You are the general manager for your favorite MLB team. Your team owner has agreed to let

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Question:

You are the general manager for your favorite MLB team. Your team owner has agreed to let you spend $50,000,000 in free agency. You are expected to spend all of that $50,000,000. Further, the team owner wants to make sure you add the most productive players possible. Essentially you are looking for the combination of players to spend $50,000,000 in total on in order to maximize wins on the field. Now you already have a full roster of players, so any free agents you will add will come at the expense of a current player on your roster. The table below shows the productivity and salaries of your players and the potential free agents you are looking at in my position. Player productivity is measured by wins above replacement (WAR). Think about this number as the additional number of wins this player brings to your team.

Position	Player on Current Roster WAR	Free Agent WAR	Free Agent Salary
Starting Pitcher	6 wins	8 wins	$30,000,000
Shortstop	4 wins	5 wins	$20,000,000
Third Baseman	2.5 wins	5 wins	$20,000,000
Relief Pitcher	1 wins	2.5 wins	$10,000,000

If you are only concerned about maximizing team performance (measured by WAR) which players should you sign in free agency? Remember you must spend all $50,000,000 in free agency. If you add a free agent you must drop the current player on your roster who is in the same position. So for instance if you add a starting pitcher in free agency you must drop from your roster the current starting pitcher on your roster. For this example you will only be able to sign one starting pitcher, one shortstop, and one third baseman. You can however sign up to three relief pitchers. So for instance you could sign one starting pitcher ($30 million) and two relief pitchers ($20 million – 2 @ 10 million each), but you could not sign two third basemen ($40 million, 2 @ $20 million each) and one relief pitcher ($10 million). Please explain your answer.