In classical, deterministic cooperative game theory, there are $N$ players that can form $2^{N}$ coalitions. Each of these coalitions is assigned a value by means of the characteristic function $v ( \cdot )$ associated with the particular game, such that $v: 2^{N} \to \mathbb{R}$. The game can be "solved" (i.e. the payoffs can be distributed in a particular manner) by means of concepts such as the shapley value or the nucleolus.
In the paper "Convexity in Stochastic Cooperative Situations", Timmer et al. extend the notion of deterministic cooperative games to stochastic cooperative games. This means that the value of each coalition is not fixed anymore. Instead, the value of a coalition can take on any number within the support of the probability density function associated with that coalition. We call this the stochastic payoff $R(S)$.
For example, we could have when $N = 2$ that ("w.p." means "with probability"):
$R(S) = \left\{ \begin{array}{lll} 1, 2, \mbox{or } 3 & \mbox{w.p. } 1/3 \mbox{ if } S = \{ 1 \} \quad ; \\ 2, 3, 4, \mbox{or } 5 & \mbox{w.p. } 1/4 \mbox{ if } S = \{ 2 \} \quad ; \\ 7 \mbox{ or } 8 & \mbox{w.p. } 1/2 \mbox{ if } S = \{ 1,2 \} \end{array} \right. $
Furthermore, all of the players have their own preferences. Some could value coalitions according to the expected value they will obtain. Others might judge coalitions on the basis of the value of a certain quantile of the cumulative distribution function. Risk-averse players would choose a low quantile, while players who are willing to take a big risk would choose a high quantile.
Now, the example I gave is rather contrived/artificial. I am wondering whether there are any classical, deterministic cooperative games that can be naturally extended to a stochastic counterpart. In particular, I am interested in stochastic cooperative games that could actually arise in real life. So perhaps in a scientific area, finance, network theory or politics. Furthermore, I aim to look into games in which the core is non-empty (so it pays of to work together in the grand coalition according to the preferences of the players).
N.B. I already asked a similar question on MSE, though that version of my question was not very precise. I hope to receive more answers over here and I hope the examples given here are more applicable to science or something else, and/or are actually a stochastic version of a known deterministic cooperative game.
