Consider a static game with complete information, where each player makes a decision $a_i$ and receives a payoff $\pi_i(a_i,a_{-i})$.
Now suppose there is a natural disaster modeled by a binary random variable $\delta$, where $\delta=1$ with probability $\theta$ and $0$ with probability $1-\theta$. So each player's expected payoff now becomes $E[\delta\pi_i(a_i,a_{-i})]=\theta\pi_i(a_i,a_{-i})$. Here, the disaster is observed after each player makes their decision.
My question is: are the equilibria with and without the natural disaster the same? It seems that they are the same since the payoff function only differs by a possible scalar. However, the disaster affects all players at the same time, thus introducing additional dependence between players. Does this dependence change the equilibrium? Thanks.
