Stable compliance

Question

I am reading a book of algorithms in my time. Here is a question to which I have my own answer, but I'm not entirely sure. What do you think? Thanks!

Question: There are 2 television network companies, even if it is assumed that A and B, each of them plans a schedule of television programs in n time slots per day. Each of them puts its n programs in these slots. While each program has a rate based on last year’s popularity, these figures differ from each other. A company wins a slot when its show is at a higher level than that of its opponent. Is there a correspondence of the schedule that A made the schedule of S and B, made the schedule of T, and (S, T) is stable, so that none of the networks can unilaterally change its own schedule and win more time intervals.

Answer 1

, , (.. , , , ).

Proof

• , , - . , . , . .

• , , .

• ( ), ( , ).

• .

• , , , , . ( )

Answer 2

Haskell:

hasStable :: Ord a => [a] -> [a] -> Bool
hasStable x y =
  score x y + score y x == 0

-- score is number of slots we win minus number of slots they win
-- in our best possible response schedule
score :: Ord a => [a] -> [a] -> Integer
score others mine =
  scoreSorted (revSort others) (revSort mine)
  where
    -- revSort is sort from biggest to smallest
    revSort = reverse . sort

scoreSorted :: Ord a => [a] -> [a] -> Integer
scoreSorted (o:os) (m:ms)
  | m > o =
    -- our best show is better then theirs
    -- we use it to beat theirs and move on
    1 + score os ms
  | otherwise =
    -- their best show is better then ours
    -- we match it with our worst show
    -- so we could make use of our big guns
    -1 + score os (m : ms)
scoreSorted _ _ = 0 -- there are no shows left

> hasStable [5,3] [4,2]
False
> hasStable [5,2] [3,4]
True

Answer 3

. , 2 . A p1 (5.0) p2 (3.0); B p3 (4.0) p4 (2.0);

: S1: p1, p2 S2: p2, p1 B : T1: p3, p4 T2: p4, p3

, : (S1, T1) (S1, T2) (S2, T1) (S2, T2)

(S1, T1) - (p1, p3) (p2, p4) 2: 0 - , B T2, : (S1, T2) - (p1, p4) (p2, p3) 1: 0

Vise versa, .

Answer 4

-, 2, . TV-1:

Show1 has a rating of 20 points. show2 has a rating of 40 points.

TV-2:

Show1 has a rating of 30 points. Show2 has a rating of 50 points.

Then he clearly shows that the match is unstable.