<p>Consider a static game with complete information, where each player makes a decision <span class="math-container">$a_i$</span> and receives a payoff <span class="math-container">$\pi_i(a_i,a_{-i})$</span>.</p>
<p>Now suppose there is a natural disaster modeled by a binary random variable <span class="math-container">$\delta$</span>, where <span class="math-container">$\delta=1$</span> with probability <span class="math-container">$\theta$</span> and <span class="math-container">$0$</span> with probability <span class="math-container">$1-\theta$</span>. So each player's expected payoff now becomes <span class="math-container">$E[\delta\pi_i(a_i,a_{-i})]=\theta\pi_i(a_i,a_{-i})$</span>. Here, the disaster is observed after each player makes their decision.</p>
<p>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.</p>

CC BY-SA 4.0
2023-05-19T15:59:48.390Z
As long as $\theta>0$, nothing changes. Just write out the inequalities defining a Nash equilibrium with and without disaster.
Michael Greinecker
CC BY-SA 4.0
2023-05-19T20:27:38.197Z
@MichaelGreinecker Thank you Michael.
Justin
<game-theory>
Game With Natural Disruptions
Justin
# Game With Natural Disruptions

(Asked by: username_0 on 2023-05-19)

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.

**Tags:** game-theory
> **Comments:**
> **username_1**: As long as $\theta>0$, nothing changes. Just write out the inequalities defining a Nash equilibrium with and without disaster.  
> **username_0**: @MichaelGreinecker Thank you Michael.  

---

## 0 answers

