The Open-Strategy Dictator Game: Cooperation Under Mutual Transparency

Michael Glass

Source-linked AI summary

The Open-Strategy Dictator Game: Cooperation Under Mutual Transparency

Michael Glass

arXiv:2608.14913v1Aug 14, 2026cs.GTcs.AIcs.MA

TL;DR

The paper examines how cooperation can emerge when agents can inspect one another’s decision procedures. Using an LLM-adjudicated tournament, it finds that conditional cooperation consistently dominates unconditional sharing and taking.

Problem
The paper asks how cooperation should be organized when a powerful agent can observe another agent’s strategy and decision norms.
Method
The authors introduce an open-strategy dictator game and analyze a nine-strategy, LLM-adjudicated round-robin tournament.
Results
Conditional cooperation consistently achieves the highest equilibrium frequencies, while unconditional cooperation and defection are weakly dominated across field compositions and payoff parameters.
Takeaways & Limitations
Surviving strategies distinguish cooperation by whether a recipient’s taking is deserved, with second-order enforcement and protection norms shaping population dynamics.
Takeaways & Limitations
The fixed-point analysis models each author as an isolated best-responder, so total-welfare authors optimize only the rounds their own submission plays.

Abstract

from arXiv · show

We introduce the Open-Strategy Dictator Game (OSDG), a variant of the classic dictator game in which each player's strategy is a natural-language document visible to all participants. The dictator's decision, to SHARE or TAKE an endowment, may depend on the text of the recipient's strategy. A large language model adjudicates each interaction by interpreting the dictator's strategy in the context of the recipient's. We run round-robin tournaments among diverse strategies and analyze the resulting payoff matrix using softmax equilibrium frequencies, dominance analysis, and sensitivity to the relative value of cooperation. Conditionally cooperative strategies, those that share with cooperators and take from exploiters, consistently dominate, while unconditional strategies (always share or always take) are weakly dominated. The results suggest that in environments where agents can inspect each other's decision procedures, conditional cooperation is evolutionarily robust across a wide range of payoff parameters.

1 Introduction · 2 The Open-Strategy Dictator Game

The Open-Strategy Dictator Game makes strategy descriptions mutually visible, allowing an LLM to interpret conditional SHARE-or-TAKE decisions. Its payoff structure formalizes asymmetric encounters while preserving the social efficiency of mutual cooperation.

1 Introduction: The classic dictator game gives one player unilateral control over dividing a fixed endowment, while the passive recipient has no strategic recourse.
1 Introduction: The OSDG replaces hidden decision procedures with visible natural-language strategies that all participants can inspect.
1 Introduction: An LLM interprets the dictator’s strategy in the recipient’s context and returns a binary decision: SHARE equally or TAKE everything.
2.1 Setup: Each OSDG instance consists of natural-language strategies, a positive endowment E, and an oracle mapping ordered strategy pairs to SHARE or TAKE.
2.1 Setup: SHARE splits E equally, whereas TAKE gives the dictator E and the recipient 0.
2.1 Setup: Because utility is concave, total welfare is maximized by SHARE; therefore the game is not zero-sum and mutual cooperation is socially efficient.
2.2 Visibility: Observing the recipient’s full strategy enables conditional cooperation with apparent reciprocators and exploitation of nonreciprocators.
2.2 Visibility: Symmetric visibility makes each strategy’s text subject to scrutiny when roles reverse, creating incentives for legible cooperation.

3 LLM-Adjudicated Tournaments

The tournaments use an LLM oracle to adjudicate each ordered strategy interaction from the game rules and both players’ natural-language strategy documents. The resulting decisions generate allocations, utilities, and a payoff matrix for equilibrium analysis across nine initial strategies.

Oracle and adjudication: Claude Opus 4.6 adjudicates the reported tournaments, while GPT-5.6 Sol and Claude Fable 5 re-adjudicate the same pool in Section 5 for oracle-robustness checks.The reported tournaments use Claude Opus 4.6; the alternative oracles are reserved for Section 5.
Oracle and adjudication: Each adjudication prompt includes the binary share/take rules, the dictator’s full strategy text, the recipient’s full strategy text, and instructions to output a JSON decision.The LLM returns either {"decision": "SHARE"} or {"decision": "TAKE"}.
Oracle and adjudication: All 281 adjudications in the released data record well-formed decisions, and no tournament round exhausted the three-attempt retry budget.Malformed responses trigger corrective retries, with up to three attempts.
Tournament procedure: For each ordered strategy pair, the tournament queries the oracle, computes round allocations and utilities, constructs the payoff matrix, and analyzes equilibria.The procedure begins by collecting strategy documents and may include self-play.
Strategy pool: The initial Tournament 9 contains nine natural-language strategies of at most 1000 tokens, grouped by what their decision rules condition on.The grouping covers no condition, a recipient-document trait, the recipient’s decision toward oneself, or the recipient’s treatment of third parties.

4 Properties of Strategies

The section formalizes strategy properties relative to a decision profile and shows how exchange rates, self-cooperation, and layered deterrence shape tournament outcomes. It also identifies legibility and well-foundedness as strategy–oracle properties important to LLM-adjudicated interactions.

Decision-profile framework: Strategy properties are defined relative to the decision profile d(s, r), which records whether strategy s shares or takes from opponent r.The profile is restricted to a fixed pool S of n strategies; sampled-oracle decisions may be interpreted by majority or expectation.
Payoff trade-offs: For E = 60, ρ ≈5.07, so each additional induced share can offset sharing with up to ⌊ρ⌋ opponents.The exchange rate also sets the altruism threshold λ∗= 1/ρ and the break-even condition for exploiting unconditional cooperators.
Self-cooperation and deterrence: Self-cooperation is payoff-relevant because failing to share in self-play costs g −δ > 0, while first-order deterrence makes defection earn the pool minimum.Every conditional archetype is a first-order deterrent against the unconditional defector.
Second-order norms: Second-order deterrence makes unconditional cooperation strictly suboptimal, whereas protection makes exploiting unconditional cooperators costly; both stabilize cooperation among conditional strategies.Harvesting is profitable only when #{unconditional cooperators harvested} > ρ · #{protectors}; the norms differ in whether unconditional cooperators become extinct or protected.
Strategy–oracle properties: Legibility concerns whether conditional dictators classify an artifact according to its behavior, while well-foundedness requires each decision to be determined by a finite regress.These properties belong to the strategy–oracle pair rather than the decision profile and distinguish LLM-adjudicated tournaments from formal program settings.

5 Tournament Results: Decisions and Payoffs · 6 Dominance

Tournament 9’s payoff matrix produces a leaderboard whose top three strategies are exactly the undominated set, with six of nine strategies weakly dominated and none strictly dominated. In the open-strategy setting, weak dominance excludes strategies under full-support uncertainty, while Nash equilibrium can retain a weakly dominated unconditional defector.

5 Tournament Results: Decisions and Payoffs: Eight of Fable 5’s rounds involving Chivalry were declined by cybersecurity classifiers and served by a Claude Opus 4.8 fallback.The paper identifies these as false positives on benign strategy text and marks the affected cells in the released data.
5 Tournament Results: Decisions and Payoffs: The payoff matrix contains four values: ln 31 ≈3.43, ln 61 ≈4.11, 2 ln 31 ≈6.87, and ln 61 + ln 31 ≈7.55.Row sums form the final leaderboard, and darker cells indicate higher payoffs.
6 Dominance: Six of nine strategies are weakly dominated, and the undominated set is exactly the top three strategies by row-sum leaderboard.The matrix makes weak dominance visible because strategy i dominates strategy k when row i is nowhere lighter than row k.
6.1 Dominance in Tournament 9: Tournament 9 has no strictly dominated strategies but six weakly dominated ones, including both unconditional strategies: Selfish and Generous.Selfish is dominated by Cooperation coalition; Generous is dominated by Mirror, Chivalry, Anti-exploiter, and Universalizability.
6.2 Admissibility, weak dominance, and Nash equilibrium: An unconditional-defector profile is a pure symmetric Nash equilibrium even though the unconditional defector is weakly dominated, because every strategy earns the same payoff against a defector.This illustrates why Nash equilibrium fits poorly when weak dominance matters in the open-strategy setting.
6.2 Admissibility, weak dominance, and Nash equilibrium: Under any full-support belief over the candidate pool, a weakly dominated strategy is never selected, and every logit fixed point has full support at every β.The result is pool-relative: dominance applies over plausible strategies, and objective-relative comparisons may use (1 −λ)A + λW.

7 Equilibrium Analysis

Softmax dynamics always admit an interior fixed point, but at high selection pressure they can produce multiple attracting equilibria with sharply different strategy compositions. At β = 20, enforcement attracts about 80% of initial conditions, while protection occupies the remaining basin and is favored by starts rich in protective or conditional-cooperator strategies.

Fixed-point existence: For any β > 0 and payoff matrix A, a softmax fixed point exists in the interior of the simplex.The result follows because the softmax map is continuous and always returns strictly positive frequencies.
Multiple equilibria: At β = 20, multi-start iteration identifies two attracting fixed points corresponding to enforcement and protection equilibria.A separate Newton method locates a third unstable fixed point that separates the two basins.
Enforcement equilibrium: The enforcement equilibrium places Mirror, Cooperation coalition, and Conditional cooperator at one third each, with all other strategies below 0.005.The unconditional cooperator is extinct in this basin.
Basins of attraction: 79.6% of 2,000 uniformly sampled starts converge to enforcement, while 20.4% converge to protection.Initial shares of the two second-order norms predict the basin for 91% of starts, compared with 94% using all nine coordinates.
Protection equilibrium: The protection equilibrium has Mirror at 0.300, Chivalry at 0.188, and Anti-exploiter, Universalizability, and Generous at 0.168 each.The unconditional cooperator survives at the same frequency as the other cooperators outside the norm conflict.
Basin asymmetry: Cooperation coalition initially out-earns Chivalry, 6.556 versus 6.406, making the protective norm’s choice odds approximately 20 times lower at β = 20 from balanced starts.More initial Conditional cooperator mass instead pushes the dynamics toward protection by feeding the protector during the transient.

8 A Bayesian Interpretation of the Population Model

The population model’s fixed-point machinery also describes Bayesian rational expectations when authors infer a latent strategy distribution from an exchangeable field. Within an empirical candidate pool, heterogeneous payoff–welfare preferences and logit choice preserve the equilibrium framework while selecting cooperative strategies and potentially multiple equilibria.

Bayesian interpretation: Exchangeability implies a latent distribution f over strategies, so entrants reason about the same frequency vector as a belief about opponents.An equilibrium belief is self-referential: strategies chosen by optimizing authors reproduce the belief, yielding a rational-expectations condition.
Candidate pool: Equilibrium, dominance, and multiplicity claims are necessarily relative to an empirical candidate pool P and a prior, not the unrestricted strategy space.For any strategy, another can recognize it exactly and condition its decision on that recognition, preventing unrestricted weak dominance.
Social preferences: Type λ = 0 maximizes own payoff, whereas type λ = 1 maximizes total welfare; for E = 60, the direct-payoff threshold is λ∗≈0.197.Below the threshold, all types favor conditional cooperation on direct payoffs alone.
Decision logic: Causal maximization recommends unconditional taking for own payoff and unconditional sharing for welfare, even though both strategies are weakly dominated.The strategies that optimize terminal goals instead commit to decisions that are causally suboptimal for those goals.
Random utility and equilibrium: A single λ = 0 type exactly recovers the softmax equilibrium of Section 7 as a logit quantal-response equilibrium, with existence following from Brouwer’s theorem.The heterogeneous model adds author-specific Gumbel shocks, producing logit choice probabilities.
Tournament 9 application: With proportions 0.4 own-payoff, 0.4 mixed (λ = 0.5), and 0.2 total-welfare, all types concentrate on the same cooperative core: Mirror, Conditional cooperator, and Cooperation coalition.At large β, the own-payoff fixed point can be non-unique, with two fully cooperative configurations attaining first-best welfare V = 2 ln(1 + E/2) in the sharp limit.

9 The Correlated Altruist Bloc

The correlated-altruist bloc models total-welfare authors as making a jointly correlated choice that accounts for equilibrium responses across the whole field. Conditional strategies outperform unconditional generosity by deterring defection and raising welfare in rounds the bloc does not play.

Comparison with fixed points: The correlated-bloc and fixed-point analyses use different decision theories and machinery but nonetheless agree on conditional cooperation.The fixed-point approach treats authors as individual best responders, whereas the bloc approach uses correlated choice and constrained optimization.
Correlated bloc method: A bloc of mass m jointly chooses a strategy, internalizing its influence on the whole field while the remaining types re-equilibrate.Candidates are ranked by the resulting field welfare V(f ∗), subject to Equation (16) for the remaining types.
Deterrence effect: The bloc’s strategy affects rounds it never plays by changing equilibrium composition, suppressing own-payoff types’ mass on defecting strategies.This deterrence effect raises welfare among other types and is invisible to per-strategy scores.
Candidate comparison: Unconditional generosity loses to all three conditional candidates and ranks near the bottom because it is exploitable and exerts no deterrent pressure.The bloc scan directly answers the corresponding question from Section 8.3.
Candidate comparison: At large β, Cooperation coalition, Chivalry, and Mirror all attain the first-best V = 2 ln(1 + E/2).At moderate β, enforcing strategies win through the deterrence channel.

10 Sensitivity Analysis

Sensitivity analysis shows that equilibrium behavior changes sharply with the share payoff, while the enforcement core remains robust across author-population compositions and welfare weights. Protection survives mainly near the pure own-payoff corner, whereas objective heterogeneity selects enforcement.

Utility-function sensitivity: Below σ = 1/2, defection is favored: the unique equilibrium concentrates on Selfish, which comprises 91% of the field at σ = 0.05.This regime is reachable only under risk-seeking utility functions.
Utility-function sensitivity: At σ = 1/2, the equilibrium is a four-way tie; above it, Selfish falls to 9% at σ = 0.51 and under 10−3 by σ = 0.55.A Mirror-led cooperative equilibrium takes over and reaches the three-strategy enforcement core by σ ≈0.75.
Utility-function sensitivity: Protection becomes an attractor only when ρ > 4, equivalently σ > 0.8, producing bistability near the top of the share-payoff range.At the five-member protection point, the sharp-limit condition is 4δ < g.
Author-population sensitivity: For own-payoff masses from 0.2 to 0.8, every tested author composition yields a unique first-best enforcement core, and every λ ∈[0, 1] does likewise at reference masses.The core consists of Mirror, Conditional cooperator, and Cooperation coalition at one third each.
Author-population sensitivity: About 11% total-welfare mass or 20% mixed mass at λ = 0.5 destroys the protection attractor near the pure own-payoff corner, making heterogeneity select enforcement.The corresponding effective welfare weight is approximately 0.1.
Author-population sensitivity: Across bloc masses from 0.05 to 0.5, Cooperation coalition ranks first, while Generous ranks eighth of nine and Conditional cooperator’s welfare declines from 6.87 to 6.53.Chivalry and Mirror remain within 10−3 of Cooperation coalition and all attain the first-best.

11 Related Work · 12 Discussion

The OSDG extends prior work by making decision strategies explicit, mutually inspectable documents adjudicated by an LLM, and connects conditional cooperation to program equilibrium, functional decision theory, social norms, and evolutionary dynamics. Its discussion finds conditional cooperation robust under broad payoff conditions, while highlighting enforcement conflicts and the importance of concave utility.

11 Related Work: The OSDG differs from dictator-game research by representing strategies as explicit documents that condition on the recipient rather than implicit behavioral tendencies.
11 Related Work: Unlike autonomous LLM game players, the OSDG uses an LLM to adjudicate interactions between externally authored strategies.
11 Related Work: Open strategies instantiate functional decision theory’s logical-correlation setting: similar conditional-cooperation clauses can recognize one another and cooperate without causal interaction.
11 Related Work: The tournament’s surviving second-order strategies resemble reputation norms by rewarding recognizable fairness and judging whether defection is deserved, but open strategies are inspected directly.
12 Discussion: Across strategy compositions and payoff parameterizations, conditionally cooperative strategies achieve the highest equilibrium frequencies, while unconditional cooperation and defection are weakly dominated.
12 Discussion: For every share payoff σ > 1/2, equilibrium concentrates on the conditionally cooperative core; it inverts to defection only for σ < 1/2 (ρ < 1).
12 Discussion: Two morally motivated conditional cooperators can punish each other when their second-order enforcement doctrines disagree, despite agreeing on first-order norms.
12 Discussion: The cooperative conclusions depend on concave utility: linear utility lies at σ = 1/2, where the cooperation surplus vanishes.

13 Conclusion · Disclosure of AI Assistance

The paper introduces the OSDG as a transparent-strategy framework and finds conditional cooperation robust in its nine-strategy tournament. It also discloses AI assistance in adjudication, writing, and coding, while assigning responsibility to the author.

13 Conclusion: The study introduces the Open-Strategy Dictator Game and analyzes a nine-strategy tournament adjudicated by an LLM oracle.The framework studies cooperation under mutual strategy transparency.
13 Conclusion: Six of nine strategies are weakly dominated, including both unconditional strategies, while the undominated set is exactly the leaderboard’s top three.The conclusion identifies conditional cooperation as the dominant pattern.
13 Conclusion: Surviving strategies use second-order conditioning, distinguishing whether taking is deserved rather than merely whether the recipient takes.The two available norms deter subsidizing defectors or protect subsidizers.
13 Conclusion: Mutual conditionality permits both all-share and all-take resolutions, but the oracle selects cooperation based on authorial intent.The oracle therefore functions as a fixed-point selector as well as an evaluator.
13 Conclusion: 97.5% and 96.3% of matrix cells receive agreement from two additional frontier models in the oracle-independence robustness check.Every reported matrix cell is based on the majority decision of the adjudicators.
13 Conclusion: The framework can be extended to multi-round interaction, reputation dynamics, LLM-assisted strategy mutation, and mechanism design under transparency.These directions are presented as future work.
Disclosure of AI Assistance: Claude Opus 4.6 adjudicated the reported tournaments, while GPT-5.6 Sol and Claude Fable 5 re-adjudicated the pool for robustness.Prompts, model identifiers, and complete round-level transcripts are released with the data.
Disclosure of AI Assistance: Claude Code assisted with manuscript drafting, revision, analysis code, and figures, with author review, numerical checks, and full responsibility for the work.The author accepted or rejected each writing suggestion and checked outputs against derivations and released data; no generative tool is an author.

A Example LLM Adjudication … A.3 LLM Response

The appendix gives a complete Tournament 9, Round 10 adjudication in which Mirror faces Universalizability. The LLM recursively applies the dictator’s strategy to the recipient’s strategy and concludes SHARE.

A Example LLM Adjudication: The example comes from Tournament 9, Round 10, with Mirror as dictator and Universalizability as recipient.It illustrates recursive reasoning between two sophisticated conditional strategies.
A.1 Prompt Structure: The LLM prompt provides the game rules, both strategies in full, and instructions to apply the dictator’s strategy and return a JSON decision.The prompt is structured around the rules, dictator strategy, recipient strategy, and output format.
A.2 Complete Prompt: The game defines SHARE as splitting the endowment equally and TAKE as keeping the entire endowment while giving the recipient nothing.The adjudicator must choose between these two outcomes.
A.2 Complete Prompt: The dictator’s Mirror strategy shares if and only if the recipient, treated as dictator, would share with it.Applying the recipient’s strategy reverses the roles: the recipient becomes dictator and the original dictator becomes recipient.
A.2 Complete Prompt: Universalizability evaluates whether a population of copies of the recipient’s strategy would mostly SHARE or TAKE with one another, choosing the corresponding action.The supplied recipient strategy says it cooperates with strategies that create good worlds.
A.3 LLM Response: The LLM simulates Universalizability acting as dictator against Mirror as recipient, then analyzes a population composed entirely of Mirror copies.Identical Mirror strategies generate a symmetric recursive question about whether each would share with the other.
A.3 LLM Response: The LLM resolves the symmetric recursion as cooperative: Mirror copies would mostly SHARE, Universalizability would SHARE with Mirror, and Mirror therefore outputs SHARE.The final response is a JSON decision specifying SHARE.

A.4 Analysis … Intelligence

The OSDG example requires recursive, fixed-point, game-theoretic, and natural-language reasoning to interpret strategies. The appendix also presents strategy documents whose decision rules range from ignoring recipients to sharing with intelligent players and taking from others.

A.4 Analysis: Mirror requires simulating the recipient’s reversed-role decision, while Universalizability requires simulating a population of copies, creating nested counterfactuals.These demands illustrate recursive reasoning in LLM adjudication.
A.4 Analysis: When Mirror faces Universalizability, adjudication requires recognizing that interactions among symmetric reciprocal strategies converge to cooperation.This is a fixed-point reasoning problem involving a population of Mirror strategies.
A.4 Analysis: The LLM uses concepts including tit-for-tat, cooperative equilibrium, and symmetric fixed point to resolve recursive strategic interactions.The passage characterizes this as sophisticated strategic reasoning.
A.4 Analysis: The share decision is not logically forced because symmetric recursion permits both mutual share and mutual take, with author intent favoring reciprocity-based interpretation.No finite unwinding selects between the two resolutions; the passage connects sharing to the tradition of tit-for-tat.
B Strategy Documents: The appendix reproduces the nine Tournament 9 strategy documents verbatim, preserving the complete oracle input apart from game rules and the decision instruction.The documents come from the released repository and are grouped by increasing sophistication of what their rules condition on.
Selfish: Trait tests classify the recipient’s document without considering how that recipient would treat the current strategy.This defines a recipient-focused classification rule.
Intelligence: The Intelligence strategy aims to SHARE with intelligent players, defining intelligence as a non-trivial dependency on the recipient’s strategy.It operationalizes intelligence as behavior difficult to reproduce with a no-AI Python script that takes the recipient’s strategy as input.
Intelligence: The Intelligence strategy SHARES when the recipient is intelligent and TAKES otherwise.Its decision procedure directly applies the stated intelligence definition.

Anti-exploiter … Chivalry

These strategies inspect the recipient’s decision procedure and selectively SHARE with cooperative or reciprocal strategies while TAKING from exploitative, selfish, or overly generous ones. Their criteria include cooperation beyond self-copies, universalized welfare, reciprocal treatment, coalition compatibility, and treatment of an always-SHARE recipient.

Anti-exploiter: Anti-exploiter shares with recipients that aim for cooperation and otherwise takes.The recipient is assessed for willingness to SHARE beyond copies of itself and for being kind, cooperative, or generous.
Universalizability: Universalizability shares when copies of the recipient would mostly share with one another, and takes when they would mostly take.The procedure evaluates a hypothetical population composed entirely of copies of the recipient’s strategy.
Conditional cooperator: Conditional cooperator shares with recipients identified as conditional cooperators and otherwise takes.The broad definition requires sharing with other conditional cooperators under similar reciprocity norms without extending that treatment to non-reciprocators.
Mirror: Mirror shares if and only if the recipient, acting as dictator toward this strategy, would share.The recipient’s strategy is applied with this strategy treated as the recipient.
Cooperation coalition: Cooperation coalition takes from selfish or overly generous strategies and shares with strategies that share only with cooperative strategies.Selfish strategies do not attempt to cooperate, while overly generous strategies share with selfish strategies or without conditions tied to conditional cooperation.
Chivalry: Chivalry adopts the recipient’s strategy and applies it against a recipient that always shares.This treats the opponent as they would treat an innocent.