There is a theory that in a choice to join a ban on Bitcoin, individual states face a prisoner's dilemma. A meaningful ban implies one or more states (the "prison") will enforce economic sanctions (at least) on other states (the "prisoners") potentially moving to Bitcoin as a reserve currency.

We assume that the prisoners who may decide to use Bitcoin are trading partners. In other words its use as a reserve currency requires a partner with whom to transact.

Ordinal utility is implied by subjective value. No outcome ties are observed, implying strong dilemma. Both symmetric and asymmetric knowledge assumptions are evaluated.

The outcome for individual Bitcoin (Sucker):

• Economic sanction.
• No trading partners (using Dollar).
• An unusable reserve currency (no trading partners).

The outcome for mutual Bitcoin (Reward):

• Economic sanction.
• Economic sanction of trading partner.
• A reserve currency not taxed via seigniorage.

The outcome for individual Dollar (Temptation):

• No economic sanction.
• Economic sanction of trading partner.
• A reserve currency taxed via seigniorage.

The outcome for mutual Dollar (Punishment):

• No economic sanction.
• No economic sanction of trading partner.
• A reserve currency taxed via seigniorage.

To be considered a prisoner's dilemma T > R > P > S must be true where:

• T > R and P > S imply that Dollar is the dominant strategy for each.
• R > P implies that mutual Bitcoin is preferred by each to mutual Dollar.

We can conclude that P > S holds, as individual sanction implies no international settlement and therefore no benefit from a foreign exchange reserve, and presumably sanctions are undesirable.

To determine if R > P and T > R hold, an objective method is required to relate only seigniorage and sanction, as presumably sanctions are undesirable. This can be obtained by the observation that Gold is subject to neither seigniorage nor sanction. In other words Gold provides the above benefits of Bitcoin without sanction. Yet Gold has not been chosen (and was previously dropped in favor of Dollar), which implies the Dollar outcome is preferred to Gold and therefore Bitcoin. As such, neither of the strategies hold. As such there is no dilemma.

To be considered a prisoner's dilemma Ti > Ri > Pi > Si must be true where:

• Tr > Rr and Pr > Sr
• Tc > Rc and Pc > Sc
• Rr > Pr and Rc > Pc

If these relations all hold then individual Dollar is preferred to Bitcoin, and mutual Bitcoin is preferred. Given that these are the same relations evaluated in the symmetric scenario, there is no dilemma.

The Gold-Bitcoin relation assumes that clearing costs, of transporting Gold and confirming Bitcoin, are negligible in the context of international settlement. Clearing requires periodic movement of only payment imbalances between states.

... any correction of an economic imbalance would be accelerated and normally it would not be necessary to wait for the point at which substantial quantities of gold needed to be transported from one country to another.

gold.org

Dollar has been preferred to Gold despite having similar weight, significantly larger size, and seigniorage. The Gold-Bitcoin relation assumes no distinction in volatility and liquidity, though Gold objectively outperforms Bitcoin in both areas. Given that Gold and Bitcoin are both stable monies, no speculative return is assumed for either. Other Gold, Bitcoin and Dollar monetary properties are assumed to be either equivalent or not relevant to state reserve currency.