Chapter 5: The Price Signal as Heuristic Field

The System 1 / System 2 Mapping

The correspondence between A* search and dual-process theory is not approximate. It is exact:

Consider what each component does. The g(n) function accumulates actual costs as the agent moves through the decision complex. This is the spreadsheet work: Maria knows she has spent $14,000 this month on rent, $6,200 on supplies, $9,800 on wages. These are recorded, verifiable, deliberate calculations. System 2 is slow because it operates on concrete data that must be retrieved, processed, and summed.

The h(n) function estimates remaining costs from the current state to the goal. This is the intuitive assessment: Maria feels that switching to a non-fair-trade supplier would “cost too much” – not in dollars, but in identity (d_7), trust (d_5), and community standing (d_6). She does not compute these costs; she perceives them, the way you perceive that a dark alley is dangerous without computing the conditional probability of assault. System 1 is fast because it operates on compressed, pre-computed estimates – heuristics – rather than on raw data.

The genius of A* search is that it combines both: f(n) = g(n) + h(n). The decision is not “trust your gut” (use only h) or “run the numbers” (use only g). It is the sum. And the sum has a mathematical property that neither component has alone: when h is admissible – when it never overestimates the true remaining cost – A* is guaranteed to find the optimal path.

The Price Signal as Scalar Heuristic

Where does the price signal fit in this architecture? Prices are the most ubiquitous heuristic in economic life. When Maria sees that fair-trade Ethiopian Yirgacheffe costs $8.20 per pound and conventional Colombian costs $4.50 per pound, the prices are providing her with a scalar estimate of the cost of transitioning between states on the decision manifold.

But prices are heuristics about d_1 only. The price of a good does not encode its fairness (d_3), the labor conditions under which it was produced (d_4), the trust relationship with the supplier (d_5), or its environmental impact (d_6). The price is a projection of the full edge weight onto the monetary dimension:

h_{\text{price}}(n) = \pi_{d_1}(w(v_n, v_{\text{goal}}))

where \pi_{d_1} is the projection onto the first coordinate.

This is why Hayek was both profoundly right and profoundly incomplete. Hayek argued that prices communicate dispersed information that no central planner could aggregate. This is correct – on d_1. Prices are an extraordinarily efficient mechanism for communicating monetary cost-to-go. But they communicate nothing about the other eight dimensions. The price of a diamond does not encode the labor conditions in the mine. The price of a house does not encode the community it displaces. The price of carbon does not encode its intergenerational cost.

The geometric framework says: the price system is a heuristic field over the economic decision complex, but it is a heuristic on the scalar projection, not on the full manifold. An “efficient market” is one where the price heuristic is admissible on d_1 – prices never overestimate the true monetary cost of reaching the goal. But a market can be efficient on d_1 and catastrophically misleading on the full manifold. Surge pricing during a crisis is d_1-efficient and d_3-catastrophic. Pharmaceutical price gouging is d_1-rational and d_7-devastating. The price signal guides the search optimally on the wrong manifold.

Admissibility

In A* search, a heuristic h(n) is admissible if it never overestimates the true cost-to-go: h(n) \leq h^*(n) for all nodes n, where h^*(n) is the true minimum remaining cost on the full manifold. An admissible heuristic guarantees that A* finds the optimal path.

Theorem 7 (Heuristic Truncation Theorem). Let h_M(n) be a moral heuristic – a function that assigns high cost to actions violating moral rules and zero cost to actions consistent with them:

h_M(n) = \sum_k \beta_k \cdot P(\text{action from } n \text{ crosses boundary } k)

If the boundary penalties \beta_k are lower bounds on the true moral cost of crossing boundary k (i.e., \beta_k \leq \beta_k^* where \beta_k^* is the actual cost including legal penalty, social sanction, and psychological distress), then h_M is admissible, and A* search with h_M is guaranteed to find the optimal path on \mathcal{E}.

Proof. By construction, h_M(n) \leq h^*(n): the true remaining cost includes the actual boundary penalties \beta_k^* plus the Mahalanobis distance to the goal, both of which are non-negative. Since \beta_k \leq \beta_k^* and the Mahalanobis component is non-negative, h_M underestimates the true cost-to-go. By the admissibility theorem for A*, the search finds the optimal path. \square

This theorem reframes the entire relationship between moral intuition and economic calculation.

Corollary 8 (Moral Rules Are Not Biases). If moral heuristics are admissible search heuristics, then following them does not produce suboptimal behavior on the full manifold. It produces behavior that appears suboptimal only when projected onto a lower-dimensional subspace (e.g., scalar utility). Calling moral heuristics “biases” is a dimensional-projection error.

The corollary is a direct challenge to the standard framing in behavioral economics. When an agent rejects an unfair offer in the ultimatum game, forgoing real money, behavioral economics catalogs this as a “bias” – a deviation from rational self-interest. The geometric framework says: the behavior is optimal on the full manifold. It appears irrational only because the economist is projecting onto d_1. The “bias” is in the economist’s projection, not in the agent’s decision.

When Heuristics Are Inadmissible

Remark 9 (When Heuristics Are Inadmissible). Not all moral heuristics are admissible. A heuristic that overestimates moral cost – e.g., excessive guilt, irrational fear of social disapproval, pathological risk aversion – is inadmissible and may cause the search to miss the optimal path. This corresponds to genuine bias: the agent avoids options that are actually available. The framework distinguishes between rational heuristic use (admissible: the “bias” is in the projection, not the behavior) and genuine cognitive distortion (inadmissible: the heuristic itself is miscalibrated).

This distinction gives behavioral economics a sharper taxonomy. Instead of the blunt classification “rational vs. biased,” we have:

1. Admissible heuristic, observed on the full manifold: The agent’s behavior is optimal. No correction needed.
2. Admissible heuristic, observed on the scalar projection: The agent’s behavior appears irrational. The error is in the observer’s model, not in the agent.
3. Inadmissible heuristic: The agent’s behavior is genuinely suboptimal. The heuristic overestimates cost and causes the agent to miss better paths. This – and only this – deserves the label “bias.”

RUNNING EXAMPLE – MARIA’S COFFEE SHOP

Maria’s System 1 tells her instantly that her landlord’s 40% rent increase is unfair. This is h(n) firing a fairness boundary penalty on d_3. Her gut says: “This is wrong. He is taking advantage of gentrification that my business helped create.”

Is this a bias? The classical economist says yes – Maria should evaluate the rent increase purely on monetary terms (d_1). If the business remains profitable at the higher rent, she should pay it; if not, she should close or relocate.

But Maria’s fairness heuristic is admissible. The true cost of the rent increase on the full manifold includes the fairness violation (d_3), the trust erosion (d_5), the signal to other landlords in the district that they can extract similar increases (d_6, d_8), and the identity cost of accepting a deal she considers exploitative (d_7). These are real costs, not cognitive errors. Maria’s System 1 is estimating them – imprecisely, but as a lower bound. The heuristic underestimates the true manifold cost. It is admissible.

Her System 2 then runs the spreadsheet (g(n)): current revenue, projected revenue at different price points, cost of relocating versus staying. The decision is f = g + h: the sum of the spreadsheet calculation and the moral-intuitive assessment. Neither alone gives the right answer.

The Jean Valjean Problem

A natural and powerful objection arises at this point. Consider the moral rule “never steal.” In the framework, this assigns \beta_{\text{theft}} = \infty to the theft boundary. But the true cost of stealing a loaf of bread to survive is finite – a small fine, a brief social sanction. So h(n) has overestimated the true cost-to-go. The heuristic is inadmissible. And A* loses its optimality guarantee. Does this not invalidate the entire mapping?

No. The objection conflates two different cost functions, and the resolution illuminates the deepest feature of the framework.

Theorem 10 (Admissibility Depends on the Manifold). Let h_\mathcal{E}^*(n) denote the true minimum remaining cost on the full manifold \mathcal{E}, and let h_1^*(n) denote the true minimum remaining cost on the scalar projection d_1 alone. A moral heuristic h_M(n) with \beta_k = \infty for some boundary k satisfies:

1. h_M(n) \leq h_E^*(n) (admissible on the full manifold) whenever the true cost of crossing boundary k on \mathcal{E} – including legal penalty, social sanction, psychological distress, identity damage, and downstream path costs on all nine dimensions – is indeed infinite or exceeds h_M(n).

2. h_M(n) > h_1^*(n) (inadmissible on the scalar projection) whenever the monetary cost alone of crossing boundary k is finite.

The heuristic is admissible or inadmissible relative to the manifold on which optimality is defined.

Proof. The true cost h_\mathcal{E}^*(n) includes all nine dimensions. For the theft boundary, the true manifold cost includes: legal penalty (d_1, d_4: loss of liberty), social sanction (d_6: reputation destruction), identity cost (d_7: “I am a thief”), rights violation (d_2: criminal record creates lasting Hohfeldian disability), and epistemic cost (d_9: future uncertainty about consequences). For most agents, the sum of these costs across all dimensions is effectively unbounded – a felony conviction forecloses entire regions of the decision manifold permanently. Thus \beta_{\text{theft}} = \infty is not an overestimate on \mathcal{E}; it is a reasonable lower bound. On the scalar projection, only the monetary component (d_1) of this cost survives, and h_M(n) > h_1^*(n). \square

The critic who objects that “never steal” overestimates the cost of theft is implicitly computing h^* on the scalar projection d_1. On that manifold, the objection is correct – and the conclusion is that Homo economicus should steal, which is empirically false. On the full manifold \mathcal{E}, \beta_{\text{theft}} = \infty is admissible because the true cost of theft on \mathcal{E} is effectively infinite for most agents in most contexts.

Jean Valjean Steals Bread

For extreme cases, the framework predicts exactly what we observe. Consider Jean Valjean stealing bread to survive. The framework predicts that:

1. h(n) and g(n) disagree. The moral heuristic says “do not steal” (h(n) = \infty for the theft path). But the accumulated cost of the non-theft path has become catastrophic: starvation drives \Delta d_1 \to -\infty on the non-theft path. System 2 (g(n)) can compute that the alternative path cost exceeds \beta_{\text{theft}} in this extreme case.

2. The decision is agonizing, not automatic. When h(n) and g(n) agree, decisions are fast and confident (most economic decisions). When they disagree – when the heuristic screams “stop” but the accumulated cost screams “you must” – the agent experiences intense moral conflict. This is precisely what we observe in Valjean’s case: the decision to steal is not casual. It is a crisis.

3. Society treats the transgression with nuance. The community recognizes that the agent’s manifold was distorted by starvation (\Delta d_1 \to -\infty on the non-theft path), making the theft boundary the lower-cost path on the true manifold even though \beta_{\text{theft}} is very large. The heuristic’s overestimate is detected precisely because System 2 (g(n)) can compute that the alternative path cost exceeds \beta_{\text{theft}} in this extreme case.

This is not a failure of the framework. It is one of its most striking predictions: moral heuristics are designed by evolution and culture to be conservative overestimates in the vast majority of cases (to prevent catastrophic boundary crossings), with the result that in rare extreme cases – cases where the non-transgression path is genuinely more costly than the transgression – the agent experiences exactly the kind of agonizing moral conflict that literature, law, and everyday experience document.

Prices as d_1-Heuristics

The price of a good or service is a scalar estimate of the d_1 cost of a transition on the economic decision manifold. When Maria sees that a pound of coffee beans costs $8.20, the price is telling her: “The monetary cost of transitioning from the state ‘I do not have these beans’ to the state ‘I have these beans’ is approximately $8.20 along d_1.”

This is a heuristic in the technical A* sense: it estimates the cost-to-go without requiring the agent to trace through the full sequence of intermediate states (sourcing, shipping, roasting, packaging, retailing). The market aggregates this information and presents it as a single number.

The Efficient Market Hypothesis as Admissibility

Hayek’s claim that prices communicate dispersed information is, in the geometric framework, the claim that the price heuristic is admissible on d_1: prices do not systematically overestimate the true monetary cost of reaching the goal. The Efficient Market Hypothesis, in its strongest form, is the claim that h_{\text{price}}(n) = h_1^*(n) – the price heuristic is not merely admissible but exact on the monetary dimension.

The framework immediately reveals the limitation: even a perfectly efficient market (exact d_1 heuristic) provides no information about the other eight dimensions. The price of a good does not encode:

• d_2: Whether its production violated property rights or contractual obligations
• d_3: Whether the price is fair relative to production cost and worker compensation
• d_4: Whether the buyer has genuine alternatives or is choosing under coercion
• d_5: Whether the seller’s claims about quality are trustworthy
• d_6: Whether the production process damaged communities or ecosystems
• d_7: Whether purchasing the good is consistent with the buyer’s identity and values
• d_8: Whether the transaction complies with the spirit (not just the letter) of regulations
• d_9: Whether the buyer has adequate information to assess what they are getting

Market failures, in this framing, are not failures of the price mechanism to clear markets. They are failures of the price heuristic to capture manifold dimensions that are relevant to the agent’s actual decision problem. An externality is a d_6 cost that the price heuristic ignores. Information asymmetry is a d_9 miscalibration. Coercive pricing is a d_4 violation invisible to d_1. Each “market failure” cataloged by welfare economics is a specific dimensional gap between the price heuristic and the full-manifold cost.

Why Some Markets “Behave Classically” and Others Don’t

The framework generates a sharp prediction: markets will approximate the classical model when the active dimensions of the decision collapse to d_1 alone, and will deviate systematically when additional dimensions are active.

Consider:

• Anonymous commodity markets (wheat futures, currency exchange): The participants are trading standardized goods with no personal relationship, no fairness norm beyond the contract, no identity stake, and no community impact. Dimensions d_3 through d_9 are effectively zero. The decision manifold collapses to d_1. Price is the only relevant heuristic. Classical prediction is accurate.

• Labor markets: Workers care about wages (d_1) but also about autonomy (d_4), trust in the employer (d_5), social status (d_6), alignment with identity (d_7), and procedural fairness (d_3). The decision manifold is high-dimensional. Price (wage) is a d_1 heuristic that misses most of the structure. Classical prediction fails systematically: workers accept lower-paying jobs that score higher on other dimensions; they refuse higher-paying jobs that violate identity or fairness constraints.

• Medical markets: Patients face decisions where d_1 (cost) interacts with d_4 (autonomy over one’s body), d_5 (trust in the physician), d_6 (impact on family), d_7 (identity – “the kind of patient I am”), and d_9 (epistemic uncertainty about outcomes). The manifold is maximally high-dimensional. Price is nearly irrelevant to the decision architecture. Classical prediction is maximally wrong: patients routinely choose more expensive treatments that preserve autonomy, maintain trust, and align with identity.

The pattern is not random. It is predicted by the dimensionality of the active manifold. Tell me which dimensions are active, and I will tell you whether the price heuristic is sufficient or misleading. This is a falsifiable claim.

Step 1: Define the Search Space

Start node v_0: Maria’s current state. - d_1: Revenue $18,900/month; costs $17,500/month; margin $1,400/month - d_3: Prices perceived as fair by customers and community - d_5: High trust with regulars and supplier - d_6: Community anchor – known gathering place for neighborhood - d_7: Identity as ethical, community-oriented business owner

Goal region G: A state where the business remains viable (d_1 positive), relationships are maintained (d_5, d_6), identity is preserved (d_7), and fairness norms are respected (d_3).

Step 2: Enumerate the Paths

Path A: Absorb the full cost increase. No price change; margin drops to roughly −2,020/month (from $1,400 to -$620 after the $3,420/month cost increase from the 18% supplier price hike on $19,000 of annual bean costs). Business becomes unsustainable within months.

Total cost: Very high. The d_1 catastrophe eventually cascades to d_6 (closure), making this path dominated despite its short-term preservation of d_3 and d_5.

Path B: Pass the full cost increase ($4.50 \to $5.50/cup). A $1.00 increase covers the cost increase and adds margin.

Total cost: The d_1 gain is offset by boundary penalties on d_3 (fairness threshold crossed for a jump over 15%) and friction on d_5 and d_6. The edge weight w_B includes \beta_{\text{fairness}} for the sharp increase.

Path C: Moderate increase ($4.50 \to $5.00/cup) + negotiate with supplier + slight menu adjustment. Raise prices by $0.50, negotiate a 5% reduction in the supplier’s increase (from 18% to 13%), and slightly reduce the proportion of the most expensive beans in the house blend.

Total cost: Low friction on all dimensions. No boundary penalties triggered. The Mahalanobis distance from v_0 to the goal state is minimized.

Path D: Switch to cheaper non-fair-trade supplier. Drop the Oakland roaster entirely and source conventional beans at 45% lower cost.

Total cost: The d_5 boundary penalty (\beta_{\text{trust}} = \infty for a complete betrayal of a six-year relationship) makes this path infinitely costly on the full manifold, despite its d_1 superiority. Homo economicus chooses Path D. Maria does not – and she is right.

Step 3: Compute f = g + h for Each Path

For each path, System 2 computes g(n) (the accumulated monetary costs and benefits) and System 1 computes h(n) (the heuristic estimate of remaining costs on all dimensions):

A* search selects Path C – the moderate increase with negotiation. This is the Bond geodesic: the minimum-cost path on the full manifold.

Step 4: What the Scalar Economist Misses

A d_1-only analysis ranks the paths: D > B > C > A. Homo economicus switches suppliers. The geometric analysis ranks them: C > B > A > D. The optimal path on the full manifold is third-best on the monetary projection. The discrepancy is not noise. It is the information content of eight dimensions that the scalar discards.

Note that Maria’s actual decision process matches the geometric prediction, not the scalar prediction. She does not run a formal optimization. She “considers the options” (generates paths), “sleeps on it” (lets System 1 process the heuristic costs), “runs some numbers” (System 2 computes g), and arrives at the moderate increase – the path that feels right and works financially. The feeling and the arithmetic are not in conflict. They are h and g, components of a single evaluation function.

The Price Field

The price system defines a scalar field h_{\text{price}}: V(E) \to \mathbb{R} over the vertices of the decision complex. At each economic state v, h_{\text{price}}(v) estimates the monetary cost of reaching the goal. This field has the properties of a classical potential field: agents move in the direction of decreasing h_{\text{price}}, and equilibrium occurs where the gradient vanishes.

Hayek’s insight is that this field is decentralized: no single agent computes it. Instead, it emerges from the aggregation of all agents’ local information through market transactions. The price field is the emergent heuristic of a distributed computation.

The Moral-Heuristic Field

But prices are not the only heuristic field on \mathcal{E}. Cultural norms, moral rules, and social expectations define additional fields:

• The fairness field h_{\text{fair}}: V(E) \to \mathbb{R} estimates the d_3 cost of reaching the goal from any state. It assigns high cost to states where the agent has taken unfair advantage or been unfairly treated.
• The trust field h_{\text{trust}}: V(E) \to \mathbb{R} estimates the d_5 cost. It assigns high cost to states where trust has been broken or relationships damaged.
• The identity field h_{\text{identity}}: V(E) \to \mathbb{R} estimates the d_7 cost. It assigns high cost to states inconsistent with the agent’s self-image.

The total heuristic is the sum over all active fields:

h(n) = h_{\text{price}}(n) + h_{\text{fair}}(n) + h_{\text{trust}}(n) + h_{\text{identity}}(n) + \cdots

This is why economic decisions “feel” multi-dimensional: the agent is simultaneously consulting multiple heuristic fields, each estimating cost on a different dimension of the manifold. System 1 integrates these estimates into a single h(n) – the “gut feeling” about a decision – which System 2 then combines with the computed g(n) to produce the total evaluation f(n).

Heuristic Calibration and Cultural Variation

Different cultures calibrate their moral-heuristic fields differently. The Mahalanobis weights in the covariance matrix \Sigma determine how strongly each heuristic field contributes to the total h(n).

In cultures with high market integration and strong rule of law (Henrich et al.’s Western undergraduate samples), the price field dominates: h_{\text{price}} receives high weight, while h_{\text{fair}} and h_{\text{trust}} receive lower weights. Economic decisions in such contexts approximate the classical model because the d_1 heuristic dominates the sum.

In cultures with low market integration and high cooperative dependence (Henrich et al.’s Lamelara whale-hunting community), the trust and social-impact fields dominate: h_{\text{trust}} and h_{\text{social}} receive high weights. Economic decisions in such contexts deviate sharply from classical predictions because the non-d_1 heuristics dominate the sum.

The cross-cultural variation in economic behavior – documented by Henrich, Boyd, Bowles, Camerer, Fehr, Gintis, and McElreath (2001, 2005) – is not random. It is predicted by the relative weights of the heuristic fields, which are in turn calibrated by the culture’s ecological and social environment. The framework does not explain this variation post hoc; it predicts it from the metric structure.

When They Agree

In most economic decisions, h(n) and g(n) point in the same direction. The gut feeling (“this is a good deal”) aligns with the arithmetic (“the numbers work”). In these cases, decisions are fast and confident. The agent does not experience conflict because the heuristic and the accumulated cost agree on the minimum-cost path.

This is the typical case, not the exception. Most grocery purchases, routine business transactions, and standard economic exchanges occur in regions of the decision manifold where the moral-heuristic field is smooth and the price field is a good approximation to the full cost. The classical model works in these regions precisely because the additional dimensions add little to the d_1 heuristic.

When They Disagree

The interesting cases are those where System 1 and System 2 conflict – where the gut says “no” but the numbers say “yes,” or vice versa.

System 1 says no, System 2 says yes. Maria’s landlord offers a deal: if she signs a ten-year lease at the 40% higher rent, he will guarantee no further increases for the decade. System 2 computes that this is financially advantageous over the long term (the net present value of the guarantee exceeds the cost of the higher rent). But System 1 fires a fairness alarm: “He is locking me into an exploitative rate by dangling stability.”

In this case, g(n) is low (good deal on d_1) but h(n) is high (fairness and trust costs). The question is whether h(n) is admissible. If the fairness alarm is tracking a real manifold cost – for instance, if signing the lease at an exploitative rate damages Maria’s negotiating position with other vendors (d_5), signals to the community that she accepts unfair treatment (d_6), or erodes her self-respect (d_7) – then the heuristic is admissible and Maria should trust it. If the alarm is a mis-calibrated overreaction to any rent increase, then the heuristic is inadmissible and Maria should override it.

System 1 says yes, System 2 says no. A customer offers to buy Maria’s entire stock of specialty beans at double the retail price for a corporate event. System 1 says: “Great opportunity!” System 2 computes: if she sells all the specialty stock, she cannot serve her regular customers for three days until the next shipment arrives. The regulars – her community, her d_6 base – will go elsewhere and some may not return.

Here h(n) is low (the deal feels good on d_1) but g(n) is high when all dimensions are traced forward. System 1’s heuristic is inadmissible because it is responding to the immediate d_1 gain without estimating the downstream costs on d_5 and d_6. This is a genuine case where deliberate System 2 analysis should override intuition – precisely because the heuristic is failing to estimate the full-manifold cost.

Marr’s Levels of Analysis

A critical clarification is necessary. The claim that economic decision-making is A* search must be understood at the correct level of David Marr’s (1982) tri-level analysis:

• Computational level (what is computed and why): A* search on the decision manifold – the agent finds the minimum-cost path given a heuristic. This is our claim.
• Algorithmic level (how it is computed): The brain may use satisficing, fast-and-frugal heuristics (Gigerenzer), accumulator models, or drift-diffusion processes that are algorithmically very different from A* but converge to the same computational-level solution. We make no claim here.
• Implementational level (the neural substrate): Which circuits, neurotransmitters, and brain regions execute the computation. We make no claim here.

The framework predicts the outcome of the computation (which path the agent selects); it does not predict the neural implementation of the selection process. Cognitive scientists who object that “the brain does not run A*” are objecting at Marr’s algorithmic level, while the framework’s claims are at the computational level. The two are compatible.

Market Design as Heuristic Engineering

If the price signal is a heuristic field, then market design is heuristic engineering: the construction of institutional structures that make the price heuristic a better approximation to the full-manifold cost.

A carbon tax adds d_6 costs to the price heuristic: the price of carbon-intensive goods now partially reflects their environmental impact, making h_{\text{price}} a better estimate of the full edge weight. Fair-trade certification adds d_3 and d_4 information to the price: consumers can distinguish goods produced under fair conditions from those that were not, improving the heuristic’s coverage beyond d_1.

Disclosure requirements (nutrition labels, financial product disclosures, environmental impact reports) are attempts to make the price heuristic less inadmissible on d_9 (epistemic status): they provide information that allows the agent to better estimate the cost-to-go on dimensions the price does not capture.

In every case, the policy intervention has the same geometric structure: it improves the admissibility of the price heuristic by incorporating information from dimensions that the unregulated price ignores. The framework provides a unifying language for diverse policy tools that appear unrelated in classical analysis.

When to Trust the Heuristic, When to Override

The hierarchy of heuristic quality (Remark 13) provides a practical decision rule:

1. If the heuristic is strictly admissible (the moral rule tracks a genuinely infinite or very large boundary cost): follow it. “Do not steal” is not a bias. “Do not exploit captive customers” is not a bias. These rules are tracking real costs that the spreadsheet cannot see.

2. If the heuristic is \varepsilon-admissible (the moral intuition slightly overestimates the true cost): follow it unless the overestimate is clearly identifiable. Tipping 20% when 15% would suffice is a small \varepsilon-overestimate of the social cost of undertipping. The suboptimality is bounded and the social-norm preservation is worth the small monetary cost.

3. If the heuristic is inadmissible (a genuine phobia, excessive guilt, or pathological risk aversion is causing the agent to avoid options that are actually available): override it. This requires System 2 to identify the heuristic failure – which is exactly what therapy, financial advising, and good mentorship do.

4. If the heuristic is gauge-variant (the intuition changes depending on how the problem is framed, not on its actual content): pause and reframe. Framing effects are the clearest signal that the heuristic is not tracking the manifold but is instead responding to surface features of the description. The remedy is to re-describe the decision in multiple frames and check for consistency.

A* Search: Formal Definition

[Established Mathematics.] A* search (Hart, Nilsson, Raphael, 1968) is a best-first graph search algorithm. Given a weighted graph with start node v_0 and goal set G:

1. Initialize the open list with v_0, setting g(v_0) = 0 and f(v_0) = h(v_0).
2. Select the node n from the open list with the lowest f(n) = g(n) + h(n).
3. If n \in G, return the path from v_0 to n. Terminate.
4. For each neighbor n' of n:
   • Compute g(n') = g(n) + w(n, n').
   • Compute f(n') = g(n') + h(n').
   • If n' is not on the open list or the new g(n') is lower, add/update n' on the open list.
5. Move n to the closed list. Go to step 2.

Theorem (A* Optimality). If h(n) \leq h^*(n) for all n (admissibility) and h is consistent (h(n) \leq w(n, n') + h(n') for all edges), then A* finds the minimum-cost path from v_0 to G.

Weighted A* and the Optimality-Tractability Tradeoff

[Established Mathematics.] Weighted A* uses f(n) = g(n) + w \cdot h(n) with w > 1. The algorithm finds a path with cost at most w times the optimal cost, while exploring at most O(N / w) nodes compared to standard A* (Pohl, 1970). The tradeoff is:

Human System 1 operates with w > 1 (weighted A*): it inflates heuristic estimates to prune the search space, trading bounded suboptimality for computational tractability. The inflation weight w is calibrated by evolution and cultural learning to be close to 1 for common decisions and larger for novel or high-stakes decisions where the heuristic is less reliable.

The Admissibility Theorem Applied to Economics

[Conditional Theorem.] Let the economic decision complex \mathcal{E} have edge weights w(v_i, v_j) = \Delta \mathbf{a}^T \Sigma^{-1} \Delta \mathbf{a} + \sum_k \beta_k \cdot \mathbf{1}[\text{boundary } k \text{ crossed}]. Let h_M(n) = \sum_k \beta_k \cdot P(\text{path from } n \text{ crosses boundary } k) be the moral heuristic. Then:

1. h_M is admissible on \mathcal{E} if and only if \beta_k \leq \beta_k^* + \Delta \mathbf{a}^T \Sigma^{-1} \Delta \mathbf{a} for all boundaries k that the optimal path crosses.

2. h_M is \varepsilon-admissible on \mathcal{E} if \beta_k \leq (1 + \varepsilon)(\beta_k^* + \Delta \mathbf{a}^T \Sigma^{-1} \Delta \mathbf{a}) for all k.

3. h_M is inadmissible on the scalar projection d_1 whenever \beta_k > 0 and the monetary cost of crossing boundary k is less than \beta_k. This is the source of the “irrationality” label: the behavior appears suboptimal on d_1 because the heuristic includes costs that the d_1 projection discards.

Simon’s Bounded Rationality, Formalized

Herbert Simon (1955) proposed that agents satisfice – they choose the first option that exceeds an aspiration level – because they lack the computational resources for full optimization. The framework formalizes this: satisficing is the limiting case of weighted A* with w \to \infty, where the agent follows the heuristic greedily and terminates at the first goal-region state encountered. The aspiration level is the boundary of the goal region G. When w = \infty, the search explores only the greedy path and accepts any state in G – exactly satisficing behavior.

The framework interpolates between Simon and the neoclassical model:

Simon was right that agents do not optimize perfectly. The neoclassical economists were right that the behavior has a rational structure. Both were describing the same algorithm at different parameter values. The framework unifies them by showing that the difference is a single parameter w that trades optimality for tractability.

Gigerenzer’s Fast-and-Frugal Heuristics

Gerd Gigerenzer’s research program (Gigerenzer & Goldstein, 1996; Gigerenzer, Todd, & the ABC Research Group, 1999) demonstrated that simple heuristics can outperform complex optimization in uncertain environments – the “less is more” effect.

In the geometric framework, Gigerenzer’s heuristics are specific implementations of h(n) that exploit environmental structure. The “recognition heuristic” (choose the recognized alternative) is an h(n) that assigns lower cost-to-go to states associated with familiar entities – admissible in environments where familiarity correlates with quality. The “take-the-best” heuristic (choose on the single most discriminating cue) is an h(n) that projects the full manifold onto the dimension with the highest diagnostic value – a dimensionally reduced heuristic that is admissible when the projection captures most of the variance.

Gigerenzer showed empirically that these heuristics work. The geometric framework shows why: they are admissible or \varepsilon-admissible projections of the full-manifold cost function, designed by evolution to exploit the statistical structure of the environments in which they evolved.