## 1. The entrepreneur chooses a level of monetizationπ

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---

### Game structure and strategies
- Timing each period t:
  - Entrepreneur chooses monetization πt and revenue sharing αt.
  - Users simultaneously decide:
    - (a) New arrivals in period t choose to join or not.
    - (b) Existing users choose to stay or leave.
  - Utilities realize.
- Strategy definitions:
  - Entrepreneur’s strategy: mapping from histories to (πt, αt).
  - Users’ strategy: mapping from histories to binary join/stay decisions.
  - Tie-breaking rules:
    - Newly arriving indifferent users join.
    - Indifferent locked-in users remain.

### Myopic revenue maximization (per-period problem) and key formulas
- Per-period optimization:
  - max αt, πt αt πt φ(μt)
  - s.t. V(μt) − k πt^2 + (1−αt) μt πt φ(μt) = ˆu
  - 1 ≥ αt ≥ 0
- Solution revenue ψ(μt, ˆu):
  - ψ(μt, ˆu) =
    - μt V(μt) + φ(μt)^2 / (4 k μt) − μt ˆu if [φ(μt) / (2 k μt)]^2 ≥ V(μt) − ˆu / k
    - φ(μt) √(V(μt) − ˆu / k) if [φ(μt) / (2 k μt)]^2 < V(μt) − ˆu / k
- Interpretation of cases:
  - First case: entrepreneur shares revenue with users (αt ∈ (0,1)).
  - Second case: entrepreneur keeps all revenue (αt = 1).
- Properties:
  - ψ is increasing in μt.
  - ψ is decreasing in ˆu.
  - Feasibility: ˆu cannot exceed what users would get if all revenue distributed to users.
  - Dependence on k (users’ aversion to monetization):
    - Small k → higher monetization and revenue sharing.
    - Large k → less monetization and no revenue sharing.
- Explicit optimal choices when feasible:
  - If φ(μ_t)^2 / (4kμ_t^2) + V(μ_t) ≥ ˆu and [φ(μ_t) / (2kμ_t)]^2 ≥ (V(μ_t)−ˆu)/k:
    - π_t = φ(μ_t) / (2kμ_t)
    - α_t = 1/2 + 2kμ_t^2 (V(μ_t)−ˆu) / φ(μ_t)^2
    - Revenue: μ_t V(μ_t) + φ(μ_t)^2 / (4kμ_t) − μ_t ˆu
  - If φ(μ_t)^2 / (4kμ_t^2) + V(μ_t) ≥ ˆu and [φ(μ_t) / (2kμ_t)]^2 < (V(μ_t)−ˆu)/k:
    - π_t = sqrt((V(μ_t)−ˆu)/k)
    - α_t = 1
    - Revenue: sqrt((V(μ_t)−ˆu)/k) φ(μ_t)
  - If φ(μ_t)^2 / (4kμ_t^2) + V(μ_t) < ˆu then entrepreneur cannot ensure utility ˆu for users.

### Growth vs. exploitation trade-off and thresholds
- Exploitation payoff if entrepreneur exploits existing users forever (starting with μ_{t−1} existing users):
  - Present value = 1/(1−δ) ψ(μ_{t−1}, −(1−δ) u)
  - Per-period utility provided to users is −(1−δ) u so discounted utility equals −u.
- Grow once then exploit (grow in period t, exploit thereafter):
  - Revenue = ψ(g(μ_{t−1}), δ u) + δ/(1−δ) ψ(g(μ_{t−1}), −(1−δ) u)
- Indifference platform size ̄μ solves:
  - 1/(1−δ) ψ(̄μ, −(1−δ) u) = ψ(g(̄μ), δ u) + δ/(1−δ) ψ(g(̄μ), −(1−δ) u)
  - ̄μ is the platform size where entrepreneur is indifferent between exploiting now vs. growing once then exploiting.
- Existence conditions:
  - ̄μ exists under economically relevant conditions (growth slows sufficiently or network effects eventually diminish).
  - Tractable example: V(μt) constant, φ(μt) linear in μt, g(μt) = μt + γ(μt) with γ(μt) strictly decreasing, strictly positive, γ(μt) → 0 as μt → ∞.

### Constructive equilibrium: grow-then-exploit and join-if-compensated
- Assumption for construction:
  - Platform sufficiently profitable at size ̄μ to provide δ u to users:
    - φ(̄μ)^2 / (4 k ̄μ^2) + V(̄μ) ≥ δ u (Inequality II.4)
- Entrepreneur strategy — Grow-then-exploit:
  - If g(μt−1) < ̄μ: set (πt, αt) to maximize ψ(μt, ˆu t) with ˆu t = 0 and μt = g(μt−1).
  - If μt−1 < ̄μ and ̄μ ≤ g(μt−1): set (πt, αt) to maximize ψ(μt, ˆu t) with ˆu t = δ u and μt = g(μt−1).
  - If ̄μ ≤ μt−1: set (πt, αt) to maximize ψ(μt, ˆu t) with ˆu t = −(1−δ) u and μt = μt−1.
- Users’ strategy — Join-if-compensated:
  - In period of arrival, join if:
    1. g(μt−1) < ̄μ and πt, αt imply ˆu t ≥ 0 for μt = g(μt−1).
    2. μt−1 < ̄μ and ̄μ ≤ g(μt−1) and πt, αt imply ˆu t ≥ δ u for μt = g(μt−1).
  - If locked in, stay if πt, αt imply ˆu t ≥ −(1−δ) u for μt ≥ μt−1.

### Equilibrium existence and failure to attract users
- Proposition 1:
  - If inequality II.4 holds, there exists a sub-game perfect Nash equilibrium in which:
    - Entrepreneur plays grow-then-exploit.
    - Users play join-if-compensated.
  - Intuition: newly arriving users are indifferent and join; locked-in users strictly prefer staying before exploitation and are indifferent at exploitation onset; profitable deviations are ruled out by departures or reduced revenue.
- If platform cannot guarantee δ u at ̄μ:
  - φ(̄μ)^2 / (4 k ̄μ^2) + V(̄μ) < δ u (Inequality II.5)
  - Entrepreneur cannot pay δ u in last growth period → no new joins.
  - Backward induction implies no users join at any period.
- Define μ (μ̄ in text) solving:
  - φ(μ)^2 / (4 k μ^2) + V(μ) = δ u (Equation II.6)
  - μ is minimum platform size required to feasibly provide δ u.
  - If ̄μ ≥ μ entrepreneur can attract users; if ̄μ < μ entrepreneur attracts no users.

### Comparative statics on locked-in effect and growth threshold
- Lemma 1:
  - μ strictly increases in u.
  - As u → ∞ it holds that μ → ∞.
- Lemma 2:
  - ̄μ strictly decreases in u.
  - As u → 0 it holds that ̄μ → ∞.
- Economic interpretation:
  - Larger locked-in effect parameter u increases required minimum platform size μ to compensate users.
  - Larger u reduces ̄μ: entrepreneur stops growing and starts exploiting earlier because temptation to exploit increases with locked-in effect.
- Corollary 1:
  - There exists u∗ such that if u > u∗ the entrepreneur is unable to attract any users; equilibrium revenue with centralized governance is 0.
  - Reason: define u∗ by μ = ̄μ; for u > u∗, ̄μ < μ and users anticipate exploitation, so none join.
  - Implication: severe commitment problem from inability to commit to not abusing locked-in users; motivates decentralized governance.

---

### 3. Utilities realize (Decentralized governance, optimality, welfare)

### Decentralized governance: voting equilibrium and consequences
- Timing and assumptions:
  - Entrepreneur chooses permanent revenue split α at t = 0.
  - At t = 1,2,... users arrive or leave; users vote on π_t each period.
  - Voting restricted to weakly undominated strategies → users vote truthfully as if pivotal.
- Proposition 2 — Voting equilibrium:
  - Every period users vote for degree of monetization
    π*_t = 1−α 2k φ(μ_t) μ_t
  - The platform grows every period.
  - The entrepreneur shares half of the revenue with the users.
- Interpretation:
  - Decentralized governance serves as effective commitment device: users are assured they will not be exploited post-lock-in, sustaining growth and continued entry.
  - Cost to entrepreneur: must share revenue (half) with users; without sharing revenue users would vote to stop monetization and entrepreneur would receive 0 revenue.
- User incentives on dilution:
  - Per-period user utility in equilibrium equals V(μ_t) + φ(μ_t) 2 8k μ_t^2.
  - Given assumption that φ(μ_t)/μ_t is non-decreasing, equilibrium utility is increasing in μ_t.
  - Network effects from new entrants compensate revenue dilution, so existing users do not block entry.

### Optimal governance: centralized vs decentralized
- Extreme case:
  - When locked-in effect u = 0, centralized governance is preferred because commitment issues disappear; centralized lets entrepreneur retain revenues.
  - Lemma 3: As u → 0, centralized governance is always preferred over decentralized governance.
- Cutoff result (Proposition 3):
  - There exists u** such that decentralized governance is preferred by the entrepreneur iff u > u**.
  - Key reasoning:
    - Entrepreneur’s revenue under decentralized governance is independent of u.
    - Entrepreneur’s revenue under centralized governance decreases in u due to:
      - Immediate effect: increasing u raises required period utility δu in last growth period (reducing revenue) and changes exploitation-period utility −(1−δ)u (increasing revenue), but concavity implies net negative effect.
      - Secondary effect: increase in u reduces maximum platform size ̄μ (Lemma 2), lowering revenue.
    - Therefore centralized revenue is decreasing in u and the two revenue functions cross exactly once.

### Welfare implications
- Pareto and utilitarian outcomes:
  - Users: centralized implementation yields user equilibrium utility 0; decentralized yields strictly positive utility → users always prefer decentralized governance.
  - Entrepreneur: prefers decentralization iff u > u** (Proposition 3).
  - Corollary 2: Decentralized governance is a Pareto improvement over centralized governance iff u > u**.
  - Utilitarian welfare:
    - If u > u** (decentralization Pareto-improving) then utilitarian welfare is higher under decentralization.
    - If u < u**, aggregate welfare change sign is model-dependent; no general if-and-only-if statement.

### Airdrops, delayed decentralization, and timing
- Empirical practice: entrepreneurs often decentralize later via airdrops rather than at launch.
- Credibility constraint: delay must be sequentially optimal or entrepreneur may renege and remain centralized to exploit locked-in users.
- Lemma 4:
  - If g(μ_t) − μ_t → 0 as μ_t → ∞ (growth slows over time) then there exists sufficiently large μ_t such that it is sequentially optimal to keep the platform centralized.
- Proposition 4:
  - If it is optimal to decentralize at t = 0, then delaying decentralization is optimal.
  - The option to decentralize later lowers threshold u** for which decentralization is optimal.
- Implication: Entrepreneurs can increase revenue by delaying decentralization via airdrops, but decentralization must occur early enough that sufficient growth potential remains.

### Equilibrium selection and coordination issues
- Multiple equilibria possible; trigger-style equilibria can discipline entrepreneur but require high coordination.
- Folk-theorem style equilibrium (sketch) is unstable under small coordination uncertainty p > 0.
- Main equilibrium (Proposition 1) is stable for all degrees of uncertainty.
- Intuition: trigger equilibria require collective punishment; small uncertainties make punishment implausible; main equilibrium requires no coordination.

### Policy-relevant discussion and concluding insights
- Large, established platforms with large locked-in user bases may be outside prescriptive scope: extracting revenues from already locked-in users can make decentralization unattractive.
- If locked-in effects grow with platform size, qualitative results unchanged: small platforms can substitute growth for commitment; large platforms make exploitation more attractive and increase decentralization value.
- Differential treatment of new vs existing users that disadvantages existing users exacerbates commitment problem and increases value of decentralization.
- Variations in decentralized governance modeling (entrepreneur control, frictions) change quantitative thresholds (u**) but not qualitative existence of a threshold above which decentralization is optimal.
- Main takeaway:
  - An entrepreneur prefers to decentralize her platform, and decentralization is a Pareto improvement, iff the locked-in effect is sufficiently large (u > u**).

---

### Appendix — existence, uniqueness, and technical conditions (selected highlights)

### Sufficient conditions for existence and uniqueness of ̄μ (Proposition 6)
- Conditions:
  1. As μ_t → ∞ it holds that g(μ_t) − μ_t → 0.
  2. ψ(g(μ_t),ˆu) − ψ(μ_t,ˆu) is decreasing in μ_t for all ˆu.
  3. sqrt(2k V′(μ_t) μ_t) < φ(μ_t) / μ_t for all μ_t > 0.
- Under these conditions ̄μ exists and is unique.

### Comparative statics and differentiability (Lemma 2 sketch)
- ψ has two smooth pieces with matching derivatives at switching point defined by (φ(μ_t)/(2kμ_t))^2 = (V(μ_t)−ˆu)/k, enabling implicit function theorem.
- Define F := ψ(g(μ_t),δu) + δ/(1−δ) ψ(g(μ_t),−(1−δ)u) − 1/(1−δ) ψ(μ_t,−(1−δ)u).
- By implicit function theorem,
  - ∂̄μ/∂u = − (∂F/∂u) / (∂F/∂μ_t) |_{̄μ,u}.
- Denominator negative; numerator becomes negative for u sufficiently large → ∂̄μ/∂u < 0.

### Decentralized governance optimization and α = 0.5
- Under decentralized voting with precommitted α or path {α_t}, users choose π_t^{∗} = (1−α_t)/(2k) · φ(μ_t)/μ_t and entrepreneur’s optimal static choice yields α^{∗} = 0.5 for all t.

*Source: wpiea2024028-print-pdf — section: "1. The entrepreneur chooses a level of monetizationπ" and "3. Utilities realize"*

### 1.  The entrepreneur chooses a level of monetizationπ

### 1.  The entrepreneur chooses a level of monetizationπ

### Game structure and strategies
- Timing each period t:
  - Entrepreneur chooses monetization πt and revenue sharing αt.
  - Users simultaneously decide:
    - (a) New arrivals in period t choose to join or not.
    - (b) Existing users choose to stay or leave.
  - Utilities realize.
- Strategy definitions:
  - A history ht up to period t maps to actions.
  - Entrepreneur’s strategy: mapping from histories to (πt, αt).
  - Users’ strategy: mapping from histories to binary join/stay decisions.
  - Tie-breaking rules:
    - Newly arriving indifferent users join.
    - Indifferent locked-in users remain.

### Myopic revenue maximization (per-period problem)
- Entrepreneur maximizes revenue given users must receive utility ˆu:
  - max αt, πt αt πt φ(μt)
  - s.t. V(μt) − k πt^2 + (1−αt) μt πt φ(μt) = ˆu
  - 1 ≥ αt ≥ 0
- Denote the solution revenue by ψ(μt, ˆu):
  - ψ(μt, ˆu) =
    - μt V(μt) + φ(μt)^2 / (4 k μt) − μt ˆu if [φ(μt) / (2 k μt)]^2 ≥ V(μt) − ˆu / k
    - φ(μt) √(V(μt) − ˆu / k) if [φ(μt) / (2 k μt)]^2 < V(μt) − ˆu / k
  - Interpretation:
    - First case: entrepreneur shares revenue with users (αt ∈ (0,1)).
    - Second case: entrepreneur keeps all revenue (αt = 1).
- Properties of ψ:
  - Increasing in μt.
  - Decreasing in ˆu.
  - Feasibility limit: ˆu cannot exceed what users would get if all revenue distributed to users.
  - Revenue-sharing depends on users’ aversion to monetization k:
    - Small k → higher monetization and revenue sharing.
    - Large k → less monetization and no revenue sharing.

### Growth vs. exploitation trade-off
- Exploitation payoff if entrepreneur exploits existing users forever (starting with μt−1 existing users):
  - Present value = 1/(1−δ) ψ(μt−1, −(1−δ) u) (Equation II.1)
  - Per-period utility provided to users is −(1−δ) u so discounted utility equals −u.
- Grow once then exploit (grow in period t, exploit thereafter):
  - Revenue = ψ(g(μt−1), δ u) + δ/(1−δ) ψ(g(μt−1), −(1−δ) u) (Equation II.2)
- Indifference point ̄μ solves:
  - 1/(1−δ) ψ(̄μ, −(1−δ) u) = ψ(g(̄μ), δ u) + δ/(1−δ) ψ(g(̄μ), −(1−δ) u) (Equation II.3)
  - ̄μ is platform size where entrepreneur is indifferent between exploiting now vs. growing once then exploiting.

### Existence and intuition for ̄μ
- ̄μ exists under economically relevant conditions (growth slows sufficiently or network effects eventually diminish).
- Tractable example: V(μt) constant, φ(μt) linear in μt, g(μt) = μt + γ(μt) with γ(μt) strictly decreasing, strictly positive, γ(μt) → 0 as μt → ∞.

### Grow-then-exploit and join-if-compensated strategies (constructive equilibrium)
- Assumption for strategy construction:
  - Platform sufficiently profitable at size ̄μ to provide δ u to users:
    - φ(̄μ)^2 / (4 k ̄μ^2) + V(̄μ) ≥ δ u (Inequality II.4)
- Definition 1 — Grow-then-exploit strategy (entrepreneur):
  - If g(μt−1) < ̄μ: set (πt, αt) to maximize ψ(μt, ˆu t) with ˆu t = 0 and μt = g(μt−1).
  - If μt−1 < ̄μ and ̄μ ≤ g(μt−1): set (πt, αt) to maximize ψ(μt, ˆu t) with ˆu t = δ u and μt = g(μt−1).
  - If ̄μ ≤ μt−1: set (πt, αt) to maximize ψ(μt, ˆu t) with ˆu t = −(1−δ) u and μt = μt−1.
- Definition 2 — Join-if-compensated strategy (users):
  - In period of arrival, join if:
    1. g(μt−1) < ̄μ and πt, αt imply ˆu t ≥ 0 for μt = g(μt−1).
    2. μt−1 < ̄μ and ̄μ ≤ g(μt−1) and πt, αt imply ˆu t ≥ δ u for μt = g(μt−1).
  - If locked in, stay if πt, αt imply ˆu t ≥ −(1−δ) u for μt ≥ μt−1.

### Equilibrium result
- Proposition 1:
  - If inequality II.4 holds (platform sufficiently profitable at ̄μ to ensure δ u), then there exists a sub-game perfect Nash equilibrium in which:
    - Entrepreneur plays grow-then-exploit.
    - Users play join-if-compensated.
  - Intuition for equilibrium:
    - Newly arriving users are indifferent and join; locked-in users strictly prefer staying before exploitation and are indifferent at exploitation onset.
    - Entrepreneur cannot profitably increase users’ utility (revenues fall) or decrease utility during growth (would prevent joining); exploitation-phase deviations that reduce utility induce departures and are unprofitable.

### Failure to attract users when compensation infeasible
- If platform cannot guarantee δ u at ̄μ:
  - φ(̄μ)^2 / (4 k ̄μ^2) + V(̄μ) < δ u (Inequality II.5)
  - Entrepreneur cannot pay δ u in last growth period; setting < δ u yields no new joins.
  - Backward induction implies no users join at any period.
- Define μ̄ (denoted μ̄ in text as μ with bar):
  - μ solves φ(μ)^2 / (4 k μ^2) + V(μ) = δ u (Equation II.6)
  - μ is the minimum platform size required to feasibly provide δ u.
  - If ̄μ ≥ μ, entrepreneur can attract users. If ̄μ < μ, entrepreneur attracts no users.

### Comparative statics on locked-in effect and growth threshold
- Lemma 1:
  - μ strictly increases in u.
  - As u → ∞ it holds that μ → ∞.
- Lemma 2:
  - ̄μ strictly decreases in u.
  - As u → 0 it holds that ̄μ → ∞.
- Economic interpretation:
  - Larger locked-in effect parameter u increases required minimum platform size μ to compensate users.
  - Larger u reduces ̄μ: entrepreneur stops growing and starts exploiting earlier because temptation to exploit increases with the locked-in effect.
- Corollary 1:
  - There exists u∗ such that if u > u∗ the entrepreneur is unable to attract any users; equilibrium revenue with centralized governance is 0.
  - Reason: define u∗ by μ = ̄μ; for u > u∗, ̄μ < μ and users anticipate exploitation, so none join.
  - Implication: severe commitment problem from lack of ability to commit to not abusing locked-in users; motivates consideration of decentralized governance.

### Transition to decentralized governance
- If entrepreneur chooses decentralized governance, each period t = 1, 2, ...:
  - 1. Users decide to join or stay/leave.
  - 2. Users collectively choose πt.
  - (Further decentralized timing and analysis follow in the next section.)

*Source: wpiea2024028-print-pdf - 1.  The entrepreneur chooses a level of monetizationπ*

### 3.  Utilities realize

### 3.  Utilities realize

### Decentralized governance: voting equilibrium and consequences
- Setup: Entrepreneur chooses permanent revenue split α at t = 0. At t = 1,2,... users arrive or leave; users vote on degree of monetization π_t and utilities realize. Voting strategies are restricted to weakly undominated strategies so users vote truthfully as if pivotal.
- Equilibrium (Proposition 2):
  - Every period the users vote for a degree of monetization
    π*_t = 1−α 2k φ(μ_t) μ_t
  - The platform grows every period.
  - The entrepreneur shares half of the revenue with the users.
- Interpretation:
  - Decentralized governance functions as an effective commitment device: users are assured their locked-in effect will not be exploited, sustaining growth and continued entry.
  - Cost to entrepreneur: she must share revenue (half) with users; without sharing revenue users would vote to stop monetization and entrepreneur would receive 0 revenue.
- User incentives regarding dilution:
  - Users’ per period utility in equilibrium equals V(μ_t) + φ(μ_t) 2 8k μ_t^2.
  - Given the assumption that φ(μ_t)/μ_t is non-decreasing, equilibrium utility is increasing in μ_t.
  - Network effects from new entrants compensate the dilution of revenue shares, so existing users do not have incentives to block entry.

### Optimal governance: centralized vs decentralized
- Extreme cases:
  - When locked-in effect u = 0, centralized governance is preferred because commitment issues disappear, both modes yield growth, but centralized lets entrepreneur retain control and revenues.
  - Lemma 3: As u → 0, centralized governance is always preferred over decentralized governance.
- Cutoff result:
  - There exists u** such that decentralized governance is preferred by the entrepreneur iff u > u** (Proposition 3).
  - Key reasoning:
    - Entrepreneur’s revenue under decentralized governance is independent of u.
    - Entrepreneur’s revenue under centralized governance decreases in u due to:
      - Immediate effect: increasing u raises the required period utility δu in last growth period (reducing revenue) and changes exploitation-period utility −(1−δ)u (increasing revenue), but overall the concavity of revenue extraction implies net negative effect.
      - Secondary effect: increase in u reduces maximum platform size ̄μ (Lemma 2), lowering revenue.
    - Therefore centralized revenue is decreasing in u and the two revenue functions cross exactly once.

### Welfare implications
- Pareto efficiency:
  - Users: in centralized implementation users’ equilibrium utility is 0; in decentralized they receive strictly positive utility — users always prefer decentralized governance.
  - Entrepreneur: prefers decentralization iff u > u** (Proposition 3).
  - Corollary 2: Decentralized governance is a Pareto improvement over centralized governance iff u > u**.
- Utilitarian welfare:
  - If decentralization is a Pareto improvement (u > u**), utilitarian welfare is higher under decentralization.
  - If u < u**, forcing decentralization produces two opposing effects: entrepreneur revenue decreases and users’ utility increases; aggregate sign depends on model parametrization — no general if-and-only-if statement.

### Airdrops and delayed decentralization
- Practice: Entrepreneurs often decentralize later via airdrops (tokens to existing users) rather than at launch.
- Credibility concern: Delay must be sequentially optimal; otherwise entrepreneur may renege and remain centralized to exploit locked-in users.
- Lemma 4: If growth slows over time, i.e., g(μ_t) − μ_t → 0 as μ_t → ∞, then there exists sufficiently large μ_t such that it is sequentially optimal to keep the platform centralized.
- Proposition 4:
  - If it is optimal to decentralize at t = 0, then delaying decentralization is optimal.
  - The option to decentralize later lowers the threshold u** for which decentralization is optimal.
- Implication: Entrepreneurs can increase revenue by delaying decentralization, but decentralization must occur early enough that sufficient growth potential remains; airdrops can rationally be used to decentralize if timed before platform becomes too large.

### Equilibrium selection and coordination issues
- Multiple equilibria exist in dynamic games; users might be able to coordinate on trigger-style equilibria that discipline the entrepreneur, but these require high coordination.
- Folk-theorem type equilibrium (sketch):
  - Users’ strategy: stay/join only if utility implied by α_t and π_t for platform of size μ_t = g(μ_{t−1}) is at least û_t = V(g(μ_{t−1})) − (1−δ)u; otherwise users punish by leaving/not joining.
  - Entrepreneur’s strategy: set α_t and π_t to deliver û_t when not punished; if punished, set utility to −(1−δ)u given zero measure users.
- Stability (Proposition 5):
  - The folk-theorem style alternative equilibrium is unstable for any degree of uncertainty p > 0 about whether others will punish.
  - The equilibrium presented in the main analysis (Proposition 1) is stable for all degrees of uncertainty.
- Intuition: The trigger-type equilibrium requires collective punishment; small uncertainties about others’ coordination make it profitable for an individual to not punish, breaking the equilibrium. The main equilibrium requires no coordination.

### Discussion and concluding insights
- Large, established platforms (e.g., platforms with billions of users) are outside the model’s prescriptive scope: existing large locked-in user bases may make decentralization unattractive because extracting revenues from already locked-in users can dominate the commitment value of decentralization.
- If locked-in effects grow with platform size, results are qualitatively unchanged: small platforms can substitute growth for commitment; large platforms make exploitation more attractive and increase decentralization value.
- Differential treatment of new vs existing users that disadvantages existing users would exacerbate the commitment problem and make decentralization more valuable.
- Variations in how decentralized governance is modeled (level of entrepreneur control, frictions) are expected to change quantitative thresholds (u**) but not the qualitative main result: there exists a threshold of the locked-in effect above which decentralization is optimal.
- Main takeaway: (i) an entrepreneur prefers to decentralize her platform, and (ii) decentralization is a Pareto improvement, iff the locked-in effect is sufficiently large (u > u**).

*Source: "3.  Utilities realize" (chapter/section) from the provided PDF.*

### 1.  If

### 1.  If

### Optimal choices for π_t and α_t and entrepreneur revenue
- Case 1: If φ(μ_t)^2 / (4kμ_t^2) + V(μ_t) < ˆu then the entrepreneur is unable to ensure utility ˆu for the users.

- Case 2: If φ(μ_t)^2 / (4kμ_t^2) + V(μ_t) ≥ ˆu and
  - (a) [φ(μ_t) / (2kμ_t)]^2 ≥ (V(μ_t)−ˆu)/k, the optimal π_t, α_t are:
    - π_t = φ(μ_t) / (2kμ_t) (A.1)
    - α_t = 1/2 + 2kμ_t^2 (V(μ_t)−ˆu) / φ(μ_t)^2 (A.2)
    - The entrepreneur’s revenue:
      - μ_t V(μ_t) + φ(μ_t)^2 / (4kμ_t) − μ_t ˆu (A.3)
  - (b) [φ(μ_t) / (2kμ_t)]^2 < (V(μ_t)−ˆu)/k, the optimal π_t, α_t are:
    - π_t = sqrt((V(μ_t)−ˆu)/k) (A.4)
    - α_t = 1 (A.5)
    - The entrepreneur’s revenue:
      - sqrt((V(μ_t)−ˆu)/k) φ(μ_t) (A.6)

### KKT derivation sketch (maximization problem and FOCs)
- Maximization problem:
  - max_{α_t,π_t} α_t π_t φ(μ_t) (A.7)
  - s.t. V(μ_t) − k π_t^2 + (1−α_t) μ_t π_t φ(μ_t) = ˆu (A.8)
  - 1 ≥ α_t ≥ 0 (A.9)
- First-order conditions and complementary slackness (Lagrangian multipliers λ_1, λ_2, λ_3):
  - ∂/∂α_t: π_t φ(μ_t) + λ_1 (−π_t μ_t φ(μ_t)) − λ_2 + λ_3 = 0 (A.10)
  - ∂/∂π_t: α_t φ_t(μ_t) + λ_1 (−2k π_t + (1−α_t) μ_t φ(μ_t)) = 0 (A.11)
  - ∂/∂λ_1: V(μ_t) − k π_t^2 + (1−α_t) μ_t π_t φ(μ_t) − ˆu = 0 (A.12)
  - Complementary slackness:
    - (1−α_t) λ_2 = 0 (A.13)
    - α_t λ_3 = 0 (A.14)
- For α_t ∈ (0,1) (so λ_2 = λ_3 = 0) the solution yields (A.15)-(A.17), with feasibility conditions:
  - α_t > 0 ⇐⇒ φ(μ_t)^2 / (4kμ_t^2) + V(μ_t) > ˆu (A.19)
  - 1 > α_t ⇐⇒ [φ(μ_t) / (2kμ_t)]^2 > (V(μ_t)−ˆu)/k (A.21)
- For α_t = 1 the solution yields π_t and revenue in (A.22)-(A.23).
- For α_t = 0 the entrepreneur’s revenue is 0; users’ utility-maximizing π_t = φ(μ_t)/(2kμ_t) and ensuring ˆu is impossible if V(μ_t) + φ(μ_t)^2 / (4kμ_t^2) < ˆu (A.25).

### Sufficient conditions for μ̄ to be well-defined (existence and uniqueness)
- Conditions considered:
  1. As μ_t → ∞ it holds that g(μ_t) − μ_t → 0
  2. ψ(g(μ_t),ˆu) − ψ(μ_t,ˆu) is decreasing in μ_t for all ˆu
  3. sqrt(2k V′(μ_t) μ_t) < φ(μ_t) / μ_t for all μ_t > 0
- Proposition 6: The conditions above are sufficient to guarantee the existence and uniqueness of ̄μ.

### Proof outline for Proposition 6 (existence and uniqueness)
- Definition of ̄μ as solution of:
  - 1/(1−δ) ψ(̄μ,−(1−δ)u) = ψ(g(̄μ),δu) + δ/(1−δ) ψ(g(̄μ),−(1−δ)u) (A.26)
- Existence:
  - At μ = 0: LHS < RHS.
  - As μ → ∞: LHS > RHS.
  - Continuity implies existence by the intermediate value theorem.
- Uniqueness:
  - Consider derivative with respect to μ of RHS − LHS (A.27), re-expressed as (A.35).
  - Show this derivative is negative using an intermediate result (Lemma 6) and Condition 2.
- Lemma 6:
  - Assumption: sqrt(2k V′(μ) μ) < φ(μ)/μ for all μ > 0 implies ∂^2 ψ / ∂μ ∂ˆu < 0 for all μ > 0.
  - Expression for ∂^2 ψ / ∂μ ∂ˆu has two cases (A.28). For the second case the condition reduces to checking inequality (A.33), and monotonicity of φ(μ)/μ (i.e., φ(μ)/μ non-decreasing) gives φ(μ) / (φ′(μ) μ) ≤ 1, leading to sufficient condition (A.34) which equals the lemma’s assumption.

### Examples illustrating the sufficient conditions
- A.B.2: General growth function and linear revenues
  - Set V(μ) constant and φ(μ) = μ with g(μ) = μ + γ(μ), where γ is strictly decreasing, strictly positive and γ(μ) → 0 as μ → ∞.
  - Condition 1 holds because γ(μ) → 0.
  - ψ(g(μ),ˆu) − ψ(μ,ˆu) is linear in μ and its derivative equals γ′(μ) ψ_μ(μ,ˆu) = γ′(μ) ψ_μ(μ,ˆu) < 0 (A.41), so Condition 2 holds.
- A.B.3: Growth that slows abruptly
  - Example growth function:
    - g(μ) = g(0) > 0 if μ = 0; g(μ) = μ if μ > 0 (A.42)
  - For arbitrary V(μ) and φ(μ) this yields ψ(g(μ),ˆu) − ψ(μ,ˆu) > 0 at μ = 0 and 0 otherwise (A.43). This allows very general V and φ while meeting existence/uniqueness needs if growth stops after the first period.
  - Note: this g includes a discontinuity; practical use requires extending g to a continuous function or adjusting the definition of ̄μ.

### Trade-offs and generalization
- The sufficient conditions capture a trade-off between the convexity of revenue function ψ (driven by V and φ) and how fast growth g(μ) slows.
  - If ψ is less convex (e.g., V constant, φ(μ) = μ), very general growth functions g are admissible.
  - If growth slows extremely fast, very general V and φ are admissible.
- A.B.4 notes these conditions are sufficient but not necessary; an alternative assumption to ensure ̄μ exists and is unique is that
  - ψ(g(μ),δu) + δ/(1−δ) ψ(g(μ),−(1−δ)u) − 1/(1−δ) ψ(μ,−(1−δ)u)
    is increasing up to some value ˜μ (A.44).

*Source: wpiea2024028-print-pdf - 1.  If*

### 2.  Strictly decreasing for anyμ > ̃μ

### 2.  Strictly decreasing for anyμ > ̃μ

### A.C Proof of Proposition 1 — one-shot deviation principle and continuity at infinity
- The one-shot deviation principle (Theorem 4.2 in Fudenberg and Tirole (1991)) is applied; the game is continuous at infinity.
- It suffices to check there is no single-period profitable deviation for entrepreneur or users.

### A.C.1 Deviations by the entrepreneur
- Consider history up to period t with mass of users μ_{t−1}.
- Case 1 (μ_{t−1} ≤ ̄μ):
  - Deviations that increase user utility are not profitable: equilibrium path unchanged and entrepreneur’s revenue is decreasing in provided utility.
  - Deviations that decrease user utility:
    - Large decrease below −(1−δ)u causes all users to leave → not profitable.
    - Small decrease keeps incumbents but deters new users; most profitable deviation is to utility level −(1−δ)u.
  - Deviation profitability check:
    - For 1 period of future growth, deviation not profitable if
      ψ(μ_{t−1},−(1−δ)u) + δ[ψ(g(μ_{t−1}),δu) + δ/(1−δ) ψ(g(μ_{t−1}),−(1−δ)u)]
      ≤ ψ(g(μ_{t−1}),δu) + δ/(1−δ) ψ(g(μ_{t−1}),−(1−δ)u).
    - Equivalently,
      1/(1−δ) ψ(μ_{t−1},−(1−δ)u) ≤ ψ(g(μ_{t−1}),δu) + δ/(1−δ) ψ(g(μ_{t−1}),−(1−δ)u).
    - Inductive argument extends to T and T+1 periods using g^{(T)} notation and condition
      1/(1−δ) ψ(g^{(T)}(μ_{t−1}),−(1−δ)u) ≤ ψ(g^{(T+1)}(μ_{t−1}),δu) + δ/(1−δ) ψ(g^{(T+1)}(μ_{t−1}),−(1−δ)u).
  - Conclusion: one-shot deviations to abuse locked-in effect are not profitable.
- Case 2 (μ_{t−1} > ̄μ):
  - Decreasing user utility not profitable (all users leave → zero revenue).
  - Increasing user utility: marginal increases do not change behavior; the smallest increase that triggers another growth is not profitable by definition of ̄μ.

### A.C.2 Deviations by a user
- Newly arriving users:
  - On-equilibrium-path: no profitable deviation; users are indifferent between joining and not joining.
  - Off-equilibrium: if history offers more utility than equilibrium, joining remains optimal; if history has reduced utility, not joining is optimal.
- Locked-in users:
  - On-equilibrium:
    - Before exploitation: remaining gives 0 utility, leaving gives −u < 0 → no profitable deviation.
    - During exploitation: users indifferent between staying and leaving → leaving not profitable.
  - Off-equilibrium:
    - Histories with increased utility: no profitable deviation.
    - Histories with reduced utility: leaving yields −u; staying yields utility smaller than −(1−δ)u for the solitary period plus −δu from leaving next period, totaling < −u → deviation not profitable.

### A.D Proof of Lemma 2 — differentiability and implicit function theorem
- Need revenue function differentiable, including at switching point where
  (φ(μ_t)/(2kμ_t))^2 = (V(μ_t)−ˆu)/k.  Two functional pieces:
  - μ_t V(μ_t) + φ(μ_t)^2/(4kμ_t) − μ_t ˆu
  - √((V(μ_t)−ˆu)/k) φ(μ_t)
- Derivatives w.r.t. ˆu:
  - −μ_t
  - −(1/2) √(k) 1/√(V(μ_t)−ˆu) φ(μ_t)
  - These are equal when (φ(μ_t)/(2kμ_t))^2 = (V(μ_t)−ˆu)/k.
- Derivatives w.r.t. μ_t simplify using same identity to equal expressions:
  - μ_t V′(μ_t) + φ′(μ_t) φ(μ_t) /(2kμ_t)
- Hence implicit function theorem applies. Define
  F := ψ(g(μ_t),δu) + δ/(1−δ) ψ(g(μ_t),−(1−δ)u) − 1/(1−δ) ψ(μ_t,−(1−δ)u).
- By implicit function theorem,
  ∂̄μ/∂u = − (∂F/∂u) / (∂F/∂μ_t) |_{̄μ,u}.
- Denominator is negative by definition of ̄μ. At (̄μ,u = 0), F > 0.
- The three parts of F are (i) decreasing and concave in u, (ii) increasing and concave, (iii) decreasing and convex.
- The derivative of F w.r.t. u must become negative at some u and then remain negative; thus numerator negative → ∂̄μ/∂u negative.

### A.E Proof of Proposition 2 — degree of monetization and entrepreneur optimization
- Degree of monetization derived from optimization:
  max_{π_t} [ V(μ_t) − k π_t^2 + (1−α) μ_t π_t φ(μ_t) ].
- Equilibrium confirmed via one-shot deviation principle:
  - No user has incentive to deviate in monetization in weakly dominant strategies.
  - All users receive strictly positive participation utility → no incentive to not join.
- Entrepreneur’s optimization at t = 0:
  max_{α} Σ_{t=1}^{∞} α/(1−α) · (1/(2k)) φ(g^{(t)}(μ_0)) / g^{(t)}(μ_0) · φ(g^{(t)}(μ_0)).
- From this, α^{∗} = 0.5.

### A.F Proof of Lemma 3 — centralized vs decentralized governance near u = 0
- At u = 0 entrepreneur ensures 0 utility for users every period.
- No value of ̄μ makes entrepreneur indifferent between growing once more and exploiting immediately; always better to grow when g(μ) − μ ≥ 0.
- At u = 0 platform grows every period as under decentralized governance.
- Choice set under centralized governance larger than decentralized → entrepreneur revenues higher under centralized governance at u = 0.
- Continuity in u implies result holds for u > 0 sufficiently close to 0.

### A.G Proof of Proposition 3 — preference ordering and effect of u on centralized revenue
- Corollary 1: decentralized governance preferred if u sufficiently large (u > u^{∗}).
- Lemma 3: centralized governance preferred if u sufficiently small.
- Entrepreneur’s revenue under decentralized governance independent of u; need centralized revenue decreasing in u.
- First-order effect of increasing locked-in effect size u (in last period of growth and exploitation) is
  δ ψ_u(μ,δu) − (1−δ) δ/(1−δ) ψ_u(μ,−(1−δ)u) = δ ψ_u(μ,δu) − δ ψ_u(μ,−(1−δ)u).
- This is negative if
  ψ_u(μ,δu) ≤ ψ_u(μ,−(1−δ)u).
- Three cases:
  1) Both sides in linear part of ψ: holds trivially.
  2) LHS linear, RHS concave: holds if
     −μ ≤ −(1/2) √(k) 1/√(V(μ) + (1−δ)u) φ(μ)
     ⇔ (φ(μ)/(2kμ))^2 < (V(μ) + (1−δ)u)/k, which is condition from lemma 5.
  3) Both in concave part: inequality reduces to u ≥ 0 → holds.
- Second-order effect: maximum platform size ̄μ decreases in u (Lemma 2), and ψ increasing in μ → decrease in ̄μ reduces entrepreneur revenue. Total effect of increase in u is negative.

### A.H Proof of Lemma 4 — decentralizing revenue vs staying centralized for large μ_t
- Revenue from decentralizing at size μ is
  Σ_{t=0}^{T} δ^{t} (1/8) φ(g^{(t)}(μ_t))^2 /(k g^{(t)}(μ_t))
  approximated by
  1/(1−δ) [ (1/8) φ(μ_t)^2 /(k μ_t) + ε(μ_t) ]
  with ε(μ_t) → 0 as μ_t → ∞ since g(μ_t) − μ_t → 0.
- Revenue from staying centralized and exploiting (μ_t large enough) is
  1/(1−δ) ψ(μ_t,−(1−δ)u).
- Not sequentially optimal to remain centralized if
  ψ(μ_t,−(1−δ)u) − (1/8) φ(μ_t)^2 /(k μ_t) > ε(μ_t).
- Show LHS positive and increasing in μ_t:
  - If ψ(μ_t,−(1−δ)u) = μ_t [ V(μ_t) + φ(μ_t)^2 /(8k μ_t^2) + (1−δ)u ] − (1/8) φ(μ_t)^2 /(k μ_t) = μ_t [ V(μ_t) + φ(μ_t)^2 /(8k μ_t^2) + (1−δ)u ] → positive and increasing.
  - If ψ in square-root part: φ(μ_t)[ √((V(μ_t)+(1−δ)u)/k) − (1/8) φ(μ_t)/(k μ_t) ] > 0 and increasing given φ(μ_t)^2/(k μ_t) < √((V(μ_t)+(1−δ)u)/k).

### A.I Proof of Proposition 4 — delay of decentralization and expanded optimality range
- First statement: optimal to delay decentralization by at least 1 period.
  - If decentralize at t = 0, compare revenue at t = 1 when centralized vs decentralized at t = 1: centralized at t = 1 can choose α_t, π_t and set ˆu_t = 0 to attract users; decentralized t = 1 yields lower revenue because decentralized user utility strictly larger and entrepreneur cannot choose α_t, π_t. Revenues from period 2 onward identical → delay increases revenue.
- Second statement: if present value centralized exceeds decentralized by small ε > 0 when deciding t = 0, delaying decentralization can make decentralizing at t = 2 optimal for ε small enough → range of locked-in effects for which decentralization is optimal is increased.

### A.J Proof of equilibrium of section III.B and Proposition 5
- A.J.1 Equilibrium of III.B:
  - Entrepreneur deviations:
    - Increasing ˆu_t lowers revenue without changing user behavior on equilibrium path → not profitable.
    - Decreasing ˆu_t causes all users to leave → 0 revenue.
    - When punished with no users, entrepreneur indifferent → no profitable deviation.
  - User deviations:
    - For users already locked-in: leaving gives −u; staying gives Σ_{t=0}^{∞} δ^{t} V(g^{(t)}(μ_t)) − u > −u for δ large enough → staying optimal.
    - Newly arriving users: Σ_{t=0}^{∞} δ^{t} V(g^{(t)}(μ_t)) − u > 0 for δ large enough → join optimal.
    - For histories with entrepreneur offering ̃u_t < ˆu_t: staying utility ̃u_t − V(g(μ_{t−1})) − δu; staying optimal iff ̃u_t > V(g(μ_{t−1})) − (1−δ)u, which cannot hold since ˆu_t = V(g(μ_{t−1})) − (1−δ)u > ̃u_t → staying not profitable.
- A.J.2 Proposition 5:
  - Instability of equilibrium with users coordinating on ˆu_t = V(g(μ_{t−1})) − (1−δ)u:
    - User prefers stay over leave if lower-bound condition holds:
      p/(1−δ) V(g(μ_{t−1})) ≥ ε, which holds for ε small enough.
  - In the equilibrium presented in main section:
    - Any reduction in utility by entrepreneur breaks user indifference and causes users to not join/leave → no profitable deviation for entrepreneur.

### A.K Extension: Pre-commitment to revenue sharing path in decentralized governance
- If entrepreneur pre-commits to full path {α_t}_{t=1}^{∞} at t = 0:
  - For any α_t, users choose monetization π_t analogous to fixed α case:
    π_t^{∗} = (1−α_t)/(2k) · φ(μ_t)/μ_t.
  - User period utility:
    V(μ_t) + (1/(4k)) [ (1−α_t) φ(μ_t)/μ_t ]^2 ≥ 0 → new users always join.
  - Entrepreneur’s maximization at t = 0:
    max_{ {α_t} } Σ_{t=1}^{∞} δ^{t} α_t (1−α_t)/(2k) · φ(μ_t)/μ_t.
  - First-order condition implies α_t = α = 0.5 for all t = 1,2,...

*Italic: IMF working paper content unit wpiea2024028-print-pdf — section: "2.  Strictly decreasing for anyμ > ̃μ"*

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*Platform Precommitment via Decentralization Working Paper No. WP/2024/028*

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_Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024028-print-pdf.pdf_