Multiplayer parallel repetition with exponential decay

Show that for any k-player game I with val(I) ≤ 1 − ε, there is ε′ = ε′(ε, I, k) > 0 such that val(I^n) ≤ (1 − ε′)^n, establishing exponential decay under parallel repetition for multiplayer games.

Background

Raz’s parallel repetition theorem provides exponential decay for 2-player games, but extending this to k ≥ 3 players remains a major open challenge. Existing techniques yield results for special classes (e.g., XOR games) or connected games, but a general theorem is lacking.

Progress here would have broad implications for hardness amplification, PCPs, and interactive proofs, and may require new analytic tools beyond current correlation and swap-norm frameworks.