For every fixed integer $k\ge2$ and fixed parameters $δ,\varepsilon\in(0,1)$ and $η\in[0,1)$, we give a deterministic polynomial-time reduction from finding a $δ$-approximate Nash equilibrium ($δ$-ANE) in a $k$-player game $G$ with payoffs in $[0,1]$ to finding an $(\varepsilon,η)$-weak approximate Nash equilibrium (weak ANE) in a game $H$ with $C$ players, where $C$ depends only on $k,δ,\varepsilon,η$, independently of the action-set sizes. The constructed game is symmetric with binary payoffs. The games and strategies are represented by complete payoff tables and probabilities, with rational entries encoded in binary. Consequently, the existence of deterministic polynomial-time weak-ANE algorithms at one fixed pair of parameters $\varepsilon\in(0,1),η\in[0,1)$ for every fixed number of players would give deterministic polynomial-time $δ$-ANE algorithms at every fixed approximation parameter $δ\in(0,1)$ and number of players. Under deterministic PPAD-ETH, combining our reduction with Rubinstein's hardness result (FOCS 2016) for bimatrix games at some fixed approximation parameter $γ_0\in(0,1)$ rules out a deterministic algorithm for $(\varepsilon,η)$-weak ANE with running time $N^{O((\log N)^{1-ξ})}$ for any fixed $ξ\in(0,1)$. This holds for every fixed $\varepsilon\in(0,1)$ and $η\in[0,1)$ at a suitable constant number $C$ of players, where $N$ is the binary encoding length of the constructed game's complete payoff table.

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