Establishing Cryptocurrency Equilibria Through Game Theory

We utilize optimization methods to determine equilibria of cryptocurrencies. A core group, the wealthy, fears the loss of assets that can be seized by a government. Volatility may be inﬂuenced by speculators. The wealthy must divide their assets between the home currency and the cryptocurrency, while the government decides the probability of seizing a fraction the assets of this group. We establish conditions for existence and uniqueness of Nash equilibria. Also examined is the separate timescale problem in which the government policy cannot be reversed, while the wealthy can adjust their allocation in reaction to the government’s designation of probability.

only about $131 billion, so that large moves in the cryptocurrency price are not 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 if the market capitalization of the cryptocurrencies increases significantly. During the dramatic round trip of Bitcoin between $6,000 to $20,000, the market capitalization of all cryptocurrencies nearly doubled in six months. Moreover, 10,000 Bitcoins were used to purchase two pizzas in the first transaction in 2010 [32]. If people gradually become more comfortable with cryptocurrencies, as they did with internet shopping, it is likely that the market capitalization could grow to a few percent of the $75 trillion Gross World Product (GWP) as the fraction of the world's savings that is under threat by government seizure, high inflation, etc., is certainly at least this fraction (as discussed further below). At this point, large price changes in cryptocurrencies would likely have an impact on the broader markets.
It is thus important to understand the factors behind the market capitalization and price of cryptocurrencies. With all other assets there is some theoretical methodology to determine the value, which is at least a first step estimating the trading price. For example, the value of a stock is assessed by measures such as the expected dividend stream (see Graham [21], [22], Luenberger [27], Bodie et al. [4], and Wolpert et. al. [39]). Even beyond these calculations, a shareholder is de facto part owner of a corporation, and shareholders can -and do -collectively exercise their rights assured by law. By contrast, a typical cryptocurrency does not assure the owner of any rights. Furthermore, there is no corporate governance at all. The "miners and developers" -whose names are usually not disclosed -get together and decide essentially on the supply (e.g., by introducing a related cryptocurrency that they term a "fork"). Unlike corporate actions in which shareholders can demand a vote, e.g., for directors via a proxy battle, it is not even clear which, if any, nation's laws apply. Thus, the absence of an intrinsic value of a cryptocurrency means that the usual traditional finance methods, such as those introduced by Graham ( [21], [22]), are inapplicable.
Our analysis begins with a game theoretic examination of the motivations of three groups that are the key players. For a core group, the basic need for a cryptocurrency arises from the inadequacy of the home currency and banking system [33]. There are also a significant number of people who are not able to in the guise of a corruption probe. The government could institute policies in which inflation is very high, e.g., the extreme example in Venezuela [20] where hyperinflation decimated any individual savings. Onerous taxes can be placed by the government on the wealthy. Thus a group of people in the world have rational reasons to replace their country's currency with one that is outside the control of their government or financial institutions, even if it presents some risk.
Once it is transferred to cryptocurrency, they would have the option of buying a 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 substantial amount 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 comfortable with or knowledgeable about cryptocurrencies. As they feel more comfortable, a greater fraction of this wealth may move into cryptocurrencies, inflating the market capitalization, perhaps to a few percent of the world's GWP.
The second group, D, represents a government that is totalitarian, at least with respect to monetary policy, so that its citizens are not free to transfer their wealth 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 the wealthy, W, who must make a decision on the fraction of their assets, denoted 1 − x held in the home currency and the remainder, x, in cryptocurrency, which presents risks of its own due to the volatility. The government, D, exhibits risk aversion as with any financial entity. After all, its existence is dependent on obtaining funds from its citizens. A third group, S, consists of the speculators 2 whose sole reason to trade is to profit from the transaction at the expense of the 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 another endeavor -while the speculators are professionals who have made thousands of trades, and make their living at the expense of novice traders. The speculators effectively determine the volatility (see Appendix). Note, however, that our analysis would be similar if the volatility were an exogenous variable that is set arbitrarily. While D can set the probability, p, with which the assets can be seized, group W can decide what fraction, x, to convert into the cryptocurrency.
We model this situation to find equilibria in two different ways. The first is to find the Nash equilibrium [28], [18], [29], [35], [13] which is defined as the point (p * , x * ) such that neither party can improve its fortunes by unilateral action. The underlying assumption is that both parties, W and D, are aware of the situation faced by the other, so that they can simultaneously self-optimize while 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 that is irrevocable during a longer time (e.g., one year) as policies (e.g., creating inflation, imposing onerous taxes) are implemented. But in doing so, D must be aware that W will self-optimize in its choice of x, knowing p. Thus, both parties are aware of the different time scales involved in anticipating the other party's decision.
The methods we present in this paper are aimed at determining the demand using optimization. The investors of the cryptocurrency do not have any clear idea how much is the right amount to pay per unit of the cryptocurrency, so that the demand will determine the trading price as discussed in Appendix A and [9]. In a setting in which there is one cryptocurrency with a fixed supply, the price will be determined as Equilibrium Trading Price = Demand in Dollars Number of Units .
Analogous methodology can be utilized for multiple cryptocurrencies.
One might be led to examining cryptocurrencies in the context of monetary policy, but the terminology "currency" is the main similarity between the cryptocurrencies and the US Dollar, Euro, Yen, etc. The differences are pro- Thus, the theory of monetary policy will be of limited use in the understanding today's cryptocurrencies. The aspect of our analysis that is closest to monetary policy involves the actions of the government, D, which must make a 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 major democratic 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 blockchain technology and its potential for increased speed and safety of transactions. The introduction of JP Morgan's JPM Coin (see Appendix A) is an example of utilization of this technology without any new economic issues, since JPM Coin would be redeemable in US Dollars. The economics ofcryptocurrencies have been discussed in terms of legal issues [24], valuation [15], security issues [5], [16] and stability [7], [23], [26] , [30] and feasibility [12]. Experiments have also been used to study cryptocurrencies and related issues [14], [19].
Our analysis can be viewed in the more general setting of an asset that is easily 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 of these. Traditionally, gold has been used as a haven, but it is not always easy to prevent theft. Nevertheless, followers have often noted spikes in gold when there is 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 functions of the three groups, modeled on portfolio theory [27], [4] whereby one seeks allocate resources to maximize return while minimizing risk. The general form for a basic utility function is U = m − d 2 σ 2 where m and σ 2 are the expectation and variance of the outcome, while the parameter d 2 quantifies the risk aversion.
The speculators, S, are assumed to have an influence on the volatility and risk. Even if they had no influence on the volatility, they are likely to profit at the expense of W, who are likely to be novices. Hence, the role of S is secondary (and discussed in the Appendix) as they create an expected loss and a variance for W.
Focusing now on the groups, W and D, we assume that W has a choice between the home currency, F, and a cryptocurrency, Y. Any money held in F faces 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 with probability 1 − p. Letting m F and σ 2 F denote the mean and variance of the investment in F, one finds, For the investment in Y, we let m Y and σ 2 Y denote the mean and variance that will be determined by the speculators (see Appendix).
The utility function for W with the fraction x ∈ [0, 1] of its assets in Y and the remainder in F can then be expressed as 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 it seizes, i.e., This can be augmented with a term (as in portfolio theory) that expresses the risk aversion. In particular, one has where d 2 D represents the risk aversion of D.

Nash Equilibria
We assume the utility functions described in Section 3, using the risk aversion form of U D above (5) . Thus, we need to find (p * , x * ) such that 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 its situation. We compute Denote the solution of (6) by x 1 (p) and that of (7) by x 2 (p) , so that The intersection of x 1 (p) and x 2 (p) determine a Nash equilibrium. We first establish sufficient conditions for at most one equilibrium, and then prove that under broad conditions, there exists a Nash equilibrium. Some of these curves for sample values of the parameters are illustrated in Figure 1. Proof.
Clearly, for p ∈ [0, 1/2] all terms are positive and the conclusion follows.
Theorem 2 If the parameters d, k and m Y satisfy . Thus there can be at most one Nash equilibrium under these conditions. When there is an intersection in the range (p * , x * ) ∈ (0, 1) 2 , we have a Nash equilibrium. It is possible to have other intersections outside of this box, but such "equilibria" are irrelevant and their behavior will lead to boundary cases such as p = 0 or p = 1.
Having determined sufficient conditions for uniqueness, we now focus on establishing existence of Nash equilibrium. Note first that the p−intercept of x 1 (p) can be on either side of x = 1 depending on the slope −2d 2 D /k. In particular, we let p c := k 2d 2 D −1 and consider the two cases separately.
then one has a Nash equilibrium, i.e., there exists then one has again a Nash equilibrium.
If in addition, equation (11) holds, then the Nash equilibrium (p * , x * ) is unique.
Proof. Recall that m Y ≤ 1. Thus, we have the inequality, We use the Intermediate Value Theorem to establish an intersection between so that by (12) one has x 2 (p c ) ≥ 0 = x 1 (p c ). Thus there exists an intersection of x 1 (p) and for p ∈ [0, 1]. Thus, an intersection of x 1 (p) and x 2 (p) for p ∈ [0, 1] must occur on the unit square provided that x 2 (1) ≥ x 1 (1) . Then the required condition is

Remark 4
The Nash equilibrium may not be unique if the condition above, i.e., of Theorem 3 is violated. An example for two Nash equilibria can be constructed with the parameters: The two equilibria are given approximately by (p * , x * ) = (0.88, 0.68) and (0.96, 0.65), as pictured in Figure 2.

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 the position of the other group. Hence, D knows that once he sets p, group W will set x =x (p) in a way that optimizes U W , and that W does not need to be concerned 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 volatility of Y) and decide on a value of p that will optimize U D (p,x (p)) . Within this setting the utility of D need not be strictly convex in order for an interior maximum (i.e. such that 0 < p < 1 and 0 < x < 1). Thus, we consider the case in which D has utility that is proportional to the amount it takes, without any risk aversion, which can be included with a bit more calculation.
We define the quantity

11
Electronic copy available at: https://ssrn.com/abstract=3359845 which arises naturally in the calculations and is a measure of the risk and expected loss from Y. Thus a higher value of A means Y is less attractive to the wealthy.
Theorem 5 Suppose that the utility functions, U W and U D , given by 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 0 ≤ A < k the optimal choice of x given p iŝ with m Y and σ 2 F given by (2) , and the optimal value of p is given by Thus the optimal point is (p, x) = (p * ,x (p * )) . The value of maximum, x * = x (p) is 0 if the right hand side of (14) is negative, and 1 if the right hand side exceeds 1.

Remark 6
Note that given p the optimal fraction of assets in the cryptocurrency is a sum of the relative variance of the home currency, i.e., σ 2 F as a fraction of σ 2 Y + σ 2 F plus the difference in expected loss from the home currency, i.e., 1 − m F minus the expected loss from the cryptocurrency, 1 − m Y scaled by a risk aversion factor. Thus the fraction invested in the cryptocurrency increases 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 utility function for U D in this type of optimization, i.e., even though D is interested in pure maximization of its revenue.
Proof. Using (3) we determine the maximum of U W for a fixed p, so that Noting that ∂ xx U W (p, x) = −2d 2 σ 2 Y + σ 2 F < 0 we see that U D is maximized byx (p) given by (14) providedx (p) ∈ [0, 1]. In the following two cases the maximum is on the boundary: Thus,x (p) interpolates between 0 and 1 by favoring Y if the relative risk of F (measured by σ 2 F σ 2 Y + σ 2 In anticipation, D optimizes U D (p, x (p)) . We thus compute, with B := σ 2 Y , This identity is equivalent to Note that A > 0 by assumption. The positive root of equation (16) is One can verify that p * ∈ [0, 1], and conclude that (p * , x * ) = (p * ,x (p * )) is the optimal point.
In other words, when A = 0 there is no risk and no expected loss in the cryptocurrency. Thus, D realizes that any nonzero value of p will result in W investing nothing in the home currency, F.
Case A = k. The quadratic numerator (16) is then Q (p) = AB − 2Bkp so that one has p * = 1/2. Case k < A ≤ 2k. By considering a small positive perturbation, δ, of A we see that Q 1 2 > 0 so that the positive region of ∂ p U D is extended toward the right as A increases.
Case A ≥ 2k. Since p ≤ 1 one has so ∂ p U D > 0 and the maximum is thus p * = 1.

Conclusion
We have examined the optimal strategies for the key parties (those with savings at risk, a dictatorial government and speculators) involved explicitly or implicitly in the formation of an equilibrium for cryptocurrencies. The second method involves different time scales in determining equilbrium that differs from the more common Nash equilibrium, in which all parties can readjust their positions continuously. As described in Section 4 this can be utilized for many realistic situations in which one entity such as a government optimizes by placing conditions such as taxes, tariffs, fees, etc., or policies that cannot be reversed or adjusted in a short time. In general, optimization in this form favors the group that can make immediate and continuous adjustments.
Each of the methods are based on parameters that can be estimated. For example, the variance of cryptocurrencies can be determined from the trading data. Parameters such as k (the fraction of assets seized) can be estimated from the policies of the government. An assessment of these quantities then leads to estimates of the amount of money that is likely to be used to purchase cryptocurrencies in the aggregate. Using the ideas summarized in [9] one can then also evaluate average price changes of cryptocurrencies as well as the total market capitalization of cryptocurrencies. The evolution of the latter is crucial in understanding the implications of instability of cryptocurrencies on other sectors of the world's economy. tant or not. There is also little understanding of the conditions under which a cryptocurrency could be either beneficial or detrimental to global society. The perspective of our paper suggests that a cryptocurrency price will vary widely depending on the demand that in turn is based on policies of countries where monetary policy and laws, in general are less developed. Together with the fact that cryptocurrencies cannot be redeemed for any asset, one cannot expect much stability. However, given a mechanism whereby a cryptocurrency is essentially backed by real assets (e.g., a structure similar to Exchange Traded Funds) one would have stability since arbitrageurs would take advantage of any discrepanicies. This could be linked of course to a single currency such as the US Dollar, but would only be a trading token in this case.
However, one can design a cryptocurrency that would essentially grow with the world's economy, unlike a commodity such as gold. A simple example would be that the cryptocurrency could be reedemable in units of the Gross World Product in terms of a basket of major currencies, so that each cryptocurrency could 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 a substitute currency that is independent of any government. As shown in our analysis, as the volatility risk would diminish, and those whose assets in the home currency are at risk would place more of their assets into this cryptocurrency. Thus the fraction, x, placed in the cryptocurrency would increase. In particular, the equilibrium point (p * , x * ) would feature x * that is larger and p * that is smaller. This would mean that the citizens have greater economic freedom, and financially totalitarian regimes would have smaller resources. In summary, the creation of a viable cryptocurrency with intrinsic value would have less volatility, and thereby reduce the fraction of savings in the home currency that is under threat by a totalitarian government, whose existence is often contingent on raising money in this manner.
In numerous experiments, prices often started well below (17) and soared far above this fundamental value, and eventually crashed. This persisted even in experiments in which the dividend payout had no randomness at all [31]. It seems difficult to deny that P a is a meaningful and useful quantity, particularly since 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 incur a 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 the experiment divided by the total of all shares. Experiments that were designed to test this concept [10,11] showed that the liquidity value has a primary role in determining the size of the bubble. In fact the peak of the bubble was often close to L. In other words, when traders pay little attention to fundamental value, or if the fundamental value is not clear, the price drifts toward the liquidity value [8]. At the opposite extreme, for short term government bonds, the calculation of fundamental value is clear, as the owner is assured of a particular sum at a particular time a few months in the future. The trading price generally trades very close to this fundamental value since there are many arbitrageurs who exploit 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 over an entity with assets (as do stocks, for example  [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 any other important quantity in modeling economics problems. One obtains some results that are not consistent with observations, and has no way to rectify the situation.
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. The perspective that fundamental value must be the trading price, renders the equivalence a tautology. As discussed above, the result is that important phenomena are left unexplained, and an even a basic understanding of the likely price evolution becomes more difficult.
Since cryptocurrencies have no fundamental value, prices will naturally drift toward the liquidity value, which will be given by the total amount of cash available 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 set by the supply (which is fixed, for example, for Bitcoin) and demand in accordance with equation (1). Thus it is a calculation of demand that is key to understanding equilibrium price.

Appendix B
1. We consider first the role of "pure" speculators who have no control of the type of trading or auction, the rules of the exchange, the enforcement of the rules, the display of orders, and the flow of information. Volatility arises endogenously due to the various trading strategies, such as trend following, and random events that motivate any of the traders. For many first time or novice traders, there is a tendency to overreact, and to chase a trend, or hop onto a fad. In the case of cryptocurrencies, which lack any fundamental value, any news is likely to result in an overreaction. Thus volatility can be expected to be high in the absence of any anchor. For example, Treasury bills offer a guaranteed payout, so that a small deviation from the certain payout due within a few months would be exploited by arbitrageurs and the price would be restored close to its intrinsic value. The speculators in many markets have a better understanding (compared to novice traders) of the factors that move prices within a short time scale. Speculators are generally believed to lower volatility [6], as they use their capital to buy when prices move unjustifyably lower. Of course, when prices exhibit very low volatility, there is no financial incentive for speculators to trade. Consequently, in an idealized setting, the short-term volatility level will be established as the minimum value at which speculators find adequate profits after costs.
2. Next we consider "speculators" in less established markets in which the rule makers, market makers, news makers are all essentially the same group.
In most developed markets such as the New York Stock Exchange (NYSE) and major commodity exchanges there are precise rules designed to promote fairness and ease of trading that have been developed over many years. An example is the NYSE rule that if there are two orders to purchase a stock, it is the higher one that prevails. Surprisingly to novice traders, this is not usually a feature of most markets. In many markets there are "market makers" who are entitled to buy the stock for their own account at a lower price, even though a higher bid has been placed by a retail customer or trader. The rules of each exchange endow the market makers and market specialists with the power to buy and sell on their own account. In many well-developed exchanges, there are rules agains "front-running" whereby insiders buy on their own accounts as they become aware of a set of large orders that are entering the market. Another example on major exchanges involves "not held" trades that are placed with the market makers but are not displayed. The intention here is that a large order to sell could prompt further selling by less informed traders. By contrast, in a less developed market environment, a market insider can place a large order (but above the market price) that will immediately lead to lower prices, whereupon he can deftly purchase. Electronic copy available at: https://ssrn.com/abstract=3359845 volatility. For example, it is well-known that trading around the clock leads to times periods of low volume so that a few trades can move prices much more than during actively traded times. On the other hand, in an exchange in which there is a single trade each day at a specified time, the maximum minus minimum price 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 information on orders is displayed. The "order book" displays the array of bids and asks for the asset in continuous time. Whether or not the order book is displayed depends upon the rules of the exchange. Also, on some exchanges, the market maker 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 complete order book.
Related to the order book are the rules under which the market maker can buy for his own account. While "front running" -the practice of buying for one's own account ahead of a large order -or "shadowing" -buying the same assets as 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 and asks are matched with some rule. In many cases purchases and sales are made through one entity that buys and sells for its own account, thereby granting itself a generous profit as the middleman. Even in large brokerages it is common for monthly statements to disclose "we make a market in this stock" that indicates the 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 remain in the home currency, F, and the other to buy the cryptocurrency with the objective of later selling in order to buy other assets such as a more reliable currency, gold, etc.
The group W experiences a loss or gain on these transactions with the speculators, group S, which itself has a non-linear utility function reflecting that fact that high volatility is good for profits up to a point after which it has a negative impact. Thus, one has the following utility function for group S: where V represents the volatility or variance, for example, and a 1 , a 2 are positive constants. Hence, there will be a maximum value V = V m that maximizes the utility of the speculators. This can be viewed as a fixed quantity from the perspective of W.
The mean, m Y , and variance, σ 2 Y of group W's investments in Y can be calculated based on V m and the other parameters that describe the trading.
In particular, we assume that there is a probability q (presumably small) that W will profit, and that their wealth will increase from 1 to 1 + r 1 V m and a probability 1 − q that it will decrease from 1 to 1 − r 2 V m where r 2 > r 1 > 0.
In other words, there is a small probabilty, q, that W will benefit by r 1 V m (as a fraction of their original wealth) and a larger probability, 1 − q, that they will lose a larger sum r 2 V m . The loss is proportional to the volatility as the professional speculators are able to exploit the ups and downs of the trading at the expense of the inexperienced W.
The mean and variance of the outcome are then In other words, there is large probability that W will take a loss on the transaction. One can consider more general probability distributions for W's profits and losses, but ultimately, the two quantities that are relevant for its utility function U W are given by m Y and σ 2 Y that one can regard as empirical observables.