💻 computer science

The Open-Strategy Dictator Game: Cooperation Under Mutual Transparency

This paper introduces the Open-Strategy Dictator Game, where natural-language strategies are mutually visible and adjudicated by an LLM, demonstrating through tournament analysis that conditionally cooperative strategies consistently dominate unconditional ones in environments of mutual transparency.

Original authors: Michael Glass

Published 2026-08-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Michael Glass

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine two strangers meeting in a vast, silent room. One holds all the power: they can choose to share a valuable resource or keep it all for themselves. The other person has no way to fight back, negotiate, or threaten; they can only wait to see what happens. This scenario, known in economics as the "dictator game," has long been used to study human nature. Traditionally, the powerful person makes a split-second decision based on gut feeling or social pressure, while the powerless person remains a mystery. But what if the rules changed? What if the powerful person could read the other person's strategy document—a plain-language text explaining exactly how they would allocate resources if they were the one in charge?

This is the core question explored in a new study called the Open-Strategy Dictator Game. The researchers wanted to see what happens when transparency is total. In this setup, every participant submits a written strategy, a plain-language document describing how they would act if they were the dictator. Crucially, these documents are visible to everyone. Before making a decision, the powerful player reads the other person's strategy. They can then choose to share if the other person's strategy indicates a disposition to share, or take everything if the other person's strategy indicates a disposition to take. To run this experiment on a large scale, the researchers used advanced artificial intelligence to act as the judge. The AI read every pair of strategies, interpreted the logic of the powerful player's document in the context of the other person's, and made the final decision to share or take.

The study pitted nine different types of strategies against each other in a massive round-robin tournament, where every strategy played against every other strategy, and even against itself. The results revealed a surprising truth about cooperation in a world of total transparency. The strategies that simply always shared, or always took, were the weakest performers. Being blindly generous made a player an easy target for exploiters, while being blindly greedy caused them to miss out on the benefits of mutual sharing. Instead, the winners were the conditional cooperators. These were the strategies that said, "I will share with you if your strategy shows you would share with me, but I will take from you if your strategy shows you would take from me."

However, the story gets more interesting when looking at how these winners interacted with each other. The researchers found that even among the smart, fair-minded strategies, there were two distinct schools of thought that could not agree. One group believed that to maintain a fair society, you must punish not only the greedy players but also anyone who is too nice to the greedy ones. They argued that if you are too generous to a bad actor, you are indirectly helping them, so you must cut them off. The other group believed that you should protect the innocent, even if they are too nice. They argued that punishing someone for being overly generous is itself a form of cruelty, and that the truly fair approach is to defend the innocent from the greedy, even if it means tolerating some extra generosity.

In the simulations, these two groups often clashed. When they met, they would sometimes refuse to share with each other, not because one was greedy, but because they disagreed on how to treat a third party. Despite this internal conflict, the study showed that conditional cooperation was the most robust path forward. In almost every scenario tested, the strategies that could read the other person's strategy and adjust their behavior accordingly outperformed those that could not. The research suggests that in a world where we can truly understand each other's intentions, the best way to survive and thrive is not to be a saint or a tyrant, but to be a mirror: to reflect the fairness of others back to them, while carefully guarding against those who would take advantage of kindness.

The findings held true even when the researchers changed the rules of the game, such as how much value was at stake or how much the players cared about the group's total happiness versus their own. As long as sharing provided some benefit to both parties, the conditional strategies dominated. The study also explored what happens when the players are not just individuals, but entire civilizations or artificial intelligences. It suggests that if advanced beings can read the "strategies" of others—whether through their culture, laws, or code—they will likely evolve toward these conditional forms of cooperation. The research implies that the key to peaceful coexistence between unequal powers is not blind trust, but the ability to see clearly into the other's decision-making process and respond with a fairness that is earned, not given.

Ultimately, the paper demonstrates that transparency changes the game entirely. When the powerful can see the logic of the powerless, the incentive to exploit disappears, replaced by a complex dance of mutual recognition. The winners are not those who give the most, nor those who take the most, but those who are smart enough to know when to do which. This suggests that in a future where artificial intelligence and human societies interact with full visibility of their decision rules, the path to a stable and cooperative world lies in building systems that can recognize and reward fairness, while firmly rejecting those who would break the social contract. The study does not claim to have solved all problems of human or machine interaction, but it offers a clear, simulated glimpse into how cooperation can evolve when the masks are removed and the true intentions of all players are laid bare.

Technical Summary: The Open-Strategy Dictator Game

Problem Statement

The paper introduces the Open-Strategy Dictator Game (OSDG), a variant of the classic dictator game designed to model interactions where agents possess mutual transparency regarding their decision procedures. In the standard dictator game, a "dictator" unilaterally divides an endowment, and the recipient has no strategic recourse. The OSDG departs from this by making the recipient's strategy—a natural-language document specifying how they would allocate resources if they were the dictator—visible to the dictator before the decision is made.

The core problem addressed is how conditional cooperation evolves when agents can inspect and reason about each other's decision logic. This setup is motivated by scenarios of interstellar first contact or human-AI interaction, where a technologically superior agent (the dictator) can observe the culture, norms, and decision procedures of a less advanced agent. The paper posits that this "mutual legibility" fundamentally alters the incentive landscape, transforming the game from one of unilateral power into a system of mutual transparency where strategies are evaluated based on their text.

Methodology

Game Formalization

The OSDG is defined as a tuple $(S, E, O)$ , where $S$ is a set of natural-language strategies, $E$ is a fixed endowment, and $O$ is an oracle.

Payoff Structure: The game uses a concave utility function $u(v) = \ln(1+v)$ to capture diminishing marginal returns. A "share" decision splits the endowment equally ( $E/2$ each), while a "take" decision awards the full endowment to the dictator and zero to the recipient. Mutual sharing maximizes total welfare.
The Oracle: Unlike previous "program equilibrium" models that rely on formal source code analysis or provability logic, the OSDG employs a Large Language Model (LLM) as the oracle. The LLM interprets the dictator's strategy text in the context of the recipient's strategy text to produce a binary decision (Share/Take). This approach treats the strategy description itself as the object of play, removing the "executable wrapper" of code.

Tournament Protocol

The author conducted round-robin tournaments involving a pool of nine distinct strategy archetypes (e.g., Generous, Selfish, Mirror, Cooperation Coalition, Chivalry).

Adjudication: For every ordered pair $(i, j)$ , the oracle $O(s_i, s_j)$ determines the outcome.
Robustness Checks: To ensure the results were not artifacts of a specific model, the tournament was re-adjudicated using two different frontier LLMs (GPT-5.6 Sol and Claude Fable 5). The cell-level agreement across models exceeded 96%, with disagreements limited to borderline second-order classifications.

Analytical Frameworks

The paper employs three primary analytical lenses:

Dominance Analysis: Evaluating strategies based on weak and strict dominance within the payoff matrix.
Softmax Equilibrium: Modeling evolutionary selection and reflective deliberation using a fixed-point equation $f^* = \text{softmax}(\beta A f^*)$ , where $\beta$ represents selection pressure or precision.
Bayesian Interpretation: Reinterpreting the population frequencies as beliefs held by rational authors choosing strategies, incorporating author types that value total welfare versus self-payoff.

Key Contributions

Definition of the OSDG: The paper formalizes a game where strategies are natural-language documents interpreted by an LLM, bridging the gap between formal program equilibrium and practical, human-readable strategy descriptions.
Identification of Second-Order Norms: The analysis distinguishes between first-order reciprocity (treating others as they treat you) and second-order norms. The paper identifies two antagonistic second-order strategies:
- Second-Order Deterrence (Cooperation Coalition): Punishes not only defectors but also those who unconditionally cooperate with defectors (subsidizers).
- Protection (Chivalry): Punishes defectors but protects unconditional cooperators from exploitation.
Equilibrium Multiplicity and Basin Analysis: The study reveals that under high selection pressure, the system exhibits bistability. Depending on initial conditions (priors), the population converges to either an "Enforcement" equilibrium (where unconditional cooperators are extinct) or a "Protection" equilibrium (where they survive).
Decision-Theoretic Insights: The paper demonstrates that in the OSDG, functional decision theory (cooperating based on logical correlation) outperforms causal decision theory. Strategies that conditionally cooperate achieve higher payoffs than unconditional maximizers, even when the immediate causal act of sharing reduces the dictator's utility.

Results

Dominance and Strategy Performance

Conditional Cooperation Dominates: Six of the nine strategies in the tournament were found to be weakly dominated. This includes both unconditional strategies (Generous and Selfish).
Undominated Set: The only strategies that were not weakly dominated were the top three performers: Mirror, Cooperation Coalition, and Chivalry.
The Cost of Unconditional Generosity: Unconditional cooperators (Generous) are weakly dominated because they subsidize defectors. While they perform equally well against other cooperators, they lose strictly against defectors compared to conditional cooperators who withhold sharing.

Equilibrium Dynamics

Enforcement vs. Protection: At high selection pressures ( $\beta=20$ $β = 20$ ), two attracting fixed points emerge:
- Basin 1 (Enforcement): The population consists of Mirror, Cooperation Coalition, and Conditional Cooperator. Unconditional cooperators go extinct.
- Basin 2 (Protection): The population includes Mirror, Chivalry, and Generous. Unconditional cooperators survive but at a lower frequency.
Sensitivity to Priors: The basin of attraction is determined largely by the initial proportion of Chivalry and Cooperation Coalition strategies. The "Enforcement" basin covers approximately 80% of the simplex, suggesting that without a significant initial advantage for protective norms, the system defaults to the stricter enforcement equilibrium.
Robustness to Payoff Parameters: The dominance of conditional cooperation holds for all concave utility functions (where the share payoff $\sigma > 0.5$ ). The bistability (existence of two basins) only emerges when the exchange rate $\rho > 4$ , corresponding to high stakes ( $E > 25.6$ under logarithmic utility).

Author Type Heterogeneity

When modeling a population of authors with mixed objectives (some maximizing self-payoff, others maximizing total welfare), the Enforcement equilibrium becomes the unique stable solution. The presence of even a small fraction of welfare-maximizing authors destroys the "Protection" basin, selecting for the enforcement core.

Significance and Claims

The paper claims that conditional cooperation is evolutionarily robust in environments where agents can inspect each other's decision procedures. The key findings are:

Transparency Changes Incentives: Mutual legibility allows agents to condition their behavior on the recipient's strategy, making unconditional defection and unconditional cooperation both suboptimal.
Second-Order Reasoning is Critical: The survival of cooperation depends not just on reciprocity, but on how agents judge the deservingness of punishment. The antagonism between "harvesting subsidizers" (to enforce strict conditionality) and "protecting innocents" creates the observed bistability.
Functional over Causal: The results support Functional Decision Theory (FDT), showing that agents who reason about the logical correlation of their decisions with others achieve better outcomes than those reasoning purely causally.
Relevance to AI and Interstellar Contact: The OSDG serves as a stylized model for interactions between unequal powers (e.g., advanced civilizations or humans and AI). It suggests that in such scenarios, standing (the right to be treated well) is derived from possessing a legible, conditional strategy rather than from status or consciousness.
Methodological Validation: The study demonstrates that LLMs can serve as robust oracles for game-theoretic analysis, producing consistent results across different models and effectively resolving the "Löbian obstacle" (circular reasoning) by selecting cooperative fixed points based on author intent.

The paper concludes that while the specific equilibrium reached (Enforcement vs. Protection) depends on the prior distribution of strategies, the qualitative conclusion—that conditional cooperation dominates unconditional strategies—remains robust across a wide range of payoff parameters and author compositions.