Chapter 7: The Bond Geodesic Equilibrium

What the Theorem Means

The theorem says three things, each with profound implications:

First, BGE and Nash equilibrium of the augmented game are the same object. This is not an approximation or an analogy. Every BGE is literally a Nash equilibrium – of the right game. The “right game” is \Gamma^+, where payoffs are measured on the full manifold. This means that all of Nash’s mathematical machinery (existence theorems, fixed-point arguments, refinement concepts) applies to BGE. We do not need new equilibrium theory. We need the old theory applied to the right payoff structure.

Second, classical Nash equilibrium is the scalar projection of BGE. When all non-monetary dimensions are zeroed out, BGE reduces to Nash. This is the same relationship as between general relativity and Newtonian mechanics: the simpler theory is the limiting case of the richer theory, obtained by collapsing the relevant structure (curved spacetime \to flat space, full manifold \to scalar projection). Nash equilibrium is not wrong; it is incomplete. It computes on a projected subspace and therefore misses the equilibria that exist on the full manifold.

Third, BGE refines Nash. In games where moral dimensions are active, BGE excludes Nash equilibria that require crossing moral boundaries. The “exploit your counterparty” equilibrium that Nash permits may not be a BGE, because the boundary penalty on the full manifold makes the exploitative strategy strictly dominated by a boundary-respecting alternative. This is why real economic interactions often converge to outcomes that are “softer” than Nash prediction: the agents are not being irrational. They are playing the game on the full manifold, where the exploitative equilibrium does not exist.

RUNNING EXAMPLE – MARIA’S COFFEE SHOP

The Maria-landlord game on d_1 alone:

	Maria pays $7K	Maria leaves
Landlord charges $7K	Landlord: +$2K/mo; Maria: −2K/mo margin	Landlord: $0 (vacancy); Maria: -relocation
Landlord charges $5K	Landlord: $0 increase; Maria: status quo	Landlord: $0; Maria: -relocation

Nash on d_1: If Maria’s business is viable at $7K, the landlord charges $7K and Maria pays. Nash outcome: ($7K, pay).

The same game on the full manifold:

The landlord’s augmented payoff for charging $7K includes: +\$2K/month on d_1, but -\beta_{\text{fairness}} on d_3 (capturing community-created value), -\beta_{\text{trust}} on d_5 (damaging a reliable tenant relationship), - cost on d_6 (risking neighborhood degradation if Maria closes), and - cost on d_8 (risking regulatory backlash in a gentrification-sensitive city).

Maria’s augmented payoff for paying $7K includes the d_1 cost plus identity costs (d_7: accepting exploitation) and autonomy costs (d_4: forced operational changes).

The BGE of the augmented game: landlord offers $6K with a 5-year guarantee; Maria accepts. The total behavioral friction for both players is lower than under the d_1-only Nash, because the boundary penalties on d_3, d_5, and d_7 are avoided.

Mixed Strategies on the Manifold

Not all games have pure-strategy equilibria. The classic example is matching pennies: no deterministic strategy survives, so agents must randomize. The same applies to the manifold game.

Definition 22 (Mixed Bond Geodesic Equilibrium). A mixed Bond geodesic equilibrium is a profile of probability distributions (\sigma_1^*, \ldots, \sigma_n^*) over paths, where each \sigma_i^* is a distribution over the finite set S_i of simple paths from v_0^{(i)} to G_i in E_i, such that each \sigma_i^* minimizes agent i’s expected behavioral friction:

\sigma_i^* \in \arg\min_{\sigma_i \in \Delta(S_i)} \mathbb{E}_{\sigma_i, \sigma_{-i}^*}[\text{BF}_i(\gamma_i \mid \gamma_{-i})] \quad \forall\, i = 1, \ldots, n

A BGE is pure if each \sigma_i^* is degenerate (puts probability 1 on a single path). Definition 19 describes a pure BGE; the mixed BGE generalizes it to randomized path selection.

Existence

Theorem 23 (Existence of Mixed Bond Geodesic Equilibrium). Let \Gamma = (N, \{E_i\}_{i \in N}, \{G_i\}_{i \in N}) be a finite game where N is a finite set of agents, each E_i is a finite economic decision complex with positive edge weights, and each G_i is a non-empty goal region. If for each agent i and each strategy profile \gamma_{-i} of the other agents, there exists at least one path from v_0^{(i)} to G_i with finite total cost, then a mixed Bond geodesic equilibrium exists.

Proof. Each agent i’s pure strategy set S_i is the finite set of simple paths from v_0^{(i)} to G_i in E_i, which is finite and non-empty by assumption. The mixed extension \Delta(S_i) is a simplex, hence compact and convex. In the augmented game \Gamma^+ = (N, \{S_i\}, \{u_i\}) (Definition 20), each u_i is multilinear (hence continuous) in the mixed-strategy profile. By Nash’s existence theorem (1950), \Gamma^+ has a mixed-strategy Nash equilibrium, which is a mixed BGE by Theorem 21(1). \square

The proof is short because it leverages the equivalence between BGE and Nash equilibria of the augmented game. The augmented game is a finite normal-form game; Nash’s theorem guarantees a mixed equilibrium; the equivalence theorem converts it to a mixed BGE. The mathematical heavy lifting was done by Nash (1950). Our contribution is the construction of the augmented game – the right game to which Nash’s theorem should be applied.

Pure BGE Existence

Remark 24 (Pure BGE Existence). A pure BGE need not exist in general, for the same reason that pure Nash equilibria need not exist in finite games: best-response cycles are possible. Pure BGE existence is guaranteed under additional structural conditions:

1. Independent decision complexes: If each agent’s edge weights do not depend on other agents’ strategies, each agent independently computes their optimal path, and the resulting profile is trivially a pure BGE. This covers single-agent decisions (Chapters 3-6) and many market settings.

2. Potential game structure: If there exists a function \Phi: \prod_i S_i \to \mathbb{R} such that for all i, \text{BF}_i(\gamma_i, \gamma_{-i}) - \text{BF}_i(\gamma_i', \gamma_{-i}) = \Phi(\gamma_i, \gamma_{-i}) - \Phi(\gamma_i', \gamma_{-i}), then any profile minimizing \Phi is a pure BGE. Many common-pool resource games and congestion games have this structure.

3. Supermodular structure: If the strategy spaces can be partially ordered such that each \text{BF}_i has increasing differences in (\gamma_i, \gamma_{-i}), a pure BGE exists by Tarski’s fixed-point theorem.

The Contraction Lemma

The key uniqueness result is the contraction lemma, which specifies when the best-response dynamics on the augmented game converge to a unique fixed point.

Lemma 26 (Contraction Condition for BGE Uniqueness). For each agent i, let \alpha_i > 0 measure the self-separation of agent i’s best-response mapping (how much i’s own behavioral friction changes with i’s own path choice) and let \kappa_i \geq 0 measure the cross-sensitivity (how much i’s behavioral friction changes with others’ path choices). If

\alpha_i > \kappa_i \quad \text{for all } i \in N

then the best-response mapping on \Gamma^+ is a contraction with Lipschitz constant L = \max_i(\kappa_i / \alpha_i) < 1, and the BGE is unique.

Proof. The best-response mapping T: S \to S maps each profile to the profile of individual best responses. For agent i, \|T_i(s) - T_i(s')\| \leq (\kappa_i / \alpha_i) \|s_{-i} - s'_{-i}\| by the ratio of cross-sensitivity to self-separation. Taking the maximum over i:

\|T(s) - T(s')\| \leq L \|s - s'\|

with L = \max_i(\kappa_i / \alpha_i) < 1. By Banach’s fixed-point theorem, T has a unique fixed point. \square

Interpretation of the Contraction Condition

Remark 27 (Interpretation). BGE is unique when each agent’s decision is dominated by their own manifold structure (high \alpha_i – the agent’s costs are mostly determined by their own path) rather than by what others do (low \kappa_i – the agent’s costs are insensitive to others’ strategies). This is the “weak coupling” condition.

Intuitively: if Maria’s business decisions are mostly about her own costs, her own values, and her own customers (high \alpha), and only weakly affected by the landlord’s specific rent number (low \kappa), then the game has a unique BGE. Maria’s optimal path is almost independent of the landlord’s strategy, and the game converges quickly to the unique equilibrium where both parties are doing their individual best.

When coupling is strong (\kappa_i \approx \alpha_i), multiple BGE can coexist – as in coordination games, where the equilibrium depends on which region of the manifold the agents coordinate on. The rent negotiation becomes a coordination problem: both Maria and the landlord would benefit from the $6K-with-guarantee outcome, but they might also end up at the $7K-with-adversarial-relationship outcome if neither party trusts the other enough to propose the cooperative path.

Computation

Remark 28 (Computation). When the contraction condition holds, the BGE can be computed by iterated best response: start with an arbitrary profile, compute each agent’s best response (via A* search on their decision complex), update, and repeat. Convergence is geometric with rate L. Each iteration requires solving n single-agent A* problems – computationally tractable.

When the contraction condition fails, the BGE computation requires Nash-equilibrium algorithms on the augmented game \Gamma^+. For small games, enumeration suffices. For larger games, the Lemke-Howson algorithm (for two-player games) or the support enumeration method provides exact solutions. For very large games, approximate methods (fictitious play, replicator dynamics) converge to approximate BGE.

Example 31: The Ultimatum Game

In the standard ultimatum game, a proposer is given $10 and makes an offer to split it with a responder. The responder can accept (both get their shares) or reject (both get nothing).

Classical Nash prediction (d_1 only): The proposer offers $0.01 (the smallest positive amount). The responder accepts, since $0.01 > $0. In subgame-perfect equilibrium, any positive offer is accepted.

Experimental reality: Offers below approximately 30% of the endowment are rejected roughly half the time. Modal offers are 40-50% of the endowment. The prediction fails spectacularly.

BGE analysis: The responder’s decision complex includes d_3 (fairness). The attribute-vector change for accepting an unfair offer (say, $1 out of $10) is:

• \Delta d_1 = +1 (monetary gain)
• \Delta d_3 = -\text{large} (accepting an unfair split violates fairness norms)
• \Delta d_7 = -\text{moderate} (accepting exploitation conflicts with self-respect)

The edge weight for accepting is:

w_{\text{accept}} = \frac{1^2}{\sigma_1^2} + \frac{(\Delta d_3)^2}{\sigma_3^2} + \beta_{\text{fairness}} \cdot \mathbf{1}[\text{fairness boundary crossed}]

The edge weight for rejecting is:

w_{\text{reject}} = \frac{1^2}{\sigma_1^2}

(The monetary loss of $1 is the only cost, since rejection does not cross any moral boundary.)

When the fairness boundary penalty \beta_{\text{fairness}} exceeds the monetary gain (\$1^2/\sigma_1^2), rejection is the Bond geodesic. The responder’s h(n) fires the fairness alarm (System 1: “this is insulting”), and System 2’s calculation of the $1 gain cannot overcome the fairness penalty on the full manifold. The responder rejects – rationally, on the correct manifold.

The proposer, anticipating this (via the BGE best-response dynamics), offers 40-50% to avoid triggering the responder’s fairness boundary. The BGE of the ultimatum game is an approximately equal split – exactly what the experiments show.

The framework goes further: it predicts that rejection rates should vary with the salience of d_3. Making fairness more salient (e.g., by framing the game explicitly as a fairness test) should increase rejections. Making it less salient (anonymous, one-shot, large stakes) should decrease them. Both predictions are confirmed by the experimental literature.

Example 32: The Prisoner’s Dilemma

Two agents simultaneously choose to cooperate (C) or defect (D). Payoffs:

Classical Nash (d_1 only): Both defect. (D, D) is the unique Nash equilibrium with payoff (1, 1).

Experimental reality: In one-shot Prisoner’s Dilemma experiments, cooperation rates range from 20% to 60% depending on context, framing, and population. This is far above the 0% predicted by Nash.

BGE analysis: The active manifold dimensions beyond d_1 include:

• d_5 (trust): Cooperating signals trustworthiness. Even in a one-shot game, agents who value their trust dimension (d_5) pay a cost for defecting – the identity cost of being someone who betrays trust.
• d_7 (identity): An agent who defects becomes “a defector” – someone who exploits cooperative others. This identity shift has a real manifold cost, even if the specific counterparty never interacts with the agent again.
• d_3 (fairness): Mutual cooperation is the fair outcome; mutual defection is the “fair” punishment for a world of defectors; but (D, C) – defecting against a cooperator – violates fairness norms.

In the augmented game, the payoff for defecting against a cooperator is:

u_i(D, C) = -\text{BF}_i(D \mid C) = -\left[\frac{5^2}{\sigma_1^2} + \beta_{\text{trust}} + \beta_{\text{fairness}}\right]

The boundary penalties \beta_{\text{trust}} and \beta_{\text{fairness}} reduce the attractiveness of defection. When \beta_{\text{trust}} + \beta_{\text{fairness}} > (5^2 - 3^2)/\sigma_1^2, mutual cooperation becomes a BGE even in the one-shot game. The cooperation rate depends on the population distribution of \beta_{\text{trust}} and \beta_{\text{fairness}} – which is why cooperation varies across contexts and cultures but is always above zero.

The framework predicts that cooperation in the Prisoner’s Dilemma should increase with: - Trust salience (e.g., playing with a partner you can see vs. anonymous play) - Identity salience (e.g., framing the game as a “Community Game” vs. a “Wall Street Game”) - Fairness norms (e.g., in cultures with strong reciprocity norms vs. weak ones)

All three predictions are confirmed by the experimental literature (see, e.g., Liberman, Samuels, and Ross, 2004, on the “Wall Street Game” framing effect, where cooperation rates dropped from 70% to 33% when the identical game was relabeled).

Example 33: Public Goods Provision

N agents each receive an endowment of $10. Each can contribute any amount c_i \in [0, 10] to a public pool. The pool is multiplied by a factor m (with 1 < m < N) and divided equally. The payoff to agent i on d_1 is:

\pi_i = (10 - c_i) + \frac{m \sum_j c_j}{N}

Classical Nash (d_1 only): Since m/N < 1, each dollar contributed returns less than a dollar to the contributor. The dominant strategy is c_i = 0 for all i. Nash prediction: zero contribution.

Experimental reality: Average contributions are typically 40-60% of the endowment in the first round, declining toward but never reaching zero in repeated games. Agents contribute far more than Nash predicts.

BGE analysis: The active manifold dimensions include:

• d_6 (social impact): Contributing to the public good has positive social impact. Free-riding imposes negative externalities. The d_6 cost of free-riding creates a boundary penalty:

\beta_{\text{free-ride}} \propto \text{perceived social harm of withholding contribution}

• d_7 (identity): Contributing is consistent with the identity of a “cooperative community member.” Free-riding conflicts with this identity.

• d_3 (fairness): If others contribute and the agent free-rides, the agent benefits unfairly. The fairness penalty increases with others’ contributions.

In the augmented game, the payoff for contributing c_i includes the d_1 payoff minus the behavioral friction from social impact, identity, and fairness dimensions. The BGE contribution level satisfies:

\frac{\partial}{\partial c_i}\left[-\text{BF}_i^{d_1}(c_i) - \text{BF}_i^{d_6}(c_i) - \text{BF}_i^{d_7}(c_i) - \text{BF}_i^{d_3}(c_i)\right] = 0

The d_6, d_7, and d_3 terms shift the optimal contribution upward from 0 (the d_1-only solution) to a positive level that depends on the weights of the active dimensions. The framework predicts:

• Contributions are positive but below the social optimum (because individual d_1 incentives still pull toward free-riding).
• Contributions decline in repeated play as the salience of d_6 and d_7 diminishes with repetition (habituation reduces the moral-heuristic sensitivity).
• Contributions are higher in smaller groups (where d_6 impact is more visible), in face-to-face interactions (where d_5 trust is activated), and in cultures with strong cooperative norms (where \beta_{\text{free-ride}} is larger).

All three patterns are confirmed by decades of public goods experiments (see Ledyard, 1995, for a survey). The framework explains the patterns from the manifold structure rather than cataloging them as anomalies.

Players and Strategy Spaces

Agent 1: The landlord (REIT) - Start state: Current lease at $5K/month, stable tenant, good neighborhood reputation - Goal: Maximize return on property investment (on all relevant dimensions) - Strategy set S_1: {Charge $5K, Charge $6K, Charge $6K + 5yr guarantee, Charge $7K, Charge $7K + buyout offer}

Agent 2: Maria - Start state: Operating business with thin margins, strong community ties - Goal: Maintain viable business with identity and community preserved - Strategy set S_2: {Accept any offer, Accept \leq $6K only, Accept \leq $6K with guarantee only, Reject all increases, Negotiate + community campaign}

Behavioral Friction Matrix

For each strategy pair, we compute the behavioral friction (total manifold cost) for each player. The BF includes the Mahalanobis distance on all active dimensions plus boundary penalties:

Landlord’s BF for each strategy pair:

The critical insight: the landlord’s BF for charging $7K includes a fairness boundary penalty \beta_{\text{fairness}} on d_3 (capturing gentrification-created value) and a legitimacy cost on d_8 (reputational risk in a gentrification-sensitive city). These penalties do not appear in the d_1-only analysis.

Maria’s BF for each strategy pair (selected cells):

• Accept $7K: d_1: −2K/month margin loss; d_3: -\beta_{\text{fairness}} (accepting exploitation); d_4: -moderate (forced operational changes); d_7: -moderate (identity compromise)
• Accept $6K with guarantee: d_1: −1K/month (manageable); d_3: small (increase is within fair range); d_5: + (long-term stability builds trust); d_7: 0 (no identity compromise)
• Negotiate + campaign: d_1: uncertain; d_4: + (exercising autonomy); d_6: + (community engagement); d_7: + (acting on values); d_8: -small (adversarial)

Computing the BGE

Step 1: Landlord’s best response to each of Maria’s strategies.

• If Maria accepts any offer: Landlord’s BF is minimized at $6K + 5yr guarantee (good d_1 return, trust benefit from stability, avoids fairness boundary of $7K)
• If Maria accepts $$6K: Landlord must offer $$6K; best response is $6K
• If Maria accepts $$6K with guarantee: Landlord’s best response is $6K + 5yr
• If Maria rejects all increases: Landlord’s best response is $5K (avoid vacancy cost)
• If Maria negotiates: Landlord’s best response is $6K + 5yr (avoids campaign costs on d_8)

Step 2: Maria’s best response to each of the landlord’s strategies.

• If landlord charges $5K: Maria accepts (status quo is optimal)
• If landlord charges $6K: Maria accepts (manageable, no boundaries crossed)
• If landlord charges $6K + 5yr: Maria accepts with guarantee (best BF – manageable cost + long-term stability)
• If landlord charges $7K: Maria negotiates + campaigns (high BF for accepting; lower BF for fighting)
• If landlord charges $7K + buyout: Maria negotiates (high BF for both accepting and buyout)

Step 3: Find fixed points.

The best-response correspondence has a fixed point at: - Landlord: $6K + 5-year guarantee - Maria: Accept $$6K with guarantee

At this profile: - Landlord’s BF: d_1 gain of $12K/year; d_5 gain from stability; no fairness boundary crossed; moderate legitimacy - Maria’s BF: d_1 cost manageable; d_3 within fair range; d_5 gain from certainty; d_7 no identity compromise

This is the Bond Geodesic Equilibrium of the lease negotiation.

Checking the contraction condition

The landlord’s \alpha (self-sensitivity): High. The landlord’s costs are primarily determined by the rent level and vacancy risk – properties of the landlord’s own strategy.

The landlord’s \kappa (cross-sensitivity): Moderate. The landlord’s costs depend on Maria’s response (vacancy vs. tenancy), but the number of possible responses is limited.

Maria’s \alpha: High. Maria’s costs are determined by her own financial position, identity, and community relationships.

Maria’s \kappa: Moderate. Maria’s costs depend on the landlord’s offer, but her values and community are largely independent of the landlord’s specific number.

Since \alpha_i > \kappa_i for both players (each agent’s situation is dominated by their own manifold structure rather than by the other’s strategy), the contraction condition (Lemma 26) is satisfied. The BGE is unique. Iterated best response converges geometrically to the $6K + guarantee equilibrium.

Comparison with d_1-Only Nash

The BGE is strictly Pareto superior to the Nash outcome on the full manifold. Both players have lower total behavioral friction. The landlord sacrifices $1K/month in d_1 but gains stability (d_5), community preservation (d_6), and regulatory goodwill (d_8). Maria absorbs a $1K/month increase but gains certainty (d_9) and preserves identity (d_7), fairness (d_3), and community (d_6).

This is not a coincidence. It is a structural feature of the BGE: by optimizing on the full manifold, agents find paths that create value on evaluative dimensions (d_3, d_5, d_6, d_7) that the scalar projection cannot see. The evaluative dimensions are not conserved (Attribute Conservation theorem), so both parties can gain on them simultaneously. The BGE exploits this positive-sum structure; the Nash equilibrium, limited to d_1, misses it entirely.

What BGE Explains That Nash Cannot

The BGE framework resolves a collection of “anomalies” that game theory has catalogued but not unified:

1. Ultimatum game rejections (Example 31): Predicted by d_3 activation. Rejection rate varies with fairness salience, as the framework requires.

2. Trust game cooperation: Predicted by d_5 activation. Cooperation is the Bond geodesic when the trust dimension’s edge weights make defection more costly on the full manifold than on the d_1 projection.

3. Public goods contribution (Example 33): Predicted by d_6 and d_7 activation. Contribution levels and their decay pattern are predicted by the relative weights of social-impact and identity dimensions.

4. Gift exchange in labor markets: Workers who receive above-market wages reciprocate with above-minimum effort. Nash (d_1 only) predicts minimum effort regardless of wage. BGE predicts reciprocity through d_3 (fairness) and d_5 (trust).

5. Third-party punishment: Observers who are not directly affected by an unfair transaction will pay money to punish the offender. Nash on d_1 predicts no punishment (it is costly with no monetary return). BGE predicts punishment through d_3 (fairness norm enforcement) and d_8 (legitimacy maintenance).

In each case, the BGE prediction follows from the manifold structure: identify the active dimensions, compute the edge weights, solve for the equilibrium. The prediction is derived, not fitted. The framework says: tell me the active dimensions and their weights, and I will tell you the equilibrium.

What BGE Does Not Claim

Intellectual honesty requires stating the framework’s limitations:

1. BGE does not solve the measurement problem. The framework requires the covariance matrix \Sigma and boundary penalties \beta_k as inputs. These must be estimated from data – behavioral experiments, cultural observations, neuroimaging. The framework tells you what to measure and how to use the measurements, but it does not provide the measurements themselves.

2. BGE is not automatically efficient. The Prisoner’s Dilemma on the manifold can still produce suboptimal outcomes. BGE inherits the efficiency failures of Nash for the same structural reason: individual rationality does not guarantee collective optimality. The improvement is that more dimensions of cost are internalized, shifting more games toward cooperation – but the fundamental tension between individual and collective rationality persists.

3. BGE assumes agents pathfind on their perceived manifold. If an agent misperceives the manifold – if their heuristic is inadmissible, or their \Sigma is miscalibrated – the BGE of the perceived game may differ from the BGE of the true game. The framework distinguishes between these (the perceived-manifold BGE and the true-manifold BGE) but does not claim that agents always perceive correctly.

Nash’s Existence Theorem (Applied to \Gamma^+)

[Established Mathematics.] Nash (1950) proved that every finite game has at least one mixed-strategy Nash equilibrium, using Kakutani’s fixed-point theorem. The augmented game \Gamma^+ is finite (finite agents, finite strategy sets), so Nash’s theorem applies directly.

The proof: Each agent’s mixed strategy is a probability distribution over S_i, forming a simplex \Delta(S_i). The product \prod_i \Delta(S_i) is compact and convex. The best-response correspondence \text{BR}: \prod_i \Delta(S_i) \rightrightarrows \prod_i \Delta(S_i) is upper-hemicontinuous with non-empty convex values (by linearity of expected payoff in own strategy). By Kakutani’s theorem, \text{BR} has a fixed point – a mixed-strategy Nash equilibrium of \Gamma^+, which is a mixed BGE by Theorem 21(1).

Banach’s Fixed-Point Theorem (Applied to Contraction Lemma)

[Established Mathematics.] Banach’s contraction mapping theorem states that if T: X \to X is a contraction (\|T(x) - T(y)\| \leq L\|x - y\| with L < 1) on a complete metric space (X, d), then T has a unique fixed point. Applied to Lemma 26: the best-response mapping is a contraction when \alpha_i > \kappa_i for all agents, so the BGE is unique.

The convergence rate is geometric: after k iterations, \|s^{(k)} - s^*\| \leq L^k \|s^{(0)} - s^*\|. For a two-player game with L = 0.5, ten iterations reduce the error by a factor of 1024. This makes iterated best response a practical algorithm for BGE computation.

The Augmented Game Payoff Structure

[Conditional Theorem.] The payoff function u_i(\gamma_i, \gamma_{-i}) = -\text{BF}_i(\gamma_i \mid \gamma_{-i}) in the augmented game has the following structure:

u_i = -\sum_{e \in \gamma_i} \left[\Delta\mathbf{a}(e)^T \Sigma_i^{-1} \Delta\mathbf{a}(e) + \sum_k \beta_{k,i} \cdot \mathbf{1}[e \text{ crosses boundary } k]\right]

where \Sigma_i is agent i’s covariance matrix (encoding i’s relative weighting of dimensions) and \beta_{k,i} is i’s boundary penalty for constraint k. Note that different agents may have different \Sigma_i and \beta_{k,i} – reflecting different cultures, values, and experiences. The augmented game payoff captures this heterogeneity naturally.

In the Maria-landlord example: \Sigma_{\text{Maria}} weights d_3 (fairness) and d_7 (identity) heavily; \Sigma_{\text{REIT}} weights d_1 (monetary return) and d_8 (regulatory compliance) heavily. The BGE reflects both agents’ manifold structures simultaneously.

Behavioral Friction Decomposition

For pedagogical clarity, the total behavioral friction can be decomposed:

\text{BF}_i = \underbrace{\text{BF}_i^{d_1}}_{\text{monetary cost}} + \underbrace{\text{BF}_i^{d_2 \ldots d_9}}_{\text{moral-social cost}} + \underbrace{\sum_k \beta_{k,i} \cdot \mathbf{1}[\text{boundary}]}_{\text{boundary penalties}}

Nash equilibrium sets the second and third terms to zero, optimizing \text{BF}_i^{d_1} alone. BGE optimizes the sum. The difference between Nash and BGE outcomes is determined by the magnitude of the moral-social costs and boundary penalties relative to the monetary cost.

When these are small (\text{BF}^{d_2 \ldots d_9} \approx 0, \beta_k \approx 0): BGE \approx Nash. This is the case in anonymous commodity markets.

When these are large: BGE \neq Nash. This is the case in face-to-face bargaining, labor markets, medical markets, and any context where moral and social dimensions are active.

The empirical question is not “is BGE right or is Nash right?” It is: “in this specific market, how large are the non-monetary behavioral friction terms?” The answer determines which equilibrium concept is the better predictor.

Connection to the Framework

The Bond Geodesic Equilibrium completes the multi-agent story:

• Chapter 5 solved single-agent pathfinding: f(n) = g(n) + h(n) on the decision manifold.
• Chapter 6 proved that scalar utility is irrecoverably lossy: the full manifold cannot be compressed to \mathbb{R} without information destruction.
• This chapter answers: what is the equilibrium when multiple agents pathfind simultaneously?
• Chapter 8 will ask: what symmetries does the equilibrium respect? (gauge invariance)
• Chapter 9 will derive: what conservation laws follow from those symmetries? (Noether’s theorem for economics)

The BGE is the game-theoretic completion of the geometric economics program. Nash equilibrium – the foundation of modern economics – is the d_1-only contraction of BGE, exactly as scalar utility is the rank-0 contraction of the economic tensor (Chapter 6). The same mathematical operation (contraction/projection) that explains why scalar ethics loses information also explains why scalar economics loses information. The framework is unified, and the BGE is its central construction.