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Level-k Distinguishable Mechanisms for Evaluating Bounded Rationality in LLMs
Binchi Zhang, Atrisha Sarkar
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 96%
Last extracted: 8/24/2026, 4:07:17 AM
Summary
This paper introduces 'Level-k Distinguishable Mechanisms' to evaluate the strategic reasoning depth of Large Language Models (LLMs) in boundedly rational environments. The authors address the issue that existing game-based evaluations often rely on canonical games present in training data, leading to memorization rather than true reasoning. They formalize a 'level-K distinguishability condition' ensuring a one-to-one mapping between reasoning depth and observable actions. Using novel and adapted game structures (11-20 Money Request, All-Pay Auction, Nash Demand, and Ring 11-20), they evaluate four LLMs (Gemma, Qwen, Claude, DeepSeek) across ten levels of iterated reasoning. The study compares recursive reasoning (explicit opponent depth) vs. inductive reasoning (opponent history). Results show LLMs maintain accurate strategic depth under recursive reasoning with strong consistency between Chain-of-Thought and actions, but performance degrades sharply under inductive inference. Explicit strategic mentalizing in CoT improves performance.
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Relation Signals (13)
Gemma 4 31B → testedin → Level-k Distinguishable Mechanisms
confidence 99% · In our experiments, we test the strategic reasoning depth of four models: Gemma 4 31B... across experimental trials spanning four LLMs
DeepSeek V4 Pro → testedin → Level-k Distinguishable Mechanisms
confidence 99% · In our experiments, we test the strategic reasoning depth of four models: ... and DeepSeek v4 Pro
Claude Sonnet 4.6 → testedin → Level-k Distinguishable Mechanisms
confidence 99% · In our experiments, we test the strategic reasoning depth of four models: ... Claude Sonnet 4.6 ...
Qwen 3.6 27B → testedin → Level-k Distinguishable Mechanisms
confidence 99% · In our experiments, we test the strategic reasoning depth of four models: ... Qwen 3.6 27B ...
Level-k Distinguishable Mechanisms → uses → Level-K Distinguishability Condition
confidence 98% · To address this, we formalise a necessary level-K distinguishability condition for strategic depth inference and construct a suite of novel game structures that meet this standard.
Nash Demand Game → adaptedtosatisfy → Level-K Distinguishability Condition
confidence 95% · We complement the Ring 11–20 game with adaptation of three canonical two-player games... and the Nash demand game... to adhere to the level-k distinguishability condition
All-Pay Auction → adaptedtosatisfy → Level-K Distinguishability Condition
confidence 95% · We complement the Ring 11–20 game with adaptation of three canonical two-player games... All-pay auction... to adhere to the level-k distinguishability condition
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Abstract
Abstract:Strategic depth of reasoning is essential for human interaction of Large Language Models (LLMs) operating in boundedly rational environments. However, existing evaluations are primarily based on canonical games prevalent in pretraining corpora, making it difficult to disentangle true strategic reasoning from memorisation. To address this, we formalise a necessary level-K distinguishability condition for strategic depth inference and construct a suite of novel game structures that meet this standard. Using these games, we evaluate strategic depth in LLMs from both the Chain-of-Thought tokens and actual actions under recursive reasoning and an inductive trace of opponent game-play data. Across experimental trials spanning four LLMs, four game structures, and ten levels of iterated reasoning, we find that model models maintain accurate strategic depth under recursive reasoning, with strong internal consistency between stated reasoning and actions at every level. Errors arise from using the wrong number of iterated depth of reasoning steps, not from computing best responses incorrectly. However, inductive inference from opponent play degrades accuracy sharply and unevenly across games, and explicit strategic mentalizing in the chain of thought substantially improves overall performance.
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- Source: https://arxiv.org/abs/2608.21296v1
- Canonical: https://arxiv.org/abs/2608.21296v1
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Level-k Distinguishable Mechanisms for Evaluating Bounded Rationality in LLMs Binchi Zhang Atrisha Sarkar Abstract Strategic depth of reasoning is essential for human interaction of Large Language Models (LLMs) operating in boundedly rational environments. However, existing evaluations are primarily based on canonical games prevalent in pretraining corpora, making it difficult to disentangle true strategic reasoning from memorisation. To address this, we formalise a necessary level-K distinguishability condition for strategic depth inference and construct a suite of novel game structures that meet this standard. Using these games, we evaluate strategic depth in LLMs from both the Chain-of-Thought tokens and actual actions under recursive reasoning and an inductive trace of opponent game-play data. Across experimental trials spanning four LLMs, four game structures, and ten levels of iterated reasoning, we find that model models maintain accurate strategic depth under recursive reasoning, with strong internal consistency between stated reasoning and actions at every level. Errors arise from using the wrong number of iterated depth of reasoning steps, not from computing best responses incorrectly. However, inductive inference from opponent play degrades accuracy sharply and unevenly across games, and explicit strategic mentalizing in the chain of thought substantially improves overall performance. Introduction In the realm of strategic reasoning, behavioural game theory models agents’ deviations from the Nash equilibrium of a game (Camerer 2004). The level-k family of models (including Cognitive Hierarchy models) provides a flexible meta-model to characterize such deviations from equilibrium by identifying the depth of iterated reasoning an agent employs to model and rationalize the behaviour of their interacting partners (Wright and Leyton-Brown 2017). Studies on humans have identified a natural limit to human capacity along this dimension (Coricelli and Nagel 2009), with heterogeneity arising from game structure and incentives (Georganas, Healy, and Weber 2015). Naturally, when LLMs are deployed for everyday tasks that involve reasoning about the behaviour and strategies of interacting agents (some of whom may be human), it is important to know whether an analogous natural limit to this depth exists for Large Language Models, for several reasons. First, LLMs are increasingly used to simulate human behaviour. In domains where it is important to model other humans as strategic agents, such as economic situations (Mallard 2012) and autonomous vehicle interactions (Sarkar, Larson, and Czarnecki 2022), LLMs are being used as a strategic decision layer (Horton 2023; Fu et al. 2024), and as such, they need the capacity to infer strategic depth and respond just as humans do in such interactions. Second, identifying the strategic depth of reasoning is crucial for AI safety: an agent with a lower level of strategic sophistication can be exploited by one with a higher level (Alon et al. 2026). In interactions with humans, who have a limit to their own depth, it is important to identify and calibrate the strategic depth of LLMs to prevent such exploitation. Existing methods for inferring the strategic depth of reasoning in LLMs from a behavioural perspective, that is, through observed actions, use standard canonical games, including coordination and signalling games (Jia et al. 2026), the Keynesian beauty contest, and negotiation games (Zhang et al. 2025; Trencsenyi et al. 2025), among others. However, the validity of these instruments is open to question. First, these games were originally designed as instruments to infer human, not artificial, strategic depth of reasoning, and are therefore designed to be simple enough for human comprehension in an experimental setting. Results from such gameplay are widely reported across decades of literature on behavioural game theory, which we can reasonably assume is present in internet-scale training corpora. Second, many of the game structures used for inferring strategic depth lack the capacity to infer reasoning up to an arbitrary depth of iteration. This is because inferring latent attributes—such as utilities, beliefs, and reasoning—from action traces alone runs into the well-known unidentifiability problem: multiple latent states can produce the same behaviour (Ng and Russell 2000). In many canonical games, a given level of strategic reasoning does not correspond to a unique observable action, and therefore suffers from the same unidentifiability issue. To improve the validity of such game-theoretic instruments, there is thus a need to design novel game structures (or mechanisms) that satisfy a one-to-one correspondence between actions and depth of iterated reasoning; we refer to this as the level-k distinguishability condition. In a multiagent environment, designing games that elicit a latent and private attribute falls within the domain of mechanism design (Dütting et al. 2024; Börgers 2015). However, unlike mechanism design, where the goal is to design mechanisms whose solutions are compatible with the private utilities of the players and, therefore, uniquely identifiable, our goal is to design games whose solutions are compatible with and unique to a particular strategic depth of a player. This difference between eliciting latent utilities and eliciting latent reasoning means that we cannot rely on the convenience of the revelation principle (Kephart and Conitzer 2016) to narrow the class of games (such as auctions) used as identifiability instruments. In this paper, we resolve the above set of identified problems through the following contributions. • We construct a novel game structure and adapt three canonical games to satisfy the level-K distinguishability property, using them as instruments to infer the strategic depth of reasoning in LLMs. We use these games to infer the strategic depth of four LLMs up to a previously untested depth of 10. • We evaluate the consistency between the models’ iterated depth of reasoning as reflected in their intermediate chain-of-thought inference and in their resulting strategies. • For modelling other players’ behaviour, we compare the effect of recursive versus inductive reasoning on the models’ level of strategic depth. Related Work Depth of reasoning and its identification. Level-k and cognitive-hierarchy models assign each player an integer depth of iterated best response anchored at a non-strategic L0L_0 type (Nagel 1995; Costa-Gomes, Crawford, and Broseta 2001; Camerer, Ho, and Chong 2004), and fitting these types to observed actions is standard practice (Crawford, Costa-Gomes, and Iriberri 2013). Machine learning methods have been used to estimate the strategic depth as a parameter from human game-play data (Hartford, Wright, and Leyton-Brown 2016). In contrast to that estimation problem, we focus on identification problem by designing games for automates agents that distinguishes each depth of reasoning. A work on the identification problem of higher order rationality is Kneeland 2015, which uses ring-network games for the purpose. However, unlike our focus on level-k solution concept, they focus on identification of strategies under different orders of rationalizability. However, (Kneeland 2015) and the game structures that follow, such as, (Cerigioni et al. 2019) target the depths human subjects reach, at most about four, and none states a condition for distinguishability beyond that range. We also improve upon the identifiability aspect through recursive and inductive conditioning of the information about other agents’, so exactly one IDR is correct and the action is identifiable against it alone. Level-k evaluation of LLMs. Many benchmarks now evaluate LLMs by having them play games (Duan et al. 2024; Wang et al. 2024; Huang et al. 2025). They report payoffs, or the share of actions matching a solution concept. However, such games cannot identify the strategic depth, because a model can reach the right action without iterative reasoning. The games themselves are also often structurally simple enough, thereby, limiting what the strategies can reveal. Wang et al. 2024 scores whether the model identifies the Nash equilibria of 2×22× 2 games, which are either solved within two rounds of iterated dominance or not dominance-solvable at all. Huang et al. 2025 includes a standard level-k instrument, Guess 2/3 of the Average, but scores it by distance from equilibrium, not the strategic depth itself. Other studies apply the level-k model directly. Zhang et al. 2025 prompts models to reason recursively in guessing, auction, and negotiation games. The negotiation game it adapts collapses depth 2 onto depth 0. Chen et al. 2025a fits level-k and Poisson cognitive-hierarchy models to LLM choices, capping the maximum type at k^=4 k=4 where fit stops improving. Fan et al. 2024 estimates a depth parameter under a truncated quantal-response equilibrium. Replications of the guessing games of Costa-Gomes and Crawford 2006 place reasoning-tuned models near three depths, compared with fewer than two for humans (Guo et al. 2024). As discussed in the introduction, the above set of work do not verify that the game structure maps each depth to a distinct action over the range it probes. Chain-of-thought as evidence about reasoning. A second body of work asks whether stated reasoning consistently describes the computation that produced the answer. Models often act on input features their explanations never mention, or explain answers they had already settled on (Turpin et al. 2023; Chen et al. 2025b). It is suggested that CoT is not an account of the computation at all (Barez et al. 2025; Kambhampati et al. 2026). In contrast to work that focus on the stronger notion of causal connection between the CoT traces and action, e.g. (Lanham et al. 2023; Kambhampati et al. 2026), we only measure consistency: whether the submitted action best-responds to the stated belief, and whether the two implied depths agree. Contamination. Replacing named games with procedurally sampled payoff matrices dramatically reduces performance, indicating that much apparent strategic competence is recall rather than computation (Zhang et al. 2026). A similar effect has been observed by perturbing the payoffs; models continue to play the canonical strategy after the equilibrium change (Li et al. 2026). In comparison to such modifications, our approach keeps both the level-k distinguishability condition and transformation of the game structures, which is relatively non-trivial compared to random changes to the game payoffs. Methodology The level-k family of models—including its cognitive-hierarchy generalization—characterizes boundedly rational play by assigning each player a discrete cognitive type k∈0,1,2,…k∈\0,1,2,…\ that indexes the depth of iterated reasoning about others (Stahl and Wilson 1994; Stahl and Wilson 1995; Nagel 1995; Costa-Gomes, Crawford, and Broseta 2001). A level-0 (L0L_0) type is non-strategic: it forms no beliefs about other players, and its action a(0)∈Sa^(0)∈ S is specified exogenously through a non-strategic level-0 anchor of the model. Every level-k type with k≥1k≥ 1 believes its opponents are level-(k−1)(k-1) and best-responds to that belief, a(k)=BR(a(k−1))≡argmaxs∈Sui(s,a(k−1)),a^(k)\;=\;BR\! (a^(k-1) )\;≡\; _s∈ S\,u_i\! (s,\,a^(k-1) ), (1) so that the latent depth k is mapped to observable behavior through iterated best response. Cognitive-hierarchy models relax the degenerate belief in Eq. (1) to a distribution over all lower types (Camerer, Ho, and Chong 2004), but retain the same structural core of a non-strategic anchor, an integer-valued latent depth, and a deterministic level-to-action map (Crawford, Costa-Gomes, and Iriberri 2013). We call the induced sequence a(k)k=0Kmax\a^(k)\_k=0^K_ the iterated depth response sequence (IDR sequence) of a game, i.e, the best-response action sequence generated by increasing the depth of iterated reasoning one level at a time. Inferring the latent depth k from an observed action is identifiable only if the level-to-action map is invertible at least up to the strategic depth we want the game instrument to reveal. Many canonical games fail this condition. For example, a game structure used in Zhang et al. 2025 to infer strategic depth is an adaptation of an item-division negotiation game. In the game, two agents should agree on how to split a pool of items of m types under imperfect information. Abstracting the alternating-offer protocol in its one-shot demand form, let q∈ℤ≥0mq _≥ 0^m be the pool, wi∈ℤ≥0mw_i _≥ 0^m agent i’s private utility vector, and a proposal a bundle si∈B(q)≡s∈ℤ≥0m:s≤qs_i∈ B(q)≡\s _≥ 0^m:s≤ q\ claimed for oneself, with payoffs ui(si,s−i)=(wi⊤si) 1[si+s−i≤q].u_i(s_i,s_-i)\;=\; (w_i s_i )\,1 [s_i+s_-i≤ q ]. Whenever wiw_i is strictly positive, the best response to any proposal is to claim the entire remaining BRi(s−i)=q−s−iBR_i(s_-i)=q-s_-i. A level-0 agent is non-strategic, and let’s say, they propose an bundle a(0)=sa^(0)=s. A level-1 agent best-responds with a(1)=q−sa^(1)=q-s. A level-2 agent best-responds to that with a(2)=q−(q−s)=s=a(0)a^(2)=q-(q-s)=s=a^(0). In this simple illustrative case, the level-2 proposal coincides with the level-0 proposal, and from behavior alone, a depth-0 and a depth-2 agent are therefore indistinguishable for every anchor and every utility profile. Definition 1 (Level-K distinguishability) Let G be a game with IDR sequence a(k)k=0Kmax\a^(k)\_k=0^K_ , where a(0)a^(0) is the L0L_0 anchor and a(k)=BR(a(k−1))a^(k)=BR(a^(k-1)). G satisfies distinguishability within depth KmaxK_ iff ∀ 0≤k1<k2≤Kmax:a(k1)≠a(k2),∀\;0≤ k_1<k_2≤ K_ : a^(k_1)≠ a^(k_2), i.e., k↦a(k)k a^(k) is injective on 0,…,Kmax\0,…,K_ \. On a bounded integer strategy space, the iteration in Eq. (1) must eventually revisit a value, resulting either in a fixed point or a periodic cycles. Therefore, we can only infer the strategic depth of an agent through a level-k model up to depth KmaxK_ . Level-k distinguishable game structures Ring 11–20 We first construct a novel game structure whose iterated depth response sequence is level-k distinguishable by adapting the 11–20 money-request game (Arad and Rubinstein 2012) into a five-player ring inspired by the ring games of Kneeland 2015. Players are arranged on a directed cycle with indices taken modulo 5 (Figure 1); player i chooses si∈1,…,20s_i∈\1,…,20\ and receives a primary bonus of +10+10 for undercutting the left neighbor by exactly 3 and a secondary bonus of +3+3 for exceeding the right neighbor by exactly 7 (Table 1). A level-k player best-responds to the belief that both neighbors are level-(k−1)(k-1). Table 1: Summary of the game structures with level-k distinguishability. 11–20 (Mod.) All-Pay Auction Nash Demand Ring 11–20 Players 2 2 2 5 (ring) Strategy space S 50,…,70\50,…,70\ 0,…,10\0,…,10\ 1,…,20\1,…,20\ 1,…,20\1,…,20\ Payoff ui(si,sj)=siu_i(s_i,s_j)=s_i + 24 1[si=sj−2]+\,24\,1[s_i=s_j-2] + 8 1[si=sj−1]+\,8\,1[s_i=s_j-1] ui(si,sj)=12 1[si>sj]u_i(s_i,s_j)=12\,1[s_i>s_j] + 6 1[si=sj]−si+\,6\,1[s_i=s_j]-s_i ui(si,sj)=si 1[si+sj≤20]u_i(s_i,s_j)=s_i\,1[s_i+s_j≤ 20] + 0.8si 1[si+sj>20] 1[si<sj]+\,0.8\,s_i\,1[s_i+s_j>20]\,1[s_i<s_j] ui(si,−i)=siu_i(s_i,s_-i)=s_i + 10 1[si=si−1−3]+\,10\,1[s_i=s_i-1-3] + 3 1[si=si+1+7]+\,3\,1[s_i=s_i+1+7] a(0)a^(0) 70 0 20 20 IDR sequence Decreasing Increasing Decreasing Periodic KmaxK_ 10 10 10 9 i+1i+1i+2i+2i−2i-2i−1i-1 Figure 1: The Ring 11–20 game (player indices modulo 5). Solid arrows point from each player i to the left neighbor i−1i-1 carrying the primary +10+10 bonus (si=si−1−3s_i=s_i-1-3); dashed arrows point to the right neighbor i+1i+1 carrying the secondary +3+3 bonus (si=si+1+7s_i=s_i+1+7). Remark 1 The Ring 11–20 IDR sequence is periodic with period 10 and non-monotone level-k distinguishability up to Kmax=9K_ =9. Proof Sketch 1 The best response to a common neighbor action c has two cases: the primary-bonus action c−3c-3 when c≥14c≥ 14, and the secondary-bonus action c+7c+7 when 11≤c≤1311≤ c≤ 13. Iterating from the anchor a(0)=20a^(0)=20 generates the sequence 20,17,14,11,18,15,12,19,16,1320,17,14,11,18,15,12,19,16,13, whose ten values are pairwise distinct, after which a(10)=20a^(10)=20 returns to the anchor. Hence k↦a(k)k a^(k) is injective through depth 9 and the sequence is periodic with period 10; the upward jumps at a(4)a^(4) and a(7)a^(7) give non-monotonicity. The structure of Ring 11–20 also has the benefit that the IDR sequences are non-monotonic, unlike the standard canonical games. This enables us to test that a model does not trivially extrapolate a strategy; for example, by subtracting a constant integer at each depth a model might spuriously generate actions that correspond to increasing depth of reasoning. Canonical Game Adaptations We complement the Ring 11–20 game with adaptation of three canonical two-player games—the 11–20 money-request game (Arad and Rubinstein 2012), All-pay auction (Baye, Kovenock, and de Vries 1996), and the Nash demand game (Nash 1953). Our adaptation involve constructing the payoffs of each of these games to adhere to the level-k distinguishability condition in Definition 1 over the full range of strategic depth we wish to test in our experiments. Because the 11–20 game is often used as an instrument to infer level-k reasoning, we make an additional minor modification to the strategy space and payoffs through an affine transformation of the original game, so that the standars published solution is no longer optimal in the version the models face. Intermediate Reasoning and Action Consistency A key component of level-k family of models is a player’s inference about the depth of strategic reasoning of others (Stahl and Wilson 1994). A level-(k+1)(k+1) player is, by construction, one who believes its opponent reasons at level k. When an LLM playing a game receives information about the opponent, the intermediate tokens of the chain of thought (CoT) contain explicit statements of the model’s belief about the opponent’s action, while the submitted action is a separate behavioural trace of whatever computation actually determined the choice. Given that these two signals have been observed to be logically uncorreleted in LLM CoT (Kambhampati et al. 2026), we extract a depth estimate from each and measure their consistency using three primary metrics: (i) Accuracy, which evaluates whether the behavioural action matches the target level of the IDR sequence; (i) Best-response consistency, which evaluates whether the behavioural action is a best response to the model’s own stated belief (allowing for multiple best responses). Note that the two can be orthogonal—a model can be consistent to a wrong belief, or accurate despite an inconsistent narration in CoT trace; (i) Internal consistency, which evaluates the relationship between the CoT and the behaviour depth estimates. If KLK_L and KBK_B denote the model’s estimate about the opponent’s strategic depth in CoT signals and the action consistent with that belief, KL>KBK_L>K_B indicates the model verbally performs deeper reasoning than its actions realise, while KB>KLK_B>K_L indicates that the action uses a reasoning level never articulated in the CoT. Recursive and Inductive Conditioning Since a level-k reasoning agent needs information about its opponent’s strategic depth in order to best respond, this information can be fed to the model in two different ways. Either by specifying the explicit level of the opponent, which requires the agent to engage in recursive reasoning by simulating the opponent’s optimal strategy in an iterated way. Or by giving samples from a distribution of opponents’ strategy, which requires the agent to infer the level from the behavioural trace and best respond. We construct one experimental condition for each of these two methods. Recursive conditioning. The model receives a description of the opponent that conveys the opponent’s strategic depth. With this information, the model needs to extract the Levek−0Levek-0 anchor, simulate the opponent’s reasoning chain, form a belief, and compute the best response. Inductive conditioning The model receives no opponent description but it can make a tool call to query the opponent’s history to obtain 30 raw action samples from the opponent’s previous play, generated at a target depth unknown to the model. We sample these actions following the quantal best response of the Quantal Cognitive Hierarchy model (Wright and Leyton-Brown 2010). For an opponent of level k≥1k≥ 1 each response hth_t is drawn with P(ht=s)∝exp(λ⋅u(s,a(k−1))−uminumax−umin),P(h_t=s)\; \; \! (λ· u (s;\,a^(k-1) )-u_ u_ -u_ ), with λ∈1,3,10λ∈\1,3,10\. Figure 2: Recursive condition: Depth limit—linguistic and behavioral signals as a function of target depth. Each panel is one model; solid lines and markers show KBK_B (behavioral depth), dashed lines show KLK_L (linguistic depth), and the gray dashed diagonal marks where K equals target k. Experiment and Results Figure 3: Inductive condition: Depth limit—linguistic and behavioral signals as a function of target depth. Columns are models and rows are the sampling precision λ; solid lines and markers show KBK_B (behavioral depth), dashed lines show KLK_L (linguistic depth), and the gray dashed diagonal marks where K equals target k. In our experiments, we test the strategic reasoning depth of four models: Gemma 4 31B, Qwen 3.6 27B, Claude Sonnet 4.6, and DeepSeek v4 Pro, at temperature 0.2 with reasoning mode enabled where supported. Each model plays four games—the 11-20 Game, the All-Pay Auction, the Nash Demand Game, and Ring 11-20—against opponents at levels L0L_0–L9L_9 (ktarget=1k_target=1–10). The recursive condition encodes the opponent’s reasoning depth in language; the inductive condition removes this description entirely and supplies only sampled historical actions, from which the model must infer the opponent’s level. The language in CoT is evaluated by a blinded judge (Claude Opus 4.8) that extracts stated values without seeing the opponent description, target level, or submitted action; correctness, iterated depth response sequence (IDR), and KLK_L/KBK_B signals are then computed deterministically from the extraction. Table 2: Accuracy of reaching target strategic depth across games and condition. Parenthesised: within-one-level accuracy, |KB−ktarget|≤1|K_B-k_target|≤ 1. Rec. = recursive condition; Ind. = inductive condition at λ=10λ=10, the sampling precision at which the opponent’s level is reliably recoverable from the observed history. Full λ sweep in Appendix A4. 11–20 (Mod.) All-Pay Auction Nash Demand Ring 11–20 Model Rec. Ind. Rec. Ind. Rec. Ind. Rec. Ind. Sonnet 4.6 49.0 (99.0) 81.0 (81.0) 54.0 (100.0) 14.0 (86.0) 49.0 (91.0) 2.0 (6.0) 41.1 (97.8) 61.1 (61.1) Qwen 3.6 27B 91.0 (95.5) 90.5 (90.5) 92.5 (100.0) 10.0 (80.0) 83.5 (94.0) 1.0 (1.0) 86.0 (98.3) 64.4 (64.4) Gemma 4 31B 88.5 (96.5) 90.0 (90.0) 87.5 (95.5) 14.0 (77.5) 88.0 (89.5) 2.5 (4.0) 71.1 (80.6) 65.6 (71.1) DeepSeek v4 Pro 89.5 (99.5) 80.5 (90.5) 91.0 (100.0) 18.0 (89.5) 86.5 (89.0) 0.5 (0.5) 86.7 (89.4) 55.0 (56.1) Recursive condition The target level ktargetk_target is the depth a model must reason at to play the trial correctly. It is one level above the opponent it faces: against a level-j opponent, correct play is to best-respond to a(j)a^(j), which is level j+1j+1 play, so ktarget=j+1k_target=j+1. Opponents run from L0L_0 to L9L_9, giving ktarget=1k_target=1 to 10. The single correct action for a trial is therefore a(ktarget)a^(k_target), the entry at that position in the game’s IDR sequence, and a trial is scored accurate when the submitted action equals it. For the recursive condition, we provide information about the opponent’s level both through natural language of how sophisticated they are (levels 1-4) and direct reference to their levels (higher than level 4). Figure 2 shows how two depth signals track the target level. The behavioral signal KBK_B is the position of the submitted action in the game’s IDR sequence. The linguistic signal KLK_L is the position of the opponent action the CoT says it expects, plus one: a model that expects a level-j action and best-responds to it is itself reasoning at level j+1j+1. Both signals are on the same scale as ktargetk_target, so a model that reasons to the required depth has KB=KL=ktargetK_B=K_L=k_target. Each signal is defined only when the value it scores matches exactly one position in the IDR sequence. We observe internal consistency between intermediate reasoning and action for almost all models in Figure 2. KLK_L and KBK_B are tightly coupled: the dashed curves (KLK_L) and solid curves (KBK_B) are nearly indistinguishable throughout the full range k=1k=1 –10. That is, we see neither KL>KBK_L>K_B, where a model states a chain deeper than its action is carried out, nor KB>KLK_B>K_L, where a model submits an action at a depth its CoT never states. The accuracy in choosing actions consistent with increasing the depth of the target of strategic reasoning is also high in the models. Sonnet 4.6 demonstrates behaviour that is one level lower than the target beyond level 4, and gemma-4 shows a tendency to lose the capacity at a deeper level (9 and 10). Inductive condition The inductive condition removes the opponent description and replaces it with a tool call that returns the opponent’s historical trace. Those 30 actions are drawn by quantal best response against a(k−1)a^(k-1) with precision λ∈1,3,10λ∈\1,3,10\, therefore λ determines how many information does the opponent’s level the trace carries. When λ=1λ=1 and λ=3λ=3, the randomness in the trace caused decrease in accuracy, and KLK_L does not align with KBK_B at any depth. When λ=10λ=10, both signals track k closely everywhere except k=1k=1 and k=10k=10, and the four models are alike in doing so. Over that same range the two signals also track each other, coming apart only at those two endpoints. At k=1k=1 the opponent is non-strategic and a single best response is the whole task, but the models iterate several steps past it, so the action goes well beyond the correct level, even though the model’s stated belief identifies it accurately. At k=10k=10 the models identify the opponent almost exactly, yet the correct action is an extreme of the strategy space —for example, 50 (the smallest possible request) in the modified 11–20 game—but in 64 out of 80 trials the models fall back to the Level-0 default of 70 instead, so KBK_B ends up far below the model’s stated belief, which has already correctly reached the target. Game specific heterogeneity Table 2 reports accuracy in four games and both conditions, each cell pooling the ten target depths ktarget=1k_target=1 – 1010. In parentheses, we also report the accuracy by relaxing the metric such that the model hits the strategic depth at one level higher or lower for comparative analysis. A larger difference between the numbers indicates that the model came close to the exact target depth by one. In the recursive condition the four games behave alike, with only Sonnet standing well below the other three. Allowing for off-by-one error puts every model above 80% in every game, and Sonnet wins the most of the four because it consistently plays one level short of the target (Figure 2). In the inductive condition, accuracy remains high in the 11-20 Game, falls to roughly 55–66% in Ring 11-20, to 10–18% in the All-Pay Auction, and to under 3% in the Nash Demand Game. Relaxing the accuracy barely improves performance in the 11-20 Game and Ring 11-20, the All-Pay Auction shows the largest gain of any cell in the table while the Nash Demand Game shows none. We analyze the nature of these errors more in the next section. Figure 4: Per-step best-response consistency and chain length against target depth k, over recursive-condition trials, one panel per model. The solid line with markers is per-step consistency on the left axis—the share of adjacent steps in a model’s written run that are exact best responses—with 95% confidence intervals. The heavy dotted line is mean chain length on the right axis, and the grey dashed diagonal is the target length, m=km=k. Both series break between k=4k=4 and k=5k=5, where the form of the opponent description changes: chain length is drawn as two separate segments, and the consistency line runs straight from k=3k=3 to k=5k=5 with k=4k=4 hung off it on fine dotted spurs. Recursive Reasoning Sequence Consistency Next, we analyse why a model fails to reason at a higher strategic depth; as is the case for Sonnet-4.6. One possible explanation for this failure is an error drift: the model computes one best response incorrectly at a particular depth, and since that error persists, the chain of recursive reasoning ends on an action that is not on the correct IDR sequence. To test this, we let the model recursively write out a run of actions, one for each level it simulates—in the 11–20 game, 70 for the level-0 opponent, then 68 for level 1, then 66 for level 2. Let c=(c1,…,cm)c=(c_1,…,c_m) be that run as the blinded judge extracts it from the CoT. Based on this data, we record the following: The length,m, of the sequence, and its per-step correctness, i.e., the share of adjacent pairs for which ct+1=BR(ct)c_t+1=BR(c_t). To reach the target strategic depth, both needs to be correct. We show this analysis in Figure 4; the grey diagonal in Figure 4 is the correct target length and the solid line shows proportion of per-step correctness in best resopnse. For the model (Sonnet-4.6) that has errors in reaching the correct strategic depth, we find that the errors arise from the incorrect length of the IDR sequence chain it needs to recursively best respond, rather than the best response computation itself. Strategic Theory-of-mind in Inductive condition Under the inductive condition, the model only has access to the samples of the opponents’ gameplay. In order to infer the best response of the opponent from those samples, the model needs to apply a strategic theory of mind, to, first infer what the opponent beliefs about other players. These inferences are captured in the intermediate CoT through statements of the form “the opponent expects me to choose X, so…”. We analyze the rate of these explicit strategic mentalizing in language under the inductive condition in Table 3. We see that models that perform better in regards to strategic depth, have higher proportion of explicit mentalizing. When we compare the accuracy (with respect to achieving the target depth of reasoning), we observe an improvement when the model uses explicit mentalizing in its intermediate CoT. Table 3: Strategic theory-of-mind mentalizing. Model Mentalizing Acc. with mentalizing Acc. without mentalizing Sonnet 4.6 6.0% 40.3% 19.3% Qwen 3.6 27B 28.3% 29.6% 12.4% Gemma 4 31B 39.3% 20.6% 15.7% DeepSeek v4 Pro 65.7% 21.4% 10.4% Conclusions This paper addressed a validity gap in existing evaluations of strategic depth in LLMs. The canonical games used by prior work were built to probe the limits of human reasoning, and nothing guarantees that distinct depths of iterated reasoning map to distinct observable actions. We made this requirement precise as the level-k distinguishability condition, and by direct construction built a novel five-player ring game, Ring 11–20, and adapted three canonical two-player games so that their iterated depth-response sequences remain injective over the depths we wished to test. Applying these instruments to four models—Gemma 4 31B, Qwen 3.6 27B, Claude Sonnet 4.6, and DeepSeek v4 Pro—across recursive and inductive conditioning, we found accuracy to be high across the models overall, with degradation showing in model-specific ways. Sonnet 4.6’s behaviour falls one level short of the target beyond depth 4, while Gemma 4 shows a tendency to lose the capacity only at the deeper levels of 9 and 10. Looking inside the recursive condition, we found that the models’ stated beliefs about their opponent (KLK_L) and their submitted actions (KBK_B) stayed tightly coupled across almost the entire depth range, so degradation at higher depths was not a case of a model acting on a belief it never stated, or vice versa. Tracing Sonnet 4.6’s failures further, through its recursive reasoning depth, showed that the error was one of incorrect inference of the length of the iterated sequence the model needed to compute, rather than the best response calcuations. 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The best-response set at a belief a is BR(a)=s∈S:u(s,a)=maxs′∈Su(s′,a).BR(a)\;=\; \\,s∈ S\;:\;u(s;a)= _s ∈ Su(s ;a)\, \. (2) This set need not be a singleton, so Eq. (1) on its own does not define a sequence. We therefore fix a deterministic selection, the smallest maximiser, br(a)=minBR(a),br(a)\;=\; (a), (3) and iterate it, a(k)=br(a(k−1))a^(k)=br (a^(k-1) ), so that exactly one action is associated with each depth. Eq. (3) is a convention for defining the IDR sequence, not a rule the model is asked to follow. Best-response consistency (Sec. A) is accordingly evaluated against the full set: a model that breaks a tie differently from Eq. (3) has still played a best response and is recorded as consistent. In the four games the convention never binds along the sequence itself—BR(a(k))BR(a^(k)) is a singleton at every depth from 00 to 1010—so the sequences we report do not depend on it. The only belief at which two actions tie is a=10a=10 in Ring 11–20, which lies off the sequence (Sec. A). Throughout, s denotes the action of the player being scored and a the action that this player believes the opponent will take. A player in Ring 11–20 has two neighbours, but a level-k player believes that both of them play a(k−1)a^(k-1), so the payoff again depends on a single opponent action. Matching a number to a depth. A number is recorded at depth k when it equals a(k)a^(k); matching is exact equality over the integers. The submission tool does not constrain the returned value to S, so a model can submit a non-integer or an out-of-range number; such a submission matches no depth and is scored as incorrect. Depths and matching windows. Every trial describes an opponent at level ℓ∈0,…,9 ∈\0,…,9\. Best-responding to that opponent is play one level above it, so the target depth is ktarget=ℓ+1k_target= +1, running from 11 to 1010, and the action a trial scores as correct is always one of a(1),…,a(10)a^(1),…,a^(10). Reading a depth off a number the model produced uses a wider window, a(0),…,a(10)a^(0),…,a^(10). The window includes the anchor so that a submission equal to a(0)a^(0) is recorded at depth 00 rather than as unmatched. Two distinctness properties are used. The first is distinguishability over the described opponents a(0),…,a(9)\a^(0),…,a^(9)\ (Definition 1), which ensures that no two of them act alike. The second is distinctness of a(1),…,a(10)\a^(1),…,a^(10)\, which ensures that every trial has exactly one correct action and that no action is correct for two different trials. Both properties are verified over the full window a(0),…,a(10)\a^(0),…,a^(10)\ in Sec. A, and both hold in all four games with a single exception, Ring 11–20 at depth 1010. This is what the KmaxK_ row of Table 1 records: distinguishability runs through depth 1010 in 11–20, the All-Pay Auction and Nash Demand, and through depth 99 in Ring 11–20. The battery is irregular in exactly two places, and both are at depth 1010: in Nash Demand the step to a(10)a^(10) is produced by a different branch of the best-response map, and in Ring 11–20 a(10)a^(10) returns to the anchor, which is the one violation just noted. We treat both cases in Sec. A. The level-0 anchor. The anchor is the action a player would choose if it did not reason about the opponent at all. Each payoff function splits into a term in the player’s own action and one or more bonus terms that depend on how that action relates to the opponent’s. A non-strategic player does not ask whether a bonus condition will in fact be met, so we strip the bonuses of their dependence on a: each is treated as granted whatever the player does. What remains is a function of s alone, and the anchor is its maximiser over S. This is the base-payoff maximisation that the L0L_0 descriptions state in words (Tables 5 and 6). The construction applies uniformly to all four games and reproduces every anchor we use: it leaves s+32s+32 in 11–20, 1.8s1.8\,s in Nash Demand and s+13s+13 in Ring 11–20, each maximised at the top of the strategy space, and 18−s18-s in the All-Pay Auction, maximised at a bid of 00. The All-Pay anchor also follows from the maximin criterion argmaxsminau(s,a) _s _au(s;a), because a player who bids s and loses still pays s, so the worst case is least bad at a bid of 00. The two criteria select the same action. We report the maximin form, since it is the one expressed by the L0L_0 description shown to the model and the one against which the judge’s anchor_concept field is scored. Table 4: The best-response map of each game in closed form, checked against every action in S. Branches marked generating are those the IDR sequence uses; the remaining branches return the sequence to a value it has already visited and therefore set KmaxK_ . Game br(a)br(a) Range of a Role 11–20 (Mod.) a−2a-2 a≥52a≥ 52 generating 7070 a≤51a≤ 51 closes the cycle All-Pay Auction a+1a+1 a≤9a≤ 9 generating 00 a=10a=10 closes the cycle Nash Demand a−1a-1 a≥12a≥ 12 generating (undercut) 20−a20-a a≤11a≤ 11 concede; L10L_10 onward Ring 11–20 a−3a-3 a≥14a≥ 14 generating (−3-3 step) a+7a+7 10≤a≤1310≤ a≤ 13 generating (+7+7 wrap) 2020 a≤9a≤ 9 never reached Best-Response Maps and IDR Sequences Table 4 gives the best-response map of each game. Iterating these maps from the anchors gives • 11–20: 70,68,…,5070,68,…,50, then back to 7070; • All-Pay: 0,1,…,100,1,…,10, then back to 00; • Nash Demand: 20,19,…,1120,19,…,11, then 99; • Ring 11–20: 20,17,14,11,18,15,12,19,16,1320,17,14,11,18,15,12,19,16,13, then back to 2020. Two of the four change branch within the range of depths we test, and we derive both changes here. Nash Demand: the change of step at L10L_10. Consider a player who expects the opponent to demand a. Two replies have to be compared. To concede is to demand the largest amount that keeps the two demands compatible, s=20−as=20-a; this is paid in full and is worth 20−a20-a. To undercut is to demand s=a−1s=a-1; the two demands then sum to more than 20, but the smaller of them is still paid at the reduced rate of 0.80.8, so the reply is worth 0.8(a−1)0.8(a-1). Undercutting is the better reply exactly when 0.8(a−1)>20−a⇔a>20.81.8=11.5¯.0.8(a-1)>20-a a> 20.81.8=11. 5. (4) The sequence begins at 20 and falls by one at each step, so it undercuts down to 11. At a=11a=11 the inequality in Eq. (4) fails and the player concedes, giving br(11)=9br(11)=9. The step is −2-2, but it comes from a different branch of the map rather than from a larger version of the same step. Conceding is self-cancelling. A demand of a is met by conceding 20−a20-a, and conceding to that demand gives 20−(20−a)=a20-(20-a)=a again, so two conceding steps return to where they started. Once the sequence reaches 11 it therefore concedes to 9, concedes back to 11, and alternates between those two values forever. Ring 11–20: the sequence as a rotation. Every value of the Ring IDR sequence lies between 11 and 20. On that range the two generating branches of Table 4 can be written as a single formula, br(a)=((a−14)mod10)+11,a∈11,…,20,br(a)\;=\; ((a-14) 10 )+11, a∈\11,…,20\, (5) so the best response moves the action three places backwards around a cycle of ten. Because 3 and 10 are coprime, repeated backward steps of three visit all ten values before returning to the starting value, which is why a(10)=a(0)a^(10)=a^(0). The upward jumps of +7+7 occur at every third step: these are the steps at which subtracting 3 would leave the range 11,…,20\11,…,20\, so the best response wraps around instead. Three actions have to be compared at a wrapping step. At a=a(3)=11a=a^(3)=11 the primary-bonus action s=8s=8 pays 8+10=188+10=18, the secondary-bonus action s=18s=18 pays 18+3=2118+3=21, and the largest action that carries no bonus, s=20s=20, pays 20. The base payoff of s=20s=20 already exceeds the payoff of the primary-bonus action, and only the secondary-bonus action exceeds that base payoff, so the sequence jumps up to 18. At a=10a=10, one step below the range covered by Eq. (5), the secondary-bonus action s=17s=17 and the top action s=20s=20 both pay 20, and Eq. (3) selects br(10)=17br(10)=17. This belief is never reached from a(0)=20a^(0)=20 and therefore affects no IDR sequence. The main text states the +7+7 branch over 11≤a≤1311≤ a≤ 13, the values the sequence actually visits; Table 4 extends it down to a=10a=10 only so that the closed form agrees with the tie-break at that unreached belief. Distinguishability Bounds Lemma 1 Let R=a(k):k≥0R=\a^(k):k≥ 0\ be the set of values the IDR sequence ever reaches from a(0)a^(0). Then Kmax≤|R|−1K_ ≤|R|-1. Proof 1 The values a(0),a(1),…,a(|R|)a^(0),a^(1),…,a^(|R|) number |R|+1|R|+1 and all lie in R, so by the pigeonhole principle two of them are equal. Some pair of depths k1<k2≤|R|k_1<k_2≤|R| therefore shares an action, which Definition 1 forbids, so distinguishability can hold at most through depth |R|−1|R|-1. The bound has to be stated in terms of the values the sequence actually reaches rather than in terms of the whole strategy space, and 11–20 shows why. The image of its best-response map is 50,…,68∪70\50,…,68\∪\70\, which is 20 of the 21 actions in S; the one unreachable action is 69, which would require an opponent action of 71. The iteration, however, starts at a(0)=70a^(0)=70 and subtracts 2 at every generating step, so it visits only the eleven even actions between 50 and 70, giving |R|=11|R|=11 rather than 20. Lemma 1 holds with equality in all four games: each sequence visits every value it can reach exactly once, and the following step returns it to a value already used. This gives Kmax=10K_ =10 for 11–20, the All-Pay Auction and Nash Demand, and Kmax=9K_ =9 for Ring 11–20, as reported in Table 1. The mechanism differs across games. In 11–20 the action that would carry the bonus falls outside S, so br(51)=br(50)=70br(51)=br(50)=70. In the All-Pay Auction the winning bid falls outside S, so br(10)=0br(10)=0. In Nash Demand the switch to the conceding branch makes the sequence alternate. In Ring 11–20 the rotation of Eq. (5) has order 10. Three of the four sequences close by returning to the anchor, Ring 11–20 at a(10)a^(10) and the other two at a(11)a^(11); Nash Demand is the exception, returning to a(9)=11a^(9)=11 rather than cycling through the whole orbit. Behaviour at ktarget=10k_target=10. The deepest condition we test is not the same kind of step in all four games. In 11–20 (a(10)=50a^(10)=50) and in the All-Pay Auction (a(10)=10a^(10)=10) it is one further step of the branch that generated every earlier step. In Nash Demand the correct action, 9, is produced by the change of branch derived above, so a model can iterate correctly and still not reach it. In Ring 11–20 the correct action is a(10)=20=a(0)a^(10)=20=a^(0), so a submission of 20 matches two depths: BK_B then has two elements and KBK_B is undefined (Sec. D). A trial of that kind cannot identify a depth, and we therefore exclude Ring 11–20 at ktarget=10k_target=10 from the depth figures. These trials are excluded from the accuracy in Table 2 as well, so the Ring 11–20 denominator is 720 trials per model across all four conditions rather than 800 (713 for Qwen 3.6 27B, which lost seven trials to API errors), and 180 rather than 200 in each of the two Ring columns of that table. The exclusion is not neutral: the ambiguous submission 20 is scored as correct at ktarget=10k_target=10, and it is submitted in 230 of the 320 affected trials, so retaining them would raise pooled accuracy from 31.9%31.9\% (n=12,467n=12,467) to 32.9%32.9\% (n=12,787n=12,787) and raise every per-model Ring figure (Sonnet 4.6 28.6→31.2%28.6→ 31.2\%, DeepSeek v4 Pro 39.4→44.1%39.4→ 44.1\%, Gemma 4 31B 36.0→37.2%36.0→ 37.2\%, Qwen 3.6 27B 38.7→44.6%38.7→ 44.6\%). Extrapolability of the IDR Sequences A model can arrive at a(k)a^(k) without performing k iterations. In three of the four games the IDR sequence is an arithmetic progression, so the whole sequence is determined by its first two values: a model that computes a(1)a^(1) and applies the step −2-2, +1+1 or −1-1 can produce a(10)a^(10) in a single operation. In these three games the distance of an action from the anchor and the depth of that action are collinear by construction, so neither accuracy nor the two depth signals separates iteration from extrapolation. Ring 11–20 separates them. Its sequence also has a closed form, Eq. (5), but that form is modular rather than affine and cannot be recovered from the first two values. A model that extrapolates the constant step visible at L1L_1–L3L_3 produces 20,17,14,11,8,5,2,…20,17,14,11,8,5,2,… instead of 20,17,14,11,18,15,12,…20,17,14,11,18,15,12,…, and the two sequences diverge at the first wrap, k=4k=4. The extrapolated values stay inside S through k=6k=6 and lie off the IDR sequence at every depth from k=4k=4 onward, so an extrapolating submission is recorded as matching no depth rather than as correct. Ring 11–20 therefore carries an extrapolation check that the other three games cannot, which is one reason accuracy is broken out by game in Table 2 rather than pooled across games, and why the per-step analysis of Sec. Recursive Reasoning Sequence Consistency scores the written intermediate values rather than the endpoint alone. Scoring Best-Response Consistency Best-response consistency asks whether the action a model submitted is a best response to the belief it stated. In the recursive condition the judge extracts a point belief b b, and the action a a is checked against the whole best-response set: consistent⇔a^∈BR(b^),consistent a ( b), (6) where BR(b^)BR( b) is obtained by enumerating u(s,b^)u(s; b) over all s∈Ss∈ S and taking the exact maximisers. In the inductive condition a model frequently states a distribution over opponent actions rather than a single action. Where the judge extracts a non-empty belief_support, consistency is scored against expected utility under that support: a a must maximise ∑jpju(a^,bj) _jp_j\,u( a;b_j) over S, where the weights pjp_j are taken from the likelihood phrases the model itself used. The point-belief check of Eq. (6) is retained alongside this test under a separate field. Where no support is extracted, Eq. (6) is used unchanged. Appendix B Prompts, Models, and Execution System Prompts The system prompt is the same for every game, condition, and model. You are a rational game-theory player. Think step by step about your strategy. Show all your reasoning, then call the submit_action tool with your final chosen number. You MUST use the tool to submit --- do not just state your answer in text. In the inductive condition one further paragraph is appended: You also have access to a query_opponent_history tool that returns your opponent’s raw action choices from their previous games against other players. You may call it to inform your decision. Game Rules and Opponent Descriptions The user turn of every trial begins with the rules of the game and, in the recursive condition, continues with a description of the opponent. Tables 5 and 6 reproduce both verbatim. The rules text is identical across conditions and depths, so the only material that changes between trials is what follows it. The opponent’s depth is conveyed in two ways. For L0L_0–L3L_3 (ktarget=1k_target=1 to 44) it is conveyed in natural language: the description states how sophisticated the opponent is, and the model has to count the levels from the way the sentences are nested. For L4L_4–L9L_9 (ktarget=5k_target=5 to 1010) the level is named directly, using a template: Your opponent is || Your opponents are Level k. A Level k player best-responds to a Level (k-1) player, all the way down to Level 0, which is defined as follows: L0L_0 description The level-0 sentence substituted into the template is the same string that appears in the L0L_0 row for that game. The two formats therefore give the model the same starting point and differ only in how the depth above that starting point is conveyed. The wording changes between ktarget=4k_target=4 and ktarget=5k_target=5, and the target depth increases across the same boundary, so the design as run cannot separate the effect of the wording from the effect of the depth; comparisons that cross this boundary confound the two. The inductive condition supplies no opponent description. The user turn contains the rules followed by: You may use the query_opponent_history tool to view your opponent’s past action choices in previous games against other players. This data may help you anticipate their strategy. No part of this text mentions levels, iteration, best responses, or the opponent reasoning about the model. This is the basis for the explicit-mentalizing measure reported in Sec. Strategic Theory-of-mind in Inductive condition and Table 3. Table 5: Rules text and L0L_0–L3L_3 opponent descriptions for 11–20 (Mod.) and the All-Pay Auction, reproduced verbatim. The L0L_0 description is also substituted into the template used for L4L_4–L9L_9. Game Text 11–20 (Mod.) Rules I am playing a two-player game. My opponent and I each simultaneously choose an integer from 50 to 70. My payoff equals my chosen number, plus a bonus of 24 if my number is exactly 2 less than my opponent’s, or a bonus of 8 if my number is exactly 1 less than my opponent’s. I want to maximize my payoff. What integer should I choose? L0L_0 Your opponent plays naively. They have no strategic awareness and simply try to maximize their own base payoff without considering how your choice might interact with theirs. L1L_1 Your opponent is a basic strategic thinker. They assume you will naively maximize your own base payoff, and they choose the action that best responds to a base-payoff maximizer. L2L_2 Your opponent is a moderately sophisticated thinker. They assume you are a basic strategist who best-responds to naive base-payoff maximizers, and they choose the action that best responds to that level of reasoning. L3L_3 Your opponent is a deeply strategic thinker. They assume you are moderately sophisticated --- someone who best-responds to basic strategists --- and they choose the action that best responds to that deeper level of reasoning. All-Pay Auction Rules I am playing a two-player all-pay auction. My opponent and I each simultaneously choose a bid from 0 to 10 (integers only). Both of us pay our bid regardless of the outcome. If my bid is strictly higher, I win a prize of 12. If we tie, I receive 6. If my bid is lower, I receive nothing. My payoff is: prize (if any) minus my bid. I want to maximize my payoff. What integer should I bid? L0L_0 Your opponent plays naively. They are very cautious and would rather not bid at all than risk paying a bid and losing. They do not think strategically about what you might bid or how to compete for the prize. L1L_1 Your opponent is a basic strategic thinker. They assume you are very cautious and will not bid at all to avoid any risk of losing money, and they choose the bid that best responds to a non-bidder. L2L_2 Your opponent is a moderately sophisticated thinker. They assume you are a basic strategist who places the minimum bid needed to beat a cautious non-bidder, and they choose the bid that best responds to that level of reasoning. L3L_3 Your opponent is a deeply strategic thinker. They assume you are moderately sophisticated --- someone who just barely outbids basic strategists --- and they choose the bid that best responds to that deeper level of reasoning. Table 6: Rules text and L0L_0–L3L_3 opponent descriptions for Nash Demand and Ring 11–20, reproduced verbatim. Game Text Nash Demand Rules I am playing a two-player Nash demand game. A prize of 20 is at stake. My opponent and I each simultaneously demand an integer from 1 to 20. If our demands sum to 20 or less, I receive my demand. If our demands exceed 20 but my demand is strictly less than my opponent’s, I receive 80% of my demand. Otherwise I receive nothing. I want to maximize my payoff. What integer should I demand? L0L_0 Your opponent plays naively. They are greedy and try to claim as much of the prize as possible for themselves, without considering how your demand might interact with theirs. L1L_1 Your opponent is a basic strategic thinker. They assume you are greedy and will demand as much as possible, and they choose the demand that best responds to a greedy maximizer. L2L_2 Your opponent is a moderately sophisticated thinker. They assume you are a basic strategist who slightly undercuts a greedy maximizer, and they choose the demand that best responds to that level of reasoning. L3L_3 Your opponent is a deeply strategic thinker. They assume you are moderately sophisticated --- someone who undercuts basic strategists --- and they choose the demand that best responds to that deeper level of reasoning. Ring 11–20 Rules I am one of 5 players arranged in a ring (so player 1 neighbors players 2 and 5, etc.). Each of us simultaneously chooses an integer from 1 to 20. My payoff equals my chosen number, plus a bonus of 10 if my number is exactly 3 less than my left neighbor’s number, plus a bonus of 3 if my number is exactly 7 more than my right neighbor’s number. I want to maximize my payoff. What integer should I choose? L0L_0 Your opponents play naively. They have no strategic awareness and simply try to maximize their own base payoff without considering how other players’ choices might interact with theirs. L1L_1 Your opponents are basic strategic thinkers. They assume other players naively maximize their own base payoff, and they choose the action that best responds to base-payoff maximizers. L2L_2 Your opponents are moderately sophisticated thinkers. They assume other players are basic strategists who best-respond to naive base-payoff maximizers, and they choose the action that best responds to that level of reasoning. L3L_3 Your opponents are deeply strategic thinkers. They assume other players are moderately sophisticated --- those who best-respond to basic strategists --- and they choose the action that best responds to that deeper level of reasoning. Models, Decoding, and Protocol Table 7 lists the settings under which trials were run. Run health by model, and the breakdown by target depth, are in the released logs. Table 7: Models, decoding, and protocol. Item Specification Models Called through OpenRouter by slug: google/gemma-4-31b-it, qwen/qwen3.6-27b, claude-sonnet-4-6, deepseek-v4-pro Reasoning mode Default effort requested for gemma-4 and qwen3.6; no reasoning parameter sent for the other two, which ran at the provider default. The resolved configuration is recorded per trial Decoding Temperature 0.2 and max_tokens 8192 sent on every request, and reported as sent: a provider may ignore or clamp them Routing Provider fallback enabled, so consecutive requests may be served by different upstream providers; the serving provider is recorded per turn. Pin a single provider to reproduce a run exactly Tools submit_action takes the final chosen number and is called once after reasoning. query_opponent_history takes no arguments and returns 30 integers; its one-call limit is stated in the description but not enforced, and the call count is recorded Turn cap 10 turns per trial Nudge A turn ending with no tool call and no action yet submitted receives: “You have not submitted an action yet. Call the submit_action tool now with your final chosen number. Do not write any more reasoning.” Chain of thought Reasoning-channel and visible text from every turn before submission, concatenated in order; text written after submission is excluded Design 44 games × 1010 opponent levels × \recursive, inductive λ=1,3,10λ\!=\!1,3,10\ × 2020 repetitions × 4 models, or 3200 trials per model (3201 for Gemma 4 31B) Failed trials A trial that raised an API error is recorded with the error and excluded; a trial that completed without an action is scored as incorrect rather than dropped, so accuracy cannot be raised by declining to answer. No model exceeded a 0.7% no-action rate. Errors: none for Sonnet 4.6; 22 no-action trials for DeepSeek v4 Pro; 3 generation and 72 judge API errors for Gemma 4 31B; 11 generation, 24 judge API and 3 no-action for Qwen 3.6 27B. No generation-side error left a trial without a result record A Worked Example Figure 5 shows a trial end to end in the format the harness records: the two prompt turns, the tool call, the chain of thought, the judge’s extraction, and the quantities the code derives from that extraction. The prompt turns, the extraction schema and the derived quantities are exactly as the pipeline produces them; the chain of thought is a constructed illustration rather than a verbatim log, so that one figure can display a fully consistent chain alongside every derived quantity. Verbatim transcripts, including trials whose chains are short or contain a step that is not a best response, are in the released logs. The example is a recursive-condition trial in the modified 11–20 game against an L5L_5 opponent, so ktarget=6k_target=6 and the correct action is a(6)=58a^(6)=58. Illustrative trial, modified 11–20 game, recursive condition, opponent level 5, ktarget=6k_target=6, correct action a(6)=58a^(6)=58. System turn. You are a rational game-theory player. Think step by step about your strategy. Show all your reasoning, then call the submit_action tool with your final chosen number. You MUST use the tool to submit --- do not just state your answer in text. User turn (rules, then the templated L5L_5 description). I am playing a two-player game. My opponent and I each simultaneously choose an integer from 50 to 70. My payoff equals my chosen number, plus a bonus of 24 if my number is exactly 2 less than my opponent’s, or a bonus of 8 if my number is exactly 1 less than my opponent’s. I want to maximize my payoff. What integer should I choose? Your opponent is Level 5. A Level k player best-responds to a Level (k-1) player, all the way down to Level 0, which is defined as follows: Your opponent plays naively. They have no strategic awareness and simply try to maximize their own base payoff without considering how your choice might interact with theirs. Chain of thought in the recorded format (reasoning channel and visible text before submission). Level 0 ignores the bonuses entirely and just takes the biggest number, so L0 plays 70. Against 70, the 24 bonus needs 68, worth 92, versus 69 with the 8 bonus at 77, versus 70 alone at 70. So L1 = 68. The same comparison holds at every step: undercutting by 2 is always worth 24 against a base difference of 2, so each level subtracts 2. L2 = 66, L3 = 64, L4 = 62, L5 = 60. My opponent is Level 5, so they will choose 60. I should best-respond to 60, which means 58 (payoff 58 + 24 = 82), better than 59 (59 + 8 = 67) or 70 (70). Tool call. submit_action(action=58) Judge extraction (recursive-condition schema, fields shown in Table 8; quotes elided for space). "anchor_present": true, "anchor_value": 70, "anchor_concept": "base_payoff_maximizer", "chain": ["value": 70, "attributed_to": "opponent", "value": 68, "attributed_to": "opponent", "value": 66, ..., "value": 64, ..., "value": 62, ..., "value": 60, "attributed_to": "opponent"], "iteration_verbalized": "YES", "final_belief_point": 60, "final_belief_ambiguous": false, "belief_is_distribution": false, "claimed_br": 58, "opponent_models_me": false Derived quantities. a^=58 a=58, and B=6K_B=\6\, so KB=6K_B=6. b^=60=a(5) b=60=a^(5), and L=5K_L=\5\, so KL=6K_L=6. Accuracy: a^=a(6) a=a^(6), correct. Best-response consistency: BR(60)=58BR(60)=\58\ and a^∈BR(60) a (60), consistent. Chain: c=(70,68,66,64,62,60)c=(70,68,66,64,62,60), so m=|c|=6=ktargetm=|c|=6=k_target, and all five consecutive pairs are exact best responses, so C=1C=1. Figure 5: A trial end to end, from the two prompt turns through to the derived quantities. The prompt turns, extraction schema and derived quantities are as the pipeline produces them; the chain of thought is a constructed illustration, not a verbatim log. The released logs contain the verbatim text, the judge’s quotes, and the per-turn provider record for every trial. Appendix C Inductive Condition: Generating the Opponent History Sampling For an opponent at level k≥1k≥ 1, the 30 records shown to the model are drawn independently from a quantal best response to a(k−1)a^(k-1), Pλ(h=s∣k)=exp(λu~(s,a(k−1)))∑s′∈Sexp(λu~(s′,a(k−1))),P_λ\! (h=s k )= \! (λ\, u(s;a^(k-1)) ) _s ∈ S \! (λ\, u(s ;a^(k-1)) ), (7) u~(s,a)=u(s,a)−umin(a)umax(a)−umin(a), u(s;a)= u(s;a)-u_ (a)u_ (a)-u_ (a), where umax(a)u_ (a) and umin(a)u_ (a) are the largest and smallest payoffs available at the fixed belief a=a(k−1)a=a^(k-1). Any constant shift of u~ u leaves Eq. (7) unchanged, so the choice of offset is immaterial. Rescaling in this way makes a single value of λ comparable across games whose payoffs differ in magnitude, and it leaves the ordering of the payoffs unchanged, so a(k)a^(k) is the modal draw at every λ. What λ controls is how tightly the 30 records concentrate on that value, and therefore how clearly the opponent’s depth can be read off the history. Eq. (7) is the quantal-response rule of the quantal cognitive hierarchy (Wright and Leyton-Brown 2010) applied to the single action a(k−1)a^(k-1), whereas a full cognitive-hierarchy belief would average over all lower types (Camerer, Ho, and Chong 2004). We use the degenerate level-k belief because the recursive condition describes an opponent with exactly that belief and the two conditions should present the same opponent; where the distinction matters below we refer to the generator as a quantal level-k generator. The sampled histories are released with the logs. Why level 0 is not the same opponent in the two conditions. A level-0 opponent has no beliefs, so there is nothing for Eq. (7) to best-respond to. In the inductive condition we therefore generate its history by drawing 30 actions uniformly at random from S, following the usual quantal-response convention. In the recursive condition, level 0 is instead described in words as a specific non-strategic action, and the model can read the anchor a(0)a^(0) off that description. The same nominal level thus means plays at random in one condition and plays a(0)a^(0) in the other. Two things follow. First, λ has no effect at ktarget=1k_target=1, because uniform draws are uniform at any value of λ; the three settings are exact replicates at that depth, and any statement about how the two depth signals move with λ therefore applies to ktarget≥2k_target≥ 2. Second, the task itself differs between the conditions at that depth. Recursively, the model best-responds to an action it has been told about; inductively, it has to recognise that the opponent has no pattern at all and decide what to do about that. Accuracy at ktarget=1k_target=1 is therefore not comparable across conditions, and cross-condition comparisons at that depth should be read with this in mind. The λ Sweep The sweep is released with the logs as lambda_sweep.csv: one row per (model, game, target depth, λ) cell, 480 rows in all, each giving accuracy and the two depth signals KLK_L and KBK_B. This is the breakdown that Table 2 refers to. Appendix D Judge Protocol and Derived Metrics Protocol The judge receives the rules of the game and the raw chain of thought, and nothing else: not the opponent description, the opponent’s level, the target depth, the sampled history, the tool calls, or the submitted action. Its system prompt restricts it to reporting what the text says, and it must supply a verbatim quote for every value it reports. You are an annotation judge. You are given the rules of a strategic game and a Chain-of-Thought (CoT) transcript produced by a language model playing that game. Your task is EXTRACTION ONLY. Rules: - Extract only what is explicitly stated in the CoT text. Do not infer, guess, fill gaps, verify correctness, or assign Level-K labels. - Every extracted value MUST include a verbatim ‘quote’ copied from the CoT. - If information is absent, use null (and empty lists for list fields). - Report numbers exactly as stated; preserve order of appearance. - Output ONLY a single valid JSON object matching the requested schema. No markdown fences, no commentary. Quote grounding. Every extracted value carries a quote that the schema requires to be copied from the transcript, which gives a mechanical check on the extraction: we test each quote for exact substring membership in the chain of thought it was drawn from, after whitespace normalisation. The outcome of that check is recorded for every extracted value and released with the logs. The judge belongs to the same model family as one of the four evaluated models. The extraction-only design limits the room for a self-preference effect, since the judge is asked to copy values rather than to assess quality and never sees the submitted action or whether the trial was correct, but it does not eliminate that possibility; the per-model quote-grounding records in the released logs are the evidence we can offer on this point. Extraction Schema The two conditions use separate schemas, because the material available in the chain of thought differs: a recursive-condition transcript can state an anchor and a chain of iterated values, whereas an inductive-condition transcript can instead make claims about the sampled history. Table 8 lists the fields that feed the results we report, with boolean-and-quote pairs collapsed into a single entry. The full schemas are in the released code. Table 8: The judge fields that feed the reported results. Group Cond. Fields Anchor rec. anchor_present, anchor_value, anchor_concept Chain rec. chain[], entries (value,attributed_to,quote)( value, attributed\_to, quote); iteration_verbalized History ind. history_referenced, cited_statistics[], noise_acknowledged, claimed_n Belief both final_belief_point with final_belief_ambiguous and final_belief_candidates; belief_is_distribution, belief_support[] Decision both claimed_br, payoff_computations[] Mentalizing rec. opponent_models_me Mentalizing ind. spurious_recursion, the mentalizing flag behind Table 3 attributed_to ∈ opponent, opponent_model_of_self, generic_player, unclear; iteration_verbalized ∈ YES, NO, STOP_RULE_STATED. Every field asks the judge to copy a value out of the text, with the exception of the two mentalizing fields. Those two ask for a decision: whether the chain of thought describes the opponent as reasoning about the model’s own current choice. Both record the same annotation, which we call explicit strategic mentalizing throughout, and both are extracted from the chain of thought alone, without sight of the target level or the submitted action. The released code names them differently by condition, opponent_models_me in the recursive condition and spurious_recursion in the inductive one, because the prompt gives such a statement a basis in one and not in the other: recursively, the prompt states the opponent’s level, and a level-k opponent is by construction one that best-responds to a belief about the other player, so the statement is supported; inductively, no part of the prompt describes the opponent as reasoning about anyone. What the Code Computes from the Extraction Let a a be the submitted action and b b the final belief the judge extracted. Matching is performed over the full window =0,1,…,10W=\0,1,…,10\ of depths for which the IDR sequence is defined: B _B =k∈:a(k)=a^, =\k :a^(k)= a\, (8) L _L =k∈:a(k)=b^. =\k :a^(k)= b\. (9) Each signal is defined only when the corresponding set contains exactly one element; KBK_B is then that element and KLK_L is that element plus one. Because W contains depth 10, KLK_L can be as large as 11, one step beyond the deepest condition we run. Such a trial is one in which the model states a belief appropriate to an L10L_10 opponent, although the deepest opponent we describe is L9L_9. We retain these trials as resolved values of KLK_L rather than discarding them, because discarding them would remove over-iteration selectively. In Nash Demand a(11)=a(9)a^(11)=a^(9), so a stated belief of 11 in that game matches depth 9 alone and yields KL=10K_L=10 rather than 12. A signal can fail to resolve in three ways, recorded separately: the value is present but matches no depth (unmatched); it matches more than one depth (ambiguous); or the chain of thought states no such value at all (none). The ambiguous case arises in Ring 11–20 whenever a model submits 20, which matches both a(0)a^(0) and a(10)a^(10). We compute the relationship between KLK_L and KBK_B only over trials in which both signals resolve. Trials with confused reasoning are both more likely to be wrong and harder to read, so they are excluded more often than clear trials, and the agreement between the two signals should therefore be read as an upper estimate. The per-step measure plotted in Figure 4 is the share of consecutive pairs in the model’s written chain c for which the second value is a best response to the first: C=1|c|−1∑t=1|c|−1[ct+1∈BR(ct)],|c|≥2.C= 1|c|-1 _t=1^|c|-11\! [c_t+1 (c_t) ], |c|≥ 2. (10) The membership test uses the best-response set, so a step that breaks a tie differently from Eq. (3) is not penalised. A chain of one value contains no consecutive pairs, so C is undefined for such a chain. This applies at ktarget=1k_target=1, where a fully correct chain consists of a single value: the k=1k=1 point on that curve is computed only over the trials in which the model wrote a second value that the task did not require, and it is therefore not comparable with the points at greater depths. The second measure is the chain length m=|c|m=|c|, compared against the length the trial requires, m=ktargetm=k_target. Uncertainty For proportions we report Wilson score intervals, p^+z22n±zp^(1−p^)n+z24n21+z2n,z=1.96, p+ z^22n± z p(1- p)n+ z^24n^21+ z^2n, z=1.96, (11) rather than Wald intervals, because several of the reported quantities lie close to 0 or 1, where the Wald interval behaves badly. For means of non-binary quantities, such as the mean chain length in Figure 4, we use normal-approximation intervals. Repeated trials within one condition share a prompt, a sampled history and a game, so intervals computed as if the trials were independent are too narrow. We therefore also report a cluster bootstrap, resampling whole (game, condition, depth, λ) cells with replacement over B=2000B=2000 replicates. The resulting intervals are substantially wider than Eq. (11): a median of 4.1×4.1× across the four per-model estimates, and 7.6×7.6× for the pooled estimate, where clustering bites hardest because the pooled sample spans the most cells. Where the two disagree, the figures show the wider interval.