Consider a game with $N$ players, each indexed by $i=1,...,N$. Every player $i$ has to choose a $J\times 1$ vector of actions $a_i\equiv (a_{i,1},...,a_{i,J})$ where each $a_{i,j}$ can be zero or one. The payoff of each player $i$ is $u_i(a_i, a_{-i})$, where $a_{-i}$ denotes the actions of the other players.
As a first best, I would like to show that this game has a pure strategy Nash equilibrium (PSNE). I'm unable to do so and I also don't want to allow for mixed strategies. As a second best, I would be happy to consider a weaker notion of equilibrium (still, in pure strategy) and show its existence.
In particular, by revealed preference, if $(a_1,...,a_N)$ is a PSNE, then the following holds:
$$ \forall i=1,...N \quad \forall j=1,...,J: \quad \text{if } a_{i,j}=1\text{, then } u_i(a_i, a_{-i})\geq u_i(a_{i, \{-j\}}, a_{-i})\\ \hspace{6.5cm} \text{if } a_{i,j}=0\text{, then } u_i(a_{i}, a_{-i})\geq u_i(a_{i,\{+j\}}, a_{-i})$$
where $a_{i, \{-j\}}$ denotes $a_i$ where $a_{i,j}=1$ is replaced by $a_{i,j}=0$; $a_{i, \{+j\}}$ denotes $a_i$ where $a_{i,j}=0$ is replaced by $a_{i,j}=1$.
Question: Is there any equilibrium notion in pure strategy such that:
it is weaker than PSNE
it implies the revealed preference inequalities reported above
its existence is typically easier to be shown (and, if you could give references on this)