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?
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 : $Eu(w + Z) =&lt; u(w)$


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.

$Eu(w+ Z) = EZ = - 5$

$Eu(w) = 0$

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.