In a particular soccer league of 20 teams, each team plays a total of 38 games over the course of a season. Teams are awarded 3 points for a win, 1 point for a draw, and no point for a loss. Lucy's favorite team scored a total of 54 points this season and she is especially happy because the sum of games won or drawn this season exceeded twice the number of games lost by 8 games. Mark's favorite team won 2 games fewer and drew 5 games more than Lucy's favorite team this season.

1. Define variables and formulate the system of equations based on the given information.

Let $W$, $D$, and $L$ be the number of games won, drawn, and lost by Lucy's favorite team respectively.
From the total number of games: $W + D + L = 38$
From the total points: $3W + D = 54$
From the relationship given: $W + D = 2L + 8$

2. Solve for the number of games lost ($L$) using substitution.

Substitute $W + D = 2L + 8$ into the total games equation:
$(2L + 8) + L = 38$
$3L + 8 = 38$
$3L = 30$
$L = 10$

3. Solve for the number of games won ($W$).

Substitute $L = 10$ into $W + D = 2L + 8$:
$W + D = 2(10) + 8 = 28$
This gives $D = 28 - W$.
Substitute $D = 28 - W$ into the points equation $3W + D = 54$:
$3W + (28 - W) = 54$
$2W + 28 = 54$
$2W = 26$
$W = 13$

1. Calculate the total points scored by Mark's team by evaluating the difference in wins and draws.

Since a win awards $3$ points and a draw awards $1$ point, the difference in points for Mark's team relative to Lucy's team is:
$\text{Difference in points} = (-2 \times 3) + (5 \times 1) = -6 + 5 = -1$
$\text{Points scored by Mark's team} = 54 - 1 = 53$

2. Compare the points to determine the better performing team.

Lucy's team scored $54$ points, whereas Mark's team scored $53$ points.
Since $54 > 53$, Lucy's team performed better.