Question

Consider a gambling game where a player pays $10 to play with a 40% chance of winning $20, 40% chance of winning $1, and a 20% chance of winning $0.

(a) If the player’s utility function is U(M) = M, what is the expected utility from playing the game? How does it compare to the player’s utility of not playing the game, i.e. having $10 for sure? Is the player risk-neutral, risk-loving, or risk-averse, and does the player play?

(b) if the player's utility function is U(M) = √M, what is the expected utility from playing the game? How does it compare to the player’s utility of not playing the game? Is the player risk-neutral, risk-loving, or risk-averse, and does the player play?

(c) If the player’s utility function is U(M) = M^2, what is the expected utility from playing the game? How does it compare to the player’s utility of not playing the game? Is the player risk-neutral, risk-loving, or risk-averse, and does the player play?

Answer 1

(a)

Expected utility =

= 0.4 U(20) + 0.40 * U(1) + 0.2 * U(0)

= 0.4 * 20 + 0.40 * 1 + 0.2 * 0

= $8.4

It is less than the player’s utility of not playing the game ($10)

and

Since, , the player is risk-neutral and the player does not play the game as the expected utility is less than $10.

(b)

Expected utility =

= 0.4 U(20) + 0.40 * U(1) + 0.2 * U(0)

= 0.4 * + 0.40 * + 0.2 *

= $2.188854

It is less than the player’s utility of not playing the game ($10)

and

Since, , the player is risk-averse and the player does not play the game as the expected utility is less than $10.

(c)

Expected utility =

= 0.4 U(20) + 0.40 * U(1) + 0.2 * U(0)

= 0.4 * 20² + 0.40 * 1² + 0.2 * 0²

= $160.4

It is greater than the player’s utility of not playing the game ($10)

and

Since, , the player is risk-loving and the player play the game as the expected utility is higher than $10.