Two players have the following utility functions:

$U_1=x_1{x}_2-10x_1-0\mathrm{.}5x_1^2$

$U_2=x_1^2{x}_2-10x_1{x}_2+20x_2-0\mathrm{.}5x_2^2$

Where $x1$ and $x2$ are the actions taken by players $1$ and $2,$ respectively.

If they play this game simultaneously, what is the Nash Equilibrium of this game? Draw their best$-$response functions and check for stability. What is the Subgame Perfect Nash Equilibrium if Player $1$ moves first?

$13.$ A group of $n$ students go to a restaurant. It is common knowledge that each student will simultaneously choose his own meal, but all students will share the total bill equally. If a student gets a meal of price $p$ and contributes $x$ towards paying the bill, his payroff will be $\sqrt[\phantom{2}]{p}-x.$

a$)$ Explain the meaning of the payoff. What is the economic intuition behind it$?$

b$)$ Find the Nash Equilibrium.

c$)$ Discuss the limiting cases $n=1$ and $n.$

d$)$ Assume now that the bill is shared by all students, but a tip of $(2+t)$ is added to the bill $($ie$.$ the final bill is $(1+t{)}^{**}$ sum of meal prices$).$ How does the tip affect the equilibrium? Are the students better off?

e$)$ Assume how that there is no tip, and the bill is shared equally: The only difference is that they Went to an upscate restaurant that serves the same meals, but the meals all cost twice as much as the one in the original restaurant. Are the students happier at the upscale restaurant?

$14.$ Analyze the utility functions below. Each function captures a certain type of person. Find the subgame perfect Nash equilibrium of a $$100$ Ultimatum Game in which both the Proposer and the Responder have the sime utility function:

a$)U_i=\sqrt[\phantom{2}]{x_i}$

b$)U_i=x_i-x_j$

c$)U_i=x_i+2ln{x}_j$

d$)U_i=$mith$(x_i,x_j)$