To level set, let me state my assumptions picking up on your example:
Public market value of assets is 10x operating earnings
Acquisition premium on those assets should be 30% (implies 13x multiple of operating earnings for take private to succeed)
Operating earnings are $100 for Firm 1 and Firm 2
No interest income
Firm 1 has cash of $800
Firm 2 has debt of $400 and cash of $200 (net debt of $200)
Firm 1 Scenario: You are assuming that company is being valued by the game at 10x P/E regardless of its cash position. I think the game is assigning it a public multiple of 10x earnings (so $1,000) PLUS the $800 of cash. Its market cap should be $1,800. The enterprise value would be calculated as Public Market Cap + Net Debt, so in this case it would be $1,000 ($1,800-$800). To acquire the firm, you should have to pay $1,300 + $800 or $2,100 in total. You could net the balance sheet cash against your purchase price to put up $1,300 to acquire 100% of the assets (and for completeness, note you get no cash from the acquired company).
Firm 2 Scenario: In your second example, you would expect the Firm 2's market cap to be 8x lower than its enterprise value because it has net debt. Enterprise value would still be $1,000, but market cap would be $800 because of the $200 of net debt ($400 debt - $200 cash). Note that while the market cap would be 8x operating earnings, it would not be 8x net income because of the interest expense. If we are getting very precise, debt might impact public market valuation in some real life circumstances but that is unusual. You should have to pay 13x operating earnings to acquire the assets, which means you would pay $1100 to acquire the equity. With the equity, you would get $200 of cash you can use to pay off $200 of debt. You would then assume or refinance the $200 in debt. If you pay off the $200 in debt with cash, the total cash you spent to acquire the operating assets of Firm 2 is $1,300. Same as Firm 1!
Premium should be applied to enterprise value only. I have been an M&A banker and work in corporate M&A now. I can assure you that in most industries companies are valued based on an Enterprise Value / EBITDA multiple, particularly for M&A purposes. From enterprise value you back out net debt to see what proceeds are to the seller. Obviously they care about equity value, but equity value is best calculated from enterprise value when using a multiple based valuation approach.