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, Iâll give you the tools youâll 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.
Letâs start with this section - the âabstract.â This section is really just a summary of the white paper. In this section, the Mechanism Designers will summarize a problem, then theyâll summarize their solution, and then - if theyâre good - theyâll also summarize the shortcomings of the paper. We may never know why the Mechanism Designers like to call this section an âabstractâ rather than a âsummaryâ - in fact, weâll probably never know why they do most of the things that they do. But by reading this paper, hopefully youâll be better able to understand what theyâre doing - and, if youâre into this kinda thing, who theyâre doing it to.
Intuitively, this paper doesnât have any shortcomings. But if I had to pick only one, itâd be that a lot of the information in this paper is deliberately wrong. Very wrong. Iâm way off on a lot of these explanations. But Iâm not being byzantine - which means âdishonestâ or âadversarialâ, by the way, and two thousand years ago it probably wouldâve been flagged by HR departments as racially offensive against the citizens of the Byzantine empire. So yeah, Iâm not being byzantine - Iâm 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 âthat definition left out a lot!â then be warned: itâs 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 X or Y).
- When sets are defined, theyâre often surrounded with brackets. For example, a set of three numbers could be written as X = \{1,2,3\}.
- Members of the set are represented with a lowercase letter (such as x or y).
There are a handful of set-related funny-looking symbols that these Mechanism Designers like to use in their white papers that you should probably learn:
-
\in means âinâ and is used to show membership in a set
â When you see x â X, that means that x is a member of the set X.
â Example: if X = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\}, then x â X means x is one of those 10 values. -
\cup means âunionâ and is meant to take two sets and combine all of their members together (removing duplicates).
â When you see X âȘ Y, that means youâre combining sets X and Y.
â Example: if X = \{1, 2, 3, 4\}, and Y = \{3, 4, 5\} then X âȘ Y = \{1,2,3,4,5\}. -
\cap means âintersectionâ and is meant to take two sets and make a new set only out of their common members.
â When you see X â© Y, that means youâre taking the intersection of sets X and Y.
â Example: if X = \{1, 2, 3, 4\}, and Y = \{3, 4, 5\} then X â© Y = \{3,4\}. -
\subset means âsub setâ - all of the elements of the first set exist in the second set.
-
\supset means âsuper setâ - all of the elements of the second set exist in the first set. Note that if you see this then youâre dealing with a particularly troublesome brand of Mechanism Designer and you should be on your guard for more trickery.
-
: (the colon) typically means âifâ and is often seen together with a funny looking upside-down A. Speaking of whichâŠ
-
\forall means âfor allâ and is meant to iterate through a set, often times in the creation of another one.
âIf youâre a programmer, whenever you see \forall think âfor loop.â
â Often times \forall and : are used to iterate through a set to create a new one.
â Example: Consider this monstrous expression: Y = \{2x, \ \forall x\in X : x>3 \}. The english translation is something along the lines of âY is a set of numbers that was created by taking each of the x's in X that were greater than 3 and then multiplying that x by 2.â For posterityâs sake, if X = \{1, 2, 3, 4, 5 \} then it follows that (aka \Rightarrow, as weâll discuss later) 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 \}. That would be the same as Y = \{2 \cdot x\in X : x>3 \} or Y = \{z \in \{2x, \forall x\in 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 âtuplesâ). Theyâre 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 now, theyâre probably day dreaming about correcting the author - presumably using made-up words like âvectorâ or âscalar.â
-
\mathbf{card}(X) or |X| is the âcardinalityâ of the set X. This is a fancy way of saying âthe number of things in X.â If X = \{ 1, 17, 31, 5\} then |X| = \mathbf{card}(X) = 4. Keep in mind that |X| is the size of X while |x| is the absolute value of x \in 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} is the set of all integers.
- \mathbb{R} is the set of all real numbers.
- \mathbb{E} isnât a set of numbers at all - it means âthe expectation ofâ and is used in probability. But it looks similar to these fancy sets, so be careful not to get it mixed up. Thatâs how they get you.
- \mathbb{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}, which means X is a subset of \mathbb{Z}, which means that all of the possible x's in X must be integers. If youâre wondering âWhy doesnât the Mechanism Designer just say that the set is made up only of integers?â the answer is because they hate you.
Stochasticisms and Probabalisticalities
Two things that Mechanism Designers simply canât get enough of are intuitions and expectations.
âIntuitivelyâŠâ or âThe intuition isâŠâ means that the Mechanism Designer is about to tell you something that they think is so obvious that they wonât explain it because only a moron would disagree. Unfortunately, for a Mechanism Designer i in the set of all people P, the set of morons M = \{ p, \forall p \in P : p \not = i \}, which, in English, means âThe set of morons is all of the people in the set of people who arenât also the Mechanism Designerâ. If you want to understand a Mechanism Designerâs intuition, your best chance is their cultureâs 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.
âThe expectation isâŠâ or â⊠in expectation.â means that the Mechanism Designer is about to do some math, and we can expect that the math will involve probabilities. Probably.
-
P(y) is the probability of event y occuring. Example: for a coinflip, P(\text{heads}) = 0.50
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P(y \cap z) is the probability of event y and event z both occuring. This is also abbreviated as P(y, z).
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P(y \cup z) is the probability of event y or event z occuring.
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P(y | z) = the probability of event y occuring if we assume event z 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}) = 0.80 and P(\text{heads} | \text{cheater bet on tails} | \text{cheater's flip}) = 0.20.
-
\mathbb{E}[X] = The expected (or probabalistic, if you want to sound smart while actually being wrong) value of X. For example, if thereâs a game with a coinflip and you get $0 for tails and $1 for heads then \mathbb{E}[\text{game}] = $0.50. Note that if we want to treat the gameâs outcomes as a set, {E}[X] = \{ {E}[\text{heads}], \ {E}[\text{tails}]\} = \{ \$0, \$1 \}, and \mathbb{E}[X] is just the average of the expected values (aka âthe expectation ofâ) the possible outcomes in X.
-
\mathbb{E}[X|y] = The value of X in expectation if we assume that y happened. For example, if thereâs a game with a coin flip and you get $0 for tails and $1 for heads, but thereâs 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}] = $0.20.
âNow 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) -
\mathbb{P}(x) is just a fancy way of saying P(x). Some Mechanism Designers insist that if P(x) is the probability of x, \mathbb{P}(x) is the probability of x in expectation, but nobody knows what that means and so we just ignore them and move on with our lives.
Fancy Math
â (Sum)
This sigma may have been an key part of your fraternity or sororityâs identity during college, but it has another, less-important use case: math.
\sum^n_{x=1}f(x) = The sum of f(x) for all x values from 1 to n.
In other words, itâs a âfor loopâ that sums the different values of f(x), with x ranging between 1 (the bottom) and n( the top).
Math Example:
\sum^4_{x=1}2x = 2+4+6+8 = 20
Note that you can replace the range notation of \sum with a set. If X = \{1, 2, 3, 4 \} then
\sum_{x \in X} 2x \ = \ 2 + 4 + 6 + 8 \ = \ 20 \ = \ 2\sum_{ X}
Mechanism Designers really like talking about whether x should start at 1 or 0, although nobody knows why. Leading experts have hypothesized that itâs a core part of their mating ritual, but the results are still inconclusive.
â (Product)
This *"pi from the uncanny valley"* is actually a product:\prod^n_{x=1}f(x) = the product of f(x) for all x values from 1 to n.
The best way to explain it is through a comparison:
\sum : \prod :: \text{addition}: \text{multiplication}
If you donât remember the : :: : comparison format from the SATs then you are beyond saving.
The "Big U" and the Binomial Coefficient
\bigcup^x_y \text{ or } \binom{n}{k}
Unless x and y are both small, normal-looking numbers, youâre about to have a really bad time. The math isnât hard, itâs 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)
Derivatives measure the rate that one thing is changing (probably x) relative to the rate another thing is changing (probably y, but maybe t if the Mechanism Designer is fully domesticated). If f(x) is a line, then its derivative is the slope of the line. In other words, itâs the rate of change of the line. If there is a line that graphed âtimeâ on the x-axis and a carâs distance from the starting point on the y-axis, then the derivative of that line would be the carâs 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 carâs rate of acceleration. This is what they teach in calc 1, but that you havenât needed to use since highschool because it only recently became cool to discuss ânovel mechanisms.â It looks like your teacher was right all along.
â« (Integral)
This squigly line is an "integral." Fun fact - there are no integrals in the bible.\int^x_yf(x) = the integral, aka âantiderivative.â
If a function f(x) creates a line on a graph, its integral is the area underneath it. Even your highschool teacher wouldâve admitted that itâs unlikely that youâll need to use integrals in your day-to-day job. After all, integrals are pretty useless unless youâre either a math teacher or having to deal with the probabilities of expected probabilities in a Mechanism Designerâs white paper. Speaking of whichâŠ
Back to Probabilities
Cumulative Distribution Function
F_X(x) is a cumulative distribution function, aka CDF. If you have a distribution X (which is a set of all possible values that x could be) then F_X(x) = \mathbb{E}[P(x > X)], which is the probability that x is greater than a randomly selected value from the set X.
Example: If X = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20 \} and x = 8, then F_X(8) = 0.30 because when you draw a random number from X there is only a 30% chance that youâll get one of the three that are less than 8 (2, 4, and 6).
Probability Density Function
f_X(x) is a probability density function, aka PDF. It is basically saying the probability that a randomly chosen value from X will equal x.
Example: If X = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20 \} then f_X(6) = 0.10 because thereâs a 10% chance that weâll draw a 6 from the set. In other words, f_X(x) = \mathbb{E}[P(x = X)] . Donât think about that equation too much - if the thought of x = X seems contradictory to you, thatâs a good sign that youâre still a healthy, normal person.
Example: X = \{2, \ 2, \ 6, 8, 10, 12, 14, 16, 18, 20 \} (note that we replaced the 4 with another 2) then f_X(2) = 0.20 and f_X(4) = 0.0.
Note that F_X(x) is very useful in analyzing auctions because if X is the set of all bids, F_X(x) is the probability that our bid x is greater than a randomly selected bid, and F_X(x)^n is the probability that our bid x is greater than n number of bids.
The calculus from the previous section comes into play because the probability density function f_X(x) is the derivative of the cumulative distribution function F_X(x), and the cumulative distribution function is the integral of the probability density function:
f_X(x) = F_X(x)\frac{d}{dx}
F_X(x) = \int^{\infty}_{- \infty} f_X(x)
Weâll 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 P, and a player in the set of players as i. Nobody knows why i was chosen over p, but it was probably so that Mechanism Designers could go around saying âi playerâ to each other and laughing at their clever inside joke. This has been going on for decades.
If the bids are referred to as b then b_i would be i's bid. If you want to compare two players, j is typically a stand-in for âthe other player,â whereas -i is the stand-in for âall players other than i.â
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 thatâs 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 i is a player (bidder) in an auction, i' might be his sworn nemesis.
- If b_i is the bid of player i, b^*_i might be the optimal bid.
âWarning: You might be wondering, âArenât all bids optimal bids? Why would player i bid an amount that isnât optimal?â but you should never ask this question out loud - itâs considered deeply inappropriate and youâll end up on a list. - If g(x) is a function that works for everyone, g_i(x) is a function that only works for special players like i.
- If t_i^2 isnât meant to be t_i squared, it may mean that itâs the second t belong to i in a sequence of t_i's. Maybe the 2 is on the bottom, but then where would we put the i to mark the t as special? This is a perfect example of the kind of difficult question that Mechanism Designers spend most of their time on.
Ï” (Epsilon)
Mechanism designers use the \epsilon (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 âteeny-tiny.â A Mechanism Designer might say something along the lines of âthe optimal bid value was market price less epsilonâ: b^*_i = v_i - \epsilon.
â (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 didnât want to copy and paste their math. Youâll typically see this only after youâve already seen the full version. For example, if youâre unlucky enough to see something like y = z + g(x^2+\mathbb{E}[Z] - \epsilon ), then later on you might see a = 2z + g(\cdot), where the \cdot is a stand-in for x^2+\mathbb{E}[Z] - \epsilon.
â (Therefore)
If the Mechanism Designer wants to prove why something is the way that it is, they might use this big arrow thingy. It means âthereforeâ or âit follows thatâŠâ. For example, if a Mechanism Designer wants to show that size of his mechanism proves that heâs 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
A good exercise is to assess whether or not you actually understood that equation. If you did, it means youâve been paying attention to the paper! Unfortunately, it also means youâre less cool in expectation. The actual english translation is âIf 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 jâs in the world who arenât 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.â
â (Approaches)
This small arrow thingy (\rightarrow) means âapproaches.â Youâll often see \rightarrow right next to \Rightarrow due to how much the Mechanism Designers like tricking people.
An example, where M_i is the set of mechanisms made by player i and F_i is the set of player i's friends:
Which, in English, means âIf the number of mechanisms made by player i approaches infinity, it follows that the number of friends that player i has approaches zero.â
Ï,Ξ,Îł,ÎŽ,Ï,Ï,Ï 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 theyâll often use lower-case greek letters. Itâs a best practice to always have a Greek alphabet available when reading a mechanismâs 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 theyâre 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. Itâs like asking your wife if sheâs had any interesting dreams recently, or if that girl at work she doesnât 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, âWith a properly designed mechanism and sealed bids, a second-price auction at equilibrium leads to more auction revenue in expectation, obviously!â 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. Itâs typically measured in expectation, but this was developed by the French so the \mathbb{E} is silent.
Example: If U(x) is a utility function, then U_i(b_i, b_{-i}) is the payoff to player i considering their bid ( b_i) and their competitorsâ bids (b_{-i}).
Bid Shading
Bid shading occurs when bidders bid less than their true value - in other words, when v_i > b^*_i. This typically happens because the bidder can make more money by bidding below their true value. Or, in the native tongue of Mechanism Designers, âThe utility function of bidders is optimized when their private valuations exceed their equilibrium bid.â. We may never know why Mechanism Designers find it so fascinating that people make more money by paying less for something than they think itâs worth.
Incentive Compatibility
Something is considered incentive compatible (IC) if the participants in an auction are willing to bid their true value. Every bidder i assigns a value v_i to the item. The mechanism is incentive compatible when v_i = b_i. Mechanisms can be strongly incentive compatible or weakly incentive compatible, but nobody really knows what that means. It probably has something to do with how big the mechanism is.
Bayesian-Nash Equilibrium
A Bayesian-Nash Equilibrium (or, as the kids say these days, a âBNEâ) exists when there are multiple rounds and each biddersâ optimal strategy (AKA their âBest Responseâ or br_i or s^*_i) stays the same for each round. In other words, they wonât benefit from using any âtrick playsâ or âbamboozles.â
Note that a mechanism with incentive compatibility is considered significantly better than a mechanism with just a bayesian-nash equilibrium. If two Mechanism Designers meet in the wild and one of them has an IC mechanism and the other has a BNE mechanism, then the wife and children of the BNE designer will instinctively join the tribe of the IC designer and the BNE designer will have to start rebuilding his mechanism and his family from scratch. Mother nature is cruel, but efficient.
Credible Neutrality
Nobody actually knows what credible neutrality means. This has led to many Mechanism Designers making up their own definitions, presumably using their intuition. Unfortunately, these different definitions of credible neutrality are not credibly neutral, which is why the term remains undefined.
Fair Exchange Problem
The fair exchange problem refers to the difficulties of getting the parties of a trade to actually do what they promised theyâd do, such as paying a bid or selling an asset at a certain price. Mechanism Designers love making people do things, so this is one of their favorite problems to solve.
When someone refers to the free look problem, know that youâre talking to an actual human being - a Mechanism Designer will always out themselves by calling it a fair exchange problem. Usage of the term optionality is a red flag but still inconclusive.
Conclusion
The conclusion section of a Mechanism Designerâs white paper is the result of feeding the rest of their paper into a LLM model and then asking it to generate a summary. This is why nobody ever actually reads it, and neither should you. Iâd also like to thank my co-author, Et Al.
