{
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   "source": [
    "<div align=\"right\" style=\"text-align: right\"><i>Peter Norvig<br>2012, 2020, 2021</i></div>\n",
    "\n",
    "# The Unfinished Game ... of Risk\n",
    "\n",
    "[Keith Devlin](https://web.stanford.edu/~kdevlin/)'s [book](https://www.amazon.com/Unfinished-Game-Pascal-Fermat-Seventeenth-Century/dp/0465018963) [*The Unfinished Game*](https://wordplay.blogs.nytimes.com/2015/12/14/devlin-unfinished-game/) describes how Fermat and Pascal discovered the rules of probability that guide gambling in games. The question they confronted was: what if a gambling game is interrupted, but one player is in the lead by a certain score. How much of the pot should the leader get?\n",
    "\n",
    "My friends and I faced a similar question when a game of [*Risk*](https://www.ultraboardgames.com/risk/game-rules.php) ran on too long (as they often do) and we were unable to finish. Player **A** had just cashed in cards and added troops to a single large force that was poised to make a sweeping attack on player **B**, whose territories were arranged in such a way that **A** could attack from one territory to the next without ever having to branch off.  We wrote down the number of **A**'s massed armies, **72**, and the number of armies in **B**'s successive territories: **22, 8, 2, 2, 2, 7, 1, 1, 3, 1, 2, 3, 5, 1.** What is the probability that **A** can capture all these territories?\n",
    "\n",
    "______\n",
    "\n",
    "# Terminology\n",
    "\n",
    "Let's explain some *Risk* rules and terminology:\n",
    "\n",
    "- A **battle** is when armies from one territory attack an enemy neighboring territory. A roll of dice determines which armies perish:\n",
    "  - The attacker will roll 3 dice if possible (but no more than the number of armies in the attacking territory minus one).\n",
    "  - The defender will roll 2 dice if possible (or only one if they have only one army remaining). \n",
    "  - The **outcome** of the battle is determined by comparing the highest die from each of the two players, with the defender losing an army if the attacker's die is higher, and the attacker losing an army if tied or lower. Then if both sides rolled at least two dice, we do the same comparison with the second highest die on each side. \n",
    "  - When a battle kills off the last defender in a territory, the attackers **occupy** the territory. They must leave behind one army, but can move the rest in.  \n",
    "\n",
    "- A **campaign** consists of a sequence of battles and occupations. We will consider only a **chain** campaign, in which attackers invade successive enemy territories in order, always moving all their remaining armies into each captured territory, never branching, and never changing strategy based on the result of a battle. The attackers **win** the campaign if they occupy all the territories. The attackers **lose** if there are any remaining defenders and only one remaining attacker, who by rule cannot attack.\n",
    "\n",
    "With that out of the way, we're ready for some Python implementation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from collections import Counter\n",
    "from functools   import lru_cache\n",
    "import itertools\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Representing the State of a Campaign\n",
    "\n",
    "I will represent the **state of a campaign** as a tuple where the first element is the number of attacking armies, and subsequent elements are the number of defenders in each successive territory. Thus we can represent the start state of our unfinished campaign as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "State = tuple\n",
    "\n",
    "start = State((72, 22, 8, 2, 2, 2, 7, 1, 1, 3, 1, 2, 3, 5, 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I'll define the following functions on states: \n",
    "- `A(state)` gives the number of attackers in a state.\n",
    "- `D(state)` gives the number of defenders in the first territory (or `0` if there are no defenders at all).\n",
    "- `allD(state)` gives the tuple of numbers of defenders in each territory.\n",
    "- `update(state, dead)` gives a new state reflecting the number of dead attacking and defending armies spewcified by the two-element tuple  `dead`. If `D(state) == D(dead)` then the attackers have killed off all the defenders in the first territory, and the new state will reflect the fact that the attackers now occupy the territory."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def A(state)    -> int:   \"Attackers\";       return state[0]\n",
    "def D(state)    -> int:   \"First defenders\"; return state[1] if allD(state) else 0\n",
    "def allD(state) -> tuple: \"All defenders\";   return state[1:]\n",
    "\n",
    "def update(state, dead) -> State:\n",
    "    \"\"\"Update the state of a campaign to reflect battle deaths.\"\"\"\n",
    "    if D(state) == D(dead):\n",
    "        return State((A(state) - A(dead) - 1, *allD(state)[1:])) # Occupy territory, leaving 1 behind\n",
    "    else:\n",
    "        return State((A(state) - A(dead), allD(state)[0] - D(dead), *allD(state)[1:])) # Can't occupy yet"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Tests\n",
    "\n",
    "assert A(start)  == 72\n",
    "assert D(start)  == 22 \n",
    "assert allD(start) == (22, 8, 2, 2, 2, 7, 1, 1, 3, 1, 2, 3, 5, 1)\n",
    "assert sum(allD(start)) == 60 \n",
    "assert len(allD(start)) == 14\n",
    "\n",
    "assert update((10, 2, 5), (0, 2)) == (9, 5)    # 2 defenders dead; occupy the first territory\n",
    "assert update((10, 2, 5), (1, 1)) == (9, 1, 5) # continue to battle\n",
    "assert update((10, 2, 5), (2, 0)) == (8, 2, 5) # continue to battle"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Rolling the Dice: Death Counts\n",
    "\n",
    "First we'll define the function `deaths`, which, given the specific sets of dice rolled by attacker and defender in a battle,  returns a `State` tuple that gives the number of attackers and defenders who perish in the battle:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "die  = (1, 2, 3, 4, 5, 6)\n",
    "Dice = list # a sorted list of die rolls\n",
    "\n",
    "def deaths(A_dice: Dice, D_dice: Dice) -> State:\n",
    "    \"\"\"How many (attacker, defender) armies perish as the result of these dice?\"\"\"\n",
    "    dead = [0, 0]\n",
    "    for a, d in zip(sorted(A_dice, reverse=True), \n",
    "                    sorted(D_dice, reverse=True)):\n",
    "        dead[a > d] += 1\n",
    "    return State(dead)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Tests: the 4 sample dice rolls from www.ultraboardgames.com/risk/game-rules.php\n",
    "\n",
    "assert deaths((6, 1, 1), (3,))   == (0, 1) # Defender loses one\n",
    "assert deaths((6, 2, 1), (3, 2)) == (1, 1) # Both lose one\n",
    "assert deaths((3, 3),    (4, 3)) == (2, 0) # Attacker loses two\n",
    "assert deaths((6,),      (5, 4)) == (0, 1) # Defender loses one\n",
    "\n",
    "assert update((10, 2, 5), deaths((6, 2, 1), (3, 2))) == (9, 1, 5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Simulation versus Full Probability Distribution\n",
    "\n",
    "My first approach to the problem was a  [**Monte Carlo simulation**](https://en.wikipedia.org/wiki/Monte_Carlo_method), in which I used random dice rolls to determine the outcome of each battle, and thus the campaign. That was good enough to estimate the probability of winning to within 1%, but it was not good enough to get a completely accurate breakdown of all possible outcomes. \n",
    "\n",
    "Therefore I decided to compute the full probability distribution for a campaign. The code becomes a bit more complex, because we have to deal with a collection of possible results for each dice roll, not just a single result. However, I think the precision gained is worth the slight code complexity. \n",
    "\n",
    "`Pdist` will be used to represent a probability distribution of the form `{state: probability, ...}`, where the probabilities are normalized so that they are all between 0 and 1, and sum to 1. (I realize that it is suggested that a `Counter`has integer counts, but that is not required, and `Counter` has a nicer API than `dict`, including the fact that it has a default value of 0. So I am not ashamed to inherit from `Counter`.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Pdist(Counter): \n",
    "    \"A Probability distribution.\"\n",
    "    def __init__(self, *args):\n",
    "        Counter.__init__(self, *args)\n",
    "        total = sum(self.values())\n",
    "        for x in self:\n",
    "            self[x] /= total # Normalize\n",
    "            assert 0.0 <= self[x] <= 1.0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now for the full probability distribution (over all dice rolls) of deaths in a battle. The input to the function `deaths_pdist` will be a state giving the number of attacking and defending armies for just this one battle. (Thus, the attackers will always be 3, 2, or 1, and the defenders will always be 2 or 1. I define it this way, rather than passing in the full state of the campaign, so that I can **cache** the results and reuse them in subsequent battles.) The return value will be a probability distribution over the number of deaths for attacker and defender. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "@lru_cache()\n",
    "def deaths_pdist(state) -> Pdist:\n",
    "    \"\"\"A probability distribution of deaths in a battle.\"\"\"\n",
    "    return Pdist(deaths(A_dice, D_dice)\n",
    "                 for A_dice in rolls(A(state))\n",
    "                 for D_dice in rolls(D(state)))\n",
    "\n",
    "def rolls(n) -> Dice: \n",
    "    \"\"\"All possible rolls of `n` dice.\"\"\"\n",
    "    return itertools.product(die, repeat=n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Below we test all six possible battle dice combinations to make sure they match the odds stated at a [*Risk* data analysis blog](http://datagenetics.com/blog/november22011/):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "assert deaths_pdist((1, 1)) == Pdist({(1, 0):   21, (0, 1):   15})\n",
    "assert deaths_pdist((2, 1)) == Pdist({(1, 0):   91, (0, 1):  125})\n",
    "assert deaths_pdist((3, 1)) == Pdist({(1, 0):  441, (0, 1):  855})\n",
    "assert deaths_pdist((1, 2)) == Pdist({(1, 0):  161, (0, 1):   55})\n",
    "assert deaths_pdist((2, 2)) == Pdist({(2, 0):  581, (1, 1):  420, (0, 2):  295})\n",
    "assert deaths_pdist((3, 2)) == Pdist({(2, 0): 2275, (1, 1): 2611, (0, 2): 2890})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Outcomes of a Campaign\n",
    "\n",
    "A `campaign` is a series of battles. We keep battling until either the attackers no longer have the two armies necessary to enter into a battle, or until there are no defenders left. The function `battle` updates a probability distribution to reflect all possible results of a single battle."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def campaign(start: State) -> Pdist:\n",
    "    \"\"\"Probability distribution for all outcomes of a campaign.\"\"\"\n",
    "    pdist = Pdist({start: 1})\n",
    "    while any(map(can_battle, pdist)):\n",
    "        pdist = battle(pdist)\n",
    "    return pdist\n",
    "\n",
    "def battle(pdist) -> Pdist:\n",
    "    \"\"\"For every possible campaign state in the Pdist, do one battle, \n",
    "    and return an updated `result` Pdist over the outcomes.\"\"\"\n",
    "    result = Pdist()\n",
    "    for state in pdist:\n",
    "        if not can_battle(state): # State carries through unchanged\n",
    "            result[state] += pdist[state] \n",
    "        else: # Replace state with all outcomes from a battle\n",
    "            pdead = deaths_pdist((min(3, A(state) - 1), min(2, D(state))))\n",
    "            for dead in pdead:\n",
    "                result[update(state, dead)] += pdist[state] * pdead[dead]\n",
    "    return result\n",
    "\n",
    "def can_battle(state) -> bool: \n",
    "    \"\"\"The attackers can battle if they have at least 2 armies, and there are defenders.\"\"\"\n",
    "    return A(state) >= 2 and allD(state)\n",
    "\n",
    "def attacker_wins(pdist) -> float: \n",
    "    \"\"\"The probability that the attackers win the campaign in this distribution.\"\"\"\n",
    "    return sum(pdist[s] for s in pdist if not allD(s))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's a small example campaign: when 4 attackers go against 2 defenders in the first territory and 1 defender in a second territory, the attackers win about 36% of the time (about 15% with 1 attacker remaining in the final territory, and 21% with 2 attackers in the final territory)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Pdist({(1, 2, 1): 0.21807067805974698,\n",
       "       (1, 1, 1): 0.12597532213712342,\n",
       "       (1, 1): 0.294669783648961,\n",
       "       (1,): 0.14620529335276633,\n",
       "       (2,): 0.21507892280140226})"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "campaign((4, 2, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.3612842161541686"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "attacker_wins(_)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Finishing the Unfinished Game\n",
    "\n",
    "What is the probability of  winning the unfinished game?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.8105485936352178"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "attacker_wins(campaign(start))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The attackers defeat all the defenders 81% of the time. \n",
    "\n",
    "# Analyzing Campaigns\n",
    "\n",
    "Let's try to see the whole picture. I'll define the **score** of a campaign as the number of attacker armies in the final territory minus the total number of defenders, minus the number of territories that the defenders hold. This score will be positive when the attackers win and negative when they lose. A score cannot be zero (the closest is that there could be one attacker and one defender left, in which case there can be no more attacks, but the score would be -1, because the defenders have one army and one territory). The function `show` plots a histogram of scores:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def show(pdist, epsilon=1e-4):\n",
    "    \"\"\"Plot and annotate a probability distribution over states.\"\"\"\n",
    "    X = [score(s) for s in pdist if pdist[s] > epsilon]\n",
    "    Y = [pdist[s] for s in pdist if pdist[s] > epsilon]\n",
    "    avg = sum(score(s) * pdist[s] for s in pdist)\n",
    "    plt.figure(figsize=(10, 5))\n",
    "    plt.title(f'Attacker wins {attacker_wins(pdist):.2%}. Average score: {avg:.1f}.')\n",
    "    plt.xlabel('Score (positive when attacker wins)')\n",
    "    plt.ylabel('Probability of score')\n",
    "    plt.bar(X, Y, width=3/4)\n",
    "    \n",
    "def score(state): return A(state) - sum(allD(state)) - len(allD(state))\n",
    "    \n",
    "show(campaign(start))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.02662110993767859"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S = campaign(start)\n",
    "Q = Pdist({score(s): S[s] for s in S})\n",
    "Q[-7]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Interesting! We see that the scores look roughly like a bell-shaped curve on the right side, but with a decidely non-normal pattern on the left side. There are gaps—scores for which the probability is zero—and there are spikes that rise above the normal curve. \n",
    "\n",
    "What's causing the gaps? We know that a score can never be 0. And in this campaign, no score can be -2. because a -2 can only be achieved with one attacker and two defenders in one territory; this campaign has only one defender in the final territory. Looking at the plot, we see gaps surrounding groups of bars, where the group sizes, reading left to right, are 1, 1, 3, 1, 2, 3, 5, and 1. Note that these are exactly the number of defenders in the final 8 territories.\n",
    "\n",
    "What's causing the spikes? To some extent they represent probability mass that has shifted over a place. But the spikes are not quite tall enough to fill in the gaps. And I don't understand the exact shape of each group.\n",
    "\n",
    "Below is an example where there is no visible gap. The normalish shape is restored, except that there is something going on with odd-versus-even scores. I believe what is happening is that all but the very last few battles will be 3-versus-2 battles, and thus will result in 2 casualties, preserving the parity of the total number of armies. The parity can only be broken in the final battles."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "show(campaign((145, 100)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The odd-even disparity can be lessened by adding a few territories where there will not be 3-versus-2 battles. The result is a very smooth curve, although not normal—it is clearer a bit fatter on the left than the right."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "show(campaign((150, 100, 1, 1, 1, 1, 1)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Analyzing Single-Territory Attacks\n",
    "\n",
    "I want to analyze a simpler situation: With *A* attackers and *D* defenders in a single territory, what is the probability the attackers win?  \n",
    "\n",
    "I will make a chart with the number of defenders varying from 1 to 60, and the number of attackers separated into eight cases (depicted as eight lines), where in each case there are Δ more attackers than defenders:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "def chart(defenders=range(1, 61), deltas=(10, 5, 2, 1, 0, -1, -2, -5)):\n",
    "    \"\"\"Plot win probability for various numbers of defenders and Δ more attackers.\"\"\"\n",
    "    plt.figure(figsize=(10, 5)); plt.grid()\n",
    "    plt.title('Each line: attackers with Δ more armies than defenders')\n",
    "    plt.xlabel('Number of Defenders'); plt.ylabel('Win Probability for Attackers')\n",
    "    for delta in deltas:\n",
    "        winprobs = [attacker_wins(campaign((max(0, d + delta), d))) for d in defenders]\n",
    "        plt.plot(defenders, winprobs, '.-', label=f'Δ = {delta}')\n",
    "    plt.legend()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "chart()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that the purple line (fourth from bottom), where the number of attackers exactly equals the number of defenders, gives a low win probability for a small attacking force, but reaches 50% for 12-on-12, and 73% for 60-on-60. The red line, where the attackers have one more army than the defenders, dips from one to two defenders but is over 50% for a 6-on-5 attack. Similarly, the green line, where the attackers have a surplus of two armies, dips sharply from 75% to 66% as the number of defenders goes from 1 to 2, dips slightly more for 3 and 4 defenders, and then starts to rise. So overall, an attacker does not need a big advantage in armies as long as there are many armies on both sides. Even when the attacker is at a disadvantage in numbers (as in the bottom grey line where the attacker has five fewer armies), the attacker can still have an advantage in win probability; with 55 against 60 the win percentage is about 57%."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}