Cryptocurrency Equilibria Through Game Theoretic Optimization
aa r X i v : . [ q -f i n . M F ] A p r Establishing Cryptocurrency Equilibria ThroughGame Theory
Carey Caginalp a,b and Gunduz Caginalp aa University of Pittsburgh, Mathematics Department, 301Thackeray Hall, Pittsburgh, PA, USA b Chapman University, Economic Science Institute, 1 UniversityDrive, Orange, CA, USAApril 16, 2019
Abstract
We utilize optimization methods to determine equilibria of cryptocur-rencies. A core group, the wealthy, fears the loss of assets that can beseized by a government. Volatility may be influenced by speculators. Thewealthy must divide their assets between the home currency and the cryp-tocurrency, while the government decides the probability of seizing a frac-tion the assets of this group. We establish conditions for existence anduniqueness of Nash equilibria. Also examined is the separate timescaleproblem in which the government policy cannot be reversed, while thewealthy can adjust their allocation in reaction to the government’s desig-nation of probability.
Cryptocurrencies have evolved into a new speculative asset form that differsfrom others in that most represent no intrinsic value; they cannot be redeemedby a financial institution for any amount [1]. The roller-coaster ride of Bit-coin prices from $6,000 to $20,000 back to $6,000, with bounces in between,all during the period from October 2017 to July 2018, was shadowed by othermajor cryptocurrencies [17]. This has been accompanied by the general feelingin government, business and academia that the speculative fever is of concernonly to those who own the cryptocurrencies. There is some justification forthis perspective as the total market capitalization of all cryptocurrencies is nowonly about $131 billion, so that large moves in the cryptocurrency price arenot likely to have a significant impact on the world’s stock and bond markets.However, this impact will present a significant risk to the world’s markets ifthe market capitalization of the cryptocurrencies increases significantly. Dur-ing the dramatic round trip of Bitcoin between $6,000 to $20,000, the marketcapitalization of all cryptocurrencies nearly doubled in six months. Moreover,10,000 Bitcoins were used to purchase two pizzas in the first transaction in 2010 In March 2019, Bitcoin hovered near $3,800. de facto part owner of a corporation, and shareholders can – and do– collectively exercise their rights assured by law. By contrast, a typical cryp-tocurrency does not assure the owner of any rights. Furthermore, there is nocorporate governance at all. The ”miners and developers” – whose names areusually not disclosed – get together and decide essentially on the supply (e.g., byintroducing a related cryptocurrency that they term a ”fork”). Unlike corporateactions in which shareholders can demand a vote, e.g., for directors via a proxybattle, it is not even clear which, if any, nation’s laws apply. Thus, the absenceof an intrinsic value of a cryptocurrency means that the usual traditional financemethods, such as those introduced by Graham ([21], [22]), are inapplicable.Our analysis begins with a game theoretic examination of the motivationsof three groups that are the key players. For a core group, the basic need for acryptocurrency arises from the inadequacy of the home currency and bankingsystem [33]. There are also a significant number of people who are not able toobtain a credit card or even open a bank account in the US, for example [38]. Inmany countries, owning large amounts of the currency can present a significantrisk. There is the possibility of expropriation by the government, sometimesin the guise of a corruption probe. The government could institute policies inwhich inflation is very high, e.g., the extreme example in Venezuela [20] wherehyperinflation decimated any individual savings. Onerous taxes can be placedby the government on the wealthy. Thus a group of people in the world haverational reasons to replace their country’s currency with one that is outside thecontrol of their government or financial institutions, even if it presents some risk.Once it is transferred to cryptocurrency, they would have the option of buyinga more reliable currency or asset in another country. We denote this group by W (the ”wealthy”). Returning to the point made above, there is a substantialamount of the world’s wealth (including individual’s whose assets are not large)that is in this situation. Many of these people, however, are not yet comfortablewith or knowledgeable about cryptocurrencies. As they feel more comfortable,a greater fraction of this wealth may move into cryptocurrencies, inflating themarket capitalization, perhaps to a few percent of the world’s GWP.The second group, D , represents a government that is totalitarian, at leastwith respect to monetary policy, so that its citizens are not free to transfer their2ealth into other, more reliable national currencies. There is a probability, p, that the government can initiate policies that will deprive citizens of a fraction k of their wealth, e.g., by printing money. This possibility is noted by thewealthy, W , who must make a decision on the fraction of their assets, denoted1 − x held in the home currency and the remainder, x , in cryptocurrency, whichpresents risks of its own due to the volatility. The government, D , exhibits riskaversion as with any financial entity. After all, its existence is dependent onobtaining funds from its citizens. A third group, S , consists of the speculators whose sole reason to trade is to profit from the transaction at the expense ofthe less knowledgeable group, W . In a typical situation, the a member of W is trading for the first or second time – having made their money in anotherendeavor – while the speculators are professionals who have made thousands oftrades, and make their living at the expense of novice traders. The speculatorseffectively determine the volatility (see Appendix). Note, however, that ouranalysis would be similar if the volatility were an exogenous variable that is setarbitrarily. While D can set the probability, p , with which the assets can beseized, group W can decide what fraction, x, to convert into the cryptocurrency.We model this situation to find equilibria in two different ways. The firstis to find the Nash equilibrium [28], [18], [29], [35], [13] which is defined as thepoint ( p ∗ , x ∗ ) such that neither party can improve its fortunes by unilateralaction. The underlying assumption is that both parties, W and D , are aware ofthe situation faced by the other, so that they can simultaneously self-optimizewhile assuming that the other party does likewise.In a later section, we utilize the more realistic assumption that while W can make immediate changes (e.g., one day), D must make a decision thatis irrevocable during a longer time (e.g., one year) as policies (e.g., creatinginflation, imposing onerous taxes) are implemented. But in doing so, D mustbe aware that W will self-optimize in its choice of x , knowing p . Thus, bothparties are aware of the different time scales involved in anticipating the otherparty’s decision.The methods we present in this paper are aimed at determining the demandusing optimization. The investors of the cryptocurrency do not have any clearidea how much is the right amount to pay per unit of the cryptocurrency, sothat the demand will determine the trading price as discussed in Appendix Aand [9]. In a setting in which there is one cryptocurrency with a fixed supply,the price will be determined asEquilibrium Trading Price = Demand in DollarsNumber of Units . (1)Analogous methodology can be utilized for multiple cryptocurrencies.One might be led to examining cryptocurrencies in the context of mone-tary policy, but the terminology ”currency” is the main similarity between the In commodities such as gold and oil, there are producers and industrial users who musttrade. In cryptocurrencies, there are no end users other than W who are trade infrequently,unlike industrial users for gold, for example, who are perpetually trading. . Major currencies are established by governments within a well-definedprocess that is governed by law. The identities of those who are responsiblefor monetary policy are known. The citizens of the country can influence therepresentatives who appoint the monetary officials. Finally, if the citizens feelthat the direction of monetary policy is not in their interest, they can elect newrepresentatives. There have been many currencies throughout the world, and itis no accident that the most viable currencies have been those of the countrieswith the most respect for the law and the voices of their citizens.Thus, the theory of monetary policy will be of limited use in the under-standing today’s cryptocurrencies. The aspect of our analysis that is closest tomonetary policy involves the actions of the government, D , which must makea decision on issues such as generating inflation (see, e.g., [2], [3]). Of course,the government in our analysis is one that is very different from the majordemocratic governments that have the more reliable currencies.Since the widespread use of cryptocurrencies is a fairly new phenomenon,the literature is also recent. Many papers have focussed on the blockchaintechnology and its potential for increased speed and safety of transactions. Theintroduction of JP Morgan’s JPM Coin (see Appendix A) is an example ofutilization of this technology without any new economic issues, since JPM Coinwould be redeemable in US Dollars. The economics ofcryptocurrencies havebeen discussed in terms of legal issues [24], valuation [15], security issues [5],[16] and stability [7], [23], [26] , [30] and feasibility [12]. Experiments have alsobeen used to study cryptocurrencies and related issues [14], [19].Our analysis can be viewed in the more general setting of an asset that iseasily traded and out of the reach of the state and other entities. However,the popularity of cryptocurrencies may indicate that there are not so many ofthese. Traditionally, gold has been used as a haven, but it is not always easy toprevent theft. Nevertheless, followers have often noted spikes in gold when thereis political uncertainty in the highly populated and less developed countries.Also, the demand for gold depends upon other factors such as industrial use. The Utility Functions of the Groups
The general framework for this section will be to write the utility functionsof the three groups, modeled on portfolio theory [27], [4] whereby one seeksallocate resources to maximize return while minimizing risk. The general formfor a basic utility function is U = m − d σ where m and σ are the expectationand variance of the outcome, while the parameter d quantifies the risk aversion.The speculators, S , are assumed to have an influence on the volatility andrisk. Even if they had no influence on the volatility, they are likely to profit atthe expense of W , who are likely to be novices. Hence, the role of S is secondary This point is probably clear to anyone who bought Bitcoin at $20 ,
000 and sold it at$6 ,
000 several months later. W .Focusing now on the groups, W and D , we assume that W has a choicebetween the home currency, F, and a cryptocurrency, Y. Any money held in Ffaces the risk that a fixed fraction k ∈ [0 ,
1] will be seized by D with probability p. Thus, the outcome will be (1 − x ) (1 − k ) with probability p and 1 withprobability 1 − p. Letting m F and σ F denote the mean and variance of theinvestment in F, one finds, m F = (1 − k ) p + 1 · (1 − p ) = 1 − kp,σ F = k p (1 − p ) . (2)For the investment in Y, we let m Y and σ Y denote the mean and variance thatwill be determined by the speculators (see Appendix).The utility function for W with the fraction x ∈ [0 ,
1] of its assets in Y andthe remainder in F can then be expressed as U W = m − d σ (3)= xm Y + (1 − x ) m F − d n x σ Y + (1 − x ) σ F + 2 x (1 − x ) Cov [ Y, F ] o . We will assume that the correlation between the two assets, Y and F, is zero,but the analysis can easily be carried out if there is a correlation.The utility function for D can be expressed in terms of the amount that itseizes, i.e., U D = (1 − x ) kp. (4)This can be augmented with a term (as in portfolio theory) that expresses therisk aversion. In particular, one has U D = (1 − x ) kp − d D p , (5)where d D represents the risk aversion of D . Nash Equilibria
We assume the utility functions described in Section 3, using the risk aversionform of U D above (5) . Thus, we need to find ( p ∗ , x ∗ ) such that ∂ x U W ( p ∗ , x ∗ ) = 0 , ∂ p U D ( p ∗ , x ∗ ) = 0 ,∂ xx U W ( p ∗ , x ∗ ) ≤ , ∂ pp U D ( p ∗ , x ∗ ) ≤ . i.e., ( p ∗ , x ∗ ) satisfies the definition of a Nash equilibrium (see e.g., [28],). Briefly,the definition ensures that at ( p ∗ , x ∗ ) neither party can unilaterally improve its5ituation. We compute0 = ∂ p U D ( p, x ) = (1 − x ) k − d D p, (6)0 = ∂ x U W ( p, x ) = m Y − m F + 2 d σ F − d (cid:0) σ Y + σ F (cid:1) x = m Y − kp + 2 d k p (1 − p ) − d (cid:0) σ Y + k p (1 − p ) (cid:1) x. (7)Denote the solution of (6) by x ( p ) and that of (7) by x ( p ) , so that x ( p ) = 1 − d D pk (8) x ( p ) = σ F + (cid:0) d (cid:1) − ( m Y − m F ) σ F + σ Y = k p (1 − p ) + (cid:0) d (cid:1) − ( m Y − kp ) k p (1 − p ) + σ Y . (9)The intersection of x ( p ) and x ( p ) determine a Nash equilibrium. We firstestablish sufficient conditions for at most one equilibrium, and then prove thatunder broad conditions, there exists a Nash equilibrium. Some of these curvesfor sample values of the parameters are illustrated in Figure 2 (see publishedversion for figures). Theorem 1
For p ∈ [0 , / one has x ′ ( p ) ≥ for all values of the parameters,so there can be at most one value of p such that x ( p ) = x ( p ) , and thus atmost one Nash equilibrium for p ∈ [0 , / . Proof.
For convenience set f ( p ) = k p (1 − p ) , c = (cid:0) d (cid:1) − (1 − m Y ) , c = (cid:0) d (cid:1) − k and c = σ Y so that x ( p ) = f ( p ) + c p − c f ( p ) + c and x ′ ( p ) = c k p + c c + ( c + c ) k (1 − p )[ f ( p ) + c ] . (10)Clearly, for p ∈ [0 , /
2] all terms are positive and the conclusion follows.
Theorem 2
If the parameters d, k and m Y satisfy c + c ≤ c i.e., (cid:0) d (cid:1) (1 − m Y ) ≤ k (11) then x ′ ( p ) ≥ for all p ∈ [0 , . Thus there can be at most one Nash equilibriumunder these conditions. roof. For p ∈ [0 , /
2] the result has been established. For p ∈ [1 / ,
1] thenumerator of (10) is c k p + c c + c k (1 − p ) = c c + c k (1 − p ) > , and the result follows.Having determined sufficient conditions for uniqueness, we now focus on es-tablishing existence of Nash equilibrium. Note first that the p − intercept of x ( p ) can be on either side of x = 1 depending on the slope − d D /k. In partic-ular, we let p c := k (cid:0) d D (cid:1) − and consider the two cases separately. Theorem 3 ( a ) If p c := k (cid:0) d D (cid:1) − < and k p c (1 − p c ) + (cid:0) d (cid:1) − ( m Y − p c k ) > , (12) then one has a Nash equilibrium, i.e., there exists ( p ∗ , x ∗ ) ∈ [0 , × [0 , suchthat x ( p ∗ ) = x ( p ∗ ) = x ∗ . ( b ) If p c := k (cid:0) d D (cid:1) − ≥ and (cid:0) d (cid:1) − ( m Y − k ) σ Y + 2 d D k ≥ then one has again a Nash equilibrium.If in addition, equation (11) holds, then the Nash equilibrium ( p ∗ , x ∗ ) isunique. Proof.
Recall that m Y ≤
1. Thus, we have the inequality,1 = x (0) > ≥ x (0) = ( m Y − / (cid:0) d σ Y (cid:1) . We use the Intermediate Value Theorem to establish an intersection between x ( p ) and x ( p ) in the unit square in ( p, x ) space.Case ( a ) . Suppose p c := k (cid:0) d D (cid:1) − < . Then x ( p c ) = k p c (1 − p c ) + (cid:0) d (cid:1) − ( m Y − p c k ) k p c (1 − p c ) + σ Y so that by (12) one has x ( p c ) ≥ x ( p c ). Thus there exists an intersectionof x ( p ) and x ( p ) at ( p ∗ , x ∗ ) ∈ [0 , × [ p c , x ( p c )] ∈ [0 , × [0 , . Case ( b ) . Suppose p c := k (cid:0) d D (cid:1) − > . Then x (1) > , and x ( p ) ∈ [0 , p ∈ [0 , x ( p ) and x ( p ) for p ∈ [0 ,
1] must occuron the unit square provided that x (1) ≥ x (1) . Then the required conditionis x (1) = (cid:0) d (cid:1) − ( m Y − k ) σ Y ≥ − d D k = x (1) . emark 4 The Nash equilibrium may not be unique if the condition above, i.e., (cid:0) d (cid:1) (1 − m Y ) ≤ k of Theorem 3 is violated. An example for two Nash equilibria can be constructedwith the parameters: k = 0 . , m Y = 0 . , d = 2 , d D = 0 . , σ Y = 0 . . The two equilibria are given approximately by ( p ∗ , x ∗ ) = (0 . , . and (0 . , . ,as pictured in Figure 1. Equilibrium with Disparate Time Scales
We consider the situation in which the wealthy, W , can decide on an allocation x immediately, (e.g., within one day), and adjust to the probability, p, while D must set p that cannot be changed for a long time e.g., one year. Thus, D lacks the opportunity to react to the value of x. Both parties are aware of theposition of the other group. Hence, D knows that once he sets p, group W willset x = ˆ x ( p ) in a way that optimizes U W , and that W does not need to beconcerned with any readjustment of p in reaction to their choice of x. Thus, D must examine U W (based on the publicly available information on the volatilityof Y) and decide on a value of p that will optimize U D ( p, ˆ x ( p )) . Within thissetting the utility of D need not be strictly convex in order for an interiormaximum (i.e. such that 0 < p < < x < D has utility that is proportional to the amount it takes, without anyrisk aversion, which can be included with a bit more calculation.We define the quantity A := 2 d σ Y + 1 − m Y (13)which arises naturally in the calculations and is a measure of the risk and ex-pected loss from Y. Thus a higher value of A means Y is less attractive to thewealthy. Theorem 5
Suppose that the utility functions, U W and U D , given by U D = (1 − x ) kpU W = m − d σ = xm Y + (1 − x ) m F − d n x σ Y + (1 − x ) σ F + 2 x (1 − x ) Cov [ Y, F ] o . are known to both parties. Assume that D sets p irrevocably to maximize U D ,while W chooses x to maximize U W based on a knowledge of p. For ≤ A < k the optimal choice of x given p is ˆ x ( p ) := m Y − m F d ( σ Y + σ F ) + σ F σ Y + σ F (14)8 ith m Y and σ F given by (2) , and the optimal value of p is given by p ∗ := σ Y k ( k − A ) s Aσ Y ( k − A ) − ! . (15) Thus the optimal point is ( p, x ) = ( p ∗ , ˆ x ( p ∗ )) . The value of maximum, x ∗ =ˆ x ( p ) is if the right hand side of (14) is negative, and if the right hand sideexceeds . Remark 6
Note that given p the optimal fraction of assets in the cryptocur-rency is a sum of the relative variance of the home currency, i.e., σ F as afraction of σ Y + σ F plus the difference in expected loss from the home currency,i.e., − m F minus the expected loss from the cryptocurrency, − m Y scaledby a risk aversion factor. Thus the fraction invested in the cryptocurrency in-creases as the expected loss and the variance of the home currency increases,and conversely. Remark 7
Note that one obtains an interior maximum with a linear utilityfunction for U D in this type of optimization, i.e., even though D is interested inpure maximization of its revenue. Proof.
Using (3) we determine the maximum of U W for a fixed p, so that0 = ∂ x U W ( p, x ) = m Y − m F + 2 d σ Y − d (cid:0) σ Y + σ F (cid:1) x. Noting that ∂ xx U W ( p, x ) = − d (cid:0) σ Y + σ F (cid:1) < U D is maximizedby ˆ x ( p ) given by (14) provided ˆ x ( p ) ∈ [0 , m Y − m F d + σ F < x ( p ) = 0 ,m Y − m F d + σ F > x ( p ) = 1 . Thus, ˆ x ( p ) interpolates between 0 and 1 by favoring Y if the relative riskof F (measured by σ F (cid:0) σ Y + σ F (cid:1) − is large in comparison with the relativelygreater expected loss in Y (scaled by the sum of the variances).In anticipation, D optimizes U D ( p, x ( p )) . We thus compute, with B := σ Y ,0 = 2 d k ∂ p U D ( p, x ( p )) = ∂ p Ap − kp B + k p (1 − p )= ( A − kp ) (cid:2) B + k p (1 − p ) (cid:3) − (cid:0) Ap − k p (cid:1) (cid:2) k (1 − p ) (cid:3) [ B + k p (1 − p )] . This identity is equivalent to Q ( p ) := AB − Bkp + k ( A − k ) p = 0 . (16)9ote that A > p ∗ = Bk ( k − A ) r AB ( k − A ) − ! . One can verify that p ∗ ∈ [0 , p ∗ , x ∗ ) = ( p ∗ , ˆ x ( p ∗ )) is theoptimal point. Remark 8
Case A = 0 . By definition (13) we see that − m Y = 0 . Note that p ∗ = 0 follows from the identity above. Using the definition and the computedvalues of m F = 1 − kp and σ F = k p (1 − p ) we write ˆ x ( p ) = m Y − kp d ( σ Y + k p (1 − p )) + k p (1 − p ) σ Y + k p (1 − p )ˆ x (0) = m Y − d σ Y = 0 . In other words, when A = 0 there is no risk and no expected loss in the cryptocur-rency. Thus, D realizes that any nonzero value of p will result in W investingnothing in the home currency, F.Case A = k. The quadratic numerator (16) is then Q ( p ) = AB − Bkp sothat one has p ∗ = 1 / . Case k < A ≤ k . By considering a small positive perturbation, δ, of A wesee that Q (cid:0) (cid:1) > so that the positive region of ∂ p U D is extended toward theright as A increases.Case A ≥ k. Since p ≤ one has Q ( p ) ≥ B ( A − k ) + k p ( A − k ) > , so ∂ p U D > and the maximum is thus p ∗ = 1 . Conclusion
We have examined the optimal strategies for the key parties (those with savingsat risk, a dictatorial government and speculators) involved explicitly or implic-itly in the formation of an equilibrium for cryptocurrencies. The second methodinvolves different time scales in determining equilbrium that differs from themore common Nash equilibrium, in which all parties can readjust their positionscontinuously. As described in Section 4 this can be utilized for many realisticsituations in which one entity such as a government optimizes by placing con-ditions such as taxes, tariffs, fees, etc., or policies that cannot be reversed oradjusted in a short time. In general, optimization in this form favors the groupthat can make immediate and continuous adjustments.Each of the methods are based on parameters that can be estimated. Forexample, the variance of cryptocurrencies can be determined from the tradingdata. Parameters such as k (the fraction of assets seized) can be estimated10rom the policies of the government. An assessment of these quantities thenleads to estimates of the amount of money that is likely to be used to purchasecryptocurrencies in the aggregate. Using the ideas summarized in [9] one canthen also evaluate average price changes of cryptocurrencies as well as the totalmarket capitalization of cryptocurrencies. The evolution of the latter is crucialin understanding the implications of instability of cryptocurrencies on othersectors of the world’s economy.Major governments have often appeared confused and lethargic in their re-sponse to cryptocurrency policy, even insofar as deciding whether it is impor-tant or not. There is also little understanding of the conditions under which acryptocurrency could be either beneficial or detrimental to global society. Theperspective of our paper suggests that a cryptocurrency price will vary widelydepending on the demand that in turn is based on policies of countries wheremonetary policy and laws, in general are less developed. Together with thefact that cryptocurrencies cannot be redeemed for any asset, one cannot ex-pect much stability. However, given a mechanism whereby a cryptocurrency isessentially backed by real assets (e.g., a structure similar to Exchange TradedFunds) one would have stability since arbitrageurs would take advantage of anydiscrepanicies. This could be linked of course to a single currency such as theUS Dollar, but would only be a trading token in this case.However, one can design a cryptocurrency that would essentially grow withthe world’s economy, unlike a commodity such as gold. A simple example wouldbe that the cryptocurrency could be reedemable in units of the Gross WorldProduct in terms of a basket of major currencies, so that each cryptocurrencycould be redeemed for one trillionth of the GWP in Dollars, Euros and Yen.Such an instrument would offer much greater stability and could be used as asubstitute currency that is independent of any government. As shown in ouranalysis, as the volatility risk would diminish, and those whose assets in thehome currency are at risk would place more of their assets into this cryptocur-rency. Thus the fraction, x, placed in the cryptocurrency would increase. Inparticular, the equilibrium point ( p ∗ , x ∗ ) would feature x ∗ that is larger and p ∗ that is smaller. This would mean that the citizens have greater economicfreedom, and financially totalitarian regimes would have smaller resources. Insummary, the creation of a viable cryptocurrency with intrinsic value wouldhave less volatility, and thereby reduce the fraction of savings in the home cur-rency that is under threat by a totalitarian government, whose existence is oftencontingent on raising money in this manner. References
The authors thank the Economic Science Institute at Chapman University andthe Hayek Foundation for their support. Discussions with Prof. Gabriele Cam-era are very much appreciated.
Appendix A: Fundamental Value and LiquidityValue
There is a temptation to stipulate that the only valuation of an asset is thetrading price, as this price reflects the preferences and values of the buyers andsellers via the intersection of the supply and demand curves. In principle thereis nothing wrong with this perspective except that important phenomena areleft unexplained, and significant risks are mischaracterized as rare or low risk.One way to examine different aspects of price or value is through the lab-oratory experiments such as the ”bubbles” experiments introduced by [36] inwhich an asset pays a dividend with expectation 24 cents at the end of each of15 periods, and is then worthless. The value of this asset at the end of Period k is clearly given in dollars by P a := 3 . − (0 . k. (17)In numerous experiments, prices often started well below (17) and soared farabove this fundamental value, and eventually crashed. This persisted even inexperiments in which the dividend payout had no randomness at all [31]. Itseems difficult to deny that P a is a meaningful and useful quantity, particularlysince it is a quantity that the trader can be assured of receiving. For example,purchasing at a trading price early in the experiment that is often below P a ensures the trader will gain a specific profit. If one ignores the intrinsic value, P a , one would conclude that the risk is the same at any price, and likely incura large loss during the course of the experiment. Also, it has been noted [8],that in these experiments, there is a third quantity with units of price per share.This is the ”liquidity value,” L, defined as the ratio of the sum of all cash in theexperiment divided by the total of all shares. Experiments that were designedto test this concept [10, 11] showed that the liquidity value has a primary role indetermining the size of the bubble. In fact the peak of the bubble was often closeto L. In other words, when traders pay little attention to fundamental value, or15f the fundamental value is not clear, the price drifts toward the liquidity value[8]. At the opposite extreme, for short term government bonds, the calculationof fundamental value is clear, as the owner is assured of a particular sum at aparticular time a few months in the future. The trading price generally tradesvery close to this fundamental value since there are many arbitrageurs whoexploit any deviations.The vast majority of cryptocurrencies do not have any redemption value,they pay no dividends, and they do not endow holders with voting power overan entity with assets (as do stocks, for example). Thus, classical finance cal-culations involving expected dividends, book value, replacement value, etc., allyield a fundamental value of zero. One exception is JP Morgan’s JPM Coin,announced in February 2019 which would be redeemable in US dollars. Theredemption price would yield the guaranteed value, which would be P a , thefundamental value, so long as the investors are confident in JP Morgan’s abilityto fulfil its commitment.Ignoring fundamental or intrinsic value often leads to disastrous practicalresults, as investors discovered with the internet stocks in 1999, or the Japanesemarket in 1990, for example, when standard calculations of stock value [21, 22]showed a large discrepancy between the trading price and the fundamental value.Similarly, in theoretical development, neglecting either the fundamental value, P a , or the liquidity value, L, will have the same consequences as overlooking anyother important quantity in modeling economics problems. One obtains someresults that are not consistent with observations, and has no way to rectify thesituation.Although one cannot calculate a positive P a for the typical cryptocurrency,people are paying for these units, so that they must see some value in it. Theperspective that fundamental value must be the trading price, renders the equiv-alence a tautology. As discussed above, the result is that important phenomenaare left unexplained, and an even a basic understanding of the likely price evo-lution becomes more difficult.Since cryptocurrencies have no fundamental value, prices will naturally drifttoward the liquidity value, which will be given by the total amount of cashavailable for the cryptocurrency (i.e., demand) divided by the number of units[10, 11].The absence of a non-zero fundamental value means that price will be setby the supply (which is fixed, for example, for Bitcoin) and demand in ac-cordance with equation (1). Thus it is a calculation of demand that is key tounderstanding equilibrium price. Appendix B
1. We consider first the role of ”pure” speculators who have no control ofthe type of trading or auction, the rules of the exchange, the enforcement ofthe rules, the display of orders, and the flow of information. Volatility arisesendogenously due to the various trading strategies, such as trend following, and16andom events that motivate any of the traders. For many first time or novicetraders, there is a tendency to overreact, and to chase a trend, or hop onto afad. In the case of cryptocurrencies, which lack any fundamental value, anynews is likely to result in an overreaction. Thus volatility can be expectedto be high in the absence of any anchor. For example, Treasury bills offera guaranteed payout, so that a small deviation from the certain payout duewithin a few months would be exploited by arbitrageurs and the price wouldbe restored close to its intrinsic value. The speculators in many markets havea better understanding (compared to novice traders) of the factors that moveprices within a short time scale. Speculators are generally believed to lowervolatility [6], as they use their capital to buy when prices move unjustifyablylower. Of course, when prices exhibit very low volatility, there is no financialincentive for speculators to trade. Consequently, in an idealized setting, theshort-term volatility level will be established as the minimum value at whichspeculators find adequate profits after costs.2. Next we consider ”speculators” in less established markets in which therule makers, market makers, news makers are all essentially the same group.In most developed markets such as the New York Stock Exchange (NYSE) andmajor commodity exchanges there are precise rules designed to promote fairnessand ease of trading that have been developed over many years. An example isthe NYSE rule that if there are two orders to purchase a stock, it is the higherone that prevails. Surprisingly to novice traders, this is not usually a featureof most markets. In many markets there are ”market makers” who are entitledto buy the stock for their own account at a lower price, even though a higherbid has been placed by a retail customer or trader. The rules of each exchangeendow the market makers and market specialists with the power to buy and sellon their own account. In many well-developed exchanges, there are rules agains”front-running” whereby insiders buy on their own accounts as they becomeaware of a set of large orders that are entering the market. Another exampleon major exchanges involves ”not held” trades that are placed with the marketmakers but are not displayed. The intention here is that a large order to sellcould prompt further selling by less informed traders. By contrast, in a lessdeveloped market environment, a market insider can place a large order (butabove the market price) that will immediately lead to lower prices, whereuponhe can deftly purchase.Novice traders usually make numerous assumptions relating to fairness onthe nature of market rules and procedures. Unfortunately, these are generallyfalse for less developed markets that cater to inexperienced traders. The wishfulthinking of new traders seeking quick riches (or escape from a currency) providesfor a healthy income for those dominating these markets in terms of making therules (if there are any at all) and using their capital to control the volatility. Formany of the cryptocurrencies, for example, it is not even clear what the rulesare, or where they would be enforced. Thus, in an under-developed market, agroup of participants that controls the rules of trading has numerous tools at itsdisposal to adjust volatility. Even the hours of trading have a strong impact on17olatility. For example, it is well-known that trading around the clock leads totimes periods of low volume so that a few trades can move prices much more thanduring actively traded times. On the other hand, in an exchange in which thereis a single trade each day at a specified time, the maximum minus minimumprice within one week is likely to be much lower than in 24 hour trading.Another feature that can influence prices is the extent to which informationon orders is displayed. The ”order book” displays the array of bids and asksfor the asset in continuous time. Whether or not the order book is displayeddepends upon the rules of the exchange. Also, on some exchanges, the marketmaker can choose to display only some of the orders. In laboratory experiments[11] it was shown that bubbles are tempered by the display of the completeorder book.Related to the order book are the rules under which the market maker canbuy for his own account. While ”front running” – the practice of buying for one’sown account ahead of a large order – or ”shadowing” – buying the same assetsas a particular trader – are banned in some of the most developed exchanges,one cannot assume that they will be prohibited universally.Of course, all of this assumes that there is a real market in which bids andasks are matched with some rule. In many cases purchases and sales are madethrough one entity that buys and sells for its own account, thereby granting itselfa generous profit as the middleman. Even in large brokerages it is common formonthly statements to disclose ”we make a market in this stock” that indicatesthe bid/ask spread is whatever the company designates as revenue for itself.From the perspective of the individual trader, the bid/ask spread, of course,adds to the cost and volatility of the transaction.
Appendix C
In examining the choice faced by W we assume that one option is to remainin the home currency, F, and the other to buy the cryptocurrency with theobjective of later selling in order to buy other assets such as a more reliablecurrency, gold, etc.The group W experiences a loss or gain on these transactions with the spec-ulators, group S , which itself has a non-linear utility function reflecting thatfact that high volatility is good for profits up to a point after which it has anegative impact. Thus, one has the following utility function for group S : U S = a V − a V where V represents the volatility or variance, for example, and a , a are positiveconstants. Hence, there will be a maximum value V = V m that maximizes theutility of the speculators. This can be viewed as a fixed quantity from theperspective of W .The mean, m Y , and variance, σ Y of group W ’s investments in Y can becalculated based on V m and the other parameters that describe the trading.In particular, we assume that there is a probability q (presumably small) that18 will profit, and that their wealth will increase from 1 to 1 + r V m and aprobability 1 − q that it will decrease from 1 to 1 − r V m where r > r > . In other words, there is a small probabilty, q , that W will benefit by r V m (asa fraction of their original wealth) and a larger probability, 1 − q , that theywill lose a larger sum r V m . The loss is proportional to the volatility as theprofessional speculators are able to exploit the ups and downs of the trading atthe expense of the inexperienced W .The mean and variance of the outcome are then m Y = q (1 + r V m ) + (1 − q ) (1 − r V m ) ,σ Y = q (1 − q ) ( r + r ) V m . In other words, there is large probability that W will take a loss on the transac-tion. One can consider more general probability distributions for W ’s profits andlosses, but ultimately, the two quantities that are relevant for its utility function U W are given by m Y and σ Y that one can regard as empirical observables.19that one can regard as empirical observables.19