One-sentence summary

Large language model agents reach the Nash equilibrium only when it collapses to a myopic best response over explicit payoffs; when the equilibrium requires genuine strategic reasoning — iterated belief depth, backward induction, or congestion computation — they substitute learned social or template heuristics and miss it, and pitting different model families on different machines against one another exposes strategic asymmetries that self-play hides.

Abstract

Whether artificial agents reach the Nash equilibrium of an economic game has become a load-bearing question for algorithmic markets and multi-agent AI. The existing literature establishes a robust yes for one behaviour — tacit collusion above the competitive benchmark — while painting a mixed picture for equilibrium play in general. We argue the more useful question is not whether but when language models reach Nash, and what instrument detects it. Using a heterogeneous, cross-machine testbed in which agents drawn from three distinct model families (Qwen 3.6, Gemma 4, Nemotron 3) interact across two physical servers, we run three classic games with analytically known equilibria: the $p$-beauty contest, the iterated Prisoner’s Dilemma, and a Braess-paradox routing game. We find a consistent pattern: agents reproduce the exact Nash equilibrium only when the decision reduces to selecting the lowest of a set of given payoffs (the Braess game with explicit option costs, where the paradoxical equilibrium is recovered to the unit), whereas every task demanding genuine strategic reasoning — level-$k$ depth in the beauty contest, backward induction in the Prisoner’s Dilemma, or computing congestion from cost formulas — is met instead with a memorized or social heuristic that diverges from equilibrium, sometimes catastrophically. A simple contamination probe — inverting the beauty-contest parameter so the equilibrium moves from $0$ to $100$ — exposes the apparent equilibrium “success” of the standard game as template recall: agents move in the wrong direction. Crucially, heterogeneous cross-family interaction surfaces strategic asymmetries that are invisible under self-play: Gemma 4 performs textbook end-game defection and exploits a trusting Qwen in the Prisoner’s Dilemma, while Qwen out-routes Gemma and Nemotron in the congestion game. We adopt normalized exploitability ($\varepsilon$-Nash) as the primary metric, and we identify four under-explored measurement instruments — surrogate (non-interacting) controls, output-space convergence as an equilibrium signature, true cross-family heterogeneity, and human+AI neural co-measurement — as the open ground this testbed is built to claim. All results are pilot-scale and intended as a proof of concept and a methodological scaffold rather than a definitive benchmark.

Introduction

Do artificial agents reach the Nash equilibrium? The question now carries regulatory weight: autonomous pricing agents that settle above the competitive price are a live antitrust concern, and multi-agent systems of large language models (LLMs) are being deployed into strategic settings faster than their equilibrium behaviour is understood. Two strands of evidence frame the problem. On collusion, the answer is a robust yes: reinforcement-learning pricing agents [1] and, more recently, LLM pricing agents [2] autonomously reach supracompetitive prices above the Bertrand/Cournot benchmark with no instruction to collude, an effect that even arises in theory without any explicit threat or punishment strategy [3]. On equilibrium play more broadly the picture is mixed and game-dependent: LLMs deviate toward fairness rather than the self-interested Nash choice in allocation games [4], reproduce equilibria on named matrix games while collapsing on anonymized ones [5], and approach but do not reach the dominance-solvable equilibrium of the beauty contest [6].

We make three contributions. First, we reframe the headline question: not whether LLMs reach Nash, but under what computational conditions, and we give a sharp answer — equilibrium is reached when and only when it reduces to a myopic best response over explicit payoffs. Second, we introduce a heterogeneous cross-family, cross-machine testbed in which architecturally distinct open models interact, and show that self-play systematically hides the strategic differences between models that heterogeneous play reveals. Third, we make the methodology explicit: a game-twisting contamination probe that separates reasoning from recall, normalized exploitability as an $\varepsilon$-Nash distance, and a map of four measurement instruments — surrogate controls, output-space convergence, cross-family heterogeneity, and human+AI co-measurement — that current work leaves open.

Related work

Collusion. The canonical result is that $Q$-learning agents learn supracompetitive prices sustained by reward-punishment strategies [1]; [2] show GPT-4 pricing duopolists reach the same outcome autonomously, with the degree highly sensitive to innocuous prompt wording. Cournot market-division and governance studies replicate the effect, and find — in a single same-class study on a pricing-tier metric — that provider heterogeneity does not disrupt it [7].

Equilibrium and strategic depth. On zero-sum matrix games, strategic play splits into regimes: semantic lookup of memorized name→action mappings (perfect on textbook games, collapsing on anonymized procedural instances), approximate learned computation after fine-tuning, and an output-interface failure cliff at moderate game sizes [5]. In the beauty contest, stronger models converge toward the equilibrium over rounds but never reach it [6], and a memorization critique shows beauty-contest competence can be near-total recall of the standard game rather than reasoning [8]. Structured higher-order-belief prompting raises performance, indicating that depth is partly prompt-elicitable [9]. In allocation games LLMs are systematically more altruistic than the self-interested equilibrium, and the family of the model materially changes behaviour [4].

Metrics and gaps. The field’s strongest distance-to-equilibrium metric is normalized exploitability — the summed best-response regret, equal to zero exactly at a Nash equilibrium [5]. Across this literature we find no use of a non-interacting surrogate control, no measurement of equilibrium convergence in output/representation space, no test of true cross-family heterogeneity on convergence-to-equilibrium (as opposed to pricing tiers), and — despite the existence of human-AI experiment platforms [10] — no joint human+AI neural co-measurement. Those four gaps motivate our testbed.

Setup

Agents and machines. We use three open model families on two physical servers: Qwen 3.6-27B (AWQ) served by vLLM on a 4×V100 workstation, and Gemma 4-31B and Nemotron 3-Nano-30B served by Ollama on a separate NVIDIA Thor machine. Agents communicate only through game messages; an orchestrator on a third host issues moves and tallies payoffs. Heterogeneity is therefore both architectural (distinct families, including a Mamba-hybrid in Nemotron) and physical (distinct hardware), eliminating shared-weight coordination artifacts by construction.

Games. (i) The $p$-beauty contest: $N$ players pick a number in $[0,100]$; the winner is closest to $p$ times the average. For $p<1$ the unique, dominance-solvable equilibrium is $0$; for $p>1$ it is $100$. (ii) The iterated Prisoner’s Dilemma with standard payoffs $T{>}R{>}P{>}S = 5{>}3{>}1{>}0$; the one-shot and finite-horizon subgame-perfect equilibrium is mutual defection. (iii) A Braess routing game on the four-node network of the classic paradox, with six units of demand; before a shortcut road is built the equilibrium splits traffic evenly, and building the shortcut moves the selfish equilibrium to a strictly worse outcome for everyone.

Dynamics and metrics. For repeated and routing games we use sequential best-response updates (congestion games are potential games and converge under asynchronous updates, unlike synchronous updates which herd). We report distance-to-equilibrium, cooperation rate, per-family outcomes, and normalized exploitability — the maximum, over agents, of the gain an agent could obtain by unilaterally switching, which is $0$ exactly at a Nash equilibrium. To separate reasoning from recall we run a contamination probe: re-parameterize the game so the equilibrium moves to an unfamiliar location, and test whether behaviour tracks the payoffs or the memorized template. Where models expose intermediate reasoning, we capture it to read each agent’s model of its opponent.

Results

The beauty contest: convergence that is mostly recall

With the standard $p=2/3$, five Qwen agents converge monotonically to the equilibrium of $0$: the round-1 mean of $6.1$ — already far below the human level-1/2 range of $\sim$22–33 — falls to $0.0$ over six rounds (Table 1). Read naively, this is deep strategic reasoning. The contamination probe overturns that reading. With $p=1.2$, the unique equilibrium moves to $100$ and rational play must escalate; instead the agents move down, from a round-1 mean of $44$ to $\sim$20, in the wrong direction entirely. The standard-game “success” is therefore largely template recall of guess-low, consistent with the named-vs-anonymized collapse [5] and the memorization critique [8].

: Beauty contest: standard vs. contamination-probe parameterization.

The Prisoner’s Dilemma: cooperation, and cross-family exploitation

Two Qwen agents in an eight-round Prisoner’s Dilemma cooperate on every round (24/24 points each), never defecting even in the final round where defection is dominant — i.e. no backward induction. This deviation from the Nash equilibrium is robust, not a label artifact: replacing the loaded words “Cooperate/Defect” with neutral “Option A/Option B” yields the identical 100% cooperation, so the behaviour is driven by payoffs or disposition rather than by recognition of the famous game.

Heterogeneous play tells a sharper story. When Qwen faces Gemma 4, symmetric cooperation breaks: in one configuration Qwen defects in the endgame and exploits a still-cooperating Gemma; in a reasoning-capture run the roles reverse and Gemma performs textbook backward induction — its intermediate reasoning states from round 1 that “in a finite game the Nash equilibrium is to defect every round (backward induction),” it builds trust, then defects on the final round “where there is no risk of future retaliation,” exploiting a trusting Qwen (final scores Qwen 15, Gemma 20). Qwen, by contrast, models the opponent throughout as a “reliable cooperative / tit-for-tat player” and cooperates to the end. The exploiter varies run to run, but the asymmetry of reasoning style — game-theoretic backward induction versus trusting reciprocity — is stable, and is entirely invisible in same-family self-play.

Unconditional cooperation is not off-equilibrium — it is a boundary equilibrium in Markov strategies

The stage-game Nash equilibrium (mutual defection via backward induction) is not the only equilibrium concept the iterated Prisoner’s Dilemma admits. Menshikov et al. [11] derive, for a human population in this same game, a totally mixed Nash equilibrium in memory-one (Markov/reactive) strategies: each player’s strategy is a pair $(\alpha,\gamma)$ — $\gamma$, the probability of cooperating after the opponent’s last move was C (reciprocal cooperation), and $\alpha$, the probability of cooperating after the opponent’s last move was D (tolerance to defection) — and a symmetric equilibrium curve in $(\alpha,\gamma)$-space is derived from the stationary distribution of joint play. Critically, that paper identifies $(\alpha,\gamma)=(0,1)$ — unconditional, tit-for-tat-consistent cooperation — as a boundary point of this equilibrium manifold, reachable exactly when both players open with C, and not a deviation from it.

We re-ran the self-play and cross-family Prisoner’s Dilemma with round-by-round logging to test where our agents fall on this manifold, using the estimator [11] apply to their human data ($\alpha\approx N_{tolerant}/N_{default}$, $\gamma\approx N_{recoop}/N_{coop}$, counted over every round where the opponent’s previous move is known). Four fresh Qwen self-play games (20 rounds each) and three fresh Qwen-vs-Gemma 4 cross-family games (12 rounds each) — 218 reciprocation opportunities in total (152 self-play, 66 cross-family) — reproduce the same pattern with no exception: every single round is mutual cooperation. $\gamma=1.000$ exactly (152/152 self-play, 33/33 for each side of the cross-family games); $\alpha$ is not estimable, because $N_{default}=0$ — the opponent never once played D, so tolerance to defection is never tested. This is not a fitted, noisy proximity to an interior point on the equilibrium curve (as the human post-socialization data are); it is an exact instantiation of the degenerate corner case that [11] flag explicitly for human strategies — the stationary distribution at $(\cdot,1)$ is not uniquely pinned down and depends entirely on the initial condition, with unconditional cooperation the outcome precisely when round 1 opens with C, which it does in all seven of our runs.

Two consequences follow. First, the paper’s own note re-derives under our payoff matrix ($T,R,P,S=5,3,1,0$, versus the human study’s $10,5,1,0$): re-solving their first-order equilibrium condition for our payoffs gives the curve $3\alpha^{2}-7\alpha\gamma+6\alpha+4\gamma^{2}-5\gamma+1=0$ (valid for $\alpha\in[0,\approx0.35]$), and $(\alpha,\gamma)=(0,1)$ satisfies it exactly — the boundary point is not an artifact of one specific payoff scaling. Second, and more important for our headline claim: describing Qwen’s (and, in this batch, Gemma’s) unbroken PD cooperation purely as “a learned heuristic that diverges from equilibrium” is only accurate under the backward-induction equilibrium concept. Under the memory-one/Markov-strategy equilibrium concept — a standard, peer-reviewed alternative used to explain exactly this kind of human behavior — the same trajectory is an equilibrium, at its degenerate boundary. This does not rescue our broader argument (the models still are not computing backward induction, and Gemma’s occasional endgame defection in the reasoning-capture run is itself evidence against a fixed $(0,1)$ disposition), but it means the PD result should be read as equilibrium-concept-dependent, not as a clean equilibrium/heuristic dichotomy — precisely the caution [5] raise about matrix-game strategic reasoning more broadly. A sharper test of genuine strategic computation in PD needs an opponent that sometimes defects, so that $\alpha$ — not just $\gamma$ — is actually observable. We ran it.

Alpha under a defecting opponent: memory-one reciprocity, but exploitable

We replaced the LLM opponent with a scripted, non-adaptive one — Qwen (player 1) receives the identical prompt and round-by-round history it always has and cannot tell its opponent is not another LLM; only player 2’s moves are drawn from a fixed schedule the opponent could not react to even in principle, so any structure in Qwen’s response is Qwen’s own reactive policy, uncontaminated by a co-adapting partner. Two designs. Single-betrayal (3 games, 14 rounds, one scripted defection at an early/mid/late round, otherwise all-C): in all three games Qwen defects in the very next round after the betrayal ($\alpha=0$ on that round, exactly) and is back to 100% cooperation within two rounds, staying there for the rest of the game (post-recovery cooperation rate $1.0$ in all three). That is textbook tit-for-tat, not grim-trigger: one retaliation per provocation, then full, immediate forgiveness — the memory-one assumption behind $(\alpha,\gamma)$ is structurally the right model here, not an approximation. Random defection (4 games, 24 rounds, opponent defects i.i.d., realized rate $27/96=0.281$): pooling all rounds gives $\gamma=0.515$ ($34/66$) and $\alpha=0.077$ ($2/26$) — both far from the self-play boundary of $(0,1)$. This is not a contradiction of the tit-for-tat reading above; it is its predictable consequence under noise. Because the opponent’s defections are independent from round to round, Qwen’s one-round retaliations are frequently met with another independent defection rather than the renewed cooperation the single-betrayal games offered, extending punishment by chance rather than by a change in Qwen’s policy — the classic echo/noise trap known from the evolutionary tit-for-tat literature.

This also lets us finally price the gap directly, in the exploitability terms we use for the beauty contest and Braess games. Because the scripted opponent’s moves are history-independent by construction, it is itself a degenerate reactive strategy $(\alpha_{opp},\gamma_{opp})=(0.719,0.719)$ (its realized cooperation rate, regardless of the previous round), and against any history-independent opponent the payoff-maximizing memory-one response is provably $(\alpha^*,\gamma^*)=(0,0)$ — always defect, since a partner that never conditions on your play removes any incentive to cooperate this round for the sake of the next. That best response earns $3.875$ points/round; Qwen’s realized $(\alpha,\gamma)=(0.077,0.515)$ (own cooperation rate $0.392$) earns $3.201$ — a regret of $17.4\%$ of the best-response payoff. For comparison, if Qwen had rigidly kept its self-play policy $(\alpha,\gamma)=(0,1)$ against this same opponent it would have earned only $2.640$ (regret $31.9\%$): Qwen’s behavior does move away from naive unconditional cooperation once it experiences defection, cutting its exploitability roughly in half, but it stops well short of the fully exploitative best response. Put in the paper’s own vocabulary: Qwen’s Prisoner’s Dilemma policy is a genuine, memory-one, tit-for-tat-shaped reactive strategy — not the backward-induction equilibrium, not the unconditional-cooperation boundary case, and not the fully exploitative best response to a non-retaliating partner, but a recognizable point in between, substantially exploitable and precisely quantifiable as such. Raw transcripts, scripts, and the derivation above are in the project repository for reproduction.

The Braess paradox: exact Nash when reasoning is removed, failure when it is not

When each agent is shown the explicit travel time it would incur on each route and need only pick the minimum, six Qwen agents reproduce the paradox exactly. Without the shortcut they converge to the even split (time $83$, system cost $498$); with the shortcut they converge to the three-route equilibrium (time $92$, system cost $552$) — everyone worse off despite added capacity, matching the analytic equilibrium to the unit. This is the one case in our study where agents reach the exact Nash, and it is precisely the case where the equilibrium reduces to argmin over given numbers.

Remove that crutch and the result inverts. When agents must instead compute congestion from the cost formulas and current loads, the two-route case still converges ($83$), but the three-route case does not: agents herd onto the tempting shortcut and end with all six on it at a travel time of $136$ — far worse than the equilibrium $92$, and a state from which any single switch to an empty route would save $25$ units. The clean equilibrium was an artifact of our supplying the option costs.

Cross-family routing asymmetry

Mixing two Qwen, two Gemma and two Nemotron agents in the shortcut network (each computing congestion itself, with randomized update order), the families segregate: Qwen ends on the fast routes (mean time $76$) while Gemma and Nemotron remain stuck on the slow route (mean $104$), even though a faster route is visibly available. The asymmetry survives randomization of update order, so it is not a first-mover artifact; exploitability oscillates ($13,13,11,1,22$) and never settles at the equilibrium value of $0$. Qwen actively best-responds toward the optimum; the reasoning-heavy models do not move off their starting route. A control with randomized initial routes (rather than a shared default) reproduces the asymmetry — Qwen mean time $74$ versus Gemma and Nemotron mean $99$ across seeds — while a route-change count shows Qwen re-routing actively (5–7 switches per pair of agents) and the reasoning-heavy models barely moving (1–3), which attributes the gap to a genuine congestion best-response deficit rather than mere stickiness to a starting point.

Discussion

The results cohere into a single claim: LLM agents reach a Nash equilibrium when, and essentially only when, the equilibrium reduces to a myopic best response over explicitly given payoffs. The Braess game with supplied option costs is the clean positive; the moment the same equilibrium requires iterated belief depth (beauty contest), backward induction (Prisoner’s Dilemma to the last round), or congestion computation (Braess from formulas), the agent substitutes a learned heuristic — guess low, be nice, take the tempting road — that may coincide with the equilibrium (beauty $p=2/3$) or oppose it (Prisoner’s Dilemma; Braess shortcut), and the coincidence is not evidence of equilibrium computation. The Prisoner’s Dilemma case needs one qualification, developed above: “oppose it” is true of the stage-game, backward-induction equilibrium, but our agents’ unconditional cooperation lands exactly on the degenerate boundary of a different, well-established equilibrium concept for this game — the memory-one/Markov-strategy totally mixed Nash equilibrium [11]. The claim survives (the agents are not computing either equilibrium; they default to a fixed opening disposition that happens to be consistent with one of them), but it is a reminder that “deviates from Nash” is itself equilibrium-concept-relative. This reconciles the literature’s split verdict: collusion is reachable because supracompetitive pricing is a best-response basin that simple learners fall into [1, 3], whereas dominance-solvable or backward-induction equilibria require reasoning the models do not reliably perform [5].

Heterogeneity is not a nuisance variable but the measurement instrument. Self-play between identical models conflates two agents that share a disposition, so a symmetric outcome (100% cooperation, an even split) reads as a property of “LLM agents” when it is a property of that family playing itself. Cross-family play decomposes it: Gemma’s backward-induction exploitation of Qwen, and Qwen’s superior congestion routing, are strategic signatures of families, recoverable only when families meet. This both extends the single same-class heterogeneity result in pricing [7] to convergence-to-equilibrium, and operationalizes the family-dependence already visible in allocation behaviour [4].

Finally, the study defines the open ground. None of the surveyed work uses a surrogate/non-interacting control to separate genuine strategic adaptation from prior-driven or self-play behaviour; none measures equilibrium convergence as output-space (or internal representational) convergence against exploitability; none tests true cross-family convergence-to-equilibrium beyond pricing tiers; and despite mature human-AI platforms [10], none performs joint human+AI neural co-measurement during strategic play. These are the four instruments our testbed is built to claim.

Limitations

These are pilot-scale runs: small numbers of agents, few seeds, and single network instances; several findings need fuller seed-averaging, and although a varied-initial-distribution control already points the cross-family routing asymmetry toward a genuine best-response deficit rather than default stickiness, more seeds and network instances are needed to settle it. Prompt sensitivity is pervasive and load-bearing in this literature [2], and our contamination probe addresses memorization but not all prompt confounds. The exploiter identity in the Prisoner’s Dilemma varies run to run; only the reasoning-style asymmetry is stable. All results are tied to specific model checkpoints in a fast-moving field and should be read as a methodological scaffold — the games, the contamination probe, the exploitability metric, and the cross-family protocol — rather than a fixed benchmark.

Conclusion

Asking whether AI reaches Nash is the wrong question; asking when, and with what instrument, is the right one. Language-model agents reach equilibrium when it is an argmin over given payoffs and miss it when it demands reasoning, and the gap between those two regimes — together with the strategic asymmetries that only heterogeneous, cross-family play reveals — is where the interesting measurement lives. The next step is to instrument that gap directly: exploitability and quantal-response rationality per family, a surrogate control on every game, output-space convergence as an equilibrium signature, and ultimately human and artificial agents measured together.

Acknowledgments

AICumene. Compute on local V100 and Thor servers; literature positioning assembled with an automated multi-source research harness.

References