White Paper GOLDEN RATIO PER LIQUIDITY - GRPL

White Paper

GRPL

GOLDEN RATIO PER
LIQUIDITY

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 Email 1: support@grpl.finance
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Table of Content
 Whitepaper

 What is GRPL? 02

 Fibonacci & Nash Equilibrium 03

 What is Golden Ratio? 04

 What is Nash Equilibrium? 09

 Equilibrium of Fibo & Nash 14

 Beyond The Nash Equlibrium 16

 Grpl Protocol 18

 Fibonacci Formula 18

 Rebase Rules 19

01

01. Introduction
What is Grpl ?

 Grpl, uses Fibonacci correction

01 levels as a formula in the Rebase
 system.
 Grpl is an elastic supply token which uses
 Fibonacci correction levels as a formula in the
 Rebase system. It pushes the natural
 mathematical will of the people to do the most
 beneficial thing for the group, based on the fibo
 Many eyes who know how to see can see, correction levels at the target price. It provides
 many minds who know how to perceive can Nash Equilibrium, or fair distribution of
 comprehend that the mathematics is a language income, this way.
 that nature uses to perform its art. Then, this We are also designed to maintain a price target
 insight penetrates the soul of the person, of one gram of gold
 becoming a part of it. You will be looking for a
 trace of it wherever and whenever you look.
 All adjustments in procurement (REBASE) are
 This trace sometimes appears as the golden
 made to achieve this goal. Oracle price; is
 ratio of Fibonacci, the embodiment of nature's
 daily moving averages price of exchanges.
 mathematics, while it is occasionally seen in
 Fibo price is the interm target price that
 John Nash's Equilibrium as the mathematical
 calculate the ratio between the last two days
 sociology of people's greed.
 oracle price. The harmony between oracle
 prices continue rebases until it reaches 0.61834
 everyday.

 02

Introduction
 Fibonacci & Nash Equilibrium

 The golden ratio is the perfect harmony where

 0,618 everyone benefits well and achieves a fair
 distribution of income. When this rebase

 Golden Ratio: Phi can be defined by
 occurs, Supply is increased or decreased for
 taking a stick and breaking it into two equal the oracle price to the price of one gram
 portions. If the ratio between these
 two portions is the same as the ratio of gold. Our real goal is to make the mass win,
 between the overall stick and the
 larger segment, the portions are said not individuals. Natural mind choice will pave
 to be in the golden ratio.
 the way for us to win for society.
 Two quantities are in the Golden
 Ratio if their ratio is the same as the
 ratio of their sum to the larger of the We preferred gold because; It is used both as a
 two quantities
 barter and a value storage tool. It is a
 traditional value that has been considered
 valuable in every period of history. It is the safe
 haven in the global economy as well as the
 most durable money against economic shocks.
 The source of price movements in gold-based.
 systems is, as a rule, not monetary. It varies
 according to the demand and supply.

 .

03

Introduction
 What is Golden Ratio ?

 It is said that a well-proportioned face must lie many conformed with this ratio. This was why,
1,1,2,3,5… in what is called a "golden rectangle" of by the time of the Renaissance, it had become
 dimensions in the ratio of approximately 1 to known as the "Divine Proportion".
Golden Ratio: Mathematically, the 1.6. Not only living forms, but also works of art
Golden Ratio appears as the limit of Mathematically, the Golden Ratio appears as
 and buildings, including the splendid domes of
several sequences, the most important
 Persia and the Athens Parthenon, are found to the limit of several sequences, the most
of which are those that satisfy the

recurrence relation s(n+1) = s(n) + s(n- adhere to this rule. The ratio became even more important of which are those that satisfy the
1) where s (n) is the nth term of the pronounced during the European Renaissance, recurrence relation s(n+1) = s(n) + s(n-1) where
sequence, whereas s(n- 1) and s(n+1)
 when Leonardo Da Vinci studied the physical s (n) is the nth term of the sequence, whereas
are respectively the preceding and
 proportions of man and portrayed them in his s(n- 1) and s(n+1) are respectively the
succeeding terms, as in the famous

Fibonacci sequence; 1, 1, 2, 3, 5, 8, 13, unfinished canvas of St Jerome along with preceding and succeeding terms, as in the
21, 34, 55, 89, 144, .... other works such as the "Mona Lisa" and the famous Fibonacci sequence; 1, 1, 2, 3, 5, 8, 13,
 "Vitruvian Man". 21, 34, 55, 89, 144, ....

 So, quite apart from the other mathematical in which any term is the additive sum of its two
 constants, Pi = 3.14, e = 2.718 (Euler Number), preceding terms, or in any other sequence
 Gamma = 0.577 (Euler Constant), and i = sqrt(- satisfying the above recurrence, even if we start
 1) which possess mathematical properties only, with any two numbers such as 5 and 2; that is
 the Golden Ratio Phi = 1.618... has an 5, 2, 7, 9, 16, 25... We always have that the
 additional aesthetic feature. Since quotient of two successive terms s(n+1)/s(n)
 mathematicians in ancient times were often approaches a certain number called Phi. To
 poets and philosophers who believed in the calculate the value of Phi from the above
 uniformity of nature, mathematics served to recurrence relation written as s(n+1)/s(n) = 1 +
 satisfy their need to understand the world s(n-1)/s(n), it turns out, in the limiting situation,
 around them and to resolve its secrets. Hence, that Phi = 1 + 1/phi, or that Phi equals half of
 they started tracking this constant in everyday the quantity 1+sqrt(5) = 1.618...
 objects such as plants and animals, and
 discovered that the proportions of

 04

Introduction
What is Golden Ratio ?

 Historians trace the Golden Ratio back to that the Egyptians had a mathematical
 Euclid, yet it appears that even before him it understanding of these constants, the ratios, not
 was governing the dimensions of monuments in the symbol, 1000 earlier. Petrie, for example,
 ancient Egypt. The most pronounced of these is noticed that the dimensions of many Egyptian
 the Great Pyramid. The dimensions of the inner tombs, especially those of a parallelepiped
 triangle (the so-called "Egyptian triangle") of structure, adhered to the ratios 1 : Phi : Phi
 Khufu's Pyramid, for instance, in Royal cubits square. The same ratio also appears in a grid
 (one cubit equalling roughly 0.524 metres), are surrounding a human body depicted in the
 (220c, 280c, 356c) i.e. in the ratios 1 : sqrt(Phi) royal tomb of Amenhotep III in the Valley of
 : Phi. That the foregoing relation is not a matter the Kings.
 of coincidence is discussed elsewhere.
 However, the Great Pyramid is not the only There was much cross-culture between the
 structure from ancient Egypt that complies with Egyptian and Greek civilisations in the cities of
 constants like Pi or Phi; Schwaller De Lubicz, the north coast during the Hellenistic era,
 who studied the temples of Upper Egypt from particularly in Alexandria where Egyptian and
 1937 to 1952, collected massive amounts of Greek scientists studied together at the
 evidence to show that the Egyptians used the Mouseion. Among them, in the third century
 Golden Ratio in many ways both in the BC, was one who was considered, par
 architecture of their temples and in their excellence, the most reputed scientist in
 drawings. antiquity, the great Euclid. Historians call him
 Euclid of Alexandria without precluding the
 So whereas, prior to De Lubicz's research, the possibility he might have been Egyptian. It was
 discovery of the "golden rule" was generally in Alexandria that Euclid wrote his opus The
 credited to the Greeks (although some Elements, still the most famous mathematical
 historians denied this), the findings of such work ever written.
 Egyptologists as De Lubicz and Fliders Petrie
 produced irrefutable proof .

05

Introduction
What is Golden Ratio ?

180o Since the invention of the Gutenberg machine
 this work, compiled in 13 volumes, has been
 There are many applications for the Golden
 Ratio varying from mathematical optimisation
 printed more than any book apart from the to architecture. We find it in other sciences too.
Holly Triangle: Records tell us that the
 Bible. (Euclid introduced the ratio, obtained In biology it controls the distribution of the
Egyptians were aware of the triangle

3:4:5 which Pythagoras himself called
 from extreme and mean ratio of three collinear leaves around the stem of a plant such that they
the "Holy Triangle". Eight pyramids points, in volume VI). The Arab mathematician receive the maximum amount of light. It was
from the fourth and fifth dynasties Al-Haggag produced the first translation of The found that the distribution of their angle of
have their inner triangle conforming to
 Elements into Arabic, and as the original Greek rotation or distances from one another followed
these ratios.
 work was subsequently lost it was only through terms as in the sequence 1/2, 1/3, 2/5, 3/8, 5/13,
 the Arabic translation that the book became 8/21, 13/34, in which the numerator a(n) of the
 known to the rest of the World.Euclid was only quotients represents the number of turns we
 one of the scientists who performed research at climb, say, around the stem in such that any
 the Mouseion. The great intellects of the day one leaf returns to a position exactly above the
 flocked to Alexandria, and among them we point where it started, whereas the denominator
 encounter such names as Eratosthenes, b(n) represents the number of leaves in
 Archimedes, Apollonius, Menelaus, Heron, between. This ratio differs from plant to plant,
 Nicomachus, Ptolemy, Diophantus, Pappus, yet it varies on average between 1/2 and 1/3.
 Galen, Theon and his daughter Hypatia. Greek The reader will notice that in the sequence both
 scholars were visiting Egypt even before the the numerators and denominators are terms in
 Mouseion was founded, including Thales, the Fibonacci sequence, in which a(n) and b(n)
 Socrates, Plato and Aristotle, and above all satisfy b(n+1) = b(n) + a(n+1). Since both
 Pythagoras who spent 22 years in Egypt about a(n+1)/a(n) and b(n+1)/b(n) tend to Phi, it
 600 BC and announced his theory only after follows that a(n+1)/b(n) tends to Phi-1 = 1/Phi.
 leaving. Records tell us that the Egyptians were As to the distribution of the leaves around the
 aware of the triangle 3:4:5 which Pythagoras stem b (n)/a(n), it tends to Phi square for at
 himself called the "Holy Triangle". Eight each turn the leaves organise themselves on the
 pyramids from the fourth and fifth dynasties stem.
 have their inner triangle conforming to these
 ratios.

 06

Introduction
 What is Golden Ratio ?

Phi Square We can elaborate further to calculate the Likewise, in rabbit reproduction, say, starting
 divergence angle of any one leaf such that no with one pair of rabbits, at the end of the first
Fibonacci curves: A fascinating two leaves would be above one another; month they will have mated but there is still
phenomenon associated with the allowing for maximum light exposure. Having one only pair. At the end of the second month
Golden Ratio is its regular appearance
 that each complete turn of any one leaf in order they produce a new pair, so now there are two
in such objects of nature as nautili and
 to return to its original position travels an angle pairs of rabbits. At the end of the third month
other shells. Such shells imitate a curve

called in mathematics the "logarithmic of a(n)x360 while passing by b(n) leaves in the original pair produces a second pair,
spiral", first investigated by Jacob between, it follows that each leaf occupies an making three pairs. At the end of the fourth
Bernoulli in the 18th century.
 angle of a(n)x360/ b(n) = 360/(Phi square) month the original pair has produced yet
 almost 137 degrees. another new pair, while the pair born two
 months before produces their first pair, making
 Thus, if there are Phi square leaves per turn (or five pairs. And so on (assuming that each pair
 equivalently 1/Phi square turns per leaf), then born consists of a male and a female rabbit).
 each leaf gets the maximum exposure to light
 while casting the least shadow on the others. A fascinating phenomenon associated with the
 This also gives the best possible area exposed Golden Ratio is its regular appearance in such
 to falling rain, or, in the case of flowers, the objects of nature as nautili and other shells.
 best possible exposure to attract insects for Such shells imitate a curve called in
 pollination. Still in biology, the Fibonacci mathematics the "logarithmic spiral", first
 sequence, of which the ratio of two successive investigated by Jacob Bernoulli in the 18th
 terms tend in the limit to Phi, often appears in century. This was the second curve in history,
 the reproduction of animals and plants. The next to the circle, to have its length calculated,
 figure depicts the increase in the number of since although it has an infinite number of
 leaves in some plants from row to row. loops, its length approaches a finite value.

 07

Introduction
What is Golden Ratio ?

 One extremely amazing appearance As to the mathematical properties of Phi, we
 of the logarithmic spiral in nature is associated summarise them as follows: Phi = 1.618..., 1/
 with raptors. Since these predatory birds must Phi = 0.618, Phi square = 2.618... and so on.
 keep the prey in sight all the time, and since For instance, Phi raised to the 11th power
 their eyes are on the side of the head, a hawk or equals 199.00502... and 1/Phi raised also to the
 eagle swivels its head to one side at an angle of 11th power equals 0.00502. Also Phi has the
 about 40 degrees and fixes its prey in this eye. peculiar continued fraction representation
 Keeping its head fixed at that 40-degree angle,
 the bird then dives in a way which keeps the Again, in terms of nested radicals: Also, by
 prey in sight in that one eye. The fixed angle of writing Phi square = 1 + Phi, Phi cube = 1 +
 the head results in the bird's following a 2Phi, Phi power 4 = 2 + 3Phi, Phi power 5 = 3
 logarithmic spiral path that converges on its + 5Phi, Phi power 6 = 5 + 8Phi, etc... we notice
 prey. that the coefficients represent two Fibonacci
 sequences. The interesting properties of the
 A special case of the logarithmic golden ratio led the academic circle to issue a
 spiral is the "golden spiral". This is drawn regular periodical called the Fibonacci
 either from the outside or the inside. In the first Quarterly.
 option, we start from a square of unit side-
 length, then extend it into a "golden rectangle"
 of base unity and height 1.618. Then we draw a
 quarter of a circle inside the square, and
 another one in the extension of radius 0.618 =
 1/Phi, etc... making sure that each time we
 isolate a square from a golden rectangle. An
 interesting feature that I found with this spiral
 is that its inner area can almost be filled up (I
 said almost) by a set of adjacent Egyptian
 triangles.

 08

Introduction
 What is Nash Equilibrium ?

Game Theory In game theory, a Nash equilibrium is An equilibrium point of such a game is a
 an array of strategies, one for each player, such strategy profile ∗ ∈ with the property that,
Nash Equilibrium: a Nash equilibrium is an that no player can obtain a higher payoff by for each player and each strategy ∈ ,
array of strategies, one for each player, switching to a different strategy while the ( ∗ ) ≥ ( , ∗ − ).
such that no player can obtain a higher
 strategies of all other players are held fixed.
payoff by switching to a different strategy
 The concept is named after John Forbes Nash That is, a strategy profile is an equilibrium
while the strategies of all other players are

held fixed. Jr. For example, if Chrysler, Ford, and GM point if no player can gain from a unilateral
 choose production levels for pickup trucks, a deviation to a different strategy. The invention
 commodity whose market price depends on and succinct formulation of this concept, along
 aggregate production, an equilibrium is an with the establishment of its existence under
 array of production levels, one for each firm, very general conditions, reshaped the landscape
 such that none can raise its profits by making a of research in economics and other social and
 different choice. Formally, an -player game behavioral sciences. Nash’s existence theorem
 consists of a set = {1, … , } of players, a set pertains to games in which the strategies 
 of strategies for each player ∈ , and a set available to each player are probability
 of goal functions distributions over a finite set of alternatives.
 ∶ 1 × ⋯ × → ℝ Typically, each alternative specifies what
 that represent the preferences of each player action to take under each and every
 over the -tuples, or profiles, of strategies circumstance that the player may encounter
 chosen by all players. A strategy profile has a during the play of the game. The alternatives
 higher goal-function value, or payoff, than are referred to as pure strategies and the
 another if and only if the player prefers it to the probability distributions over these as mixed
 other. Let = 1 ×⋯× denote the set of all strategies. Players’ randomizations, according
 strategy profiles, with generic element , and to their chosen probability distributions over
 let ( , − ) denote the strategy profile ( 1,… , their own set of alternatives, are assumed to be
 −1, , +1, … , ) obtained from by statistically independent.
 switching player ’s strategy to ∈ while
 leaving all other strategies unchanged.

 09

Introduction
What is Nash Equilibrium ?

Gain=Lost Any -tuple of mixed strategies then and for games in which each player has three
 induces a probability distribution or lottery pure strategies, proved the existence of what
Nash Equilibrium: In such a game each over -tuples of pure strategies. Provided that a would later come to be called a maxmin pair of
player has the same number of pure player’s preferences over such lotteries satisfy mixed strategies. This is a pair of strategies
strategies, the gain for one player
 certain completeness and consistency such that one player’s strategy maximizes his
equals the loss to the other, and they
 conditions—previously identified by John von own gain while his opponent simultaneously
both have the same probability of

winning whenever they use the same Neumann and Oskar Morgenstern—there exists minimizes this gain. He subsequently extended
pure strategy. Borel also formalized a real-valued function with the -tuples of pure this result to the case of five strategies per
the concept of a mixed strategy,
 strategies as its domain such that the expected player, but seems to have doubted that general
 value of this function represents the player’s existence results could be achieved. A few
 preferences over -tuples of mixed strategies. years later, and apparently unaware of Borel’s
 Given only this restriction on preferences, Nash partial results, von Neumann formalized the
 was able to show that every game has at least notion of finite zero-sum games with an
 one equilibrium point in mixed strategies. arbitrary (finite) number of players, where each
 player has an arbitrary (finite) number of pure
 Emile Borel had a precursory idea, concerning strategies. For all such games involving two
 symmetric pure conflicts of interest between players he proved the existence of a maxmin
 two parties with very few alternatives at hand. strategy pair, presented the result in Göttingen
 In 1921 he defined the notion of a finite and in 1927, and published it in 1928.
 symmetric zero-sum two-player game. In such
 a game each player has the same number of
 pure strategies, the gain for one player equals
 the loss to the other, and they both have the
 same probability of winning whenever they use
 the same pure strategy. Borel also formalized
 the concept of a mixed strategy,

 10

Introduction
What is Nash Equilibrium ?

 In comes Nash, a young doctoral student in Γ( ). That is, there exists a fixed point of the
 mathematics at Princeton University. Nash correspondence. Nash’s existence proof relies
 defined a much more general class of games on the construction of what today is called the
 and a more general equilibrium concept. He best-reply correspondence, which can then be
 allowed for any (finite) number of players, each shown to satisfy the conditions of Kakutani’s
 having an arbitrary (finite) number of pure theorem. Given any -tuple of mixed strategies,
 strategies at his or her disposal and equipped Nash defined a countering -tuple as a mixed-
 with any goal function. In particular, players strategy profile that obtains for each player the
 may be selfish, altruistic, spiteful, moralistic, highest payoff given the strategies chosen by
 fair-minded, or have any goal function other players in the original, countered -tuple.
 whatsoever. His definitions and his existence By associating with each -tuple of mixed
 result contain those of Borel and von Neumann strategies the set of all countering -tuples, one
 as special cases. Previously restricted to pure obtains a selfcorrespondence on the set of all
 conflicts of interest, game theory could now be mixed-strategy profiles. Since any -tuple of
 addressed to any (finite) number of parties with mixed strategies is a point in the product space
 arbitrary goal functions in virtually any kind of obtained by taking the Cartesian product of
 strategic interaction. Nash published this in a the individual strategy spaces , the domain
 one-page article in the Proceedings of the of this correspondence is a nonempty, compact
 National Academy of Sciences in 1950. His and convex subset of ℝ for some . In fact,
 existence proof—merely sketched in this short it is a polyhedron, the Cartesian product of
 paper—is based upon Kakutani’s fixed-point finitely many unit simplexes. Furthermore, the
 theorem (established some years earlier). correspondence thus constructed is convex-
 Kakutani’s theorem states that if a subset of valued, since a convex combination of
 ℝ is nonempty, compact and convex, and a countering -tuples must itself be a countering
 (set-valued) correspondence Γ ∶ ⇉ is -tuple.
 nonemptyvalued, convex-valued and has a
 closed graph, then there exists ∈ such that
 ∈

 11

Introduction
What is Nash Equilibrium ?

 And since the payoff functions are all This is how he did it. Consider any -tuple of
 continuous (in fact, polynomial) functions with mixed strategies , and recall that the payoff to
 closed domain, the correspondence has a closed a player at this strategy profile is ( ). Let
 graph. The existence of a fixed point follows ℎ( ) denote the payoff that player would
 from Kakutani’s theorem, and any such fixed receive if he were to switch to the pure strategy
 point is a self-countering -tuple, or an ℎ while all other players continued to use the
 equilibrium point of the game. A year later strategies specified in . Define the continuous
 Nash published an alternative existence proof function ℎ( ) = max{0, ℎ( ) − ( )}.
 in the Annals of Mathematics that instead is
 based on Brouwer’s fixed-point theorem. Since Each function value ℎ( ) represents the
 Kakutani’s theorem is derived from Brouwer’s, “excess payoff” obtained by pure strategy ℎ ∈
 Nash was more satisfied with the latter. This , as compared with the payoff obtained
 second proof has a touch of genius. It is simple under strategy profile . Letting ℎ denote the
 and intuitive in retrospect but completely probability with which pure strategy ℎ is
 unexpected beforehand. In order to use played under , the function may be used to
 Brouwer’s theorem, Nash needed to construct a obtain a new -tuple of mixed strategies, ′ ,
 self-map on the space of mixed-strategy from by setting ′ ℎ = ℎ ( ) = ℎ + ℎ( )
 profiles with the property that a strategy profile 1 + ∑ℎ ℎ( ) . This defines a self-map on
 is an equilibrium point if and only if it is a
 the space of mixed strategy profiles. As long as
 fixed point of this map. But the best-reply
 there exists a pure strategy with positive excess
 correspondence could not be used for this
 payoff, lowers the probabilities with which
 purpose, since it need not be single-valued and
 pure strategies having zero excess payoff are
 does not permit a continuous selection in
 played. It is clear that if is an equilibrium
 general.
 point, it must be a fixed point of , since no
 pure strategy ℎ can yield player a higher
 payoff, forcing ℎ( ) = 0 for all and ℎ. It is
 easily verified that the

 12

Introduction
What is Nash Equilibrium ?

 the converse is also true: if is a fixed point of This is a symmetric -player game with
 , so that ℎ( ) = 0 for all and ℎ, then must countably infinite pure-strategy sets. However,
 be an equilibrium point of the game. Nash’s existence result still applies, since no
 bid above is ever a best reply to the bids of
 To complete the proof, one need only use the others, and hence the game has the same set of
 fact that is a continuous self-map on the Nash equilibria as the finite game in which bids
 compact and convex set of mixed-strategy - are bounded from above by . Nash’s result
 tuples. This is sufficient, from Brouwer’s tells us that there must be an equilibrium in
 theorem, for the existence of a fixed point. pure or mixed strategies in this game. For
 Nash’s equilibrium concept lies at the heart of instance, if = 2 and = 5/2, then it can be
 contemporary theoretical research on strategic shown that no pure strategy equilibrium exists,
 interactions in economics and other fields. One but if each player chooses the distribution (1/5,
 especially fruitful area of application has been 3/5, 1/5) over the bids {0, 1, 2}, then neither
 to auction theory, as the following example can obtain a higher payoff by deviating
 illustrates. Many strategic interactions— unilaterally to any other strategy. Furthermore,
 including lobbying, arms races, contests, and each player’s expected payoff in equilibrium is
 wars of attrition— can be modeled as all-pay 1/4, which is lower than the 5/4 that each could
 auctions in which the highest bidder obtains an secure if they colluded to bid zero. This
 object of value but all players must pay their example illustrates that equilibrium behavior,
 bids. (If there are multiple highest bidders they while individually optimal, can cause players to
 each get the object with the same probability.) impose costs on each other that are wasteful in
 Consider an object with value > 0 and ≥ 2 the aggregate. The 1994 Sveriges Riksbank
 bidders, each of whom is constrained to bid Prize in Economic Sciences in Memory of
 from the nonnegative integers. Players submit Alfred Nobel was awarded to Nash, along with
 their bids simultaneously, without knowledge Reinhard Selten and John C. Harsanyi, for their
 of any opponent’s bid “pioneering analysis of equilibria in the theory
 of noncooperative games.”

 13

Introduction
Equilibrium of Fibo & Nash

 For the duration of its existence, economics has Write a number on the sheet of paper between 0
 been based on a theory of ‘rationality’, stating and 100. Whomever gets closest to 2/3 of the
 that people will act the ways which maximize average of the room wins $20. You can see the
 their monetary take home, regardless of all else. steps of the problem here: Even assuming that
 (Robinson Crusoe gets mentioned frequently). everyone else puts the highest possible number
 This has always been a rough approximation, (100), then 2/3 of that is 66. So, the highest you
 but for the past few decades alternatives have should ever put is 66. If you assume everyone is
 been explored via research, to see if some other that smart, you say that the highest average
 theory fits humans better than perfect could be 66, so you’d put the two thirds of 66,
 rationality. In particular, today I studied why we which is 45. If you think everyone is at least that
 act differently in ‘game’ situations than smart, you put… etc. Essentially, you go down
 rationality would predict. A game is anything steps to solve the problem. An economically
 with players and a defined strategy set. An rational person, for example, would iterate
 example would be a prisoner’s dilemma, or infinitely, and realize that the only rational
 even chess (which would be immensely hard to choice is to play 0, and everyone else would
 model). One group of economists has studied play zero, and you’d split the money. Of course,
 this via a theory of ‘cognitive hierarchy.’ They we are not rational, and the general result of the
 are analyzing games under the assumption that game ends with an average around twenty or
 we solve problems in steps. A popular game, thirty. The researchers I mentioned above have
 often used in the first meeting of game theory proposed the cognitive hierarchy theory for why
 101, is a good example of this. (This game is something like this happens, and here is how it
 called a ‘Keynesian Beauty Contest’, after an works.
 analogy that appears in Keynes’ famous book A
 General Theory of Employment, Interest, and
 Money) The game is this: everyone in a room is
 given a slip of paper, and the following
 instruction: .

 14

Introduction
Equilibrium of Fibo & Nash

 People are made such that they will be smart This is enough information to find z in our
 enough or hard working enough to go Poisson distribution. Poisson is quite cool
 through k steps of a process. Being human, because the variable z not only defines the
 however, we are obnoxiously self confident and whole process, but it is also the mean. In our
 assume that no one we are playing with is as particular manipulation, the mean is the average
 smart or smarter than us. (We think no one else number of steps a person takes when solving a
 will play k steps or k+1 steps.) It is pretty clear game. And when we have (# 0 steps) + (# 1
 from both evidence and intuition (thinking more steps) = 2 * (# 2 steps), that value
 takes energy, and we’re lazy) that the more of z becomes…. 1.618, aka the golden ratio.
 steps you go, the fewer people go that far, so the
 number of people making each number of steps
 So not only does the golden ratio give us
 declines as you go to higher and higher k. As it
 pleasant rectangles and cool looking shells, it is
 turns out, this is approximated quite well with a
 perhaps also the average number of steps a
 Poisson distribution, which is quite nice because
 human is willing to think in a given moment.
 a Poisson distribution is defined by one
 parameter, let’s call it z. Now, with our Poisson
 distribution, we want to figure out what that
 value of z might be. From a look at a decade’s
 worth of multi-step games, it would appear that
 the number of people who don’t take the time to
 think at all (and take k = 0 steps, and don’t go
 through the ‘two thirds of 100 is 66' step above),
 and add them to the number of people who only
 take k = 1 steps (who see that you can’t play
 above 66, but don’t think any further), you see
 that this sum is almost exactly double the
 number who take two steps (and play 45).

 15

Introduction
Beyond The Nash Equilibrium

 John Nash proposed that it is only if everyone This mathematical structure comes from the
Fair Income in the group follows the best path for the divine ratio, the Golden ratio. .” The golden

Distribution society, not only for themselves that it is ratio is the name given to the constant number
 possible for everyone to win (take the blonde). equal to 1.618 in mathematics, and the symbol
Mass will win: John Nash proposed that it is John Nash proposed that success of the used for its expression is "Phi".
only if everyone in the group follows the best individual is utterly intertwined with that of the
path for the society, not only for themselves
that it is possible for everyone to win (take
 community. He rejected the ‘every man for Every being in the universe is in perfect
the blonde). John Nash proposed that success himself’ paradigm and encouraged the adoption harmony. The mathematical meaning of this
of the individual is utterly intertwined with
that of the community. He rejected the ‘every
 of one involving teamwork and joint strategy. harmony is the Golden ratio. The human brain
man for himself’ paradigm and encouraged People provide optimum benefit if they Works mathematically. It is a part of this
the adoption of one involving teamwork and
joint strategy.5, 8, 13, 21, 34, 55, 89, 144,
 consider the benefit of society while providing enormous harmony in mathematical movement.
.... benefit for themselves. Yes but how ? Is it The most obvious reflection of this harmony is
 possible to apply this theory in economics? the Fibonacci formulas. In our rebase system,
 Human urges always want to get the maximum the main purpose is to reflect this perfect
 benefit. Unfair distribution of income always harmony to daily price movements and to
 occurs, as individual interests override ensure that the masses make the right decision.
 Community interests. What if an outside You will see Fibonacci's footprints in every
 intervention ensures that individuals' priority is part of our system. While you cannot make
 social benefit. And if this intervention is perfect choices in investments, we help you to
 supported by a divine mathematical formula? choose perfection.

 Lets taking the Nash equilibrium one step
 further, it is necessary to introduce a natural
 mathematical structure that will control these
 impulses that people have, and a mathematical
 method that can be beneficial for both the
 society and themselves and willingly share this
 benefit.

 16

GRPL
Grpl, uses Fibonacci correction levels as a formula in the Rebase
system. It pushes the natural mathematical will of the people to do
the most beneficial thing for the group, based on the fibo correction
levels at the target price. It provides Nash Equilibrium, or fair
distribution of income, this way. Our target price is 1 gram of gold.

FIBONACCI
All adjustments in procurement (REBASE) are made to achieve this
goal. Oracle price; is daily moving averages price of exchanges.
Fibo price is the interm target price that calculate the ratio between
the last two days oracle price. The harmony between oracle prices
continue rebases until it reaches 0.61834 everyday

NASH EQUILIBRIUM
The golden ratio is the perfect harmony where everyone benefits
well and achieves a fair distribution of income. When this rebase
occurs, Supply is increased or decreased for equal the oracle price
to the price of one gram of gold. Our real goal is to make the mass
win, not individuals. Natural mind choice will pave the way for us
to win for society.

 17

GRPL PROTOCOL
 Grpl uses two different rebase systems. First, calculations are made by basing fibo prices as
 the target price. In this way, target prices which repeat each other can be created. Actually,
 all target prices in the first level rebase protocol are made to get closer to 0,618 golden
 ratio. Increase or decrease at the circulating supply are determined by this rate. When the
 rates between target prices match the 0,618 golden ratio, second rebase formula is applied.
 The price is equalized to 1 gram of gold by increasing or decreasing the circulation supply
 in the second rebase protocol. The reason of all these rebases to reach the 1 gram of gold.
 We prefer to use gold which has been used throughout history and has a securitized value
 on its own instead of US Dollar which has unlimited supply.

 FIBONACCI FORMULA
 at all, its only drawback is that, if we want to
 know a particular number, Fₙ in the sequence,
 Fibonacci numbers are one of the most
 we need two numbers Fₙ₋₁ and Fₙ₋₂ that came
 captivating things in mathematics. They hold a
 before it; that’s just how this formula works. It
 special place in almost every mathematician’s
 is not hard to imagine that if we need a number
 heart. Throughout history, people have done a
 that is far ahead into the sequence, we will have
 lot of research around these numbers, and as a
 to do a lot of “back” calculations, which might
 result, quite a lot of interesting facts have been
 be tedious.
 discovered.Let us see how they look like
 0,1,1,2,3,5,8,13,21,34,55 ………
 Let us define a function F(x), such that it can be
 expanded in a power series like this
 Any number in this sequence is the sum of the
 previous two numbers, and this pattern is
 mathematically written as;

 where n is a positive integer greater than 1, Fₙ is
 the n−th Fibonacci number
 with F₀=0 and F₁=1.

 Now, this expression is fairly easy to understand
 and quite sufficient to produce any Fibonacci
18
 number by plugging the required value of n. If

02. Ecosystem
Rebase Rules

 Oracle Price Interm Rebase Formula
Art of Math To absorb price information from the
 If Fibo rate does not equal to 0,618, we
 outside world, the protocol utilizes a
Perfect Harmony: The harmony between apply the formula below;
 market oracle system made up of
oracle prices continue rebases until it
 whitelisted independent data providers
reaches 0.61834 everyday.
 Supply Delta = Total Supply * (Oracle
 who broadcast 24 hour volume weighted
 price – Fibo price) / Fiboprice
 average price to a single on-chain
 Aggregator.
 Lag = 10

 Fibo Price
 Supply Alpha = Supply Delta /10

 is the interm target price that calculate the New supply= Total Supply + Supply
 ratio between the last two days oracle Alpha
 price. The harmony between oracle prices
 continue rebases until it reaches 0.61834
 everyday.

 Fibo Price formula as follows,

 Fn=Fn-1 + Fn-2,

 Fn = Oracle Price (Current Day)
 Fn-1= Oracle Price (Yesterday)
 Fn-2 = Oracle Price (Two days before)

 Fibo Rate formula as follows,

 Fn-1 / Fn

 19

0.2 Ecosystem
 Rebase Rules

 Main Rebase Formula
 If Fibo rate equals to 0,618 and Oracle price <
 One Gram Main Target Price (One gram of gold in Usd)
 we apply the formula below;
 Gold
 Supply Alpha = Total Supply – ( (Oracle price
 Natural Selection of Money : We
 * Total Supply ) / Main Target Price (One
 preferred gold because; It is used both
 gram of gold in usd) )
 as a barter and a value storage tool. It
 is a traditional value that has been Lag =0
 considered valuable in every period of
 history. It is the safe haven in the
 global economy as well as the most
 If Fibo rate equals to 0,618 and Oracle price >
 durable money against economic
 shocks. The source of price Main Target Price (One gram of gold in Usd)
 movements in gold-based. systems is, we apply the formula below;
 as a rule, not monetary. It varies
 according to the demand and supply.
 Supply Alpha = Total Supply + ( (Oracle price
 * Total Supply ) / Main Target Price(One
 gram of gold in usd) )
 Lag =0

20

Golden ratio brings you to perfection!
 Business Plan 2019

“If by chance I have omitted anything more or
less proper or necessary, I beg forgiveness,
since there is no one who is without fault and
circumspect in all matters”
 Leonardo Fibonacci