Modeling the Economic Impacts of AI Openness Regulation
Authors: Tori Qiu
Benjamin Laufer
Jon Kleinberg
Hoda Heidari
Venue: NeurIPS 2025 poster ()
Keywords: game theory
AI regulation
openness
Primary area: social_and_economic_aspects_of_machine_learning
TL;DR: This paper presents a stylized model of the regulator's choice of an open-source definition  in order to evaluate which standards will establish appropriate economic incentives for foundation model developers.

Abstract: Regulatory frameworks, such as the EU AI Act, encourage openness of general-purpose AI models by offering legal exemptions for "open-source" models. Despite this legislative attention on openness, the definition of open-source foundation models remains ambiguous.  This paper presents a stylized model of the regulator's choice of an open-source definition  in order to evaluate which standards will establish appropriate economic incentives for developers. In particular, we model the strategic interactions among the creator of the general-purpose model (the generalist) and the entity that fine-tunes the general-purpose model to a specialized domain or task (the specialist), in response to the regulator. Our results characterize market equilibria -- specifically, upstream model release decisions and downstream fine-tuning efforts -- under various openness policies and present an optimal range of open-source thresholds as a function of model performance.  Overall, we identify a curve defined by initial model performance which determines whether increasing the regulatory penalty vs. increasing the open-source threshold will  meaningfully alter the generalist's model release strategy. Our model provides a theoretical foundation for AI governance decisions around openness and enables evaluation and refinement of practical open-source policies.

Modeling the Economic Impacts of AI Openness Regulation
Tori Qiu
Benjamin Laufer Cornell Tech bd1560cornel1. edu
Carnegie Mellon University torig@andrew. cmu. edu
Jon Kleinberg
Hoda Heidari
Coreell University kleinberg@cornell.cco edu
Carnegie Mellon University hhei ieeidariine waadrrrnrree....rnr cmu edu
Abstract
Regulatory frameworks, such the EU Al Act, encourage openness of general- purpose Al models by offering legal exemptions for "open-source" models. Despite this legislative attention on openness, the definition of open-source foundation models remains ambiguous. This paper models strategic interactions among the creator of general-purposs model (the generalist) and the entity that fine-tunes the general-purpose model to a specialized domain or task (the specialist), in response regulatory requirements on model openness. We present a stylized model of the regulator's choice of open-source definition to evaluate which AI openness standards will establish appropriate economic incentives for developers. Our results characterize market equilibria specifically, upstream model release decisions and downstream fine-tuning efforts - under various openness regulations and present range of effective regulatory penalties and open-source thresholds. Overall, we find the model's ' baseline performance determines when increasing the regulatory penalty the open-source threshold will significantly alter the generalist's release strategy. Our model provides a theoretical foundation for Al governance decisions around openness and enables evaluation and refinement of practical open-source policies.
Introduction
Foundation models with publicly available weights, such as Llama and Deepseek 42 13], have the potential distribute economic benefits and broaden access to AI development. However, developers of these general-purpose models have faced criticism for marketing their products as "open-source" while departing from established open-source principles and concealing essential design decisions [29]. Traditionally, open- source software (OSS) describes publicly available source code with few restrictions use, modification, and redistribution 24). Foundation models complicate the standard open-source definition, as they exhibit different degrees of openness, ranging from models with hosted API access open-weight models fully open models with public weights, code, data, and no use restrictions (38 20 Open-source definitions that transfer principles of OSS to foundation models prohibit any use restrictions or monetization of the model [41] and may require disclosure of all components, including training data (24). Industry embraces a more lenient definition, where open-weights models released under a non- pro-prroryettay license qualify as open-source, even when
commercial use restrictions exist (e.g., Llama and Mistral Large 3 33
Regulatory frameworks for foundation models have proposed various definitions of "open-source" that reflect this uncertainty. While the EU AI Act exempts open- models from disclosure
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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and documentation requirements, it disqualifies any monetized open-source Al products from using exemptions for open-source models 44]. Singapore's generative AI governance framework stipulates that open-source models should provide all source code, documentation, and data required to retrain the model from scratch [17]. Other regulations provide less clarity - the US Al Foundation Model Transparency Act mentions "open source foundation models" as possibly being exempt from data disclosure requirements, but does not define open source
[]]
This work offers a game-theoretic model of strategic interactions under various Al openness reg- ulations. A general purepose technology producer (the generalist) decides an openness level for Al technology. One or more domain specialist(s) may then adapt or fine-tune the base model depending on the openness level set by the generalist. Finally, a regulatory threshold moderates the costs associated with different openness levels. We solve the game by providing a closed-form solution for both players' strategies in Section Sections and @ analyze the sensitivity of the generalist's s openness decision, bargaining outcomes, and the resulting fine-tuning investments to changes in the threshold and penalty. Finally, to illustrate the practical application of our model, we show that the model's 's predicted equilibrium corresponds to real-world release strategies of developers and present takeaways for open-source Al regulation Section
1.1 Related Work
Open-source contributions are often framed as altruistic 4 6 because conventional valuation methods cannot appraise products with price of zero and unknown quantities due to unrestricted copying and redistribution rights [22]. However, when developers' decisions to open source are viewed as strategic responses to competitors internal production constraints, these openness decisions transform into a competitive choice to reduce costs or penetrate price-sensitive markets [39]. Dynamic models
of competition and adoption in markets with proprietary software and OSS substitutes (5] 10 39
link open-source decisions to product performance, capturing profit-driven strategy rather than appealing altruism. For foundation models in particular, recent work examines how openness affects strategic behavior throughout the AI value chain. Building on the theoretical framework of fine-tuning games, Xu al. [45] show that intermediate levels of openness can motivate early-market specialists to strategically curtail their fine-tuning efforts to outcompete subsequent specialists, and this reduced model performance leaves generalists, specialists, and end consumers worse off relative to zero openness. Wu et al. [44] similarly assess how open-source engagement with a closed-source counterpart impacts innovation for pretrained and adapted foundation models
44]
This paper makes two contributions to modeling strategic openness decisions. First, while existing models evaluate openness a binary between fully closed or fully open (39 44]
analyze how generalists strategically adjust their openness levels along a spectrum in order to capture the competitive dynamics of offering partially open models that do not clear the open-source threshold. Second, we consider how administratively imposed openness guidelines alter the generalist's 's release strategy. Xu al.'s 451 setup assumes there is a single, exogenous openness level imposed on the generalist, with the analysis centering on how this fixed level affects specialist strategies. Unlike approaches which evaluate the effects of openness on technology adoption and improvement (or fine-tuning) decisions at the specialist level, our setup simultancously models how generalists adapt their openness and compliance decisions in response to regulatory constraints.
2 Model
This section describes our model of strategic interactions between actors who contribute to the development of a general-purpor Al model. We begin by introducing the concept of openness we aim to measure, then present the model formally.
2.1 Levels of Openness
Our concept of openness, denoted w E [0, 1], projects the observed behaviors of developers onto continuum to reason about how these behaviors affect developer revenue and strategic reactions. We offer numeric score for the sake of analysis, under the assumption that the set of choices firm makes can be mapped into single range that summarizes the cunulative amount of effort invested in opening model. We rely on prior work to define how different openness practices correspond to

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points on the continuum 38, 9 and treat this one- one-dimensional measure of openness modeling abstraction.
Fully closed
Hosted access
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Figure I: Example of an openness continuum defining w E [0, 1] based on model access
38
Several works suggest approaches for converting multidimensional openness assessments of models into a spectrum, using feedback mechanisms, risk descriptions, levels of system access, and other indicators to quantify openness 9 299 38 Our work builds on these recommendations by character- izing how the evaluation of openness along such a spectrum affects the incentives of general-purposs model developers.
2.2
Game Setup
The interactions between AI developers are formalized as game between two players, which we refer to as the generalist (G) and domain specialist (D). Before the game begins, we assume that the generalist has invested in bringing the anrrral-pupooe model to an initial performance level og E R+ Performance represents coarse measure of the model's accuracy on target tasks, along with generation speed and efficiency. We further assume that regulator publishes an open-source threshold [0, 1]. which is value on some continuum of openness. The regulator's choice of 0 defines the threshold at which G receives legal exemptions and, by extension, dictates G's costs of development for an open vs. closed model. After the regulation announced, G and D make investment decisions sequentially, as proposed in Laufer et al. [26]. The generalist must decide an openness level w E [0, 1] to offer the model at given the threshold. An openness level of 0 corresponds to G choosing not to release the model to market, whereas openness level of w = indicates a fully open model with all components publicly accessible and zero use restrictions.
The specialist then brings the model to improved performance level an in the fine-tuning stage, cooperating with the generalist to increase their joint revenue. The full game with G's openness decision has the following stages:
1. G and D bargain over (ws, 01) E [0, 1). coefficient that controls the division of revenue between the two players.
2. G chooses an openness level E [0, 1] for the base model with performance app where the model considered open if w 2 and closed otherwise. openness level closer represents more closed model, whereas represents fully open model. When G abstains from releasing the model (w 0), the specialist receives zero utility.
3. Assuming G does not abstain, chooses to adopt G's model and invests in improving it to performance 0₁ 2 Cg.
We assume that players will choose abstain when their utilities are negative, although in practice, developers may tolerate short-term losses. The final revenue shared by G and D. given by a revenue function r(on): : IR+ R+ depends the model's final performance on For simplicity and unless otherwise stated, we assume r is the identity function in our analysis. In standard cooperative bargaining games 225 37 8(w, a1) represents the result of linear revenue-sharing contract between players. G receives o(w, a1) r((1) as its share of the final revenue, while D receives (1 (1- an)) r(a,). However, this model of surplus division in bilateral bargains
16 35 is challenged when an openness decision enters G's strategy. The bargaining parameter
8(w,o0 01] represents the result of licensing and subscription arrangements when dividing revenue from foundation model development. Higher openness forces G to forfeit share of the bargained revenue a(w) r((1) because can no longer charge directly for model usage or other support services and must instead replace direct revenue from model performance with indirect benefits, such as reputational premium for offering an open model. 8(us, 01) simplified to for the remainder the paper, but we treat it as endogenous players' strategic decisions due to the bargaining stage.
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If the model is open, G's payoff is fixed to c01, where e E [0, 1] is a constant that captures reputational production, where participants organize efforts and capture value without exclusive rights via property contracts, relying instead on indirect rewards like reputation and cost sharing (7]. This contrasts with the 6 parameter, which reflects direct market incentives through revenue-sharing agreement between the generalist and specialist.
gains of G. The constant accounts for "commons-basee"
2.3
Utility & Cost Functions
Utility (U)
Cost (Ø)
Ua = (ew + 8(1 - w)) r(an)= de (w)())
Φ == now + cw - w) + p < 0)
Up = (1 6(1 - w)) r(ai)= dD (a1 W. an)
ΦD = (a1 aw) + cwaw
Table 1: Utility and cost functions for the generalist and specialist. The constant parameters and Cu define costs and rewards exogenous to the game, and the value of ou is fixed because our game is only concerned with G's decision to open an existing saneral-purpose model. Openness regulation takes the form of parameters (representing stringency of the open-source definition) and p (representing the penalty for failing to meet the open-source definition).
Table T describes the cost and utility functions of both players. Each player's 's utility is calculated as their revenue share minus the costs of model improvement and hosting. G's revenue structure changes according to the openness level at which the model is offered, with reputational gains (c) influencing revenue more as openness increases and the bargaining outcome (ō) influencing revenue more as openness decreases.
@a(w 0o)
= arw
+ (1-a)(--))
pll <
1. production
2. operation
3. regulatory
Generalist Cost Function. G's cost function (Equation I can be categorized into production costs, operation costs, and regulatory costs. Production costs refer to a fixed cost the generalist faces to offer the model with performance ag openness level In our case, we use Oyj to cover the production costs of documenting and releasing the model, which scale with higher levels of openness and performance. Operation costs capture the variable expenses required including costs of inference and system maintenance. For a more closed model, G hosts specialized servers to process user requests and charges the user via pay-per-use API calls. The generalist's operation costs decrease with w because Gi can offload hosting and inference costs to users and benefit from community-but buill optimizations at higher levels of openness. For simplicity, we normalize the coefficients for production and regulatory compliance costs to one, using Cu to compare between operation costs and revenue, The constant eu quantifies the ratio of average operation costs to revenue per unit of performance for D. G pays an additional regulatory cost, p. to comply with regulatory requirements when the model closed (w < O) and does not quality for open-source exemptions.
and host the model,
Op(anlus,o) = (01 0o) )
1. production
2. operation
is composed of production costs and operation costs. As w approaches zero, D's production costs go infinity, indicating an infeasible strategy where it becomes prohibitive to fine-tune or improve a model with heavily restricted access. However, cheaper production costs for models with higher levels of openness are partially offset by the operation costs D must pay for self-hosting the model and shouldering inference costs directly. We prove results for particular set of quadratic cost functions and monotonic revenue functions, although these represent only a specific subset of possible functional forms for the broader class of utility functions. Specifically, we assume D faces quadratic costs when developing the model from performance ogg a1 since incremental improvements are more expensive at higher performance, and discount development costs for by a factor of w-1 since there are fewer charges associated with adopting and improving more open model (Table Ip.
Specialist Cost Function. The total cost for the domain specialist ΦD (Equation 2

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Specialist & Generalist Cost Sharing, In the basic version of the fine-tuning game presented by prior work [26]. model performance the only feature in G's strategy space. The cost functions for the generalist and specialist are condensed into single terms, as ØG = og 2 and ΦD = (a1 - ao)² This simplification is effective when investments in model performance represent j.joint production process with shared revenue, as development and operation costs rise with performance for the game setup explicitly models both performance and openness
generalist and specialist. However, as decision variables. Unlike the original fine-tuning game [26], where costs scale equivalently for all features (the single feature being performance), openness imposes opposite effects cost for the two players. For any given a₁, higher openness decreases production costs for the specialist but increases operation costs. These changes in cost shifted to the generalist, and higher openness increases production costs for the model but decreases operation costs for the generalist. To capture how openness induces cost shifts between players, our game explicitly decomposes cost functions into separate terms for development and operation. Appendix C has more detailed discussion of the utility functions and cost sharing.
Analysis of Equilibrium Strategies
This section characterizes the Sabbaame-pefeed equilibria strategies as a function of the game parame- ters e, Cu, 0. We provide closed-form expressions for the best response of each player under fixed revenue-sharing parameter 8. Proofs of all propositions are provided Appendix B.]
3.1 Subgame Perfect Equilibria
Proposition (Characterization D's Equilibrium Strategy). In game with quadratic costs and I 1-6
monotonic revenue function r(a1). if w
cw- then D's best-responss strategy is given by D will choose to abstain.
af = 00 + -() Ifw >
1-5 -8
2
D must choose performance level of ao for their adapted, or fine-tuned, model in the game. The condition for D's participation in Proposition | < 1-6 c.- is only satisfied when the costs of operating the model at baseline performance do not exceed D's share of r(a)). Using the specialist's best response, G's strategy can then be solved in terms of the threshold.
Proposition 2 (Characterization of G's Equilibrium Strategy). In a game with quadratic costs and monotonic revenue function r(a1). G's preferred openness w will be one of the values in the following set:
w² E
3(c - 8 (6 cc) (6-5 56-5+00 ow) 6) + (cw 5) + cw)
1
[(c- 8) (cw (6)--((6-1) + +cw))
6 - 8 8 + ((e- (t- (c (c- (c-c.)(1- ()-c))(1-)))
2
G will choose the value the set which maximizes UG. G will choose to abstain if and only if Ua < 0 for all candidates in the set.
3.2 Generalist Strategies at Equilibrium
We begin by characterizing G's openness strategies at equilibrium for a two-player game under different bargaining agreements without regulation (p and 0 has effect). The Nash bargaining solution calculates joint utility as the product of UD and Uc, the vertical monopoly (VM) solution as the sum, and the egalitarian solution as the minimum of UD and Uc-
Bargaining Effects. The value which maximizes a specific measure of joint utility is identified via grid search, as analytical solution for the equilibrium bargaining agreement is intractable with multiple game parameters and w* selected from a set. Figure visualizes and D's equilibrium strategies over initial model performance (ag) and reputational benefits (e) when regulation is
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imposed. G chooses fully open model when the bargained revenue-sharing agreement is 8 = 0, but chooses a fully closed model when ag is high and d moderate. Counterintuitvelly, G chooses intermediate openness when 8" is maximized and the bargained revenue-sharing outcome for a closed model strongly favors This intermediate openness level empowers to adopt and improve the model at low Cgg, which generates greater total revenue than would be possible under fully closed approach. Full plots of the equilibrium outcomes with variations to do, e, and Cw are in Appendix G
VM Bargaining (r. 0.1)
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Figure G's equilibrium strategies for w' (top row) and 5* (bottom row) Cu 0.1 with no regulation (p = 0). Low initial performance (om) and high reputational benefits (e) lead to partial openness. Since G unilaterally controls the model's s release strategy, can use the openness decision to remove or weaken the bargain, explaining why fully open model coincides with bargains that allocate no closed revenue G.
In general, G prefers a fully closed strategy as the initial model performance on increases, even for higher values, because the performance is high enough generate considerable subscription and licensing revenue via r(a)) without specialist improvements.
Openness- Pernnesss-Prffraancee Tradeoff. G's no-regulation equilibrium strategies corroborate empirical patterns of model release, where models with lower performance are relatively more open access and closed models outpace open-weight models by several months in terms of performance (Figure 3). The openness level chosen by G at equilibrium from our main theoretical result quantifies the observed tradeoff between performance and openness in release strategies in terms of and op, well as the value bargained by players. When ag and c are both low, G chooses intermediate performant enough to drive
openness levels reduce the specialist's costs because the model adoption at full cost under a closed approach. This outcome corresponds to proprietary decisions to release underperforming models with partial open access, such as Google releasing lower-capability open variants alongside high-performing closed models (Figure 3)
Analysis of Regulation Effects
To identify effective open-source interventions, we characterize the range of regulatory effects across initial model performances and identifies specific penalty and threshold configurations required to achieve compliance. The space of key constants 8, was systematically probed (details in Appendix G and in what follows, we establish the existence three qualitatively different regulatory outcomes.

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Figure 3: Overview of empirical model release strategies. (Left) Higher performance of a model corresponds to higher percentage of closed components, determined by Eiras et al.'s assessment of model components [14). (Right) For each generation, closed-weight models have higher performance than open-weight ones, despite open-weight models showing comparable performance closed- weight models from previous time periods. The performance gap between open- vs. closed-weight models (blue region) persists even as absolute performance improves across generations. Appendix reports the figure data.
4.1 Regulation Effects on Openness Decisions
The boundary where G's openness decision transitions from a fully closed model to partially open one represents the indifference curve where G's utility for maintaining fully closed model (w 0) is equal to the utility for opening the model. For any (p. f) combination below this curve, G will choose an openness level that meets the threshold. We provide closed-form expression for this curve that identifies which combinations of (p, 0) will drive changes to G's openness strategy. Proofs are Appendix
@ Values (o,- 0.5, c.- 0.01, 0.15)
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(p.0) choices for game parameters C, 0.01, e = 0.15 and an E {0.5, 1,5). In the region of non-compliance above the indifference curve, G keeps the model at openness level wn - 0. In the area of compliance below the indifference curve, G chooses " =
Figure 4: Indifference curves for the generalist
Proposition 3. The indifference curve separating fully closed from partially open strategies in the space of regulation profiles (p, O) is defined by p 0(5 + 0g Cw) For any 0(8 + ccccw fully closes the model.
Regulation can lead deadweight loss. The area above the indifference curve illustrates an outcome where openness regulation lowers both players' utilitics without yielding any openness
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improvements (Figure(). There are no corresponding improvements to in this region because D cannot contribute via fine-tuning efforts. Above the indifference curve, any increases to the penalty directly deduct from G's utility because the model stays fully closed. D'ss revenue share is also reduced because the bargain shifts G's favor as the penalty increased. Extreme combinations of (p. f) do not just produce deadweight loss but cause G to withdraw from the market entirely by abstaining, which represents the worst-case outcome because no positive utility can be generated.
4.2 Regulation Effects With Compliance
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Figure 5: Player utilities under various (p. 0) regulations when op 0.11, Cu. 0.05, and 0.1 with Nash bargaining. (Ug, -imppoooo regulations are enclosed in the gray region, with the lower left corner of the gray region corresponding the no-regulation equilibrium where p = and 0 has effect. For overly stringent 9 thresholds, both players' utilities decrease.
Definition 1 (Pareto-Optimal Policies). Given a vector of d utility functions u E Rd, the Pareto- optimal policy the regulatory profile (9, p) that maximizes the following optimization problem: arg max,.e To u s.l. ||||| w; 0, Vi E [d]. The Pareto-optimal policies optimize some weighting over the objectives, where the weight vector w sits on the standard d-simplex.
Regulation can lead to Pareto improvement over (w, an Uc. Un. When initial performance (on) low, there is a region of regulation where G complies with the open-source threshold but would fully close otherwise. This intermediate openness level is mutually beneficial because D makes significant model improvements that increase the total amount of surplus allocated by the bargain. Figure presents one set of parameters where region of regulation achieves Pareto improvement over (w, a1 UG, U), guiding players toward a mutually beneficial equilibrium that is unachievable without imposing penalty.
The mutually beneficial openness level is not chosen in the absence of regulation because bargaining alone does not provide credible commitment mechanism. There is no assurance that G will keep the model open rather than fully closing it to maximize its own d-share D concedes a greater revenue share. Cheap shot strategies that are individually beneficial to G are constrained by regulation; the penalty ensures G will at least open the model at the threshold under certain conditions, and D safely offer more favorable bargain to secure G's cooperation. In Figure 7 the no-regulation bargaining outcome roughly balanced (8* 0.5), but introducing a penalty p > 0 changes the bargaining outcome that 8* s 1 as the threshold increases below the indifference curve.
Regulation can encourage specialist innovation. Regulation that falls below the indifference curve redistributes utility from G to D when ag is high enough for G to sacrifice licensing and subseription revenue by choosing higher levels of openness. open-source threshold below the indifference curve transfers utility from G to D without deterring G's participation. Higher openness levels this region raise on by reducing production costs for D, but G loses closed revenue. For instance, Meta loses licensing revenue by developing and releasing
illustratlss how increasing

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open-weight models, but cheap access to advanced models allows specialists to develop derivatives like Vicuna and Alpaca
40
which can surpass the original Llama models performance.
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Figure 7: Equilibrium outcomes for 0g 1, 0.01, 0.1 with Nash bargaining create region where raising the open-source threshold improves w, du, and Up but lowers Ug. This utility transfer encourages specialist innovation, measured by a At the no-regulation equilibrium (p 0), players' decisions are w Rs 0.01 and aj = 1.

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Discussion
This work introduces an openness decision into the generalist's strategy space, modeling how this parameter affects strategic equilibria in foundation model development. While existing models of openness regulation assume compliance mandatory and all model developers make homogeneous openness decisions [45] our setup allows generalists to select a continuous openness level based on regulation intensity and model performance. The model we present therefore explains the strategic positioning of AI models along prenness-performante frontier and anticipates how adjustments to open-source thresholds will impact fine-tuning and bargaining dynamics.
5.1 Implications for Open-Source Al Regulation
Regulatory approaches openness should shift with average model performance empower innovation. Low initial performance creates opportunities for Pareto improvements through openness regulation, while high performance at most permits the strategic redistribution utility between generalists and specialists. As model capabilities evolve, regulatory strategies may need to shift to redistribution approaches that target the specialist's willingness to contribute to model performance. This targeted approach can address competitive imbalances between generalists and specialists, in environments where few generalists dominate the offerings of high-peee beggh-eeffrrmncee foundation models. Given that models may excel in certain domains while underperforming in others, regulators must also determine whether to use composite performance metrics, domain-specnt benchmarks, or weighted averages define model performance, with each approach reflecting different model development priorities.
Effective open-source thresholds require calibrated penalties. Regulators should consider enforce- ment costs regions where higher thresholds cannot influence the generalist's openness decision. For regions above the indifference curve, higher thresholds cause wasted enforcement costs on evaluation and monitoring of additional closed model components, without any corresponding improvements to model openness. Resources allocated to open-source regulation should be concentrated near the indifference curve, and enforcement approaches should scale the regulatory penalty based on specific model characteristics like baseline performance or operation costs ensure that all generalists face real compliance incentives. This prevents scenarios where developers of nigh-performing models absorb regulatory penalties a business expense rather than modifying their transparency practices.
5.2 Limitations and Future Work
Our model abstracts from several behavioral and competitive dynamics that can affect the theoretical results presented this paper. For example, regulatory thresholds can create psychological anchoring that influences behavior independent of the penalty, firms target the threshold even if the penalty is minimal. Firms may also feel competitive pressure to meet the threshold if they believe that competitors will meet it, creating norms that transcend direct cost calculation of the penalty. Modeling multi-firm competition with a cdosed-source incumbent can also reveal when established firms pivot to open-source strategies and how timing of market entry affects whether entrants adopt open-source approaches.
The bargaining process may deviate similarly from axiomatic solutions. Our analysis mainly examines Nash bargaining, which uniquely satisfies independence of utility origins and units, Pareto efficiency, symmetry, and independence of irrelevant alternatives (II. The two alternative bargaining solutions we consider (VM and Egalitarian) are also Paretian, symmetric, and independent of utility origins. However, generalists and specialists may arrive at non-standard bargaining agreements in practice which violate these properties, especially when utilities are expressed in currency and players have bounded rationality [5]
While the main analysis assumes that higher model openness and performance are generally beneficial, regulator also considers safety risks associated with higher performance or market-wide loss scenarios, such as intermediate levels of openness where consumers may be worse off [45). This complexity suggests that defining openness as multidimensional property may be valuable to capture factors beyond simple model access, such as documentation, licensing, and explainability of model outputs (28).
10

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[41] The European Parliament and the Council of the European Union. Regulation (EU) 2024/1689, 2024.
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[45] Fasheng Xu, Xiaoyu Wang, Wei Chen, and Karen Xie. The Economics of Al Foundation Models: Openness, Competition, and Governance, 2024.
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A Additional Related Work
Models of AI Regulation. Numeric thresholds dictate in practice which "covered" products subject to regulatory standards. The number floating point operations (FLOPs), the compute cost based the average market price of the cloud compute, annual sales, and the number of active monthly users appear as representative thresholds in Al-related legal measures [2]. Although the accuracies of these thresholds disputed [1], binary qualification system for rewards may yield
more aggregate effort than a continuous reward structure or rewards based on relative performance, as incentives improve performance intensify with proximity the threshold [19]. Existing works on the effect of Al requirements on market dynamics model how firms strategically respond to targeted safety regulations [27] and how reputational pressures to satisfy safety constraints facilitate entry of new developers 32
Openness & Market Strategy. Technology providers gain competitive advantage by dif- ferentiating through price and performance using one of two strategies: versioning and portfolio broadening [18. Product versioning refers the release of multiple iterations of a product within the same market segment, typically by developing closed models that compete primarily on performance quality, In contrast, portfolio broadening involves expanding product scope cover several market segments, such as introducing cheaper, open variants of models. Developers of open models can acquire greater market share by monetizing paid premium models and services at later stage (e.g.. tiered pricing and commercial licensing) or by inflating demand for complementary goods, using approaches analogous to product seeding loss leading 33
Foundation Model Economics. Paid, pretrained base models with distributed fine-tuning efforts create tiered market structure that constrains downstream improvements based access terms set by the generalist. For example, fully contractible token allocations give the specialist full freedom over how to allocate budget fine-tuning, whereas per-task token pricing limits the specialist's fine- tuning options [8]. Theoretical models of foundation model development assess how this production structure affects regulatory efforts [27 23 and market competition (36 30), while complementary works focused on provider-side strategy derive pricing recommendations for API endpoints and
training data g 43 46
that align the incentives of model developers with contributors.
B Derivation of Subgame Perfect Equilibria
B.1 Solution for Fixed Initial Performance
The game is solved by backward induction using the functions from Table
Proof of Proposition
First, Un strictly concave.
UUD
0²
(11 (1-8(1- (((1-w) r(a1) -(a1 - -w craw
dai
= dai2
r(01)
8 (1 (1- 5(1 w) 2( - an) cwll)
dan
--2w
UUg < 0 because (0, 1] by definition. Since any local maximum is global maximum, the critical point of Up gives D's best response.
17

---

af = argmax Un((n)
a
aUD
dai a==
i (1- (1-6(1-si) 001 -w)) r(ai)
- ai=aj
=0
dai
r(a))=a;
0 ((1-0( 0(1 w))ar (al 2o0 coow ma =
dan
1 5(1 - -w) 2((n) - -co)w Cuu =0
w(1 - 6(1 - ww - (wlw)
+ 000-
2
When the condition S 1-8 C -g is satisfied, the critical point is feasible and optimal. In cases where
the condition is not satisfied, UD is negative in all cases. Using the fact that UD is strictly concave, if the critical point is negative, the numerical optimum of UD must be at the constraint 0i = Cg-
Up(a1 = ao) = (1 - 8(1 - w))a - 0 Cu arow (1 dw cww)
This utility is non-negative if and only if the condition is met: (1-6+dw- cyww 0 = w 1-5
c--
D will prefer to abstain as long as the condition is not met, and the complete strategy is given by:
D's strategy: aj az 601--61-=w aww)
if w
1-6
abstain 2
cw-5" else.
When D does not abstain, the performance improvement is always non-negative (i.e., no player will strategically invest in degrading performance) because w(1 2(1-8(1 ((1-)
2
Proof of Proposition
Solve for G's best response using or *. G's optimization problem becomes:
W² = argmax Uc(ag,w)
OUg
=0
dw
-=
0
(Ew + 8(1 w))r(aa) (apw + c - w) + 2)(
8w
0
(Ew + 8(1 ww)aa - apw - cw (1 - w)
=0 w=w-
Ow
0
(ew + 8(1 - -w) w(1 8(1 - w) - cww)
+ +00) - Onw
Ow
2
(c. - (ww) (w(1 8(1 - ww) - (wwm ) + an
2
18

---

(e- [6)( 5)( w² 0w (8+ + cw))
1-8
+ +ao) + ((e - 6)w² + +o](
- +0 c,)ww) - ap +
2
2
Col w* O'= .w Cuuu++
1 8
+ao) (cw cww""( (
2 (6 cw cw) =0
w² dw - (6 + Cw) w
+ (0)) + (e - 8)(1 (E-d)(1 8)w*
+ 5(1 - 6)
7 (e - 5)(
- (e - (6 + )w)
2
2
2
w"
6w - (6 + Cw )w)
(ow (1-
8(6 + ww) a0 + Ccl
+ +a0)
(6 + Cw )wn - co + )=² =0
(e (
5)(6 + (w) - (c- -c--)(6 8) + cw)
cu cu +w)) - cu(6 +(o+ (w) w 2 + (c -5)( (1-0)w*
)
8)w**
2
2
2
00) + (c- 5)(1 - 6)w* + 8(1-5) 5)
(1
Cw - 8)
816 + +w) - -a0 to
+ a0)
+
2
2
2
2
www (1 - -))
2 + co + +cw)w" = 0
(- 2 (c- + co + cw) w²² + ((c + 6)) (ow - 66 + cw))
3
(6-8- - 1 + tw + (6 c1 5)
2
8 Solving for the roots with coefficients A = - cw) c0) cw). B ( cw)((
6) + (c., - () + c.). and C = (e - 6 1 + ca) + (6---)(1-8) gives candidate values of w". Since Uc is differentiable between the intervals [0, 0 and [0, f when p >> O, each of these critical points must be evaluated individually. For some sufficiently small positive value μ > 0, G will prefer w = to w 0 μu. It suffices to show lim Us(w
μ)) Un(w = 0). Since
Ua = (ew 5(1 w))ai an 2 Org cwan -w) pl l 0):
lim Uc(w p1) = [const] p
lim Unlw 0) = [consi]
Thus, we do not have to check the upper boundary of the closed interval in the optimization over w.
C Robustness Checks
C.1 Fixed Initial Performance with Specialist-Side Penalty
The main results in Section rely on cost functions that assign the entire regulatory burden to the generalist. However, some regulatory frameworks may also impose disclosure and safety testing requirements on the specialist for fine-tuning a closed model. For instance, EU Al Act Article 25 subjects entities making "substantial modification" to a high-risk AI system quality management and documentation requirements. To determine whether the policy recommendations remain valid under this alternative liability structure, we test the scenario where both generalists and specialists face the same regulatory cost p. The specialist's cost function with penalty is ΦD := (01 Pw + Cg + pl[w < <
Proposition 4. In a game with quadratic costs, a monotonic revenue function r(a1), and a specialist- side penalty p, if w I 8-c
E + 8 1), then D's best-response strategy given by of =
00
og + w(1-ā(1-w)
1fw > 8- 0g
+ - 1), will choose to abstain.
Proof. D's utility function differs from the one in B.1 only by the constant penalty term -p1 [w < 0.. Since this constant disappears when deriving with respect to 0i, the strict concavity of Up and the
19

---

derivation of D' best response are identical to Proposition ! Specifically, the first-order condition yields 01 og -(1-6(1-=)-aww) The main difference is the feasibility condition, the utility at the constraint instead becomes:
Un(an = =00) = 5(1 w))oo
Cw - p
= =a0 (1 - 8 + d6 - (ww)
When the condition S - (22 + d 1) is satisfied, the critical point is feasible and optimal. Since Un is strictly concave, if the critical point is negative, the numerical optimum of Un must be at the constraint a1 = ag- This utility is non-negative and only the condition met: a0(1 - + dw cww) p 2 0 wh + 1). Therefore, D's complete strategy is given by:
6-6 ao
{
ai og + w(1 (1)-6(1-)-caad if S 8-c, (P - 1),
D's strategy:
abstain
else.
G's best response with specialist-side penalty is the same in Proposition since introducing the specialist penalty only affects when abstains, not the solution for a1 We replicate the equilibrium outcomes using the same game parameters as Figures 6 and 7 with the specialist-side penalty for comparison. Figure g (corresponding to Figure and 9 (corresponding to Figure 7 following differences:
show the
1. D abstains for a broader range of (0, p) combinations above the indifference curve. Specifi- cally, the maximum penalty where stops participating participating is lower when they adapt closed model that fails to meet the threshold.
2. The additional penalty term slightly reduces D's utility payoff for choosing on ", but the reduction is negligible.
The specialist-side penalty results in the same qualitative outcomes outside of abstain regions. Below the indifference curve, lower-performing models generate Pareto improvements, while higher- performing models produce utility transfers under greater openness.
20

---

@ Values
Values
Values
10
03
0.5
0.8
0.g
94g
Abiaan
Abaain
-02
Abstn
4.2
Parcio
Heston
Reasos
0.0
0.0
0.0
0.12
0.0
0.04
0.08
0.12
0.16
P
Ug Values
U Values
U x U, Values
0.06
0.008
0.30
0.007
0.15
0.35
00006
0.04
0.20
0.6
00002
0.03
0.15
0.4
0.004
-00033
0.02
0.10
0.2
0.002
Alobie Ropan
Alaavh
Ragan
0.03
Pawir Regio
0.0
0.0
0.0
0.01
0.08
0.12
0.16
0.2
0.0
0.04
0.08
0.12
0.16
0.2
0.01
0.08
0.12
0.16
P
Figure 8: (Specialist Penalty) Under specialist penalty, equilibrium outcomes for ag = 0w
0.01, = 0.1 with Nash bargaining show Pareto improvement over utilities (w,or, on, Uc, Un) for any (p, 0) regulation in the dotted region.
Values
4, Values
8' Values
1.0
1.0
1.0
1.0
088
1.4
0.8
0.6
0.6
0.5
0.6
0.6
value
0.4
0.4
1.2
0.4
0.4
022
1.1
Abeai
022
Transer
Dilly
0.0
0.0
P
U Values
U Values
U U. Values
0.4
1.0
1.
0.6
0.3
0.5
0.8
0.6
406
0.8
0.1
0.2
0.4
0.4
0.4
0.2
0.2
0.2
04
Alexis
Abeais
Uifite
0.0
0.0
0.0
000
0.0
0.8
1.2
1.6
2.0
0.0
0.4
0.8
1.2
1.6
2,0
0.0
0.4
0.8
1.2
1.6
2.0
Figure 9: (Specialist Penalty) Under a specialist penalty, equilibrium outcomes for 1, Cu. 0.01, 0.1 with Nash bargaining create a region where raising the open-source threshold improves a1, and Up but lowers UG. This utility transfer encourages specialist innovation, measured by a1-
C.2 Log Scaling for Production Costs
As an additional robustness check, we consider alternative forms of the production costs associated with releasing the model publicly. Although quadratic revenue remains the dominant term, we explore sublinear scaling for G's production costs. Fixed costs in internal documentation and release
21

---

processes suggest that logarithmic scaling may offer a more precise estimate of production costs than linear scaling. With logarithmic production costs (an logw), G's optimization problem becomes:
+ (e- 5)( w" - -o - -6+ +w)
5)(=)
+ +ao) ((c- ()w-")""" + 8)(
1 - 8
(6 cw))w) -(6+cx)w" kow +1
2
2
Col w" :ur - -8w 2= - c
+ +au) - (ew - cww")( 8 - (6 + cw )w) =0
2
2
7 (e- - 8)( w* Ow - (6 + Cw )w
+ +00) + (E - (E-d)(1 8)w*
- (6 + C )w= + 8(1 - 6)
2
2
w² - -ōw" - (6 + Cw) )w)
+ +00) (cs - -w-w" ((1 - 5)
8(8 + )w)"" - -0ow -1 + Cal
2
(6 + Cw )m¹ - cc + )=) "2= =0
( () (w) ((-8))8 + (x)
(6 + (w)
co + (w) kw)) wr? + (e - 5){ (1 -6)w*
2 +
2
2
40) + (c- 8)(1 6)w* + 8(1 5(1-5) 5)
(1-8)w*
C 11-)) 8)
816 + ww) - aow + cxl
+ an)
2
2
2
2
ww0 (1 8)
2 + co + cw) =0
3
(5 cw) 22 + ((c- 8 + (w) + (ow - 5) + co))
2
(c-8 8 - 1 1+ t00 + (0 cw) -d)
+ +aw
2
3
(e 8- (6 (w) Wwm
) + ((e 8 + (1 8) + (cs - 5) + cwl)
2
(6 c1)(1 -5) 6) ( -00-
(e- 8 + wa)
ax.
Solving the cubic with the following coefficients and checking boundary points gives candidate values of w
A - [e d + co) + cw),
. B = (e 8 + c1 5) + (cw 8) + ow),
C = (e - 8 + cw )a0 + (6-c))(1-5)
= -
With linear scaling for production costs, G's best response is obtained from the set of candidate
values given by Aw? + Bu + C (Appendix B where A = - + cw (8 + cw), B
(e 8 cc)( 8) + (ce 6) + cw), and C (e - 8 + (w) + (8 (8-m.)(10)
Log scaling for production costs produces qualitatively similar outcomes to the main results, as the quadratic or cubic terms from revenue dominate the production cost in either case. We reproduce cquilibrium outcomes using the same parameters as Figures 6 and in Figures and IT For low performance (Ag 0.1 in Figure 6. UG increases more due to reduced release costs and the model being significantly more open, but all the core qualitative insights of the analysis are preserved.
22

---

@ Values
<, Values
Values
19
03
45
0.8
04g
04g
Abiaa
Absaie
02
Abstn
402
Parcso
Haston
Reasos
0.0
0.0
0.0
0.0
0.04
0.08
0.12
0.16
Ug Values
U Values
U x U, Values
0.06
0.008
0.30
0.05
4.35
00006
0.007
0.04
0,20
0.6
40.05
0003
0.15
0.4
0.004
0.02
0.10
4,03
0.2
Alobit Ropan
Alaavh
Rapan
Pasor Regio
0.0
0.0
0,000
0.0
0.01
0.08
0.12
0.16
0.2
0.0
0.04
0.08
0.12
0.16
0.2
0.01
0.08
0.12
0.16
02
P
P
Figure 10: (Log Scaling) Under log scaling for production costs, equilibrium outcomes for low performance (on = 0.1, cu = 0.01, = 0.1) with Nash bargaining still create a region where raising the open-source threshold improves wi, on, and Un but lowers Ua.
Values
Values
Values
1.0
1.0
1.0
1.0
0.8
1.4
088
0.6
0.6
0,6
1.3
0.6
0.6
0.4
0.4
0.4
1.2
0.4
0.4
0.2
0.2
Absten Tranater
Unitit Itauster
Inaeter
0.0
0.0
0.0
1.6
P
UP Values
U, Values
U, U. Values
1.0
0.4
1.0
0,20
value
0.6
0.6
0.1
0.2
0.4
0.4
0.10
4.4
0.2
0.2
0.2
Abebis
Abaais
4.2
Transtrr
Unitit Trandir
0.0
0.0
0.0
0.00
1.2
1.6
2.0
0.0
0.4
0.8
1.2
1.6
2.0
0.0
0.4
1.2
1.6
2.0
Figure 11: (Log Scaling) Under log scaling for production costs, equilibrium outcomes for high performance (Okg Cw 0.01, 0.1) with Nash bargaining show Pareto improvement utilities (w, a1, UG. Up) for any (p, f) regulation in the dotted region.
C.3 Alternate Division of Operation Costs
At either extreme of openness (w = 0 or 1), either the specialist generalist will assume the majority of operation costs. For the main results in Section 3 G's operation cost ca w) in order to ensure that operation costs sum to constant between the generalist and specialist:
23

---

@G., + DDr = cwar(1 w) + Cwaww Cu. Operation costs between G and D sum to constant value because least one player must pay for hosting and inference in order to generate revenue through model usage. This implies that total operation costs depend on the final performance achieved after fine-tuning, rather than baseline performance, and the generalist must bear some the operation costs associated with improved performance. As robustness check, we examine the effect of changing G's operation cost from C (1 w) to co (1 - w), meaning the generalist does not bear any additional operation costs resulting from downstream model improvements. The coefficients G's best response change from:
( cw) cw) to (6+cw)(c-8)
2
B (e 8 cw) 8) (ew 6) + cw) B (c-d)(1-A) co - - + + (=-c,)(10)) to C oo(c 5 cw).
While UG increases slightly, there are no qualitative differences in the equilibrium outcomes with Cu {0.01, 0.1, 0.5, 1) between the version with G's operation costs cw vs. c (w- We add plots for the replication of equilibrium outcomes with the same parameters as Figures 6 and with ag Figures and 13
These two setups are qualitatively indistinguishable because for models with downloadable weights, D will incur most of the operation costs for fine-tuning when high enough. If is low and the model mostly closed, the operation costs are divided in one of two ways. First, the closed model provider offers fine-tuning services, but this fine-tuning occurs the generalist's infrastructure (e.g., OpenAl's fine-tune fine-tuning APIs). Second, D covers operation costs related to fine-tuning but must choose a ag because production costs ((a} - ) w1) become prohibitive tries to improve the model when openness is extremely low. In this case, the distinction between ai vs. in G's operation costs becomes irrelevant because @1 Og.
i Values
Values
1,0
0.8
0,8
0.8
08
06
0.6
-04
0.6
40
0.4
Abaals
Abait
402
0,0
0.0
0.04
Ug Values
U, Values
U U, Values
1.0
0.30
0.8
0.8
0.8
0.35
1004
0.6
0.30
0.005
0/03
0.4
0.004
0002
0.003
0.2
0.002
Abaxh
0.01
Abian
0.05
Abvais
Reaine
0.001
0.00
00000
0.0
0.04
0.08
0.12
0.16
0.2
0.0
0.01
0.08
0.12
0.16
0.2
0.0
0.04
0.08
0.12
0.16
0.2
Figure 12: (Alternate Operation Costs) When G's operation costs are function of og instead of final performance, equilibrium outcomes for low performance (a,, 0.1, Go 0.01, e 0.1) with Nash bargaining still create a region where raising the open-source threshold improves w, and UD but lowers UG.
The results hold regardless of which player bears the additional operation costs associated with the fine-tuned model's improved performance. In practice, generalists can batch operation costs and take advantage of cost amortization across larger deployment volumes [12. but because our interest is in cost parameters the regulator can directly control, we defer the exploration of hardware and scale efficiencies to future research.
24

---

Values
Values
Values
1.0
1.0
1.0
1.0
0.8
1.4
0.8
0.6
0.6
0.6
0.6
0.6
value
0.4
0.4
0.4
-12
0.4
0.4
0.2
0.2
Abebie
1.1
0.2
Abenta
9.3
Unility Timsderr
Toanerr
Ttander
0.0
0.0
P
U Values
U Values
U U, Values
0.4
1.0
0.20
0.6
0.6
0.6
0.4
0.2
0.4
0.4
-0.10
40.6
0.2
0,2
0.2
Avasts
Abaats
Torafer
Uiilit Torrde
0.0
0.0
0.00
1.2
1.6
2.0
0.0
0.4
0.8
1.2
1.6
2.0
0.0
0.4
0.8
1.2
1.6
2.0
Figure 13: (Alternate Operation Costs) When G's operation costs are function of og instead of final performance, equilibrium outcomes for high performance (aw 1, Cu 0.01, 0.1) with Nash bargaining show Pareto improvement over utilities (w, a1, UG. Up) for any (p. 0) regulation in the dotted region.
D Model Extensions
D.1 Flexible Initial Performance
G's costs of developing the model to base performance ag dropped from &G in game, assume has already invested in achieving a fixed performance level prior to the regulation and cannot adaptively recover sunk development costs. "Development" is general label for all fixed costs related to pre-training (data licensing, labor, hardware, R&D) and post-training (alignment, evaluation). This setup resembles the decisions of model developers who have reached a performance ceiling and cannot overcome resource limitations in the short term.
Although the main results assume that G's choice of ao occurs prior to the introduction of the regulation, an alternative setup may allow simultaneously choose w and og rather than selecting an openness level given fixed performance level. This setup represents scenarios where new entrants jointly choose performance and openness levels for a model or when incumbents decide to release several variants of pre-existing model.
Propusition 5. When E 1 - 8) 2 Cu. the best response (ao", obtained by calculating w' and finding a numerically using w" achieves global maximum for Ug.
Proof. G strategy two-dimensional:
0 w
argmax
UG
doww
From Proposition I
UG can be written terms of w and ag.
Ue = (ew + 5(1 - w)) as + w(1 w(1 w) cww))
2
anw + cw cu w(l w(1 - w) - ((wي)) (1 - W) - -plp < 0).
2
25

---

Differentiating Ug with respect to G's decision variables, we get
AUG cw 8(1 - w) w + Cw - w),
daw
OUG (E- -8)( w سw (6 + )w²
+ +a0) ((e 0)w + 6)( 1-6
(6+cw)/) apt
Ow
2
w dw dws ?wwo
1-8
Cw(
+ a0) - (c - (ww)(
2 (6 + cw)ww
2
OUg
8 Cw:
dayw
aUg =6- 8 - + Gw
Owao
UGG O.
dag
From the derivation in the proof of Proposition 2 we know that aUe Aw² + BB + C, where du
A = - ( 6 + c cw). B = (c + (w) 5) + (cw 5) (w), and C (c - 8
Cw )a0 + (6-m))1-6) Therefore,
2
UUG =2A + B Ow2
8 -3 8 c5) cw)((1 5) + (cw - 8)(6 cw) (c. 6) cw).
(E 1 = 3(6 c.) 300+
---
واUg Joo" L'a OwDan
8²U dgodw d'U Ow
V²UG=
c-1 E 11-++ (E- +-8+cc)le cw)(1 -36 + cw) + (cw, 8) + c.)
0
c-1-8- - 5- Cw
Next, we show that the Hessian of UG is negative semidefinite when |e - 8) 2 c.. By inspection, the first principal minor satisfies d'Ua < 0. The criterion det V2 2 0 only holds when dag
I - 88 2 cw-
= de([v"(c)=0-((
cw)(E 8
=- 5)² + c.
Taking the square root of both sides, we get that det( V2 2 only when |e - 81 2 c.s given that c. 2 0 since it is nonnegative cost ratio. This implies that the Hessian matrix V2 is negative semidefinite when je - 1 dl 2 cwi which establishes the concavity of UG w.r.t. an and w and guarantees the joint solution achieves global maximum.
When the condition Ic 16-1-1 5 2 cu is not met, G generates almost no positive utility from releasing the model (Figures a4d
D.2 Multidimensional Performance Definition
The utility functions assume performance a one-dimensional property which permits a linear ranking of models. However, it is plausible that no single model consistently dominates across all dimensions (reasoning, general tasks, efficiency). Performance can be extended to a multidimensioonal variable capture such scenarios. Formally, this requires the closed-form derivations for both players' utility functions to be reformulated to handle vector-value performance metrics. For example, aUp
Zan
26

---

0.8
0.2
0.6
0.4
0.4
0.6
0.8
1.0
0.0
0gg
U
1.2
1.
0.8
0.8
0.8
0.8
0.6
0.4
0.2
0.0
0.6
0.4
0.2
5
1.0
0.0
Og
Figure 14: Generalist's strategy space for (atp, w) when |e e-
8 2 C.. with parameters e 0.1, 5 = 0.5, C 0.01, 1, = 0.2. (Top) G's unconstrained strategy space for azimuth 145 and azimuth = 245. (Bottom) G's strategy space when abstaining for negative utility payoffs for azimuth 145 and azimuth 245.
would become an n--imensional vector Va, UD. where each partial derivative entry represents the marginal utility with respect to a different performance dimension. This setup can encode how regulatory performance preferences diverge from the utility considerations of end users, but because this complexity distracts from the core analysis on model openness, we omit such fine- mee-grained performance definitions from the main results.
E Regulation Equilibrium Results
The area above the indifference curve in the space of (p, () choices represents when -> O, while the area below represents when G meets the open-source threshold.
Proof of Proposition 3
Uc(w = 0) 2 Up(w = 0)
8α c
p² > (e0 8(1 0))a; a00 cwai 0)
p¹ 2 e0 + 8 -00
onf
8 c.
cw + cwo
0n
an
> 0(c 8 00 + Gw)
a1
a
00
p¹ < 0
cw)
For given threshold, as long p < p², G will keep the model closed rather than meeting the threshold. pf < O, G will always open the model or above the threshold, regardless of the threshold.
D
27

---

0.0
0.0
405
40.5
1.5
1.0
1.5
-2.0
-15
2.0
2.5
-00
2.5
3.0
-25
-3.0
0.8
0.0
3
0.8
0.4
6
0.2
4
1.0
0.0
Gg
0.0
4.5
U
-1.0
-.0
0,020
-15 C.
0,015
2.0
2.5
-3.
0.005
1.0
-3.0
0,000
0,6
0.4
0.2
@
0.4
0.6
0,2
1.0
0.8
00
0.0
Figure 15: Generalist's strategy space for (ako, w) when |e 1 < c.. with parameters 0.7, 5 = 0.1, eu = 0.5, p = 1, 8 = 0.2. (Top) G's unconstrained strategy space for azimuth 145 and azimuth 245. (Bottom) G's strategy space when abstaining for negative utility payoffs for azimuth 145 and azimuth 245.
Proposition There exist games with quadratic costs and a monotonic revenue function r(on) in which there are Pareto-optimal regulations that improve all utilities (w, on, UG, UD) compared to regulation.
Proof. Given a game with Nash bargaining and the parameters og = 0.1, = 0.05, c 0.1, the no- regulation equilibrium is (w" 0, 8" 0.53, o 0.1024) with Un = 0.0480 and UG = 0.0478. Introducing a regulation 0.97, a 0.2746, Ua 0.0575, Un 0.1090) because Uo(w" 0.6) Valw" 0). This Pareto-dominates no regulation, as (w,01, UG, Un) are all strictly better. Figure visualizes the utility implications the space of possible regulation tuples (p, 0).
0.6 and 0.05 leads to the equilibrium outcome (w" = 0.6, =
Proposition 7 (Bounds for Pareto- OptimalOptimal Regulation). Let μ be an arbitrarily small constant. Given that there exist region is characterized by upper bounds of - Pmax and lower bounds of μ1, p 0 + cn e Ja p.
regulations for utilities (w, On UG. UD), the Pareto-optimal
Proof. To improve beyond the no-regulation equilibrium, any Pareto optimal regulation must create binding constraint where complies with the threshold requirement. There are three cases of the game: (1) G' already clears the required openness threshold without regulation, (2) G violates the threshold requirement and does not change strategy, or (3) G adjusts the openness level to the threshold to comply with the regulation. Case (2) is dominated by (1), the game must fall in (1) or (3). The regulation only has an effect in case (3), implying that the Pareto improvement region must lie under the indifference curve.
28

---

By Proposition 3 the lower bound on penalty for the region is given p 0( cc cw-cw)o a1 + on where μ > 0 is small constant. The upper bound on penalty is Pmax because raising the penalty at given threshold under the indifference curve does not change G's strategy. By Proposition T the constraint c-- must be satisfied if a is improved. These two constraints imply 0 v giving upper bound on 0. Further, the region of Pareto improvement is lower-bounded by 0 since any effective regulation must improve w through case (3).
Data for Figure 3
Although ELO rankings and MMLU scores used heuristics to compare model performance. they are not meant to be a comprchensive measure of overall model capability.
ELO Score is the model's arena score from the Chatbot Arena LLM Leaderboard. % Closed is calculated the same as Figure 4b from Eiras et al. [14], using Table 3's categorization of model components. The percentage is the the portion of components that received fully closed classification (CI/DI) out of the number of components where such a score is available (excluding ? and N/A labels).
Model
ELO Score
% Closed
Pythia
893
0
T5
868
Stable LM
840
0
Llama
799
0.43
DBRX
1103
0.5
Mistral-7B
1075
0.71
Falcon
1034
0.67
PaLM
1004
1.0
Llama 2
1093
0.8
Command
1180
0.8
Command R+
1215
0.8
Llama 3
1269
0.8
GPT-4
1256
0.82
Gemini
1380
0,92
Claude 2
1132
0.92
29

---

MMLU Score uses exact match accuracy for MMLU All Subjects from the HELM leaderboard
MMLU scores marked with (*) are self-reported the model documentation, rather than collected from HELM evaluations, so the comparisons may be inconsistent.
Model
MMLU Score
Weights
Release Date
Claude 3.5 Sonnet
0.873
Closed
06-2024
DeepSeek v3
0.872
Open
12-2024
Gemini 1.5 Pro
0,869
Closed
02-2024
Claude Opus
0.846
Closed
03-2024
Llama 3.1 Instruct Turbo (405 B)
0.845
Open
07-2024
GPT-4o
0.843
Closed
05-2024
Qwen2.5 Instruct Turbo (72B)
0.834
Closed
04-20024
GPT-4
0.824
Closed
03-2023
Amazon Nova Pro
0.82
Closed
12-2024
GPT-4 Turbo
0.813
Closed
11-2023
Llama 3.2 Vision Instruct Turbo (90B)
0.803
Open
09-2024
Llama 3.1 Instruct Turbo (70B)
0.801
Open
07-2024
Mistral Large 2
0.8
Open
07-2024
Gemini 2.0 Flash
0.797
Closed
12-2024
Llama (70B)
0.793
Open
04-2024
Llama 3.3 Instruct Turbo (70B)
0.791
Open
12-2024
PaLM-2 (Unicor)
0.786
Closed
05-2023
Jamba 1.5 Large
0.782
Open
08-2024
Mixtral (8x22B)
0.778
Open
04-2024
Phi-3 (14B)
0.775
Open
05-2024
Qwen1.5 (72B)
0.774
Closed
02-2024
Yi (34B)
0.762
Open
11-2023
Gemma (27B)
0.757
Closed
06-2024
Claude 3.5 Haiku
0.743
Closed
11-2024
DBRX Instruct
0.741
Open
03-2024
Gemini 1.5 Flash
0.739
Closed
05-2024
DeepSeek LLM Chat (67B)
0.725
Open
11-2023
Command R Plus
0.694
Open
08-2024
PaLM-2 (Bison)
0.692
Closed
11-2023
GPT-3.5 Turbo
0.689
Closed
03-2023
Llama (8B)
0.668
Open
04-2023
OLMo 1.7 (7B)
0.538
Open
07-2024
Chinchilla (70B)
*0.676
Closed
03-2022
PaLM (540B)
*0,693
Closed
04-2022
U-PaLM
*0,707
Closed
10-2022
Flan-PaLM
*0.752
Closed
10-2022
Llama (65B)
*0.634
Open
02-2023
Llama 2 (70B)
*0.689
Open
07-2023
Falcon (180B)
*0.706
Open
09-2023
30

---

G
Additional Figures
The equilibrium of the game is characterized by triple (w,6,a1) Here, we check the effects of varying and E {0, 0.01, 0.1. on the no-regulation equilibrium, (w, ó, an) for = 0, under different bargaining solutions. Figure uses Cw 0.1 over different ag and values.
G.1
Openness Levels
VM Bangzining (c.-
VM Bargaining (c_
Dargaining (6,
VM Bargzining 035)
0.6
3.
0.3
0.3
-02
8-
3.0
0.0
0.9
100
00
0.1
9.2
033
0.1
0.5
9.0
011
0.3
03
04
05
04
0,5
0.
01
6.7
0.3
0.4
0.5
Nash Rergaining (rum
Nash Rergzining 0.01)
Nish Rangsining (c--
Nach Rergnining
0.5
04
0.4
064
Villo
02
81
0.1
0.
-03
0,3
000
60
0.1
0.2
0.3
0.4
3.0
0.1
0.2
0.3
04
05
0.0
0.1
0.2
03
0.4
055
0.1
0.2
0.3
0.4
055
Egalitorion Bargaining (r_
Egalitorian Bargsining (c_- 0.01)
Egalitarion Bargaining
Egalltaaion Bargaining -05)
03
-0
0.0
60
0.3
0.2
8-1
11.
0.5
0.1
81
033
0.5
0/4
0.5
01.0
6.1
0.2
03
04
05
0.
01
6.7
0.7
0.4
2.5
Figure 16: Openness levels (w*) chosen by G without regulation (p = 0) under VM, Nash, and egalitarian bargaining solutions. Rows correspond to bargaining solutions, and operation costs increase from left to right: openness the region of low oop e because G avoids operation costs by fully opening, The fully open area at high Og, recedes as increases for Nash and egalitarian bargaining solutions because D's utility harmed by high operation costs for high performance when G receives the majority of revenue through 8*.
O, 0.01, 0.5. Increasing causes less intermediate values of
31

---

G.2
Bargaining Coefficients
VM Bargaining le. 0)
VM Bargaining te 0.01)
VM Bergiiting (eym 0.3)
VM Bargaining (c 0.5)
04
0.4
0.4
0.2
92
02
0.
3.1
0.1
-02
0.0
033
04
0.5
%
Nash Borgaining (c_ 0)
Nash Bergaining (c_- 0.01)
Bargaining (c.- 0.1)
Nash Dargaining (c_- 0.5)
0.5
0.5
0.3
0.2
-04
0.0
%
0g
Egaliturian Bargaining (c 0)
Egallituiaan Bargaining (c, 0.01)
Egalitaran Bargaining (c,- 0.10
Ggaliturian Dargaining (c 0.5)
0.5
0.4
-03
0.2
077
07
03
-0.4
0.1
%
Figure 17: Bargaining coefficients (6*) jointly chosen by G and D without regulation (p 0) under VM, Nash, and egalitarian bargaining solutions. Rows correspond to bargaining solutions, and operation costs increase from left right: eu 0.01, 0.1, 0.5. For all three bargaining solutions. higher operation costs push 8 to favor G more in regions where G fully closes the model. For low ap. where G fully opens model and there is less revenue, the bargain favors more.
32

---

G.3
Final Performance
VM Bangsining (c_-
VM Bargaining 0,013
Dargaining 0.3
VM Dargnining
3.
0.3
0.2
80
30
0.0
00
0.1
0.2
B-3
0.4
0.5
0.
011
0.3
0.5
04
05
0.0
03
04
0.5
00
01
6.7
0.
0.4
25
Nash Rergaining 0)
Nash Rergaining 0.01)
Nush Rargsining (c_- 0.13
Nash Rergnining 0.5)
04
0.4
0.4
00
02
0.2
004
01
0.1
0.1
0.3
00
0.1
9.2
0.3
0.4
9.0
011
0.3
0.3
0.4
0.5
0.0
0.7
03
0.4
0.5
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Figure 18: Improvement level (af) chosen by D without regulation (p = 0) under VM, Nash, and egalitarian bargaining solutions. Rows correspond to bargaining solutions, and operation costs increase from left to right: S = 0, 0.01, 0.1, 0.5. D's level of improvement decreases as operation costs increase across all bargaining solutions, and regardless whether G' fully opens closes the model.
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Reviews:
[{"review_id": "9il80b5ZMT", "reviewer": "Reviewer_t96V", "summary": "The paper presents an economic model that tries to capture the economics of open sourcing foundational models for those who build them (the generalists) and those who finetunes these into specialized models (the domain specialists) given that the government enforces some form of regulation. \n\nThe results characterize market equilibria under various openness policies and present an optimal range of regulatory penalties and open-source thresholds as a function of the foundation model performance.\n\nThe model seeks to provide theoretical foundations for AI governance decisions around openness and enables evaluation and refinement of practical open-source policies, and the discussion concludes with three implications of the analysis.", "questions": "* Regarding the implications in 5.1: Would it not be better to articulate all three of them as advises to the regulatory bodies, as is the case with the second one? For example, the first one could be formulated as something like “Regulators should adopt tailored approaches according to the maturity of the technology”? I think the results would be more accessible to regulatory bodies, who might not be as easily sold on pareto improvements. \n* Why is there not a constant term specifying the cost in the production cost term $\\alpha_0\\omega$? As I read the term, both $\\alpha_0$ and $\\omega$ are rates. When they are added together with operation and regulatory which both contains constants ($c_w$ and $p$), then this would not yield the correct result. Am I wrong? What is it that I do not understand?\n* Are there other parameters that could lead to the same behavior as described in Figure 3 than those that you give?\n* Regarding negative examples: Are there real-world examples, such as the one given in Figure 3, that are not described by the model. If yes, then why does it not capture this?\n* As far as I understand, the model only captures a single $G$ and a single $D$. How would many $G$ to many $D$ change the analysis. Would it be a worthwhile analysis? Is this future work?", "limitations": "I think this is sufficiently addressed.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "## Strengths\n* The topic is novel, and the potential impact could be high given that it could help governments in their understanding of how to regulate foundational models. Such regulations are already in place in several countries, including EU, US, and Singapore, as explained in the introduction. \n* The model is simple but still captures the main drivers, and the analysis seems to me – at least – to be sound. I especially like the discussion where the central findings are discussed, although I have some suggestions for how to improve them, see below. \n* The paper is very well structured and generally well written. \n* I really like the explanations of the costs in the equations on line 125.\n* I like how you show how the model explains the real-world example in Figure 3.\n \n## Weaknesses\n* My main gripe with the paper is the writing but not all of it. I found especially the parts where the model was introduced in detail to be hard to read because variables were not properly introduced before using them. The introduction came at a later point, but this made me very confused when reading through it the first time. I do not think it will take a lot of effort to fix this. Quite the opposite, I think it is quite easy. Examples are $\\epsilon$, $p$ and $c_W$. Utility $U$ and cost ($\\phi$) are shown in Table 1 but not mentioned in the text. I would move Table 1 later in section 2.3 and introduced all variables in the text before using them, maybe even in section 2.2. \n* I would like a discussion on negative examples. Which scenarios will the model fail to capture? \n\n## Minor comments \n* Line 23: OSS describes publicly available source code with no restrictions on use … This is not fully correct. Many OSS licenses do not restrict use, but some do, especially those that have twin licenses where academic and non-commercial use is OK but commercial use is restricted. Such licenses are not uncommon even in academia, especially in the fields of AI and ML. \n* Fully open models, as described in lines 26 and 27, should also share the code used for training and inference/reasoning. If not, they are not reproducible (which should be the main goal of open-sourcing foundation models). See for example (Gundersen, 2021) for a definition of reproducibility that requires code and data.\n * Instead of using the term “industry developers” on line 29, which targets individuals and might therefore be much less coherent, I suggest using the term “software industry” or “industry” as this indicates companies, which is how I read the sentence anyhow. \n* The comment on $\\epsilon$ on line 117 confuses more than it helps, especially as the variable is not introduced.. \n* The caption of Figure 4 is very parameter description heavy. A higher level-explanation would be nice. If possible.\n\n## Reference\nGundersen, O. E. (2021). The fundamental principles of reproducibility. Philosophical Transactions of the Royal Society A, 379(2197), 20200210.", "was_revised": true, "final_justification": "My main concern was related to clarity, which the authors state that they have resolved. Also, their responses to the questions and especially those related to limitations of the method seem nice. \n\nMy concerns have been resolved, and I think this paper should be accepted to the conference.", "rebuttal": "We appreciate the positive review and have addressed your questions below.\n\n# Questions\n> Are there other parameters that could lead to the same behavior as described in Figure 3 than those that you give?\n\nIncreasing $\\alpha_0$ under no penalty always leads to more closed behavior as long as $\\epsilon < 1$ and $c_\\omega < 1$. For example, the openness trends in Figure 2 and Appendix F continue for higher values of $\\alpha_0$ (we tested at $\\alpha_0 = 10$ and $\\alpha_0 = 100$). We require $\\epsilon < 1$ because when reputational benefits always match or exceed closed revenue ($\\epsilon=1$), $G$ will just open the model fully. \n\nThe behavior in Figure 3 also requires that  $c_\\omega < 1$ because if operation costs exceed any revenue $G$ can earn from model adoption, $G$ always fully opens the model or abstains at any performance level. $c_\\omega < 1$ is a realistic assumption because developers can price the costs of inference into model offerings via API pricing or subscription charges. \n\n> As far as I understand, the model only captures a single $G$ and a single $D$. How would many $G$ to many $D$ change the analysis. Would it be a worthwhile analysis? Is this future work?\n\nYes, this definitely seems like worthwhile analysis. We have noted modeling competition between multiple players as a future direction. \n\n**Extension to multiple $G$ with varying $\\alpha_0$ levels.** This can only be done via simulation. In the first stage of backward induction, $D$ would need to solve multiple optimization problems and choose the $G$ that maximizes their utility payoff (potentially with switching costs). Both $\\alpha_0$ and $c_\\omega$ would be specific to each $G$. \n\n**Extension to multiple $D$.** $G$’s revenue would be the sum of all $\\alpha_1$ for the $D$ who adopt their model. However, this extension is trickier because it requires modelling multiple $D$ with different valuations and priorities; e.g., one specialist might prefer cheaper operation costs if inference demand is high, while another tolerates high operation costs for better model accuracy due to domain-specific requirements. Future work could model the strategic dynamics of $G$ differentiating their product along different dimensions -- higher model openness vs. higher performance -- to target a specific subset of specialists (see: Giarratana & Fosfuri, “Product Strategies and Survival in Schumpeterian Environments: Evidence from the US Security Software Industry”).\n\nModeling multiple $G$ and $D$ would require solving a separate bargain for each ($G, D$) pairing. Since we identify $\\delta^*$ via grid search over the interval [0,1] and there are $G/D$-specific cost functions with multiple players, this would be intractable as add-on analysis for the paper.\n\n> Why is there not a constant term specifying the cost in the production cost term $\\alpha_0\\omega$? As I read the term, both $\\alpha_0$ and $\\omega$ are rates. When they are added together with operation and regulatory which both contains constants ($c_w$ and $p$), then this would not yield the correct result. Am I wrong? What is it that I do not understand?\n\nIt’s a good idea to include the constant terms more explicitly. We had analyzed the version where constants for production and regulatory costs are normalized to 1 and the $c_\\omega$ constant compares between operation costs and revenue. However, we realize we can provide more general results where these constants are any positive number. We have now included all constant terms into closed-form solutions and our numerical simulations. An important empirical direction would be estimating appropriate values for the two constants based on how regulatory requirements map to dollar value penalties, how documentation costs correspond to inference costs, etc.  \n\n# Clarifications\n\n> Many OSS licenses do not restrict use, but some do, especially those that have twin licenses where academic and non-commercial use is OK but commercial use is restricted. \n\nThese distinctions seem significant, so we have updated L23 to reflect nuances in OSS licensing. We had originally written “no use restrictions” based on the Open Source Initiative definition of OSS. \n\n> Fully open models, as described in lines 26 and 27, should also share the code used for training and inference/reasoning. If not, they are not reproducible (which should be the main goal of open-sourcing foundation models). \n\nThis has been noted in the text, with supporting examples drawn from Gundersen (2021).\n\n# Weaknesses\nWe have revised Section 2 for clarity and changed Section 5 to rearticulate findings as high-level considerations for regulatory bodies without referencing player utilities or Pareto improvements. \n\n> I would like a discussion on negative examples. Which scenarios will the model fail to capture?\n\nSee Appendix C for our justifications for omitting considerations like deployment scale. To more explicitly discuss negative examples, the discussion in Section 5.2 has been revised with more examples: “The two-player framework also overlooks important social utility considerations. While our analysis assumes that higher model openness and performance are generally beneficial, a regulator must consider safety risks associated with higher performance or market-wide loss scenarios, such as possible intermediate levels of openness where consumers may be worse off. This complexity suggests that defining openness as a multidimensional property may be valuable to capture factors beyond simple model access, such as documentation and explainability of model outputs. Additionally, modeling multi-firm competition between a closed-source incumbent and multiple generalists can reveal when established firms pivot to open-source strategies and how timing of market entry affects whether entrants adopt open-source approaches.”"}, {"review_id": "JsB9Uqyx8k", "reviewer": "Reviewer_Rs1J", "summary": "The paper proposes a two-stage game-theoretic model that links a foundation-model developer (generalist $G$) and a downstream fine-tuner (specialist $D$) who bargain over revenue, while a regulator sets an openness threshold and penalty. The game progresses in multiple stages. First, G chooses an openness level on a 0-1 continuum (Fig. 1) that governs whether the model clears the regulatory threshold and what costs it bears. Then, D decides whether (and how far) to fine-tune the model to a new performance level.\nThe authors derive closed-form expressions for D’s best response and for G’s sub-game-perfect openness choice. The authors demonstrate three policy regimes: (i) Pareto-improvement when $\\alpha_0$ is low and moderate, which push G to open the model, (ii) utility-loss zones where penalties only transfer surplus without added openness, and (iii) utility-transfer zones where tighter thresholds reroute surplus from G to D, raising $\\alpha_1$ but lowering $U_G$.\n\nThe authors claim this provides guidance for calibrating openness rules in the EU AI Act and similar regimes.", "questions": "Game theoretic analyses are a fantastic way to provide quantification and grounding for policy makers. This paper is commendable in doing exactly that. However, there are a few areas in which the authors can improve their paper.\n\nThe model presently assigns the entire regulatory burden to the generalist. Downstream fine-tuners, however, can face disclosure or safety-testing duties under both the EU Act (Article 28) and the US NIST RMF. I would like to see a specialist-side penalty term (or a brief formal argument for why it is negligible). If introducing that term alters the qualitative comparative statics, please highlight where; if it does not, you gain credibility by demonstrating robustness.\n\nThe paper assumes performance improvements exhibit diminishing returns as the base model approaches the frontier, and you treat performance as the only strategic dimension. Yet fine-tuning often delivers non-marginal gains on underperforming tasks, and many practitioners optimize for latency or parameter efficiency instead of raw benchmark scores. I ask for (a) at least one empirical learning-curve plot or citation supporting the concavity assumption, and (b) a short discussion of how a second axis such as inference cost would change the results. Demonstrated robustness to alternative tuning curves or an extended two-dimensional game would lift my novelty and insight scores; silence on the issue would pull them down.", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 4, "significance": 2, "originality": 3, "strengths_and_weaknesses": "# Strengths\nThe paper tackles a policy problem that is both immediate and under-specified: how openness rules in upcoming AI regulation will shape economic incentives along the foundation-model value chain. By allowing the \"generalist” developer, rather than the regulator, to choose an openness level endogenously, the authors capture a strategic knob that existing game-theoretic work leaves fixed. Their closed-form solutions make the comparative statics transparent, so readers can see precisely when higher openness lowers total welfare versus when it re-allocates surplus between the upstream model builder and downstream fine-tuner. Together, these features turn what could have been an abstract toy model into a practically useful lens for policymakers and practitioners who must weigh documentation burdens, revenue-sharing, and compliance costs.\n\n1. The model addresses a live regulatory gap rather than an abstract problem. The EU AI Act exemptions hinge on an ill-defined notion of \"open-source\", and industry \"open-washing” is widely discussed problem.\n2. This provides a realistic strategic lever for model developers. The relationship between producers of a general model and those building derived specialist models is under-studied. This extends the relevant game literature by letting G choose openness endogenously. Previous work treats openness as fixed.\n3. This work allows clear comparative-statics reasoning instead of pure simulation by obtaining closed-form best responses.\n\nOverall, this work can help readers reason about who bears which costs at different openness levels.\n\n# Weaknesses\nSeveral conceptual and modeling shortcuts undermine that practical promise. The paper treats \"foundation models” as uniquely troublesome for open-source definitions, yet earlier software and ML precedents raise identical dilemmas. This framing overstates novelty.\n\nMore overall, who decides what is high or low performance? These are vague notions the work depends on, but are never defined. The window of relevant performance changes over time? Once upon a time, GPT-3 was very high performing, but today we would call it low performing. The change in performance could trigger changes in openness. This analysis should minimally define performance, and to improve the modeling, introduce time variance.\n\n1. L24: \"Foundation models complicate the standard open-source definition\". The complication is not unique to foundation models. Binary-only drivers and source-available databases raised the same debates years ago. It's worth citing \"The Value of Open Source Software\" by Hoffman et. al. [0]\n2. L42-43: The discussion of specialists and fine-tuning appears could be foreshadowed earlier in the introduction. It's worth discussing fine-tuning and the game this implies before this statement.\n3. L113: \"dividing revenue from foundation model development\". The usage of the term \"foundation model\" does not add much to the discussion. Again, this phenomenon is not unique in the ML world to just foundation models. Look at Gupta et. al. for a discussion on the issues raised by the usage of \"foundation\" models in policy-related discussions [1]\n4. L125-126 and L130: \"$\\phi_G(\\omega, \\alpha_0|\\theta) = ... + c_{\\omega}\\alpha_1(1 - \\omega) + ...$\" Should this be $\\alpha_1$ or $\\alpha_0$? Currently this implies that operation costs hinge on the final performance reached after fine-tuning, not the baseline. Why does the generalist bear this cost?\n5. L125-126: Why are there not regulatory costs for the specialist? Fine-tuned models could be closed-source and/or regulated as well.\n6. L127: \"there are additional costs to documents and releasing the model publicly which scale with higher levels of openness and performance\". Sure, but this is not linearly so? There are already costs associated with documentation internally, as well as the cost of a proper internal model release process. Perhaps this is better modeled as log scaling rather than linear?\n7. L128: \"safety risk management for open models\". This is a regulatory cost, but you are trying to provide an example for production costs.\n8. L136: \"since incremental improvements are more expensive at higher performance levels...\" The entire point of fine-tuning a general model is because it was initially not performing well on the desired downstream task. Can you justify quadratic cost? Empirical fine-tuning curves often show super-linear returns for early steps. Look at literature like Xia et. al. [2]\n9. L139: \"fine-tune or improve\". There are plenty of ways to improve the performance of a closed-source model. In-context learning, RAG, tool use, etc. Maybe just say \"fine-tune\" instead?\n10. L148-149: \"model performance is the only feature in $G$'s strategy space\". This is a very limiting assumption. Generalists also worry about efficiency. Reference the debate around DeepSeek, o3, o3-mini, and new nano-style models.\n11. L157: \"higher openness increases production costs\". There is ample evidence showing this to be false. Projects like YOLO, vLLM, etc. have benefited drastically from open-souce contributions. Outside of ML, look at Docker, Redis, etc. Even within foundation models, improvements to Gemma have led to improvements in Gemini.\n\n# Checklist concerns\nFor question 5, the authors say \"no\" to open access to data and code. But, the authors reference a numerical simulation to produce their figures. Where is the code for that? The justification that \"the plots can be easily reproduced\" is hand-wavey. Ok, if it is easy, then just share the code?\n\nFor question 8, while the simulations can be trivially run on a CPU, a few extra words saying for just how long or how many CPUs would be useful.\n\n# Typos\nL221: \"effectsacross\" -> \"effects across\"\nL222: \"identifies conditions the\" -> \"identifies the conditions for\" ?\nL294: \"teh\" -> \"the\"\n\n[0] Hoffmann, Manuel, Frank Nagle, and Yanuo Zhou. \"The Value of Open Source Software.\" Harvard Business School Working Paper, No. 24-038, January 2024.\n\n[1] Gupta et al. \"Data-Centric AI Governance: Addressing the Limitations of Model-Focused Policies.\" arXiv preprint arXiv:2409.17216 (2024).\n\n[2] Y. Xia et al., \"Understanding the Performance and Estimating the Cost of LLM Fine-Tuning,\" 2024 IEEE International Symposium on Workload Characterization (IISWC), Vancouver, BC, Canada, 2024, pp. 210-223, doi: 10.1109/IISWC63097.2024.00027.", "was_revised": true, "final_justification": "Assuming the authors include the clarified results as a result of our discussion into the paper, my concerns are met and I am comfortable recommending an accept!", "rebuttal": "We thank the reviewer for engaging carefully with our work. The feedback has helped us refine explanations of our model’s assumptions. \n\n# Robustness\n\n> I would like to see a specialist-side penalty term (or a brief formal argument for why it is negligible). \n\nWe derived closed-form results and ran simulations with a specialist-side penalty ($\\phi_D$ includes a $p \\mathbb{1}[\\omega < \\theta]$ term). The qualitative takeaways remain the same, since the solutions for ${\\alpha_1}^\\*$  and $\\omega^\\*$ do not change, but the  specialist penalty increases $D$’s region of abstaining above the indifference curve because the feasibility condition becomes $\\omega \\leq \\frac{1}{\\delta - c_\\omega}(\\frac{p}{\\alpha_0} + \\delta - 1)$ instead of $\\omega \\leq \\frac{1-\\delta}{c_\\omega - \\delta}$. There’s also a slight effect on $\\delta^*$ and joint utilities above the indifference curve because $U_D$ goes down by $p$, but this does not change our Pareto improvement and utility transfer findings. We have added the results to Appx. B.\n\n> I ask for (a) at least one empirical learning-curve plot or citation supporting the concavity assumption, and (b) a short discussion of how a second axis such as inference cost would change the results. Demonstrated robustness to alternative tuning curves or an extended two-dimensional game would lift my novelty and insight scores; silence on the issue would pull them down.\n\n**(a)** We assume $U_D$ is concave in $\\alpha_1$ because higher performance should only benefit $D$ up to a critical point before further investments in fine-tuning cause utility to taper off (costs go up but performance and revenue do not) -- see: Springer et al. “Overtrained Language Models Are Harder to Fine-Tune” (ICML 2025). We don’t claim anywhere that $U_G$ is concave, but we believe most utility functions should be concave because the optimum will always be on a boundary otherwise ($\\omega^\\*=0, \\omega^\\*=1$ and ${\\alpha_1}^\\*=\\alpha_0, {\\alpha_1}^\\* = \\infty$). Concavity guarantees $D$ will adopt high-performing models as-is, choosing $\\alpha_1 = \\alpha_0$ instead of always fine-tuning (addressing your concern with L136).\n\n**(b)** We’ve moved Prop. 3 from Appx. D into the main body of the paper for an analytical way to inspect how something like inference ($c_\\omega$) affects openness choices. We’ve also  added discussion of inference costs and a 2D game (jointly optimizing $\\omega^\\*, {\\alpha_0}^\\*$) to the Appx.\n\n> Generalists also worry about efficiency. Reference the debate around DeepSeek, o3, o3-mini, and new nano-style models.\n\n\"[m]odel performance is the only feature in G's strategy space\" describes prior work (the paper which introduced the framework of fine-tuning games). Our paper presents a 2D game with *both* openness and performance in $G$’s strategy space, though the main results analyze when $G$ decides an openness level given $\\alpha_0$.\n\nWe capture efficiency through the $c_\\omega$ constant (L145). The operation cost to performance ratio should be low in the case of high-capability, small models like those you cite but closer to 1 (or even > 1) for inefficient models. Our rationale for not representing efficiency as a strategic dimension is that translating investments into efficiency improvements is dependent on external technological and resource constraints (e.g., hardware innovation, energy capacity) and *model-specific* relationships between architecture choices and inference costs (consistent with the Xia et al. reference you provided). As noted in Appx. C, efficiency is an important dimension for future work, but our paper is primarily concerned with how model openness, as a policy-relevant dimension, affects players’ choices.\n\n# Weaknesses\n\n> The window of relevant performance changes over time? Once upon a time, GPT-3 was very high performing, but today we would call it low performing. \n\nFigure 3(b) is meant to address this concern, as it shows that high vs. low performance is determined relative to the *current* generation of models.\n\n> who decides what is high or low performance? \n\nThis is relative (see above), but we’ve added clarification to Sec. 5.1: “Given that models may excel in certain domains while underperforming in others, regulators must determine whether to use composite performance metrics, domain-specific benchmarks, or capability-weighted averages to define model performance, with each approach reflecting different model development priorities.” \n\n> This analysis should minimally define performance, and to improve the modeling, introduce time variance.\n\nSec. 2.2 has been updated: “Performance represents a coarse measure of the model's accuracy on target tasks, generation speed and efficiency, and user satisfaction.” \n\nWe acknowledge that our work relies on several simplifying assumptions. This follows established practices in economic modeling, where parsimony yields more robust and clear insights. While we agree time variance is an important consideration, it substantially complicates the analysis (see response to Reviewer 1) without meaningfully strengthening our core theoretical insights about regulation. That said, we appreciate the modeling suggestions and have noted time variance as a future direction.\n\n> The paper treats \"foundation models” as uniquely troublesome for open-source definitions, yet earlier software and ML precedents raise identical dilemmas. \n\nWe understand the concern and didn't mean to suggest they’re fully unique (see L607 in Appx.). Our position is that foundation models raise the stakes of open-source definitions because (1) they implicate a broader set of capabilities and downstream uses, (2) achieving reproducibility requires more than open code and data, and (3) high inference costs amplify the advantages of open-sourcing. We have incorporated Gupta et al. as a caveat to our terminology and rewritten Sec. 1 to make this position clearer. \n\n> It's worth citing \"The Value of Open Source Software\" by Hoffman et. al. [0]\n\nHoffman et al. was cited in our related works (L50).\n\n# Cost Modeling\n\nSec. 2.3 has been rewritten to distinguish when we’re referring to the structure of a utility function (broader classes of production, regulatory, and operation costs) vs. a specific functional form (quadratic cost and monotonic revenue). \n\n> There are already costs associated with documentation internally, as well as the cost of a proper internal model release process. Perhaps this is better modeled as log scaling rather than linear?\n\nThis is a good point. We ran numerical simulations with log scaling, and the results are qualitatively the same. $G$’s best response is similar to the one derived in Appx. B, except it requires solving for roots of $A \\omega^3 + \\omega^2, C (\\text{without} -\\alpha_0) \\omega - \\alpha_0$ because $\\frac{\\partial}{\\partial \\omega}  \\alpha_0 \\log (\\omega)= \\alpha_0 \\omega^{-1}$. The quadratic/cubic terms from revenue dominate the production cost in either case.\n\n> Currently this implies that operation costs hinge on the final performance reached after fine-tuning, not the baseline. Why does the generalist bear this cost?\n\nFor a closed model that offers fine-tuning via API (e.g., GPT‑4o), the generalist handles inference costs and prices this into per-token API rates. This also ensures operation costs sum to a constant between players ($c_\\omega \\alpha_1$). \n\n> L128: \"safety risk management for open models\". This is a regulatory cost, but you are trying to provide an example for production costs.\n\nThis example has been replaced. \n\n> Can you justify quadratic cost? Empirical fine-tuning curves often show super-linear returns for early steps. \n\nWe want to make sure we’re interpreting your point correctly. Could you clarify why super-linear returns for early steps contradicts quadratic cost scaling? Our understanding is that “super-linear returns [for model performance] for early steps” is captured by a curve like $\\alpha_0 = \\text{cost}^{1/k}$, where initial investments get much more mileage for performance. Our quadratic cost claim describes the relationship $\\text{cost} = \\alpha_0^{k}$, where marginal cost for each unit of improvement gets more expensive. These are the same equations. Is it possible that we’re talking about the same relationship, just with flipped axes?\n\n> L157: \"higher openness increases production costs\". There is ample evidence showing this to be false. Projects like YOLO, vLLM, etc. have benefited drastically from open-source contributions. Outside of ML, look at Docker, Redis, etc. Even within foundation models, improvements to Gemma have led to improvements in Gemini.\n\nThe line has been changed to clarify that higher openness increases production costs only for the *particular product* being offered at higher openness. We capture the benefits of open-source contributions through revenue rather than reduced production costs -- i.e., when an open model is improved by specialists to a higher level, $G$ recoups the increased production costs through higher revenue ($r(\\alpha_1)$) because cheaper model access encourages specialists to invest more in $\\alpha_1$.\n# Checklist \nWe can upload the code as Supplementary Material. CPU runtime depends on the granularity of simulations. A simulation to produce a set of plots like those in Figures 4 and 5 with a sweep over 100 penalties, 100 thresholds, and 100 $\\delta$ values (100,000 parameter combinations) executes in 317.362s / ~5 min on a single CPU. Similarly:\n\n* 150 $p$, 150 $\\theta$, 100 $\\delta \\rightarrow$ 717.484s / ~12 min\n* 200 $p$, 200 $\\theta$, 100 $\\delta \\rightarrow$ 1470.59s / ~24 min\n\nWe were able to produce smooth plots at 200 x 200 x 100 granularity. For each plot in Appx. F, we used a sweep over 200 $\\epsilon$, 200 $\\alpha_0$, and 100 $\\delta$, which executes in 1216.88s / ~20 min on a single CPU. We did not optimize for efficiency. Item 8 in the checklist has been updated with approximate lengths of runs and # of CPUs needed."}, {"review_id": "YGbBr9oQgv", "reviewer": "Reviewer_fUP1", "summary": "In an effort to understand and predict the impacts of AI governance policy in relation to open-source general-purpose AI foundation models, the authors introduce a game-theoretic model for assessing and predicting the effects of AI governance regulation thresholds and penalties on decisions made by the producers and refiners of open-source general-purpose AI models. The approach models the effects of strategic decisions made by AI developers to adjust levels of openness along a spectrum rather than under an open or closed binary construct. To reveal possible open-source value chain dynamics, the model allows for various levels of openness to be set by the producer of the general-purpose model, and weighs impacts of other criteria and forces such as revenue-related bargaining decisions between generalists and specialists, and regulatory open-source thresholds.", "questions": "•\tDid the authors consider factoring-in the distinctions among how G, D, and regulators define model improvements or performance? As noted in the Weaknesses section, the regulatory goals addressed in the modeling should be better articulated.\n•\tDid the authors consider what group or groups are their target audience? It might be helpful for the reader to understand this to get a better sense of the ultimate goals for application of the model.", "limitations": "The authors acknowledge some limitations and caveats to their proposed framework; for example, their stated limitations acknowledge some important areas for future research and model improvement, in particular inclusion of “social utility” related criteria in the modeling.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths\n•\tThe game is valuable as a way to understand how some regulatory approaches might affect AI developers, whether regulatory approaches could work, and areas where they might fail.\n•\tThe authors advance prior work in a way that somewhat reflects today’s regulatory and AI business landscapes, including by modeling for some of the nuances and complexities present in real-world scenarios.\n•\tThe authors are clear in their explanation of the game’s role players, forces and factors affecting predicted outcomes, and how the game works.\n\nWeaknesses\n•\tAn overarching and relatively obvious weakness of this work is the game’s existence as a theoretical exercise that -- despite its addition of nuances prior work does not consider (such as incorporating impacts of openness along a spectrum rather than an open/closed binary) – does not consider several other real world factors that could limit the model’s predictive capabilities. \n•\tFor instance, as just one example, the model does not consider the impacts of venture capital funding which could allow generalists or specialists to tolerate lower revenue goals. It also does not consider political forces, or other factors that might disrupt the model.\n•\tThe authors do not adequately define the regulatory goals their model addresses or considers. It is somewhat implied that the goals the model is designed to address are “innovation,” model performance as measured by industry-designed benchmarks, and possibly competition related goals, rather than other regulatory goals that are also facilitated through open-sourcing such as model transparency and explainability related goals. But it is important to better define these regulatory goals.\n•\tThe paper does not adequately define or explain reputational benefits, how they are measured, or how the size or market power of a company might affect reputational benefits/premiums.\n•\tA minor point, but the authors could more directly explain and cite the Nash bargaining solution.", "was_revised": true, "final_justification": "Following positive engagement with the authors which improved the paper, I've increased the rating from borderline accept to accept.", "rebuttal": "Thanks for the thoughtful engagement with our work. We appreciate your positive feedback and address your questions and suggestions below.\n\n# Regulatory Goals\n\n>  It is somewhat implied that the goals the model is designed to address are “innovation,” model performance as measured by industry-designed benchmarks, and possibly competition related goals, rather than other regulatory goals that are also facilitated through open-sourcing such as model transparency and explainability related goals. \n\n*Defining Regulatory Goals*.  \n\nWe agree that the regulatory goals should be stated more explicitly. To handle this, we’ve moved the formalism of a Pareto-optimal policy (Def. 1) from the Appendix into 4.2. The regulator can choose any set of utility functions (i.e., $\\alpha_1, \\omega, U_D, U_G$) and any weighting of those utility functions to define regulatory objectives. At the same time, we avoid committing to a particular balance of objectives, since we want to illustrate the full space of possible regulatory approaches. \n\nWhile the regulator could be modeled as a third player with an explicit utility function, we intentionally refrain from defining specific regulatory preferences. This enables flexible parameterization of the model according to different objectives, rather than constraining analysis to a particular social utility function that may not suit diverse regulatory contexts. We also want to note that providing a more granular definition of regulatory utility beyond what Def. 1 offers would invite the well-documented complexity of ranking and weighting various objectives from social choice and welfare aggregation literature.\n\n*Other Open-Sourcing Goals.* \n\nOur analysis assumes $\\omega$ is a 1D variable that captures a property like model access. One could imagine using an equivalent model for reasoning about a spectrum of transparency or explainability measures. Sec. 5.2 now mentions that model transparency and explainability can be captured *in addition to* model access in future work: “While our analysis assumes that higher model openness and performance are generally beneficial, a regulator must consider safety risks associated with higher performance or market-wide loss scenarios, such as possible intermediate levels of openness where consumers may be worse off. This complexity suggests that defining openness as a multidimensional property may be valuable to capture factors beyond simple model access, such as documentation and explainability of model outputs.” \n\n# Modeling Assumptions\n\n\n> The paper does not adequately define or explain reputational benefits, how they are measured, or how the size or market power of a company might affect reputational benefits/premiums.\n\nWe roughly capture the issue of market power in $G$’s utility function by scaling the reputational premium by model performance. The $\\epsilon \\alpha_1$ term in $U_G$ means firms with superior models receive greater reputational benefits from openness.\n\nTo better explain reputational benefits, Sec. 2.2 now references extensive prior work on defining reputational benefits in the context of OSS contributions and commons-based production:\n\n* “Coase’s Penguin, or, Linux and The Nature of the Firm” (2002), which discusses reputation as an “indirect appropriation mechanism.” \n* “Why develop open-source software? The role of non-pecuniary benefits, monetary rewards, and open-source licence type” (2007), which profiles OSS developer motivations. \n* “Social Status in an Open-Source Community” (2005), which has a case study on OSS reputation incentives.\n\nIn our case, $\\epsilon$ captures reputational advantages within developer and research communities, which may aid in recruitment and third-party integration with the product. We don’t provide an exact approach for measuring $\\epsilon$, though it should in general be low (at least $< 0.5$), given that reputational benefits generally do not exceed the amount of closed revenue that can be gained from monetizing a model. \n\n> as just one example, the model does not consider the impacts of venture capital funding which could allow generalists or specialists to tolerate lower revenue goals. \n\nThis is an interesting point. We have updated L102 to explicitly state the assumption that players abstain when their utility is negative. Venture capital might lower revenue goals because it provides capital that firms can sit on, but on the other hand, it tends to push companies towards high risk and fast growth. Modeling scenarios where players tolerate lower revenues in expectation of future profits requires introducing time variance and additional parameters specifying players’ tolerance for short-term losses. This would substantially increase model complexity and require additional assumptions about discount rates and availability of capital. We have therefore added modeling time variance and multi-firm competition as an important future direction in Sec. 5.2. \n\n> It also does not consider political forces, or other factors that might disrupt the model. \n\nSocial utility considerations of the regulator give some capacity for modeling political factors -- e.g., political pressures associated with setting the penalties and thresholds too high. We allude to this in L255 when discussing “wasted enforcement costs” associated with ineffective, high thresholds. However, this could be formalized alongside Def. 1 by applying a regulator penalty for different $(\\theta, p)$ combinations:  ${\\arg\\max}_{p, \\theta}\\  w^\\top \\mathbf{u} - \\lambda \\theta p \\  \\quad s.t.\\ \\|w\\|_1=1, w_i>0, \\forall i \\in [d]$.\n\nOur paper focuses on levers that the regulator can directly control (see Appx. C for our justifications for omitting considerations like deployment scale). We abstract away from exogenous factors to isolate causal relationships between policy instruments and equilibrium outcomes. However, we acknowledge that several factors may disrupt the model’s outcomes in practice, and future work can systematically characterize these factors or add stochasticity to examine how the model performs under uncertainty or unexpected market changes. \n\n> An overarching and relatively obvious weakness of this work is the game’s existence as a theoretical exercise that -- despite its addition of nuances prior work does not consider (such as incorporating impacts of openness along a spectrum rather than an open/closed binary) – does not consider several other real world factors\n\nWe acknowledge that our work relies on several simplifying assumptions, though these abstractions serve two important purposes. First, this approach follows established practices in economic modeling, where parsimony yields more robust and interpretable insights by isolating the most direct relationships between regulation and developer choices. Second, the abstract definitions of $\\omega, \\alpha_0$ are designed to provide flexibility over how such measures are constructed, allowing the model to remain general while accounting for relative magnitude. This abstraction is meant to accommodate different measurement capabilities, rather than tying the model to particular technical definitions – i.e., $\\alpha_0$ may cover efficiency or latency considerations beyond benchmark accuracy alone.\n\n# Miscellaneous\n\n> Did the authors consider factoring-in the distinctions among how G, D, and regulators define model improvements or performance? \n\nWhen constructing the model, we did consider how regulators may consider emissions targets or safety as part of performance, while $G$ and $D$ may focus more narrowly on commercial viability or benchmarks. Our model is designed to evaluate regulator-defined objectives, so we maintain $\\alpha_0$ as a metric focused on task accuracy and efficiency and rely on the regulator’s utility function / discretion to incorporate broader environmental, safety, or equity concerns that diverge from firms' revenue-maximizing definition of performance. However, we recognize this is an important modeling consideration and have added a discussion in Sec. 5.1 about how regulators might choose different definitions of *average* performance: “Given that models may excel in certain domains while underperforming in others, regulators must determine whether to use composite performance metrics, domain-specific benchmarks, or capability-weighted averages to define model performance, with each approach reflecting different model development priorities.”\n\n> A minor point, but the authors could more directly explain and cite the Nash bargaining solution \n\nWe have added a citation on L188 and included a discussion of properties and limitations of the bargaining solutions: “Our analysis mainly examines Nash bargaining, which uniquely satisfies independence of utility origins and units, Pareto efficiency, symmetry, and independence of irrelevant alternatives. The two alternative bargaining solutions we consider (VM and Egalitarian) are also Paretian, symmetric, and independent of utility origins. However, generalists and specialists may arrive at non-standard bargaining agreements in practice which violate these properties, especially when utilities are expressed in currency and players have bounded rationality.”"}]

Author rebuttal:

Meta-review: This paper touches upon an important subject (AI governance and the expected impacts of regulation). 

After a fruitful discussion with authors, all reviewers agreed that this paper was worth accepting. Therefore I recommend acceptance.
Decision: Accept (poster)