Question

A risk-neutral state representative is choosing whether to run for a seat in Congress. He estimates he has a 1/8 chance of winning if the incumbent chooses to run again, and a 5/8 chance of winning if she does not. If he decides not to run for Congress but instead remain in the state legislature, he estimates he has a 9/10 chance of reelection. (Assume that state law prohibits running for both offices at the same time.)

If he values being in Congress twice as much as being in the state legislature, what probability of the incumbent's seeking reelection will make him indifferent between running for the state legislature and running for Congress?

Answer 1

Let R be the event of the incumbent's seeking reelection. Let $P(R)$ be the probability of the incumbent's seeking reelection.

Let WC be the chance of winning in a seat in the Congress.

the conditional probability of the representative winning given that the incumbent chooses to run again is 1/8. That is

$P(WC\mid R)=1/8$

Let $R^c$ be the event that the incumbent does not seek reelection. Let $P(R^c)=1-P(R)$ be the probability of the incumbent's not seeking reelection.

the conditional probability of the representative winning given that the incumbent does not choose to run again is 5/8. That is

$P(WC\mid R^c)=5/8$

The marginal probability of the representative winning a seat in the Congress is

$\begin{align*} P(WC)&=P(WC\cap R)+P(WC\cap R^c)\\ &=P(WC\mid R)P(R)+P(WC\mid R^c)P(R^c)\quad\text{using the formula for conditional probabilities}\\ &=\frac{1}{8}\times P(R)+\frac{5}{8}(1-P(R))\\ &=\frac{1}{8}(5-4P(R)) \end{align*}$

Let WL be the event that the representative wins a seat in the state legislature.

The probability of the representative winning a seat in the state legislature is

$\begin{align*} P(WL)=\frac{9}{10} \end{align*}$

Let u be the value the representative has of being in the state legislature and 2u be the value he has of being in Congress.

Let 0 be the value the representative has when he is not in either.

The expected value of winning a seat in the congress is

$\begin{align*} EV(\text{Congress})&=(\text{Value of winning a seat in the congress})\times (\text{Probability of winning})+\\&\qquad(\text{Value of not winning a seat in the congress})\times (\text{Probability of not winning})\\ &=2u\times P(WC)+0\times (1-P(WC))\\ &=2u\times \frac{1}{8}(5-4P(R)) \end{align*}$

The expected value of winning a seat in the state legislature is

$\begin{align*} EV(\text{State Legislature})&=(\text{Value of winning a seat in the legislature})\times (\text{Probability of winning})+\\&\qquad(\text{Value of not winning a seat in the legislature})\times (\text{Probability of not winning})\\ &=u\times P(WL)+0\times (1-P(WL))\\ &=u\times \frac{9}{10} \end{align*}$

For the representative to be indifferent  between running for the state legislature and running for Congress, the 2 expected values should be the same.

$\begin{align*} &EV(\text{Congress})=EV(\text{state legislature})\\ \implies &2u\times \frac{1}{8}(5-4P(R))=u\frac{9}{10} \\ \implies &20(5-4P(R))=72\\ \implies &80P(R)=100-72\\ \implies &P(R)=\frac{28}{80}=0.35 \end{align*}$

ans: the probability of the incumbent's seeking reelection will make him indifferent between running for the state legislature and running for Congress is 0.35