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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The pseudocode for Monte Carlo ES is inefficient because, for each state–action pair, it maintains a list of all returns and repeatedly calculates their mean. It would be more efficient to use techniques similar to those explained in Section 2.4 to maintain just the mean and a count (for each state–action pair) and update them incrementally. Describe how the pseudocode would be altered to achieve this."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Monte Carlo ES (Exploring Start) Incremental Implemention\n",
+    "\n",
+    "#### Initialize:\n",
+    "$\\:\\:\\:\\:\\:\\;\\;\\;\\;\\:\\:\\: \\space \\pi(s) \\in A(s) (arbitrairly), \\text{for all}\\: s \\in S $\n",
+    "\n",
+    "$\\:\\:\\:\\:\\:\\;\\;\\;\\;\\:\\:\\: Q(s, a) \\in R (arbitrarily), \\text{for all}\\: s \\in S, a \\in A(s)$\n",
+    "\n",
+    "$\\:\\:\\:\\:\\:\\;\\;\\;\\;\\:\\:\\: Returns (a,s) \\gets \\text{empty list}, \\text{for all}\\: s \\in S, a \\in A(s)$\n",
+    "\n",
+    "#### Loop forever (for each episode):\n",
+    "$\\:\\:\\:\\:\\:\\;\\;\\;\\;\\:\\:\\: Choose  \\: S_0 \\in S, A_0 \\in A(S_0) \\:\\text{randomly such that all pairs have probability > 0}$\n",
+    "$\\:\\:\\:\\:\\:\\;\\;\\;\\;\\:\\:\\: \\text{Generate an episode from $S_0, A_0$, following $\\pi: S_0, A_0, R_1, ..., S_{T-1}, A_{T-1}, R_T$ }$\n",
+    "\n",
+    "$\\:\\:\\:\\:\\:\\;\\;\\;\\;\\:\\:\\:  G \\gets 0$\n",
+    "\n",
+    "$\\:\\:\\:\\:\\:\\;\\;\\;\\;\\:\\:\\:  \\text{Loop for each step of episode, $t = T-1, T-2,..., 0:$}$\n",
+    "\n",
+    "$\\:\\:\\:\\:\\:\\;\\;\\;\\;\\:\\:\\:\\:\\:\\:\\:\\: G \\gets \\gamma G + R_{t+1}$\n",
+    "\n",
+    "$\\:\\:\\:\\:\\:\\;\\;\\;\\;\\:\\:\\:\\:\\:\\:\\:\\: \\text{Unless the pair $S_t, A_t$ appears in $S_0, A_0, S_1, A_1,...,S_{t-1},A_{t-1}$:}$\n",
+    "\n",
+    "$\\:\\:\\:\\:\\:\\;\\;\\;\\;\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\: N(S_t, A_t) \\gets N(S_t, A_t) + 1 $\n",
+    "\n",
+    "$\\:\\:\\:\\:\\:\\;\\;\\;\\;\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\: Q(S_t, A_t) \\gets Q(S_t, A_t) + \\frac{1}{N(S_t, A_t)}[G - Q(S_t, A_t)]$\n",
+    "\n",
+    "$\\:\\:\\:\\:\\:\\;\\;\\;\\;\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\: \\pi(S_t) \\gets \\text{argmax}_aQ(S_t, a)$\n"
+   ]
+  }
+ ],
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