Exchange Rate Determination for Cryptocurrency Mergers: A Formal Framework

2026-09-30

AI Summary

This study addresses the absence of a fair pricing framework in cryptocurrency mergers, where network effects render exchange rates endogenous. It proposes the first endogenous valuation model by employing an axiomatic function that links exchange rates, community migration, and post-merger network value. Methodologically, game theory and Metcalfe’s law are leveraged to conduct convexity analysis and establish Lipschitz control. The results demonstrate that wealth preservation is unattainable when value destruction occurs. Furthermore, this work derives threshold conditions for price ratios satisfying five-dimensional fairness, establishes necessary and sufficient conditions for wealth preservation, and characterizes the set of admissible exchange rates containing non-degenerate intervals. Crucially, it reveals that liquidity constitutes the sole factor capable of disconnecting this admissible set.

Abstract

Many of the thousands of existing cryptocurrencies suffer from declining adoption, low liquidity and weak security, and merging two of them into a single ecosystem is a natural alternative to abandonment. No rigorous framework exists, however, for determining a fair exchange rate in such a merger. Unlike that of a corporation, the value of a cryptocurrency is driven by network effects, so the exchange rate itself influences the value of the merged asset. We extend the exchange-ratio framework of Mainini, Moretto and Visetti [7] by making the merged value endogenous: the exchange rate determines community migration, migration determines (up to user overlap) the post-merger network state, and an axiomatised valuation function assigns a value to that state. The resulting synergy has no predetermined sign, and the bounds of the bargaining region become self-referential in the exchange rate. We prove that no exchange rate at which the merger destroys value preserves wealth, and that the pre-merger price ratio preserves wealth exactly when the merger does not destroy value at that rate. In that case, under explicit threshold conditions, the price ratio satisfies all five of our fairness conditions -- wealth, adoption, security, governance and liquidity preservation -- and, under strict versions of these conditions, the admissible set contains a nondegenerate interval around it, the wealth-admissible set one of explicit width. The network-value laws proposed in the literature, from the linear law to Metcalfe's and its generalised power-law forms, share a convexity property that yields the Lipschitz control behind these estimates. Liquidity emerges as the only non-wealth condition that can disconnect the admissible set.

Problem

• Cryptocurrency Merger
• Exchange Rate Determination
• Network Effects
• Fairness Conditions
• Liquidity

Innovation

• Cryptocurrency Mergers
• Exchange Rate Determination
• Endogenous Network Value
• Fairness Axioms
• Metcalfe's Law