Assignment-2.ipynb

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   "source": [
    "# The Prisoners’ Dilemma and the Feasible Set of Payoff Vectors\n",
    "The Prisoners’ Dilemma is a well-known model that highlights the conflict between social cooperation and self-interested behavior. In a two-player Prisoners’ Dilemma, the cooperative payoff possibilities can be mathematically represented by a polytope. This polytope is defined as the convex hull of the payoff vectors (4,4), (6,0), (0,6), and (0,0).\n",
    "\n",
    "If we consider the Prisoners’ Dilemma as a bargaining situation with a disagreement point d = (d1, d2), where player i cannot receive a payoff below her disagreement point, we can describe the feasible set of payoff vectors (u1, u2) for the bargaining problem as follows: <br>\n",
    "\n",
    "$u1 + 2u2 ≤ 12$ <br>\n",
    "\n",
    "$2u1 + u2 ≤ 12$ <br>\n",
    "\n",
    "$u1 ≥ d1, u2 ≥ d2$ <br>\n",
    "\n",
    "Understanding this model is not necessary, but knowing the feasible set of payoff vectors is crucial in analyzing the Prisoners’ Dilemma."
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 1\n",
    "Suppose the disagreement point is given by d = (3.5, 2). A weighted utilitarian criterion is defined as<br>\n",
    "\n",
    "$W(u1, u2) = θu1 + (1 − θ)u2$ where $θ ∈ [0, 1]$<br>\n",
    "\n",
    "The weighted utilitarian solution of the bargaining problem, defined as<br>\n",
    "\n",
    "$max$   $W(u1, u2)$ such that $(u1, u2) ∈ F$ <br>\n",
    "$u1,u2$ <br>\n",
    "\n",
    "Plot player 1’s utilitarian optimum u1(θ) as the weight θ varies in [0, 1]."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import cvxpy as cp\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "d = [3.5, 2]\n",
    "u1 = cp.Variable()\n",
    "u2 = cp.Variable()\n",
    "theta = cp.Parameter(nonneg=True)\n",
    "\n",
    "constraints = [u1 + 2*u2 <= 12, 2*u1 + u2 <= 12, u1 >= d[0], u2 >= d[1]]\n",
    "weighted = cp.Maximize(theta*u1 + (1-theta)*u2)\n",
    "problem = cp.Problem(weighted, constraints)\n",
    "\n",
    "utilitarian = []\n",
    "\n",
    "for i in range(100):\n",
    "    theta.value = i/100\n",
    "    problem.solve()\n",
    "    utilitarian.append(u1.value)\n",
    "\n",
    "plt.plot([i/100 for i in range(100)], utilitarian)\n",
    "plt.xlabel('Weight, θ')\n",
    "plt.ylabel('Player 1’s utilitarian optimum, u1')\n",
    "plt.show()"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 2\n",
    "Suppose the disagreement point is given by d = (3.5, 2). A Nash welfare criterion is defined\n",
    "as<br>\n",
    "\n",
    "$N(u1, u2) = log(u1 − d1) + log(u2 − d2)$ <br>\n",
    "\n",
    "Find the Nash bargaining solution of the bargaining problem, defined as<br>\n",
    "\n",
    "$max$    $N(u1, u2)$ such that $(u1, u2) ∈ F$ <br>\n",
    "$u1,u2$ <br>\n",
    "\n",
    "1. In addition to displaying the Nash bargaining solution, display the primal optimal value and the optimal dual variables for the constraints as well.\n",
    "2. Now fix the disagreement payoff of player 2 at d2 = 2. In the same figure, plot how both players’ payoffs in the Nash bargaining solution vary as the disagreement payoff d1 of player 1 varies over the interval [2, 5]."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Nash bargaining Solution: (4.249999934669844, 3.5000001072596256)\n",
      "Primal Optimal Value:  0.11778302005591901\n",
      "Optimal Dual Variables:  3.6170408140313444e-09 0.6666667548200322 3.202879721090295e-09 1.5363574774749631e-09\n"
     ]
    },
    {
     "data": {
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4+dm8sXgT+4/k06FZPf46uCO92zahahXb/WOOzwqBMZWUqjJ/3S4yMrP5cPV24kXo064Jw3uk0al5/aDjmUrECoExlczhvALeXrqZcfOy+WrbfhrUrMrPL2nJLV1TaVw3Meh4phKyQmBMJbF17yFemb+BSYs2svtgHq0a1+ap69ozoOPpJCZY8zdz8qwQGBPBVJXPN+4hI3M9M7/YhqpyRZvTGN6jBRe0aGDN30yFsEJgTAQ6ml/Iuyu3kJGZzYqcvdROrMKIHmnc1i2NZg1qBB3PRBkrBMZEkNz9R3h14UYmLNxA7v4jnJFck8euPpeBnVOoac3fjE/sk2VMBPhi814yMrOZvnwLRwsKueScZIb3aMFFLRta8zfjOysExgQkv6CQWV7zt8XZu6lRNZ7BXZoxtHsaZybXCjqeiSFWCIwJsz0Hj/La4k28Mn8Dm/ccolmD6jzYtzWD0ptRt7o1fzPhZ4XAmDD5Zvt+MjKzmbo0h8N5hXQ7I4mH+rehZ+vTiLfdPyZAVgiM8VFhoTLn6x1kZGbzn7U7qVYljms6NmVYjzRaN6kTdDxjgDAUAhGJB7KAzarar9i4asDLwHnALuBGVc32O5Mxftt/OI/JWTmMn5/Nhl0HaVwnkfuuOoebujSnQc2qQccz5hjh2CK4C1gNlPT1ZySwW1Vbishg4EngxjBkMsYX2Tu/Z9y8bKYsyeHAkXw6N6/HvVeeQ6+2jUmw3v8mQvlaCEQkBegLPA7cU8IkVwMPe8+nAM+KiKiqVnSW5XMmc2T5WxU9W2N+8F1BIn/ceTl74urTt10ThvdoQYdm9YKOZUyZ/N4ieAb4LVDajVCbApsAVDVfRPYCScDO0IlEZBQwCqB58+YnFeRwbjapexad1GtjkVDhtTjqdWQv3eosIH/wayS16BB0HGPKzbdCICL9gB2qukRELjmVeanqC8ALAOnp6Sf1F+qCG+4D7juVGMYc35alVH31RnitP9z4CpxxcdCJjCkXP3da9gAGiEg28BpwmYhMKDbNZqAZgIhUAeriDhobU/mc3glu/xDqnA4TroNlk4JOZEy5+FYIVPV+VU1R1TRgMPCxqt5SbLJpwFDv+fXeNLZPwlRe9ZrDiPchtTu8fSfM+TPYR9pEuLCfxiAij4rIAO/HF4EkEVmLO5j8+3DnMabCVa8HQ6ZAxyHwyRMw9U7IPxp0KmNKFZYLylR1LjDXez46ZPhhYFA4MhgTVlWqwtX/gPotYM6fYN9md9ygut1C0kQeO7HZGL+IwMX3wcB/w6aF8OKVsDs76FTG/IgVAmP81v4GuHUqHNgBYy+HnCVBJzLmGFYIjAmHtAth5GxIqAHj+sLq6UEnMuYHVgiMCZfks+H2j+C0c+H1W2H+P+2MIhMRrBAYE061kmHYDGjdDz64H2b+FgoLgk5lYpwVAmPCLaE6DHoZuv0CFr0Arw2Bo98HncrEMCsExgQhLg6uehz6/B+s+QAy+sD+bUGnMjHKCoExQepyB9z0Guxc484o2rE66EQmBlkhMCZoZ18Fw9+Dgjx3rcG3c4JOZGKMFQJjIsHpHeGOj6BuM5h4PSwt3p/RGP9YITAmUtRNcQ3r0i6Cd34OH//JTi81YWGFwJhIklgHhkyGzrfBp2PgrVGQfyToVCbKhaXpnDHmBMQnQP+/Qf00+OhRr2HdBKjRIOhkJkrZFoExkUgELvoNXPci5Cx2B5G/Wx90KhOljlsIROQj798nwxPHGHOMdtfDbe/AwZ3u9NJNi4NOZKJQWVsETUSkO+6Wk51EpHPoIxwBjYl5qd1h5IdQrTaM7wdfvhN0IhNlyjpGMBr4I5AC/AWQkHEKXOZTLmNMqIYt3f2QJ90EbwyFKx9zLSpEyn6tMWUoqxBsVdXeIjJaVR89kRmLSCLwKVDNW84UVX2o2DTNgfFAPSAe+L2qvnciyzEmZtRsCEOnuVtfznrQHTPo/RTE2zkf5tSUtWvob96/15zEvI8Al6lqB6Aj0EtEuhab5kHgDVXthLvB/T9PYjnGxI6E6nB9BvS4C7JehNdugiMHgk5lKrmyvkrkicgLQFMR+Vvxkar6q9JeqKoKFH1CE7xH8atjFKjjPa8LbClPaGNiWlwcXPGoO7303Xshozfc/AbUaRJ0MlNJlbVF0A/4GDgMLCnhcVwiEi8iy4AdwGxVXVhskoeBW0QkB3gP+GUp8xklIlkikpWbm1vWYo2JDekj4ObX4bt1MLYnbF8VdCJTSYmW4xJ2EemgqstPeiEi9YCpwC9V9YuQ4fd4Gf4iIt2AF4G2qlpY2rzS09M1KyvrZKMYE322rYSJN8CR/XDDeGjZM+hEJgKJyBJVTS9pXHkvKNslIlNFZIf3eFNEUsobQFX3AHOAXsVGjQTe8KaZDyQCDcs7X2MM0LidO6OofipMHARLxgedyFQy5S0EGcA04HTvMd0bVioRSfa2BBCR6sAVwFfFJtsI9PSmaY0rBLbvx5gTVbcpDJ8JZ14K038FHz4ChaVuWBtzjPIWgkaqmqGq+d5jHJBcxmuaAHNEZAWwGHeMYIaIPCoiA7xpfgPcISLLgUnAMC3PvipjzI8l1oGbXofzhsF/noa3boe8w0GnMpVAeU9A3ikit+D+WAPcBOw63gtUdQXQqYTho0Oefwn0KGcGY0xZ4qtAv2egfgv48CHYtwUGv2oN68xxlXeLYARwA7AN2ApcDwz3K5Qx5hSIwIV3u+sNNn/uehTt+jboVCaClbcQHFDVAaqarKqNVPUaVd3oazJjzKlpOxCGTodDu10x2Fj87G1jnPIWggUiMllEeotYcxNjKo3mF7gziqrXh/H9YdXUoBOZCFTeQnA28AJwG7BGRP5XRM72L5YxpsIknemKQdPOMHkY/OcZuwWmOUa5CoE6s1X1JuAOYCiwSEQ+8S4EM8ZEshoN4Na3oe117iDyjF9DQX7QqUyEKNdZQyKSBNwC3Apsx7WCmIZrJjcZaOFTPmNMRUlIhIFjoV6qO7107yYYNM7d58DEtPLuGpqPaw53jar2VdW3vOsJsoDn/ItnjKlQcXFw+UPunsjfzoGXesPezUGnMgErbyE4R1UfU9Wc4iNU1W5jaUxlc95QGDIZdme7hnVbVwSdyASovIWgoYiMEZH3ROTjooevyYwx/mrZE0a8DxLnWlmvmR10IhOQ8haCibg+QS2AR4BsXNsIY0xl1rgt3P4RNGgBr94IWS8FncgEoLyFIElVXwTyVPUTVR2B3a/YmOhQp4lrWNeypzubaPZoa1gXY8pbCPK8f7eKSF8R6QRY8xJjokW12jB4EqSPhMy/wpThkHco6FQmTMrbdO5PIlIX1y3077gziH7tWypjTPjFV4G+f3G7iWY96BrW3TQJatotQqLdcQuBiCQCdwItgabAi6p6aTiCGWMCIALdfwl1m8HUn7oeRUOmQMOWQSczPipr19B4IB1YCfQG/uJ7ImNM8M69BobOcLe/fPFy2DA/6ETGR2UVgjaqeouqPo9rPX1RGDIZYyJBs/Ph9tlQIwleHgArpwSdyPikrEJQdJAYVbXGJMbEmgZnwMjZ0DQd3hwJn/3FGtZFobIKQQcR2ec99gPti56LyL5wBDTGBKxGA7jtbWg3CD56FKb9EgryynyZqTyOe7BYVeNPdsbegeZPgWrecqao6kMlTHcD8DCgwHJVvflkl2mM8UmVajDw31A/DT4dA3tz4IaX3X2STaVX3usITsYR4DJV7YDrUtpLRLqGTiAiZwH3Az1U9Vzgbh/zGGNOhQhc9iAMeBayP4OXermCYCo93wqBdw+DA96PCd6j+M7FO4B/qOpu7zU7/MpjjKkgnW91p5Tu3QT/7glblgWdyJwiP7cIEJF4EVkG7ABmq2rxm6aeDZwtIpkiskBEepUyn1EikiUiWbm5uX5GNsaUx5mXwogPID4BMvrANx8EncicAl8LgaoWqGpHIAXoIiJti01SBTgLuAS4Cfi3iNQrYT4vqGq6qqYnJyf7GdkYU16ntXG3wGzYEiYNhkX/DjqROUm+FoIiqroHmAMU/8afA0xT1TxVXQ98gysMxpjKoHZjGPYenHUVvHcvfPCANayrhHwrBCKSXPTtXkSqA1fgWlmHehu3NYCINMTtKlrnVyZjjA+q1YLBE6HLKJj/LEweag3rKhk/twiaAHNEZAXu3gWzVXWGiDwqIgO8aT4AdonIl7gthvtUdZePmYwxfoiLh95PwVV/htXTYXx/OGDH8yoL0Up2lWB6erpmZWUFHcMYU5rV0+HNO6BWI3d2UfLZQScygIgsUdX0ksaF5RiBMSaGtO4Pw96FvIPw4hWQnRl0IlMGKwTGmIqXcp47o6hWI3jlGljxRtCJzHFYITDG+KN+GoycBc0ugLfugE/GWMO6CGWFwBjjn+r14Za3oP1gmPMneOcX1rAuApX3VpXGGHNyqlSFa59zWwifPOFaU9zwMlSvF3Qy47EtAmOM/0Tg0vvhmn/BhkzXsG7PxqBTGY8VAmNM+HS82e0q2rfF3Q95y9KgExmsEBhjwu2Mi91B5PhqrmHd1zODThTzrBAYY8KvUSt3emnyOfDazbDwhaATxTQrBMaYYNQ+zV14dnZvmHkfvH8/FBYEnSomWSEwxgSnak248RXo+jNY8E944zY4ejDoVDHHCoExJlhx8dDrz9DrSfjqXRjXFw7YzQrDyQqBMSYydL0TBr8KuV/B2J6Q+3XQiWKGFQJjTORo1cdrWHfYNaxb/2nQiWKCFQJjTGRp2hnu+AhqN4FXBsKySUEninpWCIwxkadecxjxAaR2g7fvhLlPWMM6H1khMMZEpur1YMib0HEIzP0zvP0/kH806FRRyc97FieKyCIRWS4iq0TkkeNMe52IqIiUePccY0yMqlIVrv4HXPoALJ8EEwbCod1Bp4o6fm4RHAEuU9UOQEegl4h0LT6RiNQG7gIW+pjFGFNZicDFv4VrX4CNC+DFq2D3hqBTRRXfCoE6B7wfE7xHSTv5HgOeBA77lcUYEwU63Ai3vQ0HtrvTS3OWBJ0oavh6jEBE4kVkGbADmK2qC4uN7ww0U9V3y5jPKBHJEpGs3Nxc/wIbYyJb2oUwcjYk1HAXnq2eEXSiqOBrIVDVAlXtCKQAXUSkbdE4EYkDngZ+U475vKCq6aqanpyc7FteY0wlkHw23P4RnHYuvH4LLPhX0IkqvbCcNaSqe4A5QK+QwbWBtsBcEckGugLT7ICxMaZMtZJh6HRo1Rfe/z3M/J01rDsFfp41lCwi9bzn1YErgK+KxqvqXlVtqKppqpoGLAAGqGqWX5mMMVGkag13y8tuv4CFz7mtg6PfB52qUvJzi6AJMEdEVgCLcccIZojIoyIywMflGmNiRVw8XPU49Pk/+OZ9d6Ob/duDTlXpiFayq/XS09M1K8s2GowxxXz9PkwZDjWSYMhkaNQ66EQRRUSWqGqJu97tymJjTHQ4pxcMnwkFee5ag3Vzg05UaVghMMZEj9M7ultg1m0KE66DpRODTlQpWCEwxkSXes1gxPuQdhG88zP4+HFrWFcGKwTGmOiTWNcdJ+h0K3z6FEz9KeQfCTpVxKoSdABjjPFFfAIM+DvUT4OPH4O9OXDjBKjRIOhkEce2CIwx0UsEfnIvXPci5CyGF6+E79YHnSriWCEwxkS/dtfDbe/AwZ0w9nLIsVPQQ1khMMbEhtTuMPJDqFbbNaz78p2gE0UMKwTGmNjRsKU7vbRxe3hjKMx71s4owgqBMSbW1GwIQ6dBmwEw6wF4714oyA86VaCsEBhjYk9Cdbh+HPS4CxaPhdduhiMHynxZtLJCYIyJTXFxcMWj0PdpWDsbMnrDvq1BpwqEFQJjTGw7fyTc/AZ8t86dUbR9VdCJws4KgTHGnHWFa1inBa5h3bcfB50orKwQGGMMQJP27haY9VNh4iD4/OWgE4WNFQJjjClSt6nbMmhxMUz7JXz0KBQWBp3Kd1YIjDEmVGIduPl16DwUPvsLvHVH1Des8/OexYkiskhElovIKhF5pIRp7hGRL0VkhYh8JCKpfuUxxphyi0+A/n+Fyx+GL6bAy9fAwe+CTuUbP7cIjgCXqWoHoCPQS0S6FptmKZCuqu2BKcBTPuYxxpjyE4ELfw3XvwSbl7gzinZ9G3QqX/hWCNQpukIjwXtosWnmqOpB78cFQIpfeYwx5qS0vc5diXxoN7x4BWxaFHSiCufrMQIRiReRZcAOYLaqLjzO5COBmX7mMcaYk9K8q+tRlFgXxvWDVVODTlShfC0Eqlqgqh1x3/S7iEjbkqYTkVuAdGBMKeNHiUiWiGTl5ub6ltcYY0qVdKbrXnp6J5g8DP7zTNQ0rAvLWUOqugeYA/QqPk5ELgceAAaoaomH5lX1BVVNV9X05ORkX7MaY0ypaia5+xqcOxA+fAjevScqGtb5edZQsojU855XB64Avio2TSfgeVwR2OFXFmOMqTAJie6OZxf+GrJegkmD4cj+oFOdEj+3CJoAc0RkBbAYd4xghog8KiIDvGnGALWAySKyTESm+ZjHGGMqRlycO7W0/19dO4qXesO+LUGnOmmilWwfV3p6umZl2W3mjDERYu2H8MYwd+ezIW9A43ZBJyqRiCxR1fSSxtmVxcYYcypaXg4j3nfXHbzUC9Z8GHSiE2aFwBhjTlXjtu700gYt4NUbICsj6EQnxAqBMcZUhDqnu4Z1LXvCjLth9kOVpmGdFQJjjKko1WrD4EmQPgIyn4E3R0De4aBTlalK0AGMMSaqxFdxt7+s3wJm/9GdTTR4krsGIULZFoExxlQ0EejxKxg0HrYuhxcju2GdFQJjjPHLudfA0OlweK/rXrphftCJSmSFwBhj/NSsizujqEYDeHkAfPFm0Il+xAqBMcb4rcEZMHI2NE2HKSPgs6cjqmGdFQJjjAmHGg3gtreh3SD46BGY/isoyAs6FWBnDRljTPhUqQYD/w310+DTMbA3xx1QTqwTaCzbIjDGmHASgcsehAHPwvpPXVuKvTmBRrJCYIwxQeh8KwyZAns3uTOKti4PLIoVAmOMCcqZl8KID0DiXSvrb2YFEsMKgTHGBOm0Nu700oYtYdKNsHhs2CNYITDGmKDVaQLD3oOzroR3fwOzHgxrwzorBMYYEwmq1YLBr8L5d8C8v8PkoZB3KCyLtkJgjDGRIi4e+oyBq/4XVk+H8f3h+53+L9avGYtIoogsEpHlIrJKRB4pYZpqIvK6iKwVkYUikuZXHmOMqRREoNvP4cZXYNsXMLYn7Fzj6yL93CI4Alymqh2AjkAvEelabJqRwG5VbQn8P+BJH/MYY0zl0bo/DJsBRw6400uzM31blG+FQJ0D3o8J3qN4c42rgfHe8ylATxERvzIZY0ylkpLuziiq1QheuQZWve3LYnw9RiAi8SKyDNgBzFbVhcUmaQpsAlDVfGAv8KO7N4jIKBHJEpGs3NxcPyMbY0xkadACRs6CMy9zz33gayFQ1QJV7QikAF1EpO1JzucFVU1X1fTk5OQKzWiMMRGven24+XVo0sGX2YflrCFV3QPMAXoVG7UZaAYgIlWAusCucGQyxhjj+HnWULKI1POeVweuAL4qNtk0YKj3/HrgY9UIatJtjDExwM821E2A8SISjys4b6jqDBF5FMhS1WnAi8ArIrIW+A4Y7GMeY4wxJfCtEKjqCqBTCcNHhzw/DAzyK4Mxxpiy2ZXFxhgT46wQGGNMjLNCYIwxMc4KgTHGxDipbGdrikgusOEkX94Q8L+V34mzXCfGcp24SM1muU7MqeRKVdUSr8itdIXgVIhIlqqmB52jOMt1YizXiYvUbJbrxPiVy3YNGWNMjLNCYIwxMS7WCsELQQcoheU6MZbrxEVqNst1YnzJFVPHCIwxxvxYrG0RGGOMKcYKgTHGxLioKAQi0kxE5ojIlyKySkTuKmEaEZG/ichaEVkhIp1Dxg0VkTXeY2jx1/qca4iXZ6WIzBORDiHjsr3hy0QkK8y5LhGRvd6yl4nI6JBxvUTka29d/j7Mue4LyfSFiBSISANvnF/rK1FEFonIci/XIyVMU01EXvfWyUIRSQsZd783/GsRuSrMue7x1ucKEflIRFJDxhWErMtpYc41TERyQ5Z/e8g4v34fy5Pr/4Vk+kZE9oSM82V9hcw/XkSWisiMEsb5+/lS1Ur/wLW87uw9rw18A7QpNk0fYCYgQFdgoTe8AbDO+7e+97x+GHN1L1oe0Lsol/dzNtAwoPV1CTCjhNfGA98CZwBVgeXFX+tnrmLT98fdw8Lv9SVALe95ArAQ6Fpsmp8Bz3nPBwOve8/beOuoGtDCW3fxYcx1KVDDe/4/Rbm8nw9U9Lo6gVzDgGdLeK2fv49l5io2/S+Bl/xeXyHzvwd4tZTfO18/X1GxRaCqW1X1c+/5fmA17n7Ioa4GXlZnAVBPRJoAV+Hup/ydqu4GZvPjO6n5lktV53nLBViAu62nr8q5vkrTBVirqutU9SjwGm7dBpHrJmBSRSy7jFyqqge8HxO8R/GzLK4GxnvPpwA9RUS84a+p6hFVXQ+sxa3DsORS1TmqetD7MVyfr/Ksr9L4+ft4ornC8vkCEJEUoC8wtpRJfP18RUUhCOVtMnXCVftQTYFNIT/neMNKGx6uXKFG4rZaiigwS0SWiMiois5UjlzdvM3omSJyrjcsItaXiNTA/YF4M2Swb+vL22xfBuzA/aEq9fOlqvnAXiAJn9dXOXKFKv75ShSRLBFZICLXVFSmE8h1nbfLaoqINPOGRcT68nahtQA+Dhns2/oCngF+CxSWMt7Xz1dUFQIRqYX7w3C3qu4LOk+R8uQSkUtxv6i/Cxl8oap2xu0y+rmI/CSMuT7H9SbpAPwdeLsil30KuYr0BzJV9buQYb6tL1UtUNWOuG/UXUSkbUXN+1SUN5eI3AKkA2NCBqeqa1dwM/CMiJwZxlzTgTRVbY/71j+eMDiB/8fBwBRVLQgZ5sv6EpF+wA5VXVIR8zsZUVMIRCQB98djoqq+VcIkm4FmIT+neMNKGx6uXIhIe9wm4dWquqtouKpu9v7dAUylgnYplCeXqu4r2oxW1feABBFpSASsL89gim22+7m+QpaxB5jDj3dX/LBeRKQKUBfYhc/rqxy5EJHLgQeAAap6JOQ1RetrHTCXEu4o6FcuVd0VkmUscJ73PPD15Tne56ui11cPYICIZON2tV4mIhOKTePv5+tEDypE4gN3EOhl4JnjTNOXYw8WL9L/HpxajzswVd973iCMuZrj9ut1Lza8JlA75Pk8oFcYczXmvxccdgE2eq+rgjuA14L/Hiw+N1y5vOnq4u5xXTNM6ysZqOc9rw58BvQrNs3POfZg3hve83M59mDeOiruYHF5cnXCHUA8q9jw+kA173lDYA0Vd9C/PLmahDy/FljgPffz97HMXN64VrgTDyQc66vYsi+h5IPFvn6+KvRNBPUALsTtH14BLPMefYA7gTu9aQT4h/dLsRJID3n9CNwf47XA8DDnGgvsDhmf5Q0/w/sPXg6sAh4Ic65feMtdjjvI2D3k9X1wZ/R8G+5c3nTDcAfIQl/r5/pqDyz1cn0BjPaGP4r7lg2QCEz2PkOLgDNCXv+At66+BnqHOdeHwPaQ9TnNG97d+z1Y7v07Msy5/hzy+ZoDtAp5vV+/j2Xm8n5+GHii2Gt9W1/FlnMJXiEI5+fLWkwYY0yMi5pjBMYYY06OFQJjjIlxVgiMMSbGWSEwxpgYZ4XAGGNinBWCGBHSOXGV1zbiNyIS541LF5G/BZ0xHMR1Ve0e4PLHeP8HY0Qk2eskuVRELirn6y/yXr9MRKqX8zVzRSRsN2IXkZ+IyOciki8i15/E64eJyLMVMS9TPlWCDmDC5pC6S+sRkUa4Lod1gIdUNQuosLbNxYlIFXX9USLBJcAB3AVnQRiFu0CqQEQGAytV9fayXhRiCPBnVS1+5WlgRCRej23FsBF3rce9FTD7ipyXKYVtEcQgdS0YRgG/EOeSoh7oInJxSM/1pSJSW0Rqietl/7m4fv8/dBsVkT96fdD/IyKTROReb/hcEXlG3H0B7hKR80TkE68h3AfiOr8iImeKyPve8M9EpJU3fJyI/Mtr8LXOy/iSiKwWkXEhy79SROZ72SZ7fYqK7k3wSEjmVuIa2d0J/Np7f8d8CxeRh0XkFW9+a0TkDm94ie9fRB4VkbtDXv+4iNzlrdMx4u6XsFJEbvTGTwNqAUtE5HfAU8DVJX27F5Ge3vpf6b3vauJ69t8APCYiE4tNnyYiX4nIRG8dTRHXmI9i0/1LXOO0H/rxi8hlIvJ2yDRXiMjUcqzfJ0Xkc2BQsc9XtqquoPQGaj8iIsPF9f9fhGu5cNLzMifBj6vj7BF5D0ropQ7sAU7j2KsZpwM9vOe1cFuNVYA63rCGuKsbBTgfd7VqIu7+AWuAe73p5gL/9J4n4L6BJ3s/34jX5x34CK/9AXAB3v0FgHG4vitFrXb3Ae1wX16WAB29LJ/itZrANewrulo0G/il9/xnwFjv+cNFGUtYHw/jrhyt7s17E3D6cd5/GvC5NzwOd3VnEnAdrpFavLd+N+K1VAj9f6D0nvyJ3rLP9n5+GdeAr2i9XF/Ca9JwV2UX/d+9VOz/It173sD7N94b3t57L1+F/P+8imvqV9b6/W0Zn7kSs5YwXRNvHSXj2pZkFl8v5Z2XPU7uYVsEprhM4GkR+RWuL0s+7g/F/4rIClzLgqa4P3A9gHdU9bC6+wdMLzav171/zwHaArPFtQB+EEjxvl12ByZ7w5/H/VEoMl3dX4GVwHZVXamqhbjWBGm4nlFtgEzv9UOB1JDXFzWtW+JNXx7vqOohVd2Ja33QpbT3r6rZwC4R6QRcCSxV1zTwQmCSuk6X24FPcEWzvM4B1qvqN97P44HydFLdpKqZ3vMJXo7ibvC+xS/F9alp463jV4BbRKQe0A3Xl6us9fs6FeMCYK6q5qq7x0VFzdeUkx0jiFEicgZQgOvL3rpouKo+ISLv4nr8ZIq79V1X3Le181Q1T1yXxMRyLOb7osUBq1S1W7EMdYA96h27KEFRd8rCkOdFP1fx8s9W1ZvKeH0B5f+sF++5orj98qW9/7G4b/aNcd/Cg1RS9h+ISAvcvvbzVXW3t4ut6H1k4Ar5YWCyquaLiHD89ft9KcNNJWNbBDFIRJKB53Cb38X/WJzpffN+EliM68RYF9cvPU/cfROKvhVmAv3F3Qu2FtCvlEV+DSSLSDdvGQkicq66ew2sF5FB3nCRkHs2l8MCoIeItPReX1NEzi7jNftxu7FKc7X3fpJwu8wWU/r7B9fuuhfuG/8H3rDPgBvF3QQlGfdtftEJvK+vgbSi9wXcituqKEvzonWM65n/n2Lj6+D+eO8VkdNw920AQFW3AFtwW2sZ3uCTWb/HJSJflTB4IXCxiCSJa0M+qIRpjI+sEMSO6t5ByVW43RuzgB/dvBu42zvIuQLIw+0imAiki8hK4Dbc/mRUdTEwDdfNcSZuF87e4jP0NvevB54UkeW44wpFp3AOAUZ6w1dxAre9VNVc3LfxSV7e+bjCdTzTgWtLOljsWYHbJbQAeMz7A1ni+w95b3NwbYGLzpyZ6s1nOe4OV79V1W0n8L4OA8Nxu8xW4raAnivHS7/G3ZBnNa5t8r+KzXc5bpfQV7jjAJnFXj8Rt3tptTf9yaxfROR8EcnB/UF/3vvMIe5+FlLC+92KOz4z38u0uqx5mYpl3UfNKRGRWqp6wDtD5VNglHr3Ha5sRORh3MHc/zuB18Th7uY2SFXX+JWtHDnScAf8T/rOaeLO3V+qqi9WWLBj598P1z45Jq5ZqUzsGIE5VS+ISBvcvubxlbUInAzvfc8ApgZZBCqCiCzB7Tb6jV/LUNUZfs3bnBrbIjDGmBhnxwiMMSbGWSEwxpgYZ4XAGGNinBUCY4yJcVYIjDEmxv1/H/e+AccvQVoAAAAASUVORK5CYII=",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import cvxpy as cp\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "d = [3.5, 2]\n",
    "u1 = cp.Variable()\n",
    "u2 = cp.Variable()\n",
    "\n",
    "constraints = [u1 + 2*u2 <= 12, 2*u1 + u2 <= 12, u1 >= d[0], u2 >= d[1]]\n",
    "welfare = cp.Maximize(cp.log(u1 - d[0]) + cp.log(u2 - d[1]))\n",
    "problem = cp.Problem(welfare, constraints)\n",
    "\n",
    "nash = []\n",
    "\n",
    "problem.solve()\n",
    "\n",
    "print(\"Nash bargaining Solution: ({}, {})\".format(u1.value, u2.value))\n",
    "print(\"Primal Optimal Value: \", problem.value)\n",
    "print(\"Optimal Dual Variables: \", constraints[0].dual_value, constraints[1].dual_value, constraints[2].dual_value, constraints[3].dual_value)\n",
    "\n",
    "d[1] = 2\n",
    "u1_vals, u2_vals, d1_vals = [], [], []\n",
    "\n",
    "for d1 in range(2, 6):\n",
    "    constraints = [u1 + 2*u2 <= 12, 2*u1 + u2 <= 12, u1 >= d1, u2 >= 2]\n",
    "\n",
    "    welfare = cp.Maximize(cp.log(u1 - d1) + cp.log(u2 - 2))\n",
    "    problem = cp.Problem(welfare, constraints)\n",
    "    problem.solve()\n",
    "    \n",
    "    u1_vals.append(u1.value), u2_vals.append(u2.value), d1_vals.append(d1)\n",
    "\n",
    "plt.plot(d1_vals, u1_vals, label=\"Player 1\")\n",
    "plt.plot(d1_vals, u2_vals, label=\"Player 2\")\n",
    "plt.xlabel(\"Disagreement payoff of player 1, d1\")\n",
    "plt.ylabel(\"Payoff\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 3 [Interior Point Algorithm Using Log Barrier Method]\n",
    "Suppose the disagreement point is given by d = (2, 1). Solve Problem 2(a) by implementing the interior point algorithm that uses log barrier for inequality constraints. Display the following:\n",
    "1. initial t and final t\n",
    "2. Nash bargaining solution\n",
    "3. primal optimal value (Maximized Nash welfare)\n",
    "4. dual optimal variables for constraints\n",
    "5. inequality constraint function values at the optimum"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***\n",
    "### Algorithm: Line Search Subroutine\n",
    "```python\n",
    "given a descent direction d for f at the point x ∈ dom f, reduction factor β ∈ (0, 1), and\n",
    "acceptable descent parameter α ∈ (0, 0.5).\n",
    "t := 1\n",
    "while f(x) − f(x + td) < α*− Df(x)(td)\n",
    "    t := βt\n",
    "endwhile\n",
    "return t as the stepsize.\n",
    "```\n",
    "***\n",
    "### Algorithm: Newton descent algorithm\n",
    "```python\n",
    "choose an initial point x ∈ dom f, and a tolerance level η for the stopping criterion.\n",
    "while norm(∇f(x)) > η\n",
    "1. choose the descent direction as the Newton direction d = ([Hf(x)]^(−1))*(−∇f(x))\n",
    "2. line search. choose a stepsize t > 0 by the backtracking line search subroutine.\n",
    "3. update. x := x + td.\n",
    "endwhile\n",
    "return x as the minimum.\n",
    "```\n",
    "***\n",
    "### Algorithm: Interior point (log barrier)\n",
    "```python\n",
    "choose an initial strictly feasible primal point x, an initial value t = t0 of the accuracy\n",
    "parameter, the accuracy multiplier μ > 1 (usually a value between 10 to 20), a tolerance\n",
    "level η for the stopping criterion\n",
    "while m/t > η (current iterate is not within η-suboptimal)\n",
    "1. Increase accuracy. t = μt\n",
    "2. Centering step. Compute the central point x(t) by solving (12.5) using Newton descent\n",
    "method and x as the initial point.\n",
    "3. Update the initial point for next iteration. x = x(t)\n",
    "endwhile\n",
    "return x as the primal optimum; and λi = −1/t*fi(x) as the i-th dual optimum\n",
    "```\n",
    "***\n",
    "```math\n",
    "Gradient -> [t/(x-2) + 1/(x+2y-12) + 2/(2x+y-12) - 1/(x-2),\n",
    "            t/(y-1) + 2/(x+2y-12) + 1/(2x+y-12) - 1/(y-1)]\n",
    "\n",
    "Hessian -> [[-t/(x-2)^2 - 1/(x+2y-12)^2 - 4/(2x+y-12)^2 + 1/(x-2)^2, -2/(x+2y-12)^2 - 2/(2x+y-12)^2],\n",
    "            [-2/(x+2y-12)^2 - 2/(2x+y-12)^2, -t/(y-1)^2 - 2/(x+2y-12)^2 - 1/(2x+y-12)^2 + 1/(y-1)^2]]\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\aditi\\AppData\\Local\\Temp\\ipykernel_24936\\1482822841.py:10: RuntimeWarning: invalid value encountered in log\n",
      "  return -(t * (np.log(x[0] - 2) + np.log(x[1] - 1)) + (-(np.log(-(x[0] + 2 * x[1] - 12)) + np.log(-(2 * x[0] + x[1] - 12)) + np.log(-(x[0] - 2)) + np.log(-(x[1] - 1)))))\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Initial and Final value of t : 1 121\n",
      "Nash bargaining Solutions:  [4.01852653 3.92441859]\n",
      "Maximized Nash welfare:  1.7754634900510857\n",
      "Inequality constraint functions:  [-0.13263629943976696, -0.03852836054130826, -2.01852652611905, -2.924418587220592]\n",
      "Dual optimal variables:  [0.0010961677639650161, 0.0003184162028207294, 0.016682037405942565, 0.024168748654715636]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "d1 = 2\n",
    "d2 = 1\n",
    "\n",
    "def func(x):\n",
    "    return np.log(x[0] - 2) + np.log(x[1] - 1)\n",
    "\n",
    "def f(x, t):\n",
    "    return -(t * (np.log(x[0] - 2) + np.log(x[1] - 1)) + (-(np.log(-(x[0] + 2 * x[1] - 12)) + np.log(-(2 * x[0] + x[1] - 12)) + np.log(-(x[0] - 2)) + np.log(-(x[1] - 1)))))\n",
    "\n",
    "def gradient(x0, t):\n",
    "    x, y = x0\n",
    "    a = x-d1\n",
    "    b = y-d2\n",
    "    c = x+2*y-12\n",
    "    d = 2*x+y-12\n",
    "    grad = np.zeros(2)\n",
    "    grad[0] = t/a + 1/c + 2/d - 1/a\n",
    "    grad[1] = t/b + 2/c + 1/d - 1/b\n",
    "\n",
    "    return grad\n",
    "\n",
    "def hessian(x0, t):\n",
    "    x, y = x0\n",
    "    a = x-d1\n",
    "    b = y-d2\n",
    "    c = x+2*y-12\n",
    "    d = 2*x+y-12\n",
    "    hess = np.zeros((2,2))\n",
    "    hess[0,0] = -t/(a)**2 - 1/(c)**2 - 4/(d)**2 + 1/(a)**2\n",
    "    hess[0,1] = -2/(c)**2 - 2/(d)**2\n",
    "    hess[1,0] = -2/(c)**2 - 2/(d)**2\n",
    "    hess[1,1] = -t/(b)**2 - 4/(c)**2 - 1/(d)**2 + 1/(b)**2\n",
    "\n",
    "    return hess\n",
    "\n",
    "d = [2, 1]\n",
    "x = [2.001, 1.001]\n",
    "t = 1\n",
    "t0 = t\n",
    "mu = 11\n",
    "tol = 0.1\n",
    "d1 = 2\n",
    "d2 = 1\n",
    "\n",
    "def lineSearchSubroutine(f, x, d, t, tv=0.001, alpha=0.499, beta=0.1):\n",
    "    while f(x, t) - f(x+tv*d, t) < -tv*alpha*np.dot(-gradient(x, t), d):\n",
    "        tv = beta*tv\n",
    "    return tv\n",
    "\n",
    "def newtonDescent(f, x, t, tol=0.0001):\n",
    "    while np.linalg.norm(gradient(x, t)) > tol:\n",
    "        d = np.dot(np.linalg.inv(hessian(x, t)), -gradient(x, t))\n",
    "        tv = lineSearchSubroutine(f, x, d, t)\n",
    "        x = x + tv*d\n",
    "    return x\n",
    "\n",
    "def interiorPoint(f, x, t, mu=11, tol=0.1):\n",
    "    while (4/t) > tol:\n",
    "        t = t*mu\n",
    "        x = newtonDescent(f, x, t)\n",
    "    return x, t\n",
    "\n",
    "x, t = interiorPoint(f, x, t)\n",
    "\n",
    "def dual_opt(x, t):\n",
    "    dual = [-1 / t * (x[0] + 2 * x[1] - 12), -1/t*(2*x[0] + x[1] - 12), -1/t*(-x[0] + d1), -1/t*(-x[1] + d2)]\n",
    "    return dual\n",
    "\n",
    "def ineq_con(x):\n",
    "    ineq = [x[0] + 2*x[1] - 12, 2*x[0] + x[1] - 12, -x[0] + d1, -x[1] + d2]\n",
    "    return ineq\n",
    "\n",
    "print(\"Initial and Final value of t :\", t0, t)\n",
    "\n",
    "print(\"Nash bargaining Solutions: \", x)\n",
    "\n",
    "print(\"Maximized Nash welfare: \", func(x))\n",
    "\n",
    "print(\"Inequality constraint functions: \", ineq_con(x))\n",
    "\n",
    "print(\"Dual optimal variables: \", dual_opt(x, t))"
   ]
  }
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