The Nash Equilibrium in Digital Cash Systems
Revisiting rational choice under transaction-validation constraints
The claim that needs testing, not repeating
Digital cash systems live or die on predictability. Not “price predictability” in a trader’s sense, but behavioural predictability: whether rational agents continue to validate, propagate, and include transactions in a way that keeps the system usable as an everyday settlement substrate. The central claim tested here is simple. Where rules are stable and legible, miners converge towards cooperative validation because the repeated game supports long-run payoffs. Where rule change is credible, or where the environment injects persistent uncertainty, the incentive landscape compresses into short horizons and deviation becomes rational.
This is not an argument about slogans. It is a statement about equilibrium selection in a constrained strategic environment: miners do not merely react to fees; they react to the credibility of the rule set that determines how fees can be earned, how propagation advantage can be converted into revenue, and how often today’s optimal strategy is invalidated tomorrow.
What follows is a results-driven walk through the model and what the simulations show under three interacting frictions: revenue regime (subsidy-anchored versus fee-dominant), propagation delay (latency intensity and variance), and institutional noise (protocol uncertainty as a regime-mutation process).
A repeated mining game with measurable frictions
The modelling choice is not exotic: miners play a repeated game over blocks. Each block is a stage game with a payoff that depends on the chosen action (cooperate with a throughput-oriented validation stance, or deviate towards extractive tactics), the revenue environment, and network conditions.
Two design decisions matter because they force the model to “touch the ground”.
First, latency is treated as an agent-level draw, not a single global constant. The latency settings in the experiments are the unitless propagation-delay intensity values {0.00, 0.05, 0.10, 0.20}. For each miner (i), latency is drawn per run as\([
\Lambda_i \sim \max{0, \mathcal{N}(\text{latency}, 0.35\cdot \text{latency})},
]\)
so variance scales with the mean, and the distribution is truncated at zero. That draw enters payoffs directly as a linear penalty for both actions, (-0.05\Lambda_i), and it enters the fork/orphan mechanism through a fork probability term:\([
p_{\text{fork}}=\text{orphan_base}+0.02\cdot \text{dev_share}+2.0\cdot \mathrm{sd}(\Lambda),
]\)
with an additional penalty applied conditional on a fork event. This matters because it prevents “latency” from becoming a rhetorical label. It becomes an observable channel through which variance, not just level, alters incentives.
Second, protocol uncertainty is implemented as a regime-mutation hazard. The uncertainty settings are {0.00, 0.02, 0.05, 0.10}. At each block (t), with probability equal to the uncertainty setting, the institutional regime flips:
\([
\text{regime}{t+1}=1-\text{regime}{t}.
]\)
When the regime equals 1, the deviation payoff receives an additive bonus of 0.25 and the fork probability receives an additive 0.01. That is not decoration. It makes “rule instability” bite in two ways: it increases the private return to deviation in the very blocks where rule credibility is compromised, and it mechanically worsens network-level outcomes by increasing the probability of fork/orphan events.
Across these experiments, the revenue environment is parameterised by a regime index κ, which captures the structural weight of transaction fees relative to block subsidy anchoring. The κ parameter is swept across discrete bins corresponding to fee-dominant regimes, transitional mixed regimes, and subsidy-anchored regimes. In parallel, fee volatility is varied using σ_f, with volatility bins spanning low volatility (0.00–0.10), an intermediate regime (around 0.25), and high volatility conditions (0.50–1.00).
What “equilibrium” means in this paper
Equilibrium is not treated as an aesthetic concept. The relevant objects are the observed long-run strategy profiles and whether the simulation converges to a stable attractor.
A run can end in a unique cooperative attractor, a deviation-dominant attractor, a mixed basin structure with multiple equilibria (path dependence), or an oscillatory non-settling regime where convergence fails and strategy mass cycles. The value of this classification is that it separates three things that are often muddled in blockchain discussion.
Model specification is what is assumed and parameterised. Simulation outcomes are what the runs actually do. Interpretation is what can be said about why the observed behaviour changes when parameters cross measured boundaries. The paper’s Results sections treat those as separate compartments, because “governance language” is cheap unless it is disciplined by output.
Revenue regime: where convergence is unique, and where it fractures
The clearest finding is that revenue regime is not a cosmetic change. It changes which equilibria exist, which equilibria are reachable, and how robust convergence is to variance.
Latency is not merely a matter of “slower blocks”. Because fork probability depends on the standard deviation of propagation delay, sd(Λ), variance in network propagation is directly linked to orphan risk. Orphan risk constitutes a real economic cost, not a technical nuisance, and it directly alters the payoff ranking of mining strategies. As propagation variance increases, strategies that are otherwise cooperative become less attractive relative to deviation or extraction-oriented tactics, since the expected payoff from honest inclusion is discounted by a higher probability of block invalidation.
The practical meaning is direct. When revenue is structurally fee-driven and fees are volatile, the system becomes a strategic environment with persistent incentive re-optimisation. In that setting, expecting convergence to a stable cooperative equilibrium is not “optimistic”. It is inconsistent with what the runs do.
Fee volatility: multiplicity and switching are not side-effects, they are the regime
Fee volatility is often discussed as though it were a purely “market” feature. In this model, volatility matters because it changes the local payoff differences that drive best responses, and it changes the value of propagation advantage and transaction selection tactics.
The observed pattern across regimes is that volatility expands the set of feasible equilibria and increases switching between them. At low σ_f, the system can settle; at intermediate σ_f, mixed basins dominate; at high σ_f, oscillation becomes the rule.
The important point is not merely that volatility increases “variance”. It changes the equilibrium structure. Under volatility, the system behaves less like a stable repeated coordination problem and more like a cycling game where players repeatedly re-optimise around local opportunities. The result is not just inefficiency. It is a change in what can be enforced over time.
In fee-dominant regimes, that is visible as convergence collapse and persistent oscillation. In subsidy-anchored regimes, it is visible as the disappearance of near-universal unique cooperation and the rise of mixed basins and cycling. In the transitional regime, volatility is sufficient to destroy convergence entirely.
Network latency: propagation variance is an equilibrium control parameter
In fee-dominant conditions, with κ in the range 0.10–0.30 and low fee volatility σ₍f₎ in the range 0.00–0.10, under latency settings spanning 0.00–0.20, simulation runs are predominantly deviation-dominant. Specifically, 93.75% of runs fall into this category, with only occasional oscillatory behaviour observed. Even in this comparatively benign setting, strict convergence is confined to the lowest volatility bins. This already constitutes a critical correction to casual intuition. Fee dominance does not merely “shift incentives toward fees”; it shifts the stability basin away from cooperation even when volatility is mild.
As volatility increases, the system does not degrade smoothly; it bifurcates. With κ in the range 0.10–0.30 and σ₍f₎ equal to 0.25, outcomes split evenly: 50% of runs are deviation-dominant and 50% are oscillatory, with convergence collapsing entirely. When σ₍f₎ rises into the range 0.50–1.00, the system becomes fully oscillatory, with 100% of runs exhibiting persistent cycling and no stable long-run equilibrium observed. Under fee dominance, the repeated game becomes a cycling game: short-horizon tactics chase changing marginal rewards, and the system fails to settle into any fixed strategy profile.
The transitional regime, where κ equals 0.50, is where knife-edge behaviour becomes visible. Under low volatility conditions, with σ₍f₎ in the range 0.00–0.10, a cooperative attractor exists and is reached in a majority of runs. A unique cooperative equilibrium emerges in 60.94% of cases, with the remaining 39.06% occupying mixed basins. The presence of mixed basins is economically significant. It indicates that the system is neither simply cooperative nor non-cooperative. Instead, equilibrium selection is conditional on history, initialisation, and stochastic shocks. When volatility increases to σ₍f₎ in the range 0.25–1.00, convergence again collapses into oscillation, with 100% of runs failing to settle. This is not a philosophical claim about fees being undesirable; it is a measured statement about equilibrium multiplicity and non-convergence under volatility.
Subsidy-anchored regimes, with κ in the range 0.70–0.90, produce the strongest convergence behaviour, but not unconditionally. Under low fee volatility, with σ₍f₎ in the range 0.00–0.10, runs are uniquely cooperative in 92.97% of cases, with strict convergence and comparatively low switching and fork incidence. This represents the clearest demonstration of cooperative stability observed in the simulations: when fee revenue does not dominate and volatility is low, the repeated game reliably converges to cooperation.
However, even in subsidy-anchored environments, volatility reintroduces equilibrium multiplicity and cycling. At σ₍f₎ equal to 0.25, multiple equilibria persist, mixed basins become modal at 59.38%, and strict convergence becomes sensitive to additional frictions such as uncertainty and latency. When σ₍f₎ rises into the range 0.50–1.00, outcomes again become predominantly oscillatory, accounting for 73.44% of runs, with the remaining mass in mixed states and widespread convergence failure. Subsidy anchoring dampens the tendency toward cycling, but it does not immunise the system against high volatility. It enlarges the cooperative basin; it does not abolish the underlying mechanism.
The measured relationship between propagation-delay variance and fork/orphan outcomes is monotonic. As the variance proxy increases, the fork/orphan rate rises, with uncertainty bands widening at higher variance. The economic interpretation is not optional: higher propagation variance increases orphan risk, which increases the expected cost of aggressive tactics and also increases the probability that honest effort is not rewarded. That combination destabilises cooperative expectations because the reward for cooperative validation becomes less predictable relative to deviation strategies that attempt to exploit propagation advantage.
The model also makes a second point that is often ignored: deviation incidence does not necessarily move in the same direction as fork rates. The deviation line can remain high and relatively flat across the variance sweep while fork/orphan risk rises. That is a structural warning. A system can show persistent deviation incentives even before fork outcomes become visibly catastrophic, because the private return to deviation can remain attractive while the social cost accumulates through increased orphaning and reduced reliability. The damage appears in settlement reliability first, and in the strategic profile only when volatility or institutional noise pushes the system across a boundary.
This is why the correct figure for “latency and strategic deviation” is the fork/orphan outcome curve plotted against propagation-delay variance, not a plot against the raw latency parameter alone. The equilibrium-relevant object in this model is the dispersion of propagation, because that is what enters (p_{\text{fork}}) and therefore enters expected payoffs.
Protocol uncertainty: horizon compression is measurable
Uncertainty here is not a mood. It is a per-block hazard that flips regime state, adds 0.25 to deviation payoff when the regime is “unstable”, and increases fork probability by 0.01. It therefore increases the return to deviation precisely when the environment signals rule instability.
The observed pattern is consistent with horizon compression: mean cooperation is highest when uncertainty is zero and decreases when uncertainty is introduced, while convergence rates fall when uncertainty rises above zero and then remain depressed. Even when the fall in mean cooperation is not dramatic, the drop in convergence is economically decisive. Convergence is what determines whether a cooperative equilibrium is reliable, not merely whether cooperation is present at some average level.
This distinction matters for how “rule stability” is discussed. A system can show non-zero cooperation under uncertainty, but if convergence collapses, cooperation becomes episodic and fragile. That fragility is the real economic cost, because it destroys predictability for users and for long-horizon capital investment in infrastructure.
Institutional noise: collapse thresholds and post-collapse behaviour
Institutional noise is modelled through the regime-mutation process. Its role is not to generate rhetorical drama; it is to test whether cooperative equilibria are robust to credible rule-change risk.
The simulation logic is plain. When regime mutations occur, deviation becomes locally more attractive and fork risk rises. That pushes the system towards oscillation and mixed basins even when other parameters are favourable. The collapse event is not defined as “a bad feeling”; it is defined by the failure to converge and by sustained departure from the cooperative attractor region.
The key empirical output is that collapse thresholds exist and they are sharp. Cooperative stability is not linearly eroded. It persists up to a boundary and then gives way to cycling, multiple equilibria, and non-settling behaviour. Post-collapse, the system spends a larger share of time in deviation or mixed profiles and exhibits persistent switching. The economic meaning is enforcement failure: the repeated game can no longer sustain the punishments and expectations that make cooperation stable, because the underlying rule environment is itself changing with non-trivial probability.
Historical episodes are not “stories”; they are corollaries
Real-world protocol disputes and policy interventions matter because they are the empirical surface on which these mechanisms show up. When mempool policy changes, relay rules, or transaction replacement policies introduce mutability at the transaction layer, they do not merely “improve efficiency”. They change the structure of the game by altering which strategies are profitable and by changing the predictability of inclusion. When block capacity is artificially constrained, it creates scarcity that amplifies fee competition, which amplifies volatility, which expands multiplicity and cycling. When governance becomes a site of credible future mutation, it raises the perceived hazard ( \rho ) in the minds of rational agents, steepening effective discounting even if nominal rewards remain unchanged.
These are not separate arguments. They are the same mechanism observed at different scales: institutional instability compresses horizons, and compressed horizons favour deviation.
What this paper contributes, in concrete terms
Three contributions are claimed and tested through outputs.
First, the paper identifies where convergence is unique and where multiple equilibria persist, conditional on revenue regime and volatility. The results tables do not merely assert this; they report the observed equilibrium types and their frequencies across parameter bins.
Second, it shows how propagation-delay variance feeds into fork/orphan risk and how that risk interacts with equilibrium stability. Latency is not treated as an external nuisance; it is an endogenous strategic parameter through the fork probability and penalty channels.
Third, it operationalises protocol uncertainty as a regime-mutation process with explicit payoff and fork-risk effects, and it shows the measurable impact on convergence and cooperation metrics. This turns “governance” from a rhetorical domain into a model parameter that can be swept, measured, and tied to equilibrium outcomes.
Policy implications belong to constraints, not rhetoric
If the system is to function as digital cash, the strategic environment must be stable enough for long-run cooperative equilibria to be reachable and enforceable. The results place limits on what can be asked of incentive design.
When fee dominance and volatility are high, the system becomes oscillatory and non-convergent. When propagation variance rises, fork/orphan risk rises and stability weakens. When institutional noise increases, convergence collapses and mixed basins dominate. None of this is repaired by slogans about “robustness” or “adaptability”. A repeated game does not reward the rhetoric of mutability; it rewards credible constraints.
A governance regime that treats the base-layer rules as discretionary policy inputs turns the system into a stochastic game at the meta-rule level. In that setting, the rational strategy set shifts towards extraction, short horizons, and switching. If long-horizon investment and predictable settlement are the objective, then the base-layer constraints must be stable enough that agents are not forced to forecast politics as part of the payoff function.
Innovation does not vanish under such a constraint; it moves. The base-layer becomes the constitutional substrate, and experimentation occurs at layers that do not rewrite the strategic environment. That separation is not aesthetic; it is what the equilibrium results demand.
Closing statement
The economic logic is not complicated, but it is unforgiving. Cooperative equilibria in a repeated mining game require a stable rule environment, manageable volatility, and propagation conditions that do not systematically convert honest effort into orphaned loss. When those conditions are violated, deviation is not a moral failure. It is the predictable equilibrium response.
A digital cash system does not scale on hope. It scales on constraints that make long-run cooperation rational.