The National Basketball League teams draft new basketball players based (partly)

Question

The National Basketball League teams draft new basketball players based (partly) on a vertical jump test. Suppose there are two types of basketball players, Good (G) and Awesome (A). The share of the A-types in the population of basketball players is 20%. Team owners believe that in the jump test an A-type jumps at least 30 inches with probability 50% and less than 30 inches with probability 50%. For a G-type player, the probabilities are 10% of jumping at least 30 inches and 90% of jumping less than 30 inches.

Suppose an A-type player would contribute $5 million per year to the team’s revenue whereas a G-type would contribute $0.5 million. What is the maximum a team owner would be willing to pay to a player who in the test jumped more than 30 inches? Denote the salary as smax.

Explanation / Answer

Pr(A - player) = 0.20

Pr( G - palyer) = 0.80

Pr(Jump more than 30 - inch l A - player) = 0.50

Pr(Jump more thna 30-inch l G - player) = 0.10

Pr(A - player l Jump more than 30 - inch) = Pr(Jump more than 30 - inch l A - player) * Pr(A - Player) / [Pr(Jump more than 30 - inch l A - player) * Pr(A - Player) + Pr(Jump more than 30 - inch l G - player) * Pr(G- Player) ]

= 0.50 * 0.20 /( 0.50 * 0.20 + 0.10 * 0.80) = 5/9

Pr(G - player l Jump more than 30 - inch) = 4/9

Here the player who in the test jumped more than 30 inches, the maximum a team owner would be willing to pay will be calculated by the expected payoff of players who can jump more than 30 inches.

Pr(Expected payoff) = Contribution of A - type player * Pr(A - player l Jump more than 30 - inch) +  Contribution of G - type player * Pr(G - player l Jump more than 30 - inch)

= 5/9 * 5 million + 4/9 * 0.5 million

= $ 3 million

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