Does the following game has pure strategy nash equilibrium

1. Green Thumb, a manufacturer of lawn care equipment, has introduced a new product. Each unit costs $150 to manufac-ture, and the introductory price is $200. At this price, the anticipated demand is normally distributed, with a mean of μ = 100 and a standard deviation of σ = 40. Any unsold units at the end of the season are unlikely to be valuable and will be disposed of in a post-season sale for $50 each. It costs $20 to hold a unit in inventory for the entire season. How many units should Green Thumb manufacture for sale? What is the expected profit from this policy? On average, how many customers does Green Thumb expect to turn away because of stocking out?

2. AnyLogo supplies firms with apparel containing their logo to be used for promotional purposes. AnyLogo has four major customers-IBM, AT&T, HP, and Cisco. During the holiday season, the logos are adorned with a Christmas motif. Demand from each firm for apparel with the Christmas motif is normally distributed, as shown in Table 13-6.

Table 13 Demand distribution for AnyLogo

Mean 5000 7000 4000 4000
SD 2000 2500 2000 2200

AnyLogo currently produces all the apparel including the logo embroidery in Sri Lanka in advance of the holiday season. Each unit costs $15 and is sold by AnyLogo for $50. Any leftover inventory at the end of the holiday season is essentially worthless and cannot be repurposed for a different company. It is thus donated by AnyLogo to charity. Holding the apparel in inventory adds another $3 to the cost per unit donated to inventory. However, the donation allows AnyLogo to recover $6 per unit in tax savings. What production quanti¬ties do you recommend for AnyLogo? What is the expected profit from the policy? On average, how much does AnyLogo expect to donate to charity each year?

3. The manager at AnyLogo is considering the purchase of high-speed embroidery machines that will allow it to embroider on demand. In this case, the apparel will be made in Sri Lanka without any logo; the logo embroidery will be post¬poned and will be done in the United States on demand. This will raise the cost per unit to $18. However, AnyLogo will not have any holiday or company-specific apparel to be dis¬posed of at the end of the season.

The apparel without logos can be sold for $18 a unit to retailers. The cost of holding inventory and shipping adds $4 to the cost of any apparel left over after the holiday season.

With all other information as in Exercise 6, do you recommend that the manager at AnyLogo implement postponement? What will be the impact of post¬ponement on profits and inventories?

4. Does the following game has pure strategy nash equilibrium? Does it has mixed strategy nash equilibrium? Please justify your answers.

U 1. -1 3, 0
D 4, 2 0, -1

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