# Game With Natural Disruptions

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.

• As long as $\theta>0$, nothing changes. Just write out the inequalities defining a Nash equilibrium with and without disaster. May 19 at 15:59
• @MichaelGreinecker Thank you Michael. May 19 at 20:27