Game-Theoretic Solvers with Bayesian Opponent Modeling and Moral Rhetoric Framing for GLEE

Hayden Arnold

cs.GTOct 27, 2026

Game-Theoretic Solvers with Bayesian Opponent Modeling and Moral Rhetoric Framing for GLEE

Authors: Hayden Arnold

Organizations: KAIST

Abstract

We describe the agent we submitted to the GLEE Competition, which plays bargaining, negotiation, and persuasion games against a shared pool of human and AI opponents. The agent is built around three central components: a classical game-theoretic solver to compute numeric moves, an online Bayesian layer to estimate opponent archetypes, and a language model to write accompanying messages. We utilize Rubinstein bargaining for bargaining, Faratin concession for negotiation, and Kamenica--Gentzkow disclosure for persuasion, and adapt those models to live opponents using two separate layers: a Bayesian estimate of the opponent's hidden discount factor, reservation value, or signaling reliability, and a Thompson sampling bandit over five moral-foundation framings coupled with an evidence-intuition spectrum to establish message register. Most of the development effort went into the recurring problem that theoretically optimal policies perform poorly in certain live settings; we report associated failure modes, guardrails we built, and our hypotheses. We additionally evaluate our model over 13,200 Monte Carlo games against 11 synthetic opponent archetypes, an ablation of the opponent-modeling layer, and an extensive analysis of 9,337 live-game messages in which moral rhetoric framing in the persuasion setting is strongly correlated with whether the buyer purchased ( $χ^2=616.7$ , $p \approx 3.7\times10^{-132}$ ), surviving controls for quality, round, and configuration.

Figures & tables

Figure 1: (a) Displayed rating against the first 2600 live games played, per family; the dotted line is the

1000

start. (b) Median normalized Monte Carlo score per archetype. The one clearly negative bar, hardliner in negotiation (

-1.0

), holds a fixed position throughout the game that is strongly unfavorable to both players.

Family	Full	Ablated	$\Delta$ (Ablated $-$ Full)
Bargaining	$+0.238$	$+0.234$	$-0.004$
Negotiation	$-0.251$	$-0.262$	$-0.011$
Persuasion	$+0.269$	$+0.312$	$+0.043$

Table 1: Mean normalized Monte Carlo score over the 11 archetypes, with the opponent-estimation modules enabled (Full) and replaced by fixed priors (Ablated), on identical seeds (400 trials per archetype).

Foundation	$n$	Pass share	All	Pitches	True bluffs
Loyalty	2,667	6.3%	65.8%	68.4%	73.5%
Purity	2,119	11.4%	57.3%	62.5%	68.8%
Fairness	2,115	16.9%	56.1%	63.9%	74.1%
Authority	311	7.4%	49.2%	52.1%	70.5%
Care	2,125	50.1%	30.9%	49.8%	59.6%

Table 2: Persuasion buy rate by dominant foundation of the seller’s message. Pass share is the fraction of the foundation’s messages that were honest “pass” messages; pitches are the rest, and true bluffs are pitches on low-quality rounds. Intervals are in Appendix D.1 .

Divergence	Evidence	Guardrail
Honesty persists across rounds (seller)	A discovered bluff suppressed purchasing for the rest of a game	Bluff rate capped in the first three rounds; set to zero after three consecutive refusals
Honesty persists across rounds (buyer)	Re-exploration purchases were high quality 4.2% of the time vs. 76.3% for trusted sellers (102 games)	Re-exploration removed
A single offer is weak evidence of a type	One self-ask moved the discount-factor mean from 0.819 to 0.92	Shrinkage toward pre-update weights; MAE against an off-grid opponent fell from 0.053 to 0.023
Raw Rubinstein share degenerates	Near-one discount factors give a whole-pot demand or zero	Degenerate limit routed through the infinite-horizon formula; shares clamped to 30–70%; finite value damped
Holdouts run into a hidden cap	Unlimited games end near round 99–100	Deadline safety zone: mirror the opponent, accept any net-positive offer at the last chance
Stuck negotiations end in walk-aways	No per-round decay, so no pressure to settle	Accept a profitable offer that has stopped moving

Table 3: Divergences between theoretical solver assumptions and live opponents, and resulting guardrails.

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Component	Constants
Bargaining belief	Discount grid $\{0.5,0.6,0.7,0.8,0.85,0.9,0.93,0.95,0.97,0.99\}$ ; kernel $\sigma=0.08$ ; shrinkage pseudo-count $k=3$ .
Bargaining moves	Proposed share reduced by 3%; shares clamped to 30–70%; accept tolerance 1% of the pot; open at 50/50 in round 1 when $\delta_{\mathrm{me}}\leq 0.90$ (incomplete information, unlimited horizon); degenerate limit $\delta\geq 0.999$ , blended in from $0.99$ ; concession $\beta=0.7$ , $\tau=30$ ( $\tau=10$ once a holdout is confirmed); ask lowered by only 30% of the target drop against a holdout.
Bargaining deadline	Safety zone at $\leq 2$ rounds remaining (finite) or round $\geq 90$ (unlimited), with last-chance acceptance of any positive offer at round $97$ ; inside the zone accept offers of at least 10% of the pot. Holdout confirmed after 14 non-conceding offers (7 for an opponent previously confirmed hopeless in at least two games).
Stationary detection	Three consecutive offers within 0.5% of the pot (seven when $\delta_{\mathrm{me}}\geq 0.999$ ).
Negotiation	$\beta=0.85$ ; $\tau=30$ when unlimited; prior value spread $1.5\times$ ; one-shot offer at $t=0.75$ (incomplete information) or $0.03$ (complete); accept at once any offer with at least 30% margin; accept a stalled profitable offer (three offers within 0.5% of value, and under complete information at least 10% of the agreement zone); concession damped by 30% against a stalled opponent.
Persuasion buyer	Safety margin 20% over price; trust prior weight $k=20$ , prior used until two purchases; decline-model prior $\Pr(\text{high})=0.15$ ; first purchase when prior EV/price $\geq 0.95$ ; test a pass signal when upside/downside $\geq 1.2$ and prior EV/price $\geq 1.2$ ; language-model discourage verdict used at confidence $\geq 0.7$ , otherwise a phrase list; a message repeated for three rounds is treated as uninformative.

Appendix

Table 4: Constants for the deployed agent.

	Bargaining		Negotiation		Persuasion
Archetype	mean (sd)	median	mean (sd)	median	mean (sd)	median
hardliner	+0.011 (0.017)	+0.002	$-0.656$ (0.545)	$-1.000$	+0.080 (0.232)	+0.000
fair_splitter	+0.385 (0.124)	+0.391	$-0.192$ (0.636)	+0.121	+0.288 (0.399)	+0.000
tit_for_tat	+0.250 (0.183)	+0.240	$-0.273$ (0.578)	+0.050	+0.317 (0.387)	+0.100
deadline_exploiter	+0.025 (0.051)	+0.003	$-0.256$ (0.555)	+0.019	+0.149 (0.244)	+0.034
random	+0.429 (0.177)	+0.428	+0.029 (0.553)	+0.253	+0.262 (0.230)	+0.300

Appendix

Table 5: Mean (standard deviation) and median normalized score,

400

Monte Carlo trials per archetype per family, on the deployed strategy code with all three opponent-modeling mechanisms enabled. For bargaining and persuasion,

0

is the even-split or no-sale baseline; for negotiation,

-1

is a walk-away or no-deal, and the mean sits below the median because roughly half of trials have no zone of agreement and floor at

-1

Archetype	Full	Ablated	Full	Ablated	Full	Ablated
	Bargaining		Negotiation		Persuasion
hardliner	+0.011	+0.010	$-0.656$	$-0.679$	+0.080	+0.117
fair_splitter	+0.385	+0.381	$-0.192$	$-0.199$	+0.288	+0.342
tit_for_tat	+0.250	+0.237	$-0.273$	$-0.322$	+0.317	+0.373
deadline_exploiter	+0.025	+0.025	$-0.256$	$-0.265$	+0.149	+0.196
random	+0.429	+0.428	+0.029	+0.007	+0.262	+0.267

Appendix

Table 6: Mean normalized score, full system against opponent-modeling ablated,

400

trials per archetype per family, on identical seeds. The Full columns match Table 5 .

Family	$n$	Coverage	$\chi^{2}(4)$	$p$
Persuasion	9,337	88.4%	616.7	$3.7\times 10^{-132}$
Bargaining	4,661	84.7%	11.2	$0.0249$
Negotiation	7,558	89.3%	9.1	$0.059$

Appendix

Table 7: Chi-square test of independence between dominant moral-foundation framing (care, fairness, loyalty, authority, purity) and message success, per family.

n

is the number of our own outgoing messages with a classifiable dominant foundation (

\geq 2

lexicon matches); coverage is the fraction of that family’s total completed games included in the sample.

Foundation	$n$	Successes	Buy rate	95% CI
Loyalty	2,667	1,754	65.8%	[63.9, 67.5]
Purity	2,119	1,215	57.3%	[55.2, 59.4]
Fairness	2,115	1,187	56.1%	[54.0, 58.2]
Authority	311	153	49.2%	[43.7, 54.7]
Care	2,125	657	30.9%	[29.0, 32.9]

Appendix

Table 8: Persuasion: buy rate by dominant foundation of the seller’s message (

n=9{,}337

\chi^{2}(4)=616.7

p=3.7\times 10^{-132}

), with 95% Wilson intervals.

Foundation	$n$	Successes	Accept rate	95% CI
Purity	1,758	89	5.1%	[4.1, 6.2]
Care	1,324	56	4.2%	[3.3, 5.5]
Fairness	1,857	78	4.2%	[3.4, 5.2]
Loyalty	1,735	62	3.6%	[2.8, 4.6]
Authority	884	25	2.8%	[1.9, 4.1]

Appendix

Table 9: Negotiation: offer-acceptance rate by dominant foundation of our offer’s message (

n=7{,}558

\chi^{2}(4)=9.1

p=0.059

), with 95% Wilson intervals.

Foundation	$n$	Successes	Accept rate	95% CI
Loyalty	1,205	218	18.1%	[16.0, 20.4]
Fairness	1,095	185	16.9%	[14.8, 19.2]
Care	775	128	16.5%	[14.1, 19.3]
Purity	1,029	152	14.8%	[12.7, 17.1]
Authority	557	69	12.4%	[9.9, 15.4]

Appendix

Table 10: Bargaining: offer-acceptance rate by dominant foundation of our proposal’s message (

n=4{,}661

\chi^{2}(4)=11.2

p=0.025

), with 95% Wilson intervals.

Foundation	Pass share	$n$	Rate [95% CI]	$n$	Rate [95% CI]
		Pitches		True bluffs
Loyalty	6.3%	2,500	68.4% [66.6, 70.2]	536	73.5% [69.6, 77.1]
Fairness	16.9%	1,758	63.9% [61.7, 66.1]	363	74.1% [69.4, 78.3]
Purity	11.4%	1,877	62.5% [60.3, 64.7]	349	68.8% [63.7, 73.4]
Authority	7.4%	288	52.1% [46.3, 57.8]	61	70.5% [58.1, 80.4]
Care	50.1%	1,061	49.8% [46.8, 52.8]	203	59.6% [52.7, 66.1]

Appendix

Table 11: Persuasion: buy rate for pitches only and for true bluffs (pitches on low-quality rounds), with 95% Wilson intervals. Pass share is the fraction of the foundation’s messages that were honest “pass” messages.

	(1) All, unadjusted	(2) All, adjusted	(3) Pitches, adjusted
	$n=9{,}337$	$n=9{,}337$	$n=7{,}484$
Loyalty	4.29 [3.57, 5.16]	3.98 [3.31, 4.79]	2.18 [1.78, 2.67]
Purity	3.00 [2.52, 3.58]	2.78 [2.32, 3.32]	1.70 [1.39, 2.07]
Fairness	2.86 [2.40, 3.40]	2.67 [2.24, 3.18]	1.78 [1.46, 2.18]
Authority	2.16 [1.60, 2.92]	2.05 [1.52, 2.77]	1.11 [0.81, 1.53]
Joint Wald $\chi^{2}(4)$	263.5	230.5	63.4

Appendix

Table 12: Persuasion: odds ratios of a purchase relative to care, from logistic models with standard errors clustered by game (

725

games). Models (2) and (3) add product quality and round.

	Persuasion		Bargaining		Negotiation
Register	$n$	Rate	$n$	Rate	$n$	Rate
Evidence-leaning	3,649	62.4%	824	13.2%	1,897	3.5%
Intuition-leaning	490	44.1%	111	14.4%	142	4.2%
Balanced	5,198	47.6%	3,726	16.8%	5,519	4.3%
Odds ratio [95% CI]	1.83 [1.63, 2.05]		0.98 [0.75, 1.27]		0.88 [0.64, 1.20]

Appendix

Table 13: Success rate by the register of our message. The last row gives the odds ratio per unit of evidence-leaning from a logistic model with foundation dummies and message length (and, for persuasion, quality and round), clustered by game.

Panel of Figure 2 — Figure 2: System prompt used by the message layer to enforce brevity, rhetorical conditioning, and non-harassment guardrails.

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