Suppose a group of people (size=N), each with an endowment $y_i$ can increase or decrease their initial endowment by gambling. Make the simplifying assumption that everybody gambles in this group and that it is a one-time game.
Note, each individual can start the gamble with however high endowment he sees fit, such that $y_1<y_2<...<y_N$.
The members of the group each agree to chip in $\gamma$ dollars from their initial endowment. A share $\gamma \bar{y}$ from the resultant fund can be accessed (only once) by any member of the group, immediately once he starts making loses ($y_{i_{game}}<y_i)$.
Note 1: the decision of players to bring at the start of the game higher or lower endowments knowing the fund would be set does not concern me modelling-wise.
Note 2: the decision to gamble more or less risky to be able to access a share of the fund does not concern me, modelling-wise.
Note 3: I wish to model this statically and not model time per se.
I am interested in writing-down the utility function for this example. I propose:
$$U=(1-p)((1-\gamma)y_i + \int_{k\geq0}^{k=K}y dy) + p (\gamma \bar{y} + \int_{k=-K}^{k<0}ydy) $$
where, recall, $(1-\gamma) y_i$ represents the endowment each player is left with after he chips in to the fund, $\int_{k\geq0}^{K}y dy$ represents all potential values of gains from gambling, from $k>\geq0$ to $K$, $\gamma \bar{y}$ being the share of the fund one can access and $\int_{k<0}^{0}ydy)$ representing the values of his end-game endowment with loses 0 to loses $k<0$. $k$ then captures the outcome of the game, with $k\in[-K,K]$.
My question is whether the utility function is correctly specified. If not, I am interested in any suggestions aimed at improving it.