Equilibria, Free Riders, and Governing the Commons

Three problems are constantly fused. They should not be.

1. Prisoner’s dilemma (PD). Two (or n) players each do better by defecting whatever the other does; both defecting is worse for both than both cooperating. Nash equilibrium is (Defect, Defect). Observed human play often includes cooperation.
2. Free rider. The public-goods form of the same structure: the individual benefits from the good whether or not she contributes; contribution is costly; rational non-contribution under-provides the good (Olson 1965).
3. Commons. A subtractable, hard-to-exclude resource (a common-pool resource, CPR). Hardin’s “tragedy” (1968) treats it as a PD that only private property or the state can solve. Ostrom (1990) shows that users often govern CPRs themselves, under design conditions Hardin did not model.

A fourth piece, which is why this paper exists as its own axis: Nash is not the only equilibrium concept, and it is a poor predictor in games where people cooperate. Halpern and Rong (2010; 2014) define perfect cooperative equilibrium (PCE): each player does at least as well as she would if the other were best-responding. In PD, both (C,C) and (D,D) are PCE; (C,C) is the unique max-PCE. Nash assumes best-response. PCE does not. That is the formal fact this program needs, without pretending it writes a constitution.

A fifth piece, which the first writing of this paper under-spent: PD is not the only game, and choosing the form is already a picture of agents under scarcity. Two apples, or an orchard. Rousseau’s stag hunt (assurance) is the orchard: the prize pays if both show up; a hare is the sure small prize. Maynard Smith and Price’s hawk–dove is contest over a resource; pure hawk is ruin. Nowak’s five rules, Skyrms on the social contract, Axelrod’s tournaments, Fehr and Gächter’s punishment, Henrich’s ultimatum games: game theory has more to say about cooperation and competition than one matrix. Mutual aid is why the economic models borrowed a biology that was not closed. This paper is why the game you write down is the same fight in formal clothes.

This paper is the apparatus. Game of Empires uses the interstate PD (security dilemma; imperial preference as a commitment to best-response) and, more often, hawk–dove over a named mix. Center libertarianism uses the domestic CPR and free-rider problems (productive commons at scale; assemblies; no listed stocks) and, at the join, a stag hunt with monitoring. Do not collapse those uses. A solution concept is not a polity. A game-form is not nature.

1. Nash, and what it assumes

Nash (1950, 1951): a strategy profile is an equilibrium if no player can raise her payoff by a unilateral deviation. The solution concept assumes that players make best responses. It does not claim that humans always do.

In one-shot PD the unique NE is mutual defection. In finitely repeated PD with a known end, backward induction yields the same (assuming common knowledge of rationality). In indefinitely repeated PD, folk theorems: almost any feasible payoff can be NE if players are patient enough (Fudenberg and Maskin 1986). Axelrod (1984) showed that tit-for-tat and similar reciprocal strategies can survive in tournaments. Reciprocity is not magic; it is a strategy that makes (C,C) incentive-compatible in the repeated game with the right continuation values.

Two mistakes follow if this is all one has:

• Treat one-shot NE as a law of social life (Hardin’s popularization; a certain right-libertarian “competition is natural” — Foundations of the Right dates that common sense; this paper dates the parable).
• Treat the folk theorem as a proof that cooperation will happen (it shows cooperation can be NE, not that it will).

Halpern and Rong start from a different observation: people sometimes cooperate even when it is not a best response — including in one-shot PD in the lab and in life. Are there rules for that play?

2. Perfect cooperative equilibrium (Halpern and Rong)

AAMAS 2010; fuller treatment arXiv:1412.6722 (Rong and Halpern 2014). Definitions, kept tight:

Best-response utility guarantee (two-player). BUGᵢ: the best payoff i can obtain if the other player best-responds. In PD with their 3 / 5 / 0 / 1 numbers, if you play C the other best-responds D and you get 0; if you play D the other best-responds D and you get 1; so BUG = 1 for each.

PCE (two-player). A profile s is a PCE if for every player i, Uᵢ(s) ≥ BUGᵢ. Each player does at least as well as she would if the other were best-responding.

PD. Both (C,C) and (D,D) are PCE. For (C,C): the cooperator gets 3, and 3 ≥ 1. For (D,D): the Nash payoff already meets the test. The unique NE is (D,D). (C,C) is not NE and is PCE.

α-PCE. Uᵢ(s) ≥ α + BUGᵢ. PCE is 0-PCE. Negative α allows “almost” cooperation.

M-PCE (max-PCE). An α-PCE for the largest α that the game admits. Every game has a Pareto-optimal M-PCE (their Theorem 12). In PD, (C,C) is the unique M-PCE — a 2-PCE on their numbers. In the Nash demand/bargaining game, no PCE exists; (50,50) is the unique M-PCE — a fairness residue, not a slogan.

CE (cooperative equilibrium, two-player). A further concept that builds in punishment: a deviation is either not profitable against best responses, or can be punished. Every Pareto-optimal M-PCE is a CE (their Theorem 13).

Properties they prove, that this program actually uses:

• A player’s PCE payoff is at least her NE payoff.
• Every dominant-strategy profile is a PCE.
• A profile that Pareto-dominates a PCE is a PCE; so if a PCE exists, a Pareto-optimal PCE exists.
• PCE and NE are incomparable: (C,C) in PD is PCE not NE; some NE are not PCE (Traveler’s Dilemma is their example).
• PCE does not always exist; M-PCE does.
• Two-player PCE and M-PCE are polynomial-time computable.

What this does not say. It does not say states will play M-PCE. It does not refute Olson. It does not replace Ostrom’s design principles with a formula. It says: the assumption that players always best-respond is an extra axiom, not a theorem about people, and there is a well-defined alternative under which (C,C) in PD is an equilibrium.

The notice, for this program: right-libertarian common sense talks as if Nash-on-PD were nature. A certain left common sense talks as if cooperation were automatic. Both skip the solution-concept choice. Imperial preference (Game of Empires) is a commitment to best-response on expansion. Domestic commons (Center) is the question whether occupants can play something closer to M-PCE, with monitoring, than a one-shot PD.

3. Free rider and Olson

Olson, The Logic of Collective Action (1965): in a latent group, the public good is non-excludable; the individual’s contribution does not decide provision; rational members free-ride. Small groups can sometimes provide through selective incentives, social sanction, or because a single member values the good enough to provide it (the “privileged group”). Large groups need organization.

This is not Hardin’s pasture. It is the public-goods PD. Productive plant at scale, national defense, agenda-setting in an all-citizen senate, and climate are free-rider problems. So is “who monitors the threshold rule without a ministry.”

Selective incentives can be benign (Ostrom’s graduated sanctions, membership goods) or extractive (the party that owns the revolution; the firm that owns the residual). Method is not free of this. Seizing the office is, among other things, a way to solve free-riding by coercion. Agorist exit has the opposite problem: who contributes to the counter-institution when defection is cheap.

4. Hardin, then Ostrom

Hardin, “The Tragedy of the Commons,” Science (1968): a pasture open to all; each herder adds animals; ruin. The popular remedy-pair: enclose (private property) or Leviathan. Hardin’s article is a parable, not a field study. He assumes open access, no communication, no rules, no monitoring.

Ostrom, Governing the Commons (1990); Nobel 2009. CPRs (irrigation, forests, fisheries, alpine meadows) are subtractable and hard to exclude, but they are not “open to all” in the successful cases. Users often craft rules. The eight design principles (Ostrom 1990; restated Cox, Arnold, and Villamayor Tomás 2010):

1. Clearly defined boundaries (who is a user; what is the resource).
2. Congruence of rules with local conditions (and of provision with appropriation).
3. Collective-choice arrangements: affected users can modify rules.
4. Monitoring, by users or accountable to them.
5. Graduated sanctions.
6. Accessible, low-cost conflict resolution.
7. Minimal recognition of the right to organize by external authorities.
8. Nested enterprises for larger systems (polycentricity).

Three-quarters of later cases Cox et al. surveyed show strong or moderate support. Failures cluster where boundaries are fuzzy, monitors are not users, sanctions are binary (none or exile), or a state overrules local rules.

Principle 7 is the hinge to Game of Empires. A pole that does not recognize the right to organize can wreck a working commons from outside. Ostrom is not a theory of interstate anarchy. Treating her as a substitute for polarity is a category error.

Principle 8 is the hinge to nested assemblies. Local commons at the base, federated coordination above — not a single Gosplan, not a single ticker. The positive institutional problem of the property triad (productive title at hired-labor scale) is a CPR problem: who is in, who monitors, what the graduated sanction is, who may not overrule. Those are open problems 8, 9, 12, 13. Naming Ostrom does not finish them. It names the literature that already refused Hardin’s dichotomy.

5. The orchard is a stag hunt

Rousseau, Discours sur l’origine et les fondements de l’inégalité (1755): the stag hunt. Two hunters; a stag if both hunt stag; a hare if one defects to the sure small prize. Two Nash equilibria: both stag (payoff-dominant), both hare (risk-dominant). The problem is assurance, not a dominant strategy to defect.

Skyrms, The Stag Hunt and the Evolution of the Social Contract (2004); earlier Evolution of the Social Contract (1996): this, not PD, is the primitive of the contract. Cooperation can be an equilibrium. The question is how a population gets there, and stays. Signaling, correlation, structure. Not a slogan.

Two apples, or an orchard. Fighting over two apples as if the pie were fixed is PD, or hawk–dove. Planting an orchard is a stag hunt: the prize (more apples, later) pays if both plant; grabbing an apple and leaving is the hare. Scarcity is the moment. Cooperation can change the next moment’s environment. Mutual aid is why that move is in nature, not only in a seminar. Society is the existence proof. A garden is the existence proof in miniature.

CenLib’s productive commons at hired-labor scale is this game with Ostrom’s monitoring, not a one-shot PD that only a ministry or a ticker can solve. Hardin wrote PD onto the pasture by deleting communication. Write stag hunt back, and the orchard is an equilibrium if the occupants can assure one another. That “if” is open problems 8, 9, 23 — not a landing.

Sen’s isolation paradox (1967) is the public-goods cousin of assurance: each would contribute if others will. Lewis (1969) on convention, after Hume: equilibria that hold because they are expected to. Schelling (1960) on focal points: some orchards get planted because the meeting-place is obvious. None of this writes a constitution. All of it is apparatus the first draft of this paper left on the table.

6. Hawk–dove, ESS, and struggle as a strategy

Maynard Smith and Price, “The Logic of Animal Conflict” (1973); Maynard Smith, Evolution and the Theory of Games (1982). Contest a resource. Hawk fights; dove displays. If the cost of injury exceeds the prize, pure hawk is not an evolutionarily stable strategy. ESS may be a mix, or dove against hawk. Chicken is the same matrix with different names.

Fascism as hawk at every level. Race, nation, person as organ. Struggle as law. That is not “applied PD.” It is a commitment to hawk when the prize is the Volk and the cost is other people. Pure hawk at all levels is ruin for someone; historically, for millions. The ecological reading of the map puts fascism at the struggle pole for this reason. Mutual aid dates the Huxley costume. This section dates the game.

Interstate polarity is often hawk–dove, not PD. Game of Empires: a pole committed to taking and keeping a named mix. Unipolar knockout is hawk that won a round. Multipolar default is mixed ESS, or repeated contest. Bipolar freeze is two hawks who have not yet paid the cost. Imperial preference is a commitment to hawk on expansion. PCE/M-PCE remain the existence proof that (C,C) is a solution concept. Poles are not using it. Principle 7 remains the wrecking condition.

Mutual aid as dove is not a mood. It can be ESS against hawk, or a mix. Kropotkin’s “as much a law as mutual struggle” is, in this formalism, the refusal of pure hawk as nature. It is not a proof of anarchist communism.

7. Nowak’s five rules, and what they do not pick

Nowak, “Five Rules for the Evolution of Cooperation,” Science (2006); Nowak and Highfield, SuperCooperators (2011). Mechanisms, not a property regime:

1. Kin selection (Hamilton 1964).
2. Direct reciprocity (Trivers 1971; Axelrod 1984).
3. Indirect reciprocity (reputation).
4. Network reciprocity (structure; not a well-mixed soup).
5. Group selection (the live fight: Sober and Wilson 1998; Okasha 2006 on levels of selection).

Bowles and Gintis, A Cooperative Species (2011): humans as ultrasocial, with altruistic punishment. Fehr and Gächter (2000): punishment sustains contribution in public-goods games. Henrich et al. (2001): ultimatum rejections vary by society; “rational defection” is not a human constant.

What this program uses. The cooperative side has a literature. Economic models that skip it are wearing Huxley. Game theory is the shared language of that literature and of the competitive side. It does not pick CenLib. It does not pick Kropotkin’s regime. It does pick against “PD is nature.” Choosing the game is the political act Mutual aid named as an unclosed biology. Open problem 41: what would count as enough of this literature to constrain a title-regime, without letting Nowak write the triad.

8. What each neighboring paper is allowed to take

Empires. Interstate PD / security dilemma, and more often hawk–dove over the mix. Imperial preference as a decision to play hawk (or Nash) on expansion. Unipolarity as a knockout, not a solved PD. AGI racing as the current cargo. PCE/M-PCE as the existence proof that (C,C) is a solution concept — and the diagnosis that poles are not using it. Principle 7 as the external wrecking condition. Inner conflict on that board is a factional PD (who defects first in a polarization); outer is this paper’s interstate game. Named hegemonies are the cargo of the outer game; energy (Follow the Oil) is one. OPEC+ is a cartel across tents, not M-PCE.

Center. Productive commons as CPR, not as “abolish property.” Occupant use-shares as an attempt at clearly defined users (principle 1). Nested assemblies as principle 8. The orchard as a stag hunt with monitoring, not a one-shot PD. Free-riding on civic time and on monitoring as Olson’s latent-group problem — hence allotted, paid service, not per-vote piecework. No listed stocks: absentee title is a way of being a user who does not monitor (violates 1 and 4). Automation without hiring: the CPR may have no human users left; Ostrom does not speak to that. Open problem 20. Hardin is the right’s preferred commons cartoon; Foundations of the Right dates that preference. Public choice and Coase sit in that file as the right’s government-failure and assignment tools — taken as constraints, not as a proof of residual-claimant innocence.

Mutual aid. Kropotkin is ecology, not game theory. Trivers is the neo-Darwinian reciprocity model; Axelrod is its tournament cousin; Nowak is the five-rule summary; Skyrms is the stag-hunt social contract. Do not cite Halpern as biology, or Kropotkin as PCE. The spectrum on the map is the same seats read as ecology; this paper is the games those seats spend.

Agency. Solution concepts are about strategy profiles, not about whether the agent “could have done otherwise.” Do not smuggle PCE into van Inwagen. Do not smuggle ESS into free will.

The map. Office × property is the hourglass. Ecology is the spectrum. Game-form is not a third figure. It is why the spectrum was drawable: struggle as hawk, orchard as stag, mixed as a mix, the party-state as commanded stag, fascism as hawk at all levels.

9. Pressure tests

• PCE existence. Not guaranteed; M-PCE is. A constitution cannot assume a PCE.
• n-player. Halpern–Rong generalize BUG by replacing best-response with NE in subgames. Interstate systems are n-player. The two-player intuition is a teaching aid, not the model of 1914 or 2026.
• Cheap talk. PCE does not by itself solve cheap talk or renegotiation. Monitoring (Ostrom 4) is still required. Stag hunt without assurance collapses to hare.
• Fairness. M-PCE’s 50/50 in Nash bargaining is a property of that game, not a distributive axiom this program has adopted.
• Lab versus poles. Undergraduate PD is not the Politburo. The transfer to great-power play is the weakest joint in the paper. State it.
• ESS is not a title-regime. Maynard Smith does not write Locke. Nowak does not write Proudhon. Group selection does not write Gosplan.
• The orchard can fail. Planting is a stag hunt, not a guarantee. Drought, defection, a pole that refuses principle 7. Existence proofs (society, garden) are not a constitution.

10. What this paper does not fake

It does not claim that switching the solution concept ends the AGI arms race. It does not claim that Ostrom’s irrigation communities scale to planetary model-weights. It does not claim that free-riding is solved by “community.” It does not treat (C,C) as mandatory. It does not treat stag hunt as the only true game of humans. It does claim that a research program which talks about cooperation, property, and polarity without PD, Olson, Ostrom, stag hunt, hawk–dove, and an explicit equilibrium concept is talking past the formal problems it has already named. Game theory has more to say than the first draft spent. The biology has not closed which game is nature. That is the joint with Mutual aid, and it is why the economic models remain irreconcilable without a costume.