An Introduction to Blockchain Mechanism Math, Terminology, and Hieroglyphics for Deeply Casual People who want to Sound Smart when Discussing White Papers with their Peers
By Alex Watts et al
Abstract: Mechanism Designers working in DeFi really like writing white papers. They like it a lot. Perhaps too much. They also like writing these papers in an ancient, hieroglyphics-based language that is difficult for a normal, socially-adjusted person to decipher. This is unfortunate because most of the math and game theory is actually quite simple once you understand their alien terminology. In this paper, I’ll give you the tools you’ll need to sound smart when discussing these white papers with your peers, which is a very important skill set because in all likelihood your peers are also trying really hard to sound smart.
Let’s start with this section - the “abstract.” This section is really just a summary of the white paper. In this section, the Mechanism Designers will summarize a problem, then they’ll summarize their solution, and then - if they’re good - they’ll also summarize the shortcomings of the paper. We may never know why the Mechanism Designers like to call this section an “abstract” rather than a “summary” - in fact, we’ll probably never know why they do most of the things that they do. But by reading this paper, hopefully you’ll be better able to understand what they’re doing - and, if you’re into this kinda thing, who they’re doing it to.
Intuitively, this paper doesn’t have any shortcomings. But if I had to pick only one, it’d be that a lot of the information in this paper is deliberately wrong. Very wrong. I’m way off on a lot of these explanations. But I’m not being byzantine - which means “dishonest” or “adversarial”, by the way, and two thousand years ago it probably would’ve been flagged by HR departments as racially offensive against the citizens of the Byzantine empire. So yeah, I’m not being byzantine - I’m just oversimplifying things to make the concepts easier to understand in expectation.
Sets
A set is a group of things.
By the way, if your first thought after reading that last sentence was “that definition left out a lot!” then be warned: it’s all downhill from here.
- Sets are important because Mechanism Designers like subjecting entire groups of people or things to their nefarious designs.
- Sets are represented with an uppercase letter (such as XX or YY).
- When sets are defined, they’re often surrounded with brackets. For example, a set of three numbers could be written as X = \{1,2,3\}X={1,2,3}.
- Members of the set are represented with a lowercase letter (such as xx or yy).
There are a handful of funny-looking symbols that these Mechanism Designers like to use in their white papers that you should probably learn:
\in∈ means “in” and is used to show membership in a set
– When you see x ∈ Xx∈X, that means that xx is a member of the set XX.
– Example: if X = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\}X={2,4,6,8,10,12,14,16,18,20}, then x ∈ Xx∈X means xx is one of those 10 values.\cup∪ means “union” and is meant to take two sets and combine all of their members together (removing duplicates).
– When you see X ∪ YX∪Y, that means you’re combining sets XX and YY.
– Example: if X = \{1, 2, 3, 4\}X={1,2,3,4}, and Y = \{3, 4, 5\}Y={3,4,5} then X ∪ Y = \{1,2,3,4,5\}X∪Y={1,2,3,4,5}.\cap∩ means “intersection” and is meant to take two sets and make a new set only out of their common members.
– When you see X ∩ YX∩Y, that means you’re taking the intersection of sets X and Y.
– Example: if X = \{1, 2, 3, 4\}X={1,2,3,4}, and Y = \{3, 4, 5\}Y={3,4,5} then X ∩ Y = \{3,4\}X∩Y={3,4}.\subset⊂ means “sub set” - all of the elements of the first set exist in the second set.
\supset⊃ means “super set” - all of the elements of the second set exist in the first set. Note that if you see this then you’re dealing with a particularly troublesome brand of Mechanism Designer and you should be on your guard for more trickery.
:: (the colon) typically means “if” and is often seen together with a funny looking upside-down A. Speaking of which…
\forall∀ means “for all” and is meant to iterate through a set, often times in the creation of another one.
–If you’re a programmer, whenever you see \forall∀ think “for loop.”
– Often times \forall∀ and :: are used to iterate through a set to create a new one.
– Example: Consider this monstrous expression: Y = \{2x, \ \forall x\in X : x>3 \}Y={2x, ∀x∈X:x>3}. The english translation is something along the lines of “YY is a set of numbers that was created by taking each of the xx's in XX that were greater than 33 and then multiplying that xx by 22.” For posterity’s sake, if X = \{1, 2, 3, 4, 5 \}X={1,2,3,4,5} then it follows that (aka \Rightarrow⇒, as we’ll discuss later) Y = \{8, 10 \}Y={8,10}.Warning: When it comes to defining sets, Mechanism Designers really like to move variables around or remove them entirely. They do this not to confuse you but to confuse each other - presumably to increase their job security. Take, for example, Y = \{2x, \ \forall x\in X : x>3 \}Y={2x, ∀x∈X:x>3}. That would be the same as Y = \{2 \cdot x\in X : x>3 \}Y={2⋅x∈X:x>3} or Y = \{z \in \{2x, \forall x\in X \} : z>6 \}Y={z∈{2x,∀x∈X}:z>6}. Watch out for anyone using linear algebra to define sets (look for the brackets, sets multiplied by sets, or sets of multiple things in parentheses known as “tuples”). They’re one of the least understandable species of Mechanism Designers and should be avoided by all but the most skilled larpers. In fact, if one of them is reading this right then now then they’re probably day dreaming about correcting the author - presumably using made-up words like “vector” or “scalar.”
\mathbf{card}(X)card(X) or |X||X| is the “cardinality” of the set XX. This is a fancy way of saying “the number of things in XX.” If X = \{ 1, 17, 31, 5\}X={1,17,31,5} then |X| = \mathbf{card}(X) = 4|X|=card(X)=4. Keep in mind that |X||X| is the size of XX while |x||x| is the absolute value of x \in Xx∈X; be vigilant, and remember that the Mechanism Designers will only win if you let them.
There are a handful of prestigious sets that you should know because Mechanism Designers really like letting you know that they know them, too:
- \mathbb{Z}Z is the set of all integers.
- \mathbb{R}R is the set of all real numbers.
- \mathbb{E}E isn’t a set of numbers at all - it means “the expectation of” and is used in probability. But it looks similar to these fancy sets, so be careful not to get it mixed up. That’s how they get you.
- \mathbb{C}C is the set of endless agony. If you see it, run.
The Mechanism Designers will typically use these fancy sets when defining a new set. For example, they might say X \subset \mathbb{Z}X⊂Z, which means XX is a subset of \mathbb{Z}Z, which means that all of the possible xx's in XX must be integers. If you’re wondering “Why doesn’t the Mechanism Designer just say that the set is made up only of integers?” the answer is because they hate you.
Stochasticisms and Probabalisticalities
Two things that Mechanism Designers simply can’t get enough of are intuitions and expectations.
“Intuitively…” or “The intuition is…” means that the Mechanism Designer is about to tell you something that they think is so obvious that they won’t explain it because only a moron would disagree. Unfortunately, for a Mechanism Designer ii in the set of all people PP, the set of morons M = \{ p, \forall p \in P : p \not = i \}M={p,∀p∈P:p≠i}, which, in English, means “The set of morons is all of the people in the set of people who aren’t also the Mechanism Designer”. If you want to understand a Mechanism Designer’s intuition, your best chance is their culture’s traditional approach of asking for an explanation after you beat them in duel with flint-lock pistols. Intuitively, you should be sure to ask quickly.
“The expectation is…” or “… in expectation.” means that the Mechanism Designer is about to do some math, and we can expect that the math will involve probabilities. Probably.
P(y)P(y) is the probability of event y occuring. Example: for a coinflip, P(\text{heads}) = 0.50P(heads)=0.50
P(y \cap z)P(y∩z) is the probability of event yy and event zz both occuring. This is also abbreviated as P(y, z)P(y,z).
P(y \cup z)P(y∪z) is the probability of event yy or event zz occuring.
P(y | z) = P(y|z)= the probability of event yy occuring if we assume event zz has already occured. For example, assume a coinflip, but this time there is a cheater using a weighted coin that can change the odds from 50-50 to 80-20. In that case, P(\text{heads} | \text{cheater bet on heads} | \text{cheater's flip}) = 0.80P(heads|cheater bet on heads|cheater's flip)=0.80 and P(\text{heads} | \text{cheater bet on tails} | \text{cheater's flip}) = 0.20P(heads|cheater bet on tails|cheater's flip)=0.20.
\mathbb{E}[X] =E[X]= The expected (or probabalistic, if you want to sound smart while actually being wrong) value of X. For example, if there’s a game with a coinflip and you get $0 for tails and $1 for heads then \mathbb{E}[\text{game}]E[game] = $0.50. Note that if we want to treat the game’s outcomes as a set, {E}[X] = \{ {E}[\text{heads}], \ {E}[\text{tails}]\} = \{ \$0, \$1 \}E[X]={E[heads], E[tails]}={$0,$1}, and \mathbb{E}[X]E[X] is just the average of the expected values (aka “the expectation of”) the possible outcomes in XX.
\mathbb{E}[X|y] =E[X|y]= The value of XX in expectation if we assume that yy happened. For example, if there’s a game with a coin flip and you get $0 for tails and $1 for heads, but there’s a cheater who is using a weighted coin that can change the odds from 50-50 to 80-20 (against you) then \mathbb{E}[\text{game} | \text{cheater's flip}]E[game|cheater's flip] = $0.20.
–Now imagine that the cheater gets to flip some of the time and you get to flip the coin the rest of the time:
\mathbb{E}[\text{game}] = \left( \mathbb{E}[\text{game} | \text{cheater's flip}] \times P(\text{cheater's flip}) \right) \ + \ \left( \mathbb{E}[\text{game} | \text{your flip}] \times P(\text{your flip}) \right) E[game]=(E[game|cheater's flip]×P(cheater's flip)) + (E[game|your flip]×P(your flip))\mathbb{P}(x)P(x) is just a fancy way of saying P(x)P(x). Some Mechanism Designers insist that if P(x)P(x) is the probability of xx, \mathbb{P}(x)P(x) is the probability of xx in expectation, but nobody knows what that means and so we just ignore them and move on with our lives.
Fancy Math
∑ (Sum)
This sigma may have been an key part of your fraternity or sorority’s identity during college, but it has another, less-important use case: math.
\sum^n_{x=1}f(x) =∑nx=1f(x)= The sum of f(x)f(x) for all xx values from 11 to nn.
In other words, it’s a “for loop” that sums the different values of f(x)f(x), with x ranging between 1 (the bottom) and nn( the top).
Math Example:
\sum^4_{x=1}2x = 2+4+6+8 = 20∑4x=12x=2+4+6+8=20
Note that you can replace the range notation of \sum∑ with a set. If X = \{1, 2, 3, 4 \}X={1,2,3,4} then
\sum_{x \in X} 2x \ = \ 2 + 4 + 6 + 8 \ = \ 20 \ = \ 2\sum_{ X} ∑x∈X2x = 2+4+6+8 = 20 = 2∑X
Mechanism Designers really like talking about whether xx should start at 1 or 0, although nobody knows why. Leading experts have hypothesized that it’s a core part of their mating ritual, but the results are still inconclusive.
∏ (Product)
This *"pi from the uncanny valley"* is actually a product:\prod^n_{x=1}f(x) =∏nx=1f(x)= the product of f(x)f(x) for all xx values from 11 to nn.
The best way to explain it is through a comparison:
\sum : \prod :: \text{addition}: \text{multiplication} ∑:∏::addition:multiplication
If you don’t remember the : :: : comparison format from the SATs then you are beyond saving.
\bigcup^x_y \text{ or } \binom{n}{k}⋃xy or (nk)
Unless xx and yy are both small, normal-looking numbers, you’re about to have a really bad time. The math isn’t hard, it’s just a real pain in the ass to write out. The one on the left is an iterator for the union of sets and the one on the right is the binomial coefficient.
d/dx (Derivatives)
Get excited because it's finally time for everyone's favorite subject: calculus!f(x)\frac{d}{dx} = f'(x) = \text{the derivative of} f(x) f(x)ddx=f′(x)=the derivative off(x)
Derivatives measure the rate that one thing is changing (probably xx) relative to the rate another thing is changing (probably yy, but maybe tt if the Mechanism Designer is fully domesticated). If f(x)f(x) is a line, then its derivative is the slope of the line. In other words, it’s the rate of change of the line. If there is a line that graphed “time” on the x-axis and a car’s distance from the starting point on the y-axis, then the derivative of that line would be the car’s velocity (the rate that the position is changing relative to the rate that time is changing). If the velocity of a car is on the y-axis, then the derivative of that line would be the car’s rate of acceleration. This is what they teach in calc 1, but that you haven’t needed to use since highschool because it only recently became cool to discuss “novel mechanisms.” It looks like your teacher was right all along.
∫ (Integral)
This squigly line is an "integral." Fun fact - there are no integrals in the bible.\int^x_yf(x) =∫xyf(x)= the integral, aka “antiderivative.”
If a function f(x)f(x) creates a line on a graph, its integral is the area underneath it. Even your highschool teacher would’ve admitted that it’s unlikely that you’ll need to use integrals in your day-to-day job. After all, integrals are pretty useless unless you’re either a math teacher or having to deal with the probabilities of expected probabilities in a Mechanism Designer’s white paper. Speaking of which…
Back to Probabilities
Cumulative Distribution Function
F_X(x)FX(x) is a cumulative distribution function, aka CDF. If you have a distribution XX (which is a set of all possible values that xx could be) then F_X(x) = \mathbb{E}[P(x > X)]FX(x)=E[P(x>X)], which is the probability that xx is greater than a randomly selected value from the set XX.
Example: If X = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20 \}X={2,4,6,8,10,12,14,16,18,20} and x = 8x=8, then F_X(8) = 0.30FX(8)=0.30 because when you draw a random number from XX there is only a 30% chance that you’ll get one of the three that are less than 8 (2, 4, and 6).
Probability Density Function
f_X(x) fX(x) is a probability density function, aka PDF. It is basically saying the probability that a randomly chosen value from XX will equal xx.
Example: If X = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20 \} X={2,4,6,8,10,12,14,16,18,20} then f_X(6) = 0.10fX(6)=0.10 because there’s a 10% chance that we’ll draw a 6 from the set. In other words, f_X(x) = \mathbb{E}[P(x = X)] fX(x)=E[P(x=X)]. Don’t think about that equation too much - if the thought of x = Xx=X seems contradictory to you, that’s a good sign that you’re still a healthy, normal person.
Example: X = \{2, \ 2, \ 6, 8, 10, 12, 14, 16, 18, 20 \} X={2, 2, 6,8,10,12,14,16,18,20} (note that we replaced the 4 with another 2) then f_X(2) = 0.20fX(2)=0.20 and f_X(4) = 0.0fX(4)=0.0.
Note that F_X(x)FX(x) is very useful in analyzing auctions because if XX is the set of all bids, F_X(x)FX(x) is the probability that our bid xx is greater than a randomly selected bid, and F_X(x)^nFX(x)n is the probability that our bid xx is greater than nn number of bids.
The calculus from the previous section comes into play because the probability density function f_X(x)fX(x) is the derivative of the cumulative distribution function F_X(x)FX(x), and the cumulative distribution function is the integral of the probability density function:
f_X(x) = F_X(x)\frac{d}{dx} fX(x)=FX(x)ddx
F_X(x) = \int^{\infty}_{- \infty} f_X(x)FX(x)=∫∞−∞fX(x)
We’ll often have to use calculus to get back and forth between how likely someone is to bid something, and how likely a bid is to be higher than another bid.
Auction Hieroglyphics
People typically refer to the set of players (so-called) in an auction as PP, and a player in the set of players as ii. Nobody knows why ii was chosen over pp, but it was probably so that Mechanism Designers could go around saying “i player” and to each other and laughing at their clever inside joke. This has been going on for decades.
If the bids are referred to as bb then b_ibi would be ii's bid. If you want to compare two players, jj is typically a stand-in for “the other player,” whereas -i−i is the stand-in for “all players other than ii.”
Symbols Over (or Under) Letters
Sometimes a Mechanism Designer might want to share a new formula with you that is similar to an existing one, but that’s just slightly different. If you see weird symbols over or under letters, it probably means the equation or variable has had something added to it to make it extra special.
Here are some examples:
- If ii is a player (bidder) in an auction, i'i′ might be his sworn nemesis.
\item If b_ibi is the bid of player ii, b^*_ib∗i might be the optimal bid.
–Warning: You might be wondering, “Aren’t all bids optimal bids? Why would player ii bid an amount that isn’t optimal?” but you should never ask this question out loud - it’s considered deeply inappropriate and you’ll end up on a list. - If g(x)g(x) is a function that works for everyone, g_i(x)gi(x) is a function that only works for special players like ii.
- If t_i^2t2i isn’t meant to be t_iti squared, it may mean that it’s the second tt belong to ii in a sequence of t_iti's. Maybe the 2 is on the bottom, but then where would we put the ii to mark the tt as special? This is a perfect example of the kind of difficult question that Mechanism Designers spend most of their time on.
ϵ (Epsilon)
Mechanism designers use the \epsilonϵ (epsilon) symbol a non-trivial amount, which is ironic because it represents a trivial amount. Trivial, by the way, is just a cooler-sounding way of saying “teeny-tiny.” A Mechanism Designer might say something along the lines of “the optimal bid value was market price less epsilon”: b^*_i = v_i - \epsilonb∗i=vi−ϵ.
⋅ (The Dot Thingy)
While this may mean multiplication, if you see it by itself inside of a function then it probably means the Mechanism Designer is being lazy and didn’t want to copy and paste their math. You’ll typically see this only after you’ve already seen the full version. For example, if you’re unlucky enough to see something like y = z + g(x^2+\mathbb{E}[Z] - \epsilon )y=z+g(x2+E[Z]−ϵ), then later on you might see a = 2z + g(\cdot)a=2z+g(⋅), where the \cdot⋅ is a stand-in for x^2+\mathbb{E}[Z] - \epsilonx2+E[Z]−ϵ.
⇒ (Therefore)
If the Mechanism Designer wants to prove why something is the way that it is, they might use this arrow thingy. It means “therefore” or “it follows that…”. For example, if a Mechanism Designer wants to show that size of his mechanism proves that he’s really smart, it might look like:
\mathbf{card}(mechanism_i) > \mathbf{card}(mechanism_{j}) \forall j\in P : j \not = i \Rightarrow F_{IQ}(iq_i) = 1 - \epsilon card(mechanismi)>card(mechanismj)∀j∈P:j≠i⇒FIQ(iqi)=1−ϵ
A good exercise is to assess whether or not you actually understood that equation. If you did, it means you’ve been paying attention to the paper! Unfortunately, it also means you’re less cool in expectation. The actual english translation is “If the number of things in the mechanism made by player i is greater than the number of things in the mechanism made by player j, for all possible player j’s in the world who aren’t player i, then it follows that the IQ of player i has a 100% chance (minus a teeny-tiny percent) of being greater than a randomly selected IQ from the distribution of all the IQs in the world.”
ϕ,θ,γ,δ,σ,ψ,τ or Other Greek Letters
Mechanism Designers love defining variables. Unlike their sworn enemies the mathematicians, Mechanism Designers really like for their variables to be exotic, and so they’ll often use lower-case greek letters. It’s a best practice to always have a Greek alphabet available when reading a mechanism’s white paper so that you can quickly check to see if the designer is referring to a variable - which is pretty normal - or using some sort of ritualistic-sacrifice-based summoning math - which is a red flag. Intuitively.
Smart-Sounding Auction Terms
ex ante and ex post
When auction players have to figure out what their bid for an item is before they know what the value of the item is, the auction is said to be ex ante. If the value of the item is known at bidding time, the auction is said to be ex post. This is one of the rare cases in which the obscure auction language is actually less wordy than what its describing. Way to go, Mechanism Designers! You did it!
First-Price and Second-Price Auctions
A first-price auction is one in which the highest bidder pays what they bid. A second-price auction is one in which the highest bidder pays what the second-highest bidder bid. Mechanism Designers really like second-price auctions because they pay the beneficiary more than a first-price would, but they’re also easier to cheat and therefore require a more robust mechanism.
Warning: Never ask a Mechanism Designer why second-price auctions are better than first-price auctions. It’s like asking your wife if she’s had any interesting dreams recently, or if that girl at work she doesn’t like has caused any more problems. If a Mechanism Designer ever brings up the subject of first-price vs second-price, just look them directly in the eyes and then tell them with a firm voice, “With a properly designed mechanism and sealed bids, a second-price auction at equilibrium leads to more auction revenue in expectation, obviously!” This response is a strictly-dominant strategy.
Sealed Bid means "private"
That's what the intuition is, anyway.Utility Function
A utility function, often called a payoff function, is how valuable someone thinks something is. It’s typically measured in expectation, but this was developed by the French so the \mathbb{E}E is silent.
Example: If U(x)U(x) is a utility function, then U_i(b_i, b_{-i})Ui(
