Conjectural Reasoning Equilibrium

• Conjectural Reasoning Equilibrium is a family of equilibrium models where agents form and act on self-confirming conjectures about opponents’ behaviors and environmental responses.
• It employs diverse formulations—from water-filling games to sequential and bounded-rational setups—each linking conjectural optimality with varying consistency requirements.
• The framework has practical implications in strategic game design, online learning algorithms, and market calibration, bridging theoretical insights with real-world applications.

Searching arXiv for papers directly related to conjectural equilibrium and neighboring concepts. Searching arXiv for "conjectural equilibrium". Conjectural Reasoning Equilibrium is best understood as an umbrella label for equilibrium notions in which agents optimize against conjectures about opponents’ behavior, residual matching patterns, or payoff-relevant environmental responses, and those conjectures are then constrained by some notion of correctness, self-confirmation, ordinal consistency, or data-consistency. The exact phrase appears explicitly in the LLM-Stackelberg setting (Zhu, 12 Jul 2025), but closely neighboring formalizations include conjectural equilibrium in water-filling games (0811.0048), consistent conjectural variations equilibrium in continuous games (Calderone et al., 2023), Conjectural Stackelberg Equilibrium in hierarchical games (Morri et al., 23 Jan 2025), Sequential Cursed Equilibrium in extensive-form games (Cohen et al., 2022), and conjecture-rationalizable stability in matching with externalities (Nicolò et al., 26 Jun 2026). This suggests that the topic is not a single canonical solution concept, but a family of belief-dependent equilibrium constructions.

1. Core structure of conjecture-based equilibrium reasoning

A recurrent formal pattern is that agents do not optimize against a fixed environment; they optimize against a conjectured response law. In the water-filling formulation, player $k$ forms a belief $\tilde s_k:\mathcal A_k\to \mathcal S_k$ about how the payoff-relevant state responds to own action, and conjectural equilibrium requires

$$\tilde s_k(a_k^*) = s_k(a_1^*,\dots,a_K^*) \quad\text{and}\quad a_k^* = \arg\max_{a_k\in \mathcal A_k} U_k(\tilde s_k(a_k),a_k).$$

In the continuous-game formulation of consistent conjectural variations equilibrium, $(c_1,c_2)$ is a conjecture profile and $x^\star$ satisfies

$$x_i^\star = c_i(x_{-i}^\star),\qquad x_i^\star = \arg\min_{x_i}\{f_i(x_i,x_{-i})\mid x_{-i}=c_{-i}(x_i)\}.$$

These formulations isolate two ingredients: optimality under a conjecture and a discipline linking conjecture to realized play (0811.0048, Calderone et al., 2023).

The literature differs mainly in what counts as a conjecture and what counts as discipline.

Concept	Conjecture object	Discipline
CE in water-filling	state response $\tilde s_k(a_k)$	self-fulfilling state at realized action
CCVE	opponent reaction map $c_{-i}(x_i)$	pointwise and local derivative consistency
SCE	information-set conjectures over coarse sets	cursed-consistency via trembles
LLM-SCRE	conjectured follower model $\tilde\mu^\xi$	KL-minimizing consistency
Matching with externalities	residual matching after deviation	iterated elimination to $\mathbf C^\infty$

This variety matters. In some models, conjectures are explicit maps from own action to others’ actions; in others they are beliefs about residual states, post-deviation matchings, or prompt-induced response distributions. A plausible implication is that “conjectural reasoning equilibrium” is better viewed as a comparative theme than as a single theorem-defined object.

2. Canonical formalizations in strategic and dynamic games

The most direct classical instance is “Conjectural Equilibrium in Water-filling Games” (0811.0048). There, a foresighted user in a frequency-selective Gaussian interference channel models experienced interference as a function of own power allocation. The conjecture takes the local affine form

$\tilde s_k:\mathcal A_k\to \mathcal S_k$0

and the user solves a modified water-filling problem against that response law. The paper proves existence of conjectural equilibrium for the investigated water-filling game, and shows that both Nash equilibrium and Stackelberg equilibrium are special cases of the generalization. In this formulation, Nash corresponds to the zero-response belief $\tilde s_k:\mathcal A_k\to \mathcal S_k$1, while Stackelberg corresponds to a conjecture matching the local value and derivative of the true induced interference response.

“Consistent Conjectural Variations Equilibrium: Characterization & Stability for a Class of Continuous Games” (Calderone et al., 2023) develops a more structural theory for two-player continuous games with $\tilde s_k:\mathcal A_k\to \mathcal S_k$2 costs and quadratic local approximation. Conjectures are affine,

$\tilde s_k:\mathcal A_k\to \mathcal S_k$3

and equilibrium consistency produces coupled Riccati-type equations,

$\tilde s_k:\mathcal A_k\to \mathcal S_k$4

The conjecture dynamics can then be rewritten as a linear fractional transformation,

$\tilde s_k:\mathcal A_k\to \mathcal S_k$5

so fixed points correspond to invariant subspaces of an associated matrix pencil. The paper gives a local characterization, a computation method via generalized eigenvalues, a local stability test through spectral ratios, and conditions guaranteeing a unique stable equilibrium.

“Learning in Conjectural Stackelberg Games” (Morri et al., 23 Jan 2025) extends the conjectural-variations logic to multi-leader single-follower Stackelberg games. Each leader $\tilde s_k:\mathcal A_k\to \mathcal S_k$6 forms conjectures about other leaders,

$\tilde s_k:\mathcal A_k\to \mathcal S_k$7

and about the follower,

$\tilde s_k:\mathcal A_k\to \mathcal S_k$8

A Conjectural Stackelberg Equilibrium is a solution of the conjectural game in which leaders minimize

$\tilde s_k:\mathcal A_k\to \mathcal S_k$9

while the follower solves its own problem. The paper is explicit that Stackelberg Equilibrium is a refinement of this notion, and that a consistency refinement imposes derivative matching between conjectures and actual response mappings. Its algorithmic contribution, COSTAL, first learns conjectures from data and then updates strategies; the resulting stochastic approximation converges almost surely to a local CSE.

The only paper in the set that uses the exact label is “LLM-Stackelberg Games: Conjectural Reasoning Equilibria and Their Applications to Spearphishing” (Zhu, 12 Jul 2025). The model inserts a reasoning layer into a leader–follower game: the sender chooses a prompt $$\tilde s_k(a_k^*) = s_k(a_1^*,\dots,a_K^*) \quad\text{and}\quad a_k^* = \arg\max_{a_k\in \mathcal A_k} U_k(\tilde s_k(a_k),a_k).$$0, the LLM generates a message $$\tilde s_k(a_k^*) = s_k(a_1^*,\dots,a_K^*) \quad\text{and}\quad a_k^* = \arg\max_{a_k\in \mathcal A_k} U_k(\tilde s_k(a_k),a_k).$$1, the receiver chooses a prompt $$\tilde s_k(a_k^*) = s_k(a_1^*,\dots,a_K^*) \quad\text{and}\quad a_k^* = \arg\max_{a_k\in \mathcal A_k} U_k(\tilde s_k(a_k),a_k).$$2, and the receiver’s LLM generates an action $$\tilde s_k(a_k^*) = s_k(a_1^*,\dots,a_K^*) \quad\text{and}\quad a_k^* = \arg\max_{a_k\in \mathcal A_k} U_k(\tilde s_k(a_k),a_k).$$3. The sender does not know the receiver’s true reasoning model, so it chooses a conjectured response model from a parameterized class

$$\tilde s_k(a_k^*) = s_k(a_1^*,\dots,a_K^*) \quad\text{and}\quad a_k^* = \arg\max_{a_k\in \mathcal A_k} U_k(\tilde s_k(a_k),a_k).$$4

An LLM-Stackelberg Conjectural Reasoning Equilibrium $$\tilde s_k(a_k^*) = s_k(a_1^*,\dots,a_K^*) \quad\text{and}\quad a_k^* = \arg\max_{a_k\in \mathcal A_k} U_k(\tilde s_k(a_k),a_k).$$5 requires sender optimality against $$\tilde s_k(a_k^*) = s_k(a_1^*,\dots,a_K^*) \quad\text{and}\quad a_k^* = \arg\max_{a_k\in \mathcal A_k} U_k(\tilde s_k(a_k),a_k).$$6 and conjectural consistency through

$$\tilde s_k(a_k^*) = s_k(a_1^*,\dots,a_K^*) \quad\text{and}\quad a_k^* = \arg\max_{a_k\in \mathcal A_k} U_k(\tilde s_k(a_k),a_k).$$7

The paper introduces the concept and its consistency equations, but does not provide a formal existence or uniqueness theorem.

3. Extensive-form, bounded-rational, and epistemic variants

“Sequential Cursed Equilibrium” (Cohen et al., 2022) is a narrower but highly relevant extensive-form formalization. In a finite extensive-form game of perfect recall, players reason conjecturally at information sets, but in a distinctive asymmetric way: they infer from information they actually observe, yet neglect the informational implications of contingencies they are merely considering hypothetically. The technical device is the “coarsest valid partition” $$\tilde s_k(a_k^*) = s_k(a_1^*,\dots,a_K^*) \quad\text{and}\quad a_k^* = \arg\max_{a_k\in \mathcal A_k} U_k(\tilde s_k(a_k),a_k).$$8 of nonterminal histories; coarse strategies are measurable with respect to $$\tilde s_k(a_k^*) = s_k(a_1^*,\dots,a_K^*) \quad\text{and}\quad a_k^* = \arg\max_{a_k\in \mathcal A_k} U_k(\tilde s_k(a_k),a_k).$$9. An assessment

$(c_1,c_2)$0

is a sequential cursed equilibrium if the conjecture system is cursed-consistent with $(c_1,c_2)$1 and $(c_1,c_2)$2 is a local best response to it. The paper proves existence for every finite game and shows that, in simultaneous Bayesian games, independently cursed equilibrium, weighted posterior cursed equilibrium, and SCE coincide; in two-player simultaneous Bayesian games, CE, WPCE, and SCE are equivalent. Its novelty lies in dynamic environments where realized and hypothetical information are processed differently.

“Learning and Selfconfirming Equilibria in Network Games” (Battigalli et al., 2018) gives a feedback-based variant. Agents repeatedly choose actions in a network game, may not know the network, and observe only realized payoff. A steady state of conjectural best-reply learning is a selfconfirming equilibrium. The payoff-relevant local state is

$(c_1,c_2)$3

and players best respond to subjective conjectures about that state. The paper characterizes the set of selfconfirming equilibria, relates it to Nash equilibrium, and shows that conjectural best-reply paths have limit points that are selfconfirming equilibria. Under observable payoffs and own-action independence of feedback about the state, SCE and NE coincide; otherwise, limited feedback permits persistent mistaken conjectures, including inactivity traps.

“Rationality and correctness in n-player games” (Bastianello et al., 2022) studies a stricter epistemic route based on correct conjectures. A conjecture $(c_1,c_2)$4 is correct if

$(c_1,c_2)$5

and rationality means best response to a conjecture. The paper observes that if every player is rational and correct, actual play is a Nash equilibrium. It then introduces mutual assumption of rationality and correctness (MARC) and proves two sharp results: MARC holds in every finite two-person zero-sum game, but does not in general hold in $(c_1,c_2)$6-player games. For conjectural reasoning, the result is limiting rather than expansive: correct-conjecture epistemic foundations are coherent in minimax environments, but not generically.

4. Learning, computation, and design

“Conjectural Online Learning with First-order Beliefs in Asymmetric Information Stochastic Games” (Li et al., 2024) supplies an online-learning architecture for subjective conjectures in asymmetric information stochastic games. COL uses a forecaster-actor-critic architecture. First-order state beliefs are

$(c_1,c_2)$7

while conjectures are opponent-strategy parameters $(c_1,c_2)$8 drawn from a finite candidate set. The posterior

$(c_1,c_2)$9

is updated by Bayesian learning, and rollout planning optimizes against the current conjectured opponent model. The paper proves that the conjectures produced by COL are asymptotically consistent with the information feedback in the sense of a relaxed Bayesian consistency, and that the empirical strategy profile induced by COL converges to Berk-Nash equilibrium if the empirical strategy profile converges.

“Steering Noncooperative Games Through Conjecture Design” (Morri et al., 12 Nov 2025) turns conjectures from beliefs into instruments. A coordinator chooses conjecture parameters $x^\star$0 and a target profile $x^\star$1 by solving a constrained problem whose stationarity condition is

$x^\star$2

Under the paper’s assumptions, equilibrium existence is guaranteed in both centralized and decentralized versions. The framework uses conjectures not only to guide the system toward desirable outcomes, but also to decouple the game into independent optimization problems, enabling efficient computation and parallelization in large-scale settings.

In applied conjectural-variation models, computation and identification issues become central. “Calibration algorithm for spatial partial equilibrium models with conjectural variations” (Baltensperger et al., 2015) gives an iterative calibration method for spatial gas-market models that keeps market power parameters in $x^\star$3, derives admissible intervals for anchor prices and demand elasticities, and updates supplier sales when the reference data are incompatible with bounded conjectural-variation mechanics. The full calibration in the European gas application required less than one minute. “Multiplicity of equilibria in conjectural variations models of natural gas markets” (Baltensperger et al., 2015) complements this by showing that the resulting equilibrium set can be polyhedral and non-singleton. In particular, trader-to-consumer gas flow $x^\star$4 is unique whenever the market-power parameter satisfies $x^\star$5, while selecting one arbitrary point from the solution space can yield misleading economic interpretations.

5. Matching, uncertainty, and broader generalizations

“Rationalizable Behavior in Matching with Externalities” (Nicolò et al., 26 Jun 2026) translates conjectural reasoning into matching markets where deviations require beliefs about residual matching. Starting from

$x^\star$6

the paper iteratively removes $x^\star$7-dominated conjectures and defines

$x^\star$8

A matching is conjecture-rationalizable stable if it is $x^\star$9-stable. The concept always exists, extends Gale–Shapley stability, and coincides with it when externalities are absent. The paper also proves that every conjecture-rationalizable stable matching is rationalizable, and shows that in matching with couples the concept yields non-empty predictions even when standard stability is vacuous. Its epistemic foundation is especially close to the present topic: rationalizability is behaviorally implied by pairwise rationality and common belief in pairwise rationality, while conjecture-rationalizable stability additionally requires belief correctness.

“Non-Cooperative Games with Uncertainty” (Konczer, 27 Feb 2025) broadens the topic from conjectures about opponents to subjective priors over uncertain parameters. Each player chooses a mixed strategy $$x_i^\star = c_i(x_{-i}^\star),\qquad x_i^\star = \arg\min_{x_i}\{f_i(x_i,x_{-i})\mid x_{-i}=c_{-i}(x_i)\}.$$0 and a subjective prior $$x_i^\star = c_i(x_{-i}^\star),\qquad x_i^\star = \arg\min_{x_i}\{f_i(x_i,x_{-i})\mid x_{-i}=c_{-i}(x_i)\}.$$1. Extended Equilibrium requires both expected-utility optimality and a regret-maximizing prior: $$x_i^\star = c_i(x_{-i}^\star),\qquad x_i^\star = \arg\min_{x_i}\{f_i(x_i,x_{-i})\mid x_{-i}=c_{-i}(x_i)\}.$$2 for all $$x_i^\star = c_i(x_{-i}^\star),\qquad x_i^\star = \arg\min_{x_i}\{f_i(x_i,x_{-i})\mid x_{-i}=c_{-i}(x_i)\}.$$3, and

$$x_i^\star = c_i(x_{-i}^\star),\qquad x_i^\star = \arg\min_{x_i}\{f_i(x_i,x_{-i})\mid x_{-i}=c_{-i}(x_i)\}.$$4

for all $$x_i^\star = c_i(x_{-i}^\star),\qquad x_i^\star = \arg\min_{x_i}\{f_i(x_i,x_{-i})\mid x_{-i}=c_{-i}(x_i)\}.$$5. Existence is proved by a Brouwer fixed-point argument. The “No Fictional Faith” theorem adds an anti-dogmatism restriction: any subjective equilibrium prior must stay non-concentrated if the parameter truly matters to a player. This is not a standard conjectural-equilibrium model over opponents’ responses, but it is a belief-endogenizing analogue of conjectural reasoning.

6. Relation to neighboring equilibrium notions and interpretive issues

A large part of the modern literature weakens correctness while preserving a role for conjectures. “M Equilibrium: A theory of beliefs and choices in games” (Goeree et al., 2018) is a set-valued solution concept in which players’ beliefs need not be correct, but must be consequentially unbiased in the sense of inducing the same ordinal payoff ranking as actual play. Formally, it requires monotonicity,

$$x_i^\star = c_i(x_{-i}^\star),\qquad x_i^\star = \arg\min_{x_i}\{f_i(x_i,x_{-i})\mid x_{-i}=c_{-i}(x_i)\}.$$6

and ordinal consistency,

$$x_i^\star = c_i(x_{-i}^\star),\qquad x_i^\star = \arg\min_{x_i}\{f_i(x_i,x_{-i})\mid x_{-i}=c_{-i}(x_i)\}.$$7

This can be read as an equilibrium of mutually compatible belief-choice rankings rather than an equilibrium of correct conjectures.

“S Equilibrium: A Synthesis of (Behavioral) Game Theory” (Goeree et al., 2023) weakens the consistency requirement further. An $$x_i^\star = c_i(x_{-i}^\star),\qquad x_i^\star = \arg\min_{x_i}\{f_i(x_i,x_{-i})\mid x_{-i}=c_{-i}(x_i)\}.$$8-equilibrium is a pair of choice and belief sets such that beliefs imply the same best option as realized choices, while suboptimal actions are bounded by a complexity parameter $$x_i^\star = c_i(x_{-i}^\star),\qquad x_i^\star = \arg\min_{x_i}\{f_i(x_i,x_{-i})\mid x_{-i}=c_{-i}(x_i)\}.$$9: $\tilde s_k(a_k)$0 This is conjectural in a weak sense: beliefs matter explicitly, but need only get the best option right.

At the opposite end, “Regularized Conventions: Equilibrium Computation as a Model of Pragmatic Reasoning” (Jacob et al., 2023) is strategically interdependent but not conjectural in the usual sense. ReCo models pragmatics as equilibrium search in a KL-regularized signaling game, operationalized via piKL-Hedge, with guarantees toward coarse correlated equilibrium of the regularized game. The paper is explicit that this is closest to a regularized equilibrium-learning model with implicit conjectures, not a canonical conjectural reasoning equilibrium.

Three interpretive points follow. First, the phrase does not identify a single agreed-upon concept; it ranges from exact self-fulfilling conjectures to ordinally disciplined subjective beliefs. Second, the consistency requirement is the real dividing line: correctness (Bastianello et al., 2022), self-confirmation (Battigalli et al., 2018), tremble-disciplined cursed-consistency (Cohen et al., 2022), KL projection (Zhu, 12 Jul 2025), relaxed Bayesian consistency (Li et al., 2024), and iterated elimination of dominated conjectures (Nicolò et al., 26 Jun 2026) are materially different. Third, dynamic structure matters. Some models distinguish sharply between conjectures about realized contingencies and conjectures about hypothetical contingencies, while others treat conjectures as static response maps or residual outcomes. The resulting family is unified less by a single definition than by a common question: how far can strategic equilibrium be reconstructed when agents reason through conjectures rather than through fully correct Bayesian models?