2d:null 2e:["$","title",null,{"children":"RICKS Mechanism Analysis | Saneel Sreeni | Observable"}] 33:T524,This analysis was inspired by [this paper](https://www.paradigm.xyz/2021/10/ricks/) from Dave White and Dan Robinson at Paradigm and Andy at Fractional Art, and fueled by my own boredom and fascination with RICKS. I decided to try and take a deeper look at the mechanisms involved and mathematically codify them. First, I wanted to calculate how many days it would take for me to accumulate the desired portion of RICKS supply given the issuance rate, initial holdings, and the total supply at the time I made my decision (assuming I could purchase every RICKS that went on the market). Second, the paper mentions the coin flip mechanism from Martingale fractionalization for deciding buyouts, where the procedure must be "EV-fair" but also must be able to make odds worse the further away a coin flip is initiated from 100% ownership. This intrigued me and I wanted to better flush this out; I ended up not only solving for the factor used to affect odds when "EV-Fair" was upheld even as the majority owner went away from 99% (a Martingale, as the EV at each buyout is always [0], but also the general exponential decay equations we could use to decrease probability (and make EV negative) in relation to the distance between majority ownership and 100% ownership, both for "EV-Fair" and non-fair assumptions. 34:T100b,Next, let us consider the Martingale fractionalization buyout posited in the paper. In the given example, a majority owner (99% of RICKS) wants to flip an unbiased coin. If it lands heads, the majority gets the remaining fraction of RICKS (1%) and thus owns the NFTs; else everyone else has their share doubled and the owner loses 1%. This is a fair process, as far as EV goes (the expected value of gain for the majority owner is 0.01 x 0.5 - 0.01 x 0.5 = 0) We can abstract this a bit. We consider the probability of a coin landing heads *p* (the probability of tails is accordingly *(1 - p)*), the percentage share that the majority owner has (as a decimal) as *x*, and we assume that when the majority owner initiates a coin flip for ownership of the NFT, the positive payout for them is the remaining share of the NFT (and they forfeit this payout accordingly). We can then reformulate the scenario posited originally as: (1) EV[*Majority Gain for Coin Flip*] = *p(1 - x) - (1-p)(1-x) = (2p-1)(1-x)* Note that the EV[*Minority Gain for Coin Flip*] is symmetric to this as the negative payoff for the majority is simply the positive payout for the minority. We can then set this EV to 0 to simulate a fair EV process (expected zero gain for both majority and minority, due to the aforementioned symmetry). Doing so yields: (2) *p(1-x) = (1-p)(1-x)* Which simplifies to: (3) *p = 1 - p* (4) *p = 0.5* We can see that if we assume that the majority owner either wins the remaining share of RICKS or forfeits the same amount to the minorities, the probability that makes this a fair coin flip, EV wise, is always 50% (a fair coin)! But wait... The paper also states: *"To avoid weirdness around the 99% boundary, we could allow a majority owner to trigger a coin flip at 98%, or 90%, or even 75%, with the caveat that they would receive worse odds the farther away they were from 99% ownership."* Intuitively, this makes sense; a majority owner with 75% of the RICKS should not be able to enjoy the same odds that a majority owner with 99% does when they stand to gain far more. At the same time, we'd like to preserve the "fairness" of the coin flip (EV of gain for both the majority and minority is 0). It turns out doing so isn't particularly difficult. We introduce an additional penalty scaling factor *a* (where *a > 1*) to our original EV calculation for the majority owner's gain. This penalty is multiplied with the negative payoff to more heavily weigh the downside of losing the flip as *(1-x)* increases, given that the negative payoff is *(1-p)(1-x)*. In other words, if a majority owner initiated a flip with 75% ownership, he stands to gain 25% of the remaining RICKS, but lose 25% * *a* RICKS, where a > 1. Thus, we get (5) EV[*Adjusted Majority Gain for Coin Flip*] = *p(1 - x) - a(1-p)(1-x)* We can also think of the *a* term as a multiplier on the odds that the majority owner loses, as some function of the remaining RICKs to be won and the probability of heads of the coin we chose. As such, we solve for *a*: (6) *(1-x)(p-a(1-p)) = 0* (7) *p - a(1-p) = 0* (8) *a = p / (1-p)* Thus, we can simply multiply by *p / (1 - p)* to penalize the additional distance from 100% ownership while still preserving the fairness of the game (EV of gain is 0). The nice part about this approach is that *p* can be arbitrarily set if the only goal is to preserve EV fairness; because the EV of gain is 0 at all *p* values, we are free to design any sort of curve that pushes *p* up and down depending on other factors (if you'd like to verify this for yourself, please feel free to use the widget below). One such possibility would be adding the following curve that reduces *p* based on the difference between 100% ownership and the majority owner's ownership: (9) *p(x) = 0.5e^{-(1.00 - x)}* *where 0.50 < x < 1.00* This paper exponentially decays *p* based on the difference between 1.00 and the decimal representing the ownership of the majority owner. Again, the game remains EV-fair, but we can take advantage of having the freedom to manipulate *p* accordingly. 35:T687,As one last point, what if we don't *care* if the game remains EV-fair? What if instead the system was meant to properly penalize and decrease EV as the majority owner increased their distance from 100%. In this case, we can remove the *a* multiplier (which moves inversely proportional to the decay function in (9)). Then, we get: (10) EV[*Majority Gain for Coin Flip*] = *p(1 - x) - (1-p)(1-x)* We substitute *p* for (9). (11) EV[*Majority Gain for Coin Flip*] = *0.5e^{-(1.00 - x)}*(1 - x) - (1 - 0.5e^{-(1.00 - x)}*)(1-x)* (12) EV[*Majority Gain for Coin Flip*] = (1 - x)(e^{-(1.00 - x)} - 1) We see that this EV decays exponentially as x decreases within the range of *0.50 < x < 1.00*. If we wanted the EV to be negative, we'd need *(2p-1)(1-x) < 0*, which holds when *p < 1/2* or *x > 1*. The latter condition will never hold in this case, so we need to select a *p(x) < 1/2* for all *0.50 < x < 1.00*. We can solve for *K* where *p(x) = Ke^{-(1.00 - x)*}. (13) *p(x) = Ke^{-(1.00 - x)*} *< 0.5* Noting that the highest value of this function occurs at its highest bound, 100%. We then simply solve x = 1: (14) *Ke⁰ < 0.5* This gives the somewhat obvious solution that: (15) *K < 0.5 * If we wanted to instead have the EV turn negative in between 0.50 and 1.00 (at 0.75), we could solve for K as such: (16) *Ke^-0.25 < 0.5* (17) K < 0.642 We can continue to extend this example to derive other values at which we want the EV of the majority owner to be negative at certain thresholds. I created a function that would enable you to test values, as well as a Desmos graph for these values.36:[] 31:{"id":"0db2cb4908697266","slug":"picklerick","title":"RICKS Mechanism Analysis","thumbnail":"bae541c63d6420221de4ac205648dd9b0e9123509409edb18116e77832acb803","access_level":"public","create_time":"$D2021-10-06T19:36:09.347Z","update_time":"$D2021-10-07T07:02:50.548Z","view_time":null,"first_public_time":"$D2021-10-06T21:29:04.404Z","fork":{"id":"f5972eaaa1db54c4","slug":"a-interactive-widget-for-reorg-ev-calculations","title":"A Interactive Widget for Reorg EV Calculations","owner":{"id":"93804c93bac92f2f","login":"saneeldcp","name":""}},"stars":0,"starred":false,"owner":{"type":"team","id":"46126c86112a2fcf","login":"ssreeni1","name":"Saneel Sreeni","avatar_url":"https://avatars-next.observableusercontent.com/c59a4aa09f21165346307a94448d3bb78c9fed30f6fbe76f83611ae974beab46","tier":"starter_2024"},"version":500,"publish_version":500,"role":null,"authored":false,"authors":[{"type":"individual","id":"a4c3db70cc09286c","login":"ssreeni1","name":"Saneel Sreeni","avatar_url":"https://avatars-next.observableusercontent.com/c59a4aa09f21165346307a94448d3bb78c9fed30f6fbe76f83611ae974beab46","tier":null}],"members":[{"type":"individual","id":"a4c3db70cc09286c","login":"ssreeni1","name":"Saneel Sreeni","avatar_url":"https://avatars-next.observableusercontent.com/c59a4aa09f21165346307a94448d3bb78c9fed30f6fbe76f83611ae974beab46","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":true,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"title":"RICKS Mechanism Analysis","cells":[{"id":0,"value":"# RICKS Mechanism Analysis","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2,"value":"## Introduction\n\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":8,"value":"$33","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":469,"value":"-------------------------------------------------------------------------------------------------------------------------------------------------","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":275,"value":"## I. TRICK Accumulator","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":52,"value":"We decide one fine day that we want *X*% of RICKS supply for an asset that we really like, and we want to know how many days, *t*, it'll take to do so. Let *S* be the initial total supply, *I* be *our* initial total supply, and let's assume that we are so committed to buying RICKS for this asset that we bid on all of them with the fastest bot at any cost to ensure we acquire all new RICKS emitted.\n\nThen it's easy to see that the proportion of the supply we have at the end of the *t* days is:\n\n(1) *(I + S((1+r)^t - 1)) / (S(1+r)^t) *\n\nWhich simplifies to:\n \n(2) *(I / S)(1 / (1+r)^t) + 1 - (1 / (1+r)^t)*\n\nSetting this equal to *X* we then can expand and simplify further to say:\n\n(3) *(X - 1) / ((I / S) - 1) = (1 / (1+r)^t)*\n\nWhich yields:\n\n(4) *(1+r)^t = ((I / S) - 1) / (X - 1)*\n\nSolving for t:\n\n(5) *t = log_(1+r) (((I / S) - 1) / (X - 1))*\n\n\nYou can play around with these values below.\n\n\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":174,"value":"### Accumulation Widget","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":179,"value":"import {slider} from \"@jashkenas/inputs\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":215,"value":"import {log, exp} from \"@fil/math\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":183,"value":"viewof issuance_rate = slider({\n min: .01,\n max: .99,\n step: .01,\n format: \".0%\",\n description:\n \"Issuance Rate, r\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":190,"value":"viewof target_proportion = slider({\n min: .01,\n max: .99,\n step: .01,\n format: \".0%\",\n description:\n \"Target proportion, X\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":195,"value":"viewof initialTotalSupply = slider({\n min: 0,\n max: 1000,\n step: .01,\n format: \",\",\n description:\n \"Initial total RICK supply, S\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":201,"value":"viewof initialSelfSupply = slider({\n min: 0,\n max: initialTotalSupply * 0.99,\n step: .01,\n format: \",\",\n description:\n \"Initial self RICK supply, I\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":208,"value":"If you started with **${initialSelfSupply} **RICKS, the initial total supply was **${initialTotalSupply}** RICKS, and the issuance rate is **${issuance_rate * 100}%** of the current supply per day, then the number of days you'd need to reach **${target_proportion * 100}%** of the RICKS supply, would be:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":230,"value":"t = log(((initialSelfSupply / initialTotalSupply) - 1) / (target_proportion - 1)) / log(1 + issuance_rate)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":329,"value":"#### Accumulation Graph","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":296,"value":"","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":455,"value":"---------------------------------------------------------------------------------------------------------------------------------------------","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":278,"value":"## II. Martingale Buyouts","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":148,"value":"$34","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":376,"value":"#### Buyout EV Calculator (Fairness)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":303,"value":"viewof p = slider({\n min: .01,\n max: 1.0,\n step: .01,\n format: \".0%\",\n description:\n \"Probability of heads\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":315,"value":"viewof x = slider({\n min: .51,\n max: 1.0,\n step: .01,\n format: \".0%\",\n description:\n \"Majority ownership %\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":325,"value":"ev_majority_winner = (p * (1 - x)) - ((1-p) * (1-x)) //set p to 50%, and try with any majority ownership %","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":311,"value":"ev_adjusted_majority_winner = (p * (1 - x)) - ((1-p) * (1-x) * (p / ( 1 - p ))) //any combination of p and majority ownership will return 0","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":390,"value":"## Removing EV Fairness and the Martingale","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":385,"value":"$35","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":395,"value":"#### Buyout EV Calculator (Negative EV)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":410,"value":"viewof K = slider({\n min: 0.00,\n max: 1.00,\n step: .01,\n format: \",\",\n description:\n \"Constant multiplier for exponential\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":420,"value":"viewof input_x = slider({\n min: 0.50,\n max: 1.00,\n step: .01,\n format: \",\",\n description:\n \"Majority owership %\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":400,"value":"expected_value_no_fair=(1-input_x)* ((2 * K) * exp(-(1.00-input_x))-1)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":495,"value":"#### Buyout EV Graph","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":422,"value":"","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W36","stdlib":"1","titling":"auto"}} 32:"$31:body"