I am coming to terms with the connections between the so-called $P$ world and the $Q$ world. In my understanding, the risk-neutral measure $Q$ induces a probability space under which investors are indifferent to risk. For example, if we have two instruments $S^{1},S^{2}$ in a one-period model with $Q_{S^{1}_{1}}=0.5\delta_{50}+0.5\delta_{100}$ and $Q_{S^{2}_{1}}=75$, i.e. the expected payoffs under $Q$ of $S^{1},S^{2}$ are identitical, then the instruments will be of equal value.
Pricing $S^{1},S^{2}$ in the $P$ world is more difficult since it is not risk-neutral such that every state of the world needs to be investigated according to the risk preference of the investor. If the investor is risk-averse, we need to discount the price by some particular factor, otherwise in the case of risk seeking agents the price will increase.
An example of the discount factors going into the calculations of prices within the $P$ world in the case of a risk averse agent would be what? I mean the risk-free rate is the same for all market participants (in theory), right?
Is this the basic idea of the difference between the $P$ and $Q$ worlds?