computer science

$1.)$ Give an implementation level description of a Turing machine. You can check for an empty tape cell as a $\_$ character. Convert from decimal to Unary: You are given a $2-$digit integer on a tape and you need to replace it with that many $1\text{'}$s$.$ They do not have to be at the beginning of the tape. Examples: $"12"->"111111111111","07",->"11111111","00"-->"\_".$

A Turing machine implementation for the conversion from decimal to unary would begin with a blank tape. The machine would begin in the leftmost cell and scan to the right until it reaches a cell containing data. It would then read the value in that cell and write the appropriate number of $15$ in the adjacent cells. Finally, it would return to the leftmost cell and repeat the process until the tape ran out.

For instance, if the input was $"12,"$ the machine would write $"1"$ in the first cell before moving to the second. It would then write $"1111"$ in the following four cells before returning to the initial cell. Then, it would put $"1"$ in the first cell and go to the second. It would then write $"1111"$ in the next four cells before returning to the first cell and repeating the process until the end of the tape is reached.

Similarly, if the input were $"07",$ the machine would write $"1"$ in the first cell before on to the second. It would then write $"1111111"$ in the following seven cells before returning to the first cell. Then, it would put $"1"$ in the first cell and go to the second. It would then write $"1111111"$ in the next seven cells before returning to the first cell and repeating the process until the end of the tape was reached.

Lastly, given the input $"00,"$ the computer would perform no action because there are no non$-$blank cells,

$2.)$ Give the state transition diagram $($formal description$)$ for the Turing machine in Problem $1.$

Question marks$]$

A mass of consumers is uniformly distributed along the interval $0,1.$ Two firms, A and B$,$ are located at points $0$ and $1$ respectively. We denote by $p_1$ the price of firm iinA,$B.$ A consume located at point xin$[0,1]$ obtains utility $U_A(x)=u-p_A-t{x}^2$ if he consumes from firm A$,$ and $U_B(x)=u-p_B-t(1-x{)}^2$ if he consumes from firm $B.$ In the following, we assume that the gross utility $u$ is sufficiently high, so that the market will be covered and all consumers will get positiv substitute $x$ for the last number of your student ID number.

$($a$)$ Find the demand function for both firms. Assume firms set their prices simultaneously. Solve for the Nash equilibrium prices, and compute the equilibrium profits. I $[6$ marks$]$

$($c$)$ Now, assume a Stackelberg timing, where firm $A$ is the leader. Explain briefly why we should not use the Nash equilibrium concept to solve this game, and solve for the equilibrium prices and profits. $[8$ marks$]$

$($d$)$ Compare the results obtained in parts $($b$)$ and $($c$)$ and explain the intuition for such difference Are the equilibria efficient? $[5$ marks$]$

$($e$)$ Suppose that there was a technology that allowed firm A to credibly commit not to change its price $($this option to commit would make firm A the Stackelberg leader, as in part c$).$ If only firm A had access to this technology, how much would firm A be willing to pay for it$?$ How would your answer change if the technology was auctioned to the best bidder $($between firms A and B$)?[4$ marks$]$ firms A and $B)?$

Assume that firms A and $B$ can perfectly discriminate between locations. That is$,$ firm i chooses a price $p_i(x)$ for each location xin$[0,1].$ Firms now compete by simultaneously choosing the pricing functions $)$ and $).$ There is no possibility of arbitrage, and if a consumer may be indifferent between firms A and B$,$ they go to the closest firm.

$($f$)$ Find the Nash$-$equilibrium pricing functions. That is$,$ you need to find the equilibrium prices of every firm in every location.

$[8$ marks$]$

Consider a consumer whose utility function and budget constraint are given by $u(x,y)=x^{0\mathrm{.}8}{y}^{0\mathrm{.}2}$ and $p_{x}x+p_{y}y=$I, respectively, where $p_x$ is the price of good $x,p_x$ is the price of good $y,$ and I is the consumer's income.

a$)$ Derive the ordinary demand functions, $x^{**}(p_x,p_y,I)$ and code to Check if two strings are Anagram or not