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@ -2210,11 +2210,11 @@
} }
}, },
"source": [ "source": [
"This says that for a casino with a bankroll of 100 million dollars, if you want to maximize your expected value, you should be willing to pay up to \$27.49 to play the game. Would you pay that much? I wouldn't, and neither would Daniel Bernoulli. \n", "This says that for a casino with a bankroll of 100 million dollars, if you want to maximize your expected value, you should be willing to pay up to \\$27.49 to play the game. Would you pay that much? I wouldn't, and neither would Daniel Bernoulli. \n",
"\n", "\n",
"## Response 2: Value of Money\n", "## Response 2: Value of Money\n",
"\n", "\n",
"Daniel Bernoulli came up with a second response to the paradox based on the idea that if you have a lot of money, then additional money becomes less valuable to you. If I had nothing, and I won \$1000, I would be very happy. But if I already had a million dollars and I won \$1000, it would be less valuable. How much less valuable? Bernoulli proposed, and [experiments confirm](https://books.google.com/books?id=1oEa-BiARWUC&pg=PA205&lpg=PA205&dq=mr+beard+oil+wildcatter+value+of+money+utility&source=bl&ots=cBDIX-rkTz&sig=GHB8-inorWrU39vA8JYV_sCtqB8&hl=en&sa=X&ved=0CCAQ6AEwAGoVChMI5fu-p8qlyAIViKWICh0XAAz5#v=onepage&q=mr%20beard%20oil%20wildcatter%20value%20of%20money%20utility&f=false), that *the value of money is roughly logarithmic.* That is, rational bettors don't try to maximize their expected monetary value, they try to maximize their *expected utility*: the amount of \"happiness\" that the money is worth.\n", "Daniel Bernoulli came up with a second response to the paradox based on the idea that if you have a lot of money, then additional money becomes less valuable to you. If I had nothing, and I won \\$1000, I would be very happy. But if I already had a million dollars and I won \\$1000, it would be less valuable. How much less valuable? Bernoulli proposed, and [experiments confirm](https://books.google.com/books?id=1oEa-BiARWUC&pg=PA205&lpg=PA205&dq=mr+beard+oil+wildcatter+value+of+money+utility&source=bl&ots=cBDIX-rkTz&sig=GHB8-inorWrU39vA8JYV_sCtqB8&hl=en&sa=X&ved=0CCAQ6AEwAGoVChMI5fu-p8qlyAIViKWICh0XAAz5#v=onepage&q=mr%20beard%20oil%20wildcatter%20value%20of%20money%20utility&f=false), that *the value of money is roughly logarithmic.* That is, rational bettors don't try to maximize their expected monetary value, they try to maximize their *expected utility*: the amount of \"happiness\" that the money is worth.\n",
"I'll write the function `util` to describe what a dollar amount is worth to a hypothetical gambler. `util` says that a dollar is worth a dollar, until the amount is \"enough\" money. After that point, each additional dollar is worth half as much (only brings half as much happiness). Value keeps accumulating at this rate until we reach the next threshold of \"enough,\" when the utility of additional dollars is halfed again. The exact details of `util` are not critical; what matters is that overall money becomes less valuable after we have won a lot of it." "I'll write the function `util` to describe what a dollar amount is worth to a hypothetical gambler. `util` says that a dollar is worth a dollar, until the amount is \"enough\" money. After that point, each additional dollar is worth half as much (only brings half as much happiness). Value keeps accumulating at this rate until we reach the next threshold of \"enough,\" when the utility of additional dollars is halfed again. The exact details of `util` are not critical; what matters is that overall money becomes less valuable after we have won a lot of it."
] ]
}, },
@ -2388,11 +2388,11 @@
} }
}, },
"source": [ "source": [
"That says we should pay up to \$13.10 to play the game, which sounds more reasonable than \$27.49.\n", "That says we should pay up to \\$13.10 to play the game, which sounds more reasonable than \\$27.49.\n",
"\n", "\n",
"# Understanding St. Petersburg through Simulation\n", "# Understanding St. Petersburg through Simulation\n",
"\n", "\n",
"Before I plunk down my \$13, I'd like to understand the game better. I'll write a simulation of the game:" "Before I plunk down my \\$13, I'd like to understand the game better. I'll write a simulation of the game:"
] ]
}, },
{ {
@ -2608,7 +2608,7 @@
} }
}, },
"source": [ "source": [
"What can we see from this? Nine of the 10 repetitions have a final expected value payoff (after 100,000 rounds) between 10 and 35. So a price around 13 dollars still seems reasonable. One outlier has an average payoff just over 100, so if you are feeling lucky you might be willing to pay more than \$13.\n", "What can we see from this? Nine of the 10 repetitions have a final expected value payoff (after 100,000 rounds) between 10 and 35. So a price around 13 dollars still seems reasonable. One outlier has an average payoff just over 100, so if you are feeling lucky you might be willing to pay more than \\$13.\n",
"\n", "\n",
"# The Ellsburg Paradox\n", "# The Ellsburg Paradox\n",
"\n", "\n",