The AI Red Queen Race

Why Frontier Labs Can Grow Explosively and Still Face Bankruptcy

\[ \newcommand{\E}{\operatorname{E}} \newcommand{\Prob}{\operatorname{P}} \newcommand{\Exp}{\operatorname{Exp}} \newcommand{\Poisson}{\operatorname{Poisson}} \]

Introduction

Frontier AI labs are growing revenue faster than any companies in history. They are also burning capital faster than any software companies in history. Both trends are real, and they are racing each other.

In a recent interview on the Dwarkesh podcast (§), Dario Amodei, CEO of Anthropic, boldly predicted we would have “a country of geniuses in a data center”. Yet in the very same interview he said

I could buy $1 trillion of compute that starts at the end of 2027. If my revenue is not $1 trillion dollars, if it’s even $800 billion, there’s no force on earth, there’s no hedge on earth that could stop me from going bankrupt. . . what if the country of geniuses comes, but it comes in mid-2028 instead of mid-2027?

You go bankrupt.

We will take the risk of bankruptcy as the central driving force for AI economics. Using this as an organizing principle, we will develop a simple model which parameterizes frontier labs’ tradeoffs and incentives. Rather than just hand waving “AI bubble” or “existential competition” or “fast takeoff”, we can make sense of exactly what commitments AI companies can rationally make. The goal is to boil the complexities of the AI marketplace down to a few essentials, to provide a conceptual map for what assumptions and constraints shape the AI landscape.

AI Economics is a Ruin Theory Problem

Amodei’s description of frontier labs resembles a gambler racing a clock. Firms make large precommitments for compute obligations in the face of uncertain but exponentially scaling revenues. Put another way, AI firms aren’t simply managing expected profits under ordinary growth uncertainty. They are trying to survive a race where infrastructure commitments are made long before the prize arrives. Each generation of technology is increasingly more capable and valuable, but only if you can get there. The key question isn’t “Is AI a world-changing technology?” but rather “Will enough revenue or capital arrive before spending commitments burn through runway?”. This turns AI economics into a first-passage problem, where firms are managing the risk of ruin.

Some insights we will develop from this perspective are:

• In a winner-take-all race, raising capital can be self-defeating. For a single firm with no rival, more runway extends the research horizon and decreases the odds of going bust. In competition, additional capital leads to increased spending to such a degree that bankruptcy horizons shorten and the odds of a breakthrough are unchanged. It’s a race to a standstill.

• The AI bubble debate is really about a single quantity \(q\): whether value from the next generation scales slower than, equal to, or faster than costs. If \(q>1\), a finite runway can fund unlimited generations of breakthroughs. Otherwise the cascade eventually runs out.

• Technology diffusion flips competition on its head. When the wedge between winning and losing is large, firms overinvest in a race to be first. When it’s small, firms underinvest and free-ride.

• Even with big upside, the best move may be patient waiting. Going slow is a bet you will survive to the long run.

• Circular investments may be more than creative financial engineering. They may be a way to internalize a competitor’s upside.

AI is Not a Normal Industry

AI capabilities are increasing exponentially with each generation. METR says the AI agent task length has a doubling rate of around seven months over the past six years (§). There does not appear to be any plateauing as capabilities increase. Epoch’s Capabilities Index, which stitches together many benchmarks, shows continuous improvement across model generations, and seems to have grown faster since 2024 (§). Some claim recursive self-improvement will lead to super-exponential or discontinuous takeoff, but as of now that remains speculative (§).

Revenue growth of leading AI labs is unprecedented even by the standard of technology companies. OpenAI went from $0 in 2022 to $20B in annualizing revenue in 2025, with Anthropic growing to $9B in the same period. (§). This quarter alone, Anthropic is set to surpass its annualized revenue from last year (§). OpenAI’s revenues are projected to be $100B in three years, a growth scale without precedent (§).

At the same time, spending commitments are growing astronomically. AI requires capital expenditures unlike anything else in the software industry. Anthropic committed to $200B in spending (§). OpenAI is targeting compute spending around $600B by 2030 (§). Data center construction significantly contributed to GDP growth in 2025, in excess of the peak of the dot com boom (§).

Frontier labs are raising capital with a mix of venture capital, project finance, and public offerings that resembles infrastructure financing more than software. SpaceX’s IPO, after absorbing xAI, was the largest ever by proceeds. (§). After several rounds of private fundraising, (§)(§), Anthropic and OpenAI have confidentially filed for record-setting IPOs (§) (§). Even Google, which is enormously profitable, announced a $80B equity capital raise because its anticipated capital expenditures for AI infrastructure are so large (§). Morgan Stanley is projecting more than $500B in AI-related debt issuance, doubling the amount from last year (§).

Frontier labs are struggling to balance compute spending with customer demand and financing constraints. A recent New York Times opinion column speculated about whether frontier labs will run out of cash before revenues match spending (§). OpenAI CFO Sarah Friar said OpenAI missed targets for users and revenues, and worried about the ability to pay for computing contracts if revenue doesn’t grow fast enough (§). On the other hand, OpenAI also reports it is unable to meet current demand for lack of compute resources (§). After a period of having to limit usage, Anthropic recently entered into a large deal with xAI to lease additional compute (§).

Core Facts

These stylized facts about AI economics motivate what follows:

1. Firms have large but not infinite financial resources, especially when compared to spending commitments
2. Revenues stem from AI capability breakthroughs which are uncertain in nature but tied to research investment
3. Spending commitments are set in advance of realized revenues
4. Revenues and expenses scale exponentially with AI breakthroughs
5. Firms are racing to capture the benefits of being first to a breakthrough. However, losers still benefit from technology diffusion.

A Toy Model

In the next section, I present a model to use as a framework for understanding AI economics. This is a toy model in the physicist’s sense. It’s deliberately stripped down and unrealistic to isolate the mechanisms of interest. While bankruptcy pressure is an essential aspect of the model, it’s not designed to make precise statements about a company’s creditworthiness. An analyst at Moody’s analyzing likelihood of debt service on Meta’s data center financing would take a very different approach.

The model will be painted with very broad strokes. The idea is to have enough elements to capture the interesting economics. However, many of the elements will be a plainly unrealistic caricature of actual economic realities faced by an actual firm. Despite taking a low-resolution approximation to much more complex economic realities, we may hope to gain insight into the fundamental economic tensions. Those insights will come in the form of characterizing regimes where different behaviors emerge. A formal model allows for comparative statics, showing the effect of different assumptions about key parameters. This type of model also lets us contrast different extreme limits.

Runway

The runway \(R\) represents the AI firm’s resources, or its ability to continue its operations. This is a stand in for cash-on-hand, private investments, debt financing, vendor or partner investments or compute commitments, and proceeds from selling public equity. Broadly speaking, \(R\) represents all potential and actual funding possible at a given stage. When a firm runs out of runway, it goes bankrupt and ceases to exist.

In reality, an AI firm is unlikely to suddenly expire from a missed cash payment. The more likely scenario is that confidence in a firm erodes, the valuation assigned by investors declines, and the firm is unable to continue to attract talent or additional capital even if, strictly speaking, it remains a going concern. But, for our purposes, we’ll say that when a firm’s runway \(R\) reaches 0, the firm no longer exists.

Spending Commitments

Dario Amodei’s quote does not draw out exactly what the compute spending is for. One category of compute spending is for inference, the cost of deploying in response to customer demand. The marginal cost of answering a chatbot query is much higher than the marginal cost of serving a webpage. Another category of compute spending is for training, the cost of creating new more capable models, including the research costs of experimenting with new approaches.

With respect to inference costs, the primary uncertainty is uncertain future demand. AI companies enter into long-term agreements with data centers, but customers may exceed capacity or not materialize. The capacity is perishable in the sense that GPU hours must be paid for whether they are used or not. This is the AI analog of the newsvendor problem (§). While there have been reports of a lack of capacity, recent deals between labs suggest there is some elasticity in the supply of near-term compute.

With respect to training costs, the uncertainty stems from when or whether there is a breakthrough in AI capability. The central economic problem is that it is unknown when the “a country of geniuses in a data center” will arrive, though higher amounts of research investment should hasten that time. Probably Amodei is thinking of both factors. Epoch AI and Exponential View’s analysis suggests OpenAI may not be profitable accounting only for inference costs (§), which suggests a focus on future capabilities rather than monetization.

We will lump all spending into two variables. First \(d\) is the fixed drag, expressed as a rate of runway spending per time. This represents the unavoidable baseline expenses in the absence of research and training. This could include non-compute overhead like payroll, administration, marketing, or ongoing net negative revenue from inference. Ultimately one should expect AI firms, as a going concern, have positive net cashflow apart from research. However, in this model, in order for a firm to be motivated to meaningfully invest there needs to be some kind of time pressure, a cost to waiting.

The spending variable is \(k\) the discretionary capital investment in research, expressed as a rate of runway spending per time. This represents training runs, data acquisition, salaries for researchers, product launches and capacity build out. This is the primary knob AI firms will tune to maximize their economic value. At each stage, companies will set their capital investment level \(k\), and the combination \(b=d+k\) represents the net burn of a company.

Research Breakthroughs

With a burn rate of \(b=d+k\), the bankruptcy horizon, the time when there is no runway left, is given by

\[ h = \frac{R}{b} = \frac{R}{d+k} \]

If there is no research breakthrough, this is the ruin time when a firm expires.

Increasing research spending has two effects¹:

• It increases the rate of cash burn, bringing the bankruptcy horizon \(h\) closer.
• It increases the rate at which a breakthrough occurs.

Specifically, take the time of a breakthrough \(T\) to be an exponentially distributed random variable with intensity \(\lambda(k)\). Standard economic considerations suggest that research production should be increasing in investment, but that increased investment has diminishing marginal returns.

Let’s assume there’s a constant elasticity \(\gamma\) for the breakthrough intensity from research investment, so \(\lambda\) takes the form \[ \lambda( k ) = a k^\gamma \]

for a constant \(a\) and \(0<\gamma < 1\).

The Chinchilla curves (§) are one suggestive source of evidence for this type of production function as they describe a power-law relationship between compute budget and model loss for large language models. Primarily, however, the main motivation of this production function is mathematical convenience.

Taking standard economic assumptions, suppose the firm is risk-neutral and its value is given by the expectation. We’ll take interest rates to be zero as the runway drag \(d\) already puts a cost on waiting. Let \(f(R)\) represent the payoff upon reaching a breakthrough with remaining runway R. If the breakthrough happens at time \(t\), then a firm will have remaining runway \(R-bt\) after the breakthrough. Below are several equivalent expressions for the value of our firm for a given research investment level \(k\).

\[ \begin{aligned} \Phi( R, k ) &= \E[ \, f(R-bT) \mathbf 1_{T<h} ] \\ &= \int_0^h \lambda e^{-\lambda t} f( R-bt)\, dt\\ &= \int_0^{\frac R {d+k}} ak^\gamma e^{-ak^\gamma t} f( R-(d+k)t)\, dt \end{aligned} \tag{1}\]

The first, written compactly in probability notation, is in terms of the random breakthrough time \(T\) distributed exponentially \(T \sim \Exp(\lambda)\). In this expression \(\mathbf 1_{T<h}\) is a random indicator function which is 1 if the breakthrough happens before the bankruptcy horizon and 0 otherwise. The second form explicitly writes out the expectation as an integral, and the final form makes all of the \(R\) and \(k\)-dependence explicit.

The central tension between accelerating bankruptcy and accelerating breakthrough is encapsulated in Equation 1.

The Exponential Treadmill

Now suppose that AI technology progresses in discrete stages \(n=0,1,\dots,N\). Each stage might represent something like a major model release: from GPT3 to GPT4. Let a large \(N\) stand in for a stage of AI technology vastly more capable than present technology. Maybe this represents the stage where post singularity humanity has colonized the light cone, or maybe, more modestly, this represents a point where AI economics transitions to something like a normal industry without exponential scaling. Sometimes it will be useful to consider the limit \(N\to \infty\). The point is that the number of stages is large enough that it creates a treadmill of exponentially more capable technology stages for the foreseeable future.

Costs scale at each technology stage with factor \(\mu\). That is, at stage \(n\) the fixed drag increases to \(\mu^nd\) and the impact on the next breakthrough of investing capital \(k\) is reduced \[ \begin{aligned} d_{\text{stage}\, n} &= \mu^n d \\ \lambda_{\text{stage}\, n}(k) &= a \left( \frac k {\mu^n}\right)^\gamma \\ \end{aligned} \]

As each stage is a scaled-up version of the last, it’s useful to work in terms of the normalized units where we scale all quantities down by \(\mu\). Normalized quantities will be represented with a tilde.

\[ \tilde R = \frac R {\mu^n} \qquad \tilde d_{\text{stage} \, n} = \frac{ d_{\text{stage}\, n} }{\mu^n} = d \qquad \tilde k = \frac k {\mu^n} \qquad \lambda_{\text{stage}\, n}(\tilde k) = a \tilde k^\gamma \]

The bankruptcy horizon is the same in both absolute and normalized units

\[ h = \frac {\tilde R} {\tilde k+ \tilde d} = \frac R {k+d} \]

The ultimate reward upon breakthrough to the final stage \(N\) is \(f(R)\). The most important properties are that \(f\) is nonnegative and increasing in \(R\), or at least non-decreasing since constant fixed rewards are also interesting to consider. For earlier stages, there is an intermediate reward. The remaining runway gets scaled by a \(\nu\), so that the relationship of runway just prior to breakthrough and just after is given by

\[ R^+ = \nu R^- \]

Proportional rewards act as a kind of “Matthew effect”(§). Since rewards scale with remaining runway, the rich get richer. Heuristically, this type of reward resembles firms getting a higher valuation multiple upon attaining a technology breakthrough. But apart from realism, the chief advantage of a proportional reward is that it simplifies the math yielding a simple recursive structure to the Bellman optimization. One can consider extensions of this model with other types of fixed payouts or rewards, and these can lead to different kinds of incentives. For example, in a competitive model, a fixed prize can incentivize gambling behavior in firms which are lagging in runway, as they try to capture a prize with small amounts of remaining runway.

Given the cost scaling and reward scaling, there is a critical parameter \[ q = \frac \nu \mu \]

which controls how the normalized runway scales \[ \tilde R^+ = \frac {R^+}{\mu^{n+1}} = \frac \nu \mu \frac{R^-}{\mu^n} = q \tilde R^- \]

In essence, \(q\) controls whether, in the new stage, a firm comes out ahead or behind accounting both for the breakthrough and the increased cost structure. The critical case \(q=1\) represents a Red Queen treadmill where firms which advance stay exactly in the same place. Whether \(q<1\), \(q=1\) or \(q>1\) has a decisive effect on the resulting economics.

Firm Value Across Stages

The value of the AI frontier lab at each stage is given by \(\Phi\) in Equation 1. By scaling and recursively stacking the value at each stage, we arrive at the value of the AI frontier lab on a research treadmill. Let \(\tilde \Phi_n(\tilde R) = \Phi(R) / \mu^n\) represent the normalized value function. Since normalized units already account for the cost scaling, the relationship between each stage takes this form in normalized units.

\[ \begin{aligned} \Phi_{\text{stage}\, n}(\tilde R) &= \max_{\tilde k\geq 0} \E[ \,\mu\, \Phi_{\text{stage}\, n+1}(q( \tilde R-\tilde bT)) \,\mathbf 1_{T<h} ] \\ &= \max_{\tilde k\geq 0} \int_0^h \lambda e^{-\lambda t}\mu \Phi_{\text{stage}\, n+1}( q (\tilde R - \tilde b t )) \, dt \end{aligned} \tag{2}\]

The factor of \(\mu\) inside the expectation reflects the effect of cost-normalizing \(\Phi\) itself.

This is the essence of the one-firm model.

The firm must optimize its research level at each stage. The research level affects the probability per time that a research breakthrough happens, but also increases the cash burn and shortens the bankruptcy horizon. Across stages, all costs scale exponentially. But the firm’s runway also scales up exponentially on each breakthrough.

Quick Model Summary

Here is a quick reference for the key elements of the model

• \(R\) represents the runway, the measure of a firm’s financial resources
• \(k\) is the rate of capital investment in research for a breakthrough
• \(d\) represents the fixed rate of drag on runway
• \(b=k+d\) is the total rate of burn of runway
• \(h = R / b\) is the bankruptcy horizon, the time when the firm expires if there is no breakthrough before then
• \(n= 0,1,\dots,N\) represents the technology stage
• \(\lambda = ak^\gamma\) is the research production function, how investment in research translates to the probability of a breakthrough per unit time.
• \(q=\nu/\mu\) is the scaling factor for normalized runway, where \(\mu\) is the cost scaling factor for each stage, and \(\nu\) is the runway scaling factor upon a breakthrough
• \(z = \lambda h = \lambda R / b\) is the research mass, a dimensionless quantity which is related to the overall probability of a breakthrough or bankruptcy at each stage.
• \(\eta\) is the fraction of a breakthrough shared by the losers of a competitive race. It represents, among other things, technology diffusion. The wedge between winners and losers is \(1-\eta\).

The Investment Problem

How much should a frontier lab with a finite runway \(R\) and cash drag \(d\) invest in research? We’ll start with the simplest cases and work to more complicated cases.

One Firm, One Breakthrough

Start by considering one firm, a frontier lab monopoly. The firm doesn’t face competition, and we will consider the optimal strategy to attain one breakthrough. Recall Equation 1, the value for a firm with reward \(f\) on a breakthrough

\[ \Phi( R, k ) = \E[ \, f(R-bT) \mathbf 1_{T<h} ] = \int_0^h \lambda e^{-\lambda t} f( R-bt)\, dt \]

Now rescale time so that it’s expressed as the fraction of the bankruptcy horizon \(h\).
\[ X = \frac T h = \frac {bT} R \]

Since \(T \sim \Exp( \lambda)\) is an exponential random variable, so is \(X \sim \Exp\left(\frac{\lambda R} b\right)\). The breakthrough intensity in terms of rescaled time \(z = \frac{\lambda R}b\) is a key quantity we’ll call the research mass.

In terms of \(X\), Equation 1 becomes

\[ \Phi( R, z ) = \E[ \, f(R(1-X)) \mathbf 1_{X<1} ] = \int_0^1 z e^{-zx } f( R(1-x))\, dx \tag{3}\]

In this form, there is a fixed horizon: 100% of runway. Also the dependence on \(k\) is entirely mediated by the dependence on \(z\). A natural condition on \(f\) is that it is increasing in \(R\) and firms which break through with more runway get at least as much reward. This is the case for “Matthew effect” proportional rewards, any fixed reward, or any combination of rewards which are increasing in \(R\). Inspecting the form of Equation 3, the value is increased by moving more probability mass to earlier times when there is more remaining runway. In short, maximizing \(\Phi\) is equivalent to maximizing the research mass \(z\).

The first-order condition on the research mass \(z = R \lambda/b\) is
\[ \frac{\lambda'(k)}{\lambda(k)} = \frac{b'(k)}{b(k)} \tag{4}\]

which, for our specification of \(\lambda\), gives \[ k^* = \frac \gamma {1-\gamma} d \tag{5}\]

Maximizing the firm value is equivalent to maximizing the research mass \(z\). Some other economically interesting quantities are related to the research mass. The probability of bankruptcy is given by \(e^{-\lambda h} = e^{-z}\), so the optimal research level minimizes the chances of bankruptcy. A breakthrough happens if and only if a bankruptcy does not happen, so this also maximizes the probability of a breakthrough. This condition also minimizes the expected amount of runway consumed, where in a bankruptcy all of \(R\) is consumed.

For a single firm seeking a single breakthrough, optimal investment:

• Minimizes the probability of bankruptcy
• Maximizes the probability of breakthrough
• Minimizes the expected cash consumed.

Returning to Equation 5, note the optimal investment is a multiple of the cash drain. The more linear the production factor (the closer \(\gamma\) is to 1) the higher the multiple. For a quadratic production function \(\gamma = ½\) the optimal investment equals the cash drain.

From this it’s clear that the cash drain is essential for avoiding degenerate solutions. In the limit \(d\to 0\), the optimal investment \(k^*\) also shrinks to 0. There is no positive investment level which is optimal; investing less is always better. Of course \(k=0\) results in no research and no breakthrough. There needs to be economic time pressure² to incentivize an AI lab to invest anything more than the bare minimum, and engage in patient waiting. This is not a quirk of our choice of intensity function, it’s a property of any \(\lambda\) which is increasing and concave downward. Competition also introduces a time pressure from the possibility of a rival’s breakthrough. However, even in competition, patient waiting and infinitesimal investment is sometimes best without a cash drain.

It’s notable that the optimal investment level in Equation 5 is independent of runway \(R\), so research mass is proportional to runway. If a firm has twice the capital resources, it will research for twice as long.

For a single firm, there are no diminishing returns to runway. A firm with twice the runway will commit to the same rate of research, and will have twice the horizon and twice the research mass.

Research on a Treadmill

When there are multiple stages of AI development, things start to get interesting. In normalized terms, for a firm on a treadmill optimal research investment is the same at every stage, and equal to the single-stage optimal level.

We’ll use the same reasoning as in the last section, working backwards stage-by-stage. First note that the value of \(\Phi\) in Equation 1 is increasing in \(R\) for any \(f\) that is increasing in \(R\). Before the final breakthrough at stage \(N-1\), the problem is identical to a one-stage problem, and has solution \(\tilde k^*\) Equation 5, translated to normalized units. The value of the firm at this optimal investment level \(\tilde \Phi_{N-1}\) is also increasing in \(R\). Now examining the form of the recursive treadmill, the investment problem at stage \(n\) given by Equation 2 is of exactly the same form as the one-stage problem if we substitute for \(f\) the appropriate expression of \(\tilde \Phi_{n+1}\). Thus the solution \(\tilde k^*\). Furthermore, the value of the firm at that stage \(\tilde \Phi_n\) is increasing in \(R\). Inductively working backward, the optimal investment \(\tilde k^*\) is the same across all stages and given by Equation 5.

Writing this in terms of nominal units, the optimal investment level at stage \(n\) is given by

\[ k_n^* = \frac {\mu^n \gamma} {1-\gamma} d \]

The research mass \(z_n = \lambda_n h_n\) is the same across all stages and attains its maximal value at \(k_n^*\).

The optimal investment levels for the \(N\)-stage model are, in cost-normalized terms, the same across all stages. Each equals the optimal investment level for one stage. This is also the level which

• Maximizes the overall probability of lasting to stage \(N\)
• Minimizes the overall probability of bankruptcy at any stage
• Maximizes the expected runway remaining at stage \(N\)

Unwinding Equation 2 across stages gives an expression for the firm’s value at the outset, in terms of initial runway \(R\). Let \(\mathbf 1_N\) be indicator function that the AI lab survives to the final stage \(N\). Let \(T_i \sim \Exp(\lambda)\) be the exponentially distributed random variables which represent the breakthrough times to stage \(i\). Recall that \(q=\nu/\mu\) is the scaling on runway. \[ V_0(R) = \mu^N \E\left[ \,f\left( q^N R - b( q^N T_1 + q^{N-1} T_2 + \dots+ q T_N)\right)\,\mathbf 1_N \right] \]

In the critical case \(q=1\) where rewards and costs scale equally, this simplifies to \[ V_0(R) = \mu^N \E\left[ \,f\left( R - b \sum T_i\right) \,\mathbf 1_N \right] \]

The condition for surviving to stage \(N\) is

\[ T_1 + \frac {T_2}{q} + \frac {T_3}{q^2} + \dots + \frac{T_N}{q^{N-1}} < \frac R b \tag{6}\]

where the \(T_i\) are i.i.d. exponential random variables with intensity \(\lambda(v)\). In the critical case \(q=1\), the probability distribution is given by the Erlang / Gamma distribution \(\text{Gamma}( N, \lambda)\).

There are three regimes with qualitatively different behavior ³. The supercritical case \(q>1\) is the supercritical transformative AI regime, where the scaling of value from each breakthrough exceeds the scaling in costs. Research breakthroughs function as a flywheel funding future research. Here with positive probability \(p_\infty\), the firm reaches arbitrarily many stages and never goes bankrupt. In the subcritical case \(q<1\) and critical case \(q=1\), as \(N\) increases the firm goes bankrupt with probability 1. However, in the subcritical case, the expected number of stages reached increases only as \(\log(R)\) with more runway, whereas in the critical case the number of stages is linear \(R\). In fact, in the critical case as \(N\to \infty\), the number of stages attained before bankruptcy is given by \(\Poisson(z)\).

When returns to research scale less than costs, AI investment is like a normal technology and additional capital has diminishing returns. In the critical regime where costs and returns scale equally, the industry is on a precarious Red Queen treadmill, but additional capital proportionally results in additional breakthroughs. When returns scale faster than costs, the AI industry is in the intelligence explosion regime, where additional capital buys an increased probability of escape.

One can view the debate about whether we are in an AI bubble as a debate about the true value of \(q\). In the subcritical case, a firm should aim to monetize on a smallish number of breakthroughs and normal valuation rules apply. In the critical case, the firm value is determined by ruin theory and survival probability. In the supercritical case, all that matters is maximizing the probability of exponential takeoff.

The Effect of Competition

Competition between rival firms is the last missing ingredient. The goal for an AI lab isn’t just to get a breakthrough before ruin, it’s to get a breakthrough before their rival. Each firm \(i\) has its own runway \(R_i\), and its own investment levels \(k_i\). All firms will have the same breakthrough intensity production function and the same fixed cash drain \(d\). ⁴ Each firm, given its breakthrough intensity, has a random clock for when a breakthrough occurs. The firm with the first clock to ring before its bankruptcy horizon wins. If a firm’s clock does not ring before its bankruptcy horizon, it goes bust.

Winners and Losers

There is a winner prize \(w(R)\) for being the AI lab which attains a breakthrough first. To study the effect of technology diffusion, assume the loser’s prize is a fraction of the winner’s prize \(\ell(R) = \eta w(R)\) where \(0\leq \eta \leq 1\). When a breakthrough happens, all research stops and all surviving firms get their respective prizes. If the breakthrough arrives after a firm has exceeded its runway, it’s bankrupt and receives nothing. In this way, \(\eta\) is a stand-in for the ability to distill or replicate the new algorithms or architectures. When \(\eta=1\) there is perfect diffusion and when \(\eta=0\) the market is winner-take-all.

There are three layers to this model. The wedge \(\eta\) between winning and losing quantifies how much winning matters. The investment policies determine the likelihood of being first, but affect cash burn. The cash burns control the bankruptcy horizons.

Specializing to a two-firm competition, the value of a firm has two parts, depending on whether the competition is live or whether the rival has gone bust. Let \(h^-=\min( h_1, h_2)\) be the shorter bankruptcy horizon. For exponential breakthrough times, \(S \sim \text{Exp}( \lambda_1+\lambda_2)\) gives the distribution of the first breakthrough time. At the time of breakthrough, this firm attains the breakthrough with probability \[ p_1 = 1-p_2 = \frac{ \lambda_1}{\lambda_1+ \lambda_2} \tag{7}\]

Thus the live competition component of firm value is:

\[ \begin{aligned} V_1^{\text{live}} &= \E[ \, (p_1 w(R_1-b_1 S) + p_2 \ell(R_1-b_1 S)) \, \mathbf 1_{S < h^-}] \\ &= \E[ \, (p_1 + \eta p_2 ) w(R_1-b_1 S) \, \mathbf 1_{S < h^-}] \\ &= \int_0^{h^-} e^{-(\lambda_1+\lambda_2)s} (\lambda_1 + \eta \lambda_2) w(R_1-b_1 s) \, ds \end{aligned} \tag{8}\]

If the firm has a longer horizon than its rival, there is a continuation value for the period between the bankruptcy horizons. Let \(T_i\sim \Exp(\lambda_i)\) be the breakthrough time for firm \(i\). The continuation component of firm value is:

\[ \begin{aligned} V^{\text{c}}_1 &= \Pr( T_2 > h^-)\E[ \, w(R_1-b_1 T_1) \mathbf 1_{ h^- \leq T_1 \leq h_1}] \\ &= e^{-\lambda_2 h^-} \int_{h^-}^{h_1}\lambda_1 e^{-\lambda_1 t} w(R_1-b_1 t) \, dt \\ &= e^{-(\lambda_1+\lambda_2)h^-} \Phi(R_1-b_1 h^-,k_1) \end{aligned} \tag{9}\]

Here \(\Phi\) is the expression in Equation 1.

Combining these terms, the first-order condition for a maximum exhibits a number of countervailing forces. Increasing research spending has the following effects, where +/- represents the effect on firm value:

• Hastening the time of a breakthrough (+)
• Increasing the probability that the firm is the winner when a breakthrough happens (+)
• Reducing the remaining runway at the time of a breakthrough time because of faster burn (-)
• Shortening the bankruptcy horizon (-)

All of these factors affect both components of value, except for the probability share which only affects the live component.

A firm’s value depends on its rival’s investment level only indirectly. A rival’s increase in research spending has the following effects, where +/- represents the effect on the firm’s value:

• Shortens the rival’s horizon and potentially the time the race is live via \(h^-\). (+)
• Potentially increases the length of the continuation period (+)
• Increases the speed of a breakthrough (+)
• Decreases the relative likelihood the rival wins (-)

At a fixed rival investment level, there may be more than one local maximum for the optimal research investment level. Numerical studies show that a strong leader may either lean into the live competition value and sprint to make the breakthrough, or go slow and coast to lean into the continuation value.

Competition isn’t always cutthroat. A firm with more runway may prefer to wait out its rival’s bankruptcy and preserve continuation value rather than dominate research and win at all costs.

Equally Capitalized Firms

Let the two rival firms be equally capitalized with equal runway \(R=R_1=R_2\). Let’s seek a candidate for Nash equilibrium which is pure and symmetric, so \(k_1=k_2\). In this case both firms will have the same bankruptcy horizon \(h=h_1=h_2\) so neither firm outlasts the other and neither has a continuation value.

A key quantity is the modified research mass, which is related to the probability either firm attains a breakthrough before a given firm reaches bankruptcy. \[ z_{c,1} = h_1 ( \lambda_1 + \lambda_2) = \frac {aR_1(k_1^\gamma + k_2^\gamma)}{d+k_1} \]

With two firms investing, there are two research breakthrough clocks. When runways and investment levels are symmetric, the modified research mass becomes the research mass of the entire industry, and it is twice as big. In other words, \(z_c(k) = 2 h \lambda = 2 z(k)\). If runway \(R\) is allocated either exclusively to one firm, or split equally \(R/2\) between two firms, the probability of a breakthrough is invariably higher for the single firm without competition. This is because a monopoly firm chooses \(k\) to maximize research mass, so a firm in competition can at best match that optimum as it optimizes its competitive value.

Competitive pressures invariably introduce some deadweight loss. With the same total runway, two firms in competition can at best match the likelihood for a breakthrough that a single firm attains without competition.

Winner Take All

Turn now to the case of brutal cutthroat competition where the winner takes all and second place counts for nothing. We’ll seek candidate solutions for a pure symmetric Nash equilibrium with \(k_1 =k_2\). There is no continuation value, and with \(\eta=0\) the live value of the firm Equation 8 becomes

\[ V_1 = p_w \E[\mathbf 1_{S < h} w(R - b_1 S) ] = p_w \Phi(R, z_c(k_1)) \]

where \(p_w\) is defined in Equation 7 and depends on \(k_1\) via the breakthrough intensity.

The central tension in winner-take-all competition is that increasing investment increases the chance of capturing the breakthrough (term represented by \(p_w\)) while decreasing the overall chance of a breakthrough before bankruptcy (the term represented by \(\Phi\)). Since \(p_w \to 0\) as \(k_1\to 0\) and \(\Phi \to 0\) as \(k_1\to \infty\), there is a maximum firm value for some intermediate investment level. For levels of \(k < k^*\) less than the single-firm optimum, one can increase both the \(p_w\) and \(\Phi\) by increasing \(k\), so the competitive optimum investment level must be greater than the single-firm optimum level. Firms in winner-take-all competition will overinvest to win the race.

The first-order condition on a symmetric solution satisfies this equation

\[ k^*_c = \frac{ \gamma \theta} { 1-\gamma \theta} d \tag{10}\]

The formula⁵ for \(\theta\) isn’t as important as its properties.

One property is that \(\theta > 1\) so \[ k^*_c = \frac{ \gamma \theta} { 1-\gamma \theta} d > \frac{\gamma}{1-\gamma}d = k^* \]

This makes precise the earlier claim that competition leads to overinvestment. Also, typically, \(\theta\) is increasing in \(R\), so more runway results in more investment. More capital leads to more deadweight loss.

There is a large class of reward functions for which \(\theta\) depends on \(R\) only via the research mass \(z\). This includes fixed reward, proportional rewards, and all rewards of the form \(w(R)=R^\alpha\). The first-order condition for the equilibrium imposes a maximum on \(z\) via \(\theta\).

\[ z < z_{\text{max}} \qquad \text{where} \qquad \theta(z_{\text{max}}) = \gamma^{-1} \]

In Equation 10, beyond \(z_{\text{max}}\), \(\theta\) gives nonsense negative investment levels. Furthermore, \(z^*\) increases as \(R\) increases, so we get a limit \(z^* \uparrow z_{\text{max}}\) as \(R\to \infty\).

As runway increases, it gets consumed by spiraling research investment at less efficient levels. More capital barely moves the needle in terms of the probability of a breakthrough. The value of extra runway is negated by research investment at diminishingly less effective levels. For \(z\) to remain bounded while \(R\) grows, \(k^*_c\) must grow fast enough to offset the extra runway.

\[ k^*_c \sim \left( \frac{R}{z_{\text{max}}} \right)^{1/(1-\gamma)} \qquad \text{as } R\to \infty \]

This is a remarkable scaling law! Investment increases without bound as runway increases, and investment scales faster than runway since \((1-\gamma)^{-1}>1\). The net effect of additional runway is to shrink the bankruptcy horizon.

For a single firm, additional runway increases the research mass and bankruptcy horizon proportionally. In winner-take-all competition, as runway gets large, additional runway barely moves the needle. The bankruptcy horizon actually shrinks.

The symmetric sprinting logic cannot continue in the extreme. For large enough investment levels, a firm is better off switching to be a survivor rather than a sprinter. Thus, the symmetric pure strategy is not globally optimal. The equilibrium may instead involve mixed or asymmetric strategies.

A large wedge between winners and losers leads firms to overinvestment, resulting in inefficiency and deadweight loss. For many kinds of rewards, very high levels of investment eventually stop producing any benefits at all. Intense competition entirely erases the gains of more capital. More money doesn’t add more bankruptcy cushion or increase the odds of breakthroughs, it’s a race to a standstill.

Perfect Technology Diffusion

Now consider the opposite case of full instantaneous technology diffusion, when the wedge is zero. Each firm benefits perfectly from the breakthrough of the other. The value of each firm is Equation 8 with \(\eta=1\).

\[ V_1 = \E[ w(R - b_1 S) \, \mathbf 1_{S < h} ] = \Phi(R, z_c(k_1)) \]

This is exactly the same as Equation 3 except the breakthrough time \(S\) is controlled by the modified research mass \(z_c\). The first-order condition Equation 4 still applies. Maximizing with respect to \(k_1\) and then setting \(k_2=k_1\) gives the optimal research level

\[ k_c^* = \frac \gamma {2-\gamma} d < \frac \gamma {1-\gamma} d = k^* \tag{11}\]

In the perfect technology diffusion case, firms underinvest as compared with a solo firm. The reason is clear: each firm wants to free-ride on the breakthrough of its rival. By investing less, it retains more of its own runway when a breakthrough happens. The generalization of Equation 11 to \(N\) firms is \(k_c^* = \gamma d / (N-\gamma)\). In the limit \(N\to \infty\) of many many firms, everyone coasts on the effort of everyone else, and no one does any research.

In the case where there is widespread diffusion of breakthroughs, competition does not add urgency, it creates a public goods problem. When the wedge between winning and losing is modest, a firm is incentivized to free-ride on other firms’ research clocks and underinvest. As the wedge grows, the incentives flip and firms race to avoid losing a breakthrough.

Rival Breakthrough Interruption

There’s another way to view AI lab value which isolates the effect of competition. At the moment of a rival’s breakthrough, the remaining value to a firm from its solo research program evaporates and is replaced with the loser’s share. So we can start with the \(\Phi\), the firm’s value absent competition, and add a correction term based on the rival’s breakthrough. As before let \(S \sim \Exp( \lambda_1+\lambda_2)\) be the time of the first breakthrough. The rival breakthrough interruption \(C_1\) is the expected forgone value⁶

\[ \begin{aligned} C(R_1,k_1, R_2, k_2) &= \E\left[ \, p_2 \, [\ell(R_1-b_1 S) - \Phi(R_1-b_1 S)] \mathbf 1_{S < h^-}\right] \\ &= \int_0^{h^-} \lambda_2 e^{-(\lambda_1+\lambda_2)s} [ \eta w(R_1-b_1 s)-\Phi(R_1-b_1 s)]\,ds \end{aligned} \]

The total value of the firm is the value of the firm on its own plus this correction \[ V_1 = \Phi(R_1,k_1) + C(R_1,k_1,R_2,k_2) \]

The single-firm solution maximizes the value of \(\Phi\). So, in the face of competition, a firm is trading off how much to reduce the value of its solo research program against managing the value of the correction. If the rival interruption correction is small at the single-firm research level \(k^*\), then competition does not alter a firm’s behavior much.

For example, a well-capitalized firm may choose to mostly ignore its rival if the rival’s bankruptcy horizon is short enough. Since it’s possible for \(C\) to be positive or negative, managing the correction can involve limiting the loss upon a rival breakthrough, or increasing the positive value accrued. A positive correction leads to free-riding behavior and underinvestment. When the rivals are close in size and have large runway, we expect the correction to be large and negative, so competition is most intense.

The rival interruption correction gives a lens into how firms change behavior in the face of competition. If the correction term is positive, then a firm will opt to free-ride and underinvest. If the correction term is modest, firms will not change their investment level much, focusing instead on the continuation value. The most intense competition and deadweight loss is when there is a large wedge between winning and losing and equally sized well-capitalized firms square off.

A Lens for Interpretation

The real value of the model is the context it provides for understanding developments in the AI industry. Let me give a few examples.

Is AI a Bubble?

If \(q>1\) and there is no limit to the number of valuable breakthroughs \(N\), then the value of an AI lab has no bound. In this case the probability of escape \(p_\infty\) is positive. The whole question of whether lofty valuations are justified boils down to investors’ beliefs about \(q\) and \(N\). The fact that investors are willing to write $100B checks for $1T valuations is itself weak evidence that the market believes \(q>1\). IPO filings and giant private rounds map to \(R\).

Even in this case, there are different regimes. If \(q\) is slightly above 1, say 1.01, then Equation 6 is approximately a sum of i.i.d. breakthrough clocks \(T_1\) and there is a concentration of probability mass around the mean. Thus \(p_\infty\) increases sharply to 1 as the research mass \(z\) crosses \(q/(q-1)\). That threshold is the natural target for runway and anything more has a small effect on \(p_\infty\). If \(q\) is significantly above 1, then the probability in Equation 6 is controlled by the first few breakthroughs. This is the regime of recursive self-improvement and fast takeoff. All that matters for investors is getting through a few generations, then the flywheel takes over.

If there’s a big wedge between being first and second, Red Queen competition complicates the picture. The race leads to overinvestment, which counteracts the effect of increased runway. We may see evidence of overinvestment in the race to build data centers, where construction is limited only by the availability of the city-scale electricity needed. Hyperscaler and data-center investment is on track toward $1.5T annually by 2030 (§), and hyperscaler capex is on track to exceed cash inflows this year (§). Consider also the phenomenon of “tokenmaxxing” (§), where internal dashboards at major labs are spurring researchers to consume more compute resources. Uber exceeded its $3B token budget for the year in April (§).

According to the model, at very high levels of investment, competition eats additional returns. Increased data center capacity and token usage isn’t by itself evidence of wasteful investment, the proof is whether the returns materialize. However, the model does suggest a modest proposal for limiting overinvestment: in order to maximize the likelihood of AGI, have a central authority manage all AI research.

The bear case for AI is that \(q\leq 1\) and returns don’t scale with costs, or that AI potential has a limit and \(N\) is relatively low. Time will tell.

What’s the Point of Circular Investments?

AI firms are taking equity stakes in their suppliers or customers or even rivals, while at the same time signing long-term contracts to provide or receive services. In a research note on the AI industry, Goldman Sachs comments on “the increasing circularity of the AI ecosystem, with model companies, infrastructure providers, and hyperscalers signing deals with each other that are blurring the boundaries between customers, suppliers, and capacity providers” (§). There are actually several kinds of “circular financing”, and the model helps us distinguish these.

With rival cross-holdings, a backer holds an equity stake in a potential upstart competitor, so it partly internalizes the rival’s win. Microsoft’s investment in OpenAI is the clearest example, with Microsoft taking an equity stake, a large commitment to supply cloud computing and a technology-sharing/IP agreement to leverage OpenAI models. Google’s and Amazon’s stakes in Anthropic are similar, providing cloud commitments or chips, in addition to making an equity investment. Rival cross-holdings are a way to reduce the wedge and increase \(\eta\), which softens the rival correction and moderates competitive overinvestment. The clearest example is Microsoft’s investment in OpenAI because it explicitly includes a technology-sharing agreement.

With cross-industry supplier stakes, firms such as Nvidia make investments across several frontier labs, taking stakes in OpenAI and Anthropic while also making commitments to provide GPUs. In this situation, runway is increased across the industry. In this case, additional runway can spur Red Queen competition and overinvestment. Rather than a way to internalize gains, Nvidia’s strategy might be an example of “commoditize your complements”, betting on the effects of competition to concentrate value at the chip layer of the supply chain. Nvidia benefits if there is overinvestment, so long as firms are solvent enough to keep buying GPUs. Whether this is a doom loop or a virtuous cycle depends on \(q\). On the other hand, the extent to which Nvidia’s GPUs are a critical supply chain constraint, chip allocation could serve as a kind of rationing which nudges the market into more efficient research investment regimes.

Finally, there is demand-side vendor financing. Here frontier labs invest in compute providers like CoreWeave while making long-term commitments to lease the compute. The neocloud compute providers use the long-term leases to secure debt financing in the capital markets. This type of arrangement is a way to increase \(R\), effectively moving debt off balance sheet onto the neocloud. The extent to which these arrangements create unavoidable drag \(d\), they also increase time pressure on firms and spur research investment via Equation 5.

What’s the role of open weight AI?

If frontier breakthroughs diffuse quickly, then being first is less valuable. The rate and extent of diffusion directly affects \(\eta\), and it moderates competition. Andrej Karpathy’s nanochat (§) is an end-to-end pipeline which reproduces GPT-2 capability (original training cost $43k) for less than $100. Epoch AI estimates algorithmic efficiency halves the required compute every 5–14 months (§). Open source models use distillation to build smaller models which match capabilities of larger models (§). Epoch AI estimates that open weight models match the capabilities of frontier models with a 4–7 month gap, though this has been uneven over time (§). Another channel is talent mobility, and there are many high-profile examples of prominent researchers rotating between labs (§), which raises effective \(\eta\) as state-of-the-art knowledge walks out the door and joins a rival.

With perfect diffusion, there’s a public goods problem, leading all firms to underinvest to maximize the value of free-riding. In the limit, no one is incentivized to invest in anything. The scale of investment by frontier labs suggests they believe the wedge for winning is quite large and significant. One way to thread the needle of revealed preferences is to note that, unlike in the model, there is a time lag for diffusion. In a multi-stage race with high \(q\), a short window to capitalize on gains may still be quite valuable even in the face of high diffusion.

What’s the deal with Apple?

Apple, one of the three largest companies by market capitalization, is investing far less in technology than its peers. In particular it seems to have little interest in developing frontier models, instead signing a $1B/year contract with Google licensing a customized version of its Gemini for its products (§). What accounts for this difference in strategy?

Apple has huge runway compared to frontier labs, only Google matches it in financial resources and revenues. The model suggests that a firm which dominates in \(R\) has a small or positive rival interruption correction, and may choose patient waiting rather than racing. By licensing from Google, it can internalize the value of breakthroughs from that firm, which raises \(\eta\). It may also believe that with its market power, it will have access to frontier models no matter who is the technology leader. Incumbents such as Google and Meta are investing significantly more in developing frontier models, which may reflect a different interruption correction, and the likelihood their business could be disrupted.

But Apple’s strategy may also reflect its views on the overall potential of AI, at least in consumer products. Apple’s strategy fails if \(q\) is large and diffusion is low and it’s not able to partner with a breakaway winner. Apple’s revealed preferences are a statement about how it views those scenarios. Apple seems to be saying AI will be more of a commodity, at least in terms of how it integrates with its products.

Weaknesses of the Model

The model here is intentionally oversimplified, and has some definite weak spots.

• There is no accounting for ongoing revenues, or value as a going concern from inference and serving models. A firm is not able to capitalize on existing research, it can only aim for bigger breakthroughs. This makes the economic decision faced by AI firms rather stark, they can only shoot for the moon.

• The economic constraints for AI labs resemble “laws of nature”, external to the behavior of any firm. Breakthrough scaling and the effectiveness of research are just exogenous facts. Competition and research doesn’t affect the cost of burn or the quality of breakthroughs. In a richer model, the value of \(q\) or \(b\) could be affected by the actions of the firms themselves. Economic feedback loops could be positive or negative, via technological or economic channels.

• Single parameters are a stand-in for a complex assortment of fairly different things. There is not a single clock for AI progress, there is progress in many different dimensions simultaneously. There is no accounting for market niches, market ecosystem, or market segments. Different kinds of progress have different kinds of value. Moreover, there are more than two kinds of costs which are inputs to economic value in all kinds of ways.

• Runway in particular is a bit squishy. As something that scales at each stage, it resembles something like market capitalization. But as something which leads to bankruptcy, it resembles something more like cash on hand.

Every model is wrong, some are useful. Please send me critiques or feedback about what would make the model more useful.

Conclusion

It may be simultaneously true that AI is the most consequential technology in history and the biggest capital misallocation. This model doesn’t resolve the debate, but it provides a map to the terrain. The ratio \(q\) determines whether the AI economy is a treadmill, a normal diminishing-returns industry or a flywheel which can self-fund into the singularity. The diffusion parameter \(\eta\) determines whether the competition is a race or a public-goods problem. And the runway \(R\) controls the stakes and the horizons.

A monopolist firm keeps its research investment fixed as a function of drag, so its horizon scales proportionally with capital. In a winner-take-all competition, the linearity breaks, and additional runway gets consumed by overinvestment. In the limit, bankruptcy horizons shrink and additional money doesn’t buy additional safety or make breakthroughs more likely. This may be why the biggest IPOs in tech history feel like an arms race. The \(q\)-trichotomy formalizes what the AI bubble debate is really about. The model doesn’t settle who is right, but it makes the question precise.

The model, however, is at best a sketch of reality. It leaves out exploration-exploitation tradeoffs, inference revenues, market niches, and economic effects of competition. Future versions could tackle how key parameters endogenously change in response to aggregate firm behavior, and seek to incorporate US-Chinese competitive dynamics. But those are extensions, not refutations of the model as it is.

The next time someone tells you AI investment is obviously rational or obviously a bubble, ask them what they think \(q\) and \(N\) are.

About the author. Ryan McCorvie builds mathematical models of systems on the edge of catastrophe. He ran quantitative risk for credit derivatives at Goldman Sachs through the 2008 crisis, forecast the pandemic for the State of California, and now writes at martingale.ai on the mathematics of the AI race. ryan@martingale.ai. \(\cdot\) linkedin

Footnotes

1. Our model resembles an inverse of the Cramer-Lundberg model. In classical ruin theory, an insurance company receives fixed premiums, but is exposed to random payouts from insurance claims. In the present case, there are fixed expenses but random rewards.

2. Another approach to put time pressure on an investment decision is to add an interest rate. Then breakthroughs which happen earlier are more valuable because of the time value of money. Qualitatively a lot of the results are the same, but the addition of an interest rate does not allow us to do the reparameterization trick which simplified a lot of the analysis. In particular, the first-order condition does not correspond to maximizing survival probability, and the resulting analysis is a bit messier.

3. This resembles the Galton-Watson branching process. In the subcritical and critical cases, the process eventually goes extinct with probability 1, but in qualitatively different ways. In the supercritical case there is some chance extinction never happens and the process survives to arbitrarily many generations.

4. It might be hoped that competitive pressure always incentivizes research and avoids degenerate investment strategies. This is not the case. Without a cash drain, when a firm has a big enough lead in runway, it may be optimal to choose the patient waiting and infinitesimal investment.

5. For the mathematically curious, \(\theta = \frac 1 2 +\frac 1 2 \left( \frac {\partial \log \Phi}{\partial \log z}\right)^{-1}\). The derivative in this expression is the elasticity of firm value with respect to research mass. This makes it clearer why for a reward \(w(R)= R^\alpha\), which has constant elasticity \(\alpha\), the term \(\theta\) doesn’t depend on \(R\) separately from \(z\). From the form of Equation 1, the elasticity must be between 0 and 1, so \(\theta > 1\).

6. As the effect on a firm is the same regardless of which rival beats it, there is a nice generalization of the correction term to several rivals. Just replace the fixed rival intensity \(\lambda_2\) with the piece-wise constant intensity formed by adding each rival’s intensity up to its bankruptcy horizon.