# Negative expected value; risk neutral choice

Suppose there are two options: (1) take a gamble with 50% chance you win \$100 and 50% chance you lose \$110 or (2) don't take the gamble at all and win/lose nothing. Would the risk-neutral take the gamble or not?

The definition of a risk-neutral individual is someone who is indifferent between a guaranteed amount and the expected outcome of the gamble, which is -\$5 in this case. But since \$0 is only guaranteed amount does that mean all individuals who would take the gamble are risk-seekers, since \$0 is greater than the expected payoff? • I guess that what you mean is: "Does this mean that the concept of risk-aversion cannot be used to establish what type of agent takes this gamble and who doesn't?". But the answer to that question is no: if you define an agent to have a convex utility function, he will have negative risk aversion, i.e. he will be "risk-loving" as you mention. So we can apply these concepts to this gamble. The risk loving guy would take the gamble, while the risk neutral and the risk averse guys would not. Commented Apr 25, 2016 at 22:45 • @Fix.B. the risk loving guy might take the gamble. Commented Apr 26, 2016 at 16:24 • @denesp can you explain why the risk loving guy might take the gamble? Commented Apr 26, 2016 at 20:36 • @LastAlchemist it depends on his exact preferences. To put it in a lazy, mildly incorrect way, he might not be risk loving 'enough'. Commented Apr 26, 2016 at 20:44 • @denesp is right. He would prefer a gamble of 100,-100 with an average 0 to the safe 0 profit, but he might not go all the way to prefer the gamble with an expected loss of 5 to the safe gamble. Commented Apr 26, 2016 at 22:18 ## 2 Answers There are three type of individuals : risk averse, risk neutral and risk loving. Individuals evaluate risky prospects such as to maximize the expected level of their utility. So, an agent is risk averse if, at any wealth level$w$, he or she dislikes every lottery$Z$with an expected payoff of zero,$EZ = 0$, so that :$Eu(w + Z) =< u(w)$• Risk neutrality :$Eu(w + Z) = u(w)$• Risk love :$Eu(w + Z) = u(w)$Assuming that$Eu(w + Z) = EZ$(expected value of the lottery$Z$) => Risk neutral agent.$Eu(w+ Z) = EZ = - 5Eu(w) = 0\$

An risk averse agent won't take the gamble. A risk neutral agent won't take the gamble. We don't know for the risk loving agent, depends on his utility function.

• Since there are individuals who voluntarily buy both insurance and lottery tickets (each transaction having negative expected value), there may be other types of individuals too, who do not mind an almost certain small loss, but would hate the possibility of a big loss even if it had a low probability and would enjoy the possibility of a big gain even if it had a very low probability Commented Jul 27, 2016 at 17:29

Risk neutral means the guy picks the thing that gives the highest expected value. EV of first bet is -5, which is less than 0, the EV of not taking the gamble. So he wont take it.

• So people who do take it are...? Commented Apr 26, 2016 at 20:42
• Not risk neutral Commented Apr 26, 2016 at 20:43
• We can classify people as risk averse, risk neutral, or risk seeking. By ruling out the second, we have two remaining possibilities. Without further information, either is possible. If we add an additional assumption to the problem statement (people like money) then risk aversion is not possible. Without this outsid assumption, no such conclusion. Hope this helps. Commented Apr 26, 2016 at 20:56
• Without the outside assumption why would a risk neutral person not accept? If he hates money this is an excellent chance to lose some (in expected value). Commented Apr 26, 2016 at 21:02
• Please consider editing your answer to make it a well rounded whole. (And then you'll get my vote.) Commented Apr 26, 2016 at 22:59