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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!

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    $\begingroup$ This question as it is right now is very confusing. It'd be better to perhaps create a very concrete example of the type of problem you are facing. (I do not understand these example probabilities you are giving at all.) $\endgroup$ – Kitsune Cavalry♦ Nov 30 '20 at 15:44
  • $\begingroup$ If you know the probabilities, you also know the expectation and the variance. $\endgroup$ – Michael Greinecker Nov 30 '20 at 15:46
  • $\begingroup$ Sorry for the confusion! I've edited it to include an example, hope it helps :) $\endgroup$ – capcapuccino Nov 30 '20 at 16:04
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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 $e>0$. Then gamble $B$ gives you the possibility of free money. Similarly, let $a=b=c=d=0$ and let $e<0$. 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.

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  • $\begingroup$ This is really helpful, thank you! $\endgroup$ – capcapuccino Dec 2 '20 at 15:09
  • $\begingroup$ @capcapuccino if you think this answer solved your problem consider accepting it $\endgroup$ – 1muflon1♦ Dec 2 '20 at 22:22
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When you discuss risk in economic terms you need payoffs you can't just look at probabilities. For example, choosing a gamble of where you get 1000000 with probability of 0.01 and -1 with probability of 0.99 is not the same as as choosing between gamble where you get 1 with probability 0.01 and -1 with probability of 0.99. Also you can't say something is better just because it has smaller probability. Getting 100000 with probability 0.01 is not worse than getting 10 with probability of 0.10 even when your utility is risk averse. Also I don't think it is fair to say that payoffs are less observable than probabilities.

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