Cake and Games

Cryptocurrencies are complex structures, supersystems of often-counterintuitive bent. To stand any chance of understanding (much less predicting) these creatures, one must first develop a working knowledge of the basic principles of protocols and game theory, as these are the primitives from which cryptoeconomic networks are built. It’s often illuminating to begin an analysis by looking at simple constructs; of these, one particularly delightful example is the art of fair cake-cutting.

Cutting The Cake

The basic premise is simple: two people want to share a desirable resource (the cake) which can be cut and divided into smaller pieces. This can be done fairly, in an envy-free way, with a very neat and simple trick: one person gets to cut the cake while the other gets to choose which half they receive. Or more concisely: one cuts, the other chooses. This straightforward, easy-to-understand procedure (i.e. game) is profound and marvelously elegant. The one rule can be described in a mere five words: one cuts, the other chooses. And yet, as long as this formula is followed, the end result is a fair distribution of cake. After all is said and done, neither party has justification to complain that they got the worse end of the deal; they are both able to meaningfully participate in the distribution process, and neither one is afforded the opportunity to take advantage of the other. Their basic self-interest, aligned within the structure of the game, creates a perfectly balanced outcome accommodating the perspective and priorities of both.

This is a beautiful result, and it is worth a moment’s meditation. To be sure, even beyond the elegance of the balance itself, there is plenty of intellectual nourishment here for a curious or inquisitive mind. Of course, caveats, nuance, and complexity can be introduced into the equation; how are the rules enforced, how is cheating prevented, how are the participants’ respective roles chosen or assigned (who cuts and who chooses?), what is the simplest or best way to generalize this to handle more than two parties, what about in cases where the goods are more or less heterogeneous or less amenable to arbitrary-precision splitting, what level of communication is available between those involved, are there any external factors or personal preferences that might play a role in or be relevant to the interaction, …? These are all questions that can be amply explored, but the core idea and the basic principle of the arrangement can be admired and appreciated even in the simple, singular case.

Auction

The Game of Auctions

Let’s flip this on its head and take a look at mechanism design, a branch of game theory concerned with the design and analysis of the games themselves. In other words, when it comes to mechanism design, one adopts the perspective of the creator of the game, and constructs the game with some sort of goal, criteria, or outcome in mind. Satoshi took the role of the mechanism designer when architecting Bitcoin, as have the creator(s) of every cryptocurrency that followed, some more carefully than others.

Once again, we will begin by looking at a simple, straightforward, easily-understandable example that nevertheless manages to achieve an impressive and lovely set of desirable properties: the Vickrey auction.

In order to fully appreciate the benefits that a Vickrey auction provides, it helps to first take a quick look at an intuitive alternative type of auction: the first-price sealed-bid auction (FPSBA), also called a blind auction. This is a straightforward arrangement where all bidders submit their bids simultaneously (without seeing each other’s bids) and the winner simply pays the price they bid. This auction format is simple, practical, and should usually result in the auctioned good being sold to the party who values it the most (i.e. it is decision efficient). However, as you might have guessed, there’s considerable room for improvement.

One thing worth noting is that bidders aren’t actually motivated to bid what they truly value the auctioned goods at, because doing so means that even if they win the auction, they don’t get any surplus utility from their purchase. In other words, they pay full price, i.e. the maximum price that they were willing to bid, regardless of how much lower the offer of the second-highest-bidder was. Participants in this sort of auction are actually incentivized to instead guess what the most anyone else is willing to pay, and (assuming they personally value the good more than that amount) to only bid the minimum possible amount higher than that.

As an example, if Alice and Bob are bidding against each other in a blind/FPSB auction, and Alice values the auctioned goods at $2.99 while Bob values them at $10, then Bob should try to bid as low as possible while still outbidding Alice. What’s more, Bob (usually) doesn’t even know what Alice values the goods at, so he would have to make his best guess at this, and then submit his bid accordingly. So in this example, if Bob knew that Alice valued the goods at $2.99, he should rationally bid $3 and thereby derive $7 worth of surplus utility (since he received goods that he deems to be worth $10, but only had to pay $3). If he guesses Alice’s amount wrong, he could wind up overpaying or even losing the auction. To make matters worse, Bob doesn’t just have to guess what Alice values the goods at, he has to guess what Alice will actually bid, in light of all of the above. The incentives are a tangled mess, and this is in the simplest of scenarios, with only two bidding participants.

Another drawback to this approach is that after the auction is finished, regardless of who won, the “fair valuation” of the goods is still not clear or well established (i.e. the market receives a weak or garbled signal), because of the mess of incentives explained above. Even though everyone submitted their bids, these bids should be expected to reliably reflect only what the bidders thought would be appropriate to bid versus the other bidders, not what they truly value the goods at. This can be said to establish a lower-bound on what the fair value of the goods might be, but unfortunately there is no guarantee that this is even particularly close to what the market of bidders genuinely values the goods at.

The final important downside worth paying attention to is the fact that the utility surplus of this sort of mechanism is not in any way maximized. In other words, whoever wins the auction doesn’t necessarily derive as much value as they otherwise might (again, due to the incentives described above). This, in turn, can have adverse effects, such as discouraging bidder interest, which could further skew or exacerbate the other mentioned problems of the arrangement.

In short, the first-price sealed-bid auction has some warts. Bidders aren’t incentivized to bid their truthful valuation, guesswork and unpredictability leak in unhelpfully, and net utility (think: total societal benefit) isn’t maximized. But with a tiny, simple tweak to the rules of our auction, we can make an altogether breathtaking difference: instead of the top bidder paying the price they bid, they pay the price the second-highest bidder bid.

Introducing: The Vickrey Auction^{^[1]}

By requiring not that the top bidder pay what they themselves bid, but instead pay whatever the second-highest bid amount was, the incentives settle into focus. Now the best move you can make is to bid your honest personal appraisal of the auctioned goods’ worth or value. There’s no advantage to be gleaned by lying or overcomplicating the bidding process; either you are outbid or you’re not, and if you bid your truthful valuation then you either don’t have to pay anything at all (net effect nil from your perspective), or you pay some amount which is necessarily less than or equal to the amount you valued the goods at, the difference representing your individual surplus utility. The best strategy is to bid what you think it’s worth.

In a Vickrey auction, the winning play is honesty.

What’s more, no one ever really “loses out” in a Vickrey auction; if you win, you either realize a surplus (woohoo!) or (in the worst-case scenario) pay exactly what you (and at least one other party) value the purchased goods at. In other words, in a sense, the Vickrey auction’s worst-case scenario is effectively identical to the best-case scenario in naive blind auctions.

When a Vickrey auction is finished, you also have a pretty good idea of what the market, in aggregate, values the auctioned goods at. After all, the participants were each faced with a situation in which the dominant strategy was also the most honest and obvious option of all. Each of them has no incentive to lie or submit a foul bid, and can only lose by doing so, while they have every incentive to submit whatever honest fair-valuation bid they can afford, and only stand to benefit (or at worst remain where they started) from the act. The bids’ valuations would therefore reflect the total market sentiment to as great and accurate of a degree as could ever be hoped for. In other words, the data from a Vickrey auction is about the best you’ll ever do when it comes to gauging market valuation.

Isn’t this wonderful? The tiniest change – making the top bidder pay the amount submitted by the second-highest bidder – has a profound effect. Multiple benefits are realized in one fell swoop. The auction suddenly boasts an aesthetic and logical incentive structure which we can concretely reason about, honest play is rewarded, and both social surplus and market data integrity are elegantly maximized. In short, as a game, the Vickrey auction is much more beautiful than its (perhaps ever-so-slightly simpler) FPSBA cousin.

This is not to say that the Vickrey auction is the end-all-be-all of auction formats, though. There are, of course, alternatives, and some have clear advantages over the basic Vickrey approach. There’s no sort of iterative price-discovery or signaling process bundled in, and (quite importantly in some cases) the Vickrey auction doesn’t maximize, nor even try to maximize, the seller’s profits. But crucially, when compared with the FPSBA setup, the Vickrey auction is dramatically superior in many ways, and all it took is one clever little adjustment to the rules of the game.

Conclusion

If you’ve made it this far, the takeaway I’d like to leave you with is that incentive-driven interactions (known in the business as “games”) fundamentally depend on their rules and setup, and in some cases very minor changes to these can yield profound (and sometimes marvelous) results.

That is to say, a careful, deliberate, prudent, or clever alignment of incentives in a game or protocol can make possible outcomes that might otherwise be non-trivial or even impossible. Any cryptoeconomic system is necessarily the result of mechanism design, so this has been a brief peek into the wide world that’s out there on the subject.^[2]