Egalitarian and Just Digital Currency Networks
EEgalitarian and Just Digital Currency Networks
Gal ShahafWeizmann Institute Ehud ShapiroWeizmann InstituteNimrod TalmonBen-Gurion University
Abstract
Cryptocurrencies are a digital medium of exchange with decentral-ized control that renders the community operating the cryptocurrency itssovereign. Leading cryptocurrencies use proof-of-work or proof-of-staketo reach consensus, thus are inherently plutocratic. This plutocracy isreflected not only in control over execution, but also in the distributionof new wealth, giving rise to “rich get richer” phenomena. Here, we ex-plore the possibility of an alternative digital currency that is egalitarianin control and just in the distribution of created wealth. Such currenciescan form and grow in grassroots and sybil-resilient way. A single currencycommunity can achieve distributive justice by egalitarian coin minting ,whereby each member mints one coin at every time step. Egalitarianminting results, in the limit, in the dilution of any inherited assets andin each member having an equal share of the minted currency, adjustedby the relative productivity of the members. Our main theorem showsthat a currency network, where agents can be members of more than onecurrency community, can achieve distributive justice globally across thenetwork by joint egalitarian minting , whereby each agent mints one coinin only one community at each timestep. Specifically, we show that asufficiently large intersection between two communities – relative to thegap in their productivity – will cause the exchange rates between theircurrencies to converge to 1:1, resulting in global distributive justice.
Money is nothing but a piece of paper; or a string of bits, perhaps. In modernhistory, fiat money is issued and controlled by rulers and governments. Follow-ing Bitcoin [15], many blockchain-based cryptocurrencies were introduced [16].Their technology and distributed protocol renders the community operating thecurrency its sovereign as, unlike in standard computer systems, there is no thirdparty that may exert control over the system, e.g., shut it down.1 a r X i v : . [ q -f i n . GN ] J u l n existing cryptocurrencies, however, most control and benefit lies in thehands of the few – their founders, early adopters, and large stakeholders (e.g.large “mining pools”) [11]. In this paper we explore the possibility of formingan egalitarian and just digital currency that may form currency networks in agrassroots manner. Our goal here is the design of a digital currency that maybe issued by all, where both control and benefit are distributed in an egalitarianway among the people participating in the creation and use of the currency.Such a currency implements distributive justice in the sense that each personenjoys an equal share of the created currency.One key challenge in this task is the presence of fake and duplicate identi-ties, aka sybils , that may be employed by their operators in order to tilt controland wealth in their favor. Indeed, justice and equality can be achieved onlyif the parties to the currency are genuine (unique and singular) agents of theparticipating people [22], thus excluding sybils. We first observe that sybils can-not penetrate small communities of people that know and trust each other andthat, indeed, trust communities can grow in a sybil-resilient way by employinggraph-based properties [19] of genuine identifiers [22], using various mechanismsas admission rules to the community [23], or utilizing some machine learningalgorithms [9].However, as we wish our medium to be scalable, our further goal is to buildthis digital currency in a grassroots way. In particular, our paper may beviewed as means for a joint, safe scale-up of such communities, concentratingon the aspect of distributive justice as we rely on the infrastructure of digitalsocial contracts [2] for equality in execution, and techniques such as mutualsureties [22] and sybil-resilient community expansion [19] for sybil resilience.The key, high-level differences between our proposed digital currency andmost existing cryptocurrencies are outlined below: • Equality:
Leading cryptocurrencies employ either proof-of-work (PoW)or proof-of-stake (PoS) systems [17]. As such, they are inherently pluto-cratic, since control over the behavior of the system is positively correlatedwith the computing power or amount of currency available to differentparties. A cryptocurrency is egalitarian if control over the execution andmodification of the currency system is shared equally among the partiesto the currency. Such equality can be guaranteed using digital social con-tracts [2] over genuine identifiers [22]. • Distributive justice:
Leading cryptocurrencies do not aim for justice,distributive or otherwise. Newly minted coins are allocated to parties withsuperior computing power (PoW) or larger amounts of currency (PoS). Acryptocurrency satisfies distributive justice if each agent enjoys an equalshare of the newly created value of the currency. Here, we formally definedistributive justice in this context and spell out conditions that give riseto it. In particular, a single currency community can achieve distributivejustice by egalitarian coin minting, where each member mints one coin See, e.g., https://bitcoinera.app/arewedecentralizedyet/ .
2n every time step. Assuming the community has only genuine membersand no sybils, egalitarian minting results, in the limit, in the dilution ofany inherited assets and in each member having an equal share of theminted currency, adjusted by the relative productivity of the members.In a currency network, where people can be members of more than onecurrency community, a joint egalitarian minting regime in which eachperson mints one coin in only one community in each timestep, allowsmarket forces to achieve distributive justice globally across the network,under conditions that we discuss. • Grassroots Sybil-Resilience:
Leading cryptocurrencies are monolithic,in that there is one community using the cryptocurrency (e.g., one blockchainin which the bitcoin transactions are recorded). Here, we aim at a grass-roots architecture that allows currency communities to form indepen-dently, allowing people from different communities to trade and exchangetheir currencies, and eventually form a currency network that serves asa joint, grassroots medium of exchange. Our method is sybil-resilient inthat sybils in a currency network affect only the currency communitiesthat harbour them. We defer the discussion on the specific sybil resilienceachieved in such networks for future work.In this paper, we first begin with a single currency and provide a formal def-inition for a just distribution among its agents. Intuitively, distributive justiceis satisfied if every member of the currency community is granted, initially, anequal share of the currency, and may trade its portion as it pleases. Formally,at every time step, the diluted balance of every agent amounts to its equal shareplus its diluted cashflow up to this point. We then present a richer notion of asymptotic justice , where distributive justice is reached in the limit. With thisnotion, distributive justice can be reached even if agents begin with different ini-tial amounts of the currency; as such it models distributive justice in the face ofunequal inheritances. To achieve asymptotic justice, the difference between thediluted balance and cashflow must converge to an equal share of the currency,but these quantities need not match at all times. We show that this notion ofjustice may be realized via egalitarian coin minting, which provides a form of
Universal Basic Income (UBI). That is, a community in which each membermints an equal amount of coins in every time step results in asymptotic justice,regardless of the initial balance of the agents and the differences in the time ofof joining the community.Envisioning the emergence of different and independent currency communi-ties, each employing their own egalitarian minting regime as describe above, wethen analyze conditions under which multiple communities may inter-operatein such a way that, jointly, genuine agents in all communities will get an equalshare of the joint created value of all currencies; that is, we set to investigatethe possibility of achieving global distributive justice in a situation where manyindependent currencies are used at once. To this end, we define the notion ofa currency network , in which several currency communities operate simultane-ously. The formal definition of a currency network is given below; in essence, it is3 tuple of communities that employ independent currencies (each coin belongsto a single currency). The network structure arises from chain payments viaagents that are members in multiple communities simultaneously. This modelis a direct generalization of credit networks [24, 8, 20, 4, 5].In order to analyze the dynamics of such networks and the economic conse-quences of such dynamics, we apply the free exchange economy model [14] forthe emergence of exchange rates among the different currencies. Based on theserates, we extend the definition of distributive justice to a currency network,and provide sufficient conditions under which distributive justice is satisfied.Importantly, these conditions rely on the currency volumes being in perfectbalance with the marginal rates of substitution among the currencies. This bal-ance requires calibration with every alternation in the network structure (i.e.,the admission of a new member, etc.), and is thus hard to maintain without africtionless and efficient trade among the currencies.With these assumptions, we extend the notion of asymptotic justice to cur-rency networks. Our main result in this setting provides sufficient conditionsunder which asymptotic justice is achieved under an egalitarian minting regime .That is, in order to obtain distributive justice in the limit, the substantial col-laboration among the different communities is expressed in jointly ensuring thatevery agent may mint one coin of only one currency at every time step. Agentsmay choose which coin to mint from the different currency communities in whichthey are members. Specifically, our main result shows that exchange rates be-tween two communities will converge to 1:1 and asymptotic justice would follow,as long as the following conditions hold:1. Agents behave myopically, in that each agent mints the highest valuedcoin at every time step;2. The network is efficient, in that agents trade coins in order to maximizetheir utilities, causing equilibria to be reached infinitely often;3. The intersections among the two communities is sufficiently large to com-pensate for the productivity gap between them.Our focus in this paper is on the economic analysis of currency networks, asdescribed above. Ultimately, we aim to implement such currencies using dig-ital social contracts, and show social contract schemes for single- and multi-currency egalitarian minting [2]. Our analysis shows how distributive justicecan be achieved globally in a network of egalitarian and grassroots digital cur-rencies. Importantly, while a distributed implementation of this model mustdeal with the asynchronous nature of the underlying communication networkamong the agents (as in digital social contracts [2]), here, for simplicity, weassume a synchronous model of computation.Finally, we discuss the relation between people and their agents, and showthat if a currency community is genuine then it can achieve distributive justiceamong its owners. In a currency network with a genuine (sybil-free) subnet,distributive justice can be achieved among all owners of the subnet.4 .1 Organization
After reviewing related work, we proceed with the notion of a single currencycommunity at Section 2, where we define initial and asymptotic distributive jus-tice, and discuss means for achieving them. We then address currency networksat Section 3, where we discuss the emergence of exchange rates via the freeexchange economy model and extend the definitions of justice to this richer set-ting. Then, at Section 4, we analyze sufficient conditions for asymptotic justicein a network under an egalitarian minting regime.
Mathematically, the main predecessor for personal currency networks are creditnetworks [4, 5, 20, 6, 10, 8], and some of the results and analyses of creditnetworks carry over to personal currency networks. The key difference betweencredit networks and our newly proposed digital currency networks is that creditnetworks assume the existence of an objective measure of value, namely, anoutside currency, whereas currency networks aim to create an objective measureof value.While credit networks inspired some cryptocurrencies, including Ripple [21]and Stellar [13], they all had to choose an external currency to peg credit to:Ripple has chosen to provide its own cryptocurrency, XRP, the production ofwhich is controlled by the Ripple Foundation (who owns the majority of mintedXRP coins), while Stellar chose to be a “stablecoin”, pegging the credit to abasket of fiat currencies.Practically, the most related cryptocurrencies are the trust-based currenciesof Circles [3] and Duniter [7]. Both create money through Universal BasicIncome (UBI) to their members. Circles is a smart contract on top of Ethereumand is still a concept under development. Duniter is a cryptocurrency with anactive community of mostly-French users; it anticipated the idea of egalitariancoin minting presented here and has a mechanism of sybil-resilience, being anindication that the conceptual and mathematical framework presented here maybe viable.A UBI-based currency community is a possibility, as demonstrated by Duniter,and is consistent with our mathematical model. Here, in particular, we studyjoint-UBI regimes, supporting the grassroots formation of multiple currencies;so we do not only concentrate on a single currency community (like Duniterand Circles), but anticipate a network, consisting of many such currencies, andstudy their joint economic behavior. Indeed, Duniter is not grassroots in thesense that it does not provide conceptual or architectural foundation for mul-tiple independent Duniter-like currency communities to form and interoperate,like we do. 5
A Currency Community
Here we first describe a cryptocurrency community that is equal and just, pro-vided it is sybil-free. We expect people to participate in a currency communityvia computational agents, and assume a one-to-one correspondence betweenpeople and the agents and refer to the computational agents as “it”. Hence,Such a sybil-free community may be simply a small-scale community in whichall agents know and trust each other, or a larger-scale community that growsin a sybil-resilient way [22, 19]. We first define such a currency community for-mally, and analyze economic properties of its dynamics, showing in particularthat distributive justice can be achieved in the limit using an egalitarian mint-ing regime in which each agent mints a single coin in each timestep. A digitalsocial contract that implements egalitarian execution of a currency communityand egalitarian minting is described elsewhere [2].
Definition 1 (Currency Community) . A Currency Community is a tuple C =( V, C, h ), where V is a set of agents, C is a set of fungible coins, and h : C −→ V is a configuration function that indicates the holder of each coin h ( c ) ∈ V .Coins are fungible in the sense defined below. We shall also use the inversefunction h − ( v ) := { c ∈ C | h ( c ) = v } to denote the coins held by agent v ∈ V .We regard the currency as a medium of exchange for goods and services.The fundamental operation in a currency is a payment , i.e., the transfer of acoin from a payer to a payee. Definition 2 (Payment) . Let C = ( V, C, h ) be a currency community and let u, v ∈ V . A payment from u to v is a transfer from u to v of a coin c ∈ C ,initially held by u . The result of such a payment, denoted by C pay ( c,u,v ) −−−−−−→ C (cid:48) , isthe currency community C (cid:48) = ( V, C, h (cid:48) ), in which: h (cid:48) ( x ) := (cid:40) v if x = c ,h ( x ) otherwise . We observe that payments are reversible.
Observation 1 (Reversibility in a Single Currency) . If C is a currency com-munity and C pay ( c,u,v ) −−−−−−→ C (cid:48) , then C (cid:48) pay ( c,v,u ) −−−−−−→ C .Proof. By Definition 2, exchanging a coin back and forth results in the initialconfiguration.
We wish to better understand the economic properties of a currency community,in particular, to explore the possibility of achieving distributive justice withinthe community. To this end, and since we envision a digital currency built withthe currency community model in its core, we take the following approach: As6he economy of a currency community takes place in a dynamic setting, whereagents trade coins with each other for goods and services, we consider currencycommunity dynamics.We assume a dynamic setting with discrete time steps, where coins may beminted periodically by the agents. We mention that this can be implementedby a digital social contract [2] among the participants. We note that while theformal model digital social contracts, as well as any feasible realization of it, areasynchronous, we nevertheless assume a synchronous setting as a simpler firststep, in particular a notion of time is needed for egalitarian coin minting.
Definition 3 (Currency Community History) . A currency network history isa sequence of currency communities C , C , C , . . . , C t = ( V t , C t , h t ), t >
0, withthe following monotonic attributes: • Agent growth : V t ⊆ V t +1 for all t ≥ • Coin growth : C t ⊆ C t +1 for all t ≥ V := (cid:83) t V t to denote all agents throughout history, and define the following. Definition 4 (Balance, Income, Revenues and Expenses) . Let C , C , C , . . . .denote a currency community history. Then, we define the following: • Balance:
The balance of agent v at time t is the number of coins held by v at that time, denoted by: b t ( v ) = | h − t ( v ) | . • Income:
The income of agent v at time t is the number of newly mintedcoins held by v , denoted by: m t ( v ) = | h − t ( v ) ∩ ( C t \ C t − ) | . • Revenue:
The revenue of agent v at time t is the number of coins in C t − that were added to v ’s account due to trade, denoted by: rev t ( v ) = | ( h − t ( v ) ∩ C t − ) \ h − t − ( v ) | . • Expenses:
The expenses of agent v at time t are the number of coinssubtracted from v ’s account due to trade, denoted by: exp t ( v ) = | h − t − ( v ) \ h − t ( v ) | . Observation 2.
For every t > we have m t ( v ) + rev t ( v ) − exp t ( v ) = b t ( v ) − b t − ( v ) . (1) Proof. As h − t − ( v ) ⊆ C t − , we have m t ( v ) + rev t ( v ) = | h − t ( v ) ∩ ( C t \ C t − ) | + | ( h − t ( v ) ∩ C t − ) \ h − t − ( v ) | = | h − t ( v ) \ h − t − ( v ) | . It follows that m t ( v ) + rev t ( v ) − exp t ( v ) equals | h − t ( v ) \ h − t − ( v ) | − | h − t − ( v ) \ h − t ( v ) | = | h − t ( v ) | − | h − t − ( v ) | = b t ( v ) − b t − ( v ) , which finishes the proof.Summing up, we conclude the following: Corollary 1.
For every t > we have b t ( v ) = b ( v ) + t (cid:88) s =1 (cid:0) m t ( v ) + rev t ( v ) − exp t ( v ) (cid:1) . (2)That is, the balance of an agent equals its initial endowment plus its incomeand cash-flow up to this point. Given the above definitions and observations, we are ready to formally defineour desired property of distributive justice, in which, intuitively, every agent isgranted an equal share of the currency value. We then demonstrate monetaryregimes which realize distributive justice. The fundamental definition of a justcurrency is the following:
Definition 5 (Distributive Justice) . A currency community history is said tobe just if for every t ≥ v ∈ V t : b t ( v ) | C t | − (cid:80) ts =1 (cid:0) rev s ( v ) − exp s ( v ) (cid:1) | C t | = 1 | V t | . That is, the difference between the diluted balance of each agent and itsdiluted cash-flow is an equal share of the currency value.8ntuitively, a just currency grants an equal share of the currency to everycommunity member, regardless of their initial endowments, while allowing themto do with their share as they please. This results in a socially just allocationof the currency, which is offset from equality only by voluntary trade.
Observation 3 (Equal Birth Grant) . Consider a currency community historywhere each agent receives a fixed number of coins when it joins the community.Formally, b ( v ) = x > for all v ∈ V and m t ( v ) = (cid:40) x if v ∈ V t \ V t − else . Such an equal birth grant regime is just, as it satisfies b t ( v ) − (cid:80) ts =1 (cid:0) rev s ( v ) − exp s ( v ) (cid:1) | C t | = b ( v ) + (cid:80) ts =1 m t ( v ) | C t | = xx · | V t | = 1 | V t | . Next, we define a relaxed notion of distributed justice.
Definition 6 (Asymptotic Justice) . A currency community history is said tobe asymptotically just , iflim t (cid:32) b t ( v ) | C t | − (cid:80) ts =1 (cid:0) rev s ( v ) − exp s ( v ) (cid:1) | C t | (cid:33) = 1 | V | . That is, the difference between the diluted balance of each agent and itsaccumulative diluted cash-flows converges – as time advances – to an equalshare of the currency’s equity.Intuitively, Definition 6 aims to capture justice “in the limit”. We note thatDefinition 6 is weaker then Definition 5, that is, a currency community thatsatisfies distributive justice is also asymptotically just.
Remark 1.
Importantly, we note that both Definitions 5 and 6 heavily relyon the currency history being monotone (see Definition 3). A formal definitionof justice in the (very realistic) case of non-monotone histories, as well as themeans for achieving it in a setting where agents may die or depart from acommunity, would be more subtle. In this paper we refrain from these questions,which include community taxes and inheritance issues, and leave them for futureresearch.As demonstrated in Observation 3, coin minting may serve as means toachieve distributive justice. In the context of asymptotic justice, we discuss anatural minting regime, termed egalitarian minting regime , in which each agentsobtain equal income in the form of new coins minted periodically.9 efinition 7 (Egalitarian Minting) . A currency community history is said toemploy egalitarian minting , if at every step every agent mints the same amountof coins. Formally, m t ( v ) = | C t \ C t − || V t | for every t > v ∈ V t .Note that egalitarian minting might be realized using a simple digital socialcontract, as demonstrated by Cardelli et al. [2]. The following lemma specifiessufficient conditions under which egalitarian minting is asymptotically just. Proposition 1.
A currency community history that employs egalitarian mint-ing with | C t | −→ ∞ and | V | = N < ∞ is asymptotically just.Proof. Fix an agent v ∈ V that had joined the community at time t (cid:48) , i.e., v ∈ V t (cid:48) \ V t (cid:48) − and fix t ≥ t (cid:48) . By Definition 7, we have t (cid:88) s =1 m s ( v ) = t (cid:88) s = t (cid:48) | C s \ C s − || V s | ≥ t (cid:88) s = t (cid:48) | C s \ C s − | N = | C t | − | C t (cid:48) − | N .
Consider a time step t (cid:48)(cid:48) with | V t (cid:48)(cid:48) | ≥ N and fix t ≥ t (cid:48)(cid:48) . We then have t (cid:88) s =1 m s ( v ) ≤ t (cid:48)(cid:48) − (cid:88) s =1 | C s \ C s − || V s | + t (cid:88) s = t (cid:48)(cid:48) | C s \ C s − || V s |≤ t (cid:48)(cid:48) · | C t (cid:48)(cid:48) | + | C t | − | C t (cid:48)(cid:48) − | N . As | C t (cid:48) | , | C t (cid:48)(cid:48) | are constant and | C t | −→ ∞ , it now follows that (cid:80) ts =1 m s ( v ) | C t | −→ N .
We thus conclude thatlim t (cid:32) b t ( v ) − (cid:80) ts =1 (cid:0) rev s ( v ) − exp s ( v ) (cid:1) | C t | (cid:33) = lim t (cid:32) b ( v ) + (cid:80) ts =1 m t ( v ) | C t | (cid:33) = 1 | V | . The claim follows.To summarize, above we showed that a single, sybil-free currency commu-nity that employs egalitarian minting is asymptotically just, namely, as timeadvances, each member indeed approaches being awarded with an equal shareof the currency, offset only by its voluntary trades. This result is a first step to-wards the goal of the next section, in which we study the economic relationshipbetween several such currency communities.10
Currency Networks
The egalitarian minting currency described in Section 2 indeed satisfies equalityand distributive justice, however only for a single, sybil-free community. Recallthat our goal in this paper is a digital currency that is not only equal and justbut also grassroots, in that it can support the bottom-up formation of multiplecurrency communities that can interoperate. Indeed, we envision that manysuch currency communities may form independently and we wish to analyzeconditions under which all agents in a network of such currency communitieswill jointly enjoy distributive justice.To study the economic interactions between different currency communities,the novel mathematical structure we study here is a currency network . In thissection we present a formal model of currency networks; in particular, we showthat they extend and generalize the well-established models of debt and creditnetworks [8, 4, 5, 20]. We then discuss the exchange rates among independentcurrencies within the network and formally define the notion of distributivejustice in this setting.
Definition 8 (Currency Network) . A currency network is a tuple of currencycommunities CN = {C , ..., C k } , C i = ( V i , C i , h i ), with disjoint sets of coins, C i ∩ C j = ∅ for every i, j ∈ [ k ]. The currency network has agents V = (cid:83) i V i ,coins C = (cid:83) i C i , and a network configuration function h : C −→ V defined by h | C i := h i .In this model, agents may be members in several communities simultane-ously. In order to grasp the network structure, it is useful to think of a currencynetwork as a labeled hypergraph CN = ( V, { V i } ki =1 , h ), where agents V = (cid:83) i V i are the vertices, and { V i } ki =1 are the hyperedges, and each vertex v ∈ V is la-beled by the coins it holds from all the communities it is a member of, h − ( v ).See Figure 1 for a visual example. We also note that the special case in whichall currency communities are of size 2 corresponds to credit networks, where theresulting hypergraph is in fact a graph, as every community is manifested as anedge.As in a single currency, the fundamental operation in a currency network isa (direct) payment , i.e., a transfer of a coin from a payer to a payee (Definition2); However, a payment of a coin of a currency can only be made among twomembers of the coin’s currency community. Still, agents in a currency networkmay be able to transact with each other via chain payments , defined below. Definition 9 (Chain Payment) . Let CN = {C , ..., C k } be a currency network, u, v ∈ (cid:83) i V i . A chain payment from u to v is a sequence of direct payments CN j pay ( c j ,u j ,u j +1) −−−−−−−−−−→ CN j +1 , from u j to u j +1 , j ∈ [0 , m − u = u , v = u m , and CN = CN .Note that it is not the same coin that is transferred among the agents par-ticipating in a chain payment. 11igure 1: A currency network containing 7 vertices v i , i ∈ [7] and 3 communi-ties. The blue hyperedge on the left ( { v , v , v } ) represents the vertices V ofcommunity C , the red hyperedge at the bottom represents the vertices V of C , and the green hyperedge on the right represents the vertices V of C . Theagent corresponding to v holds the coins c of C as well as the coin c of C ,while the agent corresponding to v holds the coin c of C . Observation 4.
A chain payment from u to v may occur as a contiguous blockof transitions if there is a path p = ( u , u , . . . , u m ) , u = u , u m = v , for whicheach u i holds a coin acceptable to u i +1 , i ∈ [0 , m − . By induction on Observation 1, we have that chain payments in currencynetworks are reversible.
Corollary 2 (Reversibility in Currency Networks) . If CN is a currency networkand CN pay ( c,u,v ) −−−−−−→ CN (cid:48) , then CN (cid:48) pay ( c,v,u ) −−−−−−→ CN . Our main aim is to explore the possibility of distributive justice within a cur-rency network. To this end, we first address the issue of exchange rates amongthe different currencies. For now, we defer the intricate question of the emer-gence of exchange rates to the next section, and provide a formal definition ofexchange rates in this setting, denoting by EX ij the amount of coins in C j thatmay be traded in CN for a single coin in C i . Definition 10 (Coin Exchange Rates) . The coin exchange rates of a currencynetwork CN = {C , ..., C k } is given by a matrix EX ∈ R k × k that satisfies: • Currency fungibility: EX ii = 1 for all 1 ≤ i ≤ k . • Arbitrage-free trade: EX ij · EX jl = EX il for all 1 ≤ i, j, l ≤ k .That is, coins within the same currency have equal value, and exchanging c ∈ C i to C j and then to C l yields the same rate as a direct exchange from C i for C l . 12 orollary 3 (Reciprocal rates) . Let EX ∈ R k × k denote a coin exchange matrixof a currency network CN = {C , ..., C k } , then every pair of indices ≤ i, j ≤ k satisfy: EX ij = 1EX ji . (3) Proof.
Straightforward from Definition 10.Given exchange rates of coins and the total number of coins of each currency,we define the equity of an agent as the value of its coins as a fraction of thetotal value of all currencies within the network.
Definition 11 (Fractional Equity of Agent) . Let EX ∈ R k × k denote a coinexchange matrix of a currency network CN = {C , ..., C k } . The fractional equityof agent v ∈ V is given by Eq ( v ) := (cid:80) i b i ( v ) · EX ij ( CN ) (cid:80) i | C i | · EX ij ( CN ) . That is, the equity of an agent is the fraction of its assets of the total valueof the network, as may be realized in currency C j . Remark 2.
We note that Definition 11 is independent of the choice of the in-dex j . To see this, multiply both the nominator and denominator by EX jl ( CN t )and apply the arbitrage free trade property (see Definition 10). Similarly to the case of a single currency community, our interpretation of dis-tributive justice relies on the dynamics in the network over time. We thusprovide the notion of a currency network history.
Definition 12 (Currency Network History) . A currency network history is asequence of currency networks CN , CN , CN , . . . , CN t = {C t , ..., C kt } , suchthat C i , C i , C i , . . . , is a currency community history for all 1 ≤ i ≤ k . Weemploy the notation V := (cid:83) i,t V it and C := (cid:83) i,t C it to denote all agents and allcoins in the network throughout history.In short, a currency network history is nothing but a synchronized set ofdistinct community histories. As coin exchange rates may vary over time, weapply the notation EX ( CN t ) to differentiate between exchange rates at differenttime periods throughout history. With the notion of network history at hand,we now extend the notion of distributive justice to a network setting as follows: Definition 13 (Distributive Justice in a Network) . A currency network history CN , CN , CN , . . . is said to be just , if for every t ≥ v ∈ V t : (cid:80) i (cid:104) b it ( v ) − (cid:80) ts =1 (cid:0) rev is ( v ) − exp is ( v ) (cid:1)(cid:105) EX ij ( CN t ) (cid:80) i | C it | · EX ij ( CN t ) = 1 | V t | . C j and diluted properly, results in each agent’sequity being an equal share of the entire currency network’s equity, at everytime step throughout history. We note that this is a straightforward extensionof Definition 5 which corresponds to the special case k = 1.Next, we present the notion of asymptotic justice, extended to a networksetting. Definition 14 (Asymptotic Justice within a Network) . A currency networkhistory CN , CN , CN , . . . is said to be asymptotically just , if for every v ∈ V :lim t (cid:80) i (cid:104) b it ( v ) − (cid:80) ts =1 (cid:0) rev is ( v ) − exp is ( v ) (cid:1)(cid:105) EX ij ( CN t ) (cid:80) i | C it | · EX ij ( CN t ) = 1 | V | . Definitions 14 and 5 for currency networks relate to each other similarlyto the way Definitions 6 and 5 for a single currency community relate to eachother. Distributive justice in a network requires that the difference between allassets of an agent and its current cash-flow, exchanged to some currency C j anddiluted properly, converges to an equal share of the currency network’s equity.Note that Definition 6 corresponds to the special case k = 1. Achieving distributive justice within a currency network requires a joint coinminting regime that is agreeable to all communities in the network. Indeed,the admission of an agent to one community in a just network must affectthe distribution of wealth in another, and the exchange rates volatility requiresjoint efforts in order to maintain distributive justice over time. The joint mintingregime required to achieve that is a natural extension of egalitarian coin mintingto the network setting.
Definition 15 (Joint Egalitarian Minting) . A currency network history is saidto employ joint egalitarian minting if, at every time step, every agent mintsexactly one coin among all currencies in the network: Formally, if (cid:80) i m it ( v ) = 1for every t > v ∈ V t .We demonstrate elsewhere a social contract for joint egalitarian minting ina currency network [2]. In the following, we explore sufficient conditions underwhich joint egalitarian minting naturally gives rise to asymptotic justice withinall agents participating in multiple currencies within the same currency network. We begin with the natural question each agent shall ask at each timestep:
Whichcoin should I mint next?
Indeed, there are many possibilities. Here we considera simple answer:
Always mint the highest-valued coin .14 efinition 16 (Most Valued Coin) . Let CN = {C , ..., C k } be a currency net-work with coin exchange rates EX ∈ R k × k . A most valued coin in this settingis an index i that maximizes EX ij over all indices 1 ≤ j ≤ k . Given an agent v ∈ V , a most valued v -coin is an index i with v ∈ V i that maximizes EX ij overall indices 1 ≤ j ≤ k .The next definition formalizes the notion of myopic behaviour under egali-tarian minting in a network. Definition 17 (Myopic Agents) . Let CN , CN , ... be a network history thatemploys joint egalitarian minting. We say that the agents in the network are myopic if in every time step t , every agent v ∈ V t mints a most valued v -coin(ties are broken arbitrarily). The relations and interactions among the currencies within a network are inher-ent to the currency network setting. In the following, we present a conceptualand mathematical framework for the analysis of these interactions which resultin exchange rates among the different currencies. We reason that any relationamong independent currencies is based upon what the currencies represent,namely actual commodities (e.g., goods and services) that may be purchasedfrom agents that accept these currencies as payment. Specifically, our analy-sis focuses on the exchange rates that emerge at equilibrium, with respect toindividual preferences over these underlying commodities. Note that the com-modities are not represented explicitly in our model; we assume their existencesolely to induce preferences on currencies, which we then take into account.Formally, given a currency network CN = {C , ..., C k } , it will be convenientto view the balances of all agents as a matrix b ∈ R n × k , where b i ( v ) is thebalance of agent v ∈ V in currency C i ( i -balance, for short). We denote the diluted balances by (cid:101) b i ( v ) = b i ( v ) | C i | , and assume that every agent v has a preferencerelation (cid:22) v over diluted portfolios (cid:101) b ( v ) = (cid:16)(cid:101) b i ( v ) (cid:17) ≤ i ≤ k ∈ [0 , k , a vector thatcorresponds to a fractional ownership in each currency in the network.This setting is generally known as a pure exchange economy (see, e.g., [12,14, 25]). We follow standard practice and assume that the preferences of agent v are expressed via a convex, continuous, and monotone linear order over [0 , k .Given an initial endowment (cid:101) b ∈ [0 , n × k , and assuming that agents may freelytrade currencies with each other, the standard solution concept in this modelis a competitive equilibrium (cid:101) b ∗ wrt. the preferences {(cid:22) v } v ∈ V that Pareto dom-inates (cid:101) b . Importantly, a competitive equilibrium establishes not only an alloca-tion (which is reflected in the balances), but also marginal rates of substitution among currencies [18]: A matrix MRS ∈ R k × k where MRS ij denotes the quan-tity of the currency C j that an agent can exchange for one (infinitesimal) unit ofcurrency C i while maintaining the same level of utility under the equilibrium (cid:101) b ∗ .The normalization of the marginal rates of substitution among currencies bythe currency volumes, naturally gives rise to exchange rates among coins within15hese currencies. As these rates are induced by individual preferences, we termthem preferences-based exchange rates, formally defined below. Definition 18 (Preferences-based rates) . Let CN ∗ be a currency network inwhich the diluted balances matrix (cid:101) b ∗ form an equilibrium under agents’ prefer-ences over the currencies. The preferences-based rates between coins in C i and C j is given by EX ij := MRS ij · | C j || C i | . (4) Remark 3.
Note the difference between the marginal rate of substitutionamong currencies (denoted by
MRS ), which relates the effective values of thetwo economies underlying the two compared currencies, and the exchange ratebetween coins (denoted by EX ). In essence, preferences-based coin exchangerates ( EX ) are the currency rates ( MRS ), normalized by the number of coins incirculation.The following observation asserts that preferences-based rate are valid coinexchange rates as specified in Definition 10.
Observation 5.
Preferences-based rates satisfy currency fungibility and arbi-trage free trade.Proof.
As marginal rates of substitution arise in equilibrium, these rates mustsatisfy both
MRS ii = 1 and MRS ij · MRS jl = MRS il , or else agents wouldbenefit from further trade. Applying Definition 18 to these equations completesthe proof.A key merit of using coins as a medium of exchange (rather than direct tradein fractions of currencies) lies in the degree of freedom manifested in currencyvolumes, as an increase in money supply causes inflation [1]. Put simply, if morecoins are issued for a single currency, this linearly impact the exchange rate ofthis currency with other currencies. Roughly speaking, our general approachbuilds upon the observation that agent choices in coin minting affect and controlthe fractions | C j || C i | , which in turn affect the coin exchange rates.We say that the volumes of all currencies are in perfect balance if the ratiobetween the number of coins of any two currencies exactly equals the differencein the marginal rate of substitution among them in equilibrium. We claim nextthat if the volumes of a pair of currencies is in perfect balance then a fixed 1:1coin exchange rate follows. Observation 6.
Let CN ∗ be a currency network in which the diluted balancesmatrix (cid:101) b ∗ forms an equilibrium under agents’ preferences, and let EX denotepreferences-based coin exchange rates. Then, if two currencies C i , C j satisfy | C i || C j | = MRS ij , it follows that EX ij = 1 .Proof. Straightforward from Definition 10.16inally, we now aim to extend the notion of exchange rates to all configura-tions in network history. To do so, we rely on the tendency of reaching equilibriawrt. agents’ preferences via voluntary mutual trade. Indeed, not all configura-tions throughout history necessarily form an equilibrium: in particular, it mighttake several time steps for agents to perform all profitable coin trades and ar-bitrages. We thus define an efficient history as such that gives rise to equilibriainfinitely often.
Definition 19 (Efficient History) . Let CN , CN , CN , . . . be a currency net-work history with agents’ individual preferences over its currencies. Such net-work history is said to be efficient if there exists an (infinite) subsequence t < t < t . . . such that CN t i is in equilibrium wrt. to these preferences.Following that line, we now extend the notion of marginal rates of substitu-tion (and consequently, also preferences-based rates) to all time periods (pos-sibly excluding a finite prefix) by defining the rate at time t as the exchangerate at t ∗ , where t ∗ is the most recent equilibrium that precedes t . That is,we assume constant rates that are updated occasionally whenever the networkreaches equilibrium. Following Observation 6, our aim is to establish 1:1 exchange rates by reachingperfect balance among currency volumes. Our approach builds upon on thedynamics of the trade within the network, as reflected in the network’s history.While individual preferences may potentially vary in time, in the following weconsider the simple scenario of fixed agents’ preferences , where {(cid:22) v } v ∈ V is fixedeventually, namely after some finite prefix of the currency history in which itmay fluctuate.With the above notions at hand, we can now state our main theorem: Theorem 1.
Let CN , CN , CN , . . . be a currency network history with 2 com-munities CN t = ( C t , C t ) that employs joint egalitarian coin minting. Assume: • Fixed agents’ preferences over the currencies. • Preference-based coin exchange rates. • An efficient network history. • Myopic agents.Then, if it holds that | V \ V || V | ≤ lim t MRS ( CN t ) ≤ | V || V \ V | , then the network history is asymptotically just. Furthermore, it also follows that lim t EX ( CN t ) = 1 . V ∩ V are the only agents that can choose which coin to mint, and, with myopic jointegalitarian minting, they would choose the more valuable coin; thus, if thereare relatively enough agents in the intersection, then, together, they would mintenough coins to set the coin exchange rate right, and asymptotic justice thenfollows. Proof.
Assuming myopic agents, the number of C -coins minted at each timestep t , equals | V t \ V t | + EX ( CN t ) ≥ · | V t ∩ V t | , where EX ( CN t ) ≥ is anindicator function that equals 1 iff C -coins are more valuable then C -coins attime t . Consequently, | C t | t = | C | + (cid:80) ts =1 (cid:0) | V s \ V s | + EX ( CN s ) ≥ · | V s ∩ V s | (cid:1) t → | V \ V | + a t t · | V ∩ V | , where a t := { ≤ s ≤ t : EX ( CN t ) ≥ } denotes the number of time stepsuntil t in which C -coins are more valuable then C -coins. Similarly, we have | C t | t → | V \ V | + (1 − a t t ) · | V ∩ V | . We conclude thatlim t | C t || C t | = | V \ V | + a t t · | V ∩ V || V \ V | + (1 − a t t ) · | V ∩ V | . (5)As | V \ V || V | ≤ lim t MRS ( CN t ) ≤ | V || V \ V | , it follows that there exists aunique 0 ≤ x ≤ t MRS ( CN t ) = | V \ V | + x · | V ∩ V || V \ V | + (1 − x ) · | V ∩ V | . (6)Now, for sufficiently large t , if a t t < x , it follows from Equations 5,6 that | C t || C t | < MRS ( CN t ), hence, EX ( CN t ) = MRS ( CN t ) · | C t || C t | > . That is, C -coins are more valuable then C -coins, thus a t +1 = a t + 1 and a t +1 t +1 > a t t . Similarly, a t t > x corresponds to time steps where C -coins are morevaluable, hence a t +1 t +1 < a t t .We conclude that for sufficiently large t , a t t is monotonically increasing whenbelow x and monotonically decreasing above x . As | a t +1 t +1 − a t t | −→
0, we concludethat this sequence converges to x . It follows that lim t | C t || C t | = lim t MRS ( CN t ),18nd thereforelim t EX ( CN t ) = lim t MRS ( CN t ) · lim t | C t || C t | = lim t MRS ( CN t ) · t MRS ( CN t ) = 1 . In order to establish asymptotic justice, it is enough to note that for suffi-ciently large t : (1) The initial endowment of each agent v (or the exact timeof joining each community) is negligible, and (2) Approximate 1:1 rates hold( EX ( CN t ) ∼ (cid:80) i (cid:104) b it ( v ) − (cid:80) ts =1 (cid:0) rev is ( v ) − exp is ( v ) (cid:1)(cid:105) EX ( CN t ) (cid:80) i | C it | · EX ( CN t ) ∼ const + t | C t | + | C t |−→ const + tt | V | = 1 | V | , which completes the proof. We provide several examples demonstrating the analysis described above.
Example 1 (Two disjoint communities) . Let V and V be two communitiesin some History. If V and V are disjoint, that is, if V ∩ V = ∅ , then thepremise of Theorem 1 boils down to MRS ( (cid:101) b ) = | V || V | , implying that asymptoticjustice holds and coin exchange rate approaches 1:1, provided that the relationsbetween the cardinality of the communities are in perfect balance with their MRS . This is exactly because each agent will mint a coin of their own currency,thus, in particular, the agents of the community with the higher productivitywill “dilute” their currency “exactly” faster.
Example 2 (Two communities with full intersection) . Conversely, in the caseof full intersection, where V = V , the premise of Theorem 1 boils down to0 ≤ MRS ( (cid:101) b ) ≤ ∞ . That is, 1:1 exchange rates are guaranteed regardless ofthe MRS .In other words, a single community that employs an egalitarian mintingregime wrt. two currencies, always satisfies asymptotic justice and eventuallyreaches 1:1 exchange rates: This is exactly because all agents are free to se-lect which coin to mint, thus would always dilute the highest per-coin-valuedcurrency.
Here we analyzed the possibility of a digital currency that realizes equality –there is not a single entity controlling the currency but all genuine agents equally19ontrol the system; distributive justice – all genuine agents (that is, not includingsybils) enjoy an equal share of the value of the digital currency; and grassroots – several independent communities may freely trade while satisfying joint dis-tributive justice. Indeed, as we envision bottom-up growth of communities, ouranalysis, modeled via currency networks, paves the way for interoperability andoffers the possibility of equality and justice at scale.In particular, our main result shows that joint egalitarian coin minting (thatis possible to implement using digital social contracts [2] and in which each agentshall mint only a single coin in each timestep) indeed may lead to pairwise 1:1 ex-change rates and thus to joint distributive justice among genuine identifiers [22]on currency networks satisfying certain conditions, most importantly sufficientintersections between different currency communities.Next we discuss some future research directions.
We analyzed joint egalitarian minting with myopic agents. Here we mentionother possibilities: • Egocentric minting:
Here, every agent mints the coin that maximizesher private preferences. (Note that this coin depends both on the agentpreferences and on the global exchange rate between coins.) • Strategic minting:
Here, agents are rational and sophisticated, in thateach agent may mint the coin that maximizes its private preferences, tak-ing other agent choices into account. • Defensive minting:
Here, in each iteration, each agent mints the cointhat it currently has the least among all currencies it is a member of. (Thisregime can be specified and thus enforced on its parties via a digital socialcontract.)We leave a detailed study of such possibilities for future work. In particular,studying – analytically or via computer simulations – which of these possibilitiesgive rise to 1:1 exchange rate, and what is the rate of convergence, are naturalfuture research directions.In particular, issues of liquidity in such networks, which could be the mainmotivation for community merges, shall be studied, as well as the extension ofTheorem 1 to networks with more than 2 communities.Most importantly is the integration of the two approaches - achieving sybil-resilient growth [19, 22] of a currency community and a currency network, usingthe notion of joint egalitarian coin minting developed here.
Acknowledgements
Ehud Shapiro is the Incumbent of The Harry Weinrebe Professorial Chair ofComputer Science and Biology. We thank the generous support of the Braginsky20enter for the Interface between Science and the Humanities. Nimrod Talmonwas supported by the Israel Science Foundation (ISF; Grant No. 630/19).
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