## Section 2: Model (from wpiea2025185-source-pdf) ## Source details **Canonical URL:** [Section 2: Model (from wpiea2025185-source-pdf)](https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025185-source-pdf.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2025/english/wpiea2025185-source-pdf.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2025/english/wpiea2025185-source-pdf.pdf.json) --- ### Overview and timing - Two agent types: investors and brokers; investors cannot trade directly and must engage a broker. - Geography: three islands with brokers 1–3 and investors li ∈ {11, 21, 22, 31, 32, 33} (one investor on Island 1, two on Island 2, three on Island 3). - All agents risk neutral and fully informed about game structure. - Timing: Stages 0–5 (Stages 1–3 crucial for coalition formation, fee setting, and investor choice; Stage 4 preference shock; Stage 5 settlement sequencing). - Settlement and broker costs summarized across cases (intra-broker, legacy inter-broker, tokenized inter-broker, token-to-legacy detokenization). ### Investors: endowments, types, and choice - Endowments and shocks: - Each investor born with one indivisible asset and money endowment η. - Stage 4: type t ∈ {a,b} with 50% probability each; type a wants to sell, type b wants to buy one asset; gains from trade normalized to 1. - Broker selection and distance cost: - Investors choose a broker before the shock and deposit asset with that broker. - Non-pecuniary distance cost τ: τ_lj = 0 if l = j; τ_lj = τ if l ≠ j. - Investor expected utility (quasilinear in η): - u^e_li = η + max{(1 − f_lj) [ 1/2 P_aj + 1/2 P_bj ], 0} − τ_lj - P_tj = probability an investor of type t is matched with opposite type conditional on choosing broker j. - Tie/participation rule: max operator implies investors trade only if net return weakly positive. - Parameter constraint (to ensure positive utility when τ incurred): - η ≥ 2 ### Brokers: settlement technologies, costs, and tokenization - Settlement cost cases: 1. Intra-broker matching: intra-broker = 0. 2. Inter-broker legacy trading: cost γ > 0. 3. Inter-broker tokenized trading (both tokenized): validation cost ε → 0 (treated as zero). 4. Token-to-legacy trades (token broker with legacy broker): cost γ + κ with κ > 0. - Tokenized market formation: - Requires joint adoption by two or three brokers. - Total setup cost s shared evenly among participating brokers; s_j = μ_j s_m where μ_j = 1 if broker j participates and m = number of participating brokers. - Tokenized markets are excludable; excluded broker cannot join unilaterally. - Cost constraints used for analysis: - 0 < κ < γ - 0 < γ + κ ≤ 1/6 ### Broker fees and competition - Brokers post fee menus f_j = (f_1j; f_2j; f_3j) and may price-discriminate by investor origin l. - Fee competition is Bertrand-like but differentiated by: 1. Distance cost τ (local market power); Broker 3 has most local market power. 2. Variation in match probabilities P_kj across coalitions/markets. 3. Heterogeneity in expected marginal costs across settlement markets. ### Broker profits and profit function - Notation: - n_lj,p = number of trades facilitated by broker j for investors from origin l on market p ∈ {I,T,L}. - n_lj = Σ_p n_lj,p ; n_j = (n_1j; n_2j; n_3j); n_j,L = Σ_l n_lj,L. - Profit function: - π_j = f′_j n_j − (γ + T · κ) · n_j,L − s_j - T = 1 if any tokenized market formed at Stage 1, else 0. - If no tokenized market, κ never paid. - If partial coalition, trades with excluded legacy broker cost γ + κ. - If Grand Coalition (all tokenize), n_j,L = 0 so (γ + T·κ)·n_j,L = 0. ### Coalition formation and equilibrium definition - Brokers bargain over coalition structures C ∈ P, where P includes {{1},{2},{3}}, {{1,2},{3}}, {{1,3},{2}}, {{1},{2,3}}, {{1,2,3}}. - Equilibrium coalition C: no alternative coalition C′ exists such that π^e_j(C′,s,γ,κ) > π^e_j(C,s,γ,κ) for all j ∈ C′. - Tokenized market forms when equilibrium contains a coalition of size > 1. - Terminology: - "No Coalition" = {{1},{2},{3}} - "Grand Coalition" = {1,2,3} - Excludability: partial coalitions can exclude a broker and create trade diversion. ### Backward induction (Stages 5 → 1) - Stage 5: brokers minimize settlement costs; sequencing preference: - intra-broker settlement preferred, then tokenized market (if available), then legacy market. - Trade sequencing summarized: - No Coalition: 1 Intra-broker, 2 Legacy market, 3 Legacy market - Grand Coalition: 1 Intra-broker, 2 Tokenized market - Partial coalition: 1 Intra-broker, 2 Tokenized market, 3 Legacy market - Stage 3: investors choose broker j_li to maximize u^e_li(C,f). - Stage 2: brokers set fee menu f_j to maximize expected profit π^e_j(C,f), anticipating Stage 3. - Stage 1: brokers bargain over coalition C anticipating subsequent stages. --- ### Section 2.4: Constraints, equilibrium characterization, and main analytic results ### Modelling constraints and parameter ranges - Constraints to exclude monopoly extremes: - τ ≤ 2/3 excludes uncontested local monopolies (large-distance-cost case). - τ ≥ 1/3 ensures local-brokers equilibrium is always feasible; model assumes status-quo bias selects that equilibrium when feasible. - Analytical constraints summarized: - τ ∈ [1/3, 2/3] - κ ∈ (0, γ) - (γ + κ) ∈ (0, 1/6) - η ≥ 2 - Tie-breaking: indifference between coalitions favors larger tokenized market formation. ### Key computed match probabilities (P^e_j) under Stay prior - For C = {{1},{2},{3}} (No Coalition), P^e_j reported as 3×3 matrix (rows l = 1,2,3; columns j = 1,2,3): - 0.55 0.69 0.70 - 0.73 0.73 0.76 - 0.69 0.71 0.71 - For C = {{1,3},{2}} (1 & 3 Coalition), P^e_j: - 0.61 0.69 0.70 - 0.74 0.66 0.77 - 0.70 0.69 0.73 - For C = {{1,2,3}} (Grand Coalition), P^e_j identical to No Coalition matrix shown above (values preserved). ### Stage 2–3 closed-form equilibrium fees (selected exact expressions preserved) - No Coalition: - f_13 = 0, f_23 = 0 - f_11 = 1.83τ − 0.29 - f_22 = 1.38τ − 0.04 - f_33 = 1.41τ + 0.02γ (with f_32 = 0.02γ under competition) - 1 & 3 Coalition: - f_13 = 0, f_23 = 0 - f_11 = 1.64τ − 0.15 - f_22 = 1.52τ − 0.17 - f_33 = 1.36τ + 0.04 - Grand Coalition: - off-diagonal f_lj = 0 for l ≠ j - f_11 = 1.83τ − 0.29 - f_22 = 1.38τ − 0.04 - f_33 = 1.41τ ### Brokers’ expected profits by coalition (exact expressions preserved) - No Coalition ({{1},{2},{3}}): - π^e_1 = τ − 0.55γ − 0.16 - π^e_2 = 2τ − 0.45γ − 0.06 - π^e_3 = 3τ − 0.58γ - 1 & 3 Coalition ({{1, 3},{2}}): - π^e_1 = τ − 0.11(γ + κ) − 0.50s − 0.09 - π^e_2 = 2τ − 0.31(γ + κ) − 0.23 - π^e_3 = 3τ − 0.20(γ + κ) − 0.50s + 0.09 - Grand Coalition ({{1,2,3}}): - π^e_1 = τ − 0.33s − 0.16 - π^e_2 = 2τ − 0.33s − 0.06 - π^e_3 = 3τ − 0.33s ### Iterated deletion and feasible coalitions - 1 & 2 Coalition and 2 & 3 Coalition never arise in equilibrium for interior parameter space (thresholds given in exact forms (A.1.23) and (A.1.24)). - Grand Coalition never forms in baseline equilibrium for interior parameter space because at least one broker prefers deviation unless s is sufficiently large: even when γ = 1/6 and κ → 0, Broker 1 deviates unless s > 0.30. (A.1.25) - Therefore only No Coalition and the 1 & 3 Coalition can form in equilibrium. ### Parameter threshold for 1 & 3 Coalition vs No Coalition - Broker 1’s exact threshold for preferring 1 & 3 over No Coalition: - s < 1/8 + 7/8 γ − 7/32 κ . (A.1.28) - Conclusion: - If s < 1/8 + 7/8 γ − 7/32 κ, the 1 & 3 Coalition forms (partial coalition). - If s ≥ 1/8 + 7/8 γ − 7/32 κ, No Coalition forms (equality: 1 & 3 Coalition forms by tie-break). ### Welfare comparisons and social optima (Lemma 2) - Aggregate welfare decomposition: - Sum of investors’ endowments: 6η - Total expected gains from trade: 4.125 (constant) - Setup cost s incurred under any coalition with tokenization - Legacy trading costs: (γ + T·κ)· n^e_L - n^e_L(No Coalition) = 13/8 - Welfare indifference between No Coalition and Grand Coalition at: - s = 13/8 γ . (A.2.1) - Therefore: - If s > 13/8 γ → welfare maximized by No Coalition. - If s < 13/8 γ → welfare maximized by Grand Coalition. - If s = 13/8 γ → welfare equal under both. ### Private vs social outcomes (Proposition 1 and Corollary 1) - Proposition 1 (excessive investment & insufficient tokenization): - Private equilibrium features excessive investment if 13/8 γ ≤ s ≤ 1/8 + 7/8 γ − 7/32 κ. - Private equilibrium features insufficient tokenization if s ≤ 13/8 γ. - Interpretation: - Excessive investment: brokers form partial (1 & 3) coalition even though No Coalition yields higher welfare. - Insufficient tokenization: socially optimal to have Grand Coalition but private equilibrium fails to form it. - Corollary 1: - Lower κ expands region for which excessive investment occurs (i.e., declining κ raises payoff to partial-coalition brokers). --- ### Policy regimes, remedies, and extensions ### Equilibrium policy thresholds and prescriptions (exact thresholds preserved) - Three policy regimes based on s: - Take no action (laissez-faire) if s > 1/8 + 7/8 γ − 7/32 κ. - Impose an interoperability mandate if s ≤ 1/8 + 7/8 γ − 7/32 κ. - Subsidize tokenized market formation if 27/20 γ < s ≤ 13/8 γ. - Interpretation: - Laissez-faire optimal when private equilibrium (No Coalition) coincides with social optimum. - Interoperability mandate can prevent excessive investment by converting partial-coalition outcomes into No Coalition or Grand Coalition where appropriate. - Subsidy targeted to bridge s from private threshold to social threshold where needed. ### Interoperability mandate — mechanics and exact conditions - Under mandate, excluded brokers can be invited to join at no cost (Stage 1.3); partial coalitions effectively become Grand Coalition if joining is profitable. - Grand Coalition feasible under mandate when all brokers prefer Grand to No Coalition; marginal condition reduces to: - s ≤ 27/20 γ (exact fraction form). (A.3.5 and aggregate reduction) - Result: interoperability mandate yields Egalitarian Grand Coalition if s ≤ 27/20 γ; otherwise No Coalition. ### Tokenization subsidy — exact subsidy needed to tip brokers - Broker 2 chooses Grand Coalition over No Coalition if: - s ≤ 27/20 γ + σ . (A.6.1) - Socially optimal Grand Coalition requires: - s ≤ 13/8 γ . (A.6.2) - Equating these gives subsidy threshold: - σ = 11/40 γ . (A.6.3) - Policy implication: a subsidy σ ≥ 11/40 γ prevents insufficient tokenization by tipping Broker 2. ### Role of regulatory taxation (σ < 0) and κ - A sufficiently large tax (σ < 0) could prevent excessive investment by offsetting trade-diversion gains motivating partial coalitions. - Reductions in token-to-legacy friction κ shrink the laissez-faire zone and expand region requiring interoperability mandates (Corollary 2). - Policy implication: reductions in κ pursued in isolation may worsen fragmentation; optimal regulation combines κ-reducing measures with interoperability mandates. ### Extensions: side-payments and homogeneous-brokers cases (key conclusions) - Allowing broker side-payments (v = (v12; v13; v23)): - Equilibrium (C, v) must maximize aggregate broker profits Π^e. - With side-payments, excessive investment never occurs because payments can offset trade-diversion incentives; however insufficient tokenization can persist because brokers’ joint profit maximization need not align with social welfare. - With side-payments, a sufficiently large tokenization subsidy can always maximize welfare (Proposition 4). - Aggregate-profit cutoff reported numerically: 7313/4640 γ = 1.58γ; with flexible side-payments equilibrium is Grand Coalition when s ≤ 7313/4640 γ and No Coalition when s > 7313/4640 γ. - Homogeneous-brokers variant (one investor per island; tightened τ ∈ {3/8, 5/8}): - Equilibrium insufficient tokenization if s ≤ 3/2 γ. - Equilibrium excessive investment if 3/2 γ ≤ s ≤ (1 − κ)/4 + 3/4 γ. - Interoperability mandate always sufficient to achieve social optimum (no subsidy required) because ex-ante symmetry removes payoff asymmetries that create partial-coalition spillovers. --- ### Conclusion: policy lessons and research directions (key points preserved) - Main findings: - Broker coalitions can enable cheaper settlement among members but generate trade-diversion frictions with excluded brokers; these forces can produce excessive investment or insufficient tokenization relative to social optimum. - Neither interoperability mandate nor public cost-sharing alone always attains social optimum across parameter space. - Interoperability mandate reduces private return to forming exclusive tokenized markets (addresses excessive investment) but can cause underinvestment when tokenization yields investor externalities. - Subsidies stimulate investment but do not alter trade-diversion incentives; combined policy (mandate + subsidy) provides both “stick” and “carrot” to achieve optimal tokenization across regimes. - Promising extensions: - Allow choice across multiple tokenized means of payment; study role of tokenized money and public alternatives. - Model richer broker business models, including monetization of transaction data on private tokenized markets. *Content summarized from Section 2 (Model), Section 2.4 (Constraints), Lemmas, Propositions, Corollaries, and related sections in the supplied chapter.* ### Section 7 concludes. Proofs can be found in the Appendix. ### 2 Model ### Overview - Two agent types: investors and brokers. Investors cannot trade directly and must engage a broker. - Geography: three islands, each with one local broker and investors distributed as: one investor on Island 1, two on Island 2, and three on Island 3. Investors denoted li ∈ {11, 21, 22, 31, 32, 33}. - All agents are risk neutral and fully informed about the game structure. - Timing of events summarized in Table 1 (Stages 0–5). Brokers face costs summarized in Table 2. - Section notes related literature and distinction via exclusive coalition formation. ### Investors - Endowments and types: - Each investor is born with one unit of a homogeneous, indivisible asset and money endowment η. - At Stage 4, a preference shock assigns type t ∈ {a,b} with 50% probability each: type a wants to sell, type b wants to buy one additional asset. - Gains from trade normalized to 1: type a nets utility gain of 1 from selling; type b nets utility gain of 1 from buying. - Broker choice and costs: - Investors choose a broker before the shock to deposit their asset and become the broker’s client. - Depositing with local broker is costless; choosing an off-island broker incurs non-pecuniary distance cost τ. τ_lj = 0 if l = j; τ_lj = τ if l ≠ j. - Brokers charge fees f_lj to facilitate matches for an investor from island l. - Investor expected utility (quasilinear in η): - u^e_li = η + max{(1 − f_lj) [ 1/2 P_aj + 1/2 P_bj ], 0} − τ_lj - P_tj is the probability an investor of type t is matched with opposite type conditional on choosing broker j. - max operator: investors only trade if net return is weakly positive. - Notes: - Equal 50% type probability stated as without loss of generality. - Investor switching cost τ interpreted as capturing broker-client relationship value. ### Brokers — Settlement technologies and costs - Settlement cost structure (Table 2) distinguishes four cases: 1. Intra-broker matching (broker’s clientele includes both type a and type b): costless settlement (intra-broker = 0). 2. Inter-broker trade where both brokers use legacy technology: cost γ > 0 to overcome settlement risk. 3. Inter-broker trade where both brokers use same tokenized market: validation cost ε → 0 (treated as zero in analysis, used as tie-breaker). 4. Trade between token broker and legacy broker: requires detokenization; costs γ + κ where κ > 0 captures detokenization time/opportunity cost. - Tokenization features: - Tokenized market creation requires joint adoption by two or three brokers. - Total setup cost s shared evenly among participating brokers. - Tokenized markets are excludable: excluded broker cannot join unilaterally. - Tokenization yields efficiency gains only if at least one other broker also tokenizes. ### Brokers — Fees - Brokers post fee menus f_j = (f_1j; f_2j; f_3j) and may price discriminate by investor origin l. - Fee competition is Bertrand-like, subject to three differentiation sources: 1. Positive τ gives local market power; Broker 3 has most local market power (most on-island investors), Broker 1 least. 2. Variation in match probabilities P_kj due to differences in market size or tokenized markets. 3. Heterogeneity in expected marginal cost of facilitating trades across settlement markets. ### Broker profits - Notation: - n_lj,p = number of trades facilitated by broker j for investors from origin l on market p ∈ {I,T,L} (intra-broker, tokenized, legacy). - n_lj = Σ_p n_lj,p (total trades for origin l via broker j). - n_j = (n_1j; n_2j; n_3j). - n_j,L = Σ_l n_lj,L (total trades by broker j on legacy market). - Profit function: - π_j = f′_j n_j − (γ + T · κ) · n_j,L − s_j - T is a dummy: T = 1 if any tokenized market formed at Stage 1, else 0. - If no tokenized market, κ never paid. - If partial coalition formed, trades with excluded legacy broker cost γ + κ. - If grand coalition (all tokenize), n_j,L = 0 so (γ + T·κ)·n_j,L = 0. - s_j = μ_j s_m where m = number of participating brokers, μ_j = 1 if broker j participates in forming tokenized market, 0 otherwise. ### Broker coalition negotiations - Stage 1: brokers can form coalitions (tokenized markets). Negotiations may be arbitrarily long. - Definitions: - J = {1,2,3} brokers. P is set of all coalition structures (partitions of J). P explicitly includes {{1},{2},{3}}, {{1,2},{3}}, {{1,3},{2}}, {{1},{2,3}}, {{1,2,3}}. - Each coalition structure C ∈ P associated with vector of expected payoffs π^e(C,s,γ,κ) ∈ R^3. - Equilibrium coalition structure C: no alternative coalition C′ ∉ C exists such that π^e_j(C′,s,γ,κ) > π^e_j(C,s,γ,κ) for all j ∈ C′. - Tokenized market formed when equilibrium contains a coalition of more than one broker. - Terminology: “No Coalition” = {{1},{2},{3}}; “Grand Coalition” = {1,2,3}. - Excludability emphasized: partial coalitions can exclude a broker. ### Optimization problems (backward induction) - Stage 5 (post shocks, known coalitions and fees): brokers minimize settlement costs. - Preference ordering: intra-broker settlement preferred (cheapest). Among remaining, token brokers prefer tokenized market over legacy market. - Table 3 trade sequencing: - No Coalition: 1. Intra-broker, 2. Legacy market, 3. Legacy market - Grand Coalition: 1. Intra-broker, 2. Tokenized market - Partial coalition: 1. Intra-broker, 2. Tokenized market, 3. Legacy market - Stage 3: each investor li chooses broker j_li to maximize u^e_li(C,f): max_j u^e_li(C,f). - Stage 2: each broker j sets fee menu f_j to maximize expected profit π^e_j(C,f): max_{f_j} π^e_j(C,f), anticipating investor choices at Stage 3 and given coalition structure C. - Stage 1: brokers anticipate subsequent stages and bargain over coalition structures C to maximize expected profits; equilibrium coalition structures determined as above. *Source: Section 2, “Model,” from the supplied PDF content.* ### 2.4 Constraints ### 2.4 Constraints ### Scope and motivation - Aim: analyze incentives of competing brokers, each with market power, to jointly form a tokenized market while excluding cases that produce broker monopolies. - Two monopoly extremes excluded: - Distance costs so large that investors are walled in at their local broker (uncontested local monopolies). - Distance costs so small that all investors converge on a single broker (single monopolist by attracting all investors). ### Constraints to exclude uncontested monopolies and single-monopolist outcomes - Constraint to exclude uncontested local monopolies: - τ ≤ 2/3 suffices to exclude uncontested monopolies (large-distance-cost case). - Constraints to prevent the single-monopolist (all investors at one broker) from arising: - Impose τ ≥ 1/3 so that the equilibrium in which all investors choose their local brokers is always feasible (self-confirming priors). - Assume when that local-brokers equilibrium is feasible, it is selected (minimal-movement prior / status-quo bias). ### Additional modelling constraints (to obtain analytical solutions) - Maximum cost of clearing legacy market transactions: - 0 < γ + κ ≤ 1/6 - Token-to-legacy settlement friction relative to legacy friction: - 0 < κ < γ - Under 0 < κ < γ, κ is never so large that it prevents feasible token-to-legacy trades. - Investor endowment constraint to ensure positive utility if non-pecuniary cost τ is incurred: - η ≥ 2 - Tie-breaking: when brokers are exactly indifferent between coalitions, tie-breaking favors larger tokenized market formation. - Summary of parameter constraints: - τ ∈ [1/3, 2/3]; κ ∈ (0, γ); (γ + κ) ∈ (0, 1/6); η ≥ 2 ### Notes on investor selection and equilibrium selection - Investors’ selection of brokers (Stage 3) can produce multiple equilibria because investors impose positive externalities on each other (pooling increases attractiveness via intra-broker trade prioritization). - The model uses a minimal-movement prior (status-quo bias) as an equilibrium selection rule. ### Key analytical equilibrium and welfare conditions (as stated) - Lemma 1 (Baseline equilibrium): - If s > 1/8 + 7/8 γ − 7/32 κ, no tokenized market forms in equilibrium. - If s ≤ 1/8 + 7/8 γ − 7/32 κ, Brokers 1 and 3 form a tokenized market (the 1 & 3 Coalition). - Interpretation: - Above the threshold, setup cost s is too high relative to benefits. - On and below the threshold, a partial coalition (1 & 3) forms, excluding Broker 2, because partial coalition yields trade-diversion benefits and higher matching probabilities for participating brokers even as γ → 0. - A larger κ reduces attractiveness of forming the partial coalition (threshold shifts down). - Lemma 2 (Optimal outcomes for welfare): - If s > 13/8 γ, welfare is maximized when no tokenized market exists. - If s < 13/8 γ, welfare is maximized by a tokenized market that includes all brokers (Grand Coalition). - If s = 13/8 γ, welfare is equal under both outcomes. - Notes: - The Grand Coalition is socially optimal below the boundary because efficiency benefits outweigh setup costs and are maximized when legacy trading is fully eliminated. - κ, η, and τ do not enter the expression in Lemma 2 for the reasons explained (partial coalitions welfare dominated; η trivially increases welfare but not the comparison; τ not incurred in equilibrium under local-brokers outcome). - Proposition 1 (Excessive investment & insufficient tokenization): - Private equilibrium features excessive investment in tokenization if 13/8 γ ≤ s ≤ 1/8 + 7/8 γ − 7/32 κ. - Private equilibrium features insufficient tokenization if s ≤ 13/8 γ. - Interpretation: - Excessive investment: brokers form a partial (1 & 3) coalition and incur s even though No Coalition would yield higher welfare. - Insufficient tokenization: socially optimal to have the Grand Coalition, but private equilibrium fails to form it (or forms too narrow a coalition). - Corollary 1 (Token-to-legacy friction and welfare): - A lower token-to-legacy settlement friction κ expands the region for which excessive investment occurs. - Intuition: lower κ raises payoff to partial-coalition brokers, making socially costly partial coalitions more likely. ### Implications for the parameter space and equilibrium selection - Partial coalitions (1 & 3) can form even when γ approaches zero, provided s is not too high, because trade-diversion and preferential access effects raise participating brokers’ fees and matching probabilities. - Heterogeneity across brokers and the unanimity requirement for private coalition formation can cause underinvestment relative to social optimum (the least amenable broker may block the Grand Coalition despite aggregate benefits). *Italicized source attribution: Content summarized from section 2.4 and related lemmas and propositions in the supplied chapter.* ### 1. Taking no action (laissez-faire) if s > ### 1. Taking no action (laissez-faire) if s > ### Equilibrium policy thresholds and prescriptions - Three distinct policy regimes (conditions on s): - Taking no action (laissez-faire) if s > 1/8 + 7/8 γ − 7/32 κ. - Imposing an interoperability mandate if s ≤ 1/8 + 7/8 γ − 7/32 κ. - Subsidizing the formation of tokenized markets if 27/20 γ < s ≤ 13/8 γ. - Proof summary: When s > 1/8 + 7/8 γ − 7/32 κ, No Coalition maximizes welfare (Proposition 1) and laissez-faire is optimal. Policies 2 and 3 follow from combining Lemmas 3, 4, and 6. ### Equilibrium regions (as described for Figure 4b) - White region: No Coalition is both the private equilibrium and optimal → policymaker takes no action. - Blue region: Private equilibrium leads to excessive investment; interoperability mandate produces No Coalition (the optimal outcome). - Below the red line: Grand Coalition is optimal; achieved by: - Interoperability mandate alone (green region), or - Interoperability mandate combined with a sufficiently large subsidy (yellow region). ### Role of taxes and regulatory stringency - A sufficiently large tax (σ < 0), representing stringent regulatory or licensing requirements, could prevent excessive investment (blue zone of Figure 3b) by offsetting gains from trade diversion that motivate Brokers 1 and 3 to form a tokenized market. - Such a tax cannot resolve insufficient tokenization, as there is no point in Figure 3b at which brokers form the Grand Coalition. ### Corollary 2 — Token-to-legacy friction and policy - Statement: A reduction in the token-to-legacy settlement friction κ shrinks the laissez-faire zone and expands the parameter space within which an interoperability mandate is needed to attain the social optimum. - Proof: Follows directly from Proposition 2. - Policy implication: Declines in κ (e.g., via faster release of underlying assets from custodians) could worsen market fragmentation if pursued in isolation. Optimal regulation should combine efforts to reduce κ with an interoperability mandate: the former makes tokenization by some brokers more likely; the latter ensures all brokers and investors can benefit from resulting efficiency gains. ### Extensions — overview - Two variations considered: 1. Allowing brokers to make side-payments to one another. 2. A setting with one investor on each island, implying ex-ante broker homogeneity. ### 6.1 Allowing side-payments - Framework modification: - Define v = (v12; v13; v23) as net side-payments vjk from broker j to broker k. - Outcome (C, v): coalition structure C and trio of net side-payments v. - Expected broker payoffs πe(C, v,s,γ,κ) ∈ R^3 given observed cost parameters s, γ, κ. - Equilibrium (C, v) requires no alternative (C′, v′) where (i) every broker in a new coalition C′ strictly improves, and (ii) every broker making positive net payments also strictly improves. - Proposition 3 (Welfare with side-payments): With broker side-payments, insufficient tokenization can occur so the private equilibrium does not always maximize welfare. - Intuition: Side-payments allow brokers to act collectively to maximize joint surplus; excessive investment never occurs because side-payments can offset trade-diversion incentives. However, brokers’ joint profit maximization need not align with social welfare due to positive externalities to investors (on Island 3) from the Grand Coalition; underinvestment can persist. - Proposition 4 (Policy implications of side-payments): With broker side-payments, a sufficiently large tokenization subsidy can always maximize welfare. ### 6.2 Homogeneous brokers - Setup changes: - Each island has a population of one investor (ex-ante identical brokers). - To preclude broker monopoly, tighten τ from τ ∈ {1/3, 2/3} in (7) to τ ∈ {3/8, 5/8}. - When one partial coalition is a feasible equilibrium, all partial coalitions are feasible; assume one of the three feasible partial-coalition equilibria is randomly selected. - Proposition 5 (Homogeneous brokers): - Equilibrium features insufficient tokenization if s ≤ 3/2 γ. - Equilibrium features excessive investment if 3/2 γ ≤ s ≤ (1 − κ)/4 + 3/4 γ. - Therefore the excessive investment zone expands as κ declines. - An interoperability mandate eliminates both excessive investment and insufficient tokenization and thus always achieves the social optimum. - Comparative note: Unlike the baseline, here an interoperability mandate alone suffices; a tokenization subsidy is never needed. Reason: ex-ante broker homogeneity removes payoff asymmetries that create spillover-markup externalities in the baseline. ### Conclusion — policy lessons and research directions - Main findings: - Coalitions of brokers can invest in tokenized markets that enable cheaper settlement among members but add friction to trading with excluded brokers; trade diversion incentives can produce excessive investment or insufficient tokenization. - Neither an interoperability mandate nor public cost-sharing always achieves the socially optimal outcome when used in isolation. - Interoperability mandate: reduces private return to forming tokenized markets (addresses excessive investment) but can cause underinvestment when tokenization generates positive externalities for investors. - Public cost-sharing (subsidy): stimulates investment by reducing brokers’ cost burden but does not alter trade diversion incentives that produce exclusive coalitions. - Combined policy: interoperability mandate plus public cost-sharing provides both a “carrot” and a “stick” to achieve optimal tokenization across parameter regimes. - Promising avenues for future work: - Allow choice across multiple tokenized means of payment; role of tokenized money and public alternatives (link to subsidy σ). - Allow richer broker business models, e.g., monetizing transaction data generated on private tokenized markets. *Source: wpiea2025185-source-pdf - 1. Taking no action (laissez-faire) if s >* ### References ### References ### Major topical clusters in the reference list - Tokenization and tokenized assets: works include "Token-Based Platform Governance" (Abadi and Brunnermeier, 2024), "Tokenisation of Government Bonds: Assessment and Roadmap" (Aldasoro et al., 2025, BIS Bulletins 107), "Tokenisation in the Context of Money and Other Assets: Concepts and Implications for Central Banks" (BIS, 2024), "Tokenised Bonds: Assessing Efficiency and Liquidity in a Nascent Market" (Born et al., 2026, Macroprudential Bulletin 33), "Tokenized Stocks" (Cong et al., 2025a), "Tokenized Reserves and Next-Generation Capital Markets" (Patel et al., 2026, IMF Notes, Forthcoming), "The DTCC Tokenization Services" No-Action Letter Request (SEC, 2025), and "Statement on Tokenized Securities" (SEC, 2026). - Distributed ledgers, blockchain, and decentralized systems: "Distributed Ledgers and the Governance of Money" (Auer, Monnet, and Shin, 2025), "Blockchain-Based Settlement for Asset Trading" (Chiu and Koeppl, 2019), "Garratt and Monnet (2023). An Impossibility Theorem on Truth-Telling in Fully Decentralised Systems" (BIS WP 1117), and "Blockchain Consensus Mechanisms and Fragmentation" (Eidan et al., 2026, BIS Bulletin 126). - Digital money, stablecoins, CBDC, and competing digital monies: "Public and Private Money Creation for Distributed Ledgers: Stablecoins, Tokenized Deposits, or Central Bank Digital Currencies?" (Chiu and Monnet, 2024, Bank of Canada Staff Working Paper 2024-35), "Stablecoins vs. Tokenized Deposits: The Narrow Banking Debate Revisited" (Huang and Keister, 2026, Staff Reports 1179, Federal Reserve Bank of New York), "Competing Digital Monies" (Frost et al., 2025, Working Paper 25-1644, Toulouse School of Economics), "Tokenization of Financial Assets: Opportunities and Risks" (IMF, 2026, Forthcoming Global Financial Stability Report, Oct. 2026, Chapter 3). - Market fragmentation, OTC and centralized market structure, and intermediation: "Market fragmentation" (Chen and Duffie, 2021), "Entry and Exit in OTC Derivatives Markets" (Atkeson, Eisfeldt, and Weill, 2015), "Over-the-Counter Markets" (Duffie, Gârleanu, and Pedersen, 2005), "A Theory of Participation in OTC and Centralized Markets" (Dugast, Uslü, and Weill, 2022), "Fragmenting Markets: Post-Crisis Bank Regulations and Financial Market Liquidity" (Duffie, 2022). - Payments, interoperability, and mobile/digital payments adoption: "Mobile Payments and Interoperability: Insights from the Academic Literature" (Bianchi et al., 2023), "Mobile Money, Interoperability, and Financial Inclusion" (Brunnermeier, Limodio, and Spadavecchia, 2023, NBER Working Paper 31696), "Integrating Fragmented Networks: Interoperability in Money and Payments" (Copestake et al., 2025, IMF WP 25/126), and "Digital Payments Interoperabillity with Naïve Consumers" (Bianchi and Rhodes, 2024, TSE Working Papers 1559). - Market microstructure, liquidity, and trading technologies: "Price Formation and Equilibrium Liquidity in Fragmented and Centralized Markets" (Biais, 1993), "Benchmarks in Search Markets" (Duffie, Dworczak, and Zhu, 2017), "Competing on Speed" (Pagnotta and Philippon, 2018), "Balance-Sheet Netting in U.S. Treasury Markets and Central Clearing" (Bowman, Huh, and Infante, 2024, FEDS Finance and Economics Discussion Series 2024-057). - Governance, platforms, and data economics: "Data, Competition, and Digital Platforms" (Bergemann and Bonatti, 2024), "Too Much Data: Prices and Inefficiencies in Data Markets" (Acemoglu et al., 2022), and "Centralized Governance in Decentralized Organizations" (Cong et al., 2025b). - Policy- and practitioner-oriented reports and technical documents: BIS Annual Economic Report chapters (BIS, 2025; BIS, 2026), IOSCO Technical Report FR/17/25 (2025), DTCC technical material on tokenizing assets (DTCC, 2026), J.P. Morgan technical report "The Future of Wealth Management" (2023), Project Pine technical report (2025, Federal Reserve Bank of New York and BIS Innovation Hub). ### Representative exact citations and identifiers (selected examples from the list) - Abadi, J. and Brunnermeier, M. (2024). Token-Based Platform Governance. Journal of Financial Economics, 162:103951. - Aldasoro, I., Cornelli, G., Frost, J., Koo Wilkens, P., Lewrick, U., and Shreeti, V. (2025). Tokenisation of Government Bonds: Assessment and Roadmap. BIS Bulletins 107, Bank for International Settlements. - Auer, R., Monnet, C., and Shin, H. S. (2025). Distributed Ledgers and the Governance of Money. Journal of Financial Economics, 167:104026. - Chiu, J. and Monnet, C. (2024). Public and Private Money Creation for Distributed Ledgers: Stablecoins, Tokenized Deposits, or Central Bank Digital Currencies? Staff Working Paper 2024-35, Bank of Canada. - Duffie, D., Gârleanu, N., and Pedersen, L. H. (2005). Over-the-Counter Markets. Econometrica, 73(6):1815–1847. - Frost, J., Rochet, J., Shin, H. S., and Verdier, M. (2025). Competing Digital Monies. Working Paper 25-1644, Toulouse School of Economics. - SEC (2025). No-Action Letter Request Related to The Depository Trust Company’s Development of the DTCC Tokenization Services. No-Action Letter, US Securities and Exchange Commission. - SEC (2026). Statement on Tokenized Securities. Staff Statement, US Securities and Exchange Commission. - IMF (2026). Tokenization of Financial Assets: Opportunities and Risks. (Forthcoming) Global Financial Stability Report, Oct. 2026, Chapter 3, International Monetary Fund. - BIS (2024). Tokenisation in the Context of Money and Other Assets: Concepts and Implications for Central Banks. Report to the G20, Bank for International Settlements. ### ONLINE APPENDIX — A Proofs (contents and computational notes) - Appendix A contains proofs omitted from the main text. - A Mathematica file performs full calculations in exact form and is available on request. - In the appendix text, quantities are rounded to two decimal places for conciseness; the Mathematica file performs all calculations in exact form and shows exact (fraction) form only for key results. - A.1 Proof of Lemma 1: - Solution approach uses backward induction through ex-ante decision stages (Stages 1–3; Stage 4 is the type realization stage; Stage 5 decisions are shown in Table 3 in Section 2.3). - Status-quo prior: investors act from the prior that all other investors choose their local brokers (“staying” or on-island). If a unique fulfilled-expectations Subgame-Perfect Nash Equilibrium (SPNE) is found, that equilibrium is selected (the “Stay Equilibrium”), per the selection criterion in Section 2.4.2. - Stage solution steps: - Stage 3: under the status-quo prior, each investor chooses a broker based on observed broker fees and expected match probabilities. - Stage 2: each broker determines its fees, internalizing effects on investor sorting at Stage 3; brokers take coalition structure as given at this stage. - Stage 1: brokers negotiate about coalition formation, foreseeing impacts of coalition structure on fees and investor sorting and comparing expected profits π_ej(C,s,γ,κ) across coalition structures to identify the coalition structure that results in Stage 1, for each combination of s, γ and κ. *References and appendix content as provided in the source PDF.* ### 4. If this backward induction results in a unique SPNE that, moreover, is consistent with ### wpiea2025185-source-pdf - 4. If this backward induction results in a unique SPNE that, moreover, is consistent with ### Stage 3 — investors’ choices and expected utilities - Investors li choose broker j based on fees f and expected match probabilities P^e_aj and P^e_bj: - u^e_li(C,f) = η + max{(1− f_lj) 1/2 P^e_aj + 1/2 P^e_bj , 0} − τ_lj . (A.1.1) - Symmetry across types a and b implies P^e_j := P^e_aj = P^e_bj and expected payoff per investor simplifies to: - u^e_l(C,f) = η + max{(1− f_lj) P^e_j(C,j_l, Φ^Stay_l), 0} − τ_lj . (A.1.2) - Distribution of investor types at Stage 4: 2^6 = 64 outcomes; binomial counts and probabilities in Table A.1: - 6a 1/64 - 5a + b 6/64 - 4a + 2b 15/64 - 3a + 3b 20/64 - 2a + 4b 15/64 - a + 5b 6/64 - 6b 1/64 ### Investor match probabilities (Table A.2) — P^e_j(C,j_l, Φ^Stay_l) - For each coalition C ∈ P, match probabilities P^e_j are computed by: 1. Probability distribution over counts of type a vs b (Table A.1). 2. Conditioning on each aggregate outcome, derive match probabilities under Stay and Move scenarios. 3. Weight second-step results by first-step probabilities to get nine P^e_j values (one per island × chosen-broker pair). - Reported P^e_j(C,j) (rows l = 1,2,3; columns j = 1,2,3): - C = {{1},{2},{3}}: - l\j: 1 2 3 → 0.55 0.69 0.70 ; 0.73 0.73 0.76 ; 0.69 0.71 0.71 - C = {{1, 2},{3}}: - l\j: 1 2 3 → 0.56 0.69 0.70 ; 0.75 0.75 0.70 ; 0.69 0.73 0.69 - C = {{1, 3},{2}}: - l\j: 1 2 3 → 0.61 0.69 0.70 ; 0.74 0.66 0.77 ; 0.70 0.69 0.73 - C = {{1},{2, 3}}: - l\j: 1 2 3 → 0.50 0.69 0.70 ; 0.66 0.74 0.77 ; 0.66 0.71 0.71 - C = {{1, 2, 3}}: - l\j: 1 2 3 → 0.55 0.69 0.70 ; 0.73 0.73 0.76 ; 0.69 0.71 0.71 ### Stay Conditions (investor incentive to remain local) — (A.1.3) and Table A.3 - General condition for investor on island l to stay at local broker l: - η + max{(1− f_ll) P^e_l(C,l, Φ^Stay_l), 0} ≥ η + max_{j≠l}{(1− f_lj) P^e_j(C,j, Φ^Stay_l), 0} − τ_lj . (A.1.3) - Explicit Stay Conditions by coalition structure (coefficients preserved exactly): - C = {{1},{2},{3}}: - (1− f_11)· 0.55 ≥ max{(1− f_12)· 0.69, (1− f_13)· 0.70, 0} − τ - (1− f_22)· 0.73 ≥ max{(1− f_21)· 0.73, (1− f_23)· 0.76, 0} − τ - (1− f_33)· 0.71 ≥ max{(1− f_31)· 0.69, (1− f_32)· 0.71, 0} − τ - C = {{1, 2},{3}}: - (1− f_11)· 0.56 ≥ max{(1− f_12)· 0.69, (1− f_13)· 0.70, 0} − τ - (1− f_22)· 0.75 ≥ max{(1− f_21)· 0.75, (1− f_23)· 0.70, 0} − τ - (1− f_33)· 0.69 ≥ max{(1− f_31)· 0.70, (1− f_32)· 0.73, 0} − τ - C = {{1, 3},{2}}: - (1− f_11)· 0.61 ≥ max{(1− f_12)· 0.69, (1− f_13)· 0.70, 0} − τ - (1− f_22)· 0.66 ≥ max{(1− f_21)· 0.74, (1− f_23)· 0.77, 0} − τ - (1− f_33)· 0.73 ≥ max{(1− f_31)· 0.70, (1− f_32)· 0.69, 0} − τ - C = {{1},{2, 3}}: - (1− f_11)· 0.50 ≥ max{(1− f_12)· 0.69, (1− f_13)· 0.70, 0} − τ - (1− f_22)· 0.74 ≥ max{(1− f_21)· 0.66, (1− f_23)· 0.77, 0} − τ - (1− f_33)· 0.71 ≥ max{(1− f_31)· 0.66, (1− f_32)· 0.71, 0} − τ - C = {{1, 2, 3}}: - (1− f_11)· 0.55 ≥ max{(1− f_12)· 0.69, (1− f_13)· 0.70, 0} − τ - (1− f_22)· 0.73 ≥ max{(1− f_21)· 0.73, (1− f_23)· 0.76, 0} − τ - (1− f_33)· 0.71 ≥ max{(1− f_31)· 0.69, (1− f_32)· 0.71, 0} − τ ### Stage 2 — brokers’ fee-setting and best responses (A.1.4) - Brokers j choose f_j to maximize: - max_{f_j} { f'_j n^e_j(C,f) − (γ + T · κ)· n^e_{j,L}(C,f) − s_j } . (A.1.4) - Key mechanism: fees trade off revenue per trade vs. expected number of trades (investor movements). - Best competing alternative for non-local brokers often is f_lj = 0 (Bertrand undercutting), because acquiring investors increases local intra-broker matches and may raise fees on local investors. - Equilibrium fees f_ll are set where each Stay Condition binds with equality. Reported closed-form fees by coalition: No Coalition ({{1},{2},{3}}): - f_13 = 0, f_23 = 0 and - f_11 = 1.83τ − 0.29 . (A.1.5) - f_22 = 1.38τ − 0.04 . (A.1.6) - f_32: Broker 2 willing to lower until f_32 = 0.02γ → f_33 = 1.41τ + 0.02γ . (A.1.7) 1 & 2 Coalition ({{1, 2},{3}}): - f_31 = f_32 = 0 → f_33 = 1.45τ − 0.07 . (A.1.8) - Broker 3 competes to f_13 = 0 → f_11 = 1.78τ − 0.25 . (A.1.9) - Competition implies f_21 = 0 → f_22 = 1.33τ . (A.1.10) 1 & 3 Coalition ({{1, 3},{2}}): - f_13 = 0, f_23 = 0 → f_11 = 1.64τ − 0.15 . (A.1.11) - f_22 = 1.52τ − 0.17 . (A.1.12) - f_31 = 0 competition → f_33 = 1.36τ + 0.04 . (A.1.13) 2 & 3 Coalition ({{2, 3},{1}}): - f_13 = 0, f_23 = 0 → f_11 = 2.00τ − 0.41 . (A.1.14) - f_22 = 1.35τ − 0.04 . (A.1.15) - f_32 = 0 competition → f_33 = 1.40τ . (A.1.16) Grand Coalition ({{1, 2, 3}}): - legacy marginal costs zero so f_lj = 0 for l ≠ j and Bertrand competition yields Stay Conditions binding: - (1− f_11)· 0.55 = max{0.69, 0.70, 0} − τ → f_11 = 1.83τ − 0.29 . (A.1.20) - (1− f_22)· 0.73 = max{0.73, 0.76, 0} − τ → f_22 = 1.38τ − 0.04 . (A.1.21) - (1− f_33)· 0.71 = max{0.69, 0.71, 0} − τ → f_33 = 1.41τ . (A.1.22) ### Summary of Stage 2–3 equilibrium properties - For each coalition structure C, there exists a unique set of fees such that: - brokers maximize expected profits; and - investors (under prior that others stay) choose to stay. - Therefore, for any C from Stage 1 there is a unique equilibrium of the subgame; brokers use these expected profits π^e_j(C) when negotiating in Stage 1. ### Stage 1 — coalition formation and iterated deletion of dominated coalition structures - Brokers’ expected profits by coalition structure (Table A.4): - C = {{1},{2},{3}}: - π^e_1 = τ − 0.55γ − 0.16 - π^e_2 = 2τ − 0.45γ − 0.06 - π^e_3 = 3τ − 0.58γ - C = {{1, 2},{3}}: - π^e_1 = τ − 0.31(γ + κ) − 0.50s − 0.14 - π^e_2 = 2τ − 0.25(γ + κ) − 0.50s - π^e_3 = 3τ − 0.56(γ + κ) − 0.14 - C = {{1, 3},{2}}: - π^e_1 = τ − 0.11(γ + κ) − 0.50s − 0.09 - π^e_2 = 2τ − 0.31(γ + κ) − 0.23 - π^e_3 = 3τ − 0.20(γ + κ) − 0.50s + 0.09 - C = {{1},{2, 3}}: - π^e_1 = τ − 0.50(γ + κ) − 0.20 - π^e_2 = 2τ − 0.17(γ + κ) − 0.50s − 0.06 - π^e_3 = 3τ − 0.33(γ + κ) − 0.50s - C = {{1, 2, 3}}: - π^e_1 = τ − 0.33s − 0.16 - π^e_2 = 2τ − 0.33s − 0.06 - π^e_3 = 3τ − 0.33s - Iterated deletion results: - 1 & 2 Coalition never arises in equilibrium (Broker 1 prefers 1 & 3; Broker 3 prefers 1 & 3 unless s > 0.72(γ + κ) + 0.46, but then Broker 1 would prefer No Coalition). - Threshold condition: s > 0.72(γ + κ) + 0.46 . (A.1.23) - 2 & 3 Coalition never arises (Broker 3 always prefers 1 & 3; Broker 1 prefers 1 & 3 unless s > 0.78(γ + κ) + 0.22, but then Broker 3 would prefer No Coalition). - Threshold condition: s > 0.78(γ + κ) + 0.22 . (A.1.24) - Grand Coalition never forms in equilibrium for interior parameter space because at least one broker always prefers to deviate. In particular, even when γ = 1/6 and κ → 0 (maximizing Grand Coalition appeal), Broker 1 deviates unless: - s > 0.30 . (A.1.25) - Therefore only No Coalition and the 1 & 3 Coalition can form in equilibrium. - Parameter thresholds for 1 & 3 Coalition vs No Coalition: - Broker 3 prefers 1 & 3 over No Coalition when: - s < 0.18 + 0.76γ − 0.4κ . (A.1.26) - Broker 1 prefers 1 & 3 over No Coalition when: - s < 0.14 + 0.88γ − 0.22κ . (A.1.27) - Broker 1’s threshold expressed exactly: - s < 1/8 + 7/8 γ − 7/32 κ . (A.1.28) - If s < 1/8 + 7/8 γ − 7/32 κ, the 1 & 3 Coalition forms; otherwise No Coalition forms. At equality Broker 1 is indifferent and 1 & 3 Coalition forms. ### Welfare comparisons (Lemma 2) — expected aggregate welfare W^e(C) - Aggregate welfare components: - Sum of investors’ endowments: 6η - Total expected gains from trade: 4.125 (constant across coalition structures) - Setup cost s incurred under all coalitions except No Coalition - Expected total cost of legacy trades: (γ + T · κ)· n^e_L where n^e_L = n^e_{1,L} + n^e_{2,L} + n^e_{3,L} - Welfare by coalition structure (Table A.5): - {{1},{2},{3}}: W^e = 6η + 4.125 − γ· n^e_L(No Coalition) - {{1, 2},{3}}: W^e = 6η + 4.125 − s − (γ + κ)· n^e_L(1 & 2) - {{1, 3},{2}}: W^e = 6η + 4.125 − s − (γ + κ)· n^e_L(1 & 3) - {{1},{2, 3}}: W^e = 6η + 4.125 − s − (γ + κ)· n^e_L(2 & 3) - {{1, 2, 3}}: W^e = 6η + 4.125 − s - No partial coalition structure ever maximizes welfare. - n^e_L(No Coalition) = 13/8, hence welfare indifference between No Coalition and Grand Coalition occurs at: - s = 13/8 γ . (A.2.1) - Therefore: - If s > 13/8 γ → welfare maximized by No Coalition. - If s < 13/8 γ → welfare maximized by Grand Coalition. - If s = 13/8 γ → welfare equal under both. ### Interoperability mandate (Lemma 3) — equilibrium and effects on coalition formation - Stages 3 and 2 remain as in baseline; brokers’ payoffs π^e_j(C) as in Table A.4 when coalition C emerges from Stage 1. - Under mandate, Stage 1 broken into sub-stages; excluded partial coalition members can be invited to join at no cost in Stage 1.3 and will accept if profitable. - Stage 1.3 outcomes: - If 1 & 2 formed, Broker 3 prefers joining: π^e_3(Grand via 1 & 2) = 3τ dominates π^e_3(1 & 2 Coalition) = 3τ − 0.56(γ + κ) − 0.14 . (A.3.1) - If 1 & 3 formed, Broker 2 prefers joining: π^e_2(Grand via 1 & 3) = 2τ − 0.06 dominates π^e_2(1 & 3) = 2τ − 0.31(γ + κ) − 0.23 . (A.3.2) - If 2 & 3 formed, Broker 1 prefers joining: π^e_1(Grand via 2 & 3) = τ − 0.16 dominates π^e_1(2 & 3) = τ − 0.50(γ + κ) − 0.20 . (A.3.3) - Stage 1.2 payoffs under mandate (Table A.6): - If partial coalitions form they effectively become Grand Coalition at Stage 1.3 with setup cost allocation, leading to modified π^e_j entries in Table A.6 (rows for partial coalitions show π^e_j with −0.50s terms). - Equilibrium under different constraints on brokers’ maximum setup investments s^max_j: - All brokers unconstrained (s^max_j ≥ 0.5 for all j): only equilibrium is No Coalition ({{1},{2},{3}}). - Two brokers unconstrained (two s^max_j ≥ 0.5, third s^max_X < 0.5): possible equilibria are {{X},{Y,Z}} and {{X},{Y},{Z}}; reasoning leads to only {{X},{Y},{Z}} feasible. - Fewer than two brokers unconstrained: - If any broker has s^max_j < 0.33 → only No Coalition possible. - If all three brokers have s^max_j ≥ 0.33 → Grand Coalition possible if: - π^e_1(Grand) ≥ π^e_1(No Coalition) → s ≤ 1.67γ . (A.3.4) - π^e_2(Grand) ≥ π^e_2(No Coalition) → s ≤ 1.36γ (fraction form s ≤ 27/20 γ). (A.3.5) - π^e_3(Grand) ≥ π^e_3(No Coalition) → s ≤ 1.76γ . (A.3.6) - Aggregate condition reduces to s ≤ 27/20 γ with Broker 2 as the marginal broker. *Source: wpiea2025185-source-pdf - 4. If this backward induction results in a unique SPNE that, moreover, is consistent with* ### 1. If they set s ### wpiea2025185-source-pdf - 1. If they set s ### Stage 1.1–1.3 broker choice and coalition outcomes - If they set s_max_X ≥ 0.5, the outcome after Stage 1.3 will be either No Coalition or a Grand Coalition for which X pays s_X = 0.5. - If they set s_max_X < 0.33, the outcome after Stage 1.3 will be No Coalition or a Grand Coalition for which X pays s_x = 0. - If they set 0.33 ≤ s_max_X < 0.5, there are three possibilities: No Coalition, a Grand Coalition to which X contributes equally (s_X = 0.33), or a Grand Coalition to which X contributes nothing (s_X = 0). - Policy 1 is dominated by the other two policies, so X will never choose it. - When no broker sets s_max_j ≥ 0.5, free-riding Grand Coalition outcomes are ruled out; Policy 2 therefore always produces No Coalition. - Under Policy 3, with no broker setting s_max_j ≥ 0.5, outcomes are either No Coalition or the Egalitarian Grand Coalition (all brokers contribute equally). ### Interoperability mandate equilibrium characterization - Brokers’ Stage 1.1 choices reduce to allowing the possibility of the Egalitarian Grand Coalition (requiring 0.33 ≤ s_max_X < 0.5) or ruling it out (s_max_X < 0.33). - Broker 1 allows the Egalitarian Grand Coalition if condition (A.3.4) holds. - Broker 2 allows it if condition (A.3.5) holds. - Broker 3 allows it if condition (A.3.6) holds. - If condition (A.3.5) holds, so must (A.3.4) and (A.3.6); thus the outcome depends only on condition (A.3.5). - Under the interoperability mandate, the Egalitarian Grand Coalition forms if s ≤ 27/20 γ, and No Coalition forms otherwise. This outcome is summarized by the green boundary in Figure 4a. ### Relation to baseline outcomes (Lemma 3, Lemma 4, Lemma 5) - Lemma 3 follows by comparing the interoperability-mandate outcome derived above to the excessive investment region in Proposition 1. - Lemma 4 follows by comparing the outcome in Section A.3.1 to the insufficient tokenization region in Proposition 1; graphically, the lower part of the insufficient tokenization region (in Figure 3b) changes to the optimal Grand Coalition outcome in Figure 4a. The boundary of this lower part is s ≤ 27/20 γ. - Lemma 5: In the upper part of the insufficient tokenization region in the baseline, No Coalition results under the interoperability mandate. The upper boundary of this part is s ≤ 13/8 γ (from Proposition 1). The lower boundary is s > 27/20 γ (from Section A.3.1). ### Tokenization subsidy and Lemma 6 - Policymaker bears σ of total setup cost s when 27/20 γ < s ≤ 13/8 γ. - Broker 2 chooses the Grand Coalition over No Coalition if: s ≤ 27/20 γ + σ. (A.6.1) - Welfare is maximized when the corresponding boundary is: s ≤ 13/8 γ. (A.6.2) - Equating (A.6.1) and (A.6.2) gives 27/20 γ + σ = 13/8 γ ⇐⇒ σ = 11/40 γ. (A.6.3) - A subsidy σ ≥ 11/40 γ prevents the insufficient tokenization that would otherwise result by tipping Broker 2 into choosing the Grand Coalition. ### Side-payments, aggregate profits, and Proposition 3 - When side-payments are possible, any equilibrium outcome (C, v) must maximize expected aggregate broker profits Π^e. - If an equilibrium (C, v) did not maximize Π^e, there would exist (C′, v′) making all brokers strictly better off, contradicting equilibrium. - Expected aggregate broker profits are maximized by: - No Coalition when s ≥ 7313/4640 γ. - Grand Coalition when s ≤ 7313/4640 γ. - Numerical equality: 7313/4640 γ = 1.58γ (as presented). - Therefore, with flexible side-payments and tie-breaking as in Section 2.4: - Equilibrium coalition C is No Coalition when s > 7313/4640 γ. - Equilibrium coalition C is the Grand Coalition when s ≤ 7313/4640 γ. - Comparing to Lemma 2: when 7313/4640 γ < s < 13/8 γ, the Grand Coalition is socially optimal but private equilibrium features No Coalition — implying insufficient tokenization and underinvestment. ### Subsidy, alignment of private and social optima, and Proposition 4 - A subsidy σ lowers brokers’ perceived s to s − σ. - The fully flexible side-payments equilibrium fails to maximize welfare only when perceived s is too high so equilibrium is No Coalition while Grand Coalition is socially optimal. - A sufficiently large subsidy that lowers perceived s into the region where equilibrium features the Grand Coalition can align equilibrium outcomes with social welfare. ### Extended model with investor types, fees, profits, and Proposition 5 - At Stage 4, investor-type distributions: - Three investors same type: probability 1/4. - Two of one type and one of the other: probability 3/4. - Table A.7 reports distribution probabilities: 3a 1/8; 2a + 1b 3/8; a + 2b 3/8; 3b 1/8. - Table A.8 investor match probabilities by coalition structure: - No Coalition: Local broker 1/2; Off-island 5/8. - Partial coalition—includes local broker: Local 5/8; Off-island 5/8. - Partial coalition—excludes local broker: Local 1/4; Off-island 5/8. - Grand Coalition: Local 1/2; Off-island 5/8. - At Stage 2, brokers optimize fees; they always compete down to zero for off-island investors and set highest fees on local investors consistent with on-island retention: - Table A.9 brokers’ optimal fees on local investors: - No Coalition: Off-island N/A; Local broker fee = 2τ − 1/4. - Partial Coalition: Coalition insider fee = 8/5 τ? (table entries as in source) and outsider fee = 4τ − 3/2 (as presented). - Grand Coalition: Coalition insider fee = 2τ − 1/4; outsider N/A. - (Preserve the table entries exactly as shown in the source.) - Table A.10 brokers’ expected profits by position: - No Coalition: Coalition insider N/A; Coalition outsider τ − γ/2 − 1/8. - Partial Coalition: Coalition insider τ − (γ+κ)/8 − s/2; Coalition outsider τ − (γ+κ)/4 − 3/8. - Grand Coalition: Coalition insider τ − s/3 − 1/8; Coalition outsider N/A. - Being part of the Grand Coalition is always dominated in private incentives: when Grand Coalition is more profitable than No Coalition (s ≤ 3/2 γ), it is necessarily less profitable than being an insider on a partial coalition. - No Coalition is equilibrium when s > 1−κ/4 + 3/4 γ (derived from τ − γ/2 − 1/8 > τ − (γ+κ)/8 − s/2). - If s ≤ 1−κ/4 + 3/4 γ then a partial coalition is the equilibrium. - Table A.11 welfare by coalition structure: - No Coalition: Welfare = 3η + 3/2 (1 − γ). - Partial Coalition: Welfare = 3η + 3/2 − s − 2/3 (γ + κ). - Grand Coalition: Welfare = 3η + 3/2 − s. - Grand Coalition maximizes welfare when s < 3/2 γ; No Coalition dominates when s > 3/2 γ. - Since private equilibrium yields a partial coalition when s < 1−κ/4 + 3/4 γ, implications: - Insufficient tokenization if s ≤ 3/2 γ. - Excessive investment if 3/2 γ ≤ s ≤ 1−κ/4 + 3/4 γ. - Interoperability mandate always achieves social optimum by precluding trade diversion through partial coalitions: - Under mandate brokers weigh Grand Coalition vs No Coalition. - Brokers prefer Grand Coalition if τ − s/3 − 1/8 ≥ τ − γ/2 − 1/8, which simplifies to s ≤ 3/2 γ — exactly the social-optimum cutoff. - Under the mandate, brokers’ preferences coincide and when Grand Coalition chosen it maximizes aggregate broker profit and welfare. - Existence notes: - Excessive investment region always exists since condition (7) implies 3/2 γ < 1−κ/4 + 3/4 γ. - Insufficient tokenization region exists from γ > 0. *Optimal Policy for Financial Market Tokenization — Working Paper No. WP/2025/185* --- _Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025185-source-pdf.pdf_