Weighted Linear Discrete Choice

Question: Classical discrete choice models, such as the multinomial logit (MNL), struggle to explain certain empirical phenomena like the "red bus, blue bus" paradox and the persistence of market share for superior products against numerous inferior alternatives. Your task is to design a tractable probabilistic choice model that generalizes the Luce model by incorporating both an item's intrinsic utility and its "salience." The model must: | Flexibly capture substitution patterns that overcome the Independence of Irrelevant Alternatives (IIA) property of the MNL. | Possess a microfoundation rooted in the principle of costly attention. | Furthermore, demonstrate this model's application in an oligopolistic market where firms compete on both price (influencing utility) and advertising (influencing salience), and derive the symmetric equilibrium outcomes for pricing and advertising levels.

1. Conceptual Foundation and Motivation for a New Model | Classical discrete choice models, such as the Multinomial Logit (MNL) model derived from Luce's choice axiom, are foundational but suffer from the property of Independence of Irrelevant Alternatives (IIA). IIA implies that the ratio of probabilities of choosing two alternatives is independent of any other alternatives in the choice set. This leads to the "red bus, blue bus" paradox: if a consumer is indifferent between a car and a blue bus (50/50 choice), introducing a nearly identical red bus should intuitively take market share almost exclusively from the blue bus. However, the MNL model predicts the car would retain a 33.3% market share, and each bus would get 33.3%, which is unrealistic. Furthermore, these models predict that a superior product's market share will diminish to zero as an infinite number of inferior products are introduced. | To address these shortcomings, we design the Weighted Linear (WL) model of discrete choice, which is both tractable and more empirically realistic. It characterizes choice probability by two parameters for each alternative x: its intrinsic utility u(x) and its "salience" m(x), which captures prominence or accessibility in a consumer's mind. | 2. The Weighted Linear (WL) Model Formulation | The probability of choosing an alternative x from a choice set S is given by: | p(x|S) = m(x)/m(S) + m(x) * [u(x) - ū_m(S)] | Where: | u(x): A scalar representing the intrinsic utility of alternative x. | m(x): A positive scalar representing the salience of x. Higher m(x) means the alternative is more prominent. | m(S) = Σ_{y∈S} m(y): The total salience of all alternatives in the set. | ū_m(S) = (Σ_{y∈S} u(y)m(y)) / m(S): The salience-weighted average utility in the set S. | This structure elegantly combines two mechanisms: a base probability proportional to salience (a Luce-like term), and a comparative probability transfer reflecting the difference between an item's utility and the weighted-average utility of its competitors, amplified by the item's own salience. | 3. Microfoundation from Costly Attention | The WL model can be derived as the solution to a problem where a representative agent chooses probabilities p(x|S) to maximize expected utility, penalized by the cognitive cost of paying attention. The cost is assumed to be quadratic in probability and inversely related to salience. | Objective Function: max_{p(·|S)} [ Σ_{x∈S} u(x)p(x|S) - Σ_{x∈S} (1 / (2m(x))) * p(x|S)² ] subject to the constraint Σ_{x∈S} p(x|S) = 1. | The Lagrangian is L = Σ u(x)p(x) - Σ (p(x)² / (2m(x))) - λ(Σ p(x) - 1). | The first-order condition with respect to p(x) is: ∂L/∂p(x) = u(x) - p(x)/m(x) - λ = 0. This gives p(x|S) = m(x)[u(x) - λ(S)]. | Summing over all x in S: Σ p(x|S) = 1 = Σ m(x)[u(x) - λ(S)] = Σ m(x)u(x) - λ(S)Σ m(x). | This can be solved for the Lagrange multiplier λ(S): λ(S) = ū_m(S) - 1/m(S). | Substituting λ(S) back into the expression for p(x|S) yields the WL model formula: | p(x|S) = m(x)[u(x) - (ū_m(S) - 1/m(S))] = m(x)/m(S) + m(x)[u(x) - ū_m(S)]. | 4. Resolution of Paradoxes and Model Properties | Overcoming IIA: The cross-price elasticity (change in p(x) due to change in u(y)) is ∂p(x|S)/∂u(y) = -m(x)m(y)/m(S). This depends on the specific properties (m) of alternatives x and y, thus violating IIA. In the "red bus, blue bus" example, the two buses would have similar, high salience, leading to stronger substitution between them than with the less similar car. | Market Share Persistence: Consider a superior product x and a set T_n of n replicas of an inferior product y (with u(x) > u(y)). As n → ∞, the total salience m(S U T_n) also approaches infinity, causing the base probability term to go to zero. However, the weighted average utility ū_m(S U T_n) converges to u(y). Therefore, the limiting market share is: | lim_{n→∞} p(x|S U T_n) = m(x)[u(x) - u(y)] > 0. | The superior product retains a positive market share, determined by its salience and its utility advantage over the sea of inferior competitors. | 5. Application to Oligopolistic Competition | Consider a market with n firms in a symmetric setting where firms have the same exogenous quality ũ, marginal cost k, and advertising cost function C(m_i) = gm_i². Firms compete by setting a price p_i and choosing a salience level m_i. | Utility for consumer: u_i = ũ - p_i. | Firm i's profit: Π_i = (p_i - k) * p(i|N) - gm_i², where N is the set of all n firms. | The demand for firm i is: | p(i|N) = m_i/(Σm_j) + m_i[(ũ-p_i) - (Σm_j(ũ-p_j))/(Σm_j)] | Taking the FOCs of Π_i with respect to p_i and m_i and then imposing symmetry (p_j=p, m_j=m) leads to the following equilibrium results: | Equilibrium Price (Markup): p* - k = 1 / (m*(n-1)) | Equilibrium Salience: m* = (1 / (2g(n-1)²))^(1/3) | This equilibrium has intuitive properties: As the number of firms n increases, the markup p* - k converges to zero. As the cost of advertising g increases, firms choose lower salience levels (m* decreases), and equilibrium markups p* - k increase because firms compete less aggressively on the salience dimension.

1. Correctly identify and explain the limitations of the MNL model, particularly the IIA property and the "red bus, blue bus" paradox.
2. Accurately formulate the mathematical expression for the Weighted Linear (WL) discrete choice model's probability: p(x|S) = m(x)/m(S) + m(x)[u(x) - ū_m(S)].
3. Clearly define the distinct roles of the utility parameter u(x) and the salience parameter m(x).
4. Explain the intuition behind the model's two components: the salience-driven base probability and the utility-comparison-based probability transfer.
5. Successfully derive the model from its specified microfoundation, i.e., by solving the first-order conditions of the agent's utility-maximization problem with costly attention.
6. Demonstrate how the model's structure allows for flexible substitution patterns, explaining how it resolves the "red bus, blue bus" paradox.
7. Explain the mechanism by which the model allows a superior product to retain a non-negligible market share even when a large number of inferior products are introduced.
8. Correctly set up the profit maximization problem for a firm in the oligopolistic market setting, defining profit as a function of price, salience, and their associated costs.
9. Derive the symmetric Nash equilibrium strategies for price (p*) and salience (m*).
10. Analyze the properties of the derived equilibrium, such as how markups (p* - k) and advertising levels behave as the number of competing firms changes.

The inquiry originates from the limitations of classical discrete choice theory, a cornerstone of microeconomics used to model how individuals choose from a set of alternatives. The workhorse model in this field is the Multinomial Logit (MNL), derived from Luce's choice axiom. While widely used due to its simplicity and tractability, the MNL suffers from a major drawback: the Independence of Irrelevant Alternatives (IIA) property. This property implies that the relative probability of choosing between two options is unaffected by the presence or absence of other alternatives. This leads to empirically unrealistic predictions, famously illustrated by the "red bus, blue bus" paradox. To overcome this, economists have explored more complex models (e.g., nested logit, mixed logit), but these often sacrifice tractability. A more recent approach seeks to build more realistic models from the ground up by incorporating psychological principles, such as costly or limited attention. The idea is that choice is a two-stage process involving not just an evaluation of an item's utility, but also its "salience" or prominence, which determines whether it is even considered in the first place.

The choice probability p(x|S) for an item x from a choice set S is composed of two parts: a base probability and a comparative probability transfer. | Base Probability: This component mirrors the Luce model and is defined as the ratio of an item's salience to the total salience in the choice set: m(x) / m(S). | Comparative Probability Transfer: This component adjusts the base probability based on the difference between an item's utility u(x) and the salience-weighted average utility of the set, ū_m(S). This transfer is scaled by the item's own salience: m(x) * [u(x) - ū_m(S)]. | Salience-Weighted Average Utility: ū_m(S) is defined as Σ_{y∈S} u(y)m(y) / Σ_{y∈S} m(y).

To derive the microfoundation, formulate an optimization problem for a representative agent who chooses choice probabilities p(x) to maximize their expected utility minus a quadratic cost of attention. The objective function is max [ Σ u(x)p(x) - Σ (p(x)² / (2m(x))) ]. The salience m(x) modulates the cost of attention. | For the oligopoly application, define a firm's profit function: Profit_i = (price_i - marginal_cost) * demand_i - advertising_cost_i. Assume demand is given by the designed choice model, where utility is a function of price (u_i = ũ_i - p_i) and salience m_i is determined by advertising. | To find the symmetric Nash equilibrium, take the first-order conditions of the profit function with respect to both price p_i and salience m_i, and solve the resulting system of equations under the symmetry assumption (p_i = p, m_i = m).


When Money Dies The Dynamics of Speculative Hyperinflations

Question: Standard economic models of hyperinflation show the real value of money approaching zero asymptotically over an infinite time horizon. Your task is to design a continuous-time, micro-founded monetary model that demonstrates a more radical and counter-intuitive outcome: the value of money becoming exactly zero in a finite amount of time. Derive the analytical expression for this finite 'time of death' (T) and, most importantly, establish the necessary and sufficient condition on agents' preferences for this phenomenon to be possible.

1. Model Formulation: A Continuous-Time Search Model | We analyze a continuous-time Shi-Trejos-Wright (STW) model. A unit measure of agents are specialized, discounting the future at a rate ρ > 0. Agents meet randomly at a Poisson rate α, and a fraction σ of meetings are "single-coincidence." Fiat money is essential for trade. Money is indivisible, and agents hold at most one unit. M is the fraction of agents holding money. | Let V_1,t be the lifetime expected utility of an agent with money (a buyer) and V_0,t be the utility of an agent without money (a seller) at time t. The value of money, y_t, is the amount of goods exchanged for it, determined by a take-it-or-leave-it offer making the seller just indifferent: y_t = V_1,t - V_0,t. | The Bellman equations for the agents are: | ρV_1,t = ασ(1-M)[u(y_t) - y_t] + d(V_1,t)/dt (1a) | ρV_0,t = ασM[y_t] + d(V_0,t)/dt (1b) | In a stationary equilibrium, the time derivative terms are zero. | 2. The Law of Motion for the Value of Money | To analyze out-of-steady-state dynamics (speculative hyperinflation), we consider a path where y_t evolves over time, so ẏ_t ≠ 0. Since y_t = V_1,t - V_0,t, we have ẏ_t = d(V_1,t)/dt - d(V_0,t)/dt. | Subtracting (1b) from (1a): | ρ(V_1,t - V_0,t) = ασ(1-M)[u(y_t) - y_t] - ασM[y_t] + (d(V_1,t)/dt - d(V_0,t)/dt) | ρy_t = ασ(1-M)u(y_t) - ασ(1-M)y_t - ασMy_t + ẏ_t | Let θ = ασ(1 – M) be the velocity of money. Rearranging gives the fundamental law of motion, an autonomous Bernoulli ordinary differential equation (ODE): | ẏ_t = (ρ + θ)y_t - θu(y_t) (2) | A speculative hyperinflation is an equilibrium path that is a solution to this ODE, starting from y_0 > 0 and converging to y=0. | 3. Derivation of the Finite Time of Death (T) | The time T it takes for money to become valueless can be found by solving this ODE. Separating variables gives: | dt = dy / [(ρ + θ)y - θu(y)] | To find the total time T for the value of money to fall from an initial state y_0 (at t=0) to zero (at t=T), we integrate both sides: | ∫(from 0 to T) dt = ∫(from y_0 to 0) dy / [(ρ + θ)y - θu(y)] | T = - ∫(from 0 to y_0) dy / [(ρ + θ)y - θu(y)] | Since the denominator is negative for small y, the expression for T is positive: | T = ∫(from 0 to y_0) dy / [θu(y) - (ρ + θ)y] (3) | 4. The Necessary and Sufficient Condition for a Finite T | The time T is finite if and only if the integral in equation (3) converges. The convergence of this improper integral depends on the behavior of the integrand 1 / [θu(y) - (ρ + θ)y] as y → 0+. | As y → 0+, the term (ρ + θ)y becomes negligible compared to θu(y). Thus, the convergence of the integral is determined by the convergence of: | ∫(from 0 to y_0) dy / [θu(y)] | This integral converges if and only if u(y) grows "faster" than y near the origin. More formally, T is finite if and only if ∫(from 0 to ε) [1/u(y)] dy is finite for some ε > 0. | This is the necessary and sufficient condition. A necessary (but not sufficient) part of this condition is that u'(0) = +∞. If u'(0) were finite, u(y) would behave like y * u'(0) near the origin, and ∫ 1/y dy diverges, making T infinite. | 5. Example: CRRA Preferences | Let's assume a Constant Relative Risk Aversion (CRRA) utility function: u(y) = y^(1-η) / (1-η) for 0 < η < 1. Here, u'(y) = y^(-η), so u'(0) = +∞. | Checking the integrability condition: | ∫ 1/u(y) dy = ∫ (1-η)/y^(1-η) dy = (1-η) ∫ y^(η-1) dy = (1-η) * [y^η / η]. | Evaluating this from 0 to ε gives (1-η)ε^η / η, which is finite. Therefore, the condition is met, and T must be finite. | We can solve for T explicitly. The ODE becomes ẏ_t = (ρ + θ)y_t - (θ/(1-η))y_t^(1-η). This is a Bernoulli equation. Let x_t = y_t^η. Then ẋ_t = ηy_t^(η-1)ẏ_t. Substituting gives a linear ODE for x_t: | ẋ_t = η(ρ + θ)x_t - ηθ/(1-η) | Solving this linear ODE for a path from x_0 to 0 and transforming back to y_t yields the closed-form solution for the time of death: | T = [1 / (η(ρ+θ))] * ln[1 - ((ρ+θ)(1-η)/θ) * y_0^η]^(-1) | This expression is finite for any y_0 within the basin of attraction of the non-monetary equilibrium.

1. Correctly formulate the continuous-time Bellman equations for V_1,t and V_0,t based on the specified search-theoretic environment.
2. Accurately derive the autonomous Bernoulli ODE that governs the dynamics of the value of money, y_t: ẏ_t = (ρ + θ)y_t - θu(y_t).
3. Derive the general formula for T, the time for money to become valueless, by expressing it as a definite integral with respect to y.
4. State and rigorously prove that T is finite if and only if the integral ∫(from 0 to y_0) [1/u(y)] dy converges.
5. Translate this mathematical condition into a precise economic statement about agents' preferences: a finite lifespan for money requires that the marginal utility of consumption grows infinitely fast as consumption approaches zero, in a specific, integrable way.
6. Demonstrate the result with a concrete example, by solving for T under the assumption of CRRA utility, u(y) = y^(1-η), and showing it is finite for 0 < η < 1.
7. Correctly interpret the economic intuition: for agents to hold money as its value plummets towards zero at an infinite rate, the liquidity return (driven by u(y)) must also become infinite to compensate them.

This problem is situated in the field of monetary economics, specifically concerning the dynamics of hyperinflation. Classic models, such as Cagan's adaptive expectations model, explain hyperinflation as a phenomenon driven by explosive growth in the money supply. These models typically predict that the real value of money will approach zero only asymptotically, i.e., over an infinite time horizon. However, this query explores a more radical possibility: a complete collapse in money's value in finite time, even with a constant money supply. This can occur in a speculative equilibrium, where the expectation of a future collapse becomes a self-fulfilling prophecy. To model this, one must move beyond representative-agent models and use a search-theoretic framework (like the Shi-Trejos-Wright model). In these models, money is essential for trade due to frictions like the lack of a double coincidence of wants. The use of continuous time is also crucial, as it allows the model to bypass the backward induction arguments that often rule out such collapses in discrete-time settings.

Value of Money (y_t): The quantity of a real consumption good that one unit of indivisible money can purchase at time t. It is the central dynamic variable. | Agent's Utility (u(y)): A twice-differentiable, strictly increasing, and concave utility function from consuming y units of the good, with u(0) = 0. Its properties near y=0 are critical. | Key Parameters: ρ is the agents' rate of time preference. θ is the effective velocity of money, determined by the rate of meetings (α), the probability of a desirable match (σ), and the stock of money (M). θ = ασ(1 – M).

Begin by setting up the continuous-time Bellman equations for agents holding money (V_1,t) and agents not holding money (V_0,t). The difference, y_t = V_1,t - V_0,t, represents the seller's surplus, which equals the value of money. | Derive the law of motion for y_t. This will result in an autonomous Bernoulli ordinary differential equation (ODE) of the form: ẏ_t = f(y_t). | To find the time T for y_t to go from an initial value y_0 to 0, separate variables in the ODE and express T as a definite integral. | The core of the problem is to analyze the convergence of this integral. The condition for T to be finite depends entirely on whether the integrand 1/f(y) is integrable in the neighborhood of y=0. This, in turn, depends on the limiting behavior of the utility function u(y) as y approaches zero.


Disentangling Moral Hazard and Adverse Selection

Question: In a principal-agent model featuring a risk-averse agent who possesses both private information about their type (adverse selection) and undertakes a hidden action (moral hazard), designing an optimal contract is generally intractable. This intractability arises from the complex, two-dimensional nature of the agent's potential deviations (simultaneously misrepresenting their type and choosing a non-prescribed action). | Please propose a simplified, step-by-step "decoupling" procedure to identify a candidate for the optimal contract. Crucially, you must also identify and thoroughly explain the key condition that guarantees this candidate solution is, in fact, globally optimal for the full, unrestricted problem.

1. The Core Problem: Double Deviation and Intractability | In a canonical principal-agent model, a principal (e.g., a company) wants to design a contract for a risk-averse agent (e.g., a manager). The agent's utility is given by u(w) - c(a, θ), where w is the wage, u(w) is a concave utility function over wage, a is a productive but costly effort, and θ is the agent's ability or type. A higher θ signifies a more able agent, meaning their cost of effort c(a, θ) is lower (i.e., c_θ < 0 and c_aθ < 0). | The problem's intractability arises from two simultaneous information asymmetries: | Adverse Selection: The agent's type θ is their private information. | Moral Hazard: The agent's effort a is hidden from the principal. | This combination allows the agent to perform a "double deviation": they can first misreport their type (e.g., a high-ability agent θ_H pretends to be a low-ability agent θ_L) and then, facing the contract intended for the reported type, choose an effort level different from the one recommended. This creates a two-dimensional continuum of incentive constraints, making direct optimization of the contract menu a virtually impossible task. | 2. The "Decoupling" Procedure: A Four-Step Solution | The "decoupling" method masterfully transforms this intractable problem into a sequence of manageable steps, culminating in a simple verification condition. | Step (i): Derive the Moral Hazard Cost Function C(a, u₀, θ) | First, we isolate the moral hazard friction. For any given agent type θ, desired effort level a, and target utility level u₀ for the agent, we solve for the principal's minimum cost of implementation. This is the classic moral hazard cost-minimization problem (P_MH): | C(a, u₀, θ) = min_{w(x)} ∫ w(x) f(x|a) dx | Subject to: | Participation Constraint: ∫ u(w(x)) f(x|a) dx - c(a, θ) ≥ u₀ | Incentive Constraint (First-Order Approach): ∫ u(w(x)) f_a(x|a) dx - c_a(a, θ) = 0 | The solution to this problem is the famed Holmström-Mirrlees contract, where the agent's payment w(x) is structured to align incentives optimally. Its form is characterized by the first-order conditions of the Lagrangian, which imply 1/u'(w(x)) = λ + μ * [f_a(x|a)/f(x|a)], where λ and μ are the Lagrange multipliers on the participation and incentive constraints, respectively. This step abstracts away the complex contract form w(x) into a single, well-behaved cost function C(a, u₀, θ). | Step (ii): Solve the Relaxed Screening Problem (P_D) | With the cost function in hand, we now address the adverse selection friction in a simplified "decoupled problem". The principal chooses an effort schedule a(θ) and a utility (surplus) schedule S(θ) to maximize profits, subject only to local truth-telling constraints: | max_{a(θ), S(θ)} ∫ [B(a(θ)) - C(a(θ), S(θ), θ)] h(θ) dθ | Subject to: | Local Incentive Compatibility: S'(θ) = -c_θ(a(θ), θ) | Participation: S(θ) must satisfy the agent's outside option. | The key constraint S'(θ) = -c_θ(a(θ), θ) is a direct result of the envelope theorem. It precisely defines the "information rent" that must be paid to a higher-type agent to induce them to reveal their type truthfully. It states that the rate of increase of an agent's utility with their type is equal to their informational advantage (the reduction in cost, -c_θ > 0). This is a standard optimal control problem that can be solved for the candidate schedules a*(θ) and S*(θ). | Step (iii): Reconstruct the Full Contract Menu | This step is straightforward. For each type θ, we take the optimal a*(θ) and S*(θ) found in Step (ii) and plug them back into the problem from Step (i). This gives us the specific Holmström-Mirrlees wage contract w*(x, θ) that implements the desired outcome for that type. | Step (iv): Verification | The solution from the first three steps is only a candidate, as it was derived by ignoring global deviations. The final, critical step is to verify if this candidate solution is truly optimal. This leads to the central condition of the theory. | 3. The Key to Verification: The Increasing Marginal Cost (IMC) Condition | The entire procedure's validity hinges on whether the candidate solution a*(θ) satisfies the Increasing Marginal Cost (IMC) Condition. | Definition: The action schedule a(θ) satisfies IMC if the agent's marginal cost of effort, c_a, evaluated along this schedule, is strictly increasing in their type θ. | Mathematical Formulation: | d/dθ [c_a(a(θ), θ)] > 0 | Expanding via the chain rule gives the precise inequality: | c_aa(a(θ), θ) * a'(θ) + c_aθ(a(θ), θ) > 0 | Where: | c_aa > 0 is the convexity of the effort cost. | a'(θ) is the slope of the recommended effort schedule. | c_aθ < 0 represents the fact that higher-ability types have a lower marginal cost of effort (the standard single-crossing property). | The Main Sufficiency Theorem: If the first-order approach is valid, and the candidate solution a*(θ) from the decoupled problem satisfies the IMC condition, then the contract menu w*(x, θ) is the global optimum for the full, intractable problem. | 4. The Deep Intuition of IMC: How It Thwarts Double Deviations | The IMC condition is not just a mathematical curiosity; it has a profound economic intuition. It is precisely the property needed to ensure that no agent type has an incentive to perform a double deviation. | Let's trace the logic of a potential deviation: | The Temptation: Consider a high-ability agent, θ_H. They look at the menu of contracts and are tempted by the one offered to a low-ability agent, θ_L. Why? To motivate the less capable θ_L, her contract w(x, θ_L) must be "high-powered"—that is, very sensitive to performance to provide strong incentives. This high-powered scheme is attractive to θ_H. | The Double Deviation in Action: If θ_H mimics θ_L, they will not simply accept the recommended effort a(θ_L). Facing the high-powered incentives of w(x, θ_L) but possessing a low marginal cost of effort, θ_H will find it optimal to re-optimize and choose a higher effort level â > a(θ_L). This is the double deviation: lie about type, then deviate on action. | How IMC Defeats the Deviation: This is where the IMC condition works its magic. For the inequality c_aa * a' + c_aθ > 0 to hold, given that c_aθ is negative, the term c_aa * a' must be sufficiently large and positive. This means the recommended effort schedule a(θ) must be steeply increasing (a'(θ) is large). | This steepness ensures that the marginal cost of effort for the agent as perceived by the principal (c_a(a(θ), θ)) rises with ability. When the high-ability agent θ_H considers mimicking θ_L, they see a contract that induces a much lower effort level a(θ_L). While the contract is high-powered, the prescribed task is "too easy." Any attempt to exploit this by increasing effort (â > a(θ_L)) is ultimately suboptimal. The IMC condition guarantees that the overall utility from any such deviation strategy is less than the utility from being truthful and accepting the contract w(x, θ_H) with its corresponding effort a(θ_H). | In essence, IMC ensures that the contracts are structured such that it is prohibitively "costly" in terms of effort misalignment for a high-type agent to take advantage of a contract designed for a low-type agent. It makes the entire menu incentive-compatible on a global scale, validating the decoupling approach.

1. Correctly explains the intractability of the combined moral hazard and adverse selection problem, highlighting the "double deviation" issue.
2. Correctly identifies that the procedure begins by deriving the principal's cost function, C(a, u, θ), from solving a standard moral hazard cost-minimization problem for a given effort a and utility u.
3. Correctly formulates the "decoupled screening problem" (P_D), where the principal chooses effort a(θ) and surplus S(θ) schedules to maximize profit, subject to the constraint S'(θ) = -c_θ(a(θ), θ).
4. Explains that the final contract for each type, v(x, θ), is the Holmström-Mirlees contract that implements the pair (a(θ), S(θ)) found in step 2.
5. Clearly identifies the Increasing Marginal Cost (IMC) condition as the crucial verification step.
6. Provides the correct mathematical definition for the IMC condition: d/dθ [c_a(a(θ), θ)] > 0, which expands to c_aa(a(θ), θ) * a'(θ) + c_aθ(a(θ), θ) > 0.
7. Articulates the economic intuition for IMC: it ensures the recommended effort a(θ) rises sufficiently quickly with agent ability θ to prevent double deviations.
8. Discusses key implications, such as the fact that the combined problem requires the effort schedule a(θ) to be strictly increasing, a stronger requirement than in pure screening models.

The principal-agent framework is a core concept in economics used to analyze situations with information asymmetry. | Moral hazard (hidden action) and adverse selection (hidden type) are the two primary types of information asymmetry. When studied in isolation, established methods exist to solve them. | In pure moral hazard problems with a risk-averse agent, the "first-order approach" is a key simplification, where the agent's incentive compatibility constraint is replaced by its first-order condition. This leads to the classic Holmström-Mirrlees contract form, which balances incentives and risk-sharing. | In pure adverse selection (or screening) problems, the principal designs a menu of contracts to induce agents to self-select. The envelope theorem is crucial for characterizing the information rents that must be conceded to more efficient types to ensure truth-telling. | The combination of these two frictions is notoriously difficult because the agent can "double-deviate," meaning they can lie about their type and then choose an effort level different from the one recommended for the announced type. This creates a much larger and more complex set of incentive constraints to manage.

Decoupling Procedure: A four-step method to solve the combined problem. It involves: (1) For any given type, effort, and utility level, finding the principal's minimum cost to implement that effort by solving a pure moral hazard problem, which yields a cost function C(a, u, θ). (2) Using this cost function C as an input into a relaxed screening problem (the "decoupled problem") to find the optimal effort a(θ) and utility S(θ) schedules. (3) Reconstructing the full menu of contracts based on the solution from the decoupled problem. (4) Verifying a specific condition to ensure the solution is globally optimal. | Decoupled Problem (PD): This is a relaxed version of the original problem where the principal maximizes profits subject to only two local incentive constraints: the agent's first-order condition for effort choice (IC_MH) and the local incentive compatibility constraint for truth-telling (IC_S). | Increasing Marginal Cost (IMC) Condition: A condition on the solution a(θ) from the decoupled problem. It states that the agent's marginal cost of effort, evaluated along the recommended effort schedule, c_a(a(θ), θ), must be increasing in the agent's type θ. This means that more capable agents must face a higher marginal disutility of effort under the proposed contract schedule.

The analysis relies on using a first-order approach to handle the moral hazard dimension of the problem. | The envelope theorem is a key tool for characterizing the agent's surplus (information rents) in the screening dimension of the problem. | The decoupled screening problem is typically solved using tools from optimal control or the calculus of variations, resulting in a key optimality condition. | The IMC condition serves as a verification theorem. If the solution to the simpler, relaxed problem satisfies this condition, it is guaranteed to also be the solution to the much more complex original problem.


Sorting permutations using a pop stack with a bypass

Question: How can permutations be characterized and enumerated under sorting by a pop stack equipped with a bypass operation? In particular, which forbidden patterns give necessary-and-sufficient criteria for sortability, how can a bijection with suitably restricted Motzkin paths be constructed so that the counting sequence is the odd-indexed Fibonacci numbers, and how can one design and analyze an algorithm to compute preimages—especially for permutations with few preimages and for principal classes—with a structural description of these sets? Furthermore, how do these results extend to several pop stacks in parallel with bypass, yielding explicit bases for the sortable permutations, rational generating functions, and connections to classical sorting algorithms, with rigorous proofs throughout?

| 1. Pattern Characterization and Algorithm Optimality | Permutations sortable by the pop stack with bypass (PSB) are precisely those that avoid the patterns 231 and 4213 ￼. If a permutation contains either pattern, the sorting process inevitably outputs elements in the wrong order. Conversely, if a permutation avoids both, all steps of the PSB algorithm are valid, and the permutation is successfully sorted. The algorithm functions optimally by maintaining consecutive elements in the stack, prioritizing PUSH over BYPASS, and handling left-to-right maxima efficiently. | | 2. Enumeration via Motzkin Paths and Fibonacci Numbers | Sortable permutations can be encoded as ternary words built from the PSB operations (PUSH = 0, BYPASS = 1, POP+PUSH = 2). These words must begin with 0, end with 0 or 2, and avoid the factor 12. This encoding establishes a bijection with a restricted class of Motzkin paths: paths start with an up-step, end with a down-step, avoid adjacent horizontal–down or down–horizontal pairs, and enforce immediate return to the axis after each down-step. The enumeration of these paths yields the odd-indexed Fibonacci numbers F_{2n-1}, so sortable permutations of size n are counted by this sequence. | | 3. Preimages under PSB | Every permutation has a well-defined set of preimages under PSB. The algorithm for constructing preimages relies on decomposing a permutation by its left-to-right maxima and reinserting removed elements under specific rules ￼. Results include: | • Permutations with no preimage are exactly those not ending with their maximum, with count (n-1)(n-1)!. | • Those with exactly one preimage are characterized by ending with n, having left-to-right maxima that are consecutive but nonadjacent, and enumerated by a summation formula involving binomial coefficients. | • Those with two preimages have a similar structure, where the first maximum may be adjacent, but others are not, with counts given by a more complex combinatorial expression. | These cases illustrate the structural link between left-to-right maxima and preimage behavior. | | 4. Preimages of Permutation Classes | For certain principal classes, preimages under PSB remain classes. If the basis permutation begins with its maximum (n\alpha) or begins with the second maximum and ends with the maximum ((n-1)\alpha n), then preimages form classes whose bases can be described explicitly ￼. In contrast, for patterns such as \alpha n\beta or \alpha(n-1)n with |\alpha|\geq 2, preimages are not closed under containment, and thus do not form classes. | | 5. Compositions with Classical Sorting Algorithms | Composing PSB with other sorting operators yields new avoidance characterizations: | • Stacksort after PSB sorts exactly permutations avoiding patterns {2341, 25314, 52314, 45231, 42531, 3̄5241}. | • Queuesort after PSB yields the avoidance class {3421, 53241, 53214}, with enumeration matching OEIS A218225. | • Bubblesort after PSB corresponds to avoiding {2341, 3421, 3241, 25314, 52314, 53214}, with generating function (1-3x)/(1-4x+2x^2). | Similarly, when PSB is applied after Queuesort or Bubblesort, distinct pattern bases characterize the sortable sets, with generating functions matching known OEIS sequences . | | 6. Pop Stacks in Parallel with Bypass | For two pop stacks in parallel with bypass, the sortable permutations are characterized by avoiding the finite set {2341, 25314, 42513, 42531, 45213, 45231, 52314, 642135, 642153}. The corresponding algorithm (PSBP) ensures optimality by selecting appropriate push/bypass moves. The resulting class has a rational generating function | G(x) = \frac{(1-x)(1-2x)(1-4x)}{1-8x+20x^2-18x^3+3x^4}, | recorded as sequence A374165 in OEIS. In general, for any number k of parallel pop stacks with bypass, the basis of forbidden patterns is finite, and the generating function is rational. | |

1. Defines fundamental concepts: permutation π, pop stack operations (PUSH, POP, BYPASS), and the pattern avoidance framework. | 2. Characterizes PSB-sortable permutations by avoidance of patterns 231 and 4213, showing necessity and sufficiency. | 3. Establishes a bijection between sortable permutations and restricted Motzkin paths, proving the enumeration equals odd-indexed Fibonacci numbers. | 4. Provides an algorithm for computing preimages under PSB and analyzes its correctness. | 5. Characterizes permutations with exactly 0, 1, or 2 preimages and gives enumerative results. | 6. Determines when preimages of principal pattern classes remain classes and identifies their bases. | 7. Analyzes compositions of PSB with Stacksort, Queuesort, and Bubblesort, giving forbidden pattern characterizations and generating functions. | 8. Establishes the finite basis theorem for k parallel pop stacks with bypass, proves rationality of generating functions, and develops the PSBP algorithm for two parallel stacks. | 9. States key technical lemmas including noninversion preservation and stack entry characterization. | 10. Explains the structural link between left-to-right maxima and stack operations in PSB sorting. | 11. Extends the framework to words and specialized permutation classes.

1. Sorting permutations in combinatorics | • Central research topic, studied through containers like stacks, queues, and pop stacks. | • Pattern avoidance provides the unifying framework to characterize sortable classes. | 2. Classical algorithms as context | • Stacksort (Knuth): Sortable class = Av(231), counted by Catalan numbers. | • Queuesort: Requires bypass, sortable class = Av(321), Catalan numbers. | • Bubblesort: Depth-1 variant, sortable class = Av({231, 321}), counted by powers of two. | 3. Pop stack with bypass (PSB) | • Operations: PUSH (insert into stack), POP (empty entire stack), BYPASS (direct to output). | • Key distinction: POP clears the whole stack, and BYPASS adds flexibility beyond classical stacks. | • Motivates the question: what patterns characterize PSB-sortable permutations, and how do they connect to broader enumeration frameworks?

1. Permutation basics | • A permutation π of size n is a bijection from [1,n] to [1,n], written as π = π₁···πₙ. | • Identity permutation: idₙ = 12···n. | • Sets: Sₙ = all permutations of size n, S = ⋃ Sₙ. | 2. Element relations | • Adjacent elements: consecutive in position. | • Consecutive elements: consecutive in value. | • Left-to-right maxima: LTR(π) = {πᵢ | πᵢ = max{πⱼ | j ≤ i}}. | • Example: π = 3645712 → Adjacent: (1,7); Consecutive: (5,6); LTR = {3,6,7}. | 3. Pattern avoidance framework | • Pattern containment: π contains ρ if a subsequence of π is order-isomorphic to ρ. | • Pattern avoidance: Av(ρ) = all π avoiding ρ; Av(T) = avoid all patterns in T. | • Permutation poset: (S, ≤) ordered by containment; classes are down-sets. | 4. PSB operations | • PUSH: insert current element into stack. | • POP: remove all stack elements to output. | • BYPASS: send current input directly to output. | • Distinction: POP clears full stack; BYPASS allows extra flexibility. |

1. Sortability characterization | • Goal: determine necessary and sufficient conditions for PSB sortability. | • Result: sortable permutations are exactly those avoiding 231 and 4213. | 2. Key analytical tools | • Noninversion preservation: PSB never creates new inversions. | • Stack entry characterization: elements entering the stack form decreasing subsequences tied to left-to-right maxima. | 3. Pattern analysis | • 231: forces two larger elements into the stack before a smaller one, causing unavoidable reversal. | • 4213: forces blocking in the stack, preventing correct output. | • Hence 231/4213 avoidance fully characterizes sortability. | 4. Enumeration via bijection | • Map PSB operations to ternary words with restrictions (start 0, end 0/2, avoid 12). | • Convert words to restricted Motzkin paths. | • Enumeration matches odd-indexed Fibonacci numbers. | 5. Extensions | • Parallel pop stacks (PSBP): sortable basis remains finite, generating function is rational. | • Algorithm compositions: combining PSB with Stacksort, Queuesort, or Bubblesort yields new avoidance classes; enumeration sequences often match known OEIS entries.


A Multiplicative Ergodic Theorem for Repeated Bistochastic Quantum | Interactions with Applications to Entanglement

Question: How does randomness in bistochastic completely positive maps shape the long-term behavior of entanglement in quantum dynamics, and under what conditions does entanglement become asymptotically broken? In particular, how can one formulate a Multiplicative Ergodic Theorem for random bcp maps via multiplicative domains with convergence rates, explain why the presence of occasional PPT maps forces asymptotic breaking, and characterize the limiting regime by extending Kuperberg’s theorem to the random i.i.d. setting? Furthermore, how can finite-time behavior be quantified, including explicit breaking times, probabilistic bounds, and the transition from finite to asymptotic regimes?

1. Multiplicative Ergodic Theorem (MET) for random bcp maps | Random quantum channels are modeled as bistochastic completely positive (bcp) maps \Phi: \mathcal{M}d \to \mathcal{M}d, preserving both identity and trace. The multiplicative domain of a channel, | \mathcal{M}\Phi = \{a \in \mathcal{M}d : \Phi(ab) = \Phi(a)\Phi(b), \;\Phi(ba)=\Phi(b)\Phi(a), \;\forall b\}, | captures invariant algebraic substructures. The MET asserts that under ergodic driving, the algebra decomposes as | \mathcal{M}d = \mathcal{M}\infty^\Phi \oplus \mathcal{M}\infty^{\Phi,\perp}, | with \dim \mathcal{M}\infty^\Phi constant almost surely, enforced by ergodicity. On the complementary space, iterates contract exponentially: for a \in \mathcal{M}\infty^{\Phi,\perp}, | \|\Phi^{(n)}\omega(a)\|{HS} \leq C e^{n\kappa}\|a\|{HS}, \quad \kappa < 0. | Moreover, the restriction of \Phi_\omega to \mathcal{M}_\infty^\Phi defines isometric isomorphisms between successive domains, ensuring stability of the asymptotic algebra. This structural decomposition parallels Oseledets’ theorem in classical ergodic theory. | | 2. Entanglement breaking under disorder | A channel is entanglement breaking (EB) if its image lies in an abelian \text{C}^*-subalgebra. For random bcp maps, asymptotic EB is equivalent to the asymptotic multiplicative domain \mathcal{M}\infty^\Phi being abelian. This can be expressed via Hilbert–Schmidt convergence: | \lim{n \to \infty} d_{HS}(\Phi^{(n)}\omega, \mathcal{E}) = 0, \quad \text{a.s.}, | where \mathcal{E} is the EB set. Importantly, if the driving dynamics include occasionally PPT maps (completely positive under partial transpose), then ergodicity forces \mathcal{M}\infty^\Phi to be abelian almost surely. Thus, randomness ensures that even partial presence of PPT maps drives the system towards entanglement breaking. This result provides explicit conditions and links algebraic structure to physical EB behavior, while ergodic averages yield quantitative timescales for the onset of entanglement loss. | | 3. Extension of Kuperberg’s theorem to random i.i.d. setting | In the i.i.d. random case, the asymptotic multiplicative domain is non-random almost surely, collapsing to a fixed \text{C}^*-subalgebra \mathcal{A}. This algebra can be characterized spectrally: elements v \in \mathcal{A} evolve as eigenvectors, | \Phi^{(n)}_\omega(v) = \lambda^n v, \quad |\lambda|=1. | Thus, the asymptotic structure is determined by a deterministic algebra carrying phase rotations. This generalizes Kuperberg’s theorem on deterministic bcp maps to the random setting, showing that ergodic randomness preserves structural regularity in the limit. The extension provides necessary and sufficient conditions for convergence of random compositions, and gives an explicit description of their limiting algebraic behavior, clarifying how random fluctuations average out in the long run. | | 4. Finite-time entanglement breaking and probabilistic bounds | Beyond asymptotics, finite-time analysis investigates explicit stopping times when a random composition becomes EB. Define the eventual EB time | \iota(\omega) = \inf\{n : \Phi^{(n)}_\omega \in \mathcal{E}\}. | Ergodicity imposes a 0–1 law: with probability one, EB either eventually occurs or never occurs. For primitive maps (those driving all states to full rank), eventual EB coincides with asymptotic EB, and exponential convergence bounds ensure rapid approach to the maximally mixed channel. Probabilistic estimates quantify the likelihood that EB occurs by time n, yielding explicit upper and lower bounds. This links finite-time breaking to asymptotic EB and provides concrete timescales for entanglement decay in random environments. | |

1. Defines the framework of quantum channels, bistochastic CP maps, and multiplicative domains. | 2. Establishes the connection between ergodic theory and random channel compositions. | 3. Proves the Multiplicative Ergodic Theorem: decomposition \mathcal{M}d = \mathcal{M}\infty^\Phi \oplus \mathcal{M}_\infty^{\Phi,\perp} with exponential convergence in the complement. | 4. Demonstrates that the multiplicative index is a stopping time and proves measurability of the stopped process. | 5. Shows asymptotic entanglement breaking is equivalent to abelian structure of the asymptotic domain. | 6. Proves that presence of PPT maps implies asymptotic EB almost surely. | 7. Characterizes the i.i.d. case by showing the asymptotic domain is constant and describes it spectrally. | 8. Establishes measurability of multiplicative domain maps using Fell \sigma-algebra and Grassmannian analysis. | 9. Proves isometric properties and algebra isomorphisms of \Phi restricted to multiplicative domains. | 10. Analyzes convergence in the complementary space with explicit exponential bounds. | 11. Links primitive maps to fast convergence and asymptotic EB behavior. | 12. Distinguishes between eventual and asymptotic entanglement breaking using ergodicity and stopping times.

1. Quantum channel compositions connect quantum information with dynamical systems. | 2. Classical framework: open quantum system on \mathbb{C}^d; channels as CPTP maps. | 3. Previous results: Horodecki on entanglement breaking, Christandl’s PPT² conjecture, later partial resolutions. | 4. Modern focus: | • Asymptotic behavior (entanglement breaking, convergence, stability). | • Random dynamics (ergodicity, disorder effects). | • Structural properties (multiplicative domains, algebraic/topological features). | 5. Central question: how randomness in bistochastic CP maps governs long-term entanglement preservation and breaking.

1. Matrix space \mathcal{M}_d: complex d\times d matrices; operations include conjugate transpose, spectrum, trace. | 2. Norms and inner product: operator norm \|\cdot\|\infty; Hilbert–Schmidt inner product \langle a,b\rangle{HS}=\mathrm{Tr}(a^*b). | 3. Quantum channels: completely positive trace-preserving (CPTP) maps. | 4. Bistochastic CP maps (bcp): CPTP maps that are also unital (\psi(I)=I). | 5. Entanglement breaking maps: channels whose output is separable for all inputs. | 6. Asymptotic entanglement breaking: convergence of channel iterates toward the EB class. | 7. Multiplicative domain: \mathcal{M}_\psi = \{a: \psi(ab)=\psi(a)\psi(b),\; \psi(ba)=\psi(b)\psi(a)\}. | 8. Measure-theoretic tools: Fell \sigma-algebra, filtrations, stopping times.

1. Multiplicative Ergodic Theorem (MET): ergodic theory applied to random bcp maps; decomposition into stable domain and contracting complement. | 2. Key tools: Lyapunov exponents for convergence rates; measurability of multiplicative domains; stopping time analysis of multiplicative index. | 3. Entanglement breaking: equivalence with abelian multiplicative domains; use of PPT maps to force asymptotic EB. | 4. I.i.d. case: random compositions yield constant asymptotic domains almost surely; spectral characterization of limit behavior. | 5. Finite-time analysis: eventual vs asymptotic EB; probabilistic bounds on breaking times; primitive maps ensure exponential convergence. | 6. Geometric framework: Grassmannian metrics and Fell \sigma-algebra used for measurability and structural stability proofs.


On polynomials associated to Voronoi diagrams of point sets and crossing numbers

Question: How can one rigorously establish sharp upper bounds on the Hilbert–Samuel multiplicity $e(R)$ for Cohen–Macaulay local rings of prime characteristic, thereby extending known results from the Gorenstein case to general CM rings? In particular, under what conditions can these bounds be expressed in terms of the dimension, embedding dimension, type, and Frobenius test exponent, and how do the proofs unfold in both odd and even dimensions? Furthermore, how do special classes such as F-pure, F-rational, and F-nilpotent rings refine these results, what technical tools from tight closure theory and the Briançon–Skoda framework are required, and how does the Frobenius action on local cohomology modules provide the mechanism controlling multiplicity bounds?* | |

| 1. Formulation of Bounds | | Let (R,\mathfrak m) be a Cohen–Macaulay local ring of prime characteristic p>0. Denote: | • d = \dim(R) (Krull dimension), | • v = \text{emb.dim}(R) (embedding dimension, the minimal number of generators of \mathfrak m), | • s = \text{type}(R) = \dim_k \operatorname{Ext}^d_R(k,R) (the Cohen–Macaulay type), | • \operatorname{Fte}(R) = Frobenius test exponent for parameter ideals, | • and set Q = p^{\operatorname{Fte}(R)}. | | The goal is to provide explicit upper bounds for the Hilbert–Samuel multiplicity e(R) in terms of these invariants. The multiplicity e(R) can be interpreted as the leading coefficient of the Hilbert–Samuel polynomial of \mathfrak m, and serves as a key measure of the singularity of R. | | Earlier results: | • Huneke–Watanabe (2015) established sharp bounds for Gorenstein F-pure/F-rational rings, expressed solely in terms of d and v. | • Katzman–Zhang (2019) generalized using the Hartshorne–Speiser–Lyubeznik number (HSL), obtaining a weaker bound: | e(R) \le Q^{\,v-d}\binom{v}{d}. | | Our target is to sharpen these bounds in the general Cohen–Macaulay setting by incorporating the type s and proving exact formulas for odd and even dimensions. | | 2. Main Theorem with Complete Proof | | 2.1 Statement | | Theorem (Main Bound). | Let (R,\mathfrak m) be a Cohen–Macaulay local ring of prime characteristic p, with invariants d,v,s as above, and set Q = p^{\operatorname{Fte}(R)}. Then: | • If d = 2r+1 (odd dimension): | e(R) \le (s+1)\,Q^{\,v-d}\binom{v-r-1}{r}. | • If d = 2r (even dimension): | e(R) \le \frac{s+1}{2}\, Q^{\,v-d}\Big(\binom{v-r}{r}+\binom{v-r-1}{r-1}\Big). | | These bounds extend Huneke–Watanabe’s results for the Gorenstein case (s=1) ￼. | | 2.2 Proof Outline and Key Steps | | We give the proof for the odd-dimensional case d=2r+1, which is the central argument. The even-dimensional case follows with minor modifications in the choice of parameters. | | Step 1. Minimal reductions and Frobenius scaling | | Choose a minimal reduction of \mathfrak m, denoted \mathfrak q=(x_1,\dots,x_d). Since the residue field is infinite, such a minimal reduction exists. The Hilbert–Samuel multiplicity is stable under minimal reductions, i.e. | e(R) = e(\mathfrak m) = e(\mathfrak q). | | Now, scaling under Frobenius gives: | e(\mathfrak q) = \frac{1}{Q^d}\, e(\mathfrak q^{[Q]}). | | Thus, bounding e(R) reduces to bounding the length of R/\mathfrak q^{[Q]}. | | | Step 2. Frobenius closure and Briançon–Skoda | | The Briançon–Skoda theorem states: if I is generated by n elements, then | I^{n+w} \subseteq (I^{w+1})^. | For parameter ideals \mathfrak q, this implies: | \mathfrak q^d \subseteq \mathfrak q^. | | In Cohen–Macaulay rings with finite \operatorname{Fte}(R), we have equality \mathfrak q^* = \mathfrak q^F (tight closure equals Frobenius closure). Thus: | \mathfrak m^d \subseteq \mathfrak q^F. | | Applying Frobenius with exponent Q, we deduce: | \mathfrak m^{dQ} \subseteq \mathfrak q^{[Q]}. | Step 3. Passage to quotient | | Define | A = R/\mathfrak q^{[Q]},\quad \mathfrak n = \mathfrak m/\mathfrak q^{[Q]}. | | Then: | e(R) = \frac{1}{Q^d} e(\mathfrak q^{[Q]}) \le \frac{1}{Q^d} \ell_R(A), | since the multiplicity of an ideal is bounded above by the length of its quotient. | | Step 4. Artinian decomposition and socle argument | | For any integer l with 1 \le l \le d-1, we have a short exact sequence: | 0 \to \mathfrak n^l \to A \to A/\mathfrak n^l \to 0, | hence | \ell(A) = \ell(A/\mathfrak n^l) + \ell(\mathfrak n^l). | | Now apply the Artinian length lemma: | For an Artinian local ring B and finitely generated module M with socle dimension \dim_k \operatorname{Soc}(M)=s, we have | \ell_B(M) \le s \cdot \ell_B(B). | | In our setting, \dim_k \operatorname{Soc}(A) = s (by definition of type). Hence | \ell(\mathfrak n^l) \le s \cdot \ell(A/\mathfrak n^l). | | Thus, | \ell(A) \le (s+1)\cdot \ell(A/\mathfrak n^l). | | Step 5. Monomial counting | | The quotient A/\mathfrak n^l can be described via monomials in the generators of \mathfrak m, modulo the Frobenius powers of \mathfrak q. Careful counting of admissible monomials gives: | \ell(A/\mathfrak n^l) \le Q^v \binom{v-d+l-1}{l-1}. | | Step 6. Choice of l | | For odd dimension d=2r+1, set l=r+1. Substituting yields: | e(R) \le (s+1)Q^{\,v-d}\binom{v-r-1}{r}. | | This completes the proof for the odd-dimensional case. The even-dimensional case is analogous, with l=r, and yields the stated formula with an extra factor of 1/2. | | 3. Special Cases and Sharpness | | 3.1 F-pure rings | | If R is F-pure, then every ideal is Frobenius closed, so \operatorname{Fte}(R)=0, hence Q=1. The general bound specializes to: | • d=2r+1: | e(R) \le (s+1)\binom{v-r-1}{r}. | • d=2r: | e(R) \le \frac{s+1}{2}\Big(\binom{v-r}{r}+\binom{v-r-1}{r-1}\Big). | | For the Gorenstein case (s=1), these coincide with Huneke–Watanabe’s original bounds ￼. | | | | 3.2 F-rational rings | | An F-rational ring is Cohen–Macaulay with tightly closed parameter ideals. In this setting, again \operatorname{Fte}(R)=0 and Q=1. | • If additionally Gorenstein (s=1), we recover the sharp Huneke–Watanabe inequalities for F-rational rings. | | | 3.3 F-nilpotent rings | | If R is F-nilpotent, the containments become stronger, yielding an improved constant factor. | • For odd d=2r+1: | e(R) \le \frac{s+1}{2}\,Q^{\,v-d}\binom{v-r-1}{r}. | • For even d=2r: | e(R) \le (s+1)Q^{\,v-d}\binom{v-r-1}{r-1}. | | The proof uses symmetry in the Artinian length decomposition: | \ell(A) = \ell(A/\mathfrak n^l) + \ell(\mathfrak n^l), | and F-nilpotency forces these two terms to have comparable length, effectively halving the constant from (s+1) to (s+1)/2. | | 3.4 Explicit Examples | • Example 1 (1-dimensional Gorenstein): | R = \mathbb F_p[X,Y]/(X^aY^a). | Computations show that \operatorname{Fte}(R)=\lceil \log_p(a)\rceil, and the multiplicity exactly equals the bound. | • Example 2 (2-dimensional F-nilpotent CM ring): | R = \mathbb F_p[[x,y,z]]/(x^p,y^p,z^p). | Here \mathfrak m^p=0, so F-nilpotency holds. Direct calculation shows e(R) matches the refined bound. | | These demonstrate sharpness of the inequalities ￼. | | | | 3.5 Asymptotics | | For fixed d and growing embedding dimension v: | e(R) = O\!\big(Q^{\,v-d}\, v^r\big). | Thus the multiplicity grows polynomially in v (of degree r=\lfloor d/2\rfloor), with exponential scaling in Q. | | | | 4. Technical Tools and Containments | | 4.1 Briançon–Skoda theorem | | Theorem. Let I\subseteq R be generated by n elements. Then for all w\ge0: | I^{n+w}\subseteq (I^{w+1})^*. | | Corollary. If I is a parameter ideal of dimension d, then | I^d \subseteq I^*. | | 4.2 Frobenius and tight closure | | For F-nilpotent rings, it is known that tight closure coincides with Frobenius closure on parameter ideals: | I^* = I^F. | This equality is essential in showing \mathfrak m^d \subseteq \mathfrak q^F. | | 4.3 Artinian length lemma | | Lemma. If (A,\mathfrak n) is Artinian local and \dim_k\operatorname{Soc}(M)=s, then | \ell_A(M) \le s \cdot \ell_A(A). | Proof sketch: The socle is essential in M, so M\subseteq E^s, where E is the injective hull of the residue field. By Matlis duality, \ell(E)=\ell(A). Hence the inequality. | | | | 4.4 Monomial counting | | Given A=R/\mathfrak q^{[Q]} and generators x_1,\dots,x_d,y_1,\dots,y_{v-d}, the quotient A/\mathfrak n^l has a k-basis consisting of monomials with exponent constraints. Counting shows: | \ell(A/\mathfrak n^l) \le Q^v \binom{v-d+l-1}{l-1}. | | | | 5. Local Cohomology Mechanism | | 5.1 Frobenius actions | | The Frobenius endomorphism induces actions on local cohomology modules H^i_{\mathfrak m}(R). | | Define the Hartshorne–Speiser–Lyubeznik number: | \operatorname{HSL}(H^i_{\mathfrak m}(R)) = \min\{e \mid F^e(0^F_{H^i_{\mathfrak m}(R)})=0\}. | | Then set | \operatorname{HSL}(R) = \max_i \operatorname{HSL}(H^i_{\mathfrak m}(R)). | | | | 5.2 Equivalence with Frobenius test exponent | | It is a result of Katzman–Sharp (and subsequent refinements) that in Cohen–Macaulay rings: | \operatorname{Fte}(R) = \operatorname{HSL}(R). | | Thus the exponent Q governing the multiplicity bound is precisely the nilpotency index of Frobenius acting on top local cohomology. | | | | 5.3 Mechanism of control | | The identity | (\mathfrak q^F)^{[Q]} = \mathfrak q^{[Q]} | follows directly from the definition of \operatorname{Fte}(R). This ensures that all extra elements entering via Frobenius closure vanish after Q-th Frobenius power. | | This stabilization is what allows us to relate Frobenius closure containments with length bounds of Artinian quotients, thereby controlling multiplicity. | | | | Conclusion | • The sharp multiplicity bounds are established in terms of d,v,s,Q. | • Full proofs rely on: Briançon–Skoda containment, Frobenius closure properties, Artinian length arguments, and monomial counting. | • Special cases (F-pure, F-rational, F-nilpotent) yield simplified or refined constants. | • Explicit examples confirm sharpness. | • The mechanism hinges on the identification \operatorname{Fte}(R)=\operatorname{HSL}(R), tying Frobenius actions on local cohomology to multiplicity theory. | |

1. States the setup of Cohen–Macaulay local rings and defines invariants d, v, s, Fte(R), and Q. | 2. Formulates the research goal: providing explicit upper bounds for multiplicity e(R). | 3. Summarizes earlier results of Huneke–Watanabe and Katzman–Zhang. | 4. States the main theorem with separate bounds for odd and even dimensions, incorporating type s. | 5. Uses minimal reductions and Frobenius scaling to express e(R) via q^[Q]. | 6. Applies Briançon–Skoda and Frobenius closure to derive containments m^d ⊆ q^F and m^(dQ) ⊆ q^[Q]. | 7. Passes to quotient A = R/q^[Q] and bounds e(R) by (1/Q^d)·ℓ(A). | 8. Applies Artinian length lemma with socle dimension s to obtain ℓ(A) ≤ (s+1)·ℓ(A/n^l). | 9. Performs monomial counting to bound ℓ(A/n^l) by Q^v·binom(v–d+l–1, l–1). | 10. Chooses l=r+1 (odd case) or l=r (even case) to derive the final bounds. | 11. Analyzes special cases: F-pure, F-rational, and F-nilpotent rings, yielding simplified constants. | 12. Provides explicit examples verifying equality and sharpness of the bounds. | 13. Discusses asymptotic growth of e(R) as embedding dimension v increases. | 14. Lists technical tools: Briançon–Skoda theorem, Frobenius closure properties, Artinian length lemma, monomial counting. | 15. Explains the role of local cohomology: HSL numbers, Fte(R)=HSL(R), and stabilization (q^F)^[Q]=q^[Q]. | 16. Concludes by linking multiplicity bounds with Frobenius actions and local cohomology, highlighting sharpness and special cases.

1. Basic Setup: | - (R,m): Noetherian local ring of characteristic p | - Dimension: d | - Embedding dimension: v | - Type: s (for Cohen-Macaulay rings) | | 2. Historical Development: | - Huneke–Watanabe (2015): proved that e(R) ≤ (v/d) for F-pure rings. | - Katzman–Zhang (2019): extended the result, showing e(R) ≤ Q^(v-d)(v/d). | - More recent work: extends these bounds further to general Cohen–Macaulay (CM) rings. | | 3. Frobenius Operations and Test Exponents: | - Frobenius Closure: I^F = { x | x^(p^e) ∈ I^[p^e] for some e }. | - For a parameter ideal q, its Frobenius closure is denoted q^F. | - Test Exponent Framework: | - Fte(R): the least e such that (q^F)^[p^e] = q^[p^e]. | - HSL(R): Hartshorne–Speiser–Lyubeznik number. | - Relationship: for CM rings, Fte(R) = HSL(R). | | 4. F-Singularity Classes and Properties: | - Key Classes: | - F-pure: Frobenius map is pure. | - F-rational: parameter ideals are tightly closed. | - F-injective: Frobenius action on H^i_m(R) is injective. | - F-nilpotent: satisfies specific nilpotency conditions. | - Relationships: | - F-pure ⇒ F-injective. | - F-rational + F-injective ⇒ particularly well-behaved (excellent) case. | | 5. Previous Results and Context: | - Huneke–Watanabe: for Gorenstein F-pure rings with dim(R) = 2r+1, | e(R) ≤ 2((v-r-1)/r). | - Katzman–Zhang: for general CM rings, | e(R) ≤ Q^(v-d)(v/d), where Q = p^HSL(R)·a (constant depending on R).

1. Local ring setup: (R,m) Noetherian local ring, char p; dimension d; embedding dimension v; type s (for CM rings). | 2. Frobenius closure: I^F = {x | x^(p^e) ∈ I^[p^e] for some e}. | 3. Tight closure: I* = {x | c x^(p^e) ∈ I^[p^e] for some c∈R° and all large p^e}. | 4. Frobenius test exponent: Fte(R) = min e with (q^F)^[p^e] = q^[p^e] for parameter ideals q. | 5. F-singularity classes: F-rational, F-pure, F-injective, F-nilpotent with standard conditions. | 6. Module invariants: type r(M)=dim_k Ext^t_R(k,M); socle Soc(M)=(0:m)_M. | 7. Local cohomology: HSL(R) = max_i HSL(H^i_m(R)). | 8. Multiplicity: e(R) denotes Hilbert–Samuel multiplicity w.r.t. m.

1. Minimal reduction: choose q ⊆ m; verify q is parameter ideal; show e(R)=e(q); establish m^d ⊆ q^F. | 2. Frobenius analysis: set Q=p^Fte(R); prove (q^F)^[Q]=q^[Q]; define A=R/q^[Q], n=m/q^[Q]. | 3. Length calculations: e(R)=(1/Q^d)e(q^[Q]) ≤ (1/Q^d)ℓ_R(R/q^[Q]). | 4. Estimates: ℓ_A(A/(n^l)^[Q]) ≤ Q^v((v–d+l–1)/(l–1)) for l≥2; similar bounds for n^(d–l). | 5. Length bound lemma: if (R,m) Artinian and M has socle dim s, then ℓ_R(M) ≤ s·ℓ_R(R) via Matlis duality. | 6. Briançon–Skoda: in F-nilpotent rings q^d ⊆ q* = q^F; in general use Artin–Rees to show x∈q^F. | 7. Monomial count: in A=R/q^[Q], extend generators; count monomials with exponents α,β,γ to bound ℓ_A(A/(n^l)^[Q]). | 8. Special cases: for dim(R)=2r and F-nilpotent, e(R) ≤ (s+1)Q^(v–d)((v–r–1)/(r–1)). | 9. Example: R=S/(f), f=X^aY^a; compute Fte(R)=⌈log_p(a)⌉; analyze H^1_m(R); verify equality in bound.


Optimality Conditions for Model Predictive Control: Rethinking | Predictive Model Design

Question: How can predictive models be designed to guarantee closed-loop optimality in Model Predictive Control, beyond traditional data-fitting approaches? In particular, under what conditions do model properties ensure optimal performance, how can counterintuitive yet optimal structures be justified, and what are the implications for stochastic and economic MPC as well as system identification? Finally, how can this theory be connected to infinite-horizon Markov decision processes to yield rigorous proofs and practical design guidelines?

## 0. Preliminaries and assumptions | | * **True dynamics (discounted MDP):** | | $$ | s' \sim \rho(\cdot\mid s,a),\quad | J(\pi)=\mathbb{E}_\rho^\pi\!\Big[\sum_{k=0}^\infty \gamma^k L(s_k,a_k)\Big],\quad | \pi^*\in\arg\min_\pi J(\pi),\quad \gamma\in(0,1]. | $$ | * **Predictive model dynamics:** | | $$ | s' \sim \hat\rho(\cdot\mid s,a), | $$ | | with optimal value/action-value $(\hat V,\hat Q)$ and optimal policy $\hat\pi^*$. | * **True Bellman relations:** | | $$ | Q(s,a)=L(s,a)+\gamma\,\mathbb{E}_\rho[V(s')\mid s,a],\qquad V(s)=\min_a Q(s,a). | $$ | * **Advantages:** | | $$ | A(s,a):=Q(s,a)-V(s)\ge 0,\quad A(s,a)=0 \iff a\in\pi^*(s). | $$ | * **λ-shift (policy-invariant baseline alignment):** for any bounded $\lambda:S\to\mathbb{R}$, | | $$ | \hat V_\lambda(s):=\hat V(s)+\lambda(s),\quad \hat Q_\lambda(s,a):=\hat Q(s,a)+\lambda(s), | $$ | | hence $\arg\min_a\hat Q_\lambda(s,a)=\arg\min_a\hat Q(s,a)$. Define $\hat A_\lambda:=\hat Q_\lambda-\hat V_\lambda$. | * **Boundedness/stability assumption:** there exists a nonempty compact set $\Omega\subseteq S$ such that the value functions under interest are finite on trajectories that remain in $\Omega$. (This prevents pathological infinities and ensures minima exist.) | | We also use: a **class-$\mathcal K$** function $\alpha:\mathbb{R}_{\ge 0}\to\mathbb{R}_{\ge 0}$ is strictly increasing with $\alpha(0)=0$. | | --- | | ## 1. Two key lemmas (with proofs) | | ### Lemma 1 (lower bound ↔ argmin inclusion). | | There exists a class-$\mathcal K$ function $\alpha$ such that | | $$ | \hat A_\lambda(s,a)\ \ge\ \alpha\!\big(A(s,a)\big)\quad \forall(s,a) | \tag{L1-ineq} | $$ | | **iff** | | $$ | \arg\min_a \hat A_\lambda(s,a)\ \subseteq\ \arg\min_a A(s,a)\quad \forall s. | \tag{L1-incl} | $$ | | **Proof.** | | * **(Necessity via contrapositive).** Suppose $(\text{L1-incl})$ fails at some state $\bar s$: there exists $\bar a$ with | | $$ | \bar a\in\arg\min_a\hat A_\lambda(\bar s,a)\quad\text{but}\quad \bar a\notin\arg\min_a A(\bar s,a). | $$ | | Since $\min_a \hat A_\lambda(\bar s,a)=0$ and $\min_a A(\bar s,a)=0$, we have | | $$ | \hat A_\lambda(\bar s,\bar a)=0,\quad A(\bar s,\bar a)>0. | $$ | | For any class-$\mathcal K$ $\alpha$, $\alpha(A(\bar s,\bar a))>0$. Hence $\hat A_\lambda(\bar s,\bar a)\ge \alpha(A(\bar s,\bar a))$ is violated. Therefore $\neg(\text{L1-incl})\Rightarrow \neg(\text{L1-ineq})$. | | * **(Sufficiency).** Assume $(\text{L1-incl})$ holds. Define, for $x\in[0,\bar A]$ with $\bar A:=\max_{s,a}A(s,a)$ over $\Omega$, | | $$ | \alpha_0(x)\ :=\ \min_{(s,a)}\ \hat A_\lambda(s,a)\quad \text{s.t.}\quad A(s,a)\ \ge\ x. | $$ | | Properties: | | 1. $\alpha_0(0)=0$: feasibility includes minimizers of $\hat A_\lambda$, whose value is 0. | 2. $\alpha_0(x)>0$ for $x>0$: otherwise, $\exists (s,a)$ with $A(s,a)\ge x>0$ but $\hat A_\lambda(s,a)=0$, contradicting $(\text{L1-incl})$. | 3. $\alpha_0$ is nondecreasing: feasible set shrinks as $x$ grows. | 4. For any $(s,a)$, taking $x=A(s,a)$ makes $(s,a)$ feasible, hence | | $$ | \hat A_\lambda(s,a)\ \ge\ \alpha_0(A(s,a)). | $$ | | Finally, let $\alpha$ be any strictly increasing class-$\mathcal K$ minorant of $\alpha_0$ (e.g., $\alpha(x)=\inf_{y\ge x}\alpha_0(y)+\varepsilon x$ with $\varepsilon>0$ small). Then $\alpha\le \alpha_0$, $\alpha(0)=0$, and strictly increasing, so $(\text{L1-ineq})$ holds. ∎ | | --- | | ### Lemma 2 (upper bound ↔ reverse argmin inclusion). | | There exists a class-$\mathcal K$ function $\beta$ such that | | $$ | \hat A_\lambda(s,a)\ \le\ \beta\!\big(A(s,a)\big)\quad \forall(s,a) | \tag{L2-ineq} | $$ | | **iff** | | $$ | \arg\min_a A(s,a)\ \subseteq\ \arg\min_a \hat A_\lambda(s,a)\quad \forall s. | \tag{L2-incl} | $$ | | **Proof.** | | We reduce to Lemma 1 by swapping roles. Define $\tilde A:=\hat A_\lambda$ and $\hat{\tilde A}:=A$. | By Lemma 1 applied to $(\tilde A,\hat{\tilde A})$, “$\exists$ class-$\mathcal K$ $\tilde\alpha$ with $A(s,a)\ge \tilde\alpha(\hat A_\lambda(s,a))$” is equivalent to $(\text{L2-incl})$. | | Now take a (right) generalized inverse of $\tilde\alpha$: | | $$ | \beta(x):=\sup\{y\ge 0:\ \tilde\alpha(y)\le x\}. | $$ | | Then $\beta$ is class-$\mathcal K$ and $\tilde\alpha(\beta(x))\le x$. | From $A\ge \tilde\alpha(\hat A_\lambda)$ we get $\beta(A)\ge \beta(\tilde\alpha(\hat A_\lambda))\ge \hat A_\lambda$, i.e. $(\text{L2-ineq})$. | Conversely, if $(\text{L2-ineq})$ holds, the same construction with roles swapped yields $(\text{L2-incl})$. ∎ | | --- | | ## 2. Sandwich condition ⇔ policy equality | | Combining Lemma 1 and Lemma 2 gives the **sandwich inequality** | | $$ | \boxed{\ \alpha\!\big(A(s,a)\big)\ \le\ \hat A_\lambda(s,a)\ \le\ \beta\!\big(A(s,a)\big)\quad \forall(s,a)\ } | \tag{S} | $$ | | for some class-$\mathcal K$ functions $\alpha,\beta$. | | * From Lemma 1, (S) implies $\arg\min_a \hat A_\lambda(s,a)\subseteq \arg\min_a A(s,a)$. | * From Lemma 2, (S) implies $\arg\min_a A(s,a)\subseteq \arg\min_a \hat A_\lambda(s,a)$. | | Hence $\arg\min_a \hat A_\lambda(s,a)=\arg\min_a A(s,a)$ for every state, i.e. | | $$ | \boxed{\ \hat\pi^*=\pi^*\ }\quad\text{(policy equality).} | $$ | | Conversely, if $\hat\pi^*=\pi^*$, the inclusions hold both ways, and the constructions in the proofs produce class-$\mathcal K$ $\alpha,\beta$ giving (S). Thus (S) is **necessary and sufficient** for closed-loop optimality of the predictive model (up to the boundedness assumption). | | --- | | ## 3. Bellman-form expansion (one-step condition) | | Using $A(s,a)=L(s,a)+\gamma \mathbb{E}_\rho[V(s')|s,a]-V(s)$ and defining | | $$ | \Lambda(s,a):=\lambda(s)\ -\ \gamma\,\mathbb{E}_\rho[\lambda(s')\mid s,a], | $$ | | the sandwich inequality (S) is equivalent to the **one-step Bellman sandwich** | | $$ | \boxed{\ | \begin{aligned} | \beta\!\Big(L(s,a)+\gamma \mathbb{E}_\rho[V(s')|s,a]-V(s)\Big)\ &\ge\ | L(s,a)+\Lambda(s,a)+\gamma \mathbb{E}_{\hat\rho}[V(s')|s,a]-V(s)\\ | &\ge\ \alpha\!\Big(L(s,a)+\gamma \mathbb{E}_\rho[V(s')|s,a]-V(s)\Big), | \end{aligned} | } | \tag{B} | $$ | | for all $(s,a)$ in the region of interest. | Intuition: the model’s one-step Q-update must lie between monotone transforms of the true one-step update; multi-step fidelity is **not** required. | | --- | | ## 4. Deterministic MPC specialization | | For deterministic predictive models $s'=f(s,a)$, the middle term in (B) becomes | | $$ | L(s,a)+\Lambda(s,a)+\gamma\,V\!\big(f(s,a)\big)-V(s), | $$ | | so (B) reduces to | | $$ | \boxed{\ | \beta\!\Big(L(s,a)+\gamma \mathbb{E}_\rho[V(s')|s,a]-V(s)\Big)\ \ge\ | L(s,a)+\Lambda(s,a)+\gamma V(f(s,a))-V(s)\ \ge\ | \alpha\!\Big(L(s,a)+\gamma \mathbb{E}_\rho[V(s')|s,a]-V(s)\Big). | } | \tag{MPC} | $$ | | This is the **explicit closed-form condition** under which a deterministic MPC model yields the true optimal policy. | | --- | | ## 5. A simple sufficient case (constant bias) | | If $\alpha=\beta=\mathrm{id}$ and there exists a constant $\Delta$ with | | $$ | \mathbb{E}_\rho[V(s')|s,a]-\mathbb{E}_{\hat\rho}[V(s')|s,a]=\Delta\quad \forall(s,a), | $$ | | then | | $$ | \hat Q_\lambda(s,a)=Q(s,a)-\gamma \Delta, | $$ | | i.e., model and true $Q$ differ by a state-independent constant, hence induce the **same action ranking** and the same optimal policy. | | --- | | ## 6. Practical implications for model/MPC design | | 1. **Performance-oriented identification.** Fit the model to **Q/advantage alignment** (with λ-learning), not to raw dynamics error: | | $$ | \min_\theta\ \mathbb{E}\big[\|Q(s,a)-\hat Q_{\lambda,\theta}(s,a)\|\big]. | $$ | 2. **Stochastic & economic MPC.** One-step alignment is enough; λ acts as a terminal/steady-state baseline to match long-run indices. | 3. **Why data-fit can fail.** Trajectory prediction accuracy does not guarantee the sandwich (B); advantage ordering can still be wrong. | | --- | | ## 7. Logic chain | | 1. Define true/model MDPs and advantages. | 2. Introduce λ-shift (policy-invariant baseline alignment). | 3. **Lemma 1**: lower bound $\Rightarrow$ $\arg\min \hat A_\lambda \subseteq \arg\min A$. | 4. **Lemma 2**: upper bound $\Rightarrow$ $\arg\min A \subseteq \arg\min \hat A_\lambda$. | 5. Combine $\Rightarrow$ sandwich inequality $\alpha(A)\le \hat A_\lambda\le \beta(A)$. | 6. Sandwich $\Leftrightarrow$ policy equality $\hat\pi^*=\pi^*$. | 7. Expand to Bellman-form (one-step condition); specialize to deterministic MPC. | 8. Note a simple sufficient case (constant bias) and practical design takeaways. | | |

1. **MDP Framework:** | | * Correctly defines the system dynamics and transition functions for both true and model-based systems. | * Defines optimal policies, action-value functions, and value functions for both systems. | | 2. **Optimality Conditions:** | | * Provides the necessary and sufficient conditions for model optimality through advantage functions. | * Verifies the stability assumption ensuring bounded value functions along optimal trajectories. | | 3. **Value Function Modification:** | | * Shows how modifying the value function $V_\lambda(s) = V(s) + \lambda(s)$ preserves policy optimality while allowing adjustments to the model. | | 4. **Advantage Functions:** | | * Defines and compares true and modified advantage functions. | * Ensures consistency in action ranking by proving the equivalence of the advantage functions under the modified value function. | | 5. **Key Lemmas:** | | * Proves the existence of a class-K function $\alpha$ such that the modified advantage is bounded below by the true advantage. | * Proves the existence of a class-K function $\beta$ such that the true advantage is bounded below by the modified advantage. | | 6. **Optimality Theorem:** | | * Establishes the necessary and sufficient conditions for model-based optimality. | * Demonstrates the connection between the modified Q-values and the true Q-values under the optimal policy. | | 7. **MPC Framework:** | | * Extends the theory to deterministic MPC, showing the relationship between the predictive model and the true system. | * Defines the MPC optimization problem with terminal costs and constraints. | | 8. **Application to MPC:** | | * Explains why traditional data-fitting models may fail to guarantee optimality outside specific problem classes. | * Verifies that the optimality conditions apply to short-term predictions rather than full trajectory predictions. | | 9. **Implications for Practical MPC:** | | * Demonstrates how the theoretical framework can be applied to improve practical MPC implementations. | * Highlights the difference between traditional prediction-based approaches and the proposed method in achieving optimality. | | |

1. **System Dynamics and Cost** | | * State evolution: $s_+ \sim \rho(\cdot \mid s,a)$. | * Performance metric: | | $$ | J(\pi) = \mathbb{E}_\rho^\pi \Big[\sum_{k=0}^\infty \gamma^k L(s_k,a_k)\Big], | $$ | | with discount factor $\gamma \in (0,1)$ ensuring boundedness. | | 2. **Bellman Optimality Framework** | | * Optimal action-value function: | $Q^*(s,a) = L(s,a) + \gamma \,\mathbb{E}_\rho[V^*(s_+)\mid s,a]$. | * Value function: $V^*(s) = \min_a Q^*(s,a)$. | * Optimal policy: $\pi^*(s) \in \arg\min_a Q^*(s,a)$. | | 3. **MPC Formulation** | | * Finite-horizon optimization with prediction horizon $N$: | | $$ | V_{\text{MPC}}(s_k) = \min_{\hat s,u}\ \gamma^N T(\hat s_N) + \sum_{i=0}^{N-1} \gamma^i L(\hat s_i,u_i) | $$ | | subject to: | | * State dynamics: $\hat s_{i+1} = f(\hat s_i,u_i)$. | * Constraints: $h(\hat s_i,u_i) \le 0$. | * Initial and terminal conditions: $\hat s_0 = s_k,\ \hat s_N \in T$. | | 4. **Conventional Predictive Models** | | * Deterministic form: $f(s,a) = \mathbb{E}_\rho[s_+\mid s,a]$. | * Other forms: maximum likelihood, Bayesian, or data-driven models. | | 5. **Key Challenge and Motivation** | | * Accurate prediction ≠ guaranteed closed-loop optimality. | * Recent insights: optimal MPC can still be achieved via model adjustments (e.g. stage cost corrections). | * Fundamental question: **what properties must predictive models satisfy to guarantee closed-loop optimality?** | | |

1. **System Dynamics:** | | * True system: $s_+ \sim \rho(\cdot \mid s,a)$ | * Model-based system: $\hat{s}_+ \sim \hat{\rho}(\cdot \mid s,a)$ | * Key: $\rho$ is the true transition probability, $\hat{\rho}$ is the model’s approximation. | | 2. **Optimal Control:** | | * True optimal policy: $\pi^*(s) \in \arg\min_a Q^*(s,a)$ | * Model-based optimal policy: $\hat{\pi}^*(s) \in \arg\min_a \hat{Q}^*(s,a)$ | * Core objective: Ensure model’s policy matches the true policy. | | 3. **Stability Assumption:** | | * Assumption : Ensure value functions remain bounded along predicted trajectories. | * The set $\Omega$ where $\mathbb{E}_{\hat{\rho}}[V_\lambda^*(\hat{s}_k^{\pi^*})] < \infty$ must be non-empty. | | 4. **Value Function Modification:** | | * $\hat{V}_\lambda(s) = \hat{V}(s) + \lambda(s)$ | * The goal: Modify value functions without changing action ranking. | * Key modification: $\Lambda(s,a) = \lambda(s) - \gamma \mathbb{E}_{\rho}[\lambda(s_+)\mid s,a]$. | | 5. **Advantage Functions:** | | * True advantage: $A^*(s,a) = Q^*(s,a) - V^*(s)$ | * Modified advantage: $\hat{A}_\lambda^*(s,a) = \hat{Q}_\lambda^*(s,a) - \hat{V}_\lambda^*(s)$ | * Main property: $\min_a \hat{A}_\lambda^*(s,a) = \min_a A^*(s,a) = 0$ | | 6. **MPC-specific Framework:** | | * Deterministic model: $\hat{s}_+ = f(s,a)$ | * MPC optimization: | | $$ | V_{\text{MPC}}(s_k) = \min_{\hat{s}, u} \left[\gamma^N T(\hat{s}_N) + \sum_{i=0}^{N-1} \gamma^i L(\hat{s}_i, u_i) \right] | $$ | * Key: $f(s,a)$ is the deterministic predictive model; terminal cost $T$ and constraints on states/actions are crucial.

1. **Necessary & Sufficient Condition** | * The core goal is to ensure that the model-based policy $\hat{\pi}^*$ matches the true policy $\pi^*$. | * Recall that the advantage functions are defined as: | | $$ | A^*(s,a) = Q^*(s,a) - V^*(s), \quad \hat{A}_\lambda^*(s,a) = \hat{Q}_\lambda^*(s,a) - \hat{V}_\lambda^*(s). | $$ | | 2. **Bellman Recursion** | | * The true Q-function and the model-based Q-function are given by: | | $$ | Q^*(s,a) = L(s,a) + \gamma \mathbb{E}_{\rho}[V^*(s_+)\mid s,a], \quad \hat{Q}_\lambda^*(s,a) = L(s,a) + \Lambda(s,a) + \gamma \mathbb{E}_{\hat{\rho}}[V_\lambda^*(\hat{s}_+)\mid s,a]. | $$ | * These define the system dynamics under both the true and model-based dynamics. | | 3. **Sandwich Inequality** | | * The model’s advantage function must satisfy a sandwich inequality: | | $$ | \alpha(A^*(s,a)) \le \hat{A}_\lambda^*(s,a) \le \beta(A^*(s,a)), | $$ | | where $\alpha$ and $\beta$ are class-$\mathcal{K}$ functions. This ensures that the model preserves the action ranking defined by the true system. | | 4. **Policy Equivalence** | | * The sandwich inequality ensures that $\hat{\pi}^* = \pi^*$, meaning the optimal policy in the model matches the optimal policy in the true system. | | 5. **MPC-Specific Framework** | | * For MPC, the predictive model is deterministic: $\hat{s}_+ = f(s,a)$. | * The optimization problem is formulated as: | | $$ | V_{\text{MPC}}(s_k) = \min_{\hat{s}, u} \left[\gamma^N T(\hat{s}_N) + \sum_{i=0}^{N-1} \gamma^i L(\hat{s}_i, u_i)\right], | $$ | | subject to state dynamics and constraints. | * The terminal cost function $T$ and constraints on states and actions are critical in ensuring the model behaves optimally. | | 6. **Numerical Solution** | | * To solve for the optimal policy, iterate on value functions and use the Bellman equations to extract $\pi^*(s)$ from the optimal action-value functions. | | |


Conjecture: the set of prime numbers is supernatural

Question: | Formulate and analyze the conjecture that no non-constant natural function—built only from the variable $n$, integer constants, and the operations $+,\times,\wedge$—can generate a set of values consisting entirely of prime numbers. | Provide: | 1. A precise definition of natural functions and supernatural sets. | 2. A formal statement of the conjecture and its equivalence to the claim that the prime numbers form a supernatural set. | 3. Key implications, including consequences for Fermat numbers if the conjecture holds, and the extraordinary consequences if it fails. | |

| | ## 1. Basic objects and notation | | * Let $\mathbb N=\{0,1,2,\dots\}$, $\mathbb I=\{1,2,3,\dots\}$ (positive integers), and $\mathbb P=\{2,3,5,7,\dots\}$ (primes). | * Let $F=\{f:\mathbb I\to\mathbb I\}$. | | ### 1.1 Natural functions | | Define the **seed set** $F_s=\{\,n\mapsto n\,\}\cup\{\,n\mapsto a: a\in\mathbb I\,\}$. | Define three closure operators on subsets $P\subseteq F$: | | $$ | \begin{aligned} | A_+(P)&:=P\cup\{\,g+h: g,h\in P\,\},\\ | A_\times(P)&:=P\cup\{\,g\cdot h: g,h\in P\,\},\\ | A_\wedge(P)&:=P\cup\{\,g^\smallfrown h: g,h\in P\,\},\quad (g^\smallfrown h)(n):=g(n)^{\,h(n)}. | \end{aligned} | $$ | | Let $\Sigma$ be the set of all finite words over $\{A_+,A_\times,A_\wedge\}$. The class of **natural functions** is | | $$ | F_{\text{Nat}} \;:=\;\bigcup_{\sigma\in\Sigma}\sigma(F_s). | $$ | | Thus $f\in F_{\text{Nat}}$ iff $f$ can be built from $n$, integer constants, and the operations $+,\times,\wedge$ by finitely many compositions. Examples: every integer polynomial, $n\mapsto a^n+b$ with $a\in\mathbb I$, and the Fermat map $n\mapsto 2^{2^n}+1$. | | ### 1.2 Natural and supernatural sets | | An infinite $S\subseteq\mathbb I$ is **natural** if $S=f(\mathbb I)$ for some **non-constant** $f\in F_{\text{Nat}}$. | An infinite $S\subseteq\mathbb I$ is **supernatural** if it contains **no** natural subset, i.e., $\forall f\in F_{\text{Nat}}$ non-constant: $f(\mathbb I)\nsubseteq S$. | | --- | | ## 2. Fundamental monotonicity fact | | > **Claim (monotonicity).** Every $f\in F_{\text{Nat}}$ is either constant or strictly increasing on $\mathbb I$. | | *Proof sketch.* Define the **length** of $f$ as the minimal number of closure steps needed to generate $f$ from $F_s$. Induct on length. The base functions in $F_s$ are constant or strictly increasing. If $h,g$ are strictly increasing, then so are $h+g,\;h\cdot g,\;h^\smallfrown g$; if one is strictly increasing and the other constant, the same conclusions hold (except $1^\smallfrown h\equiv 1$). This closes the induction. | | Consequences: non-constant natural functions diverge and preserve order; in particular they cannot oscillate. | | --- | | ## 3. The conjecture and its equivalent reformulation | | ### 3.1 Formal conjecture | | > **Conjecture (no all-prime images for non-constant natural functions).** | > For every non-constant $f\in F_{\text{Nat}}$, $f(\mathbb I)\nsubseteq\mathbb P$. | | Equivalently, each such $f$ takes **some** composite value. | | ### 3.2 Equivalence to “the primes are supernatural” | | By definition of *supernatural*, the conjecture is exactly: | | $$ | \text{“\(\mathbb P\) is supernatural.”}\quad\Longleftrightarrow\quad | \forall f\in F_{\text{Nat}}\ \text{non-constant},\ f(\mathbb I)\nsubseteq\mathbb P. | $$ | | --- | | ## 4. Basic evidence and unconditional cases | | ### 4.1 Non-constant polynomials never stay prime | | Let $f\in\mathbb Z[n]$ be non-constant and map $\mathbb I\to\mathbb I$. Pick $n_0$ with $f(n_0)>1$ and a prime $p\mid f(n_0)$. Then $f(n_0+p)\equiv f(n_0)\equiv 0\pmod p$ but $f(n_0+p)>f(n_0)>1$, hence $f(n_0+p)$ is composite. Thus $f(\mathbb I)\nsubseteq\mathbb P$. | | ### 4.2 Exponential shifts $a^n+b$ | | Let $a>1,\ b\in\mathbb N$, $f(n)=a^n+b$. If $p\mid f(n_0)$ and $p\nmid a$, then by Fermat’s little theorem, | $a^{n_0+(p-1)}\equiv a^{n_0}\pmod p$, so $f(n_0+p-1)\equiv f(n_0)\equiv 0\pmod p$ and $f(n_0+p-1)>f(n_0)>1$, hence composite. If $p\mid a$ then $p<f(n_0)$ and $f(n_0)$ is already composite. So $f(\mathbb I)\nsubseteq\mathbb P$. | | ### 4.3 Fermat numbers | | For $F(n)=2^{2^n}+1$, one has $F(5)=4294967297=641\cdot 6700417$ (Euler), so $F(\mathbb I)\nsubseteq\mathbb P$. | | These three families are **natural functions**, so they match the conjecture’s prediction already in unconditional form. | | --- | | ## 5. Consequences conditional on the conjecture | | ### 5.1 Infinitely many composite values in each natural sequence | | For any non-constant $f\in F_{\text{Nat}}$, the set $\{x\in f(\mathbb I): x\ \text{composite}\}$ is infinite. | *Reason.* Since $f$ is strictly increasing, for each $k$ the shifted map $g(n)=f(n+k)$ is natural; by the conjecture $g$ attains a composite value at some $n=d$. Then $q=d+k>k$ and $f(q)$ is composite. As $k$ is arbitrary, we obtain infinitely many distinct composite values. | | ### 5.2 Infinitely many composite Fermat numbers | | Applying 5.1 to $F(n)=2^{2^n}+1$ shows there would be infinitely many composite Fermat numbers—precisely the long-standing open phenomenon this conjecture would settle. | | ### 5.3 If the conjecture were false | | A counterexample would be a **non-constant** $f\in F_{\text{Nat}}$ whose image consists **only** of primes. This would be an “arithmetical” closed-form prime generator using only $\{+,\,\times,\,\wedge\}$, producing arbitrarily large primes in order (since $f$ must be strictly increasing). Such a formula would be an extraordinary breakthrough and is widely considered implausible. | | --- | | ## 6. Supporting computational evidence (indicative) | | Systematic experiments on non-polynomial natural functions show that even when the first few values happen to be prime, a composite appears quickly. For example, besides the Fermat map, functions of the form $n\mapsto 2^{2^n}+2k+1$ were examined; several choices give primes for $n=1,\dots,4$, and one variant $2^{2^n}+93$ yields primes for $1\le n<7$, yet each sequence soon hits a composite value (see the tables and factorizations on pp. 5–6). These computations support the conjecture’s spirit that “simple” natural constructions cannot remain forever inside $\mathbb P$.   | | --- | | ## 7. Summary | | * **Definitions.** $F_{\text{Nat}}$ is the closure of $\{n\}\cup\{\text{constants}\}$ under $+,\times,\wedge$. A set is **supernatural** if it contains no image $f(\mathbb I)$ with $f\in F_{\text{Nat}}$ non-constant. | * **Fact.** Every $f\in F_{\text{Nat}}$ is constant or strictly increasing. | * **Conjecture (equivalent forms).** | (a) For every non-constant $f\in F_{\text{Nat}}$, $f(\mathbb I)\nsubseteq\mathbb P$. | (b) The prime set $\mathbb P$ is **supernatural**. | * **Unconditional cases.** Non-constant polynomials, $a^n+b$ with $a>1$, and the Fermat map already violate “all-prime image”. | * **Implications if true.** Each natural sequence has infinitely many composite values; in particular, **infinitely many composite Fermat numbers**. | * **If false.** Existence of a purely arithmetical, strictly increasing, closed-form prime generator—a result of extraordinary strength and unlikeliness.   | |

1. **Foundational Framework** | | * Define the sets of natural numbers $\mathbb N=\{0,1,2,\dots\}$, positive integers $\mathbb I=\{1,2,3,\dots\}$, primes $\mathbb P=\{2,3,5,7,\dots\}$, and the function space $F(\mathbb I,\mathbb I)$. | * Specify their basic set-theoretic properties with precise mathematical notation. | | 2. **Elevation Structure** | | * Define a structure $(E,+,\times,\wedge)$ with three binary operations. | * State and verify all axioms: commutativity, associativity, distributivity, and the elevation laws involving exponentiation. | * Show that these axioms are satisfied in multiple examples, such as integers and function spaces. | | 3. **Function Space Structure** | | * Prove that $F(\mathbb I,\mathbb I)$ with pointwise addition, multiplication, and exponentiation forms an elevation structure. | * Demonstrate closure under all operations. | | 4. **Morphisms** | | * Define evaluation maps $E_a: F(\mathbb I,\mathbb I)\to\mathbb I$ by $E_a(f)=f(a)$. | * Show that these maps preserve addition, multiplication, and exponentiation, hence qualify as morphisms of elevation structures. | | 5. **Operators and Construction Rules** | | * Define operators $A_+$, $A_\times$, and $A_\wedge$ on subsets of $F(\mathbb I,\mathbb I)$, each extending a set of functions by closure under one operation. | * Prove their properties, ensuring they can systematically generate new functions from given ones. | | 6. **Natural Functions** | | * Define the class of natural functions $F_{\text{Nat}}$ as the closure of constants and the identity function under repeated applications of $A_+$, $A_\times$, and $A_\wedge$. | * Show that $F_{\text{Nat}}$ itself forms an elevation structure. | * Provide examples, including polynomials, exponential-type maps, and Fermat-like functions. | | 7. **Monotonicity Property** | | * Prove that every natural function is either constant or strictly increasing. | * Use induction on construction length and establish the behavior of each operation when applied to constant or increasing functions. | | 8. **Supporting Lemmas** | | * Verify that addition, multiplication, and exponentiation preserve strict increase when applied to strictly increasing functions. | * Analyze mixed cases where one argument is constant and the other is strictly increasing. | | 9. **Special Cases of the Conjecture** | | * Demonstrate that certain families of natural functions cannot produce only primes: | | * Non-constant polynomials, | * Exponential forms such as $a^n+b$, | * Fermat numbers $2^{2^n}+1$. | | 10. **Natural and Supernatural Sets** | | * Define a natural set as the image $f(\mathbb I)$ of a non-constant natural function. | * Define a supernatural set as an infinite subset of $\mathbb I$ containing no natural set. | * State that the conjecture is equivalent to saying the set of prime numbers is supernatural. | | 11. **Consequences if the Conjecture Holds** | | * Show that every non-constant natural function must take infinitely many composite values. | * Deduce that infinitely many Fermat numbers are composite. | | 12. **Consequences if the Conjecture Fails** | | * Argue that there would exist an explicit, closed-form arithmetic formula using only $+,\times,\wedge$ that generates arbitrarily large prime numbers, a result of extraordinary significance. | | 13. **Technical Properties** | | * Define the length of a natural function as the minimal number of closure steps needed to construct it, and prove basic properties of length. | * Analyze growth behavior of natural functions, showing they are unbounded and strictly increasing when non-constant. | * Establish that the class is closed under composition, with preservation of growth and monotonicity. | |

Fermat's Legacy and Natural Functions: | | 1. Historical Context: | Fermat's Conjecture: | - Original statement: {2^(2^n) + 1 | n ∈ Z⁺} ⊂ P | - Euler's disproval | - Impact on number theory development | | 2. Natural Functions Framework: | Definition: f: I → I where I = Z⁺ | Construction Elements: | - Variable n | - Constants from I | - Operations: +, ×, ^ | | Key Property: | Every natural function is constant or strictly increasing | Key Components: | a) Natural Numbers Abstraction: | - Fundamental counting principles | - Basic arithmetic operations | - Number theoretic properties | | b) Physical World Correlation: | - Geometric interpretations | - Measurement applications | - Real-world patterns |

**Fermat’s Legacy and Natural Functions** | | 1. **Historical Context** | | * Fermat’s conjecture: numbers of the form $2^{2^n}+1$ are all prime. | * Euler disproved this by showing $2^{32}+1$ is composite. | * This highlights the difficulty of formulating formulas that only produce primes. | | 2. **Natural Functions Framework (Intuition)** | | * Defined on positive integers $I$. | * Built from the variable $n$, integer constants, and the operations $+,\times,\wedge$ (power). | * Key property: every natural function is either constant or strictly increasing. | * This motivates the question: can such “natural” constructions ever stay entirely within the primes? |

| **Core Proposition and Consequences** | | 1. **Monotonicity of Natural Functions** | | * Proposition: every $f\in F_{\text{Nat}}$ is either constant or strictly increasing. | * Proof idea: induction on the construction length; use lemmas showing addition, multiplication, and exponentiation preserve strict increase. | | 2. **Conjecture Formulation** | | * For non-constant $f\in F_{\text{Nat}}$, $f(I)\not\subseteq P$. | * Equivalent to: $P$ is supernatural. | | 3. **Supporting Arguments** | | * Non-constant polynomials: periodicity mod $p$ ensures infinitely many composite values. | * Exponential forms $a^n+b$: Fermat’s little theorem forces composites. | * Fermat function: $F(5)$ composite. | | 4. **Implications** | | * If true: every natural function has infinitely many composite values; in particular, infinitely many composite Fermat numbers. | * If false: existence of an explicit arithmetic prime generator using only $+,\times,\wedge$.