Cryptoeconomics of the Loki network

Transcription

1 The problem of ncentvsng Servce Nodes n the Lok Blockchan network 1 Brendan Markey-Towler 11 July 2018 Abstract Lok s a Blockchan network orented toward the provson of prvacy-preservng servces over a network of Servce Nodes. The salent cryptoeconomc problem s how to ncentvse actors n the Lok network to operate Servce Nodes n a manner compatble wth the objectves of the Lok network, n partcular decentralsaton and prvacy. We use cryptoeconomc game theory to characterse ths problem. We derve an equaton for the ncentves to create Servce Nodes faced by actors n the Lok network, and use ths equaton to characterse a pure strategy Nash equlbrum n the provson of Servce Nodes. Wth reasonable assumptons we can dscover a condton for the desgn of the Lok network whch supports ths Nash equlbrum consstent wth the objectves of the Lok network, whch s useful n partcular for decdng the stakng requrement for actors who wsh to operate Servce Nodes. We also derve an equaton for the value a rogue actor would have to place on undermnng ths equlbrum and Lok s prvacy. We analyse ths soluton to study how t responds to changes n the parameters of the Lok network. Dsclamer Ths paper contans tables and charts whch nclude examples of a prce for the Lok cryptographc con. Those prces are examples only and are not a predcton, forecast, or representaton as to any actual lkelhood of prce movement of the Lok cryptographc con. The payments shown n the examples below are general n nature and wll only take effect f the planned hard fork occurs. Factors outsde the control of Lok could mpact what actual payments are made to Servce Nodes. Any references to the prce of Lok n ths paper are to the market prce of the Lok cryptographc con avalable on publc cryptocurrency exchanges whch choose to lst the Lok token, notng that the Lok Foundaton does not operate a cryptocurrency exchange. Those partes not operatng a Servce Node should not rely on the examples when decdng whether or not to partcpate n the Lok project. Ths document should be read together wth the Lok whtepaper and other publcatons by Lok. 1 RMIT Blockchan Innovaton Hub and School of Economcs, Unversty of Queensland 1

2 1 The cryptoeconomc problem n the Lok network Lok s a Blockchan network whch uses a hybrd proof-of-work/proof-of-servce system n whch cryptoeconomcs s employed to ncentvse the provson of prvacy-preservng servces over a network of Servce Nodes. Servce Nodes wll offer routng and data transfer servces over a network n a manner not dssmlar to nodes n TOR. Servce Node also offer Lok Servces, the frst of whch wll be a prvacy-mantanng messenger system (Lok Messenger). In order for actors n the Lok network to operate Servce Nodes they must tme lock a certan amount of Lok. Tme lockng allows for Lok cons to be locked untl the blockchan reaches a defned block-heght. Functonally tmelockng s smlar to stakng and we use the two terms nterchangeably, however funds can never be lost when tmelockng cons. The salent cryptoeconomc problem n the Lok network s how to ncentvse actors to stake suffcent Lok to operate Servce Nodes and provde servces n a manner whch s compatble wth the objectves of the Lok network. Because Lok Servce Nodes operate as routers n a novel mxnet there must be suffcently many dstnct Servce Nodes to mantan decentralsaton and thus maxmse prvacy. To analyse ths problem we frst must derve an equaton whch descrbes the ncentves faced by actors n the Lok network and use t to determne the behavour of a ratonal actor n the Lok network. Specfcally, we derve a condton whch defnes the number of Servce Nodes a ratonal actor n the Lok network wll provde. We then use ths condton to characterse equlbrum n the provson of Servce Nodes over the Lok network, whch s a pure strategy Nash equlbrum. We then analyse ths condton and equlbrum by ntroducng reasonable assumptons whch reduce the dmensonalty of the problem, and dscover a soluton to the desgn problem faced by Lok n the form of a condton for the desgn of rewards and stakng requrements for operatng Servce Nodes. We use ths soluton to defne the stakng requrement whch supports equlbrum n the provson of Servce Nodes on the Lok network, and a defnton of those stakng requrements whch are consstent wth the objectves of the Lok network. We then analyse the response of the stakng requrement whch supports equlbrum to varous parameters n the Lok network. After takng a farly hgh-level analyss of the problem, we ntroduce some specfc forms for varous parameters of the Lok network and calculate fgures for the stakng requrement whch supports varous equlbra under varous parametersatons. We then, before concludng, establsh the response of the stakng requrement to varous parameters n ths expanded numercal analyss. 2 Charactersng ncentves and equlbrum strateges The ncentve for any actor n the Lok network to become a Servce Node and provded Lok Servces s gven by the allocaton of 50% of the Lok cryptocurrency block reward to be dstrbuted among these nodes. Let us assume, as s reasonable n a game theoretc settng, that the ncentve for an actor to run n Servce Nodes s gven by profts, π. Profts for actors operatng Servce Nodes n the Lok network depend, n partcular, on the number of nodes they operate, n, and the total number of nodes, N, n the network. The revenues, r (n N), whch accrue to Servce Nodes depends on both varables, 2

3 whle the costs, nodes run thereon, so we have c (n), of runnng servers whch host Servce Nodes depend only on the number of We wll suppose that, snce the costs, natonal currences, that profts, π (n N) = r (n N) c (n) c (n), of runnng a Servce Node at present are denomnated n π (n N), are also denomnated n natonal currences. Let us therefore say that the exchange rate of natonal currences for Lok (for nstance, USD/Lok) s ε, and that the rewards accrung to an actor operatng n Servce Nodes out of a total N s, n Lok, λ (n N). Therefore the profts accrung to a Servce Node become π (n N) = ϵ λ (n N) c (n) Now n order to operate a Servce Node, an actor n the Lok network must stake the opportunty cost of stakng ths Lok s foregong a rate of return of r s Lok. Suppose that (for nstance, government bond yelds or stock market returns), such that the opportunty cost of stakng Lok to operate a Servce Nodes s r (ϵs). In ths case, an actor n the Lok network wll stake suffcent Lok to operate n Servce Nodes f π (n N) r (ϵns) and only f ths s the case and they have ns Lok ( n ϵs n natonal currences) avalable to stake. We can safely suppose ths suffcently characterses the problem of whether an actor s ncentvsed to operate Servce Nodes wthn the Lok network. The Swarm flaggng algorthm embedded n the Lok network software can reasonably be sad to ensure that the expected proft to be ganed by operatng dshonestly n the network (.e. stakng but not offerng servces) s zero, for any such actor wll quckly be flagged and removed from the network. So an actor wll only ever be ncentvsed to operate honest Servce Nodes. The economc proft obtaned from stakng suffcent Lok to operate network of Servce Nodes s N s therefore gven by n Servce Nodes when the total π (n N) r (ϵns) = ϵ λ (n N) [ c (n) + r (ϵns)] A ratonal agent who maxmses economc profts n respondng to the ncentves emboded n ths equaton wll stake Lok to operate n Servce Nodes up to the pont at whch margnal revenues are equal to margnal costs. Snce π (n N) s concave by constructon, so too wll π (n N) r (ϵns) be. Thus a ratonal agent n the Lok network wll operate n * = n : n n * nodes defned as follows [ π (n N) r (ϵns)] = 0 That s to say, f we solve the problem embedded n the equaton, a ratonal agent n the Lok network wll operate n * Servce Nodes defned as follows n * = n : ϵ n λ (n N) = r (ϵs) + n c (n) 3

4 or, equvalently, a ratonal agent n the Lok network wll operate follows n * Servce Nodes defned as n * 1 = n : s n λ (n N) = r + ϵs 1 n c (n) Equlbrum wll be reached on the Lok network when all actors n the Lok network are operatng n * Servce Nodes so defned. Such equlbra are strategc (game theoretc equlbra), because the decson of each actor n the Lok network about how many Servce Nodes to operate depends on the decson of every other actor n the Lok network. The exact form of such equlbra can be derved f we make explct the fact that the total number of nodes, N, n the network on whch the rewards, λ (n N), for operatng a Servce Node are defned, though t does create some notatonal complexty. If the set of actors n the Lok network who mght operate Servce Nodes s of nodes n the network of N Servce Nodes s I, then the total number N = n I where n s the number of Servce Nodes provded by actor n the Lok network. In equlbrum therefore, each actor,, n the Lok network operates n* Servce Nodes defned as follows ( [ *]) n* 1 = n : s n λ n c n + n = r + 1 j ϵs n (n ) j I Such equlbra as defned by ths condton are pure strategy Nash equlbra. In such pure strategy Nash equlbra, actors n the Lok network wll obtan profts, n natonal currency, of = ϵ λ c n π ( n *) * n j j I and thus rates of return on ther Servce Node stakes ( * π n 1 sn *) * n = 1 j sn j I ( n * ϵλ s *) * n j j I of [ ( n *) * n j j I ( * ) c ( n * ) ] where n * s defned as above. 3 Analyss We can make use of the ncentve structure thus establshed n the Lok network, and the behavour of ratonal agents n response to t n equlbrum to characterse a desgn problem for Lok, whch s the settng of the stakng requrement for operatng a Servce Node n the Lok network. Obvously the equlbrum establshed above has a hgh degree of dmensonalty as a mathematcal problem, so ths wll have to be reduced by exogensng certan varables. After we do ths, we can recover a soluton 4

5 to the desgn problem whch specfes a stakng requrement whch wll support a gven desred equlbrum dstrbuton of Servce Nodes operated by actors n the Lok network. 3.1 Smplfyng assumptons and exogensatons If we consder agan the problem of actors respondng to ncentves, we can ntroduce further reasonable smplfcatons whch allow for better analyss. For convenence we restate that, a ratonal agent n the Lok network wll operate n * Servce Nodes defned as follows n * 1 = n : s n λ (n N) = r + ϵs 1 n c (n) A reasonable assumpton we may ntroduce s that the margnal cost of operatng a Servce Node s constant. Ths s reasonable because of the state of server technology at the present tme means that the cost of provdng greater routng and data transfer servces on that server wll ncrease at a farly constant rate. Snce the margnal cost of operatng a Servce Node s constant, we mght say that t can be defned as a constant, c. That s, 2 c (n) = 0 c (n) 2 n (n) = c And so we have that a ratonal agent n the Lok network wll operate follows n * 1 = n : s n λ (n N) = r + ϵs c n * Servce Nodes defned as Stll, we can see that apart from the fact that the form of λ (n N) has not yet been explctly defned, that ths s an equaton n more than one varable, so for analytcal purposes we wsh to further fx certan varables and defne them as exogenous for analytcal purposes. Dong ths means that we may smply base ther values on such data as exst rather than endogenous modellng. We wll denote these exogenous varables n the standard manner by barrng ther varables. Obvously, frst, we mght allow the cost of operatng a Servce Node to be smply defned as the cost of operatng a server over the relevant tme perod of analyss,.e. c = c. For the purposes of creatng a useful model we mght also set the exchange rate to be exogenously determned rather than endogenously by a market, we can set t based on projectons from data on the exchange rate of natonal currency for Lok pror to the relevant tme perod of analyss,.e. ϵ = ε. Further, we may also set the rate of return, r, foregone by stakng Lok to operate a Servce Node by reference to government bond yelds or stock market returns pror to the relevant tme perod of analyss,.e. r = r. Introducng these exogensatons, we see that a ratonal agent n the Lok network wll operate n * Servce Nodes defned as follows n * 1 = n : s n λ (n N) = r + εs c If we collect all remanng endogenous varables to the left hand sde of ths condton, we fnd that a ratonal agent n the Lok network wll operate n * Servce Nodes defned as follows 5

6 n * = n : s 1 [ n λ (n N) εc ] = r We stll have more varables than equatons, but we have reduced the dmensonalty of the problem sgnfcantly by exogensng the varables we have and allowng them to be set relatve to relevant data for the tme perod of analyss. Specfcally, outsde of the specfc functonal form of the reward λ ( ) for operatng a Servce Node, we have two varables yet to be determned before we can determne how many Servce Nodes n * a ratonal agent wll operate n the Lok network. Before we can determne n *, we need to defne the stakng requrement, s, and the total number of Servce Nodes on the network, N. 3.2 The desgn problem For desgn purposes we may of course recast ths problem now a lttle dfferently. In equlbrum, each actor,, n the Lok network operates n* Servce Nodes defned now as follows ( [ *]) ] n* 1 = n : [ s n λ n n + n c j ε = r j I where we have selected c, ε, and r exogenously based on such relevant data as exst. If we wsh to mplement a mechansm, λ (n N), whch ncentvses ratonal agents, I, n the Lok network to operate n* nodes on the network such that there are N = n* nodes overall, we need to pck a form for the reward, λ (n N), for operatng a Servce Node and a stakng requrement, s. The specfc form of λ (n N) I s not especally mportant however and may be desgned relatvely arbtrarly. Alternatvely, of course, t mght be desgned relatve to the monetary polcy objectves of the network. What s far more mportant for the cryptoeconomc problem n the Lok network s the value of the stakng requrement, s. To formulate a soluton to ths problem of dscoverng the stakng requrements that would support an equlbrum of our choce, let us ntroduce some further reasonable assumptons. Frstly, f we suppose that and are homogenous across the set of actors n the Lok network (as s not unreasonable), c r we have that each actor n the Lok network faces a homogenous ncentve structure. In ths case, the equlbrum we would wsh to support could be a symmetrc one n whch all actors n the Lok network operate a homogenous number of Servce Nodes, n* = n * I. In ths case, therefore, we can characterse equlbrum between homogenous agents, and focus on that stakng requrement whch would support the desred operaton of Servce Nodes by the representatve agent n the Lok network. The stakng requrement, whch ratonal agents n the Lok network provde reward, ŝ, whch would support a symmetrc equlbrum under these condtons n λ (n N), for operatng a Servce Node, s gven by ŝ = r 1 n * [ λ n I n ( * c ) ] ε Servce Nodes, gven a partcular form for the 6

7 where we can alter the functon λ (n N) snce N = n * I s endogensed by the number, I, of actors I n the Lok network (whch s tself an exogenous, gven number) and the desred provson n * of Servce Nodes by the representatve agent wthn them. That the soluton to the stakng requrement problem takes ths form s vtal gven the objectves of the Lok network. Servce Nodes offer packet based routng servces on an dstrbuted overlay network, so t s vtal that no one actor have an ncentve n equlbrum to come to provde Servce Nodes beyond a certan proporton of Servce Nodes n the Lok network. Wth the soluton to the stakng requrement problem we have dscovered, we can formally characterse the stakng requrements whch are compatble wth ths objectve of the network. If ϕ s the proporton of Servce Nodes any gven actor must operate n order to undermne the prvacy n SNApps provded on the Lok network, then t s vtal that n equlbrum, a ratonal actor n the Lok network has no ncentve to provde Servce Nodes n excess of ths proporton. That s, for equlbrum compatble wth the prvacy of SNApps provded by N = n * I Servce Nodes n the Lok network (where I for any gven actor, s the number of actors n the Lok network) we must have that I n* ϕn, n the Lok network. Ths allows us to mmedately elmnate a range of stakng requrements from consderaton. If I s the number of actors n the Lok network, and n * s the desred provson of Servce Nodes by the representatve agent wthn them, then the set of acceptable stakng requrements (those whch wll support an equlbrum compatble wth Lok s objectves) s Ω = { s = ŝ n * ( I ) : n* ϕ( n * I )} for an arbtrary actor, where ŝ n * 1 ( I ) = r [ nλ n * c ( I ) ] ε. We could, alternatvely and more ntutvely, approxmate ths by pckng an arbtrary number of Servce Nodes, N, to be provded homogenously across the network and so defne follows where ŝ (n* 1 N) = r [ λ. n (n* N) εc ] Ω, the set of acceptable stakng requrements, as Ω = {s = ŝ (n* N) : n * ϕn} 3.3 Throwng down the gauntlet: the approxmate monetary cost of undermnng prvacy n Lok Wth ths analyss n hand we can actually determne how much an actor ntent on undermnng the prvacy of the Lok network must value that undermnng n monetary terms. Suppose we have selected the stakng requrement, ŝ, to support a symmetrc equlbrum n whch the I actors n the Lok network provde n * ϕn Servce Nodes. Suppose we now have an actor ρ (for rogue ) n the 7

8 Lok network. If ths actor seeks to provde approxmately ϕn n ρ Servce Nodes, then ther proft wll be, π( n ρ (n ) I [ ρ n * + n * ]) r (εn s) ρˆ = ελ ( n ρ (n ) I [ ρ n * + n * ]) c [ + r (εnρˆ s) ] Now notce that π (n N) r (ϵns) s concave by constructon, ŝ was selected to support a symmetrc equlbrum n whch the I actors n the Lok network provde n * ϕn Servce Nodes, and equlbrum s charactersed by ratonal agents maxmsng ther economc profts. At ths pont we would therefore have the followng expresson, where n ρ = n * and ρ are actng ratonally, λ ε n ( n ρ (n ) I [ ρ n * + n * ]) = c + r (εs) ˆ Now as ncreases to some n ρ ϕn, we see that, because λ ( ) s concave by constructon, that the n ρ left hand sde of ths equaton decreases whle the left hand sde remans constant, and thus does economc proft at the margn. The value of the proft foregone relatve the equlbrum ŝ was selected to support therefore wll be approxmately π ( * n n * I ) r (εn*ˆ s) π( n ρ (n ) I [ ρ n * + n * ]) + r (εnρˆ s) ( = ελ n * n * I ) [ cn * + r (εn*ˆ s) ] ελ ( n ρ (n ) I [ ρ n * + n * ]) + cn [ ρ + r (εnρˆ s) ] Gatherng lke terms, the value of the proft foregone by the rogue actor, ρ (let us call t v ρ ), relatve the equlbrum ŝ was selected to support therefore wll be approxmately gven by v ρ = λ n n I ε ( * * ) λ n In order for t to be ratonal for a rogue actor, ( ρ (n ) I [ ρ n * + n * ]) + (rs ) ˆ + c (n ρ n * ) ρ, to seek to undermne the prvacy of the Lok network, they must value that undermnng by at least v ρ n terms of natonal currency, and at least v ρ /ϵ n terms of Lok. Now what we can see here, s that the value of the proft foregone relatve to that the equlbrum, ŝ, was selected to support by the rogue actor, ρ, ncreasng the Servce Nodes they are operatng, to some, n ρ ϕn wll grow qute rapdly. Notce that the stakng requrement, ŝ, and margnal cost, c, of operatng a server s a lnear expresson (whle the concavty of λ ( ) means t wll offset ths only n decreasng ncrements) and so a rogue actor must value the undermnng of the Lok network by (lkely) substantal multples of the opportunty cost of stakng Lok to operate Servce Nodes as well as the server costs. It s therefore ncreasngly expensve n terms of profts forgone, and lkely becomes prohbtvely costly for even the most well funded malcous actors, to operate suffcent nodes n the Lok network as to undermne ts prvacy. 4 Smple analytcs of solutons to the stakng requrement desgn problem Even wthout ntroducng a specfc functonal form for effects on the stakng requrement whch wll support an equlbrum n whch λ( n * I ), we can establsh a varety of I symmetrc ratonal 8

9 agents n the Lok network provde such equlbrum s, as we have above n * ŝ = r 1 Servce Nodes. The stakng requrement whch would support [ λ n I n ( * c ) ] ε Now f we take partal dervatves of ths functon, we can establsh varous effects on the stakng requrement whch wll support an equlbrum n whch network provde n * Servce Nodes. I symmetrc ratonal agents n the Lok 4.1 The effect of a decrease n desred nodes provded by each actor n the Lok network Frstly, and most mportantly, let us examne the effect on the stakng requrement whch solves the desgn problem of a change n the number of Servce Nodes we wsh for each actor n the Lok network to operate. We have that and so, snce λ( n * I ) 1 2 (n ) ( * ) s ˆ = λ n I n* r * 2 s concave by constructon n* ( * ) r 0 & λ n I 0 s 0 Therefore, gven I symmetrc ratonal agents n the Lok network and a desred provson of n * Servce Nodes by those agents, the fewer Servce Nodes we wsh for a ratonal agent n the Lok network to have an ncentve to operate n equlbrum, the hgher we must set the stakng requrement. Ths effect s partcularly salent gven the objectves of the Lok network. As we have dscussed above, Servce Nodes act as nodes n a mxnet, and so t s vtal that no one actor have an ncentve n equlbrum to come to provde more than a certan proporton of nodes n the network. The fewer relatve to the equlbrum number of Servce Nodes any one actor provdes the more secure the prvacy of SNApps on the Lok network. The more secure we wsh the prvacy of the Lok network to be gven a partcular number of actors n the network, the hgher we must set the stakng requrement for actors n the Lok network to provde Servce Nodes. n* ˆ 4.2 The effect of changes n operatng costs, the exchange rate and opportunty costs Now let us examne the effect on the stakng requrement whch solves the desgn problem of a change n operatng costs c, exchange rate ε, and opportunty costs r. Frstly, we have for a change n operatng costs that s c ˆ = 1 rε 9

10 Opportunty costs and the exchange rate are both postve, so we have that r 0 & ε 0 c s 0 ˆ Therefore, gven I symmetrc ratonal agents n the Lok network and a desred provson of n * Servce Nodes by those agents, the greater are the costs of operatng a Servce Node, the lower the stakng requrement whch supports equlbrum among ratonal agents provdng Servce Nodes n the Lok network. Ths effect, however, s most salent because the cost of operatng a Servce Node may decreases as server costs are subject to Nelsen's Law of Internet Bandwdth and to a lesser degree 2 3 Moores law. Therefore, gven I symmetrc ratonal agents n the Lok network and a desred provson of n * Servce Nodes by those agents, a decrease n the cost of operatng Servce Nodes causes the stakng requrement whch supports equlbrum among ratonal agents provdng Servce Nodes n the Lok network to ncrease. Now take the exchange rate, rate that ε, of natonal currency for Lok. We have for a change n ths exchange s ε ˆ = c r(ε) 2 Opportunty and operatng costs are both postve, so we have that r 0 & c 0 εˆ s = c 0 r(ε) 2 Therefore, gven I symmetrc ratonal agents n the Lok network and a desred provson of n * Servce Nodes by those agents, the greater the exchange rate (the more valuable Lok s), the greater the stakng requrement ought to be. That s to say, f Lok apprecates, the greater the stakng requrement ought to be to mplement a gven equlbrum. The reason for ths s farly straghtforward f an apprecaton n Lok occurs t ncreases the value of the reward for operatng Servce Nodes and thus the ncentve to operate them, and ths must be countered to support equlbrum. Over tme, f we make the assumpton that Lok apprecates n value wth potental adopton the stakng requrement whch would support a gven equlbrum would ncrease wth t. Now, fnally, take the rate, the Lok network. We have for a change n ths rate that r, at whch opportunty costs are accrued for operatng Servce Nodes n rˆ s = 1 r 2 Now ths s an ambguous effect, for we have that [ λ n I n ( * c ) ] ε 2 "Nelsen's Law of Internet Bandwdth - Nelsen Norman Group." 5 Apr. 1998, 3 "Moores paper." 10

11 So, for a suffcently low exchange rate of natonal currency for Lok (a suffcently deprecated exchange rate) relatve to the margnal cost-beneft rato of operatng a Servce Node, an ncrease n the opportunty cost of stakng Lok and operatng a Servce Node causes the stakng requrement that would support equlbrum to decrease. On the other hand, for a suffcently hgh exchange rate of natonal currency for Lok (a suffcently apprecated exchange rate) relatve to the margnal cost-beneft rato of operatng a Servce Node, an ncrease n the opportunty cost of stakng Lok and operatng a Servce Node causes the stakng requrement that would support equlbrum to ncrease. A more ntutve way to express ths s as follows: We can see qute clearly that the cost-beneft rato n natonal currency of operatng a Servce Node n the Lok network determnes whether the stakng requrement ought to ncrease or decrease n response to a change to the opportunty cost of operatng a Servce Node. As the cost of operatng a Servce Node decreases relatve to the reward for operatng a Servce Node n terms of natonal currency, we tend away from havng to ncrease the stakng requrement to support equlbrum as the opportunty cost of operatng a Servce Node ncreases, and have to decrease the stakng requrement to support equlbrum once costs fall below rewards. 4.3 Summary, analytcs of the stakng requrement whch supports equlbrum We have dentfed the effect of a change of four varables on the stakng requrement whch mplements an equlbrum n whch I symmetrc ratonal agents n the Lok network provde n * Servce Nodes: the number of Servce Nodes we wsh for each actor to operate, operatng costs, the exchange rate of natonal currency for Lok, and opportunty costs. Were the exchange rate to ncrease (Lok to apprecate), or future operatng costs decrease, or the number of Servce Nodes we wsh for each actor to operate to decrease, the stakng requrement whch supports equlbrum ncreases and vce versa. The confoundng factor n ths analyss s the effect of opportunty costs. Were costs to decrease suffcently relatve to the exchange rate, or were the exchange rate to apprecate suffcently, we would fnd that the stakng requrement whch supports equlbrum wll has an nverse proportonal relatonshp wth the rate at whch opportunty costs of operatng a Lok Servce Node accrue. 11

12 5 Calculatons It wll be valuable now to calculate some specfc solutons to the stakng requrement desgn problem usng the cryptoeconomc soluton developed above and establsh some responses to parameters around that equlbrum pont. Frst, however, we wll need to specfy a specfc form for the revenue functon λ (n N). Lok s revenue functon depends on ts emssons curve, whch n turn depends on the reward for processng a block n ts Blockchan. The calculatons n ths secton were prepared wth an earler proposal for the emssons curve. Snce the calculatons were completed a new emssons curve has been proposed, so the reader s drected to 4 later documentaton concernng ths new curve for ther purposes. The calculatons n ths secton ought to be consdered purely n a hstorcal context and ndcatve only of the general dynamcs of the ncentve structure n Lok. 5.1 Specfyng and modellng a revenue functon The form of the Block reward B r s an nverse exponental functon n Block heght varable B h so that the Block reward s decreasng over tme, and thus the growth rate of Lok s total supply stablsng. We have that B r = α + exp e xp { β (B )} where β (B h ) s ncreasng n B h. As B r s emtted over tme by the processng of Blocks, some 45% of t s made avalable for mners, 5% s made avalable for the Lok governance pool and 50% s made avalable for Servce Nodes. Servce Nodes are not pad out on a pro-rata bass, but rather are tabulated n an ordered lst of whch the top Servce Node s pad the entre reward avalable for all Servce Nodes. That node s then replaced to the bottom of the lst, and each node moves up the lst by one entry. Roughly speakng therefore, a Servce Node wll be pad out once every N processng perods. An approprate way to model the revenue functon over the Block processng perod for any gven actor runnng n Servce Nodes on the Lok network nonetheless s as follows λ (n N) = n( 1 N B ) r where =. 5B r. The margnal revenue for an actor on the Lok network operatng n Servce Nodes B r s therefore (snce, of course, N = ) n I λ(n N) N n = n 2 N B r h 4 "LokCryptoEconomcs" 12

13 As a check that the assumptons of our analyss hold above and playng a lttle fast and loose wth notaton we can see that margnal revenue s ndeed concave n the operaton of Servce Nodes. We have that 2 (n ) 2 λ (n N) = B 2 j j r 2 = B (n ) n j ) 2 ( j I n j 2 n j ( n j ) 3 j I As t s rather dffcult to conceptualse what a negatve operaton of Servce Nodes would be, we can safely assume that 0 j I, and so we have that n j 2 (n ) 2 λ (n N) 0 Now wthn ths revenue functon we have one fnal problem before we can pck parameters relatvely freely. That s that we wsh for a gven desred equlbrum n the provson of Servce Nodes to be feasble n the sense that actors are n fact ncentvsed to operate Servce Nodes. We would observe ths beng the case so long as the stakng requrement whch would support an equlbrum s postve (were t to be negatve, the Lok network would have to reverse the stakng requrement so that stakes were provded to actors). So, for our revenue functon we must have that ŝ 0. Now snce r 0 n all but extreme stuatons, we have n ŝ 0 λ ( n * c I ) ε Inputtng the form of λ( n * I ) and pckng a symmetrc equlbrum provson of Servce Nodes n * by I actors, we see that for ŝ 0 we requre that n ( I 1) * (n* I ) 2 B c ε The relevant varable whch may ensure ths requrement s met s the asymptote r r α of the Block reward functon. As the Lok Blockchan grows, we fnd that the Block reward converges to ths asymptote, for β B B 0 B h h r = α So, nputtng ths lmt nto the requrement mmedately above, we have that for ŝ 0 we requre n * ( I 1) (n* I ) 2.5α Any partcular form for the parameters of the Block reward must mantan ths condton for the Lok network to be feasble n the sense of requrng non-negatve stakes. Ths s dffcult to guarantee unversally, so we wll suppose the followng form for our calculatons whch appears reasonable at the present tme: ε c 13

14 B r = exp exp {( ( 6 )) B h} The desgn problem now beng fully specfed for a gven symmetrc equlbrum provson of Servce Nodes n * by I actors, we may now calculate varous stakng requrements for specfc parametersatons of operatng cost c, exchange rate ε, opportunty cost r and Block heght B h. As mentoned above, the calculatons below were prepared wth an earler proposal for the emssons curve whch has now been superseded, so the reader s drected to later documentaton concernng ths 5 new curve for ther purposes. The calculatons n ths secton ought to be consdered purely n a hstorcal context and ndcatve only of the general dynamcs of the ncentve structure n Lok. 5.2 Stakng requrement under specfc parametersatons Let us suppose, to begn wth, that the equlbrum Lok wshes to acheve s a symmetrc operaton of 1 Servce Node by 8,000 actors n the network. Ths would make Lok not dssmlar to TOR n terms of the dstrbuton of routng and data transfer servces and therefore would offer (n equlbrum) prvacy-preservng Servce Node Apps of a smlar securty. These objectves establshed, let us suppose the followng parametersaton based on extant data at the present tme for the Lok network. Important Notce : the data below does not ndcate any real world prce, or predcton of the prce of Lok n the future, ths data should not be used to nfluence the Servce Node operators decson to operate a Servce Node. The market prce of Lok wll fluctuate and these fgures are used as examples only. Operatng Cost Opportunty Cost Exchange Rate Block Heght Processng Tme 400USD p.a. 3% p.a..325usd 129,600 2 mnutes Table 1: Parametersaton for equlbrum where 8,000 actors operate 1 Servce Node each Under ths baselne parametersaton, the stakng requrement whch supports such an equlbrum as s consstent wth the objectve of havng 8,000 actors provde 1 Servce Node each s Lok, or USD. Whle these actors earn less than.01usd (.03Lok) per Block processed n equlbrum, ths translates to a yearly accountng proft of some USD (379.66Lok) for operatng a Servce Node n the Lok network whch hosts routng and data transfer servces for Servce Node Apps. In equlbrum, of course, ths means that actors n the Lok network obtan an accountng return on nvestment of 3% p.a for operatng Servce Nodes. Now ths parametersaton assumes a rate of return commensurate wth the long-term average rate of return on Australan government debt at the tme of wrtng. Of course, the true opportunty cost of stakng suffcent Lok to operate Servce Nodes s lkely to be substantally hgher as Australan government debt does provde somethng of a baselne return. So, an alternatve parametersaton substtutes a rate of return more commensurate wth good commercal rates of return. Operatng Cost Opportunty Cost Exchange Rate Block Heght Processng Tme 400USD p.a. 7% p.a..325usd 129,600 2 mnutes 5 "LokCryptoEconomcs" 14

15 Table 2: Alternatve parametersaton for equlbrum where 8,000 actors operate 1 Servce Node each Under ths parametersaton, the stakng requrement whch supports such an equlbrum as s consstent wth the objectve of havng 8,000 actors provde 1 Servce Node each s 5,420.92Lok, or USD. Whle these actors agan earn less than.01usd (.03Lok) proft per Block processed n equlbrum, ths stll translates to a yearly proft of some USD (379.66Lok). In equlbrum however, of course, actors now earn an accountng return on nvestment of 7% p.a. Now n the early stages of developng the network of Servce Nodes over whch SNApps wll be provded by the Lok system, t mght be aspratonal to suppose that a network wth the securty of TOR mght be acheved n equlbrum. Instead, t mght be better to set more realstc objectves for equlbrum whch mght be revsed over tme. We mght suppose then, that ntally we mght wsh to set that stakng requrement whch supports an equlbrum n whch 1,000 actors operate 1 Servce Node each wth the followng parametersaton. Operatng Cost Opportunty Cost Exchange Rate Block Heght Processng Tme 400USD p.a. 7% p.a..325usd 129,600 2 mnutes Table 3: Parametersaton for equlbrum where 1,000 actors operate 1 Servce Node each Under ths parametersaton, the stakng requrement whch supports such an equlbrum as s consstent wth the objectve of havng 1,000 actors operate 1 Servce Node each s 166, Lok, or USD. Actors earn an accountng proft of USD (11,652Lok) and an accountng rate of return of 7% n equlbrum n ths scenaro. Clearly ths s a dffcult equlbrum to obtan gven that the stakng requrement t demands s on the order of a depost for a house, so we mght try to relax the strct requrement that each actor provde but 1 Servce Node n equlbrum. We mght wsh to set that stakng requrement whch supports and equlbrum n whch 1,000 actors operate 2 Servce Nodes each wth the followng parametersaton. Operatng Cost Opportunty Cost Exchange Rate Block Heght Processng Tme 400USD p.a. 7% p.a..325usd 129,600 2 mnutes Table 4: Parametersaton for equlbrum where 1,000 actors operate 2 Servce Nodes each Under ths parametersaton, the stakng requrement whch supports an alternatve equlbrum whch s consstent wth the objectve of havng 1,000 actors operate 2 Servce Nodes each s Lok, or 24,163.88USD. Actors earn an accountng proft of USD (23,305Lok) n ths equlbrum and an accountng rate of return of 7% p.a. Now ths s stll a dffcult stakng requrement to obtan, so n the early phases of developng the network of Servce Nodes over whch SNApps wll be provded by the Lok system t may be worthwhle to seek to allow actors to pool funds n order to stake suffcent Lok to operate Servce Nodes. But we mght also wsh to consder more generally how the stakng requrements n response to the change of varables around ths desred equlbrum. 5.3 Equlbrum responses to changng parameters Now let us suppose that we wsh to mplement an equlbrum n whch 8000 actors provde 1 Servce Node each and take our alternatve parametersaton above (specfed n Table 2) whereby we allow for an opportunty cost of 7% p.a. n equlbrum. It wll be nterestng to nvestgate the response of 15

16 the stakng requrement to changes of the objectves of the Lok network and opportunty cost allowed for under ths parametersaton. We wll run projectons for changes to the number of desred Servce Nodes operated by a gven actor n equlbrum, the number of actors overall, and the opportunty cost allowed for. We wll not run projectons for the effect of an ncreasng Block heght. The reason for ths s that t can be readly verfed that, wth the Block reward rate beng such as t s, Block heght has lttle effect on margnal revenue and thus stakng requrement untl t becomes very large ndeed, and so can be farly easly set asde. Take frst the effect of number of desred Servce Nodes operated by a gven actor n equlbrum. Ths s shown n fgure 1. What we can see s that as the number of desred Servce Nodes we wsh to be operated by any gven actor n equlbrum decreases, the stakng requrement must ncrease, and at an ncreasng rate. The reason for ths s qute obvous t must become more dffcult to stake suffcent Lok to become a Servce Node, and for ths to dsncentvse dong so by erodng return on nvestment. Fgure 1: Effect on stakng requrement of changng the number of desred Servce Nodes operated by a gven actor n equlbrum wth 8,000 actors and the parametersaton n Table 2 Over tme therefore, should the Lok network begn wth a farly relaxed atttude toward the number of desred Servce Nodes operated by a gven actor n equlbrum, and then, wth a vew to the securty of the network, tghten ths atttude, the stakng requrement whch wll support these objectves wll ncrease. Relatve to a partcular sze of the Lok network n other words, the greater the desre for ts securty n terms of provdng prvacy, the greater the stakng requrement wll have to be whch mplements those desres n the equlbrum provson of Servce Nodes. Now take the effect of an ncrease n the number of actors present n the Lok network. In fgure 2 we can see that gven a desred operaton of Servce Nodes n equlbrum, the stakng requrement decreases as more actors enter the network. The reason for ths s a lttle subtle. 16

17 Fgure 2: Effect on stakng requrement of changng the number of actors n the network when desred operaton of Servce Nodes s one per actor gven the parametersaton n Table 2 As more actors enter the network, they erode the revenues earned by any gven Servce Node across the network spread across Block processng perods. For a gven parametersaton of the system then, the stakng requrement has to decrease to ncentvse the provson of Servce Nodes by restorng the accountng rate of return. Over tme then, unless the number of actors n the network remans farly stable, we can magne that there may be countervalng pressure aganst pure securty desres on the stakng requrement to decrease to restore accountng rates of return whch ncentvse the provson of Servce Nodes on the Lok network. Now fnally take the opportunty cost whch s allowed for. There are good reasons we may wsh to allow ths to vary over tme, for n ts early years Lok may be a relatvely rsky nvestment untl the network develops more fully. To offset ths rsk, we mght wsh to set the opportunty cost allowed for to be relatvely hgh (reflectng the opportuntes forgone by allocatng resources to operatng Servce Nodes wthn the Lok network) and then decrease t over tme. The effect of ths on the stakng requrement gven a desred equlbrum n whch 8,000 actors provde 1 Servce Node each can be seen n fgure 3. 17

18 Fgure 3: Effect on stakng requrement of a change of opportunty cost allowed for gven a desred equlbrum n whch 8,000 actors provde 1 Servce Node each and the parametersaton n Table 2 As we allow for a lesser and lesser opportunty cost, the stakng requrement whch wll support an equlbrum n whch 8,000 actors provde 1 Servce Node each, gven the parametersaton n Table 2, wll ncrease. The reason for ths, agan, s a lttle subtle and nvolves nuanced cryptoeconomc reasonng. As we decrease the opportunty cost allowed for gven a desred equlbrum and parametersaton, the opportuntes for nvestng outsde of the Lok system decrease n value, and t becomes more attractve to nvest n the Lok system by operatng Servce Nodes. In order to preserve the securty of the network then by ensurng the equlbrum operaton of Servce Nodes by actors n the network does not change, the stakng requrement must ncrease to dsncentvse the operaton of more than that number of Servce Nodes by erodng return on nvestment. So over tme, provded that the number of actors on the Lok network remans farly constant, we can expect that for a gven parametersaton (here that n Table 2) and a desred equlbrum (here one n whch 8,000 provde 1 Servce Node each) the stakng requrement for Servce Nodes n the Lok network to ncrease. Ths s because of the desre to mantan securty on the network by ensurng that no one actor has an ncentve n equlbrum to operate more than a certan proporton of the Servce Nodes n the Lok network, and the decreasng allowance for opportunty costs as the Lok network stablses. The stakng requrement wll have to ncrease to dsncentvse operatng more than the desred number of Servce Nodes n equlbrum, and to offset the ncreased attractveness of the Lok network relatve to the opportuntes forgone by operatng Servce Nodes wthn t. 6 Concluson: solutons to Lok s cryptoeconomc problem Lok s a Blockchan network orented toward the provson of prvacy-preservng servces over a network of Servce Nodes. The salent cryptoeconomc problem we addressed here was how to 18

19 ncentvse Servce Nodes n a manner compatble wth the objectves of the Lok network, n partcular decentralsaton and prvacy. We used cryptoeconomc game theory to characterse ths problem and formulate a soluton to the desgn problem. An actor n the Lok network wll stake suffcent Lok to operate n Servce Nodes among N on the network f the proft π (n N) to be ganed from dong so s greater than the opportunty cost of nvestng the stake n s n terms of natonal currency (the exchange rate thereof for Lok beng ε ) to obtan a return r π (n N) r (ϵns) and only f ths s the case and they have n s Lok ( n ϵs n natonal currences) avalable to stake. So the economc proft obtaned from stakng suffcent Lok to operate network of Servce Nodes s N s gven by π (n N) r (ϵns) = ϵ λ (n N) [ c (n) + r (ϵns)] n Servce Nodes when the total where λ (n N) s the reward allocated to n Servce Nodes when there are N n the network and c (n) s the cost of operatng those nodes. A ratonal agent n the Lok network wll therefore operate n * Servce Nodes defned as follows n * 1 = n : s n λ (n N) = r + ϵs 1 n c (n) whch, after ntroducng certan reasonable assumptons whch exogense certan varables, reduces to n * = n : s 1 [ n λ (n N) εc ] = r In equlbrum, each actor n the Lok network operates n* Servce Nodes defned as follows ( [ *]) ] n* 1 = n : [ s λ n n n + n c j ε = r j I Ths equlbrum s a pure strategy Nash equlbrum. We found as a result that the stakng requrement ŝ whch would support a symmetrc equlbrum n whch ratonal agents n the Lok network provde n * Servce Nodes s gven by ŝ = r 1 [ λ n I n ( * c ) ] ε where I s the number of actors I n the Lok network. If ϕ s the proporton of Servce Nodes any gven actor must operate n order to undermne the prvacy n SNApps provded on the Lok network, we may defne the approxmate set of acceptable stakng requrements (those whch wll support an equlbrum compatble wth Lok s objectves) by pckng an arbtrary number of Servce Nodes N to be provded homogenously across the network and so defne Ω, the set of acceptable stakng requrements, as follows 19

20 Ω = {s = ŝ (n* N) : n * ϕn} where ŝ (n* 1 N) = r [ λ. To undermne the prvacy of the Lok network by provdng n (n* N) εc ] n ρ > ϕn Servce Nodes, a rogue actor ρ must value ths undermnng by at least approxmately v ρ = λ n n I ε ( * * ) λ n ( ρ (n ) I [ ρ n * + n * ]) + (rs ) ˆ + c (n ρ n * ) whch s strongly ncreasng n substantal multples of the opportunty cost of stakng Lok to operate Servce Nodes as well as the server cost. We also establshed the effects on the stakng requrement whch wll support an equlbrum n whch I symmetrc ratonal agents n the Lok network provde n * Servce Nodes of the number of Servce Nodes we wsh for each actor to operate, operatng costs, the exchange rate of natonal currency for Lok and opportunty costs. These analytcal effects notwthstandng, we also nvestgated varous numercal teratons of the desgn problem to dscover the response of the stakng requrement to varous objectves and parametersatons of the Lok network. We found that gven a partcular parametersaton based on extant condtons n the Lok network, a farly reasonable opportunty cost of 7%, and a desred equlbrum (n whch 8,000 actors provde 1 node each) whch roughly mrrors the structure of TOR, a stakng requrement of Lok, or USD. Ths generates an accountng proft of USD (379.66Lok) p.a. and, of course, an account return on nvestment of 7% p.a. n equlbrum. Ths, however, s a very strct objectve for the Lok network to begn wth, and so we consdered alternatve scenaros whch mght be adopted n the nterm. We also consdered n ths numercal context how the stakng requrement whch would support a TOR-lke equlbrum would change n response to the desred number of nodes, the overall number of actors n the network, and the opportunty cost allowed for. We found, as a result, that we can expect the stakng requrement that would support a TOR-lke equlbrum to ncrease over tme as a result of seekng to ensure the securty of the network and the declnng opportunty cost allowed for relatve to the attractveness of operatng Servce Nodes n Lok. The numercal calculatons provded n ths report were prepared wth an earler proposal for the emssons curve. Snce the calculatons were completed a new emssons curve has been proposed, so 6 the reader s drected to later documentaton concernng ths new curve for ther purposes. The numercal calculatons presented n ths report ought therefore to be consdered purely n a hstorcal context and ndcatve only of the general dynamcs of the ncentve structure n Lok. 6 "LokCryptoEconomcs" 20