Suppose a stock is valued at $S_0=2$ at time 0 and at time 1 at $S_1\in\{1,3,4\}$. Suppose the interest rate is $r=0$.

Can we determine risk-neutral probabilities for the value of $S_1$? Not quite, because $V_1=1_{S_1=4}$ for instance is not a linear function of $S_1$ in this trinomial case.

However, we can find upper and lower bound on risk-neutral probabilities.

If we draw $V_1$ as a function of $S_1$ we see that the nonlinear function is bounded above and below by certain linear functions.
If
$$
\alpha S_1+\beta\le V_1\le \gamma S_1+\delta
$$
then we can conclude that
$$
\alpha S_0+\beta\le V_0\le \gamma S_0+\delta
$$
and $V_0$ is the risk-neutral probability of the event $S_1=4$.

To get more information about $\tilde{\mathbb P}$ we can also consider a derivative security that pays off $1-V_1$.