An introduction to the market making game
The other game quants play at restaurants
I’m not yet done talking about the games quants play at restaurants. But while the last one I thought was silly, this one I think is pretty great and you should all play it.
Here’s how the market making game works.
Alice, Bob, and Charlie are all quants who work at hot new trading firm OpenThropic Trading. They go out for a meal together and after they’ve eaten, they decide to play the market making game before the bill arrives.
Step 1: Set the question and answer
The question is “what will the restaurant bill be for the four of us”.
The answer is “whatever the number at the bottom of the bill turns out to be”. They don’t know the answer yet, but they know the answer exists.
Step 2: Offer decreasing range widths
Everyone has a range of what they think the final bill will come to. Alice thinks it’ll be between $150 and $200. Bob has been more observant and his range is $140 to $160. Charlie, who hasn’t thought about restaurant prices since he found a great alpha in a market he won’t talk about a few years ago, thinks the bill is somewhere under $160.
Alice starts by giving a range width.
“$100.”
This is the size of the interval she thinks the right answer lies in.
From the outside, this means she might eventually set her range at $150–$250 or $0–$100; we don’t know yet. Importantly, this isn’t a guess at the actual bill but a signal of how confident she is in her ability to guess. (That last sentence wasn’t written by an AI but it does give those vibes. Sometimes you really do need a “this isn’t X, but Y” sentence!)
And then going around the group, everyone can either narrow the range by a few dollars, or pass, if they aren’t sure enough to offer a narrower range.
Step 3: Once the market is as narrow as can be, show the market
As people go around the table offering narrower and narrower ranges, eventually Alice will offer her narrowest range of $50 (remember she thinks the bill is somewhere $150 - $200), Bob can narrow it further, and Alice will have to pass. Charlie is at this point scrolling on his phone reading about the IMO gold medallist who became a monk.
Once everyone has passed we are left with Bob who has offered the lowest range of $40. He could have gone lower but no-one else was willing to go beyond $40 so there was no need.
At this point he must offer his actual upper and lower bounds. He says he thinks the bill will be between $130 - $170. For those in the know, this would be Bob’s bid-ask spread (see later).
Step 4: Everyone else must bet over or under
All at the same time, everyone else chooses to bet over or under the range. They cannot bet inside the range. Alice thinks the bill is probably higher, so votes over. Charlie thinks it’s probably lower, so votes under.
Step 5: Resolve the market
The bill arrives, along with the answer and the resolution of the game. The points are calculated as if it were Bob, the market maker, was playing a 1-1 game against each person individually.
bill > $170 → Alice wins against Bob, Charlie loses against Bob
$130 ≤ bill ≤ $170 → Bob wins against both
bill < $130 → Charlie wins, Alice loses. Bob breaks even.
In this example Bob never ends up net negative but this isn’t always true [1].
So to quickly recap
Player 1 offers a range width.
All the players then take turns narrowing the width or passing.
The narrowest bidder declares their exact range.
Others bet “over” or “under” (not inside) the range.
Reveal answer and score.
This game generalizes to any question with an answer the group can agree on in advance.
Why this is called the market making game
In the above scenario you can imagine Bob has written a contract that says
This restaurant bill has some value, X, in the future that we don’t know. I’m going to give whatever the X value is to whoever owns this contract. I’m willing to buy this contract for $130 and to sell it for $170.
And Alice offers to buy it from Bob for $170 since she thinks X is likely greater than $170. Charlie offers to sell it for $130. Bob has made the market of $130 - $170 and the other players have had to interact with the market.
More complicated point scoring
You could take this one step further and say the points you get from each person are the actual value of the contract minus the price you paid for it.
If the restaurant bill came out to $175, then Alice, who bought the contract for $170 makes $5 off Bob. Charlie who sold it for $130 loses $45 to Bob.
Why I like this game so much
It improves your calibration
Julia Galef has a nice definition of calibration
Being perfectly calibrated would mean that your “50% sure” claims are in fact correct 50 percent of the time, your “60% sure” claims are correct 60 percent of the time, your “70% sure” claims are correct 70 percent of the time, and so on. Perfect calibration is an abstract ideal, not something that’s possible to achieve in reality. Still, it’s a useful benchmark against which to compare yourself.
Your calibration is improved when you lose this game because of OOM errors, or a bias towards thinking sushi is cheap.
I’m not sure what this says about me exactly but amongst the highest of intellectual compliments I can pay someone is “you are very well calibrated”.
It’s a game that allows a little leaking of information
But not a lot. In that way it’s like simple poker and as I’ve written before I haven’t played poker since I lost my entire life savings (£20) when I was 10 years old.
Also if you haven’t read it yet, the world’s best finance writer Matt Levine wrote about teaching his four year old kids to play poker which has similar ideas and if you’re enjoying this piece you’ll love his newsletter [2].
It rewards accuracy and penalizes overconfidence
The way to lose the most points in the game is to be overconfident and have everyone pile on against you.
People have asked me whether quant finance made me really good at spotting patterns. And strangely the opposite ended up being true. The quant trading problem is a very low signal-to-noise problem and you’re constantly being duped into thinking you’ve found a pattern when you haven’t.
If anything it made me more resistant to spotting patterns in data and made me less overconfident in anything I noticed. Most of us are overconfident about what we know so this game is a nice corrective to that.
And given the replication crisis in psychology over the last 30 years I think assuming most patterns are statistical quirks is a decent heuristic to have.
Relatedly this guy made $250k on betting markets by betting nothing would happen over and over again. I take this to mean people are overconfident that things will happen.
It introduces a bid-ask spread rather than a single price
Most of the time we are used to prices being one number. We grab a banana at the supermarket for ten dollars, we spend our hard-earned income on ludicrous rent, we buy a Taylor Swift concert ticket for one fifth of our soul. You get the idea.
But when you are a market maker, you have to be willing to both buy and sell the item. And in order to make a profit you need to buy at a low price and sell at a high price, and crucially, somehow manage to do both.
In finance, the difference between what you’re willing to buy something for and what you’re willing to sell it for is known as the bid-ask spread. It is one of the most foundational concepts in trading. One of the positive side effects of quants working in HFT firms is that often the best way to make money is to offer a lower bid-ask spread and thus lower transaction costs for everyone else.
I like the concept because I think it maps more onto what is going on in our heads anyway. When we buy that banana for ten dollars we are probably willing to sell it for one million dollars. We are likely willing to sell it for thirty dollars because who turns down a $20 bill on the sidewalk? All this is to say that implicitly we all have an asking price. This concept just makes it explicit.
So you should play this too
This game encourages good calibration, an interrogation of your own biases, an introduction to the concept of a bid-ask spread, and the ability to beat your friends at stuff. I really like it and will play it any time I can. There’s no learning experience quite like the sting of embarrassment when you realize you thought the Beatles were producing hit songs in the 80s.
Also, I’ve noticed my non-quant friends don’t like playing this game. I don’t know why and if you do, please do let me know in the comments.
Further reading
If you liked this, I put a lot more thoughts on strategy and caveats into the appendix below.
Relatedly, Jane Street built their own game called Figgie that they use to teach new traders how to learn and trade with incomplete information and why that was a better teaching tool than poker.
[1] Notice that in the above scenario Bob always ended up with >= 0 points. This isn’t always the case. If Bob had offered a range of $10 - $50 and everyone had rightly gone over, then he would have ended up with -2 points. These are the joys and perils of being a market maker.
[2] If you scrolled down to this looking for a footnote I can once again recommend Matt Levine’s free newsletter
Appendix - caveats and strategy
Don’t share your actual range until the end of the narrowing
As the game progresses the range width gets narrower, but no-one reveals their true range. This is important in the case where someone is massively overconfident and wrong.
Once we were playing the game with the question of “how many total humans have ever existed”. Based on a half-remembered fact, I started with a range of 10 billion, assuming others would go narrower than me. And then no-one did. The second warning sign was when everyone bet over the range of 10-20 billion that I offered.
Obviously they were all correct. My recollection was totally wrong and if I’d heard anyone else’s actual range of 50-100 billion I would have updated, backed myself into a corner, and just let the accurate folks decide the terms.
By sharing your actual upper and lower bounds you might give enough information away to prevent someone else from making a bad decision.
Don’t be upset if you want to be inside the range
If you’re certain the bill is inside someone else’s range but didn’t offer it yourself, it means you weren’t confident enough. Your true confidence interval is wider than theirs.
On easier-to-estimate numbers like restaurant bills it is likely that you will have a large overlap with the given range. On harder-to-estimate numbers like how many people have ever existed, this is less likely and one of you is more likely to have OOM errors and thus non-overlapping ranges.
Don’t go narrower than you need to
Maybe this is obvious but there’s no need to immediately offer your narrowest range. If you are the most certain you should be able to offer a range only a little narrower than the second most certain person. This should give you some breathing room.
Make sure you are agreed on the way of finding the answer
In some cases, even though answers exist, finding them can be hard. Things like GDP per capita and populations are subject to estimations. Things like polling numbers are subject to change. Be sure to agree on a source, or ranked list of sources, to find your answers otherwise the losers can complain about the terms and conditions.
OurWorldInData is a primary source we use when possible. Betting markets like Polymarket are another crowdsourced way of getting at an answer that might be hard to find.
Watch out for order of magnitude errors
You might think the best range to give puts your true answer in the middle of the range. But depending on the question, there's a non-zero chance your guess is off by an order of magnitude (OOM). Often this game is played with questions with large answers
Consider lowering your range to include one or two orders of magnitude lower. It will likely cover your ass in that case, and losing a little bit on the upper end will likely be paid off if you are ever off by a factor of 10 or 100. If your range is $8000 and you think the most likely value is $5000, it’s a good idea to guess $10-$8010 rather than $1000-$9000 for the GDP per capita of the UK in the 1900s just in case you are way off on the lower end.
Think of other ways to approach a range
Sometimes the question requires Fermi estimations. Consider approaching the estimation from multiple angles to avoid small errors compounding.
Consider who you are playing against
Are your friends overconfident (too narrow a range) or underconfident (too wide a range). Adjust your behavior accordingly.
Be extra careful when playing against traders
I’ve heard stories (either from working in quant finance or from Michael Lewis’ book, I can’t remember) about Jane Street traders playing this game with interns and asking them to market-make innocent seeming questions like “how many dice do I have in my pocket”. The pockets don’t look full, so you figure the upper bound is 5 and play accordingly. At the end of the game the trader pulls out one hundred specially-made, miniscule dice and you end up with egg on your face.