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

• monetary landscape: digital (credit/debit cards) and physical (cash) payments; • optimal combination between environmentally sustainable and safe payment methods; • non-cooperative differential game with European payment data; • baseline symmetric Nash model and Stackelberg extension; • policy implications for digital financial risks and their security dynamics. 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.