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Curriculum/Chapter 2
CHAPTER 2 OF 18

The missing dimension

~28 min full text
REVIEWED TEACHING EDITION
This chapter has completed the current author review and public-source checking pass. It remains working-paper teaching material without journal peer review. For the learning sequence, return to the curriculum.
CORE LESSON

Two dimensions, eight outcomes: the system coordinate bilateral analysis omits

~9 min

The space every framework monitors

The standard bilateral representations used here ask two questions: Did party A win or lose? Did party B win or lose? The answers produce four possible outcomes. Nash bargaining1, Kalai-Smorodinsky (a bargaining rule that grows each party's share as the whole pie grows)2, Rubinstein alternating offers (the parties take turns making offers and delay is costly)3, the Shapley value (which splits joint gains by each party's average marginal contribution)4, the core (the outcomes no subgroup could beat by breaking away)5, and the nucleolus (the split that minimizes the most-dissatisfied coalition's complaint)6 are defined on party-level payoff representations7; those representations do not carry an independent system-welfare coordinate. The structural consequence is the focus of this chapter: a bilateral payoff map cannot distinguish a Hollow Win from a Win-Win-Win when the party payoffs match8.

The third coordinate

There is a third variable: W, system welfare. The eight-outcome map uses C as the binary version of that variable: C = 1 when W is preserved, and C = 0 when W is degraded. Not A's system. Not B's system. The system both parties are embedded in: the market, the benchmark, the commons, the ecosystem9, the regulatory infrastructure that makes their deal possible in the first place. When C = 1, the system survives. When C = 0, the system is degraded. The structural claim is this: C cannot be computed from A's payoff and B's payoff, not with better data and not with a smarter algorithm. The exclusion is a property of how the payoff space is built, not a measurement gap to be closed case by case. C is outside the space that A's and B's payoffs define.

The Coase boundary: non-owned support systems cannot be bargained away

The Coase objection works when a harm can be assigned to a right10, owned by a party, and traded at tolerable cost. MST begins where that condition fails. Public health, ecological capacity, fiscal capacity, benchmark integrity, market trust, and antibiotic effectiveness are shared support systems. No party owns them, so no party can sell, waive, or bargain them away on behalf of the system. This is why W is a system-welfare coordinate rather than a social-welfare function. It does not aggregate individual preferences into a ranked social choice. It asks whether the support system remains functional: whether the aquifer remains usable, antibiotics still work, the fiscal base remains intact, the benchmark remains trustworthy, or the financial system remains stable. That distinction also clarifies distribution. SAPM measures how much system capacity was depleted; distributional analysis asks who gained from that depletion and who was left carrying the loss.

The Hollow Win

Once W is carried, a privately efficient deal can be shown as a Hollow Win: (C, A, B) = (0, 1, 1). Both parties gain privately while the system they depend on degrades. This is the outcome standard bilateral analysis cannot display. A private-payoff analysis can classify the profile as mutual gain and Pareto efficient because it has no C coordinate. The completed analysis adds the system test and can classify the same profile as a Hollow Win. The Hollow Win is not a rare edge case.11 It is a structural possibility in every strategic interaction where the system coordinate is not a function of the parties' payoffs.

What the eight-outcome map reveals

The eight-outcome map expands the standard 2x2 grid by adding the system coordinate C. The eight outcomes are: WIN, WIN, WIN (C=1, A=1, B=1), all three dimensions positive and the only outcome with no structural tension; HOLLOW WIN (C=0, A=1, B=1), parties gain while the system degrades; CORROSIVE WIN-LOSE (C=0, A=1, B=0), A gains at the expense of both B and the system; CORROSIVE LOSE-WIN (C=0, A=0, B=1), B gains at the expense of both A and the system; MISERY (C=0, A=0, B=0), universal loss across all dimensions; STABLE MISERY (C=1, A=0, B=0), the system persists but both parties lose within it; SUSTAINABLE WIN-LOSE (C=1, A=1, B=0), B loses for the benefit of the system and A; SUSTAINABLE LOSE-WIN (C=1, A=0, B=1), A loses for the benefit of the system and B. A party-payoff representation can classify both (1,1,1) and (0,1,1) as mutual gain because it has no C coordinate. The eight-outcome map makes the distinction explicit.

Nash equilibrium as a special case

The Missing System Theory (MST) analyzes strategic interactions in an augmented payoff space that includes a binary indicator C of system welfare alongside private payoffs. Nash equilibrium, defined exclusively on private payoff vectors12, is the special case of MST restricted to the regime in which the system coordinate is absent. This is the projection πu. The Hollow Win becomes a Nash outcome once the analysis projects away the system coordinate. The result distinguishes solution-concept validity from welfare completeness. It gives the MST program a clean bridge to standard game theory.13 Nash is located as the strategic face of the same restriction that CAPM occupies on the pricing face14.

Pigou, Coase, Ostrom as three switches on MST

The three classical traditions of externality economics, Pigovian taxation, Coasean bargaining, and Ostromian self-governance, are not rival theories but distinct operations on a single structural defect: the exclusion of system welfare from the payoff space of every strategic interaction. Pigou imposes the system price λW, the planner-set price on system welfare, exogenously (imposed from outside the system) via a planner15. Coase is the non-binding boundary where the system coordinate is tradable at zero transaction cost. Ostrom is the endogenous repair (arising from within the system's own incentives rather than imposed from outside)16 that folds the coordinate back into incentives. The MST capture theorem predicts the planner-imposed regime fails when capture intensity φ exceeds a threshold φ*, restoring the Hollow Win. These three traditions are conventionally taught as rival paradigms. The MST framework shows they are three switches on the same structural defect.17

What changes for a reader

For a student of economics or negotiation theory, the implication is that every framework you have been taught operates in a space that cannot represent the most important variable. For a regulator, the implication is that a deal that looks privately efficient may be degrading the system that makes the deal possible. For an executive, the implication is that decision records that track only private payoffs are structurally blind to system welfare. For a researcher, the implication is that the standard payoff space is not neutral; it is an axiom that excludes the system coordinate. The practical point is narrower: once you know the coordinate is missing, you can ask whether a given interaction sits in the regime where the projection is safe or the regime where it hides a Hollow Win.

Limits of the framework

The Missing System Theory does not claim that every two-dimensional analysis is wrong. It claims that the two-dimensional space is structurally incomplete. The theorem does not provide a method for measuring W in every context; it identifies that W is not a function of A's and B's payoffs. The theorem does not replace Nash, Pigou, Coase, or Ostrom; it identifies a structural condition those frameworks do not address in this form. The theorem does not claim that every Hollow Win is avoidable; it claims that the Hollow Win is invisible to standard analysis. The theorem does not provide a complete theory of system welfare; it provides a structural result about the space in which strategic interactions are analyzed.
The eight-outcome map in detail · ~2 min
The eight-outcome map is the central analytical tool of this chapter. Each outcome is defined by three binary coordinates: C (system welfare preserved or degraded), A (party A gains or loses), B (party B gains or loses). The eight outcomes are: (1,1,1) WIN, WIN, WIN, the only outcome with no structural tension; (0,1,1) HOLLOW WIN, both parties gain while the system degrades; (0,1,0) CORROSIVE WIN-LOSE, A gains at the expense of both B and the system; (0,0,1) CORROSIVE LOSE-WIN, B gains at the expense of both A and the system; (0,0,0) MISERY, universal loss; (1,0,0) STABLE MISERY, the system persists but both parties lose; (1,1,0) SUSTAINABLE WIN-LOSE, B loses for the benefit of the system and A; (1,0,1) SUSTAINABLE LOSE-WIN, A loses for the benefit of the system and B. The critical distinction is between (1,1,1) and (0,1,1). Standard analysis cannot distinguish them because it monitors only A and B. The eight-outcome map makes the distinction visible.
  • The eight-outcome map adds the system coordinate C to the standard 2x2 grid.
  • The Hollow Win (0,1,1) is invisible to standard analysis.
  • WIN, WIN, WIN (1,1,1) is the only outcome with no structural tension.
  • Every other outcome involves some form of system degradation or asymmetric sacrifice.
Why C cannot be computed from A and B · ~2 min
The structural claim at the heart of this chapter is that C cannot be computed from A's payoff and B's payoff. This is not a data quality problem. It is a property of how the payoff space is built. The ordinary payoff space records the parties to a transaction. System welfare W is not a party to the transaction; it is the system that makes the transaction possible. The relationship between private payoffs and system welfare is not functional: the same private payoff vector can correspond to different system welfare states depending on context. A trade that generates $1M for A and $1M for B can be system-preserving in one context and system-degrading in another. The standard payoff space cannot represent this distinction because it axiomatically excludes the system coordinate.
  • C is not a function of A's and B's payoffs.
  • The same private payoff vector can correspond to different system welfare states.
  • The exclusion of C is a property of how the payoff space is built, not an oversight.
  • Better data or algorithms cannot solve a structural exclusion.
Nash equilibrium as a projection · ~2 min
The paper 'Nash Equilibrium as a Special Case of the Missing System Theory' proves that Nash equilibrium is the special case of MST restricted to the regime in which the system coordinate is absent. This is the projection πu. The Hollow Win becomes a Nash outcome once the analysis projects away the system coordinate. This result has several implications. First, it distinguishes solution-concept validity from welfare completeness: a Nash equilibrium can be valid as a solution concept while being structurally blind to system welfare. Second, it gives the MST program a clean bridge to standard game theory; the entire edifice of Nash equilibrium is not rejected but located as a special case. Third, it shows that the problem is not with Nash equilibrium itself but with the space in which it is applied. When the system coordinate is present, the analysis changes.
  • Nash equilibrium is the special case of MST when the system coordinate is projected away.
  • The Hollow Win is a Nash outcome under the projection.
  • Solution-concept validity is distinct from welfare completeness.
  • The problem is the space, not the solution concept.
Pigou, Coase, Ostrom as switches · ~2 min
Tax It, Trade It, Govern It unifies the three classical traditions of externality economics as distinct operations on a single structural defect: the exclusion of system welfare from the payoff space. Pigou imposes the system price λW, the planner-set price on system welfare, exogenously via a planner. The MST capture theorem predicts this regime fails when capture intensity φ exceeds a threshold φ*, restoring the Hollow Win. Coase is the non-binding boundary where the system coordinate is tradable at zero transaction cost, the regime where private bargaining can internalize the externality. Ostrom is the endogenous repair that folds the coordinate back into incentives through self-organized institutions. Each switch has a domain of application and a failure mode. MST supplies the unified structure: Pigou, Coase, and Ostrom are three repair strategies for the same missing-coordinate problem.
  • Pigou, Coase, and Ostrom are three switches on the same structural defect.
  • Pigou imposes the system price exogenously; capture is the predicted failure mode.
  • Coase is the zero-transaction-cost boundary where bargaining can internalize the externality.
  • Ostrom is the endogenous repair through self-governance.
The capture theorem and the planner-imposed switch · ~2 min
The MST capture theorem predicts that the planner-imposed switch (Pigou) fails when capture intensity φ exceeds a threshold φ. When capture is below the threshold, the planner can impose the system price λW, the planner-set price on system welfare, and the Hollow Win is avoided. When capture exceeds the threshold, the planner is captured by the parties who benefit from the Hollow Win18, and the system price is set at a level that does not prevent system degradation. The capture theorem is not a claim that all Pigovian taxation fails. It is a structural prediction about when it fails and why. The threshold φ depends on the institutional context, the concentration of benefits from the Hollow Win, and the cost of organizing capture. This gives the MST framework a testable empirical implication: in domains where capture intensity is high, Pigovian taxation will not prevent Hollow Wins.
  • The capture theorem predicts planner-imposed switches fail above a capture threshold φ*.
  • Below the threshold, Pigovian taxation can prevent Hollow Wins.
  • Above the threshold, the planner is captured and the system price is ineffective.
  • The threshold depends on institutional context and benefit concentration.
The Coasean boundary and zero transaction costs · ~2 min
The Coasean switch is the non-binding boundary where the system coordinate is tradable at zero transaction cost. In this regime, private bargaining can internalize the externality regardless of the initial allocation of property rights. The MST framework locates this as a special case: when transaction costs are zero, the parties can bargain to include the system coordinate in their agreement. The problem is that the Coasean boundary is rarely met in practice. Transaction costs are rarely zero. Property rights over system welfare are rarely well-defined. The Coasean switch is therefore a benchmark, not a practical policy tool. It shows what is possible in principle but does not provide a mechanism for achieving it in practice. The MST framework makes this boundary explicit: the Coasean regime is the regime where the system coordinate can be internalized through bargaining, but the conditions for that regime are stringent.
  • Coase is the zero-transaction-cost boundary where bargaining can internalize the system coordinate.
  • The conditions for the Coasean regime are rarely met in practice.
  • The Coasean switch is a benchmark, not a practical policy tool.
  • MST makes the boundary conditions explicit.
Ostromian endogenous repair · ~2 min
The Ostromian switch is the endogenous repair that folds the system coordinate back into incentives through self-organized institutions. Ostrom (1990) demonstrated that communities can govern common-pool resources (shared resources like fisheries or forests that no one owns and many can deplete) through self-organized institutions without recourse to either the state or the market. The MST framework locates this as a third switch: the community designs institutions that make system welfare a private concern for each participant. The Ostromian switch does not require a planner (Pigou) or zero transaction costs (Coase). It requires the community to have the capacity to design and enforce rules that align private incentives with system welfare. The failure mode of the Ostromian switch is institutional erosion: the rules that align incentives can be captured or eroded over time, restoring the Hollow Win. The MST framework predicts that Ostromian switches are most effective in small, stable communities with high social capital and clear boundaries.
  • Ostrom is the endogenous repair that folds the system coordinate back into incentives.
  • It does not require a planner or zero transaction costs.
  • The failure mode is institutional erosion over time.
  • Ostromian switches are most effective in small, stable communities.
The bridge to standard game theory · ~2 min
The MST program does not reject standard game theory. It locates it as a special case. Nash equilibrium, the core, the Shapley value, and other solution concepts are valid within their domain, the domain where the system coordinate is projected away. The problem is not that these solution concepts are wrong. The problem is that they are applied in contexts where the projection is not safe. The bridge between MST and standard game theory is the projection πu: the map that drops the system coordinate from the augmented payoff space. When πu is applied, the Hollow Win becomes a Nash equilibrium. When πu is not applied, the Hollow Win is revealed as a system-degrading outcome. The practical implication is that every application of standard game theory should be preceded by a question: is the system coordinate safely projected away, or is it hiding a Hollow Win?
  • MST does not reject standard game theory; it locates it as a special case.
  • The projection πu drops the system coordinate.
  • Under πu, the Hollow Win becomes a Nash equilibrium.
  • Every application should check whether the projection is safe.
The welfare theorems and the missing coordinate · ~2 min
The first and second welfare theorems are complete-markets boundaries on the MST board. The first welfare theorem states that every competitive equilibrium (a market state where prices balance supply and demand and no participant can do better) is Pareto efficient19. The second welfare theorem states that every Pareto efficient allocation can be achieved as a competitive equilibrium with appropriate lump-sum transfers (one-time wealth reallocations that shift who has what without distorting incentives)20. The MST framework shows that these theorems operate in the space where the system coordinate is projected away. Pareto efficiency is defined over private payoffs only. A Hollow Win is Pareto efficient, both parties gain, but system welfare is degraded. The welfare theorems do not guarantee system welfare. They guarantee private efficiency. The MST framework does not reject the welfare theorems. It shows their domain: they are valid when the system coordinate is safely projected away. When the system coordinate matters, the welfare theorems are incomplete.21

The eight-outcome map

OutcomeCABDescriptionStandard analysis sees
WIN, WIN, WIN111All three dimensions positive. The only outcome with no structural tension.Cooperation. Mutual gain. Pareto optimal.
HOLLOW WIN011Parties gain while the system degrades. Private optimum and systemic preservation are incompatible.Mutual gain. Win-win. Pareto efficient. Recommend acceptance.
CORROSIVE WIN-LOSE010A gains at the expense of both B and the system. Extractive and system-degrading.Competitive advantage. Market power. Dominance.
CORROSIVE LOSE-WIN001B gains at the expense of both A and the system. Mirror of Corrosive Win-Lose.Favorable terms. Negotiating leverage.
MISERY000Universal loss across all dimensions. System, A, and B all degraded.Market failure. Lose-lose. Impasse.
STABLE MISERY100System persists but both parties lose within it. Stability prevents resolution.Stalemate. Deadlock. Structural impasse.
SUSTAINABLE WIN-LOSE110B loses for the benefit of the system and A. Asymmetric sacrifice.Enforcement. Restitution.
SUSTAINABLE LOSE-WIN101A loses for the benefit of the system and B. Asymmetric sacrifice.Unilateral restraint. Concession.

Three switches on MST: Pigou, Coase, Ostrom

SwitchMechanismKey assumptionFailure mode
Pigou (imposed price)Planner imposes system price λW exogenously.Planner is uncaptured and can observe system welfare.Capture: when capture intensity φ exceeds threshold φ*, the planner sets an ineffective price.
Coase (tradable boundary)Private bargaining internalizes the externality at zero transaction cost.Property rights are well-defined and transaction costs are zero.Non-zero transaction costs or poorly defined property rights prevent bargaining.
Ostrom (endogenous repair)Community designs self-governed institutions that align private incentives with system welfare.Community has capacity to design and enforce rules.Institutional erosion: rules are captured or eroded over time.

Standard frameworks and their domain on the MST board

FrameworkDomain on MST boardWhat it cannot see
Nash equilibriumSpecial case under projection πu (system coordinate absent).Hollow Win: (0,1,1) becomes a Nash outcome under the projection.
CAPMPricing face of the same restriction Nash occupies on the strategic face.Systematic risk that is not priced because the system coordinate is excluded.[^22]
First welfare theoremComplete-markets boundary: every competitive equilibrium is Pareto efficient in private payoffs.System welfare degradation that is Pareto efficient in private payoffs.
Second welfare theoremComplete-markets boundary: every Pareto efficient allocation can be achieved as a competitive equilibrium.System welfare is not guaranteed by Pareto efficiency.
Pigovian taxationImposed-price switch on MST.Capture failure when φ > φ*.
Coasean bargainingZero-transaction-cost boundary on MST.Non-zero transaction costs and poorly defined property rights.
Ostromian governanceEndogenous repair switch on MST.Institutional erosion over time.
APPLIED EXERCISE

Identifying Hollow Wins in standard frameworks

~2 min
Select a real or hypothetical strategic interaction that is typically analyzed using a standard two-dimensional framework (Nash bargaining, Rubinstein alternating offers, or a simple 2x2 payoff matrix). Your task is to: (1) Describe the interaction in terms of the standard 2x2 grid: what are the private payoffs for A and B? (2) Identify the system that both parties depend on. What is the system welfare W? (3) Construct the eight-outcome map for this interaction. What is the value of C? (4) Determine whether the standard analysis would recommend the interaction as a mutual gain. (5) Determine whether the interaction is actually a Hollow Win (0,1,1) or a genuine WIN, WIN, WIN (1,1,1). (6) Explain what the standard analysis misses and why. (7) Identify which of the three switches (Pigou, Coase, Ostrom) might be applicable to this interaction and what the failure mode would be.
Answer key
  1. A clear description of the interaction in standard 2x2 terms.
  2. A clear identification of the system welfare W and the binary indicator C.
  3. A correctly constructed eight-outcome map with all three coordinates.
  4. A correct determination of whether the interaction is a Hollow Win or genuine WIN, WIN, WIN.
  5. A clear explanation of what the standard analysis misses.
  6. A plausible identification of the applicable switch and its failure mode.
READING PATH
  1. This is the foundational source for the chapter. It establishes the structural claim that the system coordinate W is absent by construction from any player-indexed payoff vector. It introduces the Hollow Win (0,1,1) and the eight-outcome map.
    Extract the definition of the Hollow Win, the eight-outcome map, and the structural claim that C cannot be computed from A's and B's payoffs.
  2. This source bridges MST to standard game theory. It proves that Nash equilibrium is the special case of MST under the projection πu. It shows that the Hollow Win becomes a Nash outcome under the projection.
    Extract the definition of the projection πu and the claim that Nash equilibrium is a special case of MST. Understand why solution-concept validity is distinct from welfare completeness.
  3. This source unifies the three classical traditions of externality economics as distinct operations on the same structural defect. It introduces the capture theorem and the failure modes of each switch.
    Extract the three switches, their mechanisms, key assumptions, and failure modes. Understand why Pigou, Coase, and Ostrom are not rival theories but alternatives to the same problem.
CHAPTER SYNTHESIS
QUESTION
What is the difference between the standard 2x2 grid and the eight-outcome map?
ANSWER
The standard 2x2 grid monitors only private payoffs A and B. The eight-outcome map adds the system coordinate C, producing eight outcomes instead of four. The critical distinction is between WIN, WIN, WIN (1,1,1) and HOLLOW WIN (0,1,1), which are the same cell in the standard grid.
QUESTION
Why can C not be computed from A's and B's payoffs?
ANSWER
C is not a function of A's and B's payoffs. The same private payoff vector can correspond to different system welfare states depending on context. The exclusion of C is a property of how the payoff space is built, not an oversight to be patched case by case.
QUESTION
What is the Hollow Win and why is it invisible to standard analysis?
ANSWER
The Hollow Win is (C, A, B) = (0, 1, 1): both parties gain privately while the system they depend on degrades. Standard analysis monitors only A and B, so it sees mutual gain and recommends acceptance. It cannot see that the system is being degraded.
QUESTION
How is Nash equilibrium a special case of MST?
ANSWER
Nash equilibrium is the special case of MST restricted to the regime in which the system coordinate is absent (the projection πu). The Hollow Win becomes a Nash outcome under this projection. This distinguishes solution-concept validity from welfare completeness.
QUESTION
What are the three switches on MST and what are their failure modes?
ANSWER
Pigou (imposed price) fails when capture intensity φ exceeds threshold φ*. Coase (tradable boundary) fails when transaction costs are non-zero or property rights are poorly defined. Ostrom (endogenous repair) fails when institutions erode over time.
QUESTION
What does the MST framework change for a regulator?
ANSWER
A regulator can no longer assume that a privately efficient deal is system-preserving. The regulator must ask whether the deal is a Hollow Win. The regulator must also consider which switch (Pigou, Coase, Ostrom) is applicable and what the failure mode would be.
QUESTION
What is the projection πu?
ANSWER
πu is the map that drops the system coordinate C from the augmented payoff space (C, A, B) to the projected space (A, B). Under this projection, the Hollow Win (0,1,1) becomes a Nash equilibrium (1,1).
QUESTION
What is the relationship between the welfare theorems and MST?
ANSWER
The welfare theorems are complete-markets boundaries on the MST board. They guarantee Pareto efficiency in private payoffs but do not guarantee system welfare. A Hollow Win is Pareto efficient but system-degrading.
SOURCE
Missing System Theory
SOURCE
nash-as-special-case-of-mst
SOURCE
pigou-coase-ostrom-three-switches
NOTES & REFERENCES
  1. John F. Nash, "The Bargaining Problem," Econometrica 18, no. 2 (1950): 155–162. link. ↩
  2. Ehud Kalai and Meir Smorodinsky, "Other Solutions to Nash's Bargaining Problem," Econometrica 43, no. 3 (1975): 513–518. link. ↩
  3. Ariel Rubinstein, "Perfect Equilibrium in a Bargaining Model," Econometrica 50, no. 1 (1982): 97–109. link. ↩
  4. Lloyd S. Shapley, "A Value for n-Person Games," in Contributions to the Theory of Games, Volume II, ed. H. W. Kuhn and A. W. Tucker (Princeton: Princeton University Press, 1953), 307–317. link. ↩
  5. Donald B. Gillies, "Solutions to General Non-Zero-Sum Games," in Contributions to the Theory of Games, Volume IV, ed. A. W. Tucker and R. D. Luce (Princeton: Princeton University Press, 1959), 47–85. link. ↩
  6. David Schmeidler, "The Nucleolus of a Characteristic Function Game," SIAM Journal on Applied Mathematics 17, no. 6 (1969): 1163–1170. link. ↩
  7. John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior (Princeton: Princeton University Press, 1944). link. ↩
  8. The Missing System Theory (program paper). summary. ↩
  9. Garrett Hardin, "The Tragedy of the Commons," Science 162, no. 3859 (1968): 1243–1248. link. ↩
  10. Ronald H. Coase, "The Problem of Social Cost," Journal of Law and Economics 3 (1960): 1–44. link. ↩
  11. The Hollow Win (program paper). summary. ↩
  12. John Nash, "Non-Cooperative Games," Annals of Mathematics 54, no. 2 (1951): 286–295. link. ↩
  13. Nash Equilibrium as a Special Case of the Missing System Theory (program paper). summary. ↩
  14. William F. Sharpe, "Capital Asset Prices," Journal of Finance 19, no. 3 (1964): 425–442; John Lintner, "The Valuation of Risk Assets," Review of Economics and Statistics 47, no. 1 (1965): 13–37. link. ↩
  15. A. C. Pigou, The Economics of Welfare (London: Macmillan, 1920). link. ↩
  16. Elinor Ostrom, Governing the Commons (Cambridge: Cambridge University Press, 1990). link. ↩
  17. *Pigou, Coase, and Ostrom as Three Switches on MST∗ (program paper). summary. ↩
  18. George J. Stigler, "The Theory of Economic Regulation," Bell Journal of Economics and Management Science 2, no. 1 (1971): 3–21. link. ↩
  19. Kenneth J. Arrow, "An Extension of the Basic Theorems of Classical Welfare Economics," in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability (Berkeley: University of California Press, 1951), 507–532. link. ↩
  20. Kenneth J. Arrow and Gérard Debreu, "Existence of an Equilibrium for a Competitive Economy," Econometrica 22, no. 3 (1954): 265–290. link. ↩
  21. *The Welfare Theorems as a Boundary of the Missing System framework∗ (program paper). summary. ↩
  22. CAPM as a Special Case of SAPM (program paper). summary. ↩
DIAGRAM NOTES
These notes describe diagrams planned for this chapter. The diagrams are not published yet.
DIAGRAM NOTE
The eight-outcome map: 2x2x2 grid
three-dimensional grid visualization
Show the expansion from the standard 2x2 grid (A, B) to the 2x2x2 grid (C, A, B). The standard grid has four cells. The expanded grid has eight cells. The critical distinction is between (1,1,1) and (0,1,1), which are the same cell in the standard grid.
DIAGRAM INPUTS
C coordinate (0 or 1)
A coordinate (0 or 1)
B coordinate (0 or 1)
Outcome labels for each of the eight combinations
READER CAPTION
The eight-outcome map adds the system coordinate C to the standard 2x2 grid. The standard grid cannot distinguish WIN, WIN, WIN (1,1,1) from HOLLOW WIN (0,1,1). The expanded grid makes this distinction visible.
TEXT FALLBACK
See the eight-outcome map table in this chapter's supporting data.
Missing System Theory
DIAGRAM NOTE
The projection πu: from MST to Nash
two-column causal diagram
Show how the projection πu drops the system coordinate C from the augmented payoff space, turning the Hollow Win (0,1,1) into a Nash equilibrium (1,1) in the projected space.
DIAGRAM INPUTS
Augmented payoff space (C, A, B)
Projection πu
Projected payoff space (A, B)
Hollow Win (0,1,1) in augmented space
Nash equilibrium (1,1) in projected space
READER CAPTION
The projection πu drops the system coordinate C from the augmented payoff space. Under this projection, the Hollow Win (0,1,1) becomes a Nash equilibrium (1,1). The problem is not with Nash equilibrium but with the space in which it is applied.
TEXT FALLBACK
See the standard frameworks domain table in this chapter's supporting data.
nash-as-special-case-of-mst
DIAGRAM NOTE
Three switches on the same structural defect
three-panel diagram
Show that Pigou, Coase, and Ostrom are three distinct operations on the same structural defect: the exclusion of system welfare from the payoff space. Each panel shows the defect, the switch mechanism, and the failure mode.
DIAGRAM INPUTS
Structural defect: system coordinate W excluded from payoff space
Pigou switch: planner imposes λW
Coase switch: bargaining at zero transaction cost
Ostrom switch: self-governed institutions
Failure modes: capture, non-zero transaction costs, institutional erosion
READER CAPTION
Pigou, Coase, and Ostrom are not rival theories. They are three switches on the same structural defect: the exclusion of system welfare from the payoff space. Each switch has a domain of application and a failure mode.
TEXT FALLBACK
See the three switches comparison table in this chapter's supporting data.
pigou-coase-ostrom-three-switches
WHAT TO DO NEXT
Restate the chapter claim. For policy triage, open Policy Lab; for measurement, open Domain Tables.
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© 2026 Erik Postnieks · Independent Researcher · Salt Lake City