Unsorted majorization of two lists Definition

A vector a containing n elements is said to
majorize or
dominate a vector b with n
elements iff for all values k such that 1 ≤ k ≤ n, the sum of the first
element of a↓ through the kth element of a↓ is
greater than or equal to the sum of the first through kth elements of
b↓, where v↓ represents the vector v sorted in
descending order.

That is,

                          a_1 &gt;= b_1
                    a_1 + a_2 &gt;= b_1 + b_2
              a_1 + a_2 + a_3 &gt;= b_1 + b_2 + b_3
                              ...
      a_1 + a_2 + ... + a_n-1 &gt;= b_1 + b_2 + ... + b_n-1
a_1 + a_2 + ... + a_n-1 + a_n &gt;= b_1 + b_2 + ... + b_n-1 + b_n


where a and b are sorted in descending order.

For the purpose of this challenge, we will be using a slight generalization of
majorization: we will say a list is an unsorted majorization of another if
all of the above inequalities are true without sorting a and b. (This is,
of course, mathematically useless, but makes the challenge more interesting.)

Challenge

Given an input of two distinct lists a and b of integers in the range 0
through 255 (inclusive), both lists of length n ≥ 1, output whether the first
list unsorted-majorizes the second (a > b), the second unsorted-majorizes
the first (b > a), or neither.

You may optionally require the length of the two lists to be provided as input.
The output must always be one of three distinct values, but the values
themselves may be whatever you want (please specify which values represent a
&gt; b, b > a, and neither in your answer).

Test cases for a > b:

[255] [254]
[3,2,1] [3,1,2]
[6,1,5,2,7] [2,5,4,3,7]


Test cases for b > a:

[9,1] [10,0]
[6,5,4] [7,6,5]
[0,1,1,2,1,2] [0,1,2,1,2,1]


Test cases for no majorization:

[200,100] [150,250]
[3,1,4] [2,3,3]
[9,9,9,9,9,0] [8,8,8,8,8,9]
Jelly, 10 8 6 bytes

2 bytes thanks to @orlp.

2 bytes thanks to @Dennis.

_+\ṠQS


Try it online!

1 for a&gt;b, -1 for a&lt;b, 0 for no majorization.

_+\ṠQS

_       Difference (vectorized)
 +\     Cumulative sum.
   Ṡ    Sign of every difference
    Q   Deduplicate
     S  Sum


If there were both 1 and -1 present (some cumulative sums are bigger, some smaller), then the last step would produce 0.