The evolution of fixed-supply and variable-supply currencies

The paper studies competition between a fixed-supply currency (e.g., Bitcoin) and a variable-supply currency (e.g., fiat/CBDC), asking which currency survives under evolutionary dynamics and how players allocate transaction volumes across currencies. Two kinds of players differ in their support for the currencies and may also choose which kind to be. The key hypothesis is that relative valuation of money supply expansion/contraction versus inflation/deflation determines preferences for the variable-supply currency, while opposing preferences lead to greater use of the fixed-supply currency. The study aims to model player utilities as functions of currency support, market usage shares, peer composition, and macro factors (money supply and inflation), and to derive dynamical paths for transaction shares and player-type composition using replicator dynamics. The analysis is motivated by the rise of digital currencies, heterogeneous currency properties, and the need to understand coexistence or extinction outcomes.

The literature is grouped into four strands. (1) Competition between fiat currencies and cryptocurrencies: Schilling and Uhlig (2019) show substitution depends on fee/cost asymmetries; Fernández-Villaverde and Sanches (2019) identify equilibria for competing private fiat currencies; Almosova (2018) links costs of private currency circulation to public currency inflation; Benigno et al. (2019) study global crypto vs national currencies under interest rate differentials; Rahman (2018) examines monetary policy under fiat-digital competition; Verdier (2021) finds digital currencies can crowd out deposits and raise lending rates. (2) Central bank digital currencies and cryptocurrencies: Caginalp and Caginalp (2019) analyze portfolio allocation between crypto and home currency and government confiscation; Blakstad and Allen (2018) discuss CBDC issuance considerations and crypto risks; Masciandaro (2018) models payment media evolution and implications for monetary/banking policy; Benigno (2021) argues competing currencies may limit central bank control over interest rates and inflation; Asimakopoulos et al. (2019) study substitution between government currency and cryptocurrency under shocks. (3) Cryptocurrency market dynamics: ElBahrawy et al. (2017) identify stable market share properties among cryptocurrencies; Caporale et al. (2018) show autocorrelation and inefficiency enabling abnormal profits; ElBahrawy et al. (2019) link Wikipedia attention to crypto performance; White (2014) studies market shares of Bitcoin and altcoins; Sapkota and Grobys (2021) find inefficiency between privacy and non-privacy coins; Milunovich (2018) finds weak cross-asset connectedness; Gandal and Halaburda (2016) characterize winner-take-all dynamics. (4) Game-theoretic and evolutionary approaches: Imhof and Nowak (2006) analyze stochastic frequency-dependent selection with absorbing states; Lewenberg et al. (2015) study cooperative stability in Bitcoin mining pools. These works motivate modeling strategic substitution, market share evolution, and policy trade-offs in multi-currency settings.

Model with two kinds of players (i=1,2). At time t, player i chooses the fraction p_it of its transactions in the fixed-supply currency g (and 1−p_it in variable-supply currency n). The aggregate usage share of g is p_t = p_1t q_1t + p_2t q_2t, where q_it is the fraction of players of kind i (q_1t + q_2t = 1). Player i's utility of transacting in g is proportional to three factors: its support for g over n (ζ_it), the overall market share of g (p_t), and the prevalence of its own kind (operationalized as a multiplicative factor (1 + μ q_it) raised to an exponent m_i). The utility of transacting in n is analogous but depends on (1−ζ_it), (1−p_t), (1 + μ q_it)^{m_i}, and additionally a Cobb–Douglas term in two macro factors: (a) the ratio of current to initial money supply (reflecting cumulative money printing/withdrawal), and (b) the inverse of cumulative inflation/deflation. The Cobb–Douglas elasticities are α_i for the money supply ratio and (1−α_i) for the inverse inflation factor. Player i’s overall utility u_it is the weighted combination of utilities in g and n with weights p_it and 1−p_it. Society’s utility u_t aggregates across kinds using q_it. Dynamics are governed by replicator equations: for each player kind i, dp_it/dt = k_i p_it (1−p_it) (u_igt − u_int), so that usage of g increases when its utility exceeds that of n; and for player-type composition, dq_1t/dt = h q_1t (1−q_1t) (u_1t − u_2t), so that the population drifts toward the kind with higher overall utility. Parameters include process sensitivities k_i and h. Empirical inputs: US M2 money supply and CPI inflation for 1959–2021 are used to compute the macro utility components. Simulation design explores α_i in {0.6, 0.5, 0.35, 0.2}, support ζ_it (often denoted s_it) spanning 0.01–0.99, process sensitivities k_i ∈ {0.5, 5}, h = 0.5, initial conditions p_it0 = 0.5 and q_it0 = 0.5. Analyses consider (i) simplified utilities holding market share and peer composition fixed, (ii) more realistic utilities endogenizing p_t and q_it, (iii) heterogeneous supports across kinds (S1 ≠ S2), and (iv) effects of scaling parameters μ_i affecting the weight of peer prevalence in utilities.

Utilities over time with US 1959–2021 data: with equal supports and market shares (s = p = q = 0.5) and μ_i = 0, the fixed-supply utility u_igt is constant at 0.25. The variable-supply utility u_int increases over time when α_i is high (α_i = 0.6 or 0.5), oscillates near 0.25 at α_i = 0.35, and decreases when α_i is low (α_i = 0.2). This reflects exponential M2 growth and positive inflation; higher α_i places more weight on money supply than inflation. Replicator dynamics with simplified utilities: with α_i = 0.6 and k_i = 0.5, low support s_it ≤ 0.5 leads p_it → 0 (preference for n). With s_it = 0.6, p_it peaks at 0.59 (1972) then declines toward 0; with s_it = 0.7, p_it peaks at 0.84 (1990) then declines; with s_it = 0.99, p_it → 1. With α_i = 0.2 and k_i = 0.5, high support s_it ≥ 0.6 yields p_it → 1; s_it = 0.5 dips to 0.498 (1968) then rises to 1; s_it = 0.4 dips to 0.32 (1979) then rises; s_it = 0.3 declines to 0.115 (2000), then slightly up to 0.126 (2021); s_it = 0.01 quickly falls to ≈0. Increasing process sensitivity to k_i = 5 accelerates convergence; for example with α_i = 0.2 and s_it = 0.4 the minimum reaches 0.000429 (1979), then increases. Replicator dynamics with endogenous market shares in utilities: accounting for p_t in u_igt and u_int makes convergence faster. With α_1 = 0.6, cases with s_1 = 0.7 or s_1 = 0.6 that previously trended toward n now converge to p_it → 1 (preference for g) because higher early p_t boosts u_igt. With α_1 = 0.2, a case with s_1 = 0.4 that previously recovered toward g can instead decline toward n when P_it < 0.5 for an extended period (until ~2008). Heterogeneous support across kinds (S1 ≠ S2): with α = 0.6, very low support by one kind (S1 = 0.01) and moderate support by the other (S2 ≤ 0.7) drive both toward n; when S1 = 0.01 and S2 ≈ 0.99, P1 → 0 and P2 → 1; increases in the other kind's support can flip both players' preferences, e.g., with S1 = 0.5, raising S2 from 0.6 to 0.7 changes both from preferring n to preferring g. With α = 0.2, players generally prefer g more easily, and sufficiently high support by one kind can induce the other to increase g usage (U-shaped paths for the initially lower-support kind). Evolving player-type composition q_it: the population drifts toward the kind associated with the ultimately preferred currency. For configurations where P1 → 0 (preference for n), q_1t → 1 (players prefer to be kind 1 if that kind's configuration yields higher utility); for cases with P1 → 1 (preference for g), q_1t → 0 (players prefer to be kind 2). Asymmetric peer-utility scaling: increasing μ equally for both kinds speeds dynamics (akin to higher k_i). When μ_2 > μ_1 (e.g., μ_2 = 1, μ_1 = 0), many more trajectories yield q_1t → 0, indicating a systemic shift toward kind 2 because being that kind confers higher utility independent of currency-support and market-share effects. Overall, players' transaction shares can be inverse U-shaped or U-shaped before converging to one currency, with tipping behavior sensitive to support parameters, macro-elasticities α, and peer effects.

The findings support the hypothesis that relative valuation of money supply changes versus inflation risk drives currency choice. When players place high weight on money supply expansion (high α), the variable-supply currency becomes attractive due to increased purchasing power, especially over shorter horizons and where access to new supply is feasible. Conversely, when inflation costs are weighted more (low α), fixed-supply currencies are favored as hedges against debasement. Endogenous market-share and peer-composition effects amplify early advantages: higher initial usage of a currency boosts its utility through coordination and network effects, producing winner-take-all dynamics or rapid convergence. Heterogeneous supports across player kinds can induce cross-effects where strong support by one group gradually pulls the other group toward its preferred currency. Allowing players to choose their kind shows that populations migrate toward the kind aligned with the ultimately higher-utility currency, and asymmetries in peer-related utility (μ_i) can bias the composition even further. These insights are relevant to policy: adjusting money supply and controlling inflation influence not only macro outcomes but also the competitive dynamics between currencies (including CBDCs and cryptocurrencies), potentially affecting banking intermediation, monetary policy transmission, and financial stability.

The paper develops an evolutionary game-theoretic model of competition between a fixed-supply and a variable-supply currency, with two player kinds choosing transaction allocations and, endogenously, which kind to be. Utilities combine currency support, aggregate usage, and peer prevalence, and for the variable-supply currency also a Cobb–Douglas function of money supply changes and inflation. Using US 1959–2021 M2 and inflation data, the model shows that utility for the variable-supply currency rises over time with high elasticity on money supply and falls when inflation is weighted more. Three replicator equations describe the evolution of each kind’s transaction share and the population composition. The results highlight tipping behavior, path dependence, and thresholds in supports and macro-elasticities that determine which currency dominates or whether coexistence persists. Future research can extend the framework to more than two currencies and player kinds, incorporate additional currency attributes (backing, confidentiality, transaction efficiency, security), allow purpose-specific currency use, model heterogeneous access to money supply changes, include risk and time preferences, and analyze other regions and institutional actors (e.g., regulators and governments).

The model considers only two currencies and two player kinds; broader ecosystems with multiple currencies and heterogeneous agents are not modeled. Utilities focus on money supply and inflation and omit other salient currency attributes (e.g., regulation, fees, latency, security) beyond their embedding in a generic support parameter. Access heterogeneity to newly created money, risk attitudes, and time preferences are not explicitly modeled. Results rely on US M2 and CPI inflation from 1959–2021; other jurisdictions or periods may yield different dynamics. The timing lag between money supply changes and inflation is noted but not explicitly modeled within the utility dynamics. Simulation outcomes depend on parameter choices (supports, elasticities, process sensitivities) and initial conditions, which may limit generalizability.