In program equilibrium, players submit computer programs instead of actions. These programs can read each other's source code, allowing them to verify cooperative intent and overcome dilemmas like the Prisoner's Dilemma, which is impossible in standard game theory.

To overcome brittle code-matching, AIs can use formal logic to prove cooperative intent. This is enabled by Löb's Theorem, an obscure result which allows a program to conclude "my opponent cooperates" without falling into an infinite loop of reasoning, creating a robust cooperative equilibrium.

Despite different mechanisms, advanced cooperative strategies like proof-based (Loebian) and simulation-based (epsilon-grounded) bots can successfully cooperate. This suggests a potential for robust interoperability between independently designed rational agents, a positive sign for AI safety.

A robust AI will cooperate with a simple "always cooperate" bot, making it exploitable. However, choosing to defect is risky. A sophisticated adversary could present a simple bot to test for predatory behavior, making the decision dependent on beliefs about the opponent's strategic depth.

Softmax's technical approach involves training AIs in complex multi-agent simulations to learn cooperation, competition, and theory of mind. The goal is to build a foundational, generalizable model of sociality, which acts as a 'surrogate model for alignment' before fine-tuning for specific tasks.

Stanford researchers found the largest category of AI coordination failure (42%) was "expectation failure"—one agent ignoring clearly communicated plans from another. This is distinct from "communication failure" (26%), showing that simply passing messages is insufficient; the receiving agent must internalize and act on the shared information.

A key finding is that almost any outcome better than mutual punishment can be a stable equilibrium (a "folk theorem"). While this enables cooperation, it creates a massive coordination problem: with so many possible "good" outcomes, agents may fail to converge on the same one, leading to suboptimal results.

A simple way for AIs to cooperate is to simulate each other and copy the action. However, this creates an infinite loop if both do it. The fix is to introduce a small probability (epsilon) of cooperating unconditionally, which guarantees the simulation chain eventually terminates.

To build robust social intelligence, AIs cannot be trained solely on positive examples of cooperation. Like pre-training an LLM on all of language, social AIs must be trained on the full manifold of game-theoretic situations—cooperation, competition, team formation, betrayal. This builds a foundational, generalizable model of social theory of mind.