# What to expect when you're expecting expected value

### In Going Infinite, Michael Lewis misses some of SBF's trading mistakes

*This post is a supplement to Michael Lewis’s Blind Side, an essay of mine in Asterisk Magazine’s “Mistakes” issue.*

*Going Infinite*, Michael Lewis’s new blockbuster about Sam Bankman-Fried and the collapse of his crypto exchange FTX, is heavy on funny anecdotes and implicit psychoanalysis and lighter on technical details.

This post is a dive into one of those anecdotes, from Sam’s time as a trading intern at the proprietary trading firm Jane Street Capital (disclosure: I worked as a trader at Jane Street for 2.5 years). Finance columnist Matt Levine has already covered the story in detail, so if you’re a regular Money Stuff reader you might be familiar with some of the major flaws in (Lewis’s version of) Sam’s version of events; I’ll try to cover the minor ones as well.

Lewis narrates a story about Sam’s interactions with “Asher Mellman,” a fellow member of Sam’s Jane Street internship class. Sam and Asher trade on the maximum dollar loss of any intern that day—a value that Jane Street wisely caps at $100—such that the higher the maximum loss, the more money Sam makes and Asher loses. In trader speak, Sam buys this contract for $65, and the contract will pay out the value of the maximum intern loss (guaranteed to be between $0 and $100, inclusive). So if the maximum intern loss that day is $75, Sam will profit $10 (he pays Asher the agreed-upon $65 for the contract, and Asher pays him the $75 that corresponds to that day’s maximum intern loss).

Once they’ve made the trade, Sam turns around to the other interns, and offers to pay any intern $1 to flip a coin for $98. Lewis writes:

To the Jane Street way of thinking, Sam was offering free money. A Jane Street intern had what amounted to a professional obligation to take any bet with a positive expected value. The coin toss itself was a 50-50 proposition, and so the expected value to the person who accepted Sam’s bet was a dollar: (0.5 × $98) – (0.5 × $98) + $1 = $1. The expected value of Sam’s position was even better, thanks to his side bet with Asher that paid him a dollar for every dollar over sixty-five lost that day by any intern. After the coin flip, either Sam or some other intern would have lost ninety-eight dollars: win or lose the coin flip, Sam would collect thirty-three dollars from Asher (the difference between ninety-eight and sixty-five).

Here the technical misses begin. First, if Sam is paying $1 to his fellow intern to take the coin toss, it can’t be that “win or lose the coin flip, Sam would collect thirty-three dollars from Asher.” We should expect to see an asymmetry—either Sam wins the coin flip, and the other intern loses $98 - $1 = $97 total, or Sam loses the coin flip, losing $98 + $1 = $99 total.

Second, and far more importantly, in the case where Sam loses the coin flip, *he should not “collect [thirty-four]1 dollars from Asher.” *Doing so would bring his net losses for the day from $99 to $65, violating the proposition that the max intern loss is $99, upon which Asher’s payment was based. He should instead collect from Asher $X such that $65 + $X = $99 - $X, which is found when X = $17. This would bring his net loss for the day to $99 - $17 = $82, making Asher’s payment of $17 the appropriate amount over $65 to equal the max intern loss.

The EV of Sam’s trade is thus (0.5 × $97) – (0.5 × $82) = **$7.5**.

If, instead, he’d paid two interns $1 each to flip a coin for $98, he’d have made a guaranteed $33 - $1 - $1 = **$31**. (He probably could have gotten away with paying less than $1 each, but let’s assume that’s their happy price.)

So Sam gives up on over 75% of his expected profit in a burst of impulsivity or a hunger for volatility.

Back to our narrative:

Sam won the first coin flip. But that was just the start. To maximize Asher’s pain, some intern needed to lose one hundred dollars.

I’ll pay a dollar to anyone who will flip me for ninety-nine dollars, Sam shouted.Now he had a machine for creating wagers in which both parties enjoyed positive expected value. That machine was named Asher. Interns were lined up to take the bet.

…

I’ll pay a dollar to anyone who will flip me for ninety-nine fifty,shouted Sam.The other interns clearly felt obligated to take the bets, but the mood in the room was shifting in response to Asher’s feelings. Plus the Jane Street trader who was meant to deliver the lecture had arrived and was watching the whole thing. But Sam won the third coin flip too, so to his way of thinking the gambling wasn’t yet over.

I’ll pay a dollar to anyone who’ll flip me for ninety-nine seventy-five,he shouted.It wasn’t until the fourth flip that Sam lost—and by then everyone except Sam was unsettled by Asher’s humiliation.

Sam then gets in trouble with his superiors (“‘They said the second coin flip was already one too many,’ said Sam”), and concludes dismissively that Jane Street thought he should have been more sensitive to his fellow interns.

Incidentally, Sam’s paraphrase of his Jane Street managers could have been an insight about the expected value of his coin tosses. Matt Levine points out that indeed by the second flip, they were negative expectancy for Sam. His trade with Asher was on the max intern loss, not the sum of losses across multiple interns—so the hedged portion of his own downside went away once the max loss was (approximately) achieved, and he was giving out free expectancy in exchange for volatility. Contra the nimble Bayesian that Lewis would have us believe Sam to be, we see a Sam who pattern-matches on a successful first coin flip and keeps flipping without stopping to recalculate whether his trades are still good.

Two more technical peculiarities: first, why does Sam start with $98, then $99, then $99.50, etc.? He could have simply started with $100—the additional $1 payment was hedged by his trade with Asher. Does Lewis believe that this increment was somehow necessary? Or does he just want the rhetorical flourish of escalation, and hopes that we won’t notice the oddity (pun, as always, intended)?

Second, why does Sam stop after losing the fourth coin flip? He isn’t in the red—he just won three coin flips, so there aren’t any Jane Street internship max loss restrictions stopping him from continuing to trade. Insofar as he thought flip #4 was profitable, why would losing it imply he shouldn’t go for flip #5?

Something is funny about this story. I don’t believe that Sam’s managers chastised him for his inability to treat his fellow interns with dignity and didn’t also catch that he was doing reckless, negative expectancy trades. Sam heard what he wanted to hear, ignored the rest, and went away with exactly the wrong lessons. But what’s interesting is Michael Lewis’s indiscriminate reporting of a narrative that doesn’t pass the financial fundamentals smell test, and instead reads like a story constructed from a mishmash of trading tales and optimized to convey the vibes of the Jane Street internship trading floor (which it successfully does) at the expense of getting the details right.

By uncritically passing on (his recollection of) Sam’s narrative, Lewis paints a portrait of a Sam who is singularly motivated by doing the EV-maximizing thing *despite* the social and emotional costs inflicted on others. But the real Sam in this story is not taking a hit to his reputation in order to maximize expectancy; he’s carelessly burning expectancy in the process of cultivating a reputation for punishing and humiliating his competition. By aggressively asserting an image of himself as a ruthless maximizer, he succeeds at convincing Lewis of it as well.

This $34 incorporates the $1 payment to accurately reflect what the payout would have been if calculated correctly. If you treat Sam’s loss on the coin flip as $98 instead of $99, the math for Asher’s payment comes out to $16.5 instead of $17.

Also, I think Sam's "sure-thing" bet gets him $30, not $31? Your point still holds—$30 is more than $23.50 by a good bit—but it's not quite as stark as $31 vs. $7.50.

In your improved bet, Sam pays ($1) and ($1) to the two bettors and ($65) to Asher. The losing bettor will lose $97 in total ($98 paid to the winner offset by Sam's $1) and so Asher pays Sam $97. $97 - $65 -$1 - $1 = $30, not $31. I'm sure I've made a dumb mistake but that's how it looks to me.

> Doing so would bring his net losses for the day from $99 to $65,

> violating the proposition that the max intern loss is $99

Could Sam and Asher reasonably have disagreed about the terms of the contract?

This post reads the contract as self-referential, i.e. the maximum intern losses must be computed in a way that includes the contract payout. But one could imagine reading the contract otherwise, i.e. that it should pay out the maximum amount lost by any intern *prior* to the contract paying out.

Maybe the first reading is obviously the correct one for some reason.

> The EV of Sam’s trade is thus (0.5 × $97) – (0.5 × $82) = $7.5.

Expanding on this, if Sam loses the coin flip, his accounting is as follows:

($65) to Asher for the contract

($1) to the other intern for the coin-flip bet

($98) loss to the other intern on the coin-flip bet

$82 paid by Asher to make Sam's loss the greatest intern loss, i.e. $82

Grand total: Sam loses ($82)

But if Sam wins the coin flip his accounting is as follows:

($65) to Asher for the contract

($1) to the other intern for the coin-flip bet

$98 win from the other intern on the coin-flip bet

$97 paid by Asher because the intern who lost the coin flip lost $97

Grand total: Sam wins $129

So why isn't Sam's EV (½ 𐄂 $129) - (½ 𐄂 $82) = $23.50?