% Paper 1: Exit Dominance in Monetary Coordination Games

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\begin{center}
{\LARGE\bfseries Bitcoin Exit Dominance in Monetary Coordination Games}

\bigskip

{\large Sean Hash}\\
\url{bitcoingametheory.com}\\
\href{mailto:sean@bitcoingametheory.com}{sean@bitcoingametheory.com}

\bigskip

Preprint: February 2026 (SSRN 6299081) \\ Revised: April 2026\\[4pt]
{\normalsize Working Paper}

\medskip

{\small
\textbf{JEL Codes:} C72, C73, D83, E42, E51, G11, L14\\[4pt]
\textbf{Keywords:} Bitcoin, game theory, Nash equilibrium, monetary coordination, neutral settlement, absorbing state, coordination failure, threshold strategy, digital scarcity, Schelling point, network effects
}
\end{center}

\bigskip

% ============================================================
% ABSTRACT
% ============================================================

\begin{abstract}
\noindent
In a multipolar world with no trusted monetary coordinator, how do
rational actors settle large-value transactions across trust boundaries?
We model this as a non-cooperative game---the ``Exit Game''---in which
capital allocators choose between capturable settlement systems (``Stay'')
and neutral settlement (``Exit,'' distinct from Hirschman's
organizational-decline sense). The model rests on four empirical
axioms: persistent multipolarity, rational self-interest, computational
hardness, and network effect persistence. We prove three results.
First, the payoff advantage of Exit over Stay is strictly increasing in
adoption: every term in the payoff differential favors Exit under
maintained monotonicity conditions, and each actor's adoption threshold
approaches zero under structural debasement (Theorem 1). Second, no
coalition can sustain coordinated Stay, because permissionless access
makes defection costless and the first defector captures fleeing capital
(Theorem 2). Third, the resulting equilibrium is absorbing: the monotone
adoption process converges to full adoption once a critical mass is
reached, because trust conditions required for coordinated return cannot
be re-established (Theorem 3). The model is falsifiable: six conditions are identified under
which the central claims would fail. Bitcoin is the unique asset
satisfying the necessary properties for neutral settlement---a result
proved by systematic elimination across seven asset classes in a
companion paper.
\end{abstract}

\newpage

\noindent\textbf{Revision Note---April 2026.}
This revision prepares the February 2026 preprint (SSRN 6299081) for
journal submission. All revisions sharpen presentation, widen
literature engagement, or scope claims more honestly.

\smallskip

Citations previously narrowed to what the source establishes, and
paired where a single anchor carried weight that belongs to two, are
retained from the February revision: \S2.2 (Huberman, Leshno, and
Moallemi for congestion pricing only); \S2.5 (Hirschman as
intellectual context, distinct from strategic Exit in-model); \S4.4
(Bikhchandani, Hirshleifer, and Welch as contrast, not analog); \S2.3
(Milgrom--Shannon paired with Topkis); \S3.1 (David paired with
Arthur); \S3.4 (Kyle paired with Amihud, Katz--Shapiro paired with
Farrell--Saloner).

\smallskip

The threshold-coverage condition previously stated parenthetically
in Theorem 3's proof is promoted to Assumption 5. A new monotonicity
condition (M6) makes the capturable return $R_F$ a function of
adoption $p$, consistent with every other quantity in the model;
Theorem 1's proof is extended accordingly and is strengthened under
strict inequality. Theorem 3's information-structure framing is upgraded
from complete-information + common-knowledge to public-information +
heterogeneous-priors (following Morris and Shin 1998, Carlsson and van Damme
1993), placing the model in the global-games tradition. Theorem 3's Step 4
is rewritten from Bayesian-updating-over-hidden-types to higher-order-belief
coordination breakdown. Utility is stated explicitly as mean-variance in
\S3.4. The debt-repayment claim in \S1 is tightened. The falsification table
in \S5 is reorganized into axiom falsifiers, theorem falsifiers, and scope
conditions.

\smallskip

Three adjacent literatures are now acknowledged: global games
(Carlsson and van Damme\textsuperscript{9}; Morris and
Shin\textsuperscript{27}) as the standard tool for equilibrium selection
in coordination games under incomplete information; self-fulfilling
currency crises (Obstfeld\textsuperscript{28,29}) as the intellectual
ancestor of the coordination-failure mechanism; and search-theoretic
money (Kiyotaki and Wright\textsuperscript{23}; Lagos and
Wright\textsuperscript{25}) as providing microfoundations this paper
does not re-derive. The paper's complete-information modeling choice
and equilibrium-selection framing are defended against each.

\smallskip

Section~6 names three scope deferrals: the binary Exit/Stay structure
(continuous-allocation extension deferred); deterministic monotone
convergence (stochastic stability in the Kandori--Mailath--Rob sense
deferred); and Theorem~2's single-defection result (full coalition-proof
Nash analysis in the Bernheim--Peleg--Whinston sense deferred). No
theorem, proof, or axiom has been weakened or reversed. The February
2026 preprint is preserved at
\url{bitcoingametheory.com/papers/1-exit-dominance/archive}.

\newpage

% ============================================================
% 1. INTRODUCTION
% ============================================================

\section{1. Introduction}

The global monetary system has no referee. Multiple sovereign powers
compete for economic influence, each controlling monetary instruments
that serve domestic policy objectives at the expense of foreign holders.
This creates a coordination problem that is easy to state and impossible to solve within existing institutions: in a
world where no single entity can credibly commit to monetary neutrality,
how do rational actors settle large-value transactions across trust
boundaries?

This question is structural, not conjunctural. Sovereign debt levels
have reached historical extremes: global debt reached \$313 trillion
in 2023, a debt-to-GDP ratio of 255\% (Institute of International
Finance\textsuperscript{19}). No post-1800 nation has repaid debt
exceeding 90\% of GDP primarily through fiscal austerity without
substantial growth, inflation, financial repression, or default;
these latter mechanisms are the historically observed path (Reinhart
and Rogoff\textsuperscript{31}).
The coordination problem is not whether states \emph{will} debase, but
whether any mechanism exists through which actors can settle
\emph{despite} debasement.

The analysis is stated descriptively: we characterize equilibrium
structure without predicting timing, price trajectories, or
recommending portfolio allocations. Normative implications are
nonetheless unavoidable when a coordination game has a unique
absorbing state; readers may draw their own policy implications,
which the paper does not advocate.

The literature addresses components of this problem---mining incentives
(Biais \emph{et al.}\textsuperscript{4}; Eyal and
Sirer\textsuperscript{12}), attack economics
(Budish\textsuperscript{7}), price-security dynamics
(Pagnotta\textsuperscript{30})---but does not integrate these into a
unified strategic analysis from the allocator's perspective.
Brunnermeier, James, and Landau\textsuperscript{6} identify
the coordination failure but propose institutional solutions rather than
analyzing the equilibrium outcome. This paper addresses the gap by formalizing the allocator's strategic
problem directly.

We formalize the allocator's problem as a non-cooperative game---the
``Exit Game''---and prove that the payoff advantage of Exit over Stay is
strictly increasing in adoption. Every term in the payoff differential
favors Exit under maintained monotonicity conditions. Each actor faces
an adoption threshold that approaches zero when the capturable system
delivers negative real returns---which, under structural debasement
(Assumption 1), it does.

We also prove that no coalition can sustain Stay (Theorem 2) and that
the resulting equilibrium is absorbing (Theorem 3). Bitcoin satisfies
the seven necessary properties derived here---a result established by
systematic elimination in Hash\textsuperscript{14}. A third paper (Hash\textsuperscript{15})
extends the analysis to autonomous AI agents, for whom neutral
settlement is not a preference but a structural requirement.

\newpage

% ============================================================
% 2. RELATED WORK
% ============================================================

\section{2. Related Work}

\subsection{2.1 Game Theory of Proof-of-Work Consensus}

The foundational result is Biais, Bisiere, Bouvard, and
Casamatta\textsuperscript{4}: following the longest chain constitutes
a Markov Perfect Equilibrium in a stochastic game among \emph{miners},
and subsequent work largely builds on this framework. Eyal and
Sirer\textsuperscript{12} complicate this picture by identifying
conditions under which selfish mining is profitable---but their
threshold (a miner controlling more than one-third of hashrate) has
never been approached in practice, which says more about the incentive
structure than about the theoretical vulnerability.
Budish\textsuperscript{7} argues that the marginal cost of a majority attack scales with
the block reward flow, which must in turn scale with secured value.
The model is elegant but leaves the fee-market contribution to the
security budget incompletely characterized: a network processing large
daily settlement volumes generates security budget partly independent
of token price, a gap the literature has not yet closed.

\subsection{2.2 Monetary Competition and Coordination}

Brunnermeier, James, and Landau\textsuperscript{6} is the essential
reference on digital currency competition in a multipolar world. Their
core argument, that competing monetary systems create coordination
failures traditional institutions cannot resolve, is precisely our
starting point. Where we diverge: Brunnermeier \emph{et al.}\ treat
the coordination failure as a problem to be solved by better
institutional design. We treat it as a permanent structural feature
(Assumption 1) that selects for a non-institutional solution.
Huberman, Leshno, and Moallemi\textsuperscript{18} analyze Bitcoin's fee
market as a congestion-priced payment system in which no agent sets
prices unilaterally. Their ``monopoly without a monopolist''
characterization captures the absence of a protocol-level rent
extractor, one component of the broader neutrality property we
formalize below. Their paper establishes the congestion-pricing result;
the extension to governance and seizure resistance is ours, not theirs.

Self-fulfilling currency crises (Obstfeld\textsuperscript{28,29}) are
the intellectual ancestor of our coordination-failure mechanism:
Obstfeld established that balance-of-payments crises and
fixed-exchange-rate collapses can be driven purely by coordinated
expectations under multiple equilibria. We extend this logic from
attacks on pegs to exit from the capturable-currency system entirely.
Search-theoretic models of money (Kiyotaki and
Wright\textsuperscript{23}; Lagos and Wright\textsuperscript{25})
provide microfoundations for money's role as a medium of exchange;
this paper takes competing monetary systems as given and analyzes
equilibrium selection over them rather than re-deriving medium-of-exchange
properties.

\subsection{2.3 Monotone Comparative Statics}

The monotonicity conditions (M1)--(M6) that drive our main result belong
to the framework of monotone comparative statics developed by Milgrom
and Shannon\textsuperscript{26} and Topkis\textsuperscript{34}. In their language, the payoff
differential $\Delta_i(p)$ exhibits increasing differences in $(p, \text{adoption decision})$ under our maintained conditions---precisely the
supermodularity property that guarantees monotone best responses. Our
contribution is applying this machinery to the monetary coordination
problem rather than to the firm and market settings where it originated.

\subsection{2.4 Global Games and Equilibrium Selection}

The global-games framework (Carlsson and van
Damme\textsuperscript{9}; Morris and Shin\textsuperscript{27}) is the
standard modern tool for equilibrium selection in coordination games
with strategic complementarities, particularly under private noisy
observation of fundamentals. Morris and Shin\textsuperscript{27} apply
the framework to self-fulfilling currency attacks,
delivering a unique equilibrium selected by the informational
structure rather than by common knowledge of payoffs. Our setup---heterogeneous
thresholds, strategic complementarity, coordination
over monetary standards---is within the domain this literature
addresses. \emph{Information structure:} We adopt public information with
heterogeneous priors. All actors have access to the same public facts about
the system---code, ledger, rules, payoff structure. Actors may hold different
priors over system type owing to asymmetric narrative exposure. Information is
open; interpretation is uneven. This placement in the higher-order-beliefs
tradition rather than the Harsanyi incomplete-information tradition reflects
empirical reality: monetary-system information is openly published but
unevenly interpreted, with competing narratives shaping actors' beliefs over
whether the system is capturable. The irreversibility result in Theorem 3
depends on shifts in what actors believe others believe, not on updating over
hidden facts.

\subsection{2.5 Network Effects and Focal Points}

Katz and Shapiro\textsuperscript{21,22} provide the theoretical
machinery for our network effect assumption. What matters for our
purposes is a specific implication they do not emphasize: in monetary
networks, switching costs compound because the network's value
proposition \emph{is} its adoption. Schelling\textsuperscript{32}
provides the concept of focal points, though applying
focal point theory to money requires more care than the literature
typically exercises. We argue that the focal point is determined not by
precedence but by structural resistance to the ``who benefits?'' objection.

We note the distinction between our ``Exit'' concept and Hirschman's\textsuperscript{17}
``exit'' from organizations. In Hirschman's framework, exit is one of three
responses to organizational decline (alongside voice and loyalty). Our Exit
is a strategic action in a coordination game---adopting neutral settlement---and
does not require leaving an institution, only diversifying settlement
infrastructure.

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% 3. MODEL
% ============================================================

\section{3. Model}

\subsection{3.1 Maintained Assumptions}

The analysis rests on four empirical axioms. Each is independently
assessable. All subsequent claims derive from these assumptions; rejecting
any axiom invalidates the specific claims that depend on it.

\begin{assumption}[Multipolarity]
No single entity permanently governs global economic activity. Power
distributes across competing centers with superlinear coordination costs.
\end{assumption}

\noindent\emph{Falsification:} A single coordinator achieves permanent, stable
control over global monetary policy with sublinear coordination costs (F1).

\begin{assumption}[Rational Self-Interest]
Actors optimize for self-interest. When defection from cooperative
agreements is unpunished or unpunishable, actors defect.
\end{assumption}

\noindent\emph{Falsification:} Stable cooperation persists indefinitely among
self-interested actors without enforcement mechanisms (F3).

\begin{assumption}[Computational Hardness]
Certain mathematical problems remain computationally intractable. Digital
scarcity and cryptographic custody are implementable.
\end{assumption}

\begin{assumption}[Network Effect Persistence]
Past critical mass, switching costs exceed marginal gains of alternatives.
Incumbency compounds (David\textsuperscript{10}; Arthur\textsuperscript{2}).
\end{assumption}

\begin{assumption}[Threshold Coverage]
The distribution of exit thresholds $\{p_i^*\}$ has continuous support on
$[0, 1]$. Equivalently, there is no measure-positive set of actor types
for whom Exit is strictly dominated at every adoption level.
\end{assumption}

\noindent\emph{Falsification:} A measure-positive subpopulation has $p_i^* > 1$
(never-exit types); the absorbing state is then $p^* < 1$ at the
supremum of the threshold support.

\subsection{3.2 Definitions}

\begin{definition}[Neutral Settlement]
An asset $S$ is a \emph{neutral settlement asset} if and only if $S$ is
immune to seizure, immune to debasement, and immune to political capture
by any single actor or coordinated coalition.
\end{definition}

\begin{definition}[Exit]
The action of moving capital from a capturable system to a neutral
settlement asset.
\end{definition}

\begin{definition}[Capture Surface]
For a monetary system with parameter space $\Theta$, the capture surface
for actor $A$ is $CS_A = \{\theta \in \Theta : A \text{ can unilaterally change } \theta\}$.
A system is neutral if $|CS_A| = 0$ for all $A$. The terminology adapts
``attack surface'' from security engineering to institutional capture
(\emph{cf.}\ Stigler\textsuperscript{33} on regulatory capture).
\end{definition}

\begin{definition}[Absorbing State]
A state $p^*$ is absorbing if $P(p_{t+1} = p^* \mid p_t = p^*) = 1$. In
the monetary coordination context, the transition to $p^* = 1$ destroys
the trust conditions required for reversal, making the state absorbing
in the standard Markov chain sense (\emph{cf.}\ Arthur\textsuperscript{2} on lock-in).
\end{definition}

\subsection{3.3 Necessary Properties}

From Definition 1, we derive seven necessary properties for any neutral
settlement asset. Each blocks a specific attack class:

\begin{table}[h]
\centering
\begin{tabular}{lll}
\toprule
D1 Requirement & Attack If Missing & Required Property \\
\midrule
Immune to seizure & Physical confiscation & P6: Informational security \\
Immune to seizure & Transaction censorship & P3: Permissionless access \\
Immune to debasement & Supply inflation & P5: Absolute scarcity \\
Immune to debasement & Protocol rule change & P1: Protocol security \\
Immune to capture & Governance takeover & P2: Neutrality \\
Functions as settlement & Prohibitive cost/delay & P4: Cheap finality \\
Survives future threats & Obsolescence & P7: Adaptive resilience \\
\bottomrule
\end{tabular}
\end{table}

\begin{proposition}[Necessity]
P1--P7 are individually necessary. Removing any single property enables
the corresponding attack, violating Definition 1.
\end{proposition}

\begin{proof}
By construction from the derivation table.
\end{proof}

Bitcoin is the unique asset satisfying P1--P7---the systematic
elimination across seven asset classes appears in Hash\textsuperscript{14}.

\subsection{3.4 Game Structure}

We model monetary coordination as a non-cooperative game:

\noindent\textbf{Game} $G = (N, S, u)$ where:

\begin{itemize}[nosep]
\item $N = \{1, 2, \ldots, n\}$ is the set of capital allocators.
\item $S_i = \{\text{Exit}, \text{Stay}\}$ is the strategy set for player $i$.
\item $u_i : S \to \mathbb{R}$ is the utility function for player $i$.
\end{itemize}

Let $p \in [0, 1]$ denote the fraction of total capital that has chosen
Exit. We assume $R_B$, $R_F$, $\sigma_B$, $K_A$, $R_A$, and $K_N$ are
continuous functions of $p$, bounded on $[0,1]$. Each actor $i$ has
mean-variance preferences: utility is expected return minus a
risk-aversion coefficient $\lambda_i > 0$ times return variance, a
specification standard in the portfolio-allocation and
equilibrium-selection literatures. The utility functions are:

\[
u_i(\text{Exit}, s_{-i}) = R_B(p) - \lambda_i \cdot \sigma_B(p) - \kappa_i \cdot K_A(p) - \rho_i \cdot R_A(p)
\]

\[
u_i(\text{Stay}, s_{-i}) = R_F(p) - \lambda_i \cdot \sigma_F - K_N(p)
\]

Here $K_A(p)$ is the adoption penalty---the switching cost of moving
to neutral settlement, analogous to technology adoption costs in network
economics (Katz and Shapiro\textsuperscript{21}; Farrell and Saloner\textsuperscript{13}).
$K_N(p)$ is the non-adoption penalty---the opportunity cost imposed on
non-adopters as the competing network grows, a form of negative network
externality.

\noindent\textbf{Maintained monotonicity conditions:}

\begin{enumerate}[label=(M\arabic*),nosep]
\item $R_B'(p) > 0$ --- Network effects increase return (Assumption 4)
\item $\sigma_B'(p) < 0$ --- Deeper markets reduce volatility (Kyle\textsuperscript{24}; Amihud\textsuperscript{1})
\item $K_A'(p) < 0$ --- Adoption penalty falls with adoption
\item $R_A'(p) < 0$ --- Regulatory penalty decreases as adoption normalizes compliance
\item $K_N'(p) > 0$ --- Non-adoption penalty rises as competitors exit
\item $R_F'(p) \leq 0$ --- Capturable return weakly decreasing in neutral adoption (the capturable system loses value as capital departs)
\end{enumerate}

\noindent The market-microstructure basis for (M2) is standard: deeper
markets lower the price impact of order flow (Kyle's $\lambda$
decreases with depth), and lower price impact translates to lower
realized return volatility at a fixed rate of information arrival
(Amihud\textsuperscript{1}).

% ============================================================
% 4. MAIN RESULTS
% ============================================================

\section{4. Main Results}

\subsection{4.1 Exit Dominance}

\begin{theorem}[Monotone Exit Advantage]
Under Assumptions 1--4 and maintained conditions (M1)--(M6):

\begin{enumerate}[label=(\roman*)]
\item The payoff differential $\Delta_i(p)$ is strictly increasing in $p$
for all player types.
\item For each player $i$, there exists a threshold $p_i^* \in [0, 1]$
such that Exit is the unique best response for $p > p_i^*$.
\item Under Assumption 1 ($R_F(0) < 0$), thresholds cluster near zero
for actors with typical risk preferences.
\end{enumerate}
\end{theorem}

\begin{proof}
Define the payoff differential:

\[
\Delta_i(p) = u_i(\text{Exit}) - u_i(\text{Stay})
\]
\[
= [R_B(p) - R_F(p)] - \lambda_i[\sigma_B(p) - \sigma_F] - \kappa_i \cdot K_A(p) - \rho_i \cdot R_A(p) + K_N(p)
\]

Taking the derivative with respect to $p$:

\[
\frac{d\Delta_i}{dp} = R_B'(p) - R_F'(p) - \lambda_i \cdot \sigma_B'(p) - \kappa_i \cdot K_A'(p) - \rho_i \cdot R_A'(p) + K_N'(p)
\]

Under the maintained monotonicity conditions (M1)--(M6), each term is
non-negative and at least one is strictly positive:

\begin{table}[h]
\centering
\begin{tabular}{lcl}
\toprule
Term & Sign & Reason \\
\midrule
$R_B'(p)$ & $> 0$ & (M1) \\
$-R_F'(p)$ & $\geq 0$ & (M6) \\
$-\lambda_i \cdot \sigma_B'(p)$ & $> 0$ & $\lambda_i > 0$, $\sigma_B'(p) < 0$ by (M2) \\
$-\kappa_i \cdot K_A'(p)$ & $> 0$ & $\kappa_i \geq 0$, $K_A'(p) < 0$ by (M3) \\
$-\rho_i \cdot R_A'(p)$ & $> 0$ & $\rho_i \geq 0$, $R_A'(p) < 0$ by (M4) \\
$K_N'(p)$ & $> 0$ & (M5) \\
\bottomrule
\end{tabular}
\end{table}

\noindent Therefore $d\Delta_i/dp > 0$: the advantage of Exit is strictly increasing
in $p$. Strict inequality in (M6) strictly strengthens the result; the
weak inequality stated in (M6) is sufficient to preserve it.

At $p = 0$ (no one has exited), $\Delta_i(0) < 0$ since the adoption penalty
dominates and the capturable system provides positive baseline value. At
$p = 1$ (full adoption), $\Delta_i(1) > 0$ since exit-dominant payoffs
strictly exceed legacy payoffs under universal adoption: the neutral system's
network effects fully manifest while the capturable system has been abandoned.
By continuity of $\Delta_i$ in $p$ and the intermediate value theorem, there
exists a threshold $p_i^* \in (0, 1)$ satisfying $\Delta_i(p_i^*) = 0$. For
$p > p_i^*$, $\Delta_i(p) > 0$ and Exit strictly dominates.

Under Assumption 1, $R_F(p) < 0$ in real terms at $p = 0$ (sovereign
debt overhangs at the contemporary scale historically resolve through
debasement, restructuring, or default rather than through fiscal
austerity alone; Reinhart and Rogoff\textsuperscript{31}). This ensures that even at
$p = 0$, the first term $[R_B(0) - R_F(0)]$ provides positive
contribution, lowering $p_i^*$ toward zero for many actor types.

For the marginal allocation argument: even for actors with $p < p_i^*$,
a small allocation $\varepsilon > 0$ provides $u_i(w_i = \varepsilon) > u_i(w_i = 0)$
because the first-order gain from network participation exceeds the
first-order volatility cost for sufficiently small $\varepsilon$. This
holds for a wide range of parameter values.
\end{proof}

\begin{remark}
The game matrix at the individual level:

\begin{table}[h]
\centering
\begin{tabular}{lcc}
\toprule
 & Others: Stay & Others: Exit \\
\midrule
\textbf{Player $i$: Stay} & $R_F(0) < 0$ & $R_F(p) - K_N(p)$ \\
\textbf{Player $i$: Exit} & $R_B(0) - K_A(0)$ & $R_B(p)$ \\
\bottomrule
\end{tabular}
\end{table}

Under Assumption 1 ($R_F(0) < 0$), Exit yields higher payoff in both
columns for actors whose threshold $p_i^*$ is at or below the current
adoption level.
\end{remark}

\subsection{4.2 Coordination Failure}

\begin{theorem}[Coalition Instability]
When Exit is available and unpunishable, and individual exit payoffs exceed
coalition benefits (Theorem 1), no coalition can prevent member defection and
sustain coordinated Stay.
\end{theorem}

\begin{proof}
Consider a coalition $C \subseteq N$ maintaining Stay. For this
coalition to be self-enforcing in the sense of Bernheim, Peleg, and
Whinston\textsuperscript{3}, no member $j \in C$ can profit from unilateral
defection.

By Theorem 1, $u_j(\text{Exit}) > u_j(\text{Stay})$ for $j$ when $p$ exceeds
$p_j^*$; this payoff advantage already accounts for the adoption penalty
$K_A(p)$. Property P3 (permissionless access) means the institutional cost
of defection is zero---no permission, regulatory approval, or cooperation is
required to initiate settlement. The economic advantage of exiting (net of
adoption penalties) exceeds the coalition's ability to enforce Stay.

The coalition faces a standard free-rider problem: the benefit of
collective Stay is a public good (maintained value of capturable system),
but the benefit of individual defection is a private good (captured
network effects, avoidance of debasement). Under Assumption 2, the
private good dominates.

The dynamics compound: the first actor to defect captures fleeing
capital, raising $K_N(p)$ for remaining coalition members and
accelerating further defection.

Formally:

\[
u_j(\text{defect from } C) > u_j(\text{remain in } C)
\]

for all $j \in C$, making the all-Stay profile not coalition-proof.
\end{proof}

\begin{remark}
The first-mover dynamic is an asymmetry familiar from the history of
reserve-currency transitions (Eichengreen, Mehl, and Chi\c{t}u\textsuperscript{11}):
the defecting sovereign captures fleeing capital. El Salvador adopted
Bitcoin as legal tender in 2021 (Bukele\textsuperscript{8}), cited here as
existence proof that sovereign-scale defection is feasible, not as
evidence of equilibrium trajectory. Each sovereign defection raises
$K_N(p)$ for remaining coalition members.
\end{remark}

\subsection{4.3 Absorbing State}

\begin{theorem}[Absorption]
The state $(\text{Exit}, \text{Exit})$ is absorbing: once entered, there
exists no feasible sequence of individual actions leading back to
$(\text{Stay}, \text{Stay})$.
\end{theorem}

\begin{proof}
Define the aggregate adoption process $\{p_t\}_{t \geq 0}$ on state
space $[0, 1]$.

For $p_t \geq p_c$ (critical mass threshold), we show that
$p_{t+1} \geq p_t$ with probability 1.

\smallskip
\noindent\textbf{Step 1 (No Exit-to-Stay switching).} Any actor who has
chosen Exit at time $t$ did so because $\Delta_i(p_t) \geq 0$. By Step 2
(positive drift in $p$), $p_{t+s} \geq p_t$ for all $s \geq 0$, and by
Theorem 1 $\Delta_i$ is increasing in $p$, so $\Delta_i(p_{t+s}) \geq
\Delta_i(p_t) \geq 0$ at every future date. Infrastructure lock-in
costs (custody solutions, regulatory frameworks, accounting standards)
provide additional asymmetry against reversal but are not load-bearing
for the argument. No rational actor switches from Exit to Stay.

\smallskip
\noindent\textbf{Step 2 (Positive drift).} For actors currently at Stay:
$\Delta_i(p_t)$ is increasing in $p_t$ (shown in Theorem 1). As $p_t$
increases, more actors cross their thresholds $p_i^*$, increasing
$p_{t+1}$.

\smallskip
\noindent\textbf{Step 3 (Convergence).} The process $\{p_t\}$ is monotone
non-decreasing and bounded above by 1. By the monotone convergence
theorem for bounded monotone sequences, $p_t \to p^*$ for some
$p^* \leq 1$.

Under Assumption 5 (continuous support of thresholds on $[0,1]$), if
$p^* < 1$ there exists an actor $j$ at Stay with $\Delta_j(p^*) > 0$,
contradicting stationarity at $p^*$. Therefore $p^* = 1$. Without
Assumption 5, $p^*$ equals the supremum of the support of the threshold
distribution; when a measure-positive subpopulation has $p_i^* > 1$
(never-exit types), $p^* < 1$ and the absorbing state concentrates at
that supremum rather than at unity.

\smallskip
\noindent\textbf{Step 4 (No-return via higher-order belief breakdown).}
Observed exits are public signals that shift higher-order beliefs: each
actor revises their estimate of what other actors now believe about system
capturability. Once cumulative exits cross the critical mass $p^*$, the
higher-order belief that \emph{others believe the system is capturable}
becomes common knowledge among remaining actors, and the coordination
equilibrium on the legacy system fails. Return would require
re-establishing common knowledge of non-capturability---a coordination task
that the original narrative monopoly, having been publicly falsified by
observed exits, can no longer supply. The narrative apparatus that once
coordinated on Stay is destroyed; no new enforcement mechanism (ruled out by
Assumption 1) can rebuild it. Hence the absorbing state is one-sided: entry
requires only a critical mass of exits, exit requires reconstruction of a
destroyed common-knowledge structure. The state $p = p^*$ is therefore
absorbing: $P(p_{t+1} = p^* \mid p_t = p^*) = 1$. A return cartel faces the
same coordination failure as a stay cartel (Theorem 2).
\end{proof}

\subsection{4.4 Cascade Dynamics}

\begin{proposition}[Adoption Cascades]
Given Theorems 1--3, adoption occurs in bursts (\emph{cf.}\ Bikhchandani,
Hirshleifer, and Welch\textsuperscript{5} on informational cascades; our mechanism
is payoff-driven rather than information-driven).
\end{proposition}

The cascade operates through the interaction of $K_A(p)$ and $K_N(p)$:
as $p$ increases, the adoption penalty falls and the non-adoption penalty
rises, pushing additional actors past their thresholds $p_i^*$. This
creates punctuated equilibria: periods of stability followed by rapid
adoption bursts when a critical mass of actors simultaneously crosses
their thresholds.

\noindent\textbf{Comparative statics.} By the implicit function theorem applied
to $\Delta_i(p_i^*) = 0$:

\[
\frac{dp_i^*}{d\lambda_i} = -\frac{\partial\Delta_i / \partial\lambda_i}{d\Delta_i / dp} > 0
\]

More risk-averse actors have higher thresholds (they wait longer). But
they still eventually exit because $d\Delta_i/dp > 0$ pushes the advantage
past any finite risk aversion as $p \to 1$.

% ============================================================
% 5. FALSIFICATION
% ============================================================

\section{5. Falsification}

We separate falsifiers into three bands. Axiom falsifiers break a
maintained assumption. Theorem falsifiers break a derived result while
leaving the axioms intact. Scope conditions delimit the domain of
applicability and are not Popperian falsifiers.

\begin{table}[h]
\centering
\begin{tabularx}{\textwidth}{lXl}
\toprule
ID & Condition & Breaks \\
\midrule
\multicolumn{3}{l}{\emph{Axiom falsifiers}} \\
F1 & Global coordination cost becomes sublinear & Assumption 1 \\
F3 & (Stay, Stay) is stable equilibrium when Exit exists & Assumption 2 \\
F5 & Quantum computing breaks cryptographic primitives before migration & Assumption 3 \\
\midrule
\multicolumn{3}{l}{\emph{Theorem falsifiers}} \\
F4 & Stable cartel prevents Exit indefinitely & Theorem 2 \\
\midrule
\multicolumn{3}{l}{\emph{Scope conditions}} \\
F2 & An alternative asset satisfies P1--P7 (tested in Hash\textsuperscript{14}) & Uniqueness \\
F6 & AI agents gain enforceable legal personhood (tested in Hash\textsuperscript{15}) & Limiting case \\
\bottomrule
\end{tabularx}
\end{table}

Note what does \emph{not} falsify the model: price declines, regulatory
actions against specific intermediaries, developer controversies, or
energy consumption debates. These affect adoption timing, not
equilibrium structure.

% ============================================================
% 6. LIMITATIONS
% ============================================================

\section{6. Limitations}

\noindent\textbf{Continuous strategies.} The binary Exit/Stay structure is a
modeling choice. Real actors face continuous allocation decisions---a
pension fund does not choose between 0\% and 100\% but between 0\%, 1\%,
5\%. The marginal allocation argument in \S4.1 partially addresses this.
Under continuous allocation $w_i \in [0,1]$, the equivalent claim is
that aggregate weight on neutral settlement approaches unity in the
absorbing state; formal proof of the continuous-strategy analog under
mean-variance or constant relative risk aversion (CRRA) preferences is deferred to future work.

\noindent\textbf{Stochastic stability.} Theorem 3 establishes deterministic
monotone convergence. Stochastic stability in the sense of Kandori,
Mailath, and Rob\textsuperscript{20}, under perturbed best-response dynamics with
vanishing mutation rate, is deferred. We expect the absorbing state
to be stochastically stable under standard perturbation assumptions
but do not prove it here.

\noindent\textbf{Coalition-proofness strength.} Theorem 2 establishes that no
coalition sustains Stay against individual defection. This is weaker
than coalition-proof Nash nonexistence in the full Bernheim, Peleg, and
Whinston\textsuperscript{3} sense, which requires robustness against self-enforcing
sub-coalition deviations. The stronger claim is consistent with the
paper's economic logic but is not proved here.

\noindent\textbf{Monotonicity conditions.} (M1)--(M6) are empirically motivated
but not derived from first principles. Each could be violated in
specific regimes (for example, $R_B'(p)$ could be negative during a
network congestion crisis). The model's robustness depends on these
conditions holding \emph{on average and in the long run}, not at every
instant. Formalizing this through time-averaged conditions is a
natural extension.

% ============================================================
% 7. CONCLUSION
% ============================================================

\section{7. Conclusion}

The argument rests on four axioms: multipolarity is persistent, actors
are self-interested, cryptography works, and network effects compound.
From these we derive a monotone increasing payoff advantage, coalition
instability, and absorbing dynamics. Bitcoin is the unique asset serving
as the Exit destination (Hash\textsuperscript{14}); autonomous agents
face the strongest version of this convergence pressure
(Hash\textsuperscript{15}). Six falsification conditions across the
series identify when and how these claims fail.

The framework developed here has been applied and extended in three
companion papers. Hash\textsuperscript{14} applies the seven necessary
properties as an elimination test across seven candidate settlement
assets and finds that no non-Bitcoin candidate jointly satisfies
them. Hash\textsuperscript{15} extends the Exit Game to
autonomous-agent settlement under zero-trust conditions, where
neutrality becomes a structural requirement rather than a preference.
Hash\textsuperscript{16} studies the predator-prey dynamics that
arise when neutral settlement coexists with enforcement-driven
monetary regimes. The seven properties of this paper function as the
common spine across the series; the elimination result of
Hash\textsuperscript{14} is an external structural check the framework
had to survive.

% ============================================================
% ACKNOWLEDGEMENT
% ============================================================

\section{Acknowledgement}

The author thanks participants and readers of the February 2026 preprint for feedback that informed this revision.

% ============================================================
% CONFLICT OF INTEREST
% ============================================================

\section{Conflict of Interest}

The author holds a position in Bitcoin. The author favors the Knots
implementation, which limits Bitcoin block space to financial data
only, in order to preserve the decentralization of the network and
reduce attack surfaces.

% ============================================================
% REFERENCES
% ============================================================

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% ============================================================
% APPENDIX
% ============================================================

\clearpage

\appendix

\section{Notation}

\begin{table}[h]
\centering
\begin{tabularx}{\textwidth}{lX}
\toprule
Symbol & Definition \\
\midrule
$G$ & Settlement game $(N, S, u)$ \\
$N$ & Set of capital allocators \\
$S_i$ & Strategy set: $\{\text{Exit}, \text{Stay}\}$ \\
$u_i$ & Utility function \\
$p$ & Fraction of capital at Exit \\
$p_i^*$ & Threshold for player $i$ \\
$\Delta_i(p)$ & Payoff differential: $u_i(\text{Exit}) - u_i(\text{Stay})$ \\
$R_B(p)$ & Expected real return on neutral settlement asset \\
$R_F(p)$ & Expected real return on capturable assets \\
$\sigma_B(p)$ & Volatility of neutral settlement asset \\
$\sigma_F$ & Volatility of capturable assets \\
$K_A(p)$ & Adoption penalty \\
$R_A(p)$ & Regulatory penalty (decreases as regulatory clarity improves) \\
$K_N(p)$ & Non-adoption penalty \\
$\lambda_i, \kappa_i, \rho_i$ & Risk aversion, career penalty, regulatory penalty \\
$CS_A$ & Capture surface for actor $A$: $\{\theta \in \Theta : A \text{ can unilaterally change } \theta\}$ \\
(M1)--(M6) & Monotonicity conditions \\
\bottomrule
\end{tabularx}
\end{table}

\end{document}
