Can we model risk with only probability? Sorry for the confusion! I am adding an example to see if it helps:
For example, consider a gamble A, with payoffs {a,b,c,d}, whose probability of each payoff being realized is equal (so 25% each); and another gamble B, with payoffs {a,b,c,d,e}, and the probability of each is 20%.
Now I would like to measure the risk difference between the two gambles A and B, but all {a,b,c,d,e} are unknown in the data; only the 25% and 20% probabilities, and the fact that payoffs of B include payoffs of A are known. So it would not be possible to estimate expectation or variance, and using only probability seems a lot more feasible. What will be the potential problems with using only the probabilities as the risk measurements? Is there any way to bypass using expectation or variance for this situation (the ultimate goal here is to measure marginal risk aversion)? Thanks a lot!
I don't think it even makes sense to talk about risk without specifying the payoffs.
Take your two examplary gambles and suppose that $a=b=c=d=e=0$. In that case, there is no risk involved at all. It is like flipping a coin and if it lands Heads, nothing happens, and if it lands Tails, nothing happens. This would not even be considered a gamble.
Alternatively, let $a=b=c=d=0$ and let $e&gt;0$. Then gamble $B$ gives you the possibility of free money. Similarly, let $a=b=c=d=0$ and let $e&lt;0$. Then gamble $B$ simply is a possibility to burn money with some probability. However, these two gambles $B$ are very different.
Alternatively, suppose that all $a,b,c,d,e$ are events. The first four are associated with some payoff and in event $e$ simply gamble $A$ is played. Then gamble $A$ is in fact gamble $B$. Without a specification of the events, you cannot compare the two.