"I don't know, I recall something called the Kelly criterion which says you shouldn't scale your willingness to make risky bets proportionally with available capital - that is, you shouldn't be just as eager to bet your capital away when you have a lot as when you have very little, or you'll go into the red much faster.
I think I'm misunderstanding something here. Let's say you have dollars and are looking for the optimum number of dollars to bet on something that causes you to gain dollars with probability and lose dollars with probability . The optimum number of dollars you should bet via the Kelly criterion seems to be
(assuming positive expectation; i.e. the numerator is positive), which does scale linearly with . And this seems fundamental to this post.
I think you're right, I had misunderstood! Kind of an egregious misunderstanding, too.
I'm curious how it seems to you to be fundamental to the post, maybe I missed something on that count. I'm planning on replacing 'shouldn't scale . . . proportionally' with 'shouldn't scale more than proportionally', and I don't see how that substantially changes anything, given that Jill is right, and the concept of a Kelly fraction isn't applicable when you're starting out with zero capital of your own to gamble.
I think even the scaling thing doesn't apply here because they're not insuring bigger trips: they're insuring more trips (which makes things strictly better). I'm having some trouble understanding Dennis' point.
0 trips -> 1 trip is an addition to the number of games played, but it's also an addition to the percentage of income bet on that one game - right?
Dennis is also having trouble understanding his own point, FWIW. That's how the dialogue came out; both people in that part are thinking in loose/sketchy terms and missing important points.
The thing Dennis was trying to get at by bringing up the concrete example of an optimal Kelly fraction is that it doesn't make sense for willingness to make a risky bet to have no dependence on available capital; he perceives Jill as suggesting that this is the case.
I’m too dumb to understand this post. It seems to start with some reasonable math (though the switch from per-season to per-voyage calculations makes it unnecessarily obtuse), but then introduces a bunch of names participants in a way that it’s hard to tell if they’re unreliable and incorrect or intended to be making valid points that the author assumes the reader doesn’t understand.
Not to be too cheeky, the idea is that if we understand insurance, it should be easy to tell if the characters' arguments are sound-and-valid, or not.
The obtuse bit at the beginning was an accidental by-product of how I wrote it; I admittedly nerdsniped myself trying to find the right formula.
Is it sufficient to understand that insurance only applies to the transactional monetary level, and most of the post was about other levels and considerations? Or that the characters didn't MAKE any clear arguments, just some noises about modeling that doesn't obviously apply to the question at hand (how to share/smooth the risk of variable but overall-profitable actions)?
the idea is that if we understand insurance, it should be easy to tell if the characters' arguments are sound-and-valid, or not.
Umm, the difficulty was even understanding what the arguments are. At first glance, they are mostly irrelevant to the proposal (of an insurance pool among voyage-financiers).
The characters don't live in a world where sharing or smoothing risk is already seen as a consensus-valuable pursuit; thus, they will have to be convinced by other means.
I gave their world weirdly advanced [from our historical perspective] game theory to make it easier for them to talk about the question.
Moriarty looks at the paper.
The switch from first person pronouns to discussing a third person with quotations is confusing.
Is the part where the stuff "on the paper" is in a code block visible to you yet? I was messing with the formatting for a while, I couldn't get it to display right. [Thanks for the feedback!]
I'm using Firefox. As of the time of writing after refreshing the page: from the Aristotle quote downwards the entire post is in one 'code block'. The markdown hyperlinks aren't formatting correctly, so I'm seeing the text in square brackets followed by the intended web address in plain text.
How does it look for you?
Edit: Ah, whoops, yes, that'd be the old screwed-up version of the formatting. It must not have updated on your end yet. [I'm using Firefox too.]
For me, the code block stops right before "Moriarty looks at the paper", and the Markdown hyperlinks are displaying properly.
[ cross-posted from my blog ]
"But with regard to the things that are done from fear of greater evils or or some noble object (e.g. if a tyrant were to order one to do something base, having one's parents and children in his power, and if one did the action they were to be saved, but otherwise would be put to death), it may be debated whether such actions are involuntary or voluntary. Something of the sort happens also with regard to the throwing of goods overboard in a storm; for in the abstract no one throws goods away voluntarily, but on condition of its securing the safety of himself and his crew any sensible man does so."
-- Aristotle, Nichomachean Ethics
10% chance/season of 1 voyage getting hit w/ a tropical storm [causing me to have to throw $1,000,000 worth of cargo overboard and compensate client accordingly]
Each successful voyage pays me $200,000
6 voyages / season
Depending on which voyage gets hit, a storm might put me into debt and force me to quit sailing until I can raise more capital, and in any case, it would negate my profits for the season.
I only make money in the timeline where I don't get hit at all, but this has a 90% chance of happening. I would then earn $1,200,000 x 90% = an expected profit of $1,080,000.
But I have a [ 1 - (9/10)^(1/6) ] ~= 1.7% chance of losing -$1M,
a (9/10)^(1/6)*~1.7% chance of losing net $200K - $1M = -$800K,
a (9/10)^(1/5)*~1.7% chance of losing net $400K - $1M = -$600K,
a (9/10)^(1/4)*~1.7% chance of losing net $600K - $1M = -$400K,
a (9/10)^(1/3)*~1.6% chance of losing net $800K - $1M = -$200K,
and a (9/10)^(1/2)*~1.6% chance of losing net $1M - $1M = $0.
$1,080,000
[ ~1.7% x -$1M ] ~= $17,000
[ ~1.7% x -$800K ] ~= $13,600
[ ~1.7% x -$600K ] ~= $10,200
[ ~1.7% x -$400K ] ~= $6,800
[ ~1.6% x -$200K ] ~= $3,200
[ ~1.6% x -$0 ] = $0
----------------------------------------
= a final expected profit of ~$1,029,200Moriarty looks at the paper.
Even though I have a 10% chance of one of the really bad outcomes happening to me, I still expect to profit by sailing this season.
In fact, I stand to profit big in the event of the median outcome, even though I stand to lose big in the event of the outlier outcomes.
Moriarty looks at the table. He considers the numbers.
They feel inefficient.
I expect to make $1,029,200, and the probabilities behind this figure are real, backed up by every nautical table, and if there was a table at a betting house being run on whether I'll get hit, they'd be backed up by every bookie - yet if the boat gets hit, I have nothing to show for it.
I have all this wealth - in a very valid sense, real wealth. I should be able to spend some of the money from the good timeline, on not having the losses be so devastating in the bad timelines!
He thinks.
He gets an idea.
He makes some notes on a paper, and he heads to the bar.
A few of his sailor friends are already there, and some of them are talking to other sailors. Within a few minutes he has a group with 9 other sailors together at a table. He gets them talking about this - he's good at that, he buys them all one round of drinks, and dives into the topic before they can get any drunker - and they agree, they're all in a similar situation. They'd like to be able to spend some of the money from their good timeline to mitigate the effects in their bad timelines, too. Wilhelmina says she remembers reading about this, and that it's called diminishing utility on the value of money, and that by the St. Petersburg argument it's implied for all rational agents. That seems to seal the deal for everyone that this line of thinking makes sense.
On Moriarty's suggestion, they draw up a contract to all pay $30,000 per successful voyage, so that a full $1M can be paid out to cover anyone who gets hit, and thus make the profits from their season positive, with some leeway, to be evenly distributed at the end of the season.
One hesitates to sign - Dennis. He says, "Wait, if the people who get hit once have their losses recovered, and they get to go back and continue the sailing season, doesn't that increase the total probability of being hit experienced by the group?"
Moriarty says, "Well, yes, but the expected return on each voyage is still positive, so it should still be okay."
Dennis says, "I don't know, I recall something called the Kelly criterion which says you shouldn't scale your willingness to make risky bets more than proportionally with available capital - that is, you shouldn't be more eager to bet your capital away when you have a lot versus when you have very little, or you'll go into the red much faster. It seems like maybe people who get hit once, and get the payout, should know when to hold 'em, as they say."
Jill says, "The whole point is so we can keep sailing."
Dennis says, "No, the whole point is so we can make money."
Wilhelmina says, "No, the whole point is so we can make utility.
Jill says, "Dennis, all of us are starting out with zero capital. If the idea was to not risk going into the red, shouldn't we just not sail? Like, I sort of get it, you're asking them to take one for the team, but not taking chances isn't what Kelly betting is about, it's about using the law of large numbers to reduce the volatility of your capital in random games once you get enough of it accumulated - which is the very thing we're trying to do, making this deal."
"Kelly betting is based on the assumption of a positive expected per-game return that is fixed as a percentage of the bet quantity -"
"- okay, then it doesn't apply -"
"- right, we're in a less risk-elastic scenario, meaning per-endeavor risk tolerance should grow even more slowly as a function of capital -"
"- then why are there rich sailors?"
"There aren't! There are rich sailing company owners, which is what you would expect to see under -"
Moriarty, who has been calculating the new numbers this whole time, holds his sheet up.
Next to the old failure probabilities for each voyage - 1.7%/1.7%/1.7%/1.7%/1.6%/1.6% - he's written down the new failure probabilities: 1.7%/1.7%/1.7%/1.7%/1.7%/1.7%. Next to the old expected profit - $1,029,200 - he's written down the new expected profit - $1,029,000.
"Yes", says Moriarty, "it'll cost us somewhat to make this deal, because there'll be more sailing and thus more incidents. But like I said, the expected return on each individual voyage is still positive, so we'll actually net make money by being able to sail more!"
"Utility", mumbles Wilhelmina.
Dennis fidgets. "We probably won't actually run into much problem if we just do this with the ten of us, but I'm just thinking - if we take this to its logical conclusion - the market for overseas transportation is calibrated to expect the quantity of supply - amortized over the season - that is available, conditional on N incidents taking N sailors out of the market - roughly, that is, at least for sailors of around our means - for some particular N, that we'd be reducing. What we're doing will not only increase merchandise loss, but also increase total market supply without increasing service quality, causing prices to go down. Sure, we might not end up nominally making less, at the very least because of inflation, but in the long run, if everyone adopts this policy, are we sure we're not just shooting ourselves in the foot?"
Moriarty can't help but be a little irritated. "Would you rather be shot in the foot out at sea? That is, are you really that indifferent to suffering steep costs under high-volatility conditions, versus suffering marginal costs under low-volatility conditions?"
"I'll get shot in the foot out at sea either way", protests Dennis. "The sailing industry still goes into the same amount of net debt to its consumer market as a result of that one storm whether you guys pay it off for me and let me keep sailing, or I bow out of the game for the season and save up to pay it off myself."
George says "Well, yeah, but you're suffering high opportunity costs while you're scrounging to pay off debt incurred to your client, because you're taking time out of the labor market."
"For sailing?" Dennis says. "What I'm saying is, I don't think the extra money is really there in the market anyway."
"Sure it is," says Moriarty. "Clients are willing to pay $200,000 for their cargo being delivered on time!"
"Yeah, under the assumption that product losses will be fully recouped," says Dennis. "The market price doesn't reflect clients' true aversion to that risk. If it did, it would be lower."
"Can I ask what's probably a stupid question?" says Xena. All heads turn to the strange tall woman in the cloak that looks like it's woven with tiny threads of bronze. "For context," she says, "I'm not from around here." There's an abashedly delayed chorus of assent. Xena says, "This 'debt' thing - I've been warned, many times now, not just by you guys, by everyone I've asked for information since I arrived in this country - that I'll have to guard against it, as a contract sailor here."
"I've had no reason to doubt everyone's word for that, even though I don't understand, and what you - Moriarty - proposed tonight seemed like a reasonable scheme for guarding against it, as far as I can tell, so I've kept listening and nodding along. And I'm to the point of signing, if everyone else gets on board, because" - she gestures around - "this whole thing has the vibe of a true prospective market-breaking conspiracy, not a plant to trick a hapless foreigner out of her earnings, and I can't see how what you" - glance at Moriarty - "describe wouldn't work.
"But this debt thing. I haven't been able to get a straight answer from anybody on -" Xena's face twists with frustration "- anything! What it is, why we're doing it, how it works." She looks around. "Admittedly, I have some other questions, too, but that's the main one."
After that the conversation goes very differently.
Later that night, Moriarty fumes and shifts about in bed. If Xena had not been such an impressive and well-dressed woman, he thinks, or so articulate, then indeed it would have been easy for the company to laugh off her question and continue on with securing Dennis's signature.
But as it happened, no one could seem to give her a satisfying answer.
They'd talked about education, defense, and the problem of distributing naturally illiquid assets - she'd talked about "on-the-job-training" and ISAs, and a decision theory implying that you shouldn't pay racketeers, and some really complicated things about capital and rent-seeking, and "virtual LVT".
It was all unproven bullshit, as far as Moriarty was concerned - either foreigner religion, or tall tales flat out. But it had dissolved the company, and the prospective contract, and Moriarty's chances to insure himself against losing $1M on the season and risking his childrens' futures again. Xena had just kept getting more and more confused and skeptical, and everyone else had gotten infected with it, sounding less and less like they were trying to convince her and more and more like they were trying to convince themselves. In the end, the only contract that had gotten signed was a three-way marriage contract between her, Dennis, and Wilhelmina, who had all left the table to sail back immediately to Xena's home country. Dennis had left saying that, if he could get out of his student debt, he might be able to actually use what he'd learned studying for his degree. Wilhelmina hadn't had any debt, if she was to be believed, but had left saying that in Xena's society, she might be able to actually finance her speculative wastewater treatment idea. Xena had left predicting ill for Moriarty's venture and his people's entire way of life. Maybe so, thought Moriarty, but we've still got to keep living it.
Afterward, George had confessed to Moriarty that he'd considered asking to get in on the marriage contract and sail back with Xena, too, but he'd had a hunch his intelligence wasn't high enough to qualify him, and had chosen not to risk the embarrassment.
"Will you sign our contract?" Moriarty had asked. George had made noises with a faraway look in his eye, about writing down and trying to implement pieces of Xena's debt-free society among their own people. Moriarty, out of spite, had bought George a coffee and given him a pad of paper, and prayed for his machinations to go as well as they deserved. No amount of idealism, thinks Moriarty, can make you that much bronze for cloaks, and goes to sleep.