## Symbolic Representation

To solve the given problem, we first need to convert the natural language description into a symbolic representation.

Let's define the variables:
- $x_1$ = Number of regular games
- $x_2$ = Number of collector's edition games

The objective is to maximize profit. The profit per regular game is $20, and the profit per collector's edition game is $30. Therefore, the objective function can be represented as:

Maximize: $20x_1 + 30x_2$

The constraints given are:
1. The store can sell at most 100 video games of either type per month: $x_1 + x_2 \leq 100$
2. The store wants to spend at most $4000, with each regular game costing $30 and each collector's edition game costing $50: $30x_1 + 50x_2 \leq 4000$
3. Non-negativity constraints: $x_1 \geq 0, x_2 \geq 0$

## Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'regular games'), ('x2', 'collector\'s edition games')],
    'objective_function': '20*x1 + 30*x2',
    'constraints': [
        'x1 + x2 <= 100',
        '30*x1 + 50*x2 <= 4000',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Gurobi Code in Python

```python
import gurobipy as gp

# Create a new model
model = gp.Model("VideoGameStore")

# Define variables
x1 = model.addVar(name="regular_games", lb=0, vtype=gp.GRB.INTEGER)  # Number of regular games
x2 = model.addVar(name="collectors_edition_games", lb=0, vtype=gp.GRB.INTEGER)  # Number of collector's edition games

# Objective function: Maximize profit
model.setObjective(20*x1 + 30*x2, gp.GRB.MAXIMIZE)

# Constraints
model.addConstr(x1 + x2 <= 100, name="total_games_sold")  # At most 100 games sold
model.addConstr(30*x1 + 50*x2 <= 4000, name="total_cost")  # Total cost not exceeding $4000

# Solve the model
model.optimize()

# Print the solution
if model.status == gp.GRB.OPTIMAL:
    print(f"Optimal solution found: {model.x[0]} regular games, {model.x[1]} collector's edition games")
    print(f"Max Profit: ${model.objVal}")
else:
    print("No optimal solution found")
```