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Beretta_2025_Elsevier_j.frl.2026.109582
Journal article

Cash or card – combat or coexistence? : a non-cooperative differential game approach

  • Bartesaghi, Paolo ORCID Department of Economics, Management and Quantitative Methods, Università degli Studi di Milano, Italy
  • Beretta, Edoardo ORCID Istituto di economia politica (IdEP), Facoltà di scienze economiche, Università della Svizzera italiana, Svizzera
  • Desogus, Marco Department of Law, Università degli Studi di Sassari, Italy
  • Korn, Ralf ORCID Department of Mathematics, Rheinland-Pfälzische Technische Universität, Kaiserslautern, German
  • 2026
Published in:
  • Finance Research Letters. - 2026, vol. 92, p. 109582
Cash-cashless equilibrium
Differential games
Financial sustainability
Monetary policy
Nash equilibrium
Payment systems
English The present letter analyses the current evolution of the monetary landscape mostly characterized by means of payments based on digitization (e.g., credit/debit cards) to the disadvantage of physical ones (e.g., cash). In parallel, pleas for a more sustainable lifestyle are increasingly heard by the banking and financial sector providing payment solutions, which should be also perceived to be sufficiently secure against frauds. For instance, fraud losses due to identity theft are at a cross-country level on the surge for credit-card payments while cash is considered less practical and connected with operational and logistical challenges (e.g., production and issuance, return, sorting and destruction). In the light of these developments and trade-offs, we derive the optimal combination of (digital versus physical) means of payments to pay in an environmentally sustainable and safe way. By means of a non-cooperative differential game, this letter provides a baseline symmetric Nash model supplemented by a Stackelberg extension capturing realistic leader-follower dynamics and calibrates both frameworks to European payment data. We derive closed-form solutions and discuss coexistence conditions for optimal payment systems and suggest directions for future, empirically more sophisticated extensions.
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Language
  • English
Classification
Economics
License
CC BY
Open access status
hybrid
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Persistent URL
https://n2t.net/ark:/12658/srd1334728
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