The Infinite Treadmill: Why Endless Innovation Produces Zero Gains

The Infinite Arms Race: What Happens When Machines Keep Inventing New Weapons Against Each Other

In the spring of 2016, a computer program named AlphaGo sat across the board from Lee Sedol, an 18-time world champion of Go. Sedol had devoted his life to mastering this ancient game, meditating over its 2,500-year history of human strategic thought. Within hours, AlphaGo had not just defeated him—it had discovered entirely new strategies that human players had never conceived. Moves that masters called "not human," "beautiful," and "divine." The machine had learned everything humanity knew about Go, then surpassed all of it.

This was supposed to be a triumph of artificial intelligence. But what if it's also a warning?

A new theoretical paper by physicist and complexity theorist Matteo Marsili argues that when two opponents can keep inventing new strategies—without limits on what they can create—the result isn't a winner and a loser. It's an endless race that leaves both players exhausted, their gains shrinking toward nothing. The winner is not the cleverest or the strongest, but whoever runs out of resources first. And the stakes aren't just games.

Marsili applies this framework to something far more consequential than Go: the ongoing technological arms race in modern warfare, where AI-driven innovation is creating an ever-expanding landscape of new weapons, tactics, and threats. His analysis suggests that unless international agreements can constrain this dynamic, both sides may find themselves trapped in a competition that benefits no one—where every innovation simply triggers a counter-innovation, and the final prize is an empty board.

The Science

To understand Marsili's argument, we need to start with a deceptively simple concept: the zero-sum game.

A zero-sum game is any competitive interaction where one player's gain is exactly the other player's loss. Classical economics often describes trade as positive-sum (both parties benefit), but zero-sum situations pervade conflict: chess, tennis, military battles, and—under certain assumptions—market competitions where one firm's market share comes directly at another's expense. The formal study of such games stretches back to John von Neumann and Oskar Morgenstern's 1944 masterpiece Theory of Games and Economic Behavior, which established the mathematical framework that armies, corporations, and governments still use to model strategic conflict.

In its simplest form, a zero-sum game between two players can be represented as a matrix of payoffs. Each cell in the matrix shows what player one wins (or loses) if she plays a particular action while her opponent plays a particular response. Player one wants to maximize her payoff; player two wants to minimize it. The "value" of the game—denoted V—represents the expected payoff when both players play optimally. It's the equilibrium point where neither player can unilaterally improve their situation.

The critical innovation Marsili proposes is to make this matrix not fixed, but expandable. In traditional game theory, the set of possible strategies is given. Chess has 12 pieces and 64 squares; poker has 52 cards and a fixed set of betting actions. But what happens when players can introduce entirely new strategies? When a military can develop weapons that didn't previously exist? When an AI can discover novel approaches no human ever considered?

Marsili models this as "open-ended innovation": each player can expand their strategy set by drawing new options from some distribution of possible moves. This transforms the game from a static matrix into a dynamic, growing arena. Player one's strategy set expands from n₁ options to n₁ + δn₁; player two's expands from n₂ to n₂ + δn₂. The innovation process itself has a cost—research, development, experimentation—but crucially, that cost can approach zero. AI systems can generate new strategies automatically. In existential conflicts, nations redirect their entire economies toward military technology. In such cases, the marginal cost of innovation becomes vanishingly small.

To analyze this scenario rigorously, Marsili draws on the framework of large random zero-sum games developed by Berg and Engel in the late 1990s. The idea is this: imagine a game where the payoff matrix is filled with random numbers—each entry independently drawn from some distribution with zero mean and finite variance. Such a matrix represents a "generic" competitive situation where neither player has an inherent structural advantage. As the matrix grows large (n₁, n₂ → ∞, with their ratio x = n₁/n₂ held finite), the game's value becomes a "self-averaging" quantity: it converges to a predictable function that depends only on the ratio of strategy counts, not on the specific random draws.

This self-averaging property is crucial because it means the results don't depend on the details of any particular competitive landscape—they're robust features of large, complex strategic environments. Whether we're talking about geopolitical conflicts, market competitions, or AI versus AI, the mathematics captures the essential dynamics.

The value of such a random zero-sum game takes a specific form:

$$V(n_1,n_2)=\frac{1}{\sqrt{n_2}}u\left(\frac{n_1}{n_2}\right)$$

where u(x) is a function that encodes how the game's value depends on the relative sizes of each player's strategy arsenal. The key insight is that V scales as the inverse square root of the opponent's strategy count: adding strategies hurts you (lower V) but helps your opponent. This creates an asymmetry that Marsili exploits to reveal the arms race dynamic.

The analysis rests on several mathematical properties of u(x) that hold generically for random zero-sum games. First, u(1) = 0: when both players have equal numbers of strategies, the game is fair (the value is zero). Second, V(n₁, n₂) = −V(n₂, n₁): if you swap which player we call "first," the payoff flips sign (which makes sense—if I win $10, you lose $10). Third, and most importantly, V is a concave function of each player's strategy count—meaning that each additional strategy provides smaller marginal benefits than the last. This captures the intuitive idea of diminishing returns to innovation, which Marsili explicitly connects to empirical findings in the economics of R&D.

What They Found

The core result emerges from analyzing how one player's innovation affects the other player's incentives. When player one adds new strategies—increasing n₁ by a small amount δn₁—two things happen. First, her own payoff increases (because more strategies give her more options). Second, and crucially, this changes the marginal payoff of innovation for player two.

Marsili shows that:

$$\delta \frac{\partial V}{\partial n_2}=-\left[\frac{1}{2}{u}^{\prime }(x)+x{u}^{\prime \prime }(x)\right]\frac{\delta n_1}{n_1^{3/2}}$$

is positive under generic conditions—specifically, whenever u(x²) is concave when plotted against √x. In plain English: when your opponent innovates, their innovation makes it easier for you to gain from innovating as well.

This is the paradox at the heart of the paper. In ordinary competitive situations, you'd expect your opponent's success to hurt you. But here, the mechanism is subtler. When your opponent develops new strategies, the landscape of competition shifts. New vulnerabilities emerge. New opportunities arise. The game becomes more complex, and in that complexity, your own innovations become more valuable—not just absolutely, but relatively. Their innovation opens up new angles of attack that didn't exist before.

The result is a positive feedback loop. Player one's innovation increases player two's marginal payoff from innovation. This induces player two to innovate. Player two's innovation, in turn, increases player one's marginal payoff, spurring further innovation. The process feeds on itself, like two people running on an accelerating treadmill—each step makes the next step more necessary, but no step actually gets you anywhere.

Marsili formalizes this with a pair of equations that describe the equilibrium point of the arms race. Innovation continues as long as the marginal benefit exceeds the marginal cost. The equilibrium is reached when:

$$\frac{1}{\sqrt{n_2^{\ast }}}{u}^{\prime }\left(\frac{n_1^{\ast }}{n_2^{\ast }}\right)=c_1\text{and}\frac{1}{\sqrt{n_1^{\ast }}}{u}^{\prime }\left(\frac{n_2^{\ast }}{n_1^{\ast }}\right)=c_2$$

where c₁ and c₂ are the marginal costs of innovation for players one and two respectively.

And here's the key insight: as these innovation costs approach zero—as they do when innovation can be automated, or when existential threats drive all resources toward military technology—the equilibrium strategy counts diverge toward infinity. The game never ends. Open-ended innovation proceeds endlessly.

But here's the rub: while the arms race never stops, the payoffs both players receive approach zero. The game becomes increasingly "flat." Each player's slice of the pie shrinks as the game grows, and in the limit of costless innovation, the pie itself vanishes. Both players invest more and more in an ever-escalating competition that yields less and less return.

The model also reveals an interesting asymmetry in how the arms race plays out. The ratio x* = n₁*/n₂*—the relative sizes of each player's strategy arsenals—depends on the ratio of innovation costs. The player with lower marginal costs ends up with more strategies and, paradoxically, gains a positive payoff. But because the total payoff scales as 1/√n for large n, even the "winner" of this race ends up with vanishingly small gains.

Berg and Engel's analysis (the basis for Marsili's theoretical framework) shows how game value varies with strategy count. The main figure demonstrates how the equilibrium payoff decreases as the number of strategies grows—capturing the dynamic where more innovation, paradoxically, means less gain for everyone. The inset shows the function u(x) plotted against √x, illustrating the concave relationship that drives the feedback loop.

The result bears a striking resemblance to the "Red Queen dynamics" of evolutionary biology—species running as fast as they can just to stay in the same place. But in Marsili's formulation, the Red Queen's observation becomes literal: both players run faster and faster, but the ground beneath them is shrinking.

Why This Changes Things

The implications of Marsili's analysis extend far beyond game theory textbooks. They cut to the heart of how we should think about AI in warfare, the Ukraine conflict, and the future of international security.

Consider what's happening in Ukraine. The war has become a laboratory for military innovation at a pace perhaps unmatched in human history. Low-cost drones—sophisticated enough to be guided by AI, cheap enough to be produced in garages and repurposed factories—have demonstrated capabilities that dwarf their cost. Ukraine's defense industry has been reoriented almost entirely toward producing and refining unmanned systems. Russia, facing similar pressures, has responded with its own technological escalation.

This dynamic maps precisely onto Marsili's model. Each new drone capability—longer range, better targeting, swarm coordination—represents an innovation that expands one side's strategy set. Each innovation triggers a counter-innovation from the other side. The cost of innovation has plummeted as both sides have mobilized their tech sectors, their hobbyist communities, and their wartime industrial bases. The result is the kind of endless arms race that Marsili's mathematics predicts.

And the outcome? Both sides are spending enormous resources on a competition that delivers diminishing returns. Ukraine's drone capabilities in 2024 are vastly more sophisticated than in 2022—but so are Russia's countermeasures. The strategic balance hasn't shifted dramatically; instead, both sides have invested in an escalating dynamic where each advance is quickly neutralized.

Marsili draws a parallel to AlphaGo. When the program defeated human champions, it did so by discovering entirely new categories of strategy—moves that violated centuries of human intuition about the game. But here's what the triumph narrative often omits: AlphaGo didn't just beat humans once. It kept evolving. Later versions of the system defeated earlier versions. The innovation process never stopped. And the payoffs—measured in win rate, not money—approached a limit where further advances yielded diminishing value.

The analogy isn't perfect, but it illuminates the core dynamic. When innovation is open-ended, the goalposts keep moving. There's no stable equilibrium where either side can say "we've won this dimension of the competition." The competition itself becomes the defining feature of the landscape.

Marsili's analysis also speaks to the "democratization" of warfare that has accompanied the rise of cheap drones and AI tools. Traditional military advantages—large standing armies, expensive equipment, sophisticated industrial bases—have been partially eclipsed by small, cheap, numerous systems that can be built with commercially available components. This lowers the barriers to entry in ways that are genuinely new. Terrorist groups and criminal organizations can now access capabilities that previously required nation-state resources.

In Marsili's framework, this is precisely the condition that drives the infinite arms race: low innovation costs. When anyone can innovate, when the marginal cost approaches zero, the feedback loop he describes becomes impossible to escape. The treadmill accelerates without limit.

The comparison to other military technologies is instructive. Chemical and biological weapons, Marsili notes, have been banned by international agreement. These are technologies where the innovation dynamic he describes could have played out without constraint—and the international community decided, collectively, that the risks outweighed the potential benefits of unrestricted development. Autonomous weapons systems—AI-driven killing machines that select and engage targets without human oversight—are at a similar crossroads. The regulatory discussions are happening, but meaningful agreements remain elusive.

Marsili's theoretical contribution suggests a powerful reason to pursue such agreements: without them, both sides may be locked into a competition that benefits no one. The arms race doesn't produce a winner. It produces exhaustion.

This reframes the debate about AI in warfare. The usual arguments focus on risks of accidents, loss of human control, and the potential for catastrophic errors. Those arguments remain valid. But Marsili adds another: even if we could ensure that AI weapons work perfectly—that they never malfunction, never act unpredictably, never kill civilians—the innovation dynamic itself may be sufficient reason to limit their development. An endless arms race with diminishing returns is not a future anyone should accept voluntarily.

There's also a broader lesson about complexity and strategy. Traditional game theory assumes that the strategic landscape is given—that players choose among fixed options. Marsili's extension shows that when players can transform the landscape itself, the dynamics change qualitatively. This has implications far beyond military applications: in corporate competition, in political maneuvering, in any arena where actors can shape the rules of the game as they play it.

The mathematics of large random games suggests that these "meta-strategic" dynamics tend toward the same equilibrium regardless of the specific context. The feedback loop Marsili identifies is robust, not fragile. It emerges from the structure of competition itself, not from the particular details of any application.

What's Next

Marsili is explicit that his note is a first step, not a final word. The analysis assumes random strategy payoffs, which simplifies the mathematics but may not capture all the structure of real-world conflicts. In practice, some innovations may be more valuable than others; the distribution of potential strategies is not uniform. Extending the framework to more realistic payoff distributions is an obvious direction for future work.

The paper also focuses on a symmetric situation—two players, each responding to the other. Real-world conflicts often involve more than two parties, with complex alliances and shifting coalitions. How the multi-player case modifies the arms race dynamic remains an open question. One might expect even more instability, as each player's innovation triggers responses from multiple opponents simultaneously.

There's also the question of what "innovation" actually means in different contexts. In Go, an innovation is a new move pattern. In warfare, it's a new weapon, tactic, or doctrine. These are categorically different, and the costs and timescales of innovation vary enormously. Marsili's model treats innovation as a continuous, marginal process, but real innovation often comes in lurches—breakthroughs that suddenly expand the strategic landscape in discontinuous ways. Understanding how such discontinuous innovation interacts with the arms race dynamic is another avenue for future research.

The empirical question is equally important. Does the historical record of military technology—nuclear weapons, aircraft, submarines, drones—show the diminishing-returns dynamic Marsili predicts? Or are there cases where innovation has produced lasting advantages that didn't trigger countervailing innovation? The theory suggests that such advantages should erode over time, as the feedback loop kicks in. Testing this against historical data would be valuable.

For policy, the implications are both urgent and uncertain. Marsili's analysis provides a theoretical framework for understanding why international agreements on autonomous weapons matter—but it doesn't specify what those agreements should look like. Total bans (like those on chemical weapons) are one option. Limits on specific capabilities are another. Arms control agreements that accept some level of development while restricting others represent a third possibility. The mathematics tells us the race is counterproductive; it doesn't tell us how to stop running.

One thing is clear: the window for meaningful regulation may be closing. Marsili notes that the cost of innovation in military AI continues to fall, driven by advances in commercial technology that can be repurposed for warfare. As the marginal cost approaches zero, the feedback loop he describes becomes increasingly powerful. Early intervention—before the dynamic becomes entrenched—may be more tractable than late-stage rollback.

The Ukraine war, in this light, is not just a regional tragedy. It's a data point in the experiment Marsili is theorizing about. The innovations emerging from that conflict—drone swarms, AI-guided artillery, electronic warfare—represent the kind of open-ended technological development his model analyzes. Whether the international community can translate this evidence into effective governance remains to be seen.

Marsili himself closes with a modest hope: that these notes can contribute to the debate about autonomous weapons. The debate is happening—the references in his paper cite multiple international initiatives, Nobel laureate declarations, and UN working groups. But contribution is not resolution. The arms race continues. The game is still being played.

The final word belongs to the mathematics. When innovation costs are low enough, open-ended competition produces a unique equilibrium: an infinite race toward zero. It's a result that challenges our intuitions about competition, innovation, and progress. We tend to assume that more innovation is better—that advancing technology gives advantage, that running faster helps you win. Marsili's analysis suggests a different possibility: that in certain conditions, the race itself is the trap. Both players run faster and faster, and the finish line recedes at the same pace.

The question for policymakers, strategists, and citizens is whether we can recognize this dynamic before we're too deep in it to escape. The treadmill is already spinning. The only question is whether we'll step off voluntarily—or run until we fall.