When cheaper innovation keeps rivals racing without a lasting advantage

A race to keep up

In the preprint Open-ended innovation arm-race in zero-sum games, Matteo Marsili and Leandro Chaves Rêgo investigate a particular kind of competition: two opponents can continually introduce new strategies, but whatever one gains in the contest, the other loses.

Their central result is conditional. Within their large random-game model, the number of strategies at the innovation stopping point grows without bound as both players’ marginal innovation costs approach zero. If those costs shrink while their ratio stays finite, the model also implies that equilibrium payoffs approach zero. More options need not produce a lasting advantage.

The authors connect this mechanism to AI-enabled competition, warfare, cybersecurity and litigation. Those connections motivate the analysis; the paper does not empirically test innovation races in those settings.

Turning innovation into new options

The model starts with a two-player zero-sum game. Each player has a collection of possible actions and can choose a mixed strategy: a probability distribution over those actions. A payoff matrix specifies what player one receives for each pairing of actions. Player two receives the negative of that amount.

The game’s value is its equilibrium payoff under optimal play. Informally, each player chooses how to act while accounting for the opponent’s best response.

Innovation expands the available action set, adding a row or column to the payoff matrix. The payoffs associated with new options are drawn randomly. This represents an open-ended supply of possibilities rather than a fixed catalogue waiting to be exhausted. Randomly generated options are not all necessarily used: the strategies selected in equilibrium depend on the options already available.

The analysis considers very large action sets whose sizes grow proportionally. Drawing on earlier work on random games, the authors use an approximation in which the game’s value depends on the ratio of the players’ repertoire sizes and decreases in scale as those repertoires expand. The assumed payoff distribution has zero mean and finite variance.

Why the follower responds differently

Adding an option cannot worsen a player’s equilibrium position, because the player can leave it unused. The question is how much another option is worth—and how the opponent’s innovation changes that marginal benefit.

For the random-game solution examined in the paper, innovation by the more advanced player increases the less advanced player’s marginal benefit from innovating. Innovation by the less advanced player has the opposite effect on the leader: it reduces the leader’s marginal benefit from another innovation. At equal repertoire sizes, this cross-effect vanishes.

Here, “advanced” means having more available strategies. It is not a separate measure of technological quality, resources or institutional strength.

The authors relate this asymmetry to catch-up incentives in economic models of research competition. But their setup differs from markets where innovation can create additional aggregate value. In this contest, one player’s strategic gain is exactly the opponent’s loss.

The importance of innovation costs

Each player is assumed to innovate while the marginal payoff gain exceeds a marginal cost per additional strategy. The paper characterizes stopping points by equating those marginal gains and costs.

In this analysis, the player with the lower marginal innovation cost ends up with the larger repertoire and a positive game payoff. Its opponent receives an equal negative payoff.

The open-ended result concerns a limit, not the claim that every cheap innovation process literally continues forever. As both marginal costs approach zero with a finite cost ratio, both stopping-point repertoires diverge. Meanwhile, the equilibrium payoff advantage becomes vanishingly small.

A payoff near zero means little advantage within this game. It does not mean little innovation effort, little expenditure or little harm outside the model. The payoff matrix measures the adversarial contest, not a full accounting of its consequences.

What the model leaves unresolved

The result rests on substantial simplifications: two players, strictly opposed payoffs, randomly drawn new options and a large-repertoire approximation. Diminishing marginal returns are also important; the paper establishes concavity near equal repertoire sizes and relies on the earlier random-game solution for broader behavior.

The analysis compares equilibrium values and marginal incentives. It does not simulate a detailed sequence of discoveries or establish how quickly actual opponents would innovate.

The authors suggest that automation could bring innovation costs closer to the low-cost regime their model studies. That is an interpretation, not a measured finding. The preprint’s narrower contribution is to identify a mechanism through which cheaper innovation can sustain competitive expansion without delivering a durable payoff advantage.