neal in /usr/local/lib/python3.7/dist-packages (0.5.9)\n",
            "Requirement already satisfied: numpy<2.0.0,>=1.19.1 in /usr/local/lib/python3.7/dist-packages (from dwave-neal) (1.21.6)\n",
            "Requirement already satisfied: dimod>=0.9.11 in /usr/local/lib/python3.7/dist-packages (from dwave-neal) (0.10.17)\n",
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          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "from dimod import Binary, BQM, ExactSolver\n",
        "import matplotlib.pyplot as plt\n",
        "import dimod, re\n",
        "import neal"
      ],
      "metadata": {
        "id": "8hNFlbqWkRRu"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "#Contracting QUBO Matrix\n",
        "Q = {('p1', 'q1'):-3, ('p1', 'q2'):2, ('p2', 'q1'):2, ('p2', 'q2'):-3}\n",
        "\n",
        "bqm = BQM(Q, vartype='BINARY')\n",
        "\n",
        "# constraints on probability\n",
        "bqm.add_linear_equality_constraint(\n",
        "    [('p1',1), ('p2', 1)], constant=-1, lagrange_multiplier=10\n",
        ")\n",
        "bqm.add_linear_equality_constraint(\n",
        "    [('q1',1), ('q2', 1)],  constant=-1, lagrange_multiplier=10\n",
        ")\n",
        "\n",
        "# inequality constraints\n",
        "# might be irrelevant for this problem\n",
        "bqm.add_linear_inequality_constraint(\n",
        "    [('q1', 2), ('q2', -1)], lb=-1, ub=2, lagrange_multiplier=1, label = 'constr1'\n",
        ")\n",
        "bqm.add_linear_inequality_constraint(\n",
        "    [('q1', -1), ('q2', 1)], lb=-1, ub=1, lagrange_multiplier=1, label = 'constr2'\n",
        ")\n",
        "bqm.add_linear_inequality_constraint(\n",
        "    [('p1', 1), ('p2', -1)], lb=-1, ub=1, lagrange_multiplier=1, label = 'constr3'\n",
        ")\n",
        "bqm.add_linear_inequality_constraint(\n",
        "    [('p1', 1), ('p2', 2)], lb=-1, ub=2, lagrange_multiplier=1, label = 'constr4'\n",
        ")\n",
        "\n",
        "sampler = neal.SimulatedAnnealingSampler()\n",
        "sampleset = sampler.sample(bqm, num_reads=5000)\n",
        "sampleset.lowest()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "AZmOuL8j64Go",
        "outputId": "7b7befb9-ce09-45a9-f4b1-3b3527623b54",
        "cellView": "code"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "/usr/local/lib/python3.7/dist-packages/dimod/binary/binary_quadratic_model.py:760: UserWarning: Did not add constraint constr1. This constraint is feasible with any value for state variables.\n",
            "  f'Did not add constraint {label}.'\n",
            "/usr/local/lib/python3.7/dist-packages/dimod/binary/binary_quadratic_model.py:760: UserWarning: Did not add constraint constr2. This constraint is feasible with any value for state variables.\n",
            "  f'Did not add constraint {label}.'\n",
            "/usr/local/lib/python3.7/dist-packages/dimod/binary/binary_quadratic_model.py:760: UserWarning: Did not add constraint constr3. This constraint is feasible with any value for state variables.\n",
            "  f'Did not add constraint {label}.'\n"
          ]
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "SampleSet(rec.array([([0, 1, 0, 1, 0, 0], -3., 1), ([1, 0, 1, 0, 0, 1], -3., 1),\n",
              "           ([1, 0, 1, 0, 1, 0], -3., 1), ...,\n",
              "           ([1, 0, 1, 0, 1, 0], -3., 1), ([1, 0, 1, 0, 0, 1], -3., 1),\n",
              "           ([1, 0, 1, 0, 1, 0], -3., 1)],\n",
              "          dtype=[('sample', 'i1', (6,)), ('energy', '<f8'), ('num_occurrences', '<i8')]), Variables(['p1', 'p2', 'q1', 'q2', 'slack_constr4_0', 'slack_constr4_1']), {'beta_range': [0.07701635339554948, 18.420680743952367], 'beta_schedule_type': 'geometric'}, 'BINARY')"
            ]
          },
          "metadata": {},
          "execution_count": 3
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "sampleset.lowest()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "LA35guSHOipD",
        "outputId": "0833cbb3-4e5f-4b16-d85b-1bcf4bebc9fe"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "SampleSet(rec.array([([0, 1, 0, 1, 0, 0], -3., 1), ([1, 0, 1, 0, 0, 1], -3., 1),\n",
              "           ([1, 0, 1, 0, 1, 0], -3., 1), ...,\n",
              "           ([1, 0, 1, 0, 1, 0], -3., 1), ([1, 0, 1, 0, 0, 1], -3., 1),\n",
              "           ([1, 0, 1, 0, 1, 0], -3., 1)],\n",
              "          dtype=[('sample', 'i1', (6,)), ('energy', '<f8'), ('num_occurrences', '<i8')]), Variables(['p1', 'p2', 'q1', 'q2', 'slack_constr4_0', 'slack_constr4_1']), {'beta_range': [0.07701635339554948, 18.420680743952367], 'beta_schedule_type': 'geometric'}, 'BINARY')"
            ]
          },
          "metadata": {},
          "execution_count": 4
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# plot results\n",
        "\n",
        "def plot_sampleset(sampleset):\n",
        "    '''Plots results of sampling in a bar style.\n",
        "        Arg: sampleset: a set of samples prodused by DWave's sampler\n",
        "    '''\n",
        "        \n",
        "    keep_vars =['p1', 'p2', 'q1', 'q2']\n",
        "    sampleset = dimod.keep_variables(sampleset, keep_vars)\n",
        "    sampleset = sampleset.aggregate()\n",
        "\n",
        "    df = sampleset.to_pandas_dataframe()\n",
        "\n",
        "    strategies = {}\n",
        "    for i, row in df.iterrows():\n",
        "        strategy = str(df['p1'][i])+str(df['p2'][i])+str(df['q1'][i])+str(df['q2'][i])\n",
        "        if len((re.findall('1', strategy))) <=2:\n",
        "            strategies[strategy]=row['num_occurrences']\n",
        "    \n",
        "    myList = strategies.items()\n",
        "    myList = [('p=(' + item[0][:1]+','+ item[0][1:2]+'), q=('+item[0][2:3]+','+ item[0][3:] +')', item[1]) for item in myList]\n",
        "    myList = sorted(myList) \n",
        "    x, y = zip(*myList) \n",
        "\n",
        "    plt.bar(x,y, color='Blue')\n",
        "    plt.title('Frequency of pure NE points')\n",
        "    plt.xlabel('Pure Strategy')\n",
        "    plt.ylabel('Frequency')\n",
        "    plt.savefig('Frequency_2strategies.png', dpi=600)\n",
        "    plt.show()\n",
        "\n",
        "\n",
        "plot_sampleset(sampleset=sampleset)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 295
        },
        "id": "Cdt6eIDyogsQ",
        "outputId": "ab2b2687-41b9-42d4-f88a-e6458d0caf6c"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": "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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Two players Game with three pure Nash Equilibrium points"
      ],
      "metadata": {
        "id": "T_tIDvGDp7uZ"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "We shall explore the application of quantum annealers on the following biological example discussed in the short [overview on game theory](https://www.britannica.com/science/game-theory/The-prisoners-dilemma). The table below ellustrates the payoffs when two male birds confront each other when competing for resources. The birds can behave as 'hawks' fighting for the resources or as 'doves' avoiding any serious harm. In any case, birds will lose the competition since a species containing only 'hawks' would have a high casuality rate, where spicies containing 'doves' only can be vulnerable to an invasions or mutations. Hence, the third type of behavior called 'bourgeois' would be more stable then pure 'hawks' or pure 'dove' strategy. A 'bourgeois' can act like a 'hawk' or a 'dove' depending on an external situation. \n",
        "\n",
        "![picture](https://cdn.britannica.com/56/91956-050-AAAD7DA0/retreat-behaviour-attack-Bourgeois-strategy-population-individuals.jpg)"
      ],
      "metadata": {
        "id": "D6p7-bBTqlLz"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "So, what combinations of types are stable and most likely to evolve? Let $P$ represent the strategy choise of the first player with the payoff matrix $M$. In this case the strategy vector represents a collection of choises $(hawk, dove, bourgeois)$. The payoff can be thought of as a number of offspring a species produces.  \n",
        " \n",
        "\\begin{align}\n",
        "        \\ P = \\begin{pmatrix}\n",
        "        \\ p_1  \\\\\n",
        "        \\ p_2 \\\\\n",
        "        \\ p_3 \\\\\n",
        "        \\end{pmatrix}.\n",
        "    \\end{align}\n",
        "\n",
        "  \\begin{align}\n",
        "        M=\\begin{pmatrix}\n",
        "\\ -5 & \\ 10  & \\ 2.5\\\\\n",
        "\\ 0 & \\ 2 & \\  1 \\\\\n",
        "\\ -2.5 & \\ 6 & \\ 5 \\\\\n",
        "        \\end{pmatrix}.\n",
        "    \\end{align} \\\\\n",
        "\n",
        "And the $Q$ matrix represents the choise of strategy of the second player with the payoff matrix $N$\n",
        "\\begin{align}\n",
        "        \\ Q = \\begin{pmatrix}\n",
        "        \\ q_1  \\\\\n",
        "        \\ q_2 \\\\\n",
        "        \\ q_3 \\\n",
        "        \\end{pmatrix}.\n",
        "    \\end{align} \n",
        "\n",
        "  \\begin{align}\n",
        "        \\ N = \\begin{pmatrix}\n",
        "        \\ -5 & \\ 0 & \\ -2.5 \\\\\n",
        "\\ 10 & \\ 2 & \\ 6 \\\\\n",
        "\\ 2.5 & \\ 1 & \\ 5 \\\\\n",
        "        \\end{pmatrix}.\n",
        "    \\end{align}\n",
        "\n",
        "Adding tow matrices and considering to minimize the objective function, we get\n",
        "\n",
        "\n",
        "$ min: -F = -1 * \\begin{pmatrix}\n",
        "        \\ p_1 & p_2 & p_3\\end{pmatrix} * \\begin{align}\n",
        "                                    \\begin{pmatrix}\n",
        "                                  \\ -10 & \\ 10 & \\ 0 \\\\\\\\ 10 & \\ 4 & \\ 7 \n",
        "                                  \\\\\\\\ 0 & \\ 7 & \\ 10 \\ \n",
        "                                  \\end{pmatrix}\\end{align} * \\begin{pmatrix} \\ q_1  \\\\\\ q_2 \\\\\\ q_3 \\end{pmatrix} + \\alpha + \\beta$\n",
        "\n",
        "subject to: \n",
        "\n",
        "$ -5q_1 + 10q_2 + 2.5q_3 - \\alpha \\leq 0 $ \\\\\n",
        "\n",
        "$ 2q_2 + q_3 - \\alpha \\leq 0 $ \\\\\n",
        "\n",
        "$ -2.5q_1 + 6q_2 + 5q_3 - \\alpha \\leq 0 $ \\\\\n",
        "\n",
        "$ -5p_1 -2.5p_3 - \\beta \\leq 0 $ \\\\\n",
        "\n",
        "$ 10p_1+ 2p_2 + 6p_3 - \\beta \\leq 0 $ \\\\\n",
        "\n",
        "$ 2.5p_1+ p_2 + 5p_3 - \\beta \\leq 0 $ \\\\\n",
        "\n",
        "$ p_1 + p_2 + p_3 - 1 = 0$ \\\\\n",
        "\n",
        "$ q_1 + q_2 + q_3 - 1 = 0$ \\\\\n",
        "\n",
        "The condition $p_i,q_i \\geq 0$ is satisfied since $p$ and $q$ are binary: $ (p,q \\in (0,1)) $\n",
        "\n"
      ],
      "metadata": {
        "id": "gQbzzmOirHtL"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "#Constracting QUBO Matrix\n",
        "Q = {('p1', 'q1'):10, ('p1', 'q2'):-10, ('p2', 'q2'):-4, ('p2', 'q3'):-7, ('p3', 'q3'):-10}\n",
        "\n",
        "bqm = BQM(Q, vartype='BINARY')\n",
        "\n",
        "# constraints on probability\n",
        "bqm.add_linear_equality_constraint([('p1',1), ('p2', 1), ('p3', 1)], constant=-1, lagrange_multiplier=10)\n",
        "bqm.add_linear_equality_constraint([('q1',1), ('q2', 1), ('q3', 1)],  constant=-1, lagrange_multiplier=10)\n",
        "\n",
        "# # inequality constraints\n",
        "bqm.add_linear_inequality_constraint([('q1', -2), ('q2', 4), ('q3', 1)], lb=-2, ub=4, lagrange_multiplier=1, label = 'constr1')\n",
        "bqm.add_linear_inequality_constraint([('q2', 2), ('q3', 1)], lb=1, ub=2, lagrange_multiplier=1, label = 'constr2')\n",
        "bqm.add_linear_inequality_constraint([('q1', -5), ('q2', 12), ('q3', 10)], lb=-5, ub=12, lagrange_multiplier=1, label = 'constr3')\n",
        "bqm.add_linear_inequality_constraint([('p1', -2), ('p3', -1)], lb=-2, ub=-1, lagrange_multiplier=1, label = 'constr4')\n",
        "bqm.add_linear_inequality_constraint([('p1', -5), ('p2', 1), ('p3', 3)], lb=-5, ub=3, lagrange_multiplier=1, label = 'constr5')\n",
        "bqm.add_linear_inequality_constraint([('p1', 5), ('p2', 2), ('p3', 10)], lb=2, ub=10, lagrange_multiplier=1, label = 'constr6')\n",
        "\n",
        "sampler = neal.SimulatedAnnealingSampler()\n",
        "sampleset = sampler.sample(bqm, num_reads=5000)\n",
        "sampleset.lowest()"
      ],
      "metadata": {
        "id": "Z-2vGUCp6Ot9",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "bb0c8d08-eedc-4fcd-cbce-44696132e850"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "SampleSet(rec.array([([1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0], -10., 1),\n",
              "           ([1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0], -10., 1),\n",
              "           ([0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], -10., 1),\n",
              "           ...,\n",
              "           ([0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], -10., 1),\n",
              "           ([1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1], -10., 1),\n",
              "           ([1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0], -10., 1)],\n",
              "          dtype=[('sample', 'i1', (24,)), ('energy', '<f8'), ('num_occurrences', '<i8')]), Variables(['p1', 'p2', 'p3', 'q1', 'q2', 'q3', 'slack_constr1_0', 'slack_constr1_1', 'slack_constr1_2', 'slack_constr2_0', 'slack_constr3_0', 'slack_constr3_1', 'slack_constr3_2', 'slack_constr3_3', 'slack_constr3_4', 'slack_constr4_0', 'slack_constr5_0', 'slack_constr5_1', 'slack_constr5_2', 'slack_constr5_3', 'slack_constr6_0', 'slack_constr6_1', 'slack_constr6_2', 'slack_constr6_3']), {'beta_range': [0.002484398496630628, 9.210340371976184], 'beta_schedule_type': 'geometric'}, 'BINARY')"
            ]
          },
          "metadata": {},
          "execution_count": 6
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "\n",
        "def plot_sampleset3(sampleset):\n",
        "    '''Plots results of sampling in a bar style.\n",
        "        Arg: sampleset: a set of samples prodused by DWave's sampler\n",
        "    '''\n",
        "    \n",
        "    keep_vars =['p1', 'p2', 'p3', 'q1', 'q2', 'q3']\n",
        "    sampleset = dimod.keep_variables(sampleset, keep_vars)\n",
        "    sampleset = sampleset.aggregate()\n",
        "\n",
        "    df = sampleset.to_pandas_dataframe()\n",
        "\n",
        "    strategies = {}\n",
        "    for i, row in df.iterrows():\n",
        "    #     print(row['num_occurrences'])\n",
        "    #     break\n",
        "        strategy = str(df['p1'][i])+str(df['p2'][i])+str(df['p3'][i])+str(df['q1'][i]) +str(df['q2'][i])+str(df['q3'][i])\n",
        "        if len((re.findall('1', strategy))) <=2:\n",
        "            strategies[strategy]=row['num_occurrences']\n",
        "\n",
        "    myList = strategies.items()\n",
        "    myList = [('p=('+item[0][:1]+','+ item[0][1:2]+','+item[0][2:3]+'), q=('+item[0][3:4]+','+ item[0][4:5]+','+ item[0][5:] +')', item[1]) for item in myList]\n",
        "    myList = sorted(myList) \n",
        "    x, y = zip(*myList) \n",
        "\n",
        "    # plt.bar(x,y, color='Blue')\n",
        "    plt.bar(x,y, color='Blue')\n",
        "    plt.title('Frequency of pure NE points')\n",
        "    plt.xlabel('Pure Strategy')\n",
        "    plt.xticks(rotation = 60, horizontalalignment=\"right\", rotation_mode=\"anchor\") # Rotates X-Axis Ticks by 60-degrees\n",
        "    plt.ylabel('Frequency')\n",
        "    plt.savefig('Frequency_3strategies.png', dpi=600)\n",
        "    plt.show()\n",
        "\n",
        "\n",
        "plot_sampleset3(sampleset=sampleset)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 378
        },
        "id": "PGLMC-hFo4wz",
        "outputId": "4ae8cbca-b5e0-4a5a-9ae6-b42876909cf6"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        ""
      ],
      "metadata": {
        "id": "35YwKaP0Lib-"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "# 8 by 8 Example\n"
      ],
      "metadata": {
        "id": "id2cFKDe4_xQ"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Consider an example with eight strategies for each\n",
        "player, generated a from a dynamic, finite automata version\n",
        "of the game Prisoner’s Dilemma in which the notion of long\n",
        "term stability of a Nash equilibrium is of interest. This notion\n",
        "is referred to as evolutionarily stable strategy (ESS) because\n",
        "as the game evolves over time, an ESS Nash equilibrium will\n",
        "be robust against invasion by mutant strategies. This game is\n",
        "described by the following payoff matrices:"
      ],
      "metadata": {
        "id": "BOFbFIVdZs4H"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "![WhatsApp Image 2022-04-22 at 1.57.23 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      ],
      "metadata": {
        "id": "lHx32eXU5DBC"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "The matrix $M$ for the first player with the strategy vector $P = p_i$ where $ 1 \\le i  \\le 8, i \\in \\mathbf{Z}$:\n",
        "\n",
        "\\begin{align}\n",
        "        \\ M = \\begin{pmatrix}\n",
        "        \\ 2 & \\ 3 & \\ 2 & \\ 2 & 3 & \\ 2 & \\ 2 & 3\\\\\n",
        "        \\ -1 & \\ 0 & \\ 0 & \\ 0 & -1/2 & \\ -1/2 & \\ -1/2 & -1\\\\\n",
        "        \\ 2 & \\ 0 & \\ 2 & \\ 2 & 0 & \\ 2 & \\ 2 & -1 \\\\\n",
        "        \\ 2 & \\ 0 & \\ 2 & \\ 2 & 2/3 & \\ 2 & \\ 2 & 2 \\\\\n",
        "        \\ -1 & \\ 3/2 & \\ 0 & \\ 2/3 & 2 & \\ 2 & \\ 2 & 2 \\\\\n",
        "        \\ 2 & \\ 3/2 & \\ 2 & \\ 2 & 2 & \\ 2 & \\ 2 & 2\\\\\n",
        "        \\ 2 & \\ 3/2 & \\ 2 & \\ 2 & 2 & \\ 2 & \\ 2 & 2\\\\\n",
        "        \\ -1 & \\ 3 & \\ 3 & \\ 2 & 2 & \\ 2 & \\ 2 & 2\\\\\n",
        "        \\end{pmatrix}.\n",
        "    \\end{align}\n",
        "\n",
        "\n",
        "And the matrix $N$ for the second player  with the strategy vector $Q = q_j$ where $ 1 \\le j  \\le 8, i \\in \\mathbf{Z}$:\n",
        "  \\begin{align}\n",
        "        \\ N = \\begin{pmatrix}\n",
        "        \\ 2 & \\ -1 & \\ 2 & \\ 2 & -1 & \\ 2 & \\ 2 & -1\\\\\n",
        "        \\ 3 & \\ 0 & \\ 0 & \\ 0 & 3/2 & \\ 3/2 & \\ 3/2 & 3\\\\\n",
        "        \\ 2 & \\ 0 & \\ 2 & \\ 2 & 0 & \\ 2 & \\ 2 & 3 \\\\\n",
        "        \\ 2 & \\ 0 & \\ 2 & \\ 2 & 2/3 & \\ 2 & \\ 2 & 2 \\\\\n",
        "        \\ 3 & \\ -1/2 & \\ 0 & \\ 2/3 & 2 & \\ 2 & \\ 2 & 2 \\\\\n",
        "        \\ 2 & \\ -1/2 & \\ 2 & \\ 2 & 2 & \\ 2 & \\ 2 & 2\\\\\n",
        "        \\ 2 & \\ -1/2 & \\ 2 & \\ 2 & 2 & \\ 2 & \\ 2 & 2\\\\\n",
        "        \\ 3 & \\ -1 & \\ -1 & \\ 2 & 2 & \\ 2 & \\ 2 & 2\\\\\n",
        "        \\end{pmatrix}.\n",
        "  \\end{align}\n",
        "\n",
        "\n",
        "  Summing up the matrices for both players and setting up in the QUBO format:\n",
        "\n",
        "\n",
        "$min: \n",
        "\\begin{align} -1*(P)* \n",
        "\\begin{pmatrix}\n",
        "    \\ 4 & \\ 2 & \\ 4 & \\ 4 & 2 &  \\ 4 & \\ 4 & 2 \\\\\n",
        "    \\ 2 & \\ 0 & \\ 0 & \\ 0 & 1 &  \\ 1 & \\ 1 & 2 \\\\\n",
        "    \\ 4 & \\ 0 & \\ 4 & \\ 4 & 0 &  \\ 4 & \\ 4 & 2 \\\\\n",
        "    \\ 4 & \\ 0 & \\ 4 & \\ 4 & 4/3 &\\ 4 & \\ 4 & 4 \\\\\n",
        "    \\ 2 & \\ 1 & \\ 0 & \\ 4/3 & 4 &\\ 4 & \\ 4 & 4 \\\\\n",
        "    \\ 4 & \\ 1 & \\ 4 & \\ 4 & 4 &  \\ 4 & \\ 4 & 4\\\\\n",
        "    \\ 4 & \\ 1 & \\ 4 & \\ 4 & 4 &  \\ 4 & \\ 4 & 4\\\\\n",
        "    \\ 2 & \\ 2 & \\ 2 & \\ 4 & 4 &  \\ 4 & \\ 4 & 4\\\\\n",
        "\\end{pmatrix} *(Q) + \\alpha + \\beta \n",
        "\\end{align} $"
      ],
      "metadata": {
        "id": "amBhySEz5JU9"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "#Constracting QUBO Matrix\n",
        "Q = {('p1','q1'):-12, ('p1','q2'):-6, ('p1','q3'):-12, ('p1','q4'):-12, ('p1','q5'):-6, ('p1','q6'):-12, ('p1','q7'):-12, ('p1', 'q8'):-6,\n",
        "     ('p2','q5'):-3, ('p2','q6'):-3, ('p2','q7'):-3, ('p2', 'q8'):-3,\n",
        "     ('p3','q3'):-12, ('p3','q4'):-12, ('p3','q6'):-12, ('p3','q7'):-12, ('p3', 'q8'):-6,\n",
        "     ('p4','q4'):-12, ('p4','q5'):-4, ('p4','q6'):-12, ('p4','q7'):-12, ('p4', 'q8'):-12,\n",
        "     ('p5','q5'):-12, ('p5','q6'):-12, ('p5','q7'):-12, ('p5', 'q8'):-12,\n",
        "     ('p6','q6'):-12, ('p6','q7'):-12, ('p6', 'q8'):-12,\n",
        "     ('p7','q7'):-12, ('p7', 'q8'):-12,\n",
        "     ('p8', 'q8'):-12}\n",
        "\n",
        "bqm = BQM(Q, vartype='BINARY')\n",
        "\n",
        "# constraints on probability\n",
        "bqm.add_linear_equality_constraint([('p1',1), ('p2', 1), ('p3', 1), ('p4',1), ('p5', 1), ('p6', 1), ('p7',1), ('p8', 1)], constant=-1, lagrange_multiplier=100)\n",
        "bqm.add_linear_equality_constraint([('q1',1), ('q2', 1), ('q3', 1), ('q4',1), ('q5', 1), ('q6', 1), ('q7',1), ('q8', 1)],  constant=-1, lagrange_multiplier=100)\n",
        "\n",
        "# # inequality constraints\n",
        "bqm.add_linear_inequality_constraint([('q1',2), ('q2',3), ('q3',2),('q4',2), ('q5', 3), ('q6', 2), ('q7', 2), ('q8', 3)], lb=2, ub=3, lagrange_multiplier=1, label = 'constr1')\n",
        "bqm.add_linear_inequality_constraint([('q1',-2), ('q5', -1), ('q6', -1), ('q7', -1), ('q8', -2)], lb=-2, ub=0, lagrange_multiplier=1, label = 'constr2')\n",
        "bqm.add_linear_inequality_constraint([('q1',2), ('q3',2),('q4',2), ('q6', 2), ('q7', 2), ('q8', -1)], lb=-1, ub=2, lagrange_multiplier=1, label = 'constr3')\n",
        "bqm.add_linear_inequality_constraint([('q1',3), ('q3',3),('q4',3), ('q5', 1), ('q6', 3), ('q7', 3), ('q8', 3)], lb=0, ub=3, lagrange_multiplier=1, label = 'constr4')\n",
        "bqm.add_linear_inequality_constraint([('q1',-6), ('q2',9),('q4',4), ('q5', 12), ('q6', 12), ('q7', 12), ('q8', 12)], lb=-6, ub=12, lagrange_multiplier=1, label = 'constr5')\n",
        "bqm.add_linear_inequality_constraint([('q1',4), ('q2',3), ('q3',4),('q4',4), ('q5', 4), ('q6', 4), ('q7', 4), ('q8', 4)], lb=3, ub=4, lagrange_multiplier=1, label = 'constr6')\n",
        "bqm.add_linear_inequality_constraint([('q1',4), ('q2',3), ('q3',4),('q4',4), ('q5', 4), ('q6', 4), ('q7', 4), ('q8', 4)], lb=3, ub=4, lagrange_multiplier=1, label = 'constr7')\n",
        "bqm.add_linear_inequality_constraint([('q1',-1), ('q2',3), ('q3',3),('q4',2), ('q5', 2), ('q6', 2), ('q7', 2), ('q8', 2)], lb=-1, ub=4, lagrange_multiplier=1, label = 'constr8')\n",
        "\n",
        "bqm.add_linear_inequality_constraint([('p1',2), ('p2',-1), ('p3',2), ('p4',2), ('p5',-1), ('p6',2), ('p7',2), ('p8',-1)], lb=-1, ub=2, lagrange_multiplier=1, label = 'constr9')\n",
        "bqm.add_linear_inequality_constraint([('p1',2), ('p5',1), ('p6',1), ('p7',1), ('p8',2)], lb=0, ub=2, lagrange_multiplier=1, label = 'constr10')\n",
        "bqm.add_linear_inequality_constraint([('p1',2), ('p3',2), ('p4',2), ('p6',2), ('p7',2), ('p8',3)], lb=0, ub=3, lagrange_multiplier=1, label = 'constr11')\n",
        "bqm.add_linear_inequality_constraint([('p1',2), ('p3',2), ('p4',2), ('p5',1), ('p6',2), ('p7',2), ('p8',2)], lb=0, ub=2, lagrange_multiplier=1, label = 'constr12')\n",
        "bqm.add_linear_inequality_constraint([('p1',18), ('p2',-3), ('p4',2), ('p5',12), ('p6',12), ('p7',12), ('p8',12)], lb=-3, ub=18, lagrange_multiplier=1, label = 'constr13')\n",
        "bqm.add_linear_inequality_constraint([('p1',4), ('p2',-1), ('p3',4), ('p4',4), ('p5',4), ('p6',4), ('p7',4), ('p8',4)], lb=-1, ub=4, lagrange_multiplier=1, label = 'constr14')\n",
        "bqm.add_linear_inequality_constraint([('p1',4), ('p2',-1), ('p3',4), ('p4',4), ('p5',4), ('p6',4), ('p7',4), ('p8',4)], lb=-1, ub=4, lagrange_multiplier=1, label = 'constr15')\n",
        "bqm.add_linear_inequality_constraint([('p1',3), ('p2',-1), ('p3',-1), ('p4',2), ('p5',2), ('p6',2), ('p7',2), ('p8',2)], lb=-1, ub=3, lagrange_multiplier=1, label = 'constr14')\n",
        "\n",
        "sampler = neal.SimulatedAnnealingSampler()\n",
        "sampleset = sampler.sample(bqm, num_reads=10000)"
      ],
      "metadata": {
        "id": "SU9j7mPF5AMU"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "print(\"Best solution {}\". format(sampleset.lowest()))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "KXTGe3vKda24",
        "outputId": "ea5e8b74-3ccb-4d5f-eec5-dc93224d4d86"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Best solution     p1 p2 p3 p4 p5 p6 p7 p8 q1 q2 q3 q4 q5 ... slack_constr9_1 energy num_oc.\n",
            "0    1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "1    1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "2    1  0  0  0  0  0  0  0  1  0  0  0  0 ...               0  -12.0       1\n",
            "3    1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "4    1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "5    1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "6    1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "7    1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "8    1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "9    1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "10   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "11   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "12   1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "13   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "14   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "15   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "16   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "17   1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "18   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "19   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "20   1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "21   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "22   1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "23   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "24   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "25   1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "26   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "27   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "28   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "29   1  0  0  0  0  0  0  0  1  0  0  0  0 ...               0  -12.0       1\n",
            "30   1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "31   1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "32   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "33   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "34   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "35   1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "36   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "37   1  0  0  0  0  0  0  0  1  0  0  0  0 ...               0  -12.0       1\n",
            "38   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "39   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "40   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "41   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "42   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "43   1  0  0  0  0  0  0  0  1  0  0  0  0 ...               0  -12.0       1\n",
            "44   1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "45   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "46   1  0  0  0  0  0  0  0  1  0  0  0  0 ...               0  -12.0       1\n",
            "47   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "48   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "49   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "50   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "51   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "52   1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "53   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "54   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "55   1  0  0  0  0  0  0  0  1  0  0  0  0 ...               0  -12.0       1\n",
            "56   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "57   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "58   1  0  0  0  0  0  0  0  1  0  0  0  0 ...               0  -12.0       1\n",
            "59   1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "60   1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "61   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "62   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "63   1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "64   1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "65   1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "66   1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "67   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "68   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "69   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "70   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "71   1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "72   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "73   1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "74   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "75   1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "76   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "77   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "78   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "79   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "80   1  0  0  0  0  0  0  0  1  0  0  0  0 ...               0  -12.0       1\n",
            "81   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "82   1  0  0  0  0  0  0  0  1  0  0  0  0 ...               0  -12.0       1\n",
            "83   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "84   1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "85   1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "86   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "87   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "88   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "89   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "90   1  0  0  0  0  0  0  0  1  0  0  0  0 ...               0  -12.0       1\n",
            "91   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "92   1  0  0  0  0  0  0  0  1  0  0  0  0 ...               0  -12.0       1\n",
            "93   1  0  0  0  0  0  0  0  1  0  0  0  0 ...               0  -12.0       1\n",
            "94   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "95   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "96   1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "97   1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "98   1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "99   1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "100  1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "101  1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "102  1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "103  1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "104  1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "105  1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "106  1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "107  1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "108  1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "109  1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "110  1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "111  1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "112  1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "113  1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "114  1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "115  1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "116  1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "117  1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "118  1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "119  1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "120  1  0  0  0  0  0  0  0  1  0  0  0  0 ...               0  -12.0       1\n",
            "121  1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "122  1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "123  1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "124  1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "125  1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "126  1  0  0  0  0  0  0  0  1  0  0  0  0 ...               0  -12.0       1\n",
            "127  1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "128  1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "129  1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "130  1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "131  1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "132  1  0  0  0  0  0  0  0  1  0  0  0  0 ...               0  -12.0       1\n",
            "133  1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "134  1  0  0  0  0  0  0  0  1  0  0  0  0 ...               0  -12.0       1\n",
            "135  1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "136  1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "137  1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "138  1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "139  1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "140  1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "141  1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "142  1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "143  1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "144  1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "145  1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "146  1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "147  1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "148  1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "149  1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "150  1  0  0  0  0  0  0  0  1  0  0  0  0 ...               0  -12.0       1\n",
            "151  1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "152  1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "153  1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "154  1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "155  1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "156  1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "157  1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "158  1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "159  1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "160  1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "161  1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
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            "238  1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "239  1  0  0  0  0  0  0  0  1  0  0  0  0 ...               0  -12.0       1\n",
            "240  1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "241  1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "242  1  0  0  0  0  0  0  0  1  0  0  0  0 ...               0  -12.0       1\n",
            "243  1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "244  1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "245  1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "246  1  0  0  0  0  0  0  0  0  0  1  0  0 ...               0  -12.0       1\n",
            "247  1  0  0  0  0  0  0  0  0  0  0  1  0 ...               0  -12.0       1\n",
            "248  1  0  0  0  0  0  0  0  0  0  0  0  0 ...               0  -12.0       1\n",
            "['BINARY', 249 rows, 249 samples, 52 variables]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "def plot_sampleset(sampleset, num_sample):\n",
        "    '''Plots results of sampling in a bar style.\n",
        "        Arg: sampleset: a set of samples prodused by DWave's sampler\n",
        "             num_sample: the number of samples received from DWave's sampler'''\n",
        "    \n",
        "    keep_vars =['p1', 'p2', 'p3', 'p4', 'p5', 'p6', 'p7', 'p8', 'q1', 'q2', 'q3',\n",
        "                'q4', 'q5', 'q6', 'q7', 'q8']\n",
        "    sampleset = dimod.keep_variables(sampleset, keep_vars)\n",
        "    sampleset = sampleset.aggregate()\n",
        "\n",
        "    df = sampleset.to_pandas_dataframe()\n",
        "\n",
        "    strategies = {}\n",
        "    for i, row in df.iterrows():\n",
        "        strategy = str(df['p1'][i])+str(df['p2'][i])+str(df['p3'][i]) \\\n",
        "                  +str(df['p4'][i])+str(df['p5'][i])+str(df['p6'][i]) \\\n",
        "                  +str(df['p7'][i])+str(df['p8'][i])+str(df['q1'][i]) \\\n",
        "                  +str(df['q2'][i])+str(df['q3'][i])+ str(df['q4'][i])+str(df['q5'][i]) \\\n",
        "                  +str(df['q6'][i])+str(df['q7'][i])+str(df['q8'][i])\n",
        "        if len((re.findall('1', strategy))) <=2:\n",
        "            strategies[strategy]=row['num_occurrences']\n",
        "\n",
        "     # drop lower that 50\n",
        "    strategies = {key:value for key, value in strategies.items() if value >= 50 }\n",
        "    \n",
        "    myList = strategies.items()\n",
        "    myList = [('p=('+item[0][:8]+'), q=('+item[0][8:] +')', item[1]) for item in myList]\n",
        "    myList = sorted(myList) \n",
        "    x, y = zip(*myList) \n",
        "\n",
        "    plt.figure(figsize=(30,10))\n",
        "    plt.margins(x=0.01)\n",
        "    plt.bar(x,y, color='Blue')\n",
        "    plt.xticks(fontsize=20)\n",
        "    plt.yticks(fontsize=20)\n",
        "    plt.title('Frequency of pure NE points based on ' +str(num_sample) + ' samples', fontsize=20)\n",
        "    plt.xlabel('Pure Strategy', fontsize=20)\n",
        "    plt.xticks(rotation = 90, horizontalalignment=\"right\", rotation_mode=\"anchor\")\n",
        "    plt.ylabel('Frequency', fontsize=20)\n",
        "    # plt.tight_layout()\n",
        "    plt.savefig('Frequency_3strategies.png', dpi=600, bbox_inches = \"tight\")\n",
        "    plt.show()\n",
        "\n"
      ],
      "metadata": {
        "id": "ktSIIJE85SfS"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "plot_sampleset(sampleset=sampleset, num_sample=10000)"
      ],
      "metadata": {
        "id": "V-4Yegcn5Vdj",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 934
        },
        "outputId": "f06ab8d6-688e-4a01-cad8-2f0132f4d860"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 2160x720 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        ""
      ],
      "metadata": {
        "id": "wV2e4rF5dm6a"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        ""
      ],
      "metadata": {
        "id": "TH4QDASbeZyy"
      },
      "execution_count": null,
      "outputs": []
    }
  ]
}