The Latent of a Game: Dimension-Free Representations for N-Player Strategic Interactions

We define the Latent of an N-player game as the element of a graded Hilbert tensor algebra that encodes the complete interaction structure of the game's payoff functions, organized by interaction order. The grade-\(r\) component captures irreducible \(r\)-player interactions. We prove the Interaction Decay Theorem: if the payoff functions extend holomorphically to a Bernstein poly-ellipse with parameter \(\rho > 1\), then the grade-\(r\) interaction component has norm bounded by \(C\rho^{-r}\). The effective interaction order is \(R^ = \lceil \log(1/\varepsilon)/\log\rho \rceil\), independent of the number of players \(N\). We prove the Truncated Equilibrium Theorem: Nash equilibria of the grade-\(R^\) truncated game approximate equilibria of the full game within \(\varepsilon\) in the strategy metric, with convergence rate \(O(\rho^{-R^*})\). We introduce \(\rho\) as a game complexity measure that classifies individual game instances on a continuous spectrum from trivially compressible (\(\rho \gg 1\)) to fundamentally intractable (\(\rho \leq 1\)), providing finer resolution than worst-case PPAD-completeness. We decompose the Shapley value by interaction grade and show that the exponential decay theorem makes Shapley computation tractable for smooth cooperative games. We characterize mean-field games as the grade-1 truncation of the Latent, with explicit error bounds for the mean-field approximation. We apply the framework to Cournot oligopolies (exact grade-\(d\) representation for degree-\(d\) inverse demand), congestion games (grade-2 exactness for affine latency), and smooth auction mechanisms. All results specialize to the core Latent Theorem (Nagy, 2026) when the game has a single player.