I am in a discussion with a friend and have a question regarding the use of weights.
He presented me with the equation below. It represents an individual's payoff for taking a particular action, say going shopping. The individual determines that going shopping has a value, $x$. Furthermore, the individual assigns a probability, $\theta$, that he will survive the drive to the store and receive $x$, and $\theta \in[0,1]$. There is also a cost associated with going shopping and in this case, the cost, $C$, is a function of effort, $e$. Additionally, $\lambda$ is a weight and is $\in [0,1]$ and $\lambda$ can be considered how much the individual is excited by, $x$. For example, if the person is very excited by $x$ then $\lambda=1$. Now, assume that someone wants to offer this individual a sum of money for not going shopping, this offer, $g$ would need to incorporate the above information and,
$$g=\frac{\lambda\theta x+C(e)(1-\lambda)}{\lambda}$$
Here are two perspectives regarding $\lambda$ in this expression:
Perspective 1: The mathematical use of $\lambda$ is appropriate since it's a weight and as such the method is to apply $\lambda$ to $\lambda\theta x+C(e)(1-\lambda)$ and then divide by $\lambda$.
Perspective 2: Yes, $\lambda$ is a weight, however since we are not seeking the weighted mean, dividing by $\lambda$ is not appropriate. The expression, $\lambda\theta x+C(e)(1-\lambda)$ is fine as is. Also, since $\lambda$ can $=0$, there is a circumstance under which there is no defined solution.
I have other issues with the equation, but my primary concern is how the weights were used.
Both he and I are rusty on our math and can use some help.
Here is how an economist would look at the problem, based on your description. Going shopping yields a random reward, given by $\lambda x$ with probability $\theta$ and by death (or hospitalisation?) with probability $1 - \theta$. Assuming expected utility and risk neutrality, we have an expected benefit $\theta \lambda x + (1 - \theta) D$, where $D$ is the "benefit" of death. Moreover, there is a (certain) cost $C(e)$, to be subtracted from the expected benefit in order to know the value of the offer $$g = \theta \lambda x + (1 - \theta) D - C(e)$$
You may want to ask your friend to micro-found his/her claim that "$\lambda$ can be considered how much the individual is excited by $x$". A simple utility function $u(x)$ seems to be a more natural approach.