I do not understand the notion risk premium. Let us suppose that John goes to the city by car, but he is thinking about not paying for the parking. If he is caught in the act he must pay the fine. How to interpret the risk premium for this game (not paying the parking)?

Let $$\pi$$ be the fine that needs to be paid besides the parking fee (say, $$\phi$$) if he gets caught. Further, he may or may not get caught for not paying. So let $$p$$ be the probability perceived by John about whether he will be caught. Let $$u(x)$$ be utility function of paying for amount $$x$$, with $$u(0)=0$$, $$u'(x)<0$$.
If he pays the parking fee $$\phi$$ he has the assured utility of $$u(\phi)$$. However, if does not pay for parking, his expected utility is $$p \cdot u(\phi+\pi)$$
Now risk premium is defined as the minimum amount by which the return on risky choice must exceed the risk free choice to make the individual indifferent between the two choices. Let $$\pi_m$$ be such that: $$u(\phi) = p\cdot u(\phi+\pi_m)$$
So, for John, $$\pi_m$$ becomes the risk premium for this game. If $$\pi>\pi_m$$ then $$u(\phi) = u(\phi+\pi_m)>u(\phi+\pi)$$ and it would be irrational for him to not pay for parking and risk getting caught. If opposite, he takes decides to not pay for parking and risk getting caught.
• @Laura: It means that $x$ the minimum amount of fine that should be imposed by authorities to induce him to pay for parking. If the fine is negligible (very small x) then no-one would be willing to pay for parking. If the individual is extremely risk averse even small $x$ is enough to induce him to pay for parking Jan 5 '21 at 9:23