Edit: It appears that the code for the Fomo-3D game is not fully verified as of 8/2. Check this article for details.
FOMO3D is a blockchain game that currently has $12 million in ETH at stake, with all of it locked up in a very bizarre-looking set of rules. But upon closer inspection, the rules aren’t so strange. In fact, it turns out to resemble a scaled-up version of a classic game in behavioral game theory. They teach it at Harvard Business School.
This game, underlying FOMO3D (F3D), is called “The War of Attrition” and it can produce some insanely irrational outcomes. Fortunately, variants of the War of Attrition have been extensively studied by economists and evolutionary biologists. Here are some core findings:
Part two of this article gets a little more technical about how smart contracts can solve War of Attrition games. But let’s describe it here using the first lesson you get in a Harvard Business School professor’s negotiations class (the game was invented by Martin Shubick.)
At HBS, professor Max Bazerman auctions off a normal $20 to the highest bidder, but with one catch: the second-place bidder has to pay too. Consider what you would do in this situation. Max waves a $20 bill, and offers it for sale starting at the low price of $1. Would you bid? Take a second and think about it.
Did you bid? Of course you did! Who wouldn’t pay $1 to get $20? So we all bid, and we follow the same logic next round — after all, who wouldn’t pay $2 to get $20?
But eventually something funny happens: “Who wouldn’t pay $2 to get $20?” gets bid up until it becomes “who wouldn’t pay $19 dollars to get $20?”. Which then, of course, becomes “who wouldn’t pay $20 dollars to get $20?”. And this is where you can really feel the force of the “second bidder pays too” issue.
Let’s say you’ve bid $19. If the auction stops now, you’ll get the $20 and make $1. But let’s not forget about Competitive Carl, the second highest bidder. Carl bid $18 right before you bid, and so if the auction ends now he has to pay anyway. For Carl, the choice is to bid for a chance to break even or simply choose to lose $18.
So Carl bids.
The high bid is now $20
It’s your turn.
Do you bid $21 for a $20 bill in this situation?
If you don’t bid, you’ll pay $19 and get nothing. If you bid $21 you at least might get the $20!
That’s the hook: at each turn it feels like you’re paying an additional dollar to get $20. And because of that you’ll bid more than the $20, and so will Carl, and so on. Eventually one of you will have to walk to to the front of the class and pay some crazily high amount for a $20 bill.
It’s no different at Harvard, where everyone who’s been foolish enough to play this game has gotten a raw deal on the $20. In fact, there are documented cases of business people paying $2000 for the $20 in this game.¹ I’ll quote from Northwestern professor J. Keith Murnhigan, who describes how that escalation happened in a class he taught in Hong Kong:
“[Even at $400] there was no pause in the bidding at all. Members of the class were screaming for the bidders to stop, but amid the general tumult, they took no heed. When the bids reached $700 my knees were shaking…
When we reached $2,000 they finally stopped bidding. The class was in a total uproar. Everyone was stunned.
The two warring bidders owed the professor a total of $4,000 for the $20 bill. The “winner” was a CEO who described his strategy as an ego-driven disaster. The loser said that everything was blur to him, and that his pulse and blood pressure remained up for another hour after the class ended.
Meanwhile, the professor who made the 200x return on a $20 described getting a “ sporty, imported car.”²
The point is that this is a deeply human sort of insanity. The people who didn’t bid got to watch this drama play out, surely feeling smug. We should mention that, technically, they didn’t make the right decision either (game theory can be weird like that.³) But they’re pretty much right. You can’t do consistently better than $0.
You can’t do better than $0, but humans manage to do consistently worse. Humans overbid, sometimes ruinously as we saw. The various behavioral factors like loss aversion, inequity aversion, negative reciprocity, etc. take over. But to explain the truly extreme results like $2000, we probably have to reach for the big guns like Rene Girard’s theory of mimetic competition or something.⁴
Either way we are in the land of behavioral game theory, where utility functions get weird.
But back to FOMO3D. In F3D, you buy a “key” to win the prize (currently $12m) which gets handed out in 24 hours. As long as nobody else buys a key before time runs out, you’ll win the prize. But if someone else buys one, the timer is bumped higher and the whole process starts again with them in the lead.
At its core FOMO3D appears to be a “War of Attrition” game like the $20 auction is.⁵ The crucial part is that the amount you “lose” when someone else buys a key gets added to the $12m. As a result of that you can “get it back” by re-bidding and winning. This is the familiar psychological mechanism that we saw with the $20 auction games.
More precisely, the dynamics of F3D are probably better described as an all-pay auction with an endogenous prize value, but the takeaway is the same:
So how do you win attrition games? One way is to parametrize the behavioral utility functions of your opponents and set a mixed strategy bid profile that leaves them indifferent between bidding and not bidding. Don’t worry if you didn’t get that — you can also just not play and get the same expected result ($0).
Beyond that, having only taken a cursory look at the code, we can offer only general ideas. But the good news is that it seems to us like a commitment strategy should be available, and smart contracts are incredible tools for this. For instance, someone could write a publicly viewable contract that funds a bot bidding in the final second before the prize is released, and this would seem to eliminate the incentives of those bidding before or after (since those bids would be guaranteed to fail.)⁶
The analogous version for our $20 game is as follows:
Imagine if Clever Christina wrote an auto-executing program, funded by a smart contract, that would automatically outbid anyone in the class up to $1 million. She sets it to bid first at $1 then she publicly destroys the private key. What is your optimal response? It’s to do nothing.
Such a contract would have to take better account of the specifics of F3D players and devs and their long-run and “meta-game” payoff functions, but the basic idea seems correct. But in theory, Christina could get the same commitment even if she wrote the contract such that it refunds the remainder of the $1 million back into her wallet conditional on her winning.
But even if it’s a money-losing gambit, it may be worth it for the community to fund a Christina anyway. So long as we assume that F3D imposes negative externalities on the Ethereum network (and perhaps on the reputation of Blockchain more broadly) then ending it, even at some cost, could provide a net benefit to some person or entity.
Another solution, suggested by Justin Drake, is that miners could collude in such a way to “win” the game by updating the block such that they win. It’s a clever idea, but we suspect that this is a less desirable outcome since that would, itself, produce a negative externality in the possible loss of credibility for Ethereum.
At any rate, we think it’s reasonable to expect more of these games given the success of F3D. It will be especially interesting if Smart Contracts can be the cure. I discuss this further and get a bit more specific in an informal “part 2”, here.
 “A Very Extreme Case of the Dollar Auction”, Murnighan (2002)
 That’s a real quote but the prof actually donated the money to charity.
 To see why bidding $0 isn’t optimal, let’s assume that everyone plays that way. Well obviously, if everyone plays $0 then you should bid $1 and make $19! A Nash Equilibrium needs to have no incentives to defect.
But what if you bid $20? If you do that you make $0, but nobody else has a reason to bid. So trading identical $20s works but keeping the same $20 doesn’t. Game Theory is cool.
Also, down here in the fine print, let’s note that there should also be a mixed strategy equilibrium that yields an expected payoff of $0 and makes your opponent(s) indifferent over her choices. And furthermore, I should be clear that the HBS game doesn’t make the second-place bidder pay all of their bids sum(bi…bn), they instead pay max(bi…bn).
 In Girard’s theory, the $20 would become vastly more valuable to each participant simply because it was valued by a rival. Funny enough, in the professor’s account of these events, it turns out that the prof was even scapegoated for “changing the rules of the game.” (He forced students to bid in increments of $50 once the price reached a certain amount.)
 We are omitting some bells and whistles WLOG (as far as we can tell). For instance, there is also a dividend option in FOMO3D, but this appears to be a secondary phenomenon. That is, you wouldn’t buy a dividend unless you thought people would bid so this is a “greater fool” strategy. It’s as if the Northwestern Professor told the other profs in the department that he’d cut them in on his profits. It’s only a good idea if Clever Christina doesn’t achieve the common knowledge smart contract solution. If he did you’d lose $19.
Thanks to redditor u/questionablepolitics for reminding us to include this.
 This language is a bit loose. I get 1% more technical about how to do this and what to consider in my second article, here.