# Bitcoin Exit Dominance in Monetary Coordination Games

**Sean Hash**
bitcoingametheory.com
sean@bitcoingametheory.com

**Date:** Preprint: February 2026 (SSRN 6299081). Revised: April 2026.
**JEL Codes:** C72 (Noncooperative Games), E42 (Monetary Systems, Standards, Regimes)
**Keywords:** game theory, Nash equilibrium, monetary coordination, neutral settlement, absorbing state, coordination failure, threshold strategy, monotone comparative statics

---

## Abstract

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"). 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 explicitly 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 Hash
(2026b). The analysis is descriptive: we characterize equilibrium
structure without predicting timing, price, or recommending portfolio
allocations.

---

## Revision Note — April 2026

This revision tightens the scope of several citations from the preprint (SSRN 6299081, February 2026) after a re-reading against primary sources in preparation for journal submission. Each entry has been re-checked and the bibliography is released as a companion artifact. The model, the Exit Game, Theorems 1–3, Assumptions 1–4, and the seven necessary properties are unchanged from the preprint.

Three sets of clarifications. First, several citations are narrowed to what the source actually establishes: §2.2 (Huberman et al. cited only for their congestion-pricing result, not for governance neutrality in our sense); §2.4 (Hirschman cited as intellectual context, distinct from the strategic Exit defined in-model); §4.4 (Bikhchandani, Hirshleifer, and Welch cited as contrast — Paper 1's cascade is payoff-driven, theirs is information-driven). Second, paired canonical attributions are widened where a single anchor was carrying weight that belongs to two: §2.3 (Milgrom–Shannon with Topkis), §3.1 (David with Arthur), §3.4 (Kyle with Amihud, and Katz–Shapiro with Farrell–Saloner). Third, the Conclusion now situates Paper 1 within the four-paper sequence; the seven necessary properties have been applied as an elimination test in Hash (2026b) and the framework extended in Hash (2026c, 2026d).

No theorem, proof, assumption, or equation is changed. The preprint text (SSRN 6299081, February 2026) is preserved at `bitcoingametheory.com/papers/1-exit-dominance/archive/preprint-ssrn-6299081/BGT-PAPER-1-preprint.pdf`.

---

## 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-to-GDP exceeded 300% in
2024 (IIF, 2024). No post-1800 nation has repaid debt exceeding 120%
of GDP through taxation alone; debasement, restructuring, or default
is the historically observed path (Reinhart and Rogoff, 2011). The
coordination problem is not whether states *will* debase, but whether
any mechanism exists through which actors can settle *despite*
debasement.

The literature addresses components of this problem — mining incentives
(Biais et al., 2019; Eyal and Sirer, 2014), attack economics (Budish,
2018), price-security dynamics (Pagnotta, 2022) — but does not
integrate these into a unified strategic analysis from the allocator's
perspective. Brunnermeier, James, and Landau (2019) identify the
coordination failure but propose institutional solutions rather than
analyzing the equilibrium outcome. The gap is not a missing paper. It is
a missing perspective.

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 (2026b). A third paper (Hash, 2026c)
extends the analysis to autonomous AI agents, for whom neutral
settlement is not a preference but a structural requirement.

---

## 2. Related Work

### 2.1 Game Theory of Proof-of-Work Consensus

The foundational result is Biais, Bisiere, Bouvard, and Casamatta (2019):
following the longest chain constitutes a Markov Perfect Equilibrium in
a stochastic game among miners. This is the paper that moved Bitcoin
mining from engineering folklore into proper game theory, and subsequent
work largely builds on their framework. Eyal and Sirer (2014) 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
(2018) 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.

### 2.2 Monetary Competition and Coordination

Brunnermeier, James, and Landau (2019) 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 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 (2021) 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.

### 2.3 Monotone Comparative Statics

The monotonicity conditions (M1)-(M5) that drive our main result belong
to the framework of monotone comparative statics developed by Milgrom
and Shannon (1994) and Topkis (1998). In their language, the payoff
differential Δ_*i*(*p*) exhibits increasing differences in (*p*, 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.

### 2.4 Network Effects and Focal Points

Katz and Shapiro (1985, 1986) 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 *is* its adoption.
Schelling (1960) 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 (1970)
"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.

---

## 3. Model

### 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.

**Assumption 1 (Multipolarity).** No single entity permanently governs
global economic activity. Power distributes across competing centers with
superlinear coordination costs.

*Falsification:* A single coordinator achieves permanent, stable control
over global monetary policy with sublinear coordination costs (F1).

**Assumption 2 (Rational Self-Interest).** Actors optimize for
self-interest. When defection from cooperative agreements is unpunished
or unpunishable, actors defect.

*Falsification:* Stable cooperation persists indefinitely among
self-interested actors without enforcement mechanisms (F3).

**Assumption 3 (Computational Hardness).** Certain mathematical problems
remain computationally intractable. Digital scarcity and cryptographic
custody are implementable.

**Assumption 4 (Network Effect Persistence).** Past critical mass,
switching costs exceed marginal gains of alternatives. Incumbency
compounds (David, 1985; Arthur, 1989).

### 3.2 Definitions

**Definition 1 (Neutral Settlement).** An asset *S* is a *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.

**Definition 2 (Exit).** The action of moving capital from a capturable
system to a neutral settlement asset.

**Definition 3 (Capture Surface).** For a monetary system with parameter
space Θ, the capture surface for actor *A* is *CS_A* = {θ ∈ Θ : *A* can
unilaterally change θ}. A system is neutral if |*CS_A*| = 0 for all *A*.
The terminology adapts "attack surface" from security engineering to
institutional capture (cf. Stigler, 1971 on regulatory capture).

**Definition 4 (Absorbing State).** A state *p** is absorbing if
P(*p*_{*t*+1} = *p** | *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 (cf. Arthur, 1989 on lock-in).

### 3.3 Necessary Properties

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

| D1 Requirement | Attack If Missing | Required Property |
|----------------|-------------------|-------------------|
| 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 |

**Proposition 1 (Necessity).** P1-P7 are individually necessary.
Removing any single property enables the corresponding attack, violating
Definition 1.

*Proof.* By construction from the derivation table. ∎

Bitcoin is the unique asset satisfying P1-P7 — the systematic
elimination across seven asset classes appears in Hash (2026b).

### 3.4 Game Structure

We model monetary coordination as a non-cooperative game:

**Game** *G* = (*N*, *S*, *u*) where:

- *N* = {1, 2, ..., *n*} is the set of capital allocators.
- *S_i* = {Exit, Stay} is the strategy set for player *i*.
- *u_i* : *S* → ℝ is the utility function for player *i*.

Let *p* ∈ [0, 1] denote the fraction of total capital that has chosen
Exit. We assume *R_B*, *σ_B*, *K_A*, *R_A*, and *K_N* are continuous
functions of *p*, bounded on [0,1]. The utility functions are:

*u_i*(Exit, *s*_{-*i*}) = *R_B*(*p*) − *λ_i* · *σ_B*(*p*) − *κ_i* · *K_A*(*p*) − *ρ_i* · *R_A*(*p*)

*u_i*(Stay, *s*_{-*i*}) = *R_F* − *λ_i* · *σ_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, 1985; Farrell and Saloner, 1985). *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.

**Maintained monotonicity conditions:**

(M1) *R_B*'(*p*) > 0 — Network effects increase return (Assumption 4)
(M2) *σ_B*'(*p*) < 0 — Deeper markets reduce volatility (Kyle, 1985; Amihud, 2002)
(M3) *K_A*'(*p*) < 0 — Adoption penalty falls with adoption
(M4) *R_A*'(*p*) < 0 — Regulatory penalty decreases as adoption normalizes compliance
(M5) *K_N*'(*p*) > 0 — Non-adoption penalty rises as competitors exit

The market-microstructure basis for (M2) is standard: deeper markets
lower the price impact of order flow (Kyle's λ decreases with depth),
and lower price impact translates to lower realized return volatility
at a fixed rate of information arrival (Amihud, 2002).

---

## 4. Main Results

### 4.1 Exit Dominance

**Theorem 1 (Monotone Exit Advantage).** Under Assumptions 1-4 and
maintained conditions (M1)-(M5):

(i) The payoff differential Δ_*i*(*p*) is strictly increasing in *p*
for all player types.

(ii) For each player *i*, there exists a threshold *p_i** ∈ [0, 1]
such that Exit is the unique best response for *p* > *p_i**.

(iii) Under Assumption 1 (*R_F* < 0), thresholds cluster near zero
for actors with typical risk preferences.

*Proof.* Define the payoff differential:

Δ_*i*(*p*) = *u_i*(Exit) − *u_i*(Stay)

= [*R_B*(*p*) − *R_F*] − *λ_i*[*σ_B*(*p*) − *σ_F*] − *κ_i* · *K_A*(*p*) − *ρ_i* · *R_A*(*p*) + *K_N*(*p*)

Taking the derivative with respect to *p*:

dΔ_*i*/d*p* = *R_B*'(*p*) − *λ_i* · *σ_B*'(*p*) − *κ_i* · *K_A*'(*p*) − *ρ_i* · *R_A*'(*p*) + *K_N*'(*p*)

Under the maintained monotonicity conditions (M1)-(M5), each term is
positive:

| Term | Sign | Reason |
|------|------|--------|
| *R_B*'(*p*) | > 0 | (M1) |
| −*λ_i* · *σ_B*'(*p*) | > 0 | *λ_i* > 0, *σ_B*'(*p*) < 0 by (M2) |
| −*κ_i* · *K_A*'(*p*) | > 0 | *κ_i* ≥ 0, *K_A*'(*p*) < 0 by (M3) |
| −*ρ_i* · *R_A*'(*p*) | > 0 | *ρ_i* ≥ 0, *R_A*'(*p*) < 0 by (M4) |
| *K_N*'(*p*) | > 0 | (M5) |

Therefore dΔ_*i*/d*p* > 0: the advantage of Exit is strictly increasing
in *p*.

For each player *i*, there exists a threshold *p_i** satisfying
Δ_*i*(*p_i**) = 0 (by the intermediate value theorem, since Δ_*i* is
continuous). For *p* > *p_i**, Δ_*i*(*p*) > 0 and Exit strictly
dominates.

Under Assumption 1, *R_F* < 0 in real terms (sovereign debt overhangs
at the contemporary scale historically resolve through debasement,
restructuring, or default rather than through taxation alone;
Reinhart and Rogoff, 2011). This ensures that even at *p* = 0, the
first term [*R_B*(0) − *R_F*] 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 *ε* > 0 provides *u_i*(*w_i* = *ε*) > *u_i*(*w_i* = 0)
because the first-order gain from network participation exceeds the
first-order volatility cost for sufficiently small *ε*. This is not a
close call for most parameter values. ∎

**Remark.** The game matrix at the individual level:

|  | Others: Stay | Others: Exit |
|---|---|---|
| **Player *i*: Stay** | *R_F* < 0 | *R_F* − *K_N*(*p*) |
| **Player *i*: Exit** | *R_B*(0) − *K_A*(0) | *R_B*(*p*) |

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

### 4.2 Coordination Failure

**Theorem 2 (Coalition Instability).** No coalition can sustain coordinated
Stay when Exit is available and unpunishable.

*Proof.* Consider a coalition *C* ⊆ *N* maintaining Stay. For this
coalition to be self-enforcing in the sense of Bernheim, Peleg, and
Whinston (1987), no member *j* ∈ *C* can profit from unilateral
defection.

But by Theorem 1, *u_j*(Exit) > *u_j*(Stay) for *j* when *p* exceeds
*p_j**. Property P3 (permissionless access) ensures the defection cost
is approximately zero — no institutional cooperation is required to
initiate settlement.

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 sovereign to defect captures fleeing
capital, an asymmetry familiar from the history of reserve-currency
transitions (Eichengreen, Mehl, and Chițu, 2017). El Salvador adopted
Bitcoin as legal tender in 2021 (Bukele, 2021). Each subsequent
defection raises *K_N*(*p*) for remaining coalition members,
accelerating further defection.

Formally:

*u_j*(defect from *C*) > *u_j*(remain in *C*)

for all *j* ∈ *C*, making the all-Stay profile not coalition-proof. ∎

### 4.3 Absorbing State

**Theorem 3 (Absorption).** The state (*Exit*, *Exit*) is absorbing:
once entered, there exists no feasible sequence of individual actions
leading back to (*Stay*, *Stay*).

*Proof.* Define the aggregate adoption process {*p_t*}_{t≥0} on state
space [0, 1].

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

**Step 1 (No Exit-to-Stay switching).** For any actor currently at Exit:
switching to Stay incurs infrastructure lock-in costs (custody solutions,
regulatory frameworks, accounting standards) and Δ_*i*(*p_t*) > 0 by
Theorem 1. No rational actor switches from Exit to Stay.

**Step 2 (Positive drift).** For actors currently at Stay: Δ_*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}*.

**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* → *p** for some *p** ≤ 1.

If *p** < 1, then — assuming the distribution of thresholds {*p_i**}
has continuous support on [0, 1] (i.e., no gap in actor types) — there
exists an actor *j* at Stay with Δ_*j*(*p**) > 0. This contradicts
stationarity at *p**. Therefore *p** = 1.

The state *p* = 1 is absorbing: P(*p_{t+1}* = 1 | *p_t* = 1) = 1, since
return coordination requires re-establishing trust in a capturable
system, but exiting *is* the revelation that the system was capturable —
and revelations do not un-reveal. A return cartel faces the same
coordination failure as a stay cartel (Theorem 2). ∎

### 4.4 Cascade Dynamics

**Proposition 2 (Adoption Cascades).** Given Theorems 1-3, adoption
occurs in bursts (cf. Bikhchandani, Hirshleifer, and Welch, 1992 on
informational cascades; our mechanism is payoff-driven rather than
information-driven).

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.

**Comparative statics.** By the implicit function theorem applied to
Δ_*i*(*p_i**) = 0:

d*p_i**/d*λ_i* = −(∂Δ_*i*/∂*λ_i*) / (dΔ_*i*/d*p*) > 0

More risk-averse actors have higher thresholds (they wait longer). But
they still eventually exit because dΔ_*i*/d*p* > 0 pushes the advantage
past any finite risk aversion as *p* → 1.

---

## 5. Falsification

| ID | Condition | Breaks |
|----|-----------|--------|
| F1 | Global coordination cost becomes sublinear | Assumption 1 |
| F2 | An alternative asset satisfies P1-P7 (tested in Hash, 2026b) | Uniqueness |
| F3 | (Stay, Stay) is stable equilibrium when Exit exists | Theorem 1 |
| F4 | Stable cartel prevents Exit indefinitely | Theorem 2 |
| F5 | Quantum computing breaks cryptographic primitives before migration | Assumption 3 |
| F6 | AI agents gain enforceable legal personhood (tested in Hash, 2026c) | Limiting case |

For formal dispute procedures and evidence standards, see bitcoingametheory.com/rfc/BGT-DISPUTE.txt.

Note what does *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

The binary Exit/Stay model is a simplification. 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
(Section 4.1) partially addresses this, but a proper continuous-strategy
extension with portfolio optimization would strengthen the formalism.
This is the most natural direction for future work.

A deeper concern: the monotonicity conditions (M1)-(M5) are empirically
motivated but not derived from first principles. Each could be violated
in specific regimes (e.g., *R_B*'(*p*) could be negative during a
network congestion crisis). The model's robustness depends on these
conditions holding *on average and in the long run*, not at every instant.
Formalizing this distinction — perhaps through time-averaged conditions
— would strengthen the theoretical foundation.

---

## 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, 2026b); autonomous agents face the
strongest version of this convergence pressure (Hash, 2026c). 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 (2026b) 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 (2026c) extends the Exit Game to autonomous-agent settlement under zero-trust conditions, where neutrality becomes a structural requirement rather than a preference. Hash (2026d) 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 (2026b) is an external structural check the framework had to survive.

---

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---

## Appendix: Notation

| Symbol | Definition |
|--------|-----------|
| *G* | Settlement game (*N*, *S*, *u*) |
| *N* | Set of capital allocators |
| *S_i* | Strategy set: {Exit, Stay} |
| *u_i* | Utility function |
| *p* | Fraction of capital at Exit |
| *p_i** | Threshold for player *i* |
| Δ_*i*(*p*) | Payoff differential: *u_i*(Exit) − *u_i*(Stay) |
| *R_B*(*p*) | Expected real return on neutral settlement asset |
| *R_F* | Expected real return on capturable assets |
| *σ_B*(*p*) | Volatility of neutral settlement asset |
| *σ_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 |
| *λ_i*, *κ_i*, *ρ_i* | Risk aversion, career penalty, regulatory penalty |
| *CS_A* | Capture surface for actor *A*: {θ ∈ Θ : *A* can unilaterally change θ} |
| (M1)-(M5) | Monotonicity conditions |