When a neural network’s memories become a game’s equilibria

A neural network can reach a stable configuration without recovering the memory it was meant to retrieve. A game can reach an equilibrium without producing the outcome its designer intended. In Neural networks and games, economist Philip R. Neary of Royal Holloway University of London shows that, for particular mathematical models, these are two descriptions of the same problem.

The paper, submitted to arXiv on November 3, 2026, is a theoretical preprint. Its contribution is a formal correspondence and illustrative examples, not an empirical demonstration of improved memory systems or economic mechanisms.

Two interpretations of the same interactions

The neural-network model is a symmetric binary Hopfield network. Each neuron occupies one of two states, represented by −1 and +1. Connections have the same weight in both directions, there are no self-connections, and each neuron has a bias. A neuron's local signal combines that bias with the weighted states of its neighbours.

When selected to update, the neuron takes the sign of this signal. If the signal is zero, it keeps its current state.

Now reinterpret each neuron as a player choosing between two actions. The player's payoff combines an individual preference with pairwise interactions with neighbours. Choosing the action that maximises this payoff gives exactly the same update rule. An indifferent player, like a neuron receiving zero signal, keeps the existing action.

The correspondence rests on a shared function. The network has an energy function; the game has an exact potential, meaning that a player's unilateral payoff change equals the change in that potential. Here, the potential is simply the negative of the network's energy.

Consequently, a state-changing update lowers energy and raises potential. A configuration in which no neuron changes is a pure-strategy Nash equilibrium: no player can gain by switching action alone.

This equivalence has a specific scope. It concerns symmetric binary Hopfield networks and binary-action exact-potential games with quadratic potentials—functions built from individual terms and pairwise interactions. It is not a claim about all neural networks or all games.

Storing a memory means designing an equilibrium

To store prescribed patterns, a network designer chooses weights and biases that make those patterns stable. In the corresponding game, the same task means choosing interaction coefficients and individual payoff parameters that make the desired action profiles equilibria.

Neary interprets this as a mechanism-design problem, but highlights an important difference from socially motivated design: the desired patterns need not maximise anyone's welfare. They are simply the patterns the designer wants to store.

There is also a distinction between weak and strict stability. If indifference is allowed, setting every weight and bias to zero makes every configuration stable. That technically stores the requested patterns, but does not distinguish them from any others. Requiring strict stability gives a linear-programming feasibility problem that may have no solution.

A stable answer can be the wrong answer

The paper illustrates the difficulty with four units arranged along a path. The designer wants two memories: all four states positive and all four negative. The chosen interactions make both strict equilibria, but also admit an unintended stable pattern: two negative states followed by two positive states.

This third pattern is a “spurious memory.” It is also the unique maximiser of the game's potential. Thus, even selecting the highest-potential equilibrium would select a pattern the designer did not request. Guaranteeing that desired memories are stable is weaker than guaranteeing that they are the only stable configurations.

Retrieval adds another complication. Start from the all-negative target with only the final unit changed to positive. If unit four updates before unit three, the target is recovered. If unit three updates first, the system reaches the spurious pattern. Under uniform random asynchronous updating, each outcome has probability one-half.

Nearness is not reliability

The starting pattern differs from the target in just one position. That distance measures corruption, not the probability of successful retrieval. Neary further notes that examples can make retrieval probability arbitrarily close to zero even when the starting configuration is one change from a strict equilibrium.

Under the random asynchronous dynamics considered, convergence to an equilibrium does not guarantee convergence to the intended one. Initial conditions and update order matter.

The paper's proposed opportunity is therefore conceptual: use a common mathematical language to connect questions about incentives and equilibrium with questions about storage and retrieval. Whether that connection produces useful new methods remains a prospect, rather than a demonstrated result.