## _wp1337

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### Fraud and market participation — motivation and high-level findings
- Recent international economic turmoil prompted a re-evaluation of effectiveness of financial regulatory systems and enforcement for financial stability.
- The paper studies how institutional characteristics of the securities industry (notably demutualization and proliferation of trading venues) affect enforcement, investor market participation, and welfare.
- Key concerns:
  - Conflict of interest when exchanges perform regulatory functions while operating as for-profit competitors may trigger a "race to the bottom" in market surveillance to attract trading activity and minimize regulatory costs.
  - Fragmentation of order flow across multiple markets complicates detection and deterrence of fraud and raises questions about responsibility for trading activity across market centers.
- Empirical/contextual observations preserved from source:
  - Since the Stockholm Stock Exchange demutualized in 1993, a considerable number of stock exchanges followed similar steps; examples: CME and Nasdaq went public in 2002; NYSE demutualized in 2006.
  - SEC imposed sanctions on a SRO for failing to pursue wrongdoing (Securities and Exchange Commission, Release No. 34-51163, February 9, 2005).
  - From 2006 to September 2011, 25% and 11% of the number of controversies served by FINRA involved misrepresentation or omission of the broker and unsuitability cases, respectively.
- Core comparative result:
  - With complete contracts and investors perfectly observing monitoring technology, SRO competition (versus monopoly) enhances welfare: it reduces transaction fees and monitoring resources while preserving maximum market participation and preventing broker misconduct.
  - With incomplete contracts (investors uncertain about SRO monitoring effectiveness), competition among exchanges reduces total welfare relative to monopoly by inducing relaxation of enforcement policies and increasing the number of frauds:
    - Higher fraud probability reduces investor participation and total trading volume.
    - Increased number of investigations creates resource waste and legal/opportunity costs.
  - Results are robust to extensions with sophisticated investors: when investors can influence broker venue choice, SROs compete on fees to gain market share; reduced revenues force SROs to relax enforcement, yielding an equilibrium observationally equivalent to competition through enforcement policies.

### Model — agents, information, and objectives
- Agent classes:
  - Investors: continuum of risk-neutral investors of measure 1.
  - Brokers: continuum of risk-neutral brokers of measure 1; brokers have private information on transaction cash flows.
  - Two competitive and identical SROs: for-profit, profit-maximizing entities that own and manage trading venues and set enforcement policies and fees to attract trading volume.
- Transaction and contracting primitives:
  - After random matching, each investor offers a take-it-or-leave-it contract to the broker. Offer is extended only if investor's expected payoff is at least the reservation utility .
  - Broker outside option is 0; broker accepts any feasible contract.
  - Transaction cash flow W is a binary random variable with support  = {w, w̄} ⊂ R+ with w <  < w̄, independent of SRO where transaction takes place.
  - Realization w̄ is costlessly observable only by the broker and takes the value w̄ with probability ϱ.
  - Broker chooses trading venue indicator ϕ ∈ {1,2} with equilibrium market share q_i for SRO i.
  - Broker reports r(w) to investor; reports outside support are trivially false.
  - Contract menu Z: functions z(r;i) define investor cash contingent on report r and venue i.
- Broker incentive issues:
  - Brokers may misreport realization of W; each SRO devotes monitoring resources m to verify reports and can enforce penalty x ≥ 0 on brokers per pre-specified rules.
- SROs' actions and objectives:
  - Each SRO announces enforcement policy (m_i, x_i) and trading fee t_i before investor-broker contract negotiation.
  - Probability of detecting a fraud p(m_i) = max{1− = m_i; 0}, where = 0 represents the efficiency level of the monitoring technology (as printed in the source).
  - Monitoring cost C_i(m) is increasing; linear form C_i(m) = c m used where c is cost per unit monitoring.
  - If fraud detected, SRO i charges penalty x_i(r;w) returned to investor net of a fraction 
 lost in legal and opportunity costs.
  - SROs are profit maximizing and compete to attract trading volume.

### Timing of the game (enumerated)
- 1. Each SRO simultaneously chooses and commits to an enforcement policy (m_i, x_i) and a trading fee t_i.
- 2. Based on announced (m_i, x_i, t_i)_{i=1,2}, the investor either offers a contract z to a broker or takes reservation utility .
- 3. If offer rejected investor gets  and broker gets 0. If accepted, broker chooses where to execute transaction (probability q_i).
- 4. Broker executes transaction on SRO i, privately observes W, chooses report r and returns z(r;i) to investor.

### Key analytical mechanisms and constraints
- Broker feasibility constraint (BF): z(r;i) ≤ w − t_i; ∀w, r ∈ {w, w̄}, i ∈ {1,2}.
- Low-state misreporting never optimal: r(w) = w.
- Lemma 1: Given β > w, broker compensation for reporting low cash flow on SRO_i is zero: z_i = w − t_i.
- SROs devote zero monitoring resources to investigate high cash-flow reports: m = 0 and p(r = w̄) = 0.
- Broker incentive compatibility constraint (BIC) (general form):
  - (BIC) : w − t_i − z̄_i ≥ p_i(w) max{w − w̄ − x_i; 0} + [1 − p_i(w)](w − w̄); i = 1,2.
  - With x_i = w − w̄ and probability of not detecting fraud =  = m_i, (BIC) simplifies to:
    - (BIC) : φ_i(z_i) ≡ w − z_i − t_i − (w − w̄) m_i ≥ 0.
  - φ_i captures the contract’s incentive power; φ_i decreasing in z, increasing in broker compensation, and increasing in m.

### The Race to the Bottom — competitive equilibrium with complete contracts (summary)
- Assumptions: homogeneous investors, complete contracts, equilibrium with participation and no fraud.
- Investor minimizes broker compensation subject to (BIC).
- Proposition 3: Given m_i ≥ , and ~q_i = ~q(m_i; m_i?), optimal contract z_i = w − θ̄_i − t_i where θ̄_i = m_i (w − w̄).
- Investor participation constraint (CIR):
  - (CIR) : E[W] − Σ_{i=1,2} ~q_i [ t_i + (1 − ϱ)( (w − w̄) / m_i ) ] ≥ β.
- Broker expected utility simplifies to:
  - U_i = (1 − ϱ)(w − w̄) / m_i.
  - Broker prefers SRO_i iff m_i < m_i?.
  - Probability of executing on SRO_i:
    - q_i = { 1 if m_i < m_i?; 1/2 if m_i = m_i?; 0 if m_i > m_i? }.
- SRO_i maximization (anticipating responses):
  - max_{t_i, m_i} q_i(m_i; m_i?) [ t_i − ϱ m_i c ]
  - subject to (CIR): q_1 t_1 + q_2 t_2 + (1 − ϱ)(w − w̄) [ q_1 / m_1 + q_2 / m_2 ] ≤ ~β, where ~β ≡ E[W] − β > 0.
- Proposition 4: Given β > w and ~β > max{ p (w − w̄) c ; 2ϱ c }, if a Nash equilibrium where SROs get zero profits and ~q_i = q_i exists, it is symmetric (q_i = 1/2, t_1 = t_2 = t, m_1 = m_2 = m) and must satisfy:
  - t = ϱ c m
  - m = [ ~β / (2 ϱ c) − sqrt{ ~β / (2 ϱ c) ^2 − (1 − ϱ) ( ϱ c (w − w̄) ) } ].
- Consequences:
  - Competition drives SRO profits to zero.
  - Competition operates through monitoring resources: SROs reduce m to gain market share, increasing brokers’ compensation and reducing investor expected profits.
  - To sustain participation SROs must reduce transaction fees t (ultimately paid by investors).

### Monopoly setup and equilibrium (Proposition 5)
- Monopolist SRO problem:
  - max_{m;t} t − ϱ m c
  - s.t. (CIR) : t + (1 − ϱ)(w − w̄)/m = ~β.
- Proposition 5 (under monopoly, given c  ~β^2 / (4(1−ϱ)ϱ (w − w̄)) as printed):
  - the optimal enforcement policy and transaction fee chosen by the SRO are:
    - m = sqrt{ (w − w̄) (1 − ϱ) / (ϱ c) } (as printed in source form)
    - t = ~β − ϱ c m.
- Graphical comparison (Figure 1, Panels A and B) described:
  - Region below dark solid line satisfies CIR; dotted lines are SRO iso-profits; monopoly maximizes profit at M; competition pushes to symmetric zero-profit point N where monitoring and fees are lowest without making unit profits negative.
- Broker compensation and rent distribution:
  - Broker compensation is decreasing in detecting probability.
  - Brokers receive higher compensation under competition than under monopoly.
  - Competition transfers monopolist rent to brokers rather than investors.

### Welfare comparison (monopoly vs competition)
- Welfare measure W obtained by summing surpluses/profits of investors, brokers, and SROs:
  - W = E[W] − β − ϱ m in both competition and monopolist case.
- Since m_m > m_c (monopolist monitors more), welfare difference:
  - W_c − W_m = ϱ c (m_m − m_c) > 0 (implying monopoly yields higher welfare when contracts incomplete as discussed later).
- Interpretation:
  - Monopolist SRO over-monitors brokers, reducing their information rent to its advantage.
  - In the complete-contract benchmark no frauds occur and full participation is obtained under both monopoly and competition; competition can be welfare enhancing if monopolist over-monitors.

### Fraud and market participation — uncertainty over the monitoring technology (incomplete contracts)
- Relaxation: investors do not perfectly observe monitoring technology parameter  which is private to SROs and brokers.
- Investor j observes random variable with distribution F(x;j); investors are sorted by optimism j ∈ [0,1].
- Uniform distribution formulation used:
  - sU( −2a(1−j); + 2a j ) with F(y;j) = (y + 2a(1−j)) / (2a) for y in [ −2a(1−j); + 2a j ] and j in [0;1].
  - This gives ~  j = +a(2j − 1), increasing in j.
- New features:
  - Some investors find it too costly to offer contracts that are incentive compatible with probability 1.
  - Partial market participation and brokers may cheat in equilibrium.
- Investor expected return when high cash-flow realizes but misreport possible simplifies to (1−ϕ) x ~p_{ji} + z_i, where ~p_{ji} = E_j[max{1 −  = m_i; 0} g_{> _i(z)}].
- Investor j problem (general form):
  - max_{z_i} F_j(_i(z_i)) z_i + [1 − F_j(_i(z_i))] f( (1 − ϕ) x ~p_{ji} + z_i )
  - s.t. ~p_{ji} = E_j[ max{1 −  = m_i; 0} g_{> _i(z)} ], _i(z_i) = (w − z_i − t_i)/(w − w̄) m_i.
- Proposition 6 (general solution) — given 8ϕ ∈ (0;1) and m_i ≥ + 2a:
  - investor j's optimal high-cash-flow repayment:
    - z_{ji} = w − _j(m_i) − t_i.
  - broker high cash-flow compensation:
    - _{ji} = (w − w̄) min [ 1 +  + ~  j − a (1 + ϕ) m_i ; ~  j + a m_i ].
- Comparative statics:
  - Broker compensation inversely related to monitoring resources (as in perfect knowledge case).
  - Legal and opportunity cost ϕ increases broker compensation.
  - Investors with high uncertainty may optimally bear fraud risk: F_j(_i) < 1.
- Corollary 7 (broker compensation under uncertainty):
  - _{ji} = (w − w̄) [ 1 +  + ~  j − a (1 + ϕ) m_i ] ; if F_j(_i) < 1
  - _{ji} = (w − w̄) [ ~  j + a m_i ] ; if F_j(_i) = 1
- Participation mass n increases in m_i and decreases in t_i.
- Proposition 8: mass of investors participating:
  - n = min{ max{ 1/2 +  n −  2a ; 0 } ; 1 } with  n = B − p B^2 − C (coefficients B and C functions of m_i and t_i).
- Mass of contracts implying fraud when high cash flow realizes:
  - _i = min{ max{ 1 − m_i −  2a ; 0 } ; n g; i = 1;2 }.
  - _i decreases in m_i and increases in uncertainty a.
- SRO objective accounting for partial participation and fraud investigation costs:
  - max_{m_i; t_i} q[ n( t_i − ϱ c m_i ) − (1 − ϱ) _i c m_i ].
- Proposition 9 (symmetric Nash equilibrium under conditions ϕ ∈ (0;1), n < 1, _i < n, _i > 0):
  - equilibrium symmetric: q_i = 1/2, t_1 = t_2 = t, m_1 = m_2 = m.
  - equilibrium condition:
    - n( t − ϱ c m ) = (1 − ϱ)  c m with  = 1 − m −  2a and n = function of t and m (see source).
- Interpretation:
  - SROs earn zero profit in equilibrium but incur higher costs because of frauds: term (1 − ϱ)  c m represents expected investigation costs associated with frauds.
  - Investor participation n < 1 reduces market volume and revenues but does not reduce ratio of frauds over total transactions.

### Monopolist solution under incomplete contracts (Proposition 10) and calibrated comparison
- Monopolist solves same problem as (20) with fixed market share; frauds and participation depend only on m and t.
- Under stated parameter conditions (as printed), monopolist solution:
  - m = sqrt{ (1 − ϱ)(w − w̄) + 2a  / (ϱ c) } (as printed in source)
  - t = ~β − ϱ c m
  - z_{j} = w − ( ~  j + a m ) (w − w̄) − t; which implies n = 1 and  = 0 under those conditions.
- Interpretation:
  - Monopolist internalizes effect of enforcement and fees on participation; sets m and t to preserve full participation (n = 1) and eliminate fraud ( = 0) under parameter restrictions.
  - Greater uncertainty (higher a) increases m and reduces SRO profits; monopolist chooses tougher enforcement and lower t to preserve participation.
- Calibrated comparison (Table 1 highlights):
  - Detecting Probability: 0.82 (Monopoly) | 0.75 (Competition) | - 8.5
  - Transaction Fee over E[W]: 0.55% | 0.41% | -25.5
  - Total Volume (n): 10.95 | 5 | - 4.5
  - Total Frauds: 0 | 0.059 | ñ
  - SROs' Profits: 0.124 | 0 | ñ
  - Brokers' Average Profits: 0.152 | 0.524 | 24.5*10
  - Investors' Actual Average surplus: 0.287 | 0.047 | -16.4
  - Investors' Expected Average surplus: 0.287 | 0.046 | -16.4
  - Total Monitoring Costs: 0.438 | 0.421 | - 3.9
  - Total Welfare: 0:563 | 0:545 | - 3.2
- Calibration inputs (as described in text):
  - Spread between high and low cash flow: w − w̄ = 2:5% (printed as w = w1 = 2:5%)
  - Reservation utility: 1% higher than the low state  = w1 = 1%
  - Low state probability ϱ = 20%
  - Fix = 10 and set c such that unit monitoring cost cm is 0.8% of average transaction value when detecting probability is 50%; implies c = 0:04
  - Legal costs set at 6% of the penalty (ϕ = 6%)
  - Noise a = 10%
  - Total transaction surplus ~β = 1
- Calibrated implications:
  - Detecting probability falls by about 9% under competition (0.82 → 0.75).
  - Transaction fees fall by 25% (0.55% → 0.41%).
  - Investors’ actual average surplus falls from 0.287 to 0.047; expected surplus 0.287 to 0.046 (large drop).
  - Brokers’ average profits rise from 0.152 to 0.524 (more than threefold).
  - Under competition about 6% of total transactions involve a fraud; with detecting probability 0.75 only about 4.4% of frauds detected.
  - Total monitoring costs fall modestly (~4%).
  - Total welfare falls under competition by about 3.2% relative to monopoly.

### Mechanism driving welfare differences
- Under complete contracts: competition reduces both transaction fees and monitoring and can be welfare enhancing because monopolist over-monitors.
- Under incomplete contracts: competition reduces enforcement and, because fraud arises in equilibrium:
  - Lower investor participation → lower total trading volume.
  - More misreports → more investigations → waste of monitoring, legal, and opportunity costs.
  - Net effect: total welfare reduced under competition relative to monopoly.

### Extension: Sophisticated investors and numerical results
- Sophisticated investors anticipate broker venue choice and strategically choose contracts to induce broker behavior; SROs respond by reducing transaction fees to attract volume.
- Proposition 11 (for a = 0) characterizes optimal contracts (1, 2) and q1 depending on relative t1 and t2 and monitoring levels; q1 equals 1, 1/2 or 0 depending on relative t1 and t2.
- Proposition 12: there exists a unique symmetric equilibrium q = 1/2, t1 = t2 = t, m1 = m2 = m satisfying:
  - ϱ c m + (1 − ϱ)(w − w̄)/m = ~β
  - t = ϱ c m
  - Result: same equilibrium values as basic model with naive investors.
- Numerical simulation (comparison naive vs sophisticated, Table 2):
  - Detecting Probability: 0.75 (Sophisticated) ; 0.75 (Naive)
  - Transaction Fee over Average Transaction Value: 0.39% ; 0.41%
  - Total Volume (n): .96 ; .955
  - Total Frauds: 0.055 ; 0.059
  - SROs' Profits: 0 ; 0
  - SRO 1 Market Share: 50% ; 50%
  - Brokers' Average Profits: 0.523 ; 0.524
  - Investors' Expected Average surplus: 0.048 ; 0.046
  - Total Monitoring Costs: 0.424 ; 0.421
  - Welfare: 0.548 ; 0.545
- Parameterization used in simulations (as printed):
  - w = w1 = 2:5%
  - β = w1 = 1%
  - ϱ = 20%
  - θ = 10 (printed as = 10)
  - c = 0:04
  - ϕ = 6%
  - a = 10%
- Comparative insight: results for sophisticated heterogeneous investors are qualitatively the same as for naive investors; most quantitative differences attributed to numerical approximation errors.

### Main conclusions and policy implications
- Competition among for-profit SROs reduces transaction fees and monitoring resources:
  - Lower fees benefit investors directly, but reduced monitoring increases brokers’ information rents and fraud incidence when investor beliefs about monitoring effectiveness are imperfect.
- Under complete contracts competition can be welfare enhancing by correcting monopolist over-monitoring.
- Under incomplete contracts (realistic informational frictions) competition can reduce total welfare relative to monopoly because:
  - Reduction in enforcement leads to fraud in equilibrium → lower participation → lower volume.
  - More misreports increase investigations and legal/opportunity costs.
- Institutional implications:
  - The organizational form and competitive structure of exchanges/SROs matter for enforcement incentives; demutualization and fragmentation can have unintended adverse effects on market integrity.
  - Policymakers should account for investors' uncertainty about enforcement effectiveness and the interaction between fees, monitoring investments, and venue competition when assessing market structure reforms.
- Robustness: results persist with investor sophistication and whether competition operates through fees or enforcement.

*Source: Excerpt from the document titled "1. The Monopolist Solution" and associated sections (PDF chapter/section: _wp1337).*

### 1. The Monopolist Solution . . . . . . . . . . . . . . . . . . . . . . . .   12

### 1. The Monopolist Solution

### Fraud and market participation — motivation and high-level findings
- Recent international economic turmoil prompted a re-evaluation of the effectiveness of Önancial regulatory systems and enforcement for Önancial stability.
- The paper studies how institutional characteristics of the securities industry (notably the evolution from mutually-owned exchanges to for-proÖt competitors and the proliferation of trading venues) can affect enforcement of regulation, investor market participation, and welfare.
- Key concerns motivating the analysis:
  - Conflict of interest when exchanges perform regulatory functions while operating as for-proÖt competitors may trigger a "race to the bottom" in market surveillance to attract trading activity and minimize regulatory costs.
  - Increased fragmentation of order áow across multiple markets complicates detection and deterrence of fraud and raises questions about responsibility for trading activity across market centers.
- Empirical/contextual observations preserved from source:
  - Since the Stockholm Stock Exchange demutualized in 1993, a considerable number of stock exchanges followed similar steps; examples: CME and Nasdaq went public in 2002; NYSE demutualized in 2006.
  - SEC imposed sanctions on a SRO for failing to pursue wrongdoing, possibly related to the firm's role in SRO revenue generation (Securities and Exchange Commission, Release No. 34-51163, February 9, 2005).
  - From 2006 to September 2011, 25% and 11% of the number of controversies served by FINRA involved misrepresentation or omission of the broker and unsuitability cases, respectively.
- Core comparative result (summary of main analytical findings):
  - With complete contracts and investors perfectly observing monitoring technology, SRO competition (versus monopoly) enhances welfare: it reduces transaction fees and monitoring resources while preserving maximum market participation and preventing broker misconduct.
  - With incomplete contracts (investors uncertain about SRO monitoring effectiveness), competition among exchanges reduces total welfare relative to monopoly by inducing relaxation of enforcement policies and increasing the number of frauds. Consequences:
    - Higher fraud probability reduces investor participation and total trading volume.
    - Increased number of investigations creates resource waste and legal/opportunity costs.
  - Results are robust to extensions with sophisticated investors: when investors can influence broker venue choice, SROs compete on fees to gain market share; reduced revenues force SROs to relax enforcement, yielding an equilibrium observationally equivalent to competition through enforcement policies.

### Model — agents, information, and objectives
- Agent classes:
  - Investors: continuum of risk-neutral investors of measure 1.
  - Brokers: continuum of risk-neutral brokers of measure 1; brokers have private information on transaction cash áows.
  - Two competitive and identical SROs: for-proÖt, proÖt-maximizing entities that own and manage trading venues and set enforcement policies and fees to attract trading volume.
- Transaction and contracting primitives:
  - After random matching, each investor offers a take-it-or-leave-it contract to the broker. The offer is extended only if the investor's expected pay-o§ is at least the reservation utility .
  - Broker outside option is assumed to be 0, so the broker accepts any feasible contract.
  - Transaction cash áow W is a binary random variable with support  = {w, w} ⊂ R+ with w <  < w, independent of SRO where transaction takes place.
  - Realization w is costlessly observable only by the broker and takes the value w with probability .
  - The broker chooses where to execute the transaction (SRO i = 1 or i = 2). Ex post this choice is perfectly observable and contractible, captured by a trading venue indicator  ∈ {1,2} with probability distribution {q_i} (equilibrium q_1 is SRO 1 market share).
  - After the transaction, the broker reports r(w) to the investor. Since support of W is common knowledge, r:  →  and any report r ∉  is trivially false.
  - Contract menu Z: functions z(r;i) ∈ Z define investor cash contingent on broker report r(w) and trading venue i; z(r;i) is investor's state-contingent net trading proÖt.
- Broker incentive issues:
  - Brokers may misreport realization of W; each SRO devotes monitoring resources m to verify reports and can enforce penalty x ≥ 0 on brokers per pre-specified rules.
- SROs' actions and objectives:
  - Each SRO announces and commits to enforcement policy (m_i, x_i) and trading fee t_i before investor-broker contract negotiation.
  - Probability of detecting a fraud p(m_i) depends on monitoring resources m_i. The source's convenient formulation:
    - p(m_i) = maxf1 =m_i;0g, where 1= 0 represents the e¢ ciency level of the monitoring technology. (As printed in the source.)
  - Monitoring cost function C_i(m) is increasing; for simplicity the linear form C_i(m) = c m is used, where c is cost per unit monitoring.
  - If fraud detected, SRO i charges penalty x_i(r;w) (function of reported cash áow r(w) and actual w); the penalty is returned to investor net of a fraction 
 lost in legal and opportunity costs.
  - SROs are proÖt maximizing and compete to attract trading volume.

### Timing of the game (enumerated as in source)
- 1. Each SRO simultaneously chooses and commits to an enforcement policy (m_i, x_i) and a trading fee t_i.
- 2. Based on announced enforcement policies and transaction fees (m_i, x_i, t_i)_{i=1,2}, the investor can either o§er a contract z to a broker or take reservation utility .
- 3. If the investor o§ers the contract and the broker rejects it, investor gets  and broker gets 0. If accepted, the broker chooses where to execute the transaction (by choosing probability q_i).
- 4. The broker executes the transaction on SRO i and privately observes realization of W. He chooses report r and returns to the investor z(r;i).

### Methodology and relation to literature
- The model extends and adapts DeMarzo et al. (2005) to a competitive SRO setting and relies on costly-state-veriÖcation frameworks (Townsend (1979), Mookherjee and Png (1989)).
- Enforcement modeled in a two-tier delegated auditing / hierarchical contracting framework (as in DeMarzo et al. (2005) and Faure-Grimaud et al. 1999).
- Distinctions from prior work:
  - Unlike DeMarzo et al. (2005), SROs here are proÖt-maximizing and competitive rather than a single monopolist; the paper shows competition can reduce enforcement rigor and investor participation.
  - Contrast with Santos and Scheinkman (2001): that paper finds competition does not necessarily produce a race to the bottom when credit quality is observable; this paper finds competition likely produces less rigorous enforcement when investors hold non-contractible uncertainty about monitoring effectiveness.

### Contributions and implications (policy-relevant points)
- Two main contributions:
  - Demonstrates that competition among for-proÖt SROs competing for trading volume can penalize enforcement quality, lowering investor participation and increasing fraud relative to monopoly.
  - Extends the literature on self-regulation by showing that costs of self-regulation associated with monopoly power apply even when SROs are independent for-proÖt competitors.
- Implications for institutional design of securities regulation:
  - The organizational form and competitive structure of exchanges/SROs matter for enforcement incentives; demutualization and fragmentation can have unintended adverse effects on market integrity.
  - Policymakers should account for investors' uncertainty about enforcement effectiveness and for the interaction between fees, monitoring investments, and venue competition when assessing market structure reforms.

*Source: Excerpt from the document titled "1. The Monopolist Solution" (PDF chapter/section).*

### 5. Given the reportr, the SRO where the trade took place devotesm(r)resources to

### _wp1337 - 5. Given the reportr, the SRO where the trade took place devotesm(r)resources to

### D. The Broker-Investor Contract: General Results
- Setup
  - Investor chooses a contract z ∈ Z, z: {1,2} → R, where z(r;i) is the cash transfer from the broker to the investor for trading venue i and broker report r.
  - Broker observes enforcement policies (m_i, x_i), trading venue indicator ϕ, and cash-flow W, and transfers transaction fee t_i to SRO_i.
  - Broker feasibility constraint (BF): (BF) : z(r;i) ≤ w − t_i; ∀w, r ∈ {w, w̄}, i ∈ {1,2}:(1)
  - Assumption: broker transfer to investor is higher in the high cash-flow state than in the low cash-flow state, i.e., z̄_i > z_i.
  - From the above, it is never optimal to misreport the low cash-flow: r(w) = w.

- Lemmas and implications
  - Lemma 1: Given β > w, for each investor-broker pair, the broker’s compensation for trading on SRO_i and reporting the low cash flow, w, is zero: z_i = w − t_i.
  - Consequence: SROs devote zero monitoring resources to investigate high cash-flow reports, m = 0 and p(r = w) = 0.
  - Two mechanisms deter misreporting when high cash flow occurs: the “carrot” (positive compensation for truthful report) and the “stick” (penalty and probabilistic verification by SROs).
  - Broker incentive compatibility constraint (BIC) (general form):
    - (BIC) : w − t_i − z̄_i ≥ p_i(w) max{w − w̄ − x_i; 0} + ... + [1 − p_i(w)](w − w̄); i = 1,2:(2)
    - LHS denotes compensation when high state is reported truthfully: θ̄_i ≡ w − t_i − z̄_i.
    - RHS denotes expected return from misreporting (weighted avg of detected and undetected outcomes).

  - Lemma 2: Given β > w, both SROs optimally set the penalty to its maximum: x_i = w − w̄.
  - Using x_i = w − w̄ and probability of not detecting a fraud = = m, (BIC) simplifies to:
    - (BIC) : φ_i(z_i) ≡ w − z_i − t_i − (w − w̄) m_i ≥ 0:(3)
    - φ_i captures the contract’s incentive power; φ_i is decreasing in z, increasing in broker compensation, and increasing in m. Knowledge of the monitoring technology, , is crucial.

### II. The Race to the Bottom
- Context and assumptions
  - Homogeneous investors and complete contracts (revelation principle applies).
  - Equilibrium considered: investors participate and there is no fraud.
  - Investor subjective probability of trading on SRO_i is ~q_i; rule: ~q_i = 1 for m_i < m_i? and ~q_i = 1/2 for m_i = m_i?.
  - Investors do not design contracts strategically to induce broker venue choice (they do not internalize inducement effect).

A. The Investor Problem
- Investor minimizes broker compensation subject to BIC (equation 3).
- Proposition 3: Given m_i ≥ , and ~q_i = ~q(m_i; m_i?), the optimal contract is
  - z_i = w − θ̄_i − t_i; i = 1,2;(4)
  - where θ̄_i = m_i (w − w̄); i = 1,2:(5)
- Notes:
  - Transaction fee is ultimately paid by the investor.
  - Broker compensation is inversely related to the effectiveness of monitoring (1 − = m_i), i.e., the probability of detecting a fraud.
- Investor participation constraint (CIR):
  - (CIR) : E[W] − Σ_{i=1,2} ~q_i [ t_i + (1 − ϱ)( (w − w̄) / m_i ) ] ≥ β;(6)
  - where E[W] = ϱ w̄ + (1 − ϱ) w is the expected transaction value and ~β ≡ E[W] − β > 0 by construction.

B. The Broker Problem
- Broker expected utility U_i from trading on SRO_i (general expression):
  - U_i ≡ ϱ (w − z_i − t_i) + (1 − ϱ)[ p_i( max{ w − z_i(r(w)) − x_i(r(w); w) − t_i ; 0 } ) + (1 − p_i)( w − z_i(r(w)) − t_i ) ]:(7)
- Applying revelation principle, x_i(w;w) = 0, x_i(w̄;w) = w − w̄, and z_i(w) = 0 yields simplified U_i:
  - U_i = (1 − ϱ)(w − w̄) / m_i:(8)
- Implication:
  - Broker prefers SRO_i iff m_i < m_i?.
  - Probability of executing a transaction on SRO_i:
    - q_i = { 1 if m_i < m_i?; 1/2 if m_i = m_i?; 0 if m_i > m_i? }:(9)

C. The SRO Problem and the Equilibrium
- SRO_i chooses (m_i; x_i; t_i) anticipating investor and broker responses; objective is to maximize expected profits = profit per transaction × trading volume.
  - Profit per transaction = t_i − ϱ m_i c (expected cost of monitoring E[mc] = ϱ m_i c).
- SRO maximization problem (with ~q_i = q_i):
  - max_{t_i, m_i} q_i(m_i; m_i?) [ t_i − ϱ m_i c ]
  - s.t. (CIR): q_1 t_1 + q_2 t_2 + (1 − ϱ)(w − w̄) [ q_1 / m_1 + q_2 / m_2 ] ≤ ~β;
  - where ~β ≡ E[W] − β > 0.
- Proposition 4: Given β > w and ~β > max{ p (w − w̄) c ; 2ϱ c }, if a Nash equilibrium where SROs get zero profits and ~q_i = q_i exists, then it is symmetric (q_i = 1/2, t_1 = t_2 = t, m_1 = m_2 = m) and must satisfy:
  - t = ϱ c m(10)
  - m = [ ~β / (2 ϱ c) − sqrt{ ~β / (2 ϱ c) ^2 − (1 − ϱ) ( ϱ c (w − w̄) ) } ] :(11)
- Consequences and interpretation
  - Competition for trading volume drives SRO profits to zero.
  - When investors are naive, competition operates through monitoring resources: SROs reduce m to gain/maintain market share.
  - Reduction in m increases brokers’ compensation, reducing investors’ expected profits.
  - To sustain investor participation, SROs must also reduce transaction fees t (since t ultimately is borne by investors).

*Source: _wp1337 - 5. Given the reportr, the SRO where the trade took place devotesm(r)resources to (IMF PDF content provided)*

### 1.  The Monopolist Solution

### 1.  The Monopolist Solution

### Monopoly setup and equilibrium (Proposition 5)
- Monopolist SRO problem (as written):
  - max_{m;t} tmc
  - s:t:(CIR) :t+ (1)(ww) =m~:(12)
- Proposition 5 (under monopoly, given c
~
2
4(1)(ww)
):
  - the optimal enforcement policy and transaction fee chosen by the SRO are
    - m=
r
(ww)
(1) 
c
    - t=    ~cm:(13)

- Graphical comparison (Figure 1, Panels A and B):
  - Region below dark solid line satisfies the CIR constraint.
  - Dotted lines: SRO iso-profits; lowest iso-profit line is zero profit locus.
  - Monopoly: SRO maximizes profits at M where tangent iso-profit line is highest feasible profit.
  - Competition: drives SRO profits to zero; symmetric equilibrium lies at intersection of zero-profit line and CIR (points B and N).
  - Point B is not a Nash equilibrium: an SRO can increase unit profits and market share by reducing monitoring resources.
  - Competition pushes equilibrium to point N where monitoring resources and fees are at the lowest, and no incentive to reduce monitoring further because unit profits would become negative.

- Broker compensation and rent distribution:
  - Broker compensation (solid grey line) is decreasing in detecting probability.
  - Brokers receive higher compensation under competition than under monopoly.
  - Competition transfers monopolist rent to brokers rather than investors.

### Welfare comparison (monopoly vs competition)
- Welfare measure W obtained by summing surpluses/profits of investors, brokers, and SROs:
  - W = E[W]m in both competition and monopolist case.
- Since m is relatively high in the monopolist case, welfare difference:
  - W_cW_m = c(m_mm_c) > 0.
- Interpretation:
  - Monopolist SRO over-monitors brokers, reducing their information rent to its advantage.
  - Result intuition: in equilibrium no frauds occur and full participation is obtained under both monopoly and competition.

---

### III.  Fraud and Market Participation — uncertainty over the monitoring technology

### Setup: private monitoring parameter and investor beliefs
- Relax assumption: investors do not perfectly observe monitoring technology parameter , which is private to SROs and brokers.
- Investor j observes a random variable  with distribution F(x;j).
- Unbiased average unconditional expectation:
  - Z_{0}^{1} Z_{+1}^{-1} x f(x;j) d x d j = :
- Investor j's expectation over  denoted ~ 
  - ~ 
j
 = R_{+1}^{-1} x F(x;j) dx.
- Investors sorted by optimism: investor0 most optimistic, investor1 most pessimistic; ~ 
0
 < ~ 
0
0
 for j0 < j00 in [0;1].
- Median investor unbiased: ~ 
j=1/2 = .
- Convenient uniform distribution formulation:
  - sU( 2a(1j); + 2aj), such that
  - F(y;j)   =
y + 2a(1j)
2a
;(14)
  - for y in [ 2a(1j); + 2aj] and j in [0;1].
  - This gives ~ 
j
 = +a(2j1), increasing in j.

### A.  The Investor Problem
- New features:
  - Some investors find it too costly to offer contracts that are incentive compatible with probability 1.
  - Partial market participation; brokers may cheat in equilibrium.
- Investor's expected return when high cash-flow realizes but misreport possible:
  - [(1
)x+z_i] ~p_{ji} + z_i (1~p_{ji}); i = 1;2;
  - where ~p_{ji} = E_j[max{1	=m_i;0} g_{> _i(z)}] is probability fraud detected by SRO i conditional on investor j's information set.
  - Simplifies to (1
) x ~p_{ji} + z_i.
- Investor j problem:
  - max_{z_i} F_j(_i(z_i)) z_i + [1F_j(_i(z_i))] f( (1
) x ~p_{ji} + z_i ) (15)
  - s:t: ~p_{ji} = E_j[ max{1	=m_i;0} g_{> _i(z)} ]
  - _i(z_i) = (wz_it_i)/(ww) m_i
- Trade-off: reduce broker compensation vs increase probability of misreport.
- Proposition 6 (general solution) — conditions:
  - 8
2(0;1) and given m_i  + 2a,
  - investor j's optimal high-cash-flow repayment:
    - z_{ji} = w_j(m_i)t_i :(16)
  - broker high cash-flow compensation from trading on SRO i:
    - _{ji} = (ww) min [ 1 + + ~ 
j
 a (1 +
) m_i ; ~ 
j
 + a m_i ] :(17)
- Comparative statics:
  - Broker compensation inversely related to monitoring resources (as in perfect knowledge case recovered when a!0).
  - Presence of legal and opportunity costs, 
, increases broker compensation.
- Investors with high uncertainty may optimally bear fraud risk: F_j(_i) < 1.
- Overoptimistic investors (j < 1/2) may offer contracts that are actually not incentive compatible because their belief of offering an incentive compatible contract decreases with j while the incentive power of the contract increases.

- Corollary 7 (broker compensation under uncertainty):
  - _{ji} = (ww) [ 1 + + ~ 
j
 a (1 +
) m_i ] ; i§F_j(_i) < 1
  - _{ji} = (ww) [ ~ 
j
 + a m_i ] ; i§F_j(_i) = 1

- Participation mass and its determinants:
  - Mass of participating investors is increasing in m_i and decreasing in t_i.
  - Proposition 8: mass of investors participating:
    - n = min{ max{ 1/2 +  
n
  2a ; 0 } ; 1 } :(18)
    - with  
n
 = B p B^2C, while coefficients B and C are functions of m_i and t_i (see appendix).
  - Interpretation:  
n
 is a cut-off for expected monitoring efficiency; when ~ 
j
 >  
n
 investor j is pessimistic and will not participate.

### B.  The Broker's misconduct
- Mass of contracts implying fraud when high cash flow realizes:
  - For investors with F(_i;j) = 1, contracts incentive compatible by construction.
  - Focus on investors with F(_i;j) < 1: over-optimistic investors may offer non-incentive-compatible compensation.
  - Mass of such contracts:
    - _i = min{ max{ 1 m_i  2a ; 0 } ; n g; i = 1;2 } :(19)
- Comparative statics:
  - _i decreases in monitoring resources m_i and increases in uncertainty a.
- Additional notes:
  - All investors willing to take some risk only if m_i < + 2a / 
, for all j in [0;1]. If 
 = 0 no investor will be sure of offering an incentive compatible contract.
  - With full participation, simple relation between average investor's probability of offering a compatible contract and actual mass of non-compatible contracts  exists; legal costs 
 disconnect the average investor's belief and actual broker behavior.

### C.  The SRO Problem and Equilibrium (Proposition 9)
- SRO objective accounting for partial participation and fraud investigation costs:
  - max_{m_i; t_i} q[ n( t_i   c m_i )  (1) _i c m_i ] :(20)
  - where q, n, and _i are functions of m_i and t_i, parameterized with respect to m_i? and t_i? (see equations 9 and 18-19).
- Proposition 9 (existence and properties of symmetric Nash equilibrium given 
 2 (0;1) and conditions n < 1, _i < n, and _i > 0):
  - equilibrium is symmetric: q_i = 1/2, t_1 = t_2 = t, and m_1 = m_2 = m.
  - equilibrium condition:
    - n( t   c m ) = (1)  c m
    - with  = 1 m   2a and n =  n_t ( t  ( t   c m ) ), where
      - n_t =  1 + (1)(ww)/2 p B^2C m B = m + a(2  2 1) C = m^2 + 4 a^2 h 1 2 = 2 + (1
) ( w    t ) / ( (1)(ww) ) i m + a / 2 .
- Interpretation and implications:
  - SROs still earn zero profit in equilibrium, but incur higher costs because of frauds.
  - Term (1)  c m represents expected investigation costs associated with frauds.
  - Investor participation n < 1, implying reduced market volume and revenues.
  - Overoptimistic investors are last to prefer outside option; as participation falls, mass of fraud _i not affected until n = _i (once n = _i no contract is incentive compatible).
  - Conclusion: reduction in investor participation never reduces the ratio of frauds over total transactions.

* _wp1337 - 1.  The Monopolist Solution_*

### 1.  The Monopolist Solution under Incomplete Contracts

### 1.  The Monopolist Solution under Incomplete Contracts

### Monopolist equilibrium (Proposition 10)
- Setting: monopolist SRO solves the same problem as in (20) with fixed market share; frauds and participation depend only on choices of m and t.
- Proposition condition (as stated):
  - Given
    ~
    2
    1
    >4(ww)
    ( +3)
    2
    +2a
    c
    and
   
 >
    2a
    2a +
    q
    (1)(ww)
    +2a
    c
- Monopolist solution (as stated):
  - m = r
    (1)(ww)
    + 2a
    c
  - t =    ~c m
  - z
    j
    = w
    ~

    j
    + a
    m
    (ww)t;
  - which implies n = 1 and  = 0.
- Interpretation:
  - The monopolist SRO fully internalizes the effect of its enforcement policy and transaction fee on investor participation.
  - Market share no longer enters the monopolist problem.
  - Under the stated parameter conditions, transaction fees and monitoring resources are set so that the last investor (the most pessimistic) still participates.
  - Greater uncertainty (higher a) increases m and reduces SRO profits: higher a lowers investor expected profits, which SRO offsets by tougher enforcement (higher m) and lower transaction fees t to preserve full participation.
  - Because fraud generates extra investigation costs, the monopolist optimally chooses monitoring resources to guarantee incentive compatibility for the most optimistic investor.

### Calibrated comparison: Monopoly vs. Competition (Table 1)
- Table 1 entries (Column 1: Monopoly; Column 2: Competition; Column 3: Ratio % = Column 2 over 1 in percent)
  - Detecting Probability: 0.82 | 0.75 | - 8.5
  - Transaction Fee over E[W]: 0.55% | 0.41% | -25.5
  - Total Volume (n): 10.95 | 5 | - 4.5
  - Total Frauds: 0 | 0.059 | ñ
  - SROs' Profits: 0.124 | 0 | ñ
  - SRO 1 Market Share: ñ | 50% | ñ
  - Brokers' Average Profits: 0.152 | 0.524 | 24.5*10
  - Investors' Actual Average surplus: 0.287 | 0.047 | -16.4
  - Investors' Expected Average surplus: 0.287 | 0.046 | -16.4
  - Total Monitoring Costs: 0.438 | 0.421 | - 3.9
  - Total Welfare: 0:563 | 0:545 | - 3.2
- Notes on calibration (as described in text):
  - Spread between high and low cash flow: w = w1 = 2:5%
  - Reservation utility: 1% higher than the low state  = w1 = 1%
  - Low state probability  = 20%
  - Fix = 10 and set cost parameter c such that unit monitoring cost cm is 0.8% of the average value of a transaction when detecting probability is 50%; implies c = 0:04
  - Legal costs set at 6% of the penalty
  - Noise a = is set equal to 10%
  - Total transaction surplus is ~ = 1
- Key calibrated comparisons and implications:
  - Under competition detecting probability falls by about 9% (0.82 to 0.75), reflecting lower monitoring resources.
  - Transaction fees fall by 25% (0.55% to 0.41%).
  - Investors’ profits fall dramatically (from 0.287 to 0.047 actual surplus; expected surplus 0.287 to 0.046), an almost 85% drop in investor profits cited in text (from 0.287 to 0.047).
  - Brokers increase average profits more than 3-fold (0.152 to 0.524).
  - Reduction in enforcement outweighs fee benefits from investors’ viewpoint, causing reduced participation and a ~5% fall in total transaction volume (n = :95 noted elsewhere).
  - Under competition about 6% of total transactions involve a fraud; with detecting probability 0.75, only about 4.4% of frauds will be actually detected.
  - Total monitoring costs fall only about 4% (0.44 to 0.42) because lower per-investigation resources are offset by increased number of investigations due to frauds.
  - Total welfare falls under competition by about 3.2% relative to monopoly: W_m = 0:56 > W_c = 0:55.
- Additional calibration remarks:
  - When a is sufficiently high, Proposition 10 conditions fail and even monopoly may have participation < 1 and frauds; competition further reduces participation and increases frauds, preserving qualitative results.

### Mechanism driving welfare differences
- Under complete contracts: competition reduces both transaction fees and monitoring and is welfare enhancing because monopolist over-monitors.
- Under incomplete contracts (this paper): competition reduces enforcement and, due to fraud in equilibrium, leads to:
  - Lower investor participation → lower total trading volume.
  - More misreports → more investigations → waste of resources (monitoring, legal and opportunity costs).
  - Net effect: total welfare is reduced under competition relative to monopoly.

### Extension: Sophisticated investors (Sections IV, A)
- Setup: investors anticipate broker’s trading venue choice and strategically choose contracts to induce broker behavior; SROs respond by reducing transaction fees to attract volume.
- Robustness: results do not depend on whether competition operates through transaction fees or enforcement policies.
- Key results (Proposition 11):
  - For a sophisticated investor with no uncertainty regarding SRO monitoring technology (a = 0), the optimal contract (1, 2) and q1 are as stated in the proposition (reproduced here as in source):
    - 
      
      1
      ;
      2
      
      =
      8
      >
      <
      >
      :
      
      (
      ww) max
      i
      f

      m
      i
      g;0
      
      ;t
      1
      < t
      2
      
      (
      ww) max
      i
      f

      m
      i
      g;(ww) max
      i
      f

      m
      i
      g
      
      ;  t
      1
      =t
      2
      
      0;(
      ww) max
      i
      f

      m
      i
      g
      
      ;0;t
      1
      > t
      2
    - q1 equals 1, 1/2 or 0 depending on relative t1 and t2 (see equation (23) in source).
  - Intuition: sophisticated investors differentiate compensation to induce brokers to trade on SRO with higher monitoring or lower fees; brokers’ compensation lower bound equals maximum profit broker could obtain by misreporting on each SRO.
- Symmetric equilibrium with sophisticated investors (Proposition 12):
  - There exists a unique symmetric equilibrium with q = 1/2, t1 = t2 = t, m1 = m2 = m, where m and t must satisfy:
    - cm + (1)(ww)

      m
      =    ~
    - t = cm
  - Result: same equilibrium values as basic model with naive investors.
  - Intuition: competition (through monitoring or fees) drives SRO profits to zero; CIR (investor participation constraint) binds and SROs minimize monitoring resources, boosting broker compensation and reducing investor surplus.
  - Introducing uncertainty over a yields similar results (numerical description in Appendix B referenced).

### Main conclusions (Section V)
- Model extends hierarchical costly state verification framework to competition among SROs.
- Competition reduces transaction fees and monitoring resources:
  - Lower fees benefit investors, but reduced monitoring increases brokers’ information rents.
- Under complete contracts competition can be welfare enhancing (monopolist over-monitors).
- Under incomplete contracts competition reduces total welfare relative to monopoly because:
  - Reduction in enforcement leads to fraud in equilibrium → lower participation → lower volume.
  - More misreports increase number of investigations and legal/opportunity costs.
- Results are robust to investor sophistication: whether SROs compete via fees or enforcement yields observationally equivalent equilibria with relaxed enforcement and higher broker profits.

### Appendices and technical remarks
- Appendix 1: proofs of lemmas and propositions; key technical points include:
  - Revelation principle and BIC determine z_i when ~q_i > 0.
  - CIR always binds in equilibrium; SRO profits must be zero in competitive symmetric equilibrium.
  - Existence and uniqueness arguments for symmetric equilibrium.
- Appendix 2: case of imperfect knowledge of monitoring technology (a > 0) with sophisticated investors:
  - Low-cash flow contract remains z_i = wt_i and penalty at maximum.
  - Investor objective cannot be written in closed form because E(q_j | information) depends on investor's distribution over a and possible covariance terms, making closed-form expressions intractable; cast problem in general maximization form subject to broker strategies and announced enforcement/fees.

*Source: _wp1337 - 1.  The Monopolist Solution under Incomplete Contracts*

### 1.  The SROsíProblem

### 1.  The SROsíProblem

### Model setup and SRO objective
- SROichooses an enforcement policy and transaction fee (p_i; x_i; t_i), anticipating customer and broker responses and taking the actions of SRO_{-i} (m_i^?; x_i^?; t_i^?) as given.
- Unit (per transaction) profits = transaction fee minus expected monitoring cost in case the high state is reported, E_m c.
- Total profits = unit profits × SROís trading volume.
- Define the set of broker-investor pairs for which a contract is signed, given enforcement policy and transaction fee (m_i; t_i; m_i^?; t_i^?), as:
  - N = { j : E_j(m_i; t_i; m_i^?; t_i^?) ≥ 0 } ⊆ [0,1], where E_j is investor j expected profit. N summarizes the investorsí participation constraint (CIR) and is affected by enforcement policy and transaction fees chosen by the two SROs.
- Define the set of contracts that are not incentive compatible:
  - B_i = { j : ψ_j(z) ≤ 0 } ⊆ [0,1].
- Let I_S be the indicator function of a generic set S and o represent the SRO information set.
- With linear monitoring cost C(m) = c m, the SROi problem is:
  - max_{m;t} E_o [ ∫_0^1 q_j^i(m_i; t_i; m_i^?; t_i^?)[ t - (1 + I_{B_i}(j)) c m ] I_N(j) dj ]  (equation (25))

### Solution approach
- The problem does not have a closed form solution.
- Authors report results of a simulation exercise comparing equilibria with sophisticated heterogeneous investors versus naive investors.
- Numerical algorithm details:
  - Two-dimension best response functions approximated by Chebyshev polynomials of 12th order.
  - A fixed point is searched in a neighborhood of the naive solution.
  - Algorithm is computationally demanding; most differences between sophisticated and naive results are due to approximation errors of the algorithm.

### Key simulation results (Table 2: Naive vs Sophisticated)
- Detecting Probability: 0.75 (Sophisticated) ; 0.75 (Naive)
- Transaction Fee over Average Transaction Value: 0.39% (Sophisticated) ; 0.41% (Naive)
- Total Volume (n): .96 (Sophisticated) ; .955 (Naive)
- Total Frauds: 0.055 (Sophisticated) ; 0.059 (Naive)
- SROsí Profits: 0 (Sophisticated) ; 0 (Naive)
- SRO 1 Market Share: 50% (Sophisticated) ; 50% (Naive)
- Brokersí Average Profits: 0.523 (Sophisticated) ; 0.524 (Naive)
- Investorsí Actual Average surplus: n.a. (Sophisticated) ; 0.047 (Naive)
- Investorsí Expected Average surplus: 0.048 (Sophisticated) ; 0.046 (Naive)
- Total Monitoring Costs: 0.424 (Sophisticated) ; 0.421 (Naive)
- Welfare: 0.548 (Sophisticated) ; 0.545 (Naive)
- Definition: The average investor surplus is defined as the difference between average investor expected (actual) profits and the reservation profit.

### Parameterization used in simulations
- w = w-1 = 2:5% (note: printed as w = w1 = 2:5%)
- β = w-1 = 1% (printed as  = w1 = 1%)
- ρ = 20% (printed as  = 20%)
- θ = 10 (printed as = 10)
- c = 0:04
- φ = 6% (printed as 
 = 6%)
- a = = 10%  (printed as a= = 10%)

### Comparative insights and interpretation
- In the parameterized exercise, results for sophisticated heterogeneous investors are qualitatively the same as for naive investors.
- This suggests that, even when heterogeneous investor beliefs are introduced, the means of competition (monitoring or transaction fees) is not relevant for equilibrium determination — consistent with results previously shown for homogeneous investors.
- Most quantitative differences between sophisticated and naive cases are attributed to numerical approximation errors.

### Figures (descriptive summary)
- Figure 1: "Competitive versus Monopolistic Equilibrium"
  - Panel A: Equilibrium in the space (m; t) — top panel; shows Transaction Fee, Broker’s Compensation, Monitoring; labels N (competitive equilibrium) and M (monopoly equilibrium).
  - Panel B: Equilibrium in the space (p; t) — lower panel; shows Transaction Fee, Broker’s Compensation, Detecting Probability; labels N and M.
  - Axes and curves include Isoprofits, CIR, Equilibrium markers; Broker’s compensation and monitoring plotted against transaction fee and detecting probability.
- Figure 2: "Investor Contracts"
  - Panel A: Investor‘s Optimal Contract Choice — high-state contract across investors; x-axis orders investors by j with most “optimistic” on left.
  - Panel B: Investor‘s Expected Earnings − CIR — expected earnings across investors compared with reservation utility β.
  - Panel C: Investor‘s Probability of an Incentive Compatible Contract — subjective probability of having offered a truth-telling compatible contract, across investors (F(υ_j)).
  - Panel D: Investor‘s Optimal Incentive Compatibility Power of the Contract — compares ψ_j with threshold; light and dark thick solid lines represent contracts that are and are not incentive compatible, respectively; thick dashed line represents investors that do not participate.

*Source: _wp1337 - 1.  The SROsíProblem (PDF)._*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp1337.pdf_