This question is from the Dutta’s book
Each of the two roommmates simultaneously volunteer an time between 0 and 3 hours to clean their house. (Their choice doesn’t need to be integer) if roommate 1 volunteers x hours and the roommate 2 volunteers y hours , then the payoff to roommate 1 is $4\sqrt{x+y}-x$ the payoff to roommate 2 is $4\sqrt{x+y}-y$.
I need to find the best response functions and all the Nash equilibria.
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My attempt:
First I found the BR for R1
$$max_{\{x\}} \{4\sqrt{x+y}-x\}$$
$$\partial U_1/\partial x = 2(x+y)^{-1/2} -1=0$$ $$BR_1(y)=x(y)=4-y$$
Similarly I found the BR for R2 as follows
$$BR_2(x)=y(x)=4-x$$
So the Nash equilibrium is
$$x=y=2$$
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I only find only one Nash equilibrium. Are there any other Nash equilibria ? I am not sure for my solution. I guess there is a tricky point in this question, but I cannot see. Please share your ideas with me. Thank you.
Your calculation shows that for every $s \in [0,4]$, the pair $(x,y)=(s,4-s)$ is a pure Nash equilibrium, and these are the only pure Nash equlibria.
There are no other mixed Nash equilibria. Given any distribution $\mu$ on $[0,4]$, there is a unique $y=y(\mu)$ that maximizes $\int 4\sqrt{x+y}-y \, d\mu(x)$, namely the unique solution of $$0=\psi(y)=\int \bigl(2(x+y)^{-1/2} -1 \bigr)\, d\mu(x) \,.$$ Thus the only best response to $\mu$ is the measure concentrated at $y=y(\mu)$. Finally, the only best response to $y$ is $4-y$.