move to advanced_tutorials/agent_versus_non_agent_regret.ipynb

Author: rahulsavani

…s/04_agent_versus_non_agent_regret.ipynb …ials/agent_versus_non_agent_regret.ipynbdoc/tutorials/04_agent_versus_non_agent_regret.ipynb renamed to doc/tutorials/advanced_tutorials/agent_versus_non_agent_regret.ipynb doc/tutorials/04_agent_versus_non_agent_regret.ipynb renamed to doc/tutorials/advanced_tutorials/agent_versus_non_agent_regret.ipynb

@@ -5,7 +5,7 @@
    "id": "96dfe427-942a-47e9-8f1f-91854989b8c8",
    "metadata": {},
    "source": [
-    "# 3) Agent and standard notions of extensive form games\n",
+    "# Agent and standard notions of extensive form games\n",
     "\n",
     "The purpose of this tutorial is to explain the notions of `MixedBehaviorProfile.agent_max_regret` and `MixedBehaviorProfile.agent_liap_value`, and the corresponding solvers `Gambit.nash.enumpure_agent_solve` and `Gambit.nash.liap_agent_solve`. These notions are only relevant for *extensive-form games*, and so `agent_max_regret` and \n",
     "`agent_liap_value` are only available for `MixedBehaviorProfile`s and not for `MixedStrategyProfile`s."
@@ -23,7 +23,7 @@
     "A player's regret is 0 if they are playing a mixed (including pure) best response; otherwise it is positive and \n",
     "is the different between the best response payoff (achievable via a pure strategy) against the other players' and what the player actually gets as payoff in this profile.\n",
     "\n",
-    "Let's see an example."
+    "Let's see an example taken from [Myerson (1991)](#references)."
    ]
   },
   {
@@ -375,7 +375,9 @@
    "id": "c88d08e2-33bf-48ad-b71f-4a0c19929fdc",
    "metadata": {},
    "source": [
-    "The method `Gambit.nash.liap_solve` essentially looks for *local* minima of the function from profiles to the Liapunov value. The set of Nash equilibria are exactly the *global* minima of this function, which is why `liap_solve` may not return a Nash equilibrium."
+    "As we have seen, both the maximum regret and Liapunov value of a profile are non-negative and zero if and only if the profile is a Nash equilibrium. When positive, one can think of both notions as describing how close one is to an equilibrium.\n",
+    "\n",
+    "Based on this idea, the method `Gambit.nash.liap_solve` looks for *local* minima of the function from profiles to the Liapunov value. The set of Nash equilibria are exactly the *global* minima of this function, where the value is 0, but `liap_solve` may terminate at a non-global, local minimum, which is not a Nash equilibrium."
    ]
   },
   {
@@ -667,7 +669,7 @@
    "id": "c4eeb65f",
    "metadata": {},
    "source": [
-    "To conclude, we note that, for most use cases, the standard non-agent versions are probably what a user wants. The agent versions have applications in the area of \"equilibrium refinements\"; for more details see [Myerson (1991)](#references)."
+    "To conclude, we note that, for most use cases, the standard non-agent versions are probably what a user wants. The agent versions have applications in the area of \"equilibrium refinements\", in particular for \"sequential equilibria\"; for more details see Chapter 4, \"Sequential Equilibria of Extensive-Form Games\", in [Myerson (1991)](#references)."
    ]
   },
   {