N-Bounded Foresight Equilibrium Models

• The paper formalizes bounded rationality by constraining agents to optimize over a finite N-period horizon before defaulting to fixed forecasts.
• It employs computational methods such as Bellman recursion, fixed-point theorems, and backward induction to capture forecast errors and time inconsistency.
• The models are applied in contexts from mean field games in epidemiology to dynamic macroeconomies, interpolating between myopic and fully rational behaviors.

N-bounded foresight equilibrium models formalize bounded rationality in dynamic and strategic environments by constraining agents to optimize over only the next $N$ periods—limiting prospective reasoning or “lookahead”—before defaulting to fixed forecasts or continuation values. This modeling paradigm has been developed across numerous theoretical traditions, including mean field games, general equilibrium (macroeconomic) models, and extensive-form games, to account for realistic computational, informational, and cognitive limits. Central instances include receding-horizon mean field equilibria, N-bounded foresight equilibrium (N-BFE) in dynamic heterogeneous-agent economies, and sight-bound backward induction and epistemic solution concepts in extensive games. These models generalize classical rational expectations or Nash equilibrium by interpolating between myopic (N=0) and classically rational (N→∞) behaviors, capturing time inconsistency, forecast errors, limited attention, and endogenous belief inaccuracies.

1. Formal Definitions and Core Structures

The defining feature of N-bounded foresight models is the imposition of a finite predictive or optimization horizon:

• Mean Field Games (MFGs): An agent at time $t$ forms optimal plans over a truncated future window of size $N$, implementing actions according to an RH (receding-horizon) policy $$\pi^{*,N}_{t} = (\pi^{*,N}_{t,t'},\ t'\in[t, t+\min(N, T-t)])$$, determined by solving a fixed-point maximizing expected return on that window. The equilibrium requires consistency of agents’ strategies, induced mean-field evolution, and policy implementation only based on current state and immediate lookahead (Eich et al., 2024).
• Dynamic Economies (N-BFE): Each agent solves a Bellman recursion over states $(x_{i,t},z_{t},s_{t})$, optimizes over $N$ periods using full future-relevant stochastic process realizations, and freezes subsequent forecasts at a (possibly stationary) continuation law $s^C$. The equilibrium is a collection of policy functions and population laws consistent with these restricted-horizon decisions and with the endogenously generated flows of cross-sectional distributions up to depth $N$ (Islah et al., 23 Feb 2025).
• Extensive-Form and Preference-Sight Trees: A sight function $s_N$ assigns to each information set (history) the set of branches within $N$ steps, yielding a restricted view of the game. Equilibrium concepts such as Sight-Compatible Backward Induction (SCBI) and Sight-Compatible Epistemic Solutions (SCES) require that each player's action at each node is obtained by optimal backward induction within her visible $N$-truncated subtree, possibly involving higher-order beliefs about others’ horizons (Turrini, 2015, Liu, 2016).

2. Equilibrium Notions and Existence Results

The relevant equilibrium concepts—receding-horizon MFG equilibrium, N-BFE, SCBI, and SCES—modify standard conditions as summarized below.

Model Framework	Equilibrium Concept	Existence Overview
MFG (Discrete time)	RH–NE (Receding-Horizon Nash)	Exists by Brouwer’s theorem for each finite $N$; contractiveness guarantees uniqueness (Eich et al., 2024)
Heterogeneous Econ.	N-BFE (N-Bounded Foresight)	Schauder–Tychonoff fixed-point guarantees existence under compactness and continuity (Islah et al., 23 Feb 2025)
Extensive-Form	SCBI, SCES (sight-compatible)	Always exist for finite games; SCES computable in PTIME (Turrini, 2015, Liu, 2016)

Existence proofs rely on fixed-point theorems for mapping policy and mean-field or population distributions, under compactness and continuity. Uniqueness arises under contractiveness of the best-response mapping, typically achieved via regularization or weak coupling.

3. Computational Algorithms and Complexity

Efficient solution methods exploit the finite-horizon structure:

• MFGs: Generalized Fixed-Point Iteration (GFPI) and Generalized Fictitious Play (GFP); each iteration involves backward Bellman recursion on the length-$N$ horizon followed by policy and mean-field updates. Parallelization across initial times is possible; convergence is achieved under contractiveness or standard monotonicity, with geometric rate for GFPI, and $O(1/\tau)$ exploitability decay in continuous-time GFP (Eich et al., 2024).
• Dynamic Economies: Practical algorithms discretize the state space and approximate action sets, guess initial population law vectors, and apply value- or policy-iteration on a tree of depth $N$ with fixed continuation values. Complexity is exponential in $N$ but vastly reduced relative to full-horizon models, avoiding intractable infinite-branching (Islah et al., 23 Feb 2025).
• Extensive-Form Games: SCES is computed recursively via nested beliefs and backward induction within the sight-limited trees. For each history, one builds up solution “continuations” and selects root actions via locally optimal best-response. PTIME complexity is guaranteed for finite trees, with overall cost $O((|H|\log|H|)^2)$, matching classical backward induction (Turrini, 2015).

4. Logical and Fixed-Point Characterization

Preference-sight trees and sight-bounded equilibrium concepts admit formal characterization in modal and fixed-point logics:

• A preference-sight modal logic $\mathcal{L}$ encodes local preferences, sight, and reachability, allowing internal statements of properties such as sight-consistency, equilibria, and equivalence to full backward induction.
• The solution operator SCBI is a greatest fixed-point over terminal nodes, characterized as $\nu\Phi$ where $\Phi$ enforces the local move/best-response rule at every visible node. More expressive versions (LFP(FO)) define entire equilibrium relations in first-order logic with least and greatest fixed-points (Liu, 2016).
• Multi-agent extensions assign each player her own sight function and preferences, yielding equilibrium sets as intersections of player-specific local-optimum paths.

5. Comparative Statics, Forecast Errors, and Time Inconsistency

N-bounded foresight generates distinctive behavioral and equilibrium comparative statics:

• Forecast errors: Equilibrium forecast errors, measured as $\mathrm{FE}_j(N)$, are zero for $j\leq N$ (window of actual lookahead) and become positive for $j>N$, determined by the variation of the frozen continuation law and underlying non-stationarity. As $N\to\infty$ (rational expectations), forecast errors vanish (Islah et al., 23 Feb 2025).
• Volatility and non-stationarity: Increasing $N$ raises the standard deviation of endogenous state variables (e.g., $\mathrm{std}(\bar K)$ in Krusell–Smith–type economies), reflecting that more foresight reveals (and amplifies) aggregate future risk, in contrast to pure risk aversion channels (Islah et al., 23 Feb 2025).
• Time inconsistency: Agents’ optimal plans are time-inconsistent: at each period, re-solving the problem with a receding horizon and frozen tail means original plans are not committed beyond the window $N$. There is no binding ex-ante plan across time, even if every agent is rational within her window (Islah et al., 23 Feb 2025, Eich et al., 2024).
• Limited foresight and optimality: Larger foresight does not always guarantee better outcomes; with discrete evaluations, intermediate $N$ may lead to suboptimal pruning or local maxima, emphasized in both extensive-form and preference-sight models (Liu, 2016).

6. Applications and Illustrative Examples

Key model instantiations include:

• Epidemic Mean Field Games: Receding-horizon equilibrium in Susceptible-Infected-Susceptible (SIS) models yields bounded-rational quarantine policies rapidly, displaying different aggregate dynamics as a function of planning horizon (Eich et al., 2024).
• Dynamic Macro Models: In Krusell–Smith economies with AR(1) aggregate shocks, N-BFE models demonstrate how equilibrium volatility and agent responsiveness scale with N, paralleling bounded-memory or finite-lookahead rational expectations (Islah et al., 23 Feb 2025).
• Extensive Games and Heuristics: In perfect-information games, N-bounded sight models support epistemic analysis of heuristic search, multi-level opponent modeling, and generate solution concepts that interpolate between classical backward induction and purely local myopic heuristics (Turrini, 2015, Liu, 2016).

7. Relation to Classical Paradigms and Model Equivalence

N-bounded foresight equilibrium models interpolate between extremes:

• As $N=0$, agents are fully myopic and their response is stationary: in dynamic economies, the outcome is a stationary solution; in games, only immediate payoffs are considered.
• As $N\to\infty$, agents recover classical rational expectations (dynamic models), Nash or backward induction equilibrium (games), and all nonlinear and time-inconsistent features disappear.
• Intermediate $N$ captures the tradeoff between computational tractability, predictive complexity, and plausible cognitive constraints, with model behavior sharply dependent on the foresight parameter.

In extensive-form games, full generality is achieved only when two conditions—sight-reachability and local optimality—are satisfied, ensuring coincidence between SCBI and BI solutions. Failure of either admits pathologies or exclusive equilibria only attainable with enlarged foresight (Liu, 2016).

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Principal references: "Bounded Rationality Equilibrium Learning in Mean Field Games" (Eich et al., 2024), "Bounded Foresight Equilibrium in Large Dynamic Economies with Heterogeneous Agents and Aggregate Shocks" (Islah et al., 23 Feb 2025), "Computing rational decisions in extensive games with limited foresight" (Turrini, 2015), and "Preference at First Sight" (Liu, 2016).