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@belteshassar
Created June 24, 2017 11:30
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My analysis of The Riddler ☕ Coffee Challenge ☕
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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# The Riddler ☕ Coffee Challenge ☕ \n",
"\n",
"Below is the analysis behind my strategy in the [2017-06-23 Riddler](https://fivethirtyeight.com/features/can-you-drink-more-coffee-than-your-coworkers/). A quick recap of the rules:\n",
"\n",
"> [T]he coffee pot holds one gallon of coffee, and workers fill their mugs from it in sequence. Whoever takes the last drop has to make the next pot, no ifs, ands or buts. Every worker in the office is trying to take as much coffee as he or she can while minimizing the probability of having to refill the pot. Also, this pot is both incredibly heavy and completely opaque, so it’s tough to tell how much remains. That means a worker can’t keep pouring until she sees or feels just a drop left. Anyone stuck refilling the pot becomes so frustrated that they throw their cup to the ground in frustration, so they get no coffee that round.\n",
"\n",
"The aim is to choose a strategy (a number of gallons to try to take from the pot) that maximises total coffee consumed.\n",
"\n",
"## Nash Equilibrium\n",
"\n",
"While I don't think that the game has a Nash equilibrium under the rules above, we can tweak the rules very slightly to get one. Consider a similar game, where you lose if you try to take strictly more than the remaining amount. A Nash equilibrium (probably the only one) of this game is that every player goes for 1 gallon each time.\n",
"\n",
"To see that this is a Nash equilibrium, note that under this strategy, each player has a 50 % chance of getting 1 gallon. Every other strategy will make you win less coffee with the same probability.\n",
"\n",
"Now you may say that this is just a theoretical excercise. Clearly, in the real world, not everyone will go for the full pot. Therefore, I have dedicated the rest of this analysis to maximising the amount of coffee when the coworkers follow a distribution of strategies.\n",
"\n",
"## Simulations\n",
"\n",
"First, let's define a function for simulating many rounds to the coffee pot under a set of strategies. The strategies will be given as an array of numbers between 0 and 1."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"%matplotlib inline\n",
"import matplotlib.pyplot as plt\n",
"import numpy as np"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"def simulate(strategies, n=10000):\n",
" \"\"\"\n",
" Simulate `n` rounds to the coffee pot with the strategies given\n",
" by `strategies`. Plots a histogram of strategies and total\n",
" amount of coffee consumed per player.\n",
" \"\"\"\n",
" num_players = len(strategies)\n",
" gains = np.zeros(num_players)\n",
" order = np.arange(num_players)\n",
" \n",
" coffee_pot = 1.0\n",
"\n",
" for i in range(1, n + 1):\n",
" np.random.shuffle(order)\n",
" for j in order:\n",
" coffee_pot -= strategies[j]\n",
" if coffee_pot > 0:\n",
" gains[j] += strategies[j]\n",
" else:\n",
" coffee_pot = 1.0\n",
" if (i % 200) == 0:\n",
" print(f'\\r{i} out of {n} completed.', end='')\n",
"\n",
" fig, ax = plt.subplots(1, 2, figsize=(12, 5))\n",
" ax[0].hist(strategies, bins=20)\n",
" ax[0].set_title('Strategies')\n",
" ax[1].scatter(strategies, gains/n)\n",
" ax[1].set_title('Avg coffee consumed')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"First, let's confirm that the code gives the right answer for the theoretical case outlined above."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"scrolled": true
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"10000 out of 10000 completed."
]
},
{
"data": {
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qhlGLfqOqXj6qbCPwzqq6KsnjgSuTXDzGupK0TbHlWZLmrwOAtVV1c1X9FDgTOLzPilV1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\nkiRJUk//F2h/Lrf0WPuyAAAAAElFTkSuQmCC\n",
"text/plain": [
"<matplotlib.figure.Figure at 0x7f7ab6c90438>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"num_players = 5000\n",
"strategies = np.full(num_players, 0.999)\n",
"simulate(strategies)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now we can start testing a number of different strategy distributions. Let's start with a uniform distribution."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"10000 out of 10000 completed."
]
},
{
"data": {
"image/png": 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Ky731hTcMkVrBJeAkqctFRB9wKXACsB5YHhGLM/P2mmb3AL+XmY9FxEnA5cDc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\nYII=\n",
"text/plain": [
"<matplotlib.figure.Figure at 0x7f7ab368fd68>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"strategies = np.random.uniform(size=num_players)\n",
"simulate(strategies)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Here's a beta distribution with $\\alpha = \\beta = 3$."
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"10000 out of 10000 completed."
]
},
{
"data": {
"image/png": 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MfLI4f3Vmvgg8GBGrgUOA22sWtdQAjd4629U0NAmHAKsz8wGAiLiaUlv8SpKcmb8uK78t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\nqcL/D47Pc1HBSqheAAAAAElFTkSuQmCC\n",
"text/plain": [
"<matplotlib.figure.Figure at 0x7f7ab32a2550>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"strategies = np.random.beta(3, 3, size=num_players)\n",
"simulate(strategies)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's see what happens if people decide to be cautious. Here's a beta distribution with $\\alpha=5, \\beta=1$. I've added a player at 0.999 for illustrative ourposes."
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"10000 out of 10000 completed."
]
},
{
"data": {
"image/png": 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E7BERyyPi1uL77lXlz4yI9RGxLiKOnnz4kiR1hgM/cHWpcnee8+omRyKpmSbb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\n",
"text/plain": [
"<matplotlib.figure.Figure at 0x7f7ab31a1ba8>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"strategies = np.concatenate([np.random.beta(1, 5, size=num_players), [0.999]])\n",
"simulate(strategies)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"At last, we've found a situation when the optimum is not to be as greedy as possible. However, this relies on a lot of players choosing a bad strategy, so I think that this distribution is unlikely to occur.\n",
"\n",
"To round off, let's take a look a mixture of two beta distributions."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"10000 out of 10000 completed."
]
},
{
"data": {
"image/png": 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h4KCa1x9YbNtJZl4KXApwzDHHtHomEUlNMpX5nO+58JQWVKImuRj4QGbuiIhxD4yIs4Cz\nAObPn19CaZLK8pGvrmfL1u2lna/ThlaMZsKQnJnnAucCRMTxwF9l5tKIWAm8A7iw+P6V4iWrgc9H\nxEXAAcBC4PvNL13qTC4Kog7SSKfFMcDVRUDeB3h9RGzLzEr9m9m5IfWWVi8nPZY5A+2fA7kR05kn\n+UJgVUS8C7gXOB0gM9dHxCqqY962Ae/NzPL+NJEkjbgVWBgRh1ANx2cCf1h7QGYeMvI4Ij4LfG20\ngCypt5Q5rVu989/Q/jmQGzGpkJyZNwI3Fo9/Crx2jOMuAC6YZm2SpGnIzG0R8T5gDdAHfKboyHhP\nsf+SthYoqXTt6j0e0e6lpifDFfckqYdl5vXA9XXbRg3HmfnOMmqSVK6yZ60YzUB/X1eMQ65lSJYk\nSepRKypDXLn2vslNQdYkfRFsz2TenAGWnXhYVwVkMCRLkiT1jE7oNe6mIRXjMSRLkiT1gJG5jsuc\nyq3eXrt1x8wVjWhoWWpJkiR1tpVrNrQ1IPf3Beed2h0zVzTCnmRJkqQutqIyxFW33M/2LH/k8Syq\nSy4f0KXjjsdjSJYkSepS7ZzveOmi+Xx08ZFtOXcZDMmSJEldpjI4zPmr17N5S3vmO543Z6CnAzIY\nkiVJkrpKZXCYc1atY0cbF4dfduJh7Tt5SQzJkiRJHazdq+TV66UZLMZjSJYkSepQlcFhln3xDrZu\nL7/bOAICduqxHujv66kZLMZjSJYkSeogIwuCPLh5C7OKVevKNrIgSG0tvTiDxXgMyZIkSR2ifkGQ\ndgTk2uEUi186b8aE4nouJiJJktQhzl+9vq0Lgsyk4RQTsSdZkiSpzSqDw5xzzTp2lHzegf5ZPKe/\nj82/3DrjhlNMxJAsSZLUBu2ctWKegXhChmRJkqSSjNwIN7x5S1vO3zcr+MQfvMRw3ABDsiRJUgnq\nb8orm73Hk2NIliRJarHK4DDvX3VHW6dz0+QYkiVJkpqsMjjM+avXs3lL+1bJ658FK//AgDxVhmRJ\nkqQmWlEZ4oq197W1hjkD/aw773VtraHbOU+yJElSk1QGh9sekAf6+zj/Dc51PF32JEuSJDVBZXCY\nv1y1rq01eHNe8xiSJbXcguXXTel191x4SpMrkaTWqAwO85fXrKPs2/L6+4KVb3FKt1YwJEuSJE1R\nO2/Qs9e4tQzJkiRJk9Tu2SsCuGn5a9py7pnCkCxJkjQJlcFhln3hDrbuKH/O4xEHzBlo27lnCkOy\nJEnSONq9lHS9gf4+lp14WLvL6HmGZEmSpDFUBodZ9sU72Lq9/F7jvoBPnH40ACvXbODBzVs4wHHI\npTEkS5Ik1akMDvORr67n8V+2b8W833jewLNh2FBcPkOyJElSjcrgMOesWkcbhxwD8GCHDO+YqQzJ\nkiRJNc6+pr0Lgozw5rz2MiRLY5jqAhiSpO7TrpvzBvpn8ebfPpAv3T7Mlq3ba7Z7c167GZIlSdKM\ntqIyxBVr7yv1nHvt1s95px7x7FjjYw7e25vzOowhWZIkzTjtvDHv4jOO/rUAvPil8wzFHcaQLEmS\nel67V8iD6hCKj73pSMNwlzAka0ZwfLEkzVydsELenIF+zn/DEQbkLmJIliRJPW3lmg2lBeT+WXDG\nK+bz7R9ucnxxlzMkS5KknlT2EIvjDt2bK9/9ylLOpdYzJEuSpJ5T9owVSxfN56OLjyztfGo9Q7Ik\nSeoZlcFhPvTlIZ56ZvvEBzfBPIdT9CxDsiRJ6nplT+nm0IreZ0iWJEldrayhFc5QMbMYkiVJUlcq\nc9xxAOvOe10p51JnMCRLkqSuUp33eB1bd5R3zgPmDJR3MnUEQ7IkSep4lcFhVq7ZwPDmLaWfe6C/\nj2UnHlb6edVes9pdgCRJ0ngqg8Oce+1QKQF57h6zufiMo5k3Z4CgOnuFS0nPTPYkS1IPi4iTgP8J\n9AGfzswL6/afBvwtsAPYBpydmd8tvVBpHCvXbGDL1tZP6TZ3j9nc8qETAAzFMiRLUq+KiD7gk8AJ\nwAPArRGxOjPvqjnsW8DqzMyIOApYBbyo/GqlqtphFX0RbM/WLiftIiAay4QhOSKeA3wH2LU4/ouZ\neV5E7A1cAywA7gFOz8zHi9ecC7wL2A78eWauaUn1kqTxvALYmJk/AYiIq4HTgGdDcmb+oub43YHW\nJhJpHCPDKkZ6jVsZkA3HmkgjPclPA6/JzF9ERD/w3Yj4F+BNwLcy88KIWA4sBz4QEYcDZwJHAAcA\n34yIF2ZmOUvfSJJGzAPur3n+AHBs/UER8UbgY8B+wCnllCb9Spk35dUOqZDGM2FIzswERnoa+ouv\npNobcXyx/XLgRuADxfarM/Np4O6I2Ei1N+PmZhYuSWqOzPwy8OWI+B2q45N/b7TjIuIs4CyA+fPn\nl1egepar5KmTNTQmuRjXdjvwAuCTmXlLRMzNzIeKQx4G5haP5wFra17+QLFNklSuYeCgmucHFttG\nlZnfiYjfjIh9MvOxUfZfClwKcMwxxzgsQ9NSGRxm2RfvYOv21v9XWrjf7txwzvEtP496S0NTwGXm\n9sw8mmoD+4qIeHHd/mSS49gi4qyIuC0ibtu0adNkXipJasytwMKIOCQiZlMdCre69oCIeEFERPH4\nZVTvP/lp6ZVqRqkMDnP2NetaHpDnDPRz8RlHG5A1JZOa3SIzN0fEt4GTgEciYv/MfCgi9gceLQ5r\nqOfCHglJaq3M3BYR7wPWUJ0C7jOZuT4i3lPsvwR4M/D2iNgKbAHOKDo+pJYYCcit5E15aoZGZrfY\nF9haBOQBqlMJfZxqb8Q7gAuL718pXrIa+HxEXET1xr2FwPdbULskaQKZeT1wfd22S2oef5xqmy6V\notUB+Z4LvfdUzdFIT/L+wOXFuORZwKrM/FpE3Aysioh3AfcCpwMUvRSrqE4xtA14rzNbqFkWLL+u\n3SVIkqbg2Atu4JEnn2npOZYu8oZSNU8js1vcCbx0lO0/BV47xmsuAC6YdnWSJKkrVec8vpMtW3e0\n/Fx9Ebz12IMcYqGmcsU9SZLUNJXBYT7wpTt5eltrw7HBWK1mSJYkSU1RGRzm/V+4g+07Wnfv5167\n9TP44de17P2lEYZkSZI0bSsqQ1yx9r6WnmOgv4/zTj2ipeeQRhiSJUnStCy57GZu+vHPWnqOvXbr\n57xTj2DxS12fTOUwJEuSpEmrDA6zcs0Ghjdvafm5nPdY7WBIliRJk7KiMsSVa++b3FK7DVi6aD7H\nHLw3K9ds4MHNWzhgzgDLTjzM3mO1hSFZkiQ1rDI43JKxxxefcfSzYdhQrE5gSJYkSROqDA5z/ur1\nbN6ytenvvXTRfIOxOo4hWZIkjasyOMyyL9zB1iZP7eZcx+pkhmRJkvRrWnljnjfiqRsYkiVJ0k6q\nS0oPsWXr9qa95+6z+7jgjUc6rEJdw5AsqWMtWH7dlF53z4WnNLkSaWb5yFfXNy0gB7DEnmN1IUOy\nJEl61orKEI//cvo35w30z+JjbzrKnmN1LUOyJEkzXDPHHwfw9zXTuUndypAsSdIMtqIy1LR5j/v7\ngpVveYkBWT1hVrsLkCRJ7dHMhUF2n91nQFZPsSdZkqQZ6uxr1jXlfS52eIV6kCFZkqQZZsllN3PT\nj3/WlPfaa7d+A7J6kiFZkqQZojI4zPtXrWN7kxbO6+8Lzjv1iOa8mdRhDMmSJPW4yuAwH/ryEE89\nM725j/fctY89Bmbz4OYtHDBngGUnHmYvsnqWIVmSpB5WGRzm/V+4g+07ptd97LhjzTTObiFJUg/7\nqyYE5DkDjjvWzGNIliSpB1UGh1n4wevYNs2APCvg/Dc47lgzj8MtJEnqMZXB4aZM77bXbv2cd+oR\n9iJrRjIkS5LUIyqDw5xzzTp2TOM9jjt0b6589yubVpPUrQzJkiT1gGb0HhuQpV9xTLIkSV2uGQF5\n6aL5BmSphj3JkiR1sRWVIa5Ye9+UX+/UbtLoDMmSJHWp6S4vvXTRfAOyNAZDsiRJXWg6AXmeq+VJ\nEzIkS5LUZVZUhqYckB1eITXGG/ckSeoilcHhKY1BDgzI0mTYkyxJUpc44aIb+dGjT03ptX9vQJYm\nxZAsSVKHm+4iId6gJ02eIVltsWD5de0uQZK6wnSmePMGPWnqDMmSJHWoqY4/BrjnwlOaXI00s3jj\nniRJHeqvv3jHlF63dNH8JlcizTz2JEuS1IFOuOhGntmek3pNAEsWzeeji49sTVHSDGJIliSpg0z1\nJj2nd5Oay5AsSVKHqAwOc/Y16yb9OmevkJrPMcmSJHWA6QRkh1dIzWdPsiRJbTaVad4W7rc7N5xz\nfGsKkmRPsiRJ7TSVad4CDMhSixmSJUlqo8kOsdgl4G7nQJZazpAsST0sIk6KiA0RsTEilo+yf0lE\n3BkRQxHxvYh4STvqnKmOOu/rkzp+6aL5bPyYAVkqgyFZknpURPQBnwROBg4H3hoRh9cddjfwu5l5\nJPC3wKXlVjlzHXXe13ni6e0NH+8NelK5DMmS1LteAWzMzJ9k5jPA1cBptQdk5vcy8/Hi6VrgwJJr\nnJEmG5AX7re7AVkqmSFZknrXPOD+mucPFNvG8i7gX1pakTjhohsnHZC9SU8qn1PAaVoWLL+u3SVI\naoKIeDXVkPyqcY45CzgLYP78+SVV1jsqg8O8f9U6JrPStAFZap8Je5Ij4qCI+HZE3BUR6yPiL4rt\ne0fEDRHxo+L7XjWvObe4SWRDRJzYyguQJI1pGDio5vmBxbadRMRRwKeB0zLzp2O9WWZempnHZOYx\n++67b9OL7WUjC4VMJiDP3WO2AVlqo0aGW2wD3p+ZhwOLgPcWN34sB76VmQuBbxXPKfadCRwBnAR8\nqrh5RJJUrluBhRFxSETMpto2r649ICLmA9cCb8vM/2xDjTPCB6+9c1LHz91jNrd86IQWVSOpEROG\n5Mx8KDP/vXj8JPADqmPaTgMuLw67HFhcPD4NuDozn87Mu4GNVG8ekSSVKDO3Ae8D1lBtu1dl5vqI\neE9EvKc47MPA86l2aKyLiNvaVG7POuGiG/nl1h0NH7900XwDstQBJjUmOSIWAC8FbgHmZuZDxa6H\ngbnF43lU75AeMeqNIo5tk6TWy8zrgevrtl1S8/iPgT8uu66Z4kUfup7/msQYC2exkDpHw7NbRMRz\ngS8BZ2fmE7X7MjOBSYy0cmybJKm3nXDRjZMKyM/pC8cgSx2koZAcEf1UA/KVmXltsfmRiNi/2L8/\n8GixvaEbRSRJ6mU/evSpho+du8dsfnjB61tYjaTJamR2iwD+N/CDzLyoZtdq4B3F43cAX6nZfmZE\n7BoRhwALge83r2RJkjrbZKbHXLjf7o5BljpQI2OSjwPeBgxFxLpi2weBC4FVEfEu4F7gdIDippBV\nwF1UZ8Z4b2Y2Pmu6JEld7AXnNh6Q99y1zyEWUoeaMCRn5neBGGP3a8d4zQXABdOoS5KkrnPsBTew\nrcFhyHvu2sedHzmptQVJmjJX3JMkqQmOOu/rDS83/Zy+MCBLHa7h2S0kSdLoTrjoxoYD8i6BN+lJ\nXcCQLEnSNE1mJouNHzulhZVIahZDsiRJU1QZHJ7UTBYXn3F0C6uR1EyGZEmSpqAyOMzZ16yb+MDC\n0kXzWfzSX1uAVlKHMiRLkjQFkw3ILjctdRdnt5AkaZKOveCGho+9+Iyj7UGWupA9yZIkTUJlcJhH\nnnymoWMNyFL3sidZwOSWUJWkmazRYRbHHbq3AVnqYvYkS5LUoKPO+3rDx1757le2sBJJrWZIliSp\nASsqQw0vGOJUb1L3MyRLkjSBFZUhrlh7X0PHOg5Z6g2GZEmSJmBAlmYeQ7IkSeNodBzywv12NyBL\nPcSQLEnSGJZcdnND45Dn7jGbG845vvUFSSqNU8D1GKdyk6TmqAwOc9OPfzbhcQHc8qETWl+QpFLZ\nkyxJ0iganQ/5753JQupJhmRJkuq86EPXN3ScC4ZIvcuQLElSjcrgMP+1PSc8bumi+S4YIvUwQ7Ik\nSTUaGWax5659fHTxkSVUI6ldDMmSJBUavfn5zo+c1OJKJLWbIVmSJOAF5zYWkF1yWpoZDMmSpBlv\nyWU3s23iYcjsEnijnjRDGJIlSTNao/MhA2z82CktrkZSpzAkS5JmtEbnQ3aYhTSzGJIlSTPWkstu\nbug450Myvy9KAAAM3ElEQVSWZh5DsiRpxmp0mIXzIUszjyFZkjQjNbqqnsMspJnJkCxJmnFWVIYa\nXlXPYRbSzLRLuwvQ6Bqd0F6SNHlXrL1vwmMuPuNoA7I0g9mTLEmaURodZmFAlmY2Q7IkacaoDA43\nNMzCcciSDMmSpBmjkTmRF+63u73IkgzJkqSZ4QXnTnyvxy4BN5xzfOuLkdTxDMmSpJ53wkU3sm3i\nURYuOy3pWc5uIannTHV2mHsuNCD1qh89+lS7S5DUZexJliT1tGMvuKGh47xZT1ItQ7IkqWetqAzx\nyJPPTHjccYfu7c16knZiSJYk9axGFg057tC9ufLdryyhGkndxJAsSepJlcHhCY8JMCBLGpUhWZLU\nkxqZE3nJovklVCKpGxmSJamHRcRJEbEhIjZGxPJR9r8oIm6OiKcj4q/aUWMrnHDRjRMes3TRfD66\n+MjWFyOpKzkFnCT1qIjoAz4JnAA8ANwaEasz866aw34G/DmwuA0ltkwjU74ZkCWNx55kSepdrwA2\nZuZPMvMZ4GrgtNoDMvPRzLwV2NqOAluhkSnfFu63ewmVSOpmhmRJ6l3zgPtrnj9QbOtZx15wQ0NT\nvrn0tKSJGJIlSQ2JiLMi4raIuG3Tpk3tLmdUjQRkFw2R1AhDsiT1rmHgoJrnBxbbpiQzL83MYzLz\nmH333XfaxTXbUed9fcJjXDREUqMMyZLUu24FFkbEIRExGzgTWN3mmlqiMjjME09vH/eYpYvmOyey\npIZNOLtFRHwG+H3g0cx8cbFtb+AaYAFwD3B6Zj5e7DsXeBewHfjzzFzTksq7yILl17W7BEkzUGZu\ni4j3AWuAPuAzmbk+It5T7L8kIn4DuA3YE9gREWcDh2fmE20rfAoamRPZ2SwkTUYjU8B9FvhH4HM1\n25YD38rMC4t5N5cDH4iIw6n2VBwBHAB8MyJemJnj/3nfJQy7krpNZl4PXF+37ZKaxw9THYbRtRqd\nE1mSJmPC4RaZ+R2q82jWOg24vHh8Ob+aX/M04OrMfDoz7wY2Up2CSJKklnBOZEmtMNUxyXMz86Hi\n8cPA3OLxjJtuSJLUPvYiS2qVad+4l5kJ5GRf1w1TCUmSOteKytCEvci77jLLXmRJUzLVkPxIROwP\nUHx/tNje8HRDnT6VkCSps1259r4Jj/n4m48qoRJJvWiqIXk18I7i8TuAr9RsPzMido2IQ4CFwPen\nV6IkSTtbctnNE36EuXTRfOdEljRljUwBdxVwPLBPRDwAnAdcCKyKiHcB9wKnAxRTC60C7gK2Ae/t\nlZktJEmdoTI4zE0/rr+ffGcOs5A0XROG5Mx86xi7XjvG8RcAF0ynKEmSxtLInMgOs5A0Xa64J0nq\nGpXBiVfVvviMox1mIWnaDMmSpK5xzqrxe5EdhyypWQzJkqSuUBkcZscEd+s5DllSsxiSJUldYdkX\nJh6LLEnNYkiWJHW8FZUhtu4Y/5iLzzi6nGIkzQiGZElSx/v8LRMvHOJYZEnNZEiWJHW8icYiL100\nv5xCJM0YhmRJUkdbctnN4+7vn+UNe5Kaz5AsSepoE62ut/IPHIssqfkMyZKkjjVRL7LzIktqFUOy\nJKkjLbns5gl7kR1mIalVDMmSpI6zojI0YUA+7tC9S6pG0kxkSJYkdZwr10485duV735lCZVImqkM\nyZKkjlIZHGaCGd+c8k1SyxmSJUkdZaLlp487dG/HIktqOUOyJKljVAaHx11+un+WwywklcOQLEnq\nGB/68tC4+50TWVJZDMmSpI6wojLEU89sH3N/gHMiSyqNIVmS1HaVweEJZ7RY4s16kkpkSJYktd3K\nNRvGndGif5YLh0gqlyFZktR2w5u3jLvfsciSymZIliS11ZLLbh53/9JF8x2LLKl0u7S7gHZYsPy6\ndpcgSaI6Fnm85aeXLprvMAtJbWFPsiSpbc5fvX7c/QZkSe1iSJYktc3mLVvH3NcXUWIlkrQzQ7Ik\nqSO99diD2l2CpBnMkCxJaosVlfFX13OohaR2MiRLkko30eIhcwb6S6xGkn6dIVmSVLqJFg85/w1H\nlFaLJI3GkCxJKt2D4ywestdu/c6LLKntDMmSpNIdMGdg1O0BnHeqvciS2m9GLiYiSWqPyuAwK9ds\nYHjzFgJ2GnIRwBJX15PUIQzJkqRSVAaHOffaIbZs3Q5UA/JIUJ43Z4BlJx5mQJbUMQzJkqRSrFyz\n4dmAPGIkIN+0/DXtKUqSxuCYZElSKca6WW+8m/gkqV0MyZKkUox1s95Y2yWpnQzJkqRSLDvxMAb6\n+3baNtDfx7ITD2tTRZI0NsckS5JKMXJT3so1G3hw8xYO8GY9SR3MkCxJKs3il84zFEvqCg63kKQe\nFhEnRcSGiNgYEctH2R8R8Q/F/jsj4mXtqFOSOk1X9yQvWH5du0uQpI4VEX3AJ4ETgAeAWyNidWbe\nVXPYycDC4utY4J+K75I0o9mTLEm96xXAxsz8SWY+A1wNnFZ3zGnA57JqLTAnIvYvu1BJ6jSGZEnq\nXfOA+2ueP1Bsm+wxkjTjGJIlSQ2JiLMi4raIuG3Tpk3tLkeSWsqQLEm9axg4qOb5gcW2yR4DQGZe\nmpnHZOYx++67b1MLlaROY0iWpN51K7AwIg6JiNnAmcDqumNWA28vZrlYBPw8Mx8qu1BJ6jRdPbuF\nJGlsmbktIt4HrAH6gM9k5vqIeE+x/xLgeuD1wEbgl8AftateSeokhmRJ6mGZeT3VIFy77ZKaxwm8\nt+y6JKnTOdxCkiRJqtOykDzRKk+SJElSp2pJSK5Z5elk4HDgrRFxeCvOJUmSJDVbq3qSG1nlSZIk\nSepIrQrJruAkSZKkrtW22S0i4izgrOLpLyJiQwMv2wd4rHVVtV2vXx94jb2gZ68vPv7sw8le48FN\nL6bD3X777Y9FxL2TeEnP/r+ht68Nevv6evnaoLevbzrX1lCb3aqQPOEKTpl5KXDpZN40Im7LzGOm\nX15n6vXrA6+xF/T69cHMuMbpysxJLbnXyz/TXr426O3r6+Vrg96+vjKurVXDLRpZ5UmSJEnqSC3p\nSR5rladWnEuSJElqtpaNSR5tlacmmNTwjC7U69cHXmMv6PXrg5lxjWXr5Z9pL18b9Pb19fK1QW9f\nX8uvLaorkkqSJEka4bLUkiRJUp2OC8kTLWcdVf9Q7L8zIl7Wjjqno4FrXFJc21BEfC8iXtKOOqej\n0WXJI+LlEbEtIt5SZn3T1cj1RcTxEbEuItZHxL+VXeN0NfD/9HkR8dWIuKO4xj9qR51TFRGfiYhH\nI+I/xtjf9W1N2Xq9/e7ltts2u3vb7F5uq9veTmdmx3xRvcnvx8BvArOBO4DD6455PfAvQACLgFva\nXXcLrvG/AXsVj0/uxWusOe5fqY5df0u7627yv+Ec4C5gfvF8v3bX3YJr/CDw8eLxvsDPgNntrn0S\n1/g7wMuA/xhjf1e3NR36f6Zrf6a93HbbZndvm93rbXW72+lO60luZDnr04DPZdVaYE5E7F92odMw\n4TVm5vcy8/Hi6Vqq80x3k0aXJf8z4EvAo2UW1wSNXN8fAtdm5n0AmdmL15jAHhERwHOpNrzbyi1z\n6jLzO1RrHku3tzVl6/X2u5fbbtvs7m2ze7qtbnc73WkhuZHlrLt9yevJ1v8uqn8ldZMJrzEi5gFv\nBP6pxLqapZF/wxcCe0XEjRFxe0S8vbTqmqORa/xH4LeAB4Eh4C8yc0c55ZWi29uasvV6+93Lbbdt\ndve22TO9rW5pm9K2Zak1sYh4NdWG9lXtrqUFLgY+kJk7qn/c9pxdgN8GXgsMADdHxNrM/M/2ltVU\nJwLrgNcAhwI3RMT/ycwn2luW1F492nbbZncv2+op6rSQPOFy1g0e08kaqj8ijgI+DZycmT8tqbZm\naeQajwGuLhrbfYDXR8S2zKyUU+K0NHJ9DwA/zcyngKci4jvAS4BuaXAbucY/Ai7M6sCwjRFxN/Ai\n4PvllNhy3d7WlK3X2+9ebrtts7u3zZ7pbXVL25ROG27RyHLWq4G3F3c0LgJ+npkPlV3oNEx4jREx\nH7gWeFuX/hU74TVm5iGZuSAzFwBfBP60SxpbaOz/6VeAV0XELhGxG3As8IOS65yORq7xPqq9LkTE\nXOAw4CelVtla3d7WlK3X2+9ebrtts7u3zZ7pbXVL25SO6knOMZazjoj3FPsvoXpX7euBjcAvqf6F\n1DUavMYPA88HPlX81b4tM49pV82T1eA1dq1Gri8zfxARXwfuBHYAn87MUaew6UQN/hv+LfDZiBii\nemfxBzLzsbYVPUkRcRVwPLBPRDwAnAf0Q2+0NWXr9fa7l9tu2+zubbN7va1udzvtinuSJElSnU4b\nbiFJkiS1nSFZkiRJqmNIliRJkuoYkiVJkqQ6hmRJkiSpjiFZkiRJqmNIliRJkuoYkiVJkqQ6/xdy\nnCKAb9DDnwAAAABJRU5ErkJggg==\n",
"text/plain": [
"<matplotlib.figure.Figure at 0x7f7ab2fd4748>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"strategies = np.concatenate([\n",
" np.random.beta(2, 2, size=num_players//2),\n",
" np.random.beta(5, 2, size=num_players//2),\n",
"])\n",
"simulate(strategies)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Conclusion\n",
"\n",
"The optimal strategy should in most cases be to try to take arbitrily close to 1 gallon, in other words betting on being the first in line to a new pot."
]
}
],
"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": "3.6.1"
}
},
"nbformat": 4,
"nbformat_minor": 2
}
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