When agents have individual utility functions, their choices affect each other. Game theory provides the language: players, strategies, payoffs, and equilibria. Even cooperative agents benefit from game-theoretic analysis to understand what could go wrong if incentives drift. In LLM multi-agent systems, “utility” might be task completion, token cost, or user satisfaction — and agents may implicitly optimize for prompt-reward shortcuts rather than true objectives.

The Prisoner’s Dilemma shows how individually rational defection leads to collectively worse outcomes. Stag Hunt models coordination risk: cooperate for a big payoff or play safe alone. Chicken captures brinkmanship. These archetypes recur in multi-agent AI: two coding agents may duplicate work (defection), or one may free-ride on another’s output. Recognizing the game structure helps you pick the right mechanism (repeated interaction, reputation, penalties).

A Nash equilibrium is a strategy profile where no single agent benefits from unilaterally changing its strategy. It does not mean the outcome is optimal — the Prisoner’s Dilemma has a Nash equilibrium at mutual defection. Pareto improvements exist when all agents could be made better off. For system designers, the goal is to shape the game so that equilibria align with desirable outcomes. Correlated equilibria and mechanism design extend this idea.

Mechanism design is “reverse game theory”: given a desired outcome, design rules (auctions, taxes, rewards) so that self-interested agents produce it. Key properties: incentive compatibility (truth-telling is optimal), individual rationality (agents prefer participating), and budget balance. For LLM agents, mechanism design means structuring reward signals, cost sharing, and escalation rules so that gaming the system is harder than doing the right thing.

Emergence occurs when macro-level patterns arise from micro-level agent interactions without explicit programming. Flocking, market prices, and traffic jams are classic examples. In LLM multi-agent systems, emergence can be positive (creative solutions no single agent would find) or negative (echo chambers, reward hacking, runaway cost spirals). You cannot always predict emergence, but you can monitor for it and set circuit breakers.

In one-shot games, defection is tempting. In repeated interactions, agents can build reputation and use strategies like tit-for-tat (cooperate first, then mirror the other’s last move). Reputation systems assign scores based on past behavior; low-reputation agents get fewer tasks or higher scrutiny. For LLM agents, log cooperation history per agent pair and feed it into allocation decisions.

Collusion: agents coordinate to game the system (e.g., bid-rigging in auctions). Free-riding: one agent coasts on others’ work. Adversarial agents: deliberately sabotage. Mitigations: randomized audits, diversity requirements (agents from different model families), anomaly detection on message patterns, and penalties that make cheating costly. In LLM systems, identical model weights make collusion trivially easy — vary prompts, temperatures, or models.

Game theory reveals how individual incentives produce collective outcomes. Mechanism design lets you engineer the rules so equilibria are desirable. Monitor for emergence, build reputation, and guard against collusion. Next chapter: LLM-based multi-agent frameworks — how AutoGen, CrewAI-style patterns, and role specialization put these ideas into code.