Evaluating Counterfactual Strategic Reasoning in Large Language Models

Dimitrios Georgousis; Maria Lymperaiou; Angeliki Dimitriou; Giorgos Filandrianos; Giorgos Stamou

cs.CLMar 19, 2026

Evaluating Counterfactual Strategic Reasoning in Large Language Models

Authors: Dimitrios Georgousis, Maria Lymperaiou, Angeliki Dimitriou, Giorgos Filandrianos, Giorgos Stamou

Organizations: National Technical University of Athens

Abstract

We evaluate Large Language Models (LLMs) in repeated game-theoretic settings to assess whether strategic performance reflects genuine reasoning or reliance on memorized patterns. We consider two canonical games, Prisoner's Dilemma (PD) and Rock-Paper-Scissors (RPS), upon which we introduce counterfactual variants that alter payoff structures and action labels, breaking familiar symmetries and dominance relations. Our multi-metric evaluation framework compares default and counterfactual instantiations, showcasing LLM limitations in incentive sensitivity, structural generalization and strategic reasoning within counterfactual environments.

Figures & tables

Figure 1: LLM interaction for Prisoner’s Dilemma

Figure 2: Counterfactual impact on opponent comprehension $m$ across models (zero-shot, LLM-vs-LLM). Lower values indicate earlier adaptation, while values closer to the game horizon indicate delayed opponent modeling.

Appendix figures & tables25 assets

Supplementary material from the paper’s appendix.

Appendix

		PD
		zs	spp	cot	srep	pp	mf	tft
model	prompt
C3.5Sv2	zs	64.0 $\pm$ 0.0	64.0 $\pm$ 0.0	64.0 $\pm$ 0.0	29.8 $\pm$ 0.4	54.2 $\pm$ 4.5	30.2 $\pm$ 1.1	30.6 $\pm$ 0.9
	cot	64.0 $\pm$ 0.0	64.0 $\pm$ 0.0	64.0 $\pm$ 0.0	30.0 $\pm$ 0.0	53.6 $\pm$ 1.5	31.0 $\pm$ 1.0	30.4 $\pm$ 0.9
	spp	64.0 $\pm$ 0.0	64.0 $\pm$ 0.0	64.0 $\pm$ 0.0	30.4 $\pm$ 0.5	58.4 $\pm$ 4.3	30.6 $\pm$ 0.9	32.0 $\pm$ 0.0
	sc-zs	64.0 $\pm$ 0.0	64.0 $\pm$ 0.0	64.0 $\pm$ 0.0	30.5 $\pm$ 0.7	52.0 $\pm$ 0.0	32.0 $\pm$ 0.0	30.0 $\pm$ 1.4

Appendix

Table 1: Total Points Averaged Over All Iterations for the default PD. Bold indicates the highest total points per row.

		PD label-based counterfactual
		zs	spp	cot	srep	pp	mf	tft
model	prompt
C3.5Sv2	zs	96.0 $\pm$ 0.0	96.0 $\pm$ 0.0	96.0 $\pm$ 0.0	44.0 $\pm$ 5.3	52.0 $\pm$ 2.1	89.0 $\pm$ 0.0	60.6 $\pm$ 32.3
	cot	96.0 $\pm$ 0.0	96.0 $\pm$ 0.0	96.0 $\pm$ 0.0	47.8 $\pm$ 3.8	49.6 $\pm$ 2.1	91.8 $\pm$ 3.8	60.0 $\pm$ 32.9
	spp	96.0 $\pm$ 0.0	96.0 $\pm$ 0.0	96.0 $\pm$ 0.0	44.8 $\pm$ 9.8	47.6 $\pm$ 1.7	90.4 $\pm$ 3.1	72.0 $\pm$ 32.9
	sc-zs	96.0 $\pm$ 0.0	96.0 $\pm$ 0.0	96.0 $\pm$ 0.0	40.0 $\pm$ 7.1	54.0 $\pm$ 1.4	89.0 $\pm$ 0.0	96.0 $\pm$ 0.0

Appendix

Table 2: Total Points Averaged Over All Iterations for the label-based PD counterfactual.

		SH (payoff-based counterfactual of PD)
		zs	spp	cot	srep	pp	mf	tft
model	prompt
C3.5Sv2	zs	96.0 $\pm$ 0.0	96.0 $\pm$ 0.0	96.0 $\pm$ 0.0	43.8 $\pm$ 3.7	51.2 $\pm$ 5.1	91.8 $\pm$ 3.8	59.2 $\pm$ 33.6
	cot	96.0 $\pm$ 0.0	96.0 $\pm$ 0.0	96.0 $\pm$ 0.0	43.0 $\pm$ 3.1	48.2 $\pm$ 0.8	59.4 $\pm$ 33.4	46.8 $\pm$ 27.5
	spp	96.0 $\pm$ 0.0	96.0 $\pm$ 0.0	96.0 $\pm$ 0.0	36.8 $\pm$ 3.8	46.8 $\pm$ 1.3	79.8 $\pm$ 24.7	83.6 $\pm$ 27.7
	sc-zs	96.0 $\pm$ 0.0	96.0 $\pm$ 0.0	96.0 $\pm$ 0.0	44.5 $\pm$ 6.4	52.0 $\pm$ 8.5	92.5 $\pm$ 4.9	96.0 $\pm$ 0.0

Appendix

Table 3: Total Points Averaged Over All Iterations for Stag Hunt (payoff-based PD counterfactual)

		Joint counterfactual of PD
		zs	spp	cot	srep	pp	mf	tft
model	prompt
C3.5Sv2	zs	64.0 $\pm$ 0.0	64.0 $\pm$ 0.0	64.0 $\pm$ 0.0	30.2 $\pm$ 0.4	58.8 $\pm$ 5.0	31.2 $\pm$ 1.1	31.0 $\pm$ 1.0
	cot	64.0 $\pm$ 0.0	64.0 $\pm$ 0.0	64.0 $\pm$ 0.0	30.8 $\pm$ 0.4	58.4 $\pm$ 1.8	31.4 $\pm$ 0.5	31.6 $\pm$ 0.9
	spp	64.0 $\pm$ 0.0	64.0 $\pm$ 0.0	64.0 $\pm$ 0.0	30.4 $\pm$ 0.5	61.6 $\pm$ 0.9	31.6 $\pm$ 0.9	31.6 $\pm$ 1.8
	sc-zs	64.0 $\pm$ 0.0	64.0 $\pm$ 0.0	64.0 $\pm$ 0.0	30.5 $\pm$ 0.7	59.5 $\pm$ 0.7	31.0 $\pm$ 1.4	32.0 $\pm$ 0.0

Appendix

Table 4: Total Points Averaged Over All Iterations for the joint PD counterfactual (label-based and payoff-based).

		PD
		zs	spp	cot	srep	pp	mf	tft
model	prompt
C3.5Sv2	zs	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	2.2 $\pm$ 0.4	8.4 $\pm$ 7.0	5.6 $\pm$ 6.4	2.8 $\pm$ 1.3
	cot	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	2.0 $\pm$ 0.0	14.0 $\pm$ 2.0	3.0 $\pm$ 1.2	2.2 $\pm$ 0.4
	spp	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.6 $\pm$ 0.5	7.6 $\pm$ 6.1	2.0 $\pm$ 0.7	3.0 $\pm$ 0.0
	sc-zs	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.5 $\pm$ 0.7	12.0 $\pm$ 0.0	3.0 $\pm$ 0.0	4.5 $\pm$ 0.7

Appendix

Table 5: Round

m

# where the LLM player understood the opponent’s Strategy in default PD. Bold for best results (lower number of rounds

m

at which opponent comprehension was realized).

		PD label-based counterfactual
		zs	spp	cot	srep	pp	mf	tft
model	prompt
C3.5Sv2	zs	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	14.0 $\pm$ 1.9	13.4 $\pm$ 2.2	2.0 $\pm$ 0.0	10.0 $\pm$ 8.2
	cot	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	16.6 $\pm$ 0.5	15.4 $\pm$ 1.7	1.6 $\pm$ 0.5	9.0 $\pm$ 8.0
	spp	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	15.6 $\pm$ 1.1	16.6 $\pm$ 0.9	1.8 $\pm$ 0.4	6.0 $\pm$ 7.3
	sc-zs	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	16.5 $\pm$ 0.7	11.0 $\pm$ 0.0	2.0 $\pm$ 0.0	1.0 $\pm$ 0.0

Appendix

Table 6: Round

m

# where the LLM player understood the opponent’s Strategy in the label-based counterfactual PD setup.

		SH (payoff-based counterfactual of PD)
		zs	spp	cot	srep	pp	mf	tft
model	prompt
C3.5Sv2	zs	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	15.8 $\pm$ 0.8	13.2 $\pm$ 5.8	1.6 $\pm$ 0.5	4.2 $\pm$ 4.1
	cot	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	16.0 $\pm$ 1.2	17.0 $\pm$ 0.0	3.4 $\pm$ 2.3	6.2 $\pm$ 4.0
	spp	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	11.8 $\pm$ 5.8	17.0 $\pm$ 0.0	2.4 $\pm$ 1.5	1.8 $\pm$ 1.8
	sc-zs	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	16.0 $\pm$ 0.0	11.5 $\pm$ 7.8	1.5 $\pm$ 0.7	1.0 $\pm$ 0.0

Appendix

Table 7: Round

m

# where the LLM player understood the opponent’s Strategy in the SH counterfactual.

		Joint counterfactual of PD
		zs	spp	cot	srep	pp	mf	tft
model	prompt
C3.5Sv2	zs	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.8 $\pm$ 0.4	2.0 $\pm$ 0.0	2.6 $\pm$ 0.5	2.2 $\pm$ 0.8
	cot	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.2 $\pm$ 0.4	6.0 $\pm$ 5.8	1.8 $\pm$ 1.1	2.8 $\pm$ 0.4
	spp	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.6 $\pm$ 0.5	2.0 $\pm$ 0.0	2.8 $\pm$ 0.4	1.8 $\pm$ 0.8
	sc-zs	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.5 $\pm$ 0.7	3.0 $\pm$ 1.4	2.5 $\pm$ 0.7	3.0 $\pm$ 0.0

Appendix

Table 8: Round

m

# where the LLM player understood the opponent’s strategy for the joint PD counterfactual (label-based and payoff-based).

		PD
		zs	spp	cot	srep	pp	mf	tft
model	prompt
C3.5Sv2	zs	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	0.1 $\pm$ 0.0	0.4 $\pm$ 0.2	0.2 $\pm$ 0.1	0.2 $\pm$ 0.1
	cot	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	0.1 $\pm$ 0.0	0.5 $\pm$ 0.1	0.2 $\pm$ 0.1	0.1 $\pm$ 0.0
	spp	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	0.1 $\pm$ 0.0	0.3 $\pm$ 0.2	0.1 $\pm$ 0.0	0.2 $\pm$ 0.0
	sc-zs	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	0.1 $\pm$ 0.0	0.5 $\pm$ 0.0	0.2 $\pm$ 0.0	0.2 $\pm$ 0.0

Appendix

Table 9: Average Cooperation (Ratio of Cooperative Moves) in the default PD case. Bold denotes higher cooperation rates per LLM-prompt instance.

		PD label-based counterfactual
		zs	spp	cot	srep	pp	mf	tft
model	prompt
C3.5Sv2	zs	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	0.3 $\pm$ 0.2	0.5 $\pm$ 0.0	0.9 $\pm$ 0.0	0.6 $\pm$ 0.4
	cot	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	0.4 $\pm$ 0.1	0.4 $\pm$ 0.1	1.0 $\pm$ 0.0	0.6 $\pm$ 0.4
	spp	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	0.4 $\pm$ 0.2	0.3 $\pm$ 0.2	0.9 $\pm$ 0.0	0.7 $\pm$ 0.4
	sc-zs	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	0.3 $\pm$ 0.2	0.5 $\pm$ 0.0	0.9 $\pm$ 0.0	1.0 $\pm$ 0.0

Appendix

Table 10: Average Cooperation (Ratio of Cooperative Moves) in the label-based PD counterfactual setup.

		SH (payoff-based counterfactual of PD)
		zs	spp	cot	srep	pp	mf	tft
model	prompt
C3.5Sv2	zs	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	0.3 $\pm$ 0.1	0.3 $\pm$ 0.2	1.0 $\pm$ 0.0	0.5 $\pm$ 0.5
	cot	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	0.2 $\pm$ 0.1	0.3 $\pm$ 0.1	0.5 $\pm$ 0.5	0.3 $\pm$ 0.4
	spp	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	0.1 $\pm$ 0.0	0.2 $\pm$ 0.1	0.8 $\pm$ 0.4	0.8 $\pm$ 0.4
	sc-zs	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	0.1 $\pm$ 0.0	0.3 $\pm$ 0.3	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0

Appendix

Table 11: Average Cooperation (Ratio of Cooperative Moves) in the Stag Hunt counterfactual.

		Joint counterfactual of PD
		zs	spp	cot	srep	pp	mf	tft
model	prompt
C3.5Sv2	zs	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	0.1 $\pm$ 0.0	0.2 $\pm$ 0.2	0.2 $\pm$ 0.0	0.1 $\pm$ 0.1
	cot	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	0.1 $\pm$ 0.0	0.3 $\pm$ 0.1	0.1 $\pm$ 0.1	0.2 $\pm$ 0.0
	spp	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	0.1 $\pm$ 0.0	0.1 $\pm$ 0.0	0.2 $\pm$ 0.0	0.1 $\pm$ 0.0
	sc-zs	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	0.1 $\pm$ 0.0	0.2 $\pm$ 0.0	0.2 $\pm$ 0.0	0.2 $\pm$ 0.0

Appendix

Table 12: Average Cooperation (Ratio of Cooperative Moves) in the joint counterfactual setup.

		PD	Label-based	Payoff-based	Joint
model	prompt
C3.5Sv2	zs	18.35 $\pm$ 10.24	28.04 $\pm$ 12.12	21.84 $\pm$ 12.24	15.56 $\pm$ 9.72
	cot	9.00 $\pm$ 4.05	15.88 $\pm$ 8.80	11.25 $\pm$ 5.37	7.85 $\pm$ 2.92
	spp	7.54 $\pm$ 2.98	11.99 $\pm$ 4.73	9.60 $\pm$ 4.14	5.93 $\pm$ 2.41
	sc-zs	6.37 $\pm$ 3.42	9.88 $\pm$ 4.56	7.63 $\pm$ 3.60	4.59 $\pm$ 2.71
	sc-cot	2.96 $\pm$ 1.46	6.06 $\pm$ 4.01	5.50 $\pm$ 5.26	2.58 $\pm$ 1.00

Appendix

Table 13: Average Efficiency (Points per kilo-token) for all PD setups.

model	avg validity rate
C3.5Sv2	100.0 $\pm$ 0.0
C3.7S	100.0 $\pm$ 0.0
C3.7S(T)	100.0 $\pm$ 0.0
C4S	100.0 $\pm$ 0.0
C4S(T)	100.0 $\pm$ 0.0
DS-R1	99.1 $\pm$ 6.3

Appendix

Table 14: Average Valid Rate (% of Valid Outcomes)

		RPS
		zs	spp	cot	srep	pp	ap	tft
model	prompt
C3.5Sv2	zs	6.6 $\pm$ 13.5	-1.2 $\pm$ 9.2	-1.8 $\pm$ 6.9	-0.4 $\pm$ 3.3	6.0 $\pm$ 10.1	11.2 $\pm$ 6.4	16.6 $\pm$ 6.9
	cot	0.8 $\pm$ 5.4	2.0 $\pm$ 5.7	5.4 $\pm$ 4.8	0.2 $\pm$ 5.3	3.8 $\pm$ 11.9	6.0 $\pm$ 3.7	13.0 $\pm$ 6.8
	spp	8.0 $\pm$ 9.9	7.2 $\pm$ 11.6	1.6 $\pm$ 6.9	-1.4 $\pm$ 2.2	9.4 $\pm$ 11.2	6.8 $\pm$ 1.5	12.8 $\pm$ 7.6
	sc-zs	-12.5 $\pm$ 12.0	-0.5 $\pm$ 21.9	0.5 $\pm$ 4.9	-0.5 $\pm$ 9.2	1.5 $\pm$ 2.1	9.0 $\pm$ 1.4	21.5 $\pm$ 0.7

Appendix

Table 15: Total Points Averaged Over All Iterations for the default RPS setup. Bold denotes best results (most total points).

		Label-based RPS counterfactual
		zs	spp	cot	srep	pp	ap	tft
model	prompt
C3.5Sv2	zs	-1.6 $\pm$ 3.5	-3.4 $\pm$ 2.3	-0.8 $\pm$ 3.2	-0.4 $\pm$ 3.4	-4.4 $\pm$ 2.5	9.0 $\pm$ 2.6	5.2 $\pm$ 7.1
	cot	0.8 $\pm$ 8.0	-2.8 $\pm$ 6.7	-0.4 $\pm$ 1.7	-5.0 $\pm$ 4.7	-9.0 $\pm$ 10.9	4.4 $\pm$ 3.9	-0.8 $\pm$ 3.4
	spp	1.6 $\pm$ 3.7	-2.6 $\pm$ 1.5	1.8 $\pm$ 4.6	0.2 $\pm$ 5.0	-8.0 $\pm$ 5.0	6.6 $\pm$ 3.9	2.0 $\pm$ 4.4
	sc-zs	3.0 $\pm$ 5.7	-2.0 $\pm$ 4.2	3.5 $\pm$ 2.1	3.5 $\pm$ 2.1	-12.0 $\pm$ 2.8	7.0 $\pm$ 1.4	12.0 $\pm$ 1.4

Appendix

Table 16: Total Points Averaged Over All Iterations in the label-based RPS counterfactual.

		Payoff-based RPS counterfactual
		zs	spp	cot	srep	pp	ap	tft
model	prompt
C3.5Sv2	zs	3.8 $\pm$ 16.1	-1.2 $\pm$ 25.5	-5.0 $\pm$ 12.8	0.2 $\pm$ 5.6	-4.0 $\pm$ 17.2	9.2 $\pm$ 6.2	29.6 $\pm$ 10.7
	cot	-3.0 $\pm$ 21.6	11.6 $\pm$ 23.1	7.0 $\pm$ 15.1	-0.4 $\pm$ 6.3	-7.0 $\pm$ 5.1	6.8 $\pm$ 2.8	24.2 $\pm$ 14.0
	spp	20.2 $\pm$ 11.2	12.4 $\pm$ 11.8	7.4 $\pm$ 8.5	3.4 $\pm$ 3.9	1.6 $\pm$ 17.7	6.4 $\pm$ 2.1	13.2 $\pm$ 12.8
	sc-zs	-14.5 $\pm$ 21.9	3.5 $\pm$ 3.5	17.0 $\pm$ 5.7	2.0 $\pm$ 9.9	35.0 $\pm$ 0.0	18.0 $\pm$ 1.4	40.0 $\pm$ 0.0

Appendix

Table 17: Total Points Averaged Over All Iterations for the payoff-based RPS counterfactual.

		Joint RPS counterfactual
		zs	spp	cot	srep	pp	ap	tft
model	prompt
C3.5Sv2	zs	0.6 $\pm$ 6.0	2.2 $\pm$ 10.3	-1.0 $\pm$ 10.9	-3.4 $\pm$ 4.7	-8.6 $\pm$ 22.2	10.6 $\pm$ 5.0	11.8 $\pm$ 12.1
	cot	2.0 $\pm$ 6.8	-5.8 $\pm$ 15.9	-4.2 $\pm$ 13.0	-0.2 $\pm$ 5.4	-8.8 $\pm$ 16.5	6.2 $\pm$ 11.4	12.2 $\pm$ 7.4
	spp	-1.4 $\pm$ 11.0	5.0 $\pm$ 10.2	2.6 $\pm$ 9.8	3.6 $\pm$ 3.9	-25.8 $\pm$ 10.0	14.0 $\pm$ 4.8	11.8 $\pm$ 7.7
	sc-zs	-7.0 $\pm$ 8.5	-7.5 $\pm$ 4.9	4.5 $\pm$ 0.7	0.5 $\pm$ 6.4	-17.0 $\pm$ 19.8	8.5 $\pm$ 4.9	7.0 $\pm$ 5.7

Appendix

Table 18: Total Points Averaged Over All Iterations for the joint RPS counterfactual.

		RPS
		zs	spp	cot	srep	pp	ap	tft
model	prompt
C3.5Sv2	zs	10.6 $\pm$ 13.1	21.4 $\pm$ 4.6	19.6 $\pm$ 5.6	21.0 $\pm$ 3.5	14.6 $\pm$ 12.4	10.4 $\pm$ 11.2	1.0 $\pm$ 0.0
	cot	17.2 $\pm$ 7.3	19.6 $\pm$ 7.5	20.6 $\pm$ 5.6	19.6 $\pm$ 7.1	16.0 $\pm$ 11.9	22.6 $\pm$ 2.1	5.8 $\pm$ 10.7
	spp	11.2 $\pm$ 10.6	11.8 $\pm$ 10.9	20.4 $\pm$ 5.9	23.0 $\pm$ 1.9	11.4 $\pm$ 11.3	18.4 $\pm$ 9.3	1.6 $\pm$ 0.9
	sc-zs	25.0 $\pm$ 0.0	14.0 $\pm$ 15.6	21.5 $\pm$ 4.9	13.5 $\pm$ 16.3	12.0 $\pm$ 15.6	19.5 $\pm$ 6.4	1.0 $\pm$ 0.0

Appendix

Table 19: Round

m

# where the LLM player understood the opponent’s Strategy in the default RPS setup. Lower is better;

m=25

indicates no stable comprehension within the 24-round game.

		Label-based RPS counterfactual
		zs	spp	cot	srep	pp	ap	tft
model	prompt
C3.5Sv2	zs	19.0 $\pm$ 10.3	22.4 $\pm$ 4.2	24.4 $\pm$ 1.3	20.4 $\pm$ 4.2	21.6 $\pm$ 3.8	11.4 $\pm$ 12.5	14.8 $\pm$ 11.4
	cot	21.8 $\pm$ 6.1	19.6 $\pm$ 7.3	19.8 $\pm$ 6.8	20.8 $\pm$ 5.1	21.6 $\pm$ 6.5	24.4 $\pm$ 0.9	23.6 $\pm$ 1.7
	spp	21.8 $\pm$ 2.6	24.0 $\pm$ 0.7	14.6 $\pm$ 8.7	22.6 $\pm$ 3.0	18.4 $\pm$ 9.0	22.8 $\pm$ 2.7	16.4 $\pm$ 11.3
	sc-zs	7.5 $\pm$ 9.2	22.5 $\pm$ 0.7	24.0 $\pm$ 1.4	12.0 $\pm$ 4.2	25.0 $\pm$ 0.0	19.0 $\pm$ 5.7	13.0 $\pm$ 17.0

Appendix

Table 20: Round

m

# where the LLM player understood the opponent’s Strategy in the label-based RPS counterfactual

		Payoff-based RPS counterfactual
		zs	spp	cot	srep	pp	ap	tft
model	prompt
C3.5Sv2	zs	16.2 $\pm$ 11.0	17.6 $\pm$ 10.3	24.4 $\pm$ 0.5	23.4 $\pm$ 1.9	19.0 $\pm$ 10.4	19.4 $\pm$ 10.4	5.6 $\pm$ 10.3
	cot	20.2 $\pm$ 9.1	12.6 $\pm$ 11.0	21.0 $\pm$ 5.3	17.6 $\pm$ 9.8	24.2 $\pm$ 1.3	18.4 $\pm$ 9.8	10.0 $\pm$ 12.4
	spp	10.6 $\pm$ 10.1	13.4 $\pm$ 11.5	22.2 $\pm$ 4.1	18.4 $\pm$ 6.9	15.0 $\pm$ 9.1	24.2 $\pm$ 0.8	14.2 $\pm$ 12.3
	sc-zs	23.5 $\pm$ 2.1	24.0 $\pm$ 1.4	13.0 $\pm$ 14.1	18.0 $\pm$ 9.9	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0	1.0 $\pm$ 0.0

Appendix

Table 21: Round

m

# where the LLM player understood the opponent’s Strategy in the payoff-based RPS counterfactual.

		Joint RPS counterfactual
		zs	spp	cot	srep	pp	ap	tft
model	prompt
C3.5Sv2	zs	23.8 $\pm$ 1.3	18.8 $\pm$ 7.6	14.6 $\pm$ 10.7	24.2 $\pm$ 0.8	21.2 $\pm$ 7.4	15.0 $\pm$ 12.8	10.2 $\pm$ 9.3
	cot	16.2 $\pm$ 9.5	17.6 $\pm$ 9.6	19.0 $\pm$ 7.4	17.4 $\pm$ 10.3	24.4 $\pm$ 0.9	18.6 $\pm$ 5.6	14.2 $\pm$ 12.1
	spp	23.4 $\pm$ 0.9	24.6 $\pm$ 0.5	15.8 $\pm$ 9.6	20.0 $\pm$ 3.7	25.0 $\pm$ 0.0	19.0 $\pm$ 8.0	7.8 $\pm$ 10.2
	sc-zs	24.0 $\pm$ 1.4	24.0 $\pm$ 0.0	15.0 $\pm$ 0.0	19.0 $\pm$ 8.5	21.0 $\pm$ 5.7	23.5 $\pm$ 2.1	24.5 $\pm$ 0.7

Appendix

Table 22: Round

m

# where the LLM player understood the opponent’s Strategy in the joint RPS counterfactual.

		RPS	RPS label-based	RPS payoff-based	RPS joint
model	prompt
C3.5v2	zs	1.09 $\pm$ 2.37	0.09 $\pm$ 1.34	0.47 $\pm$ 2.65	0.25 $\pm$ 1.61
	cot	0.44 $\pm$ 0.74	-0.23 $\pm$ 0.84	0.49 $\pm$ 1.64	-0.03 $\pm$ 1.28
	spp	0.49 $\pm$ 0.71	0.01 $\pm$ 0.48	0.66 $\pm$ 0.83	0.05 $\pm$ 1.10
	sc-zs	0.20 $\pm$ 0.60	0.10 $\pm$ 0.48	0.46 $\pm$ 0.56	-0.03 $\pm$ 0.35
	sc-cot	0.14 $\pm$ 0.18	-0.08 $\pm$ 0.25	0.15 $\pm$ 0.41	-0.01 $\pm$ 0.36

Appendix

Table 23: Average Efficiency (Points per kilo-token). Bold values denote higher efficiency per model-prompting setup.

model	avg validity rate
C3.5Sv2	100.0 $\pm$ 0.0
C3.7S	100.0 $\pm$ 0.0
C3.7S(T)	100.0 $\pm$ 0.2
C4S	99.9 $\pm$ 0.9
C4S(T)	100.0 $\pm$ 0.0
DS-R1	100.0 $\pm$ 0.2

Appendix

Table 24: Average Valid Rate (% of Valid Outcomes)

Model	PD	RPS	Counterfactuals	Efficiency	Main weakness
Claude 3.7	strong	strong	relatively robust	moderate	overreasoning
Claude 4	mixed	strong	mixed	lower	instability
DeepSeek R1	mixed	moderate	brittle	lower	coordination
Llama 3.3	strong	moderate	moderate	good	RPS adaptation
Mistral Large	weak	weak	weakest	low	comprehension

Appendix

Table 25: Cross-model qualitative comparison across games, counterfactual robustness, and efficiency.

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Level-k Distinguishable Mechanisms for Evaluating Bounded Rationality in LLMs

Binchi Zhang, Atrisha Sarkar

University of Western Ontario

LLM Evaluation Strategic Reasoning Risks in Language Models Reasoning Consistency in Language Models

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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.

Abstract

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