At a company, 20 employees are making contributions for a retirement gift.Each person is choosing how many dollars to contribute from the interval[0,10]. The payoff to person i is bi X xi - xi, where bi > 0 is the “warm glow”he receives from each dollar he contributes, and he incurs a personal cost of 1.a. Assume bi < 1 for all i. Find all Nash equilibria. How much is collected?b. Assume bi > 1 for all i. Find all Nash equilibria. How much is collected?c. Assume bi = 1 for all i. Find all Nash equilibria. How much is collected?Now suppose the manager of these 20 employees has announced that shewill contribute d > 0 dollars for each dollar that an employee contributes.The warm glow effect to employee i from contributing a dollar is now bi X(1 + d) because each dollar contributed actually results in a total contribution of 1 + d. Assume bi = 0.1 for i = 1, . . . , 5; bi = 0.2 for i = 6, . . . , 10; bi = 0.25 for i = 11, . . . , 15; and bi = 0.5 for i = 16, . . . , 20.d. What value must the manager choose for d in order to get her employees to contribute $100?e. What value must the manager choose for d in order to raise $750 intotal from both her employees and her own matching contribution?
At a company, 20 employees are making contributions for a retirement gift.
Each person is choosing how many dollars to contribute from the interval
[0,10]. The payoff to person i is bi X xi - xi, where bi > 0 is the “warm glow”
he receives from each dollar he contributes, and he incurs a personal cost of 1.
a. Assume bi < 1 for all i. Find all Nash equilibria. How much is collected?
b. Assume bi > 1 for all i. Find all Nash equilibria. How much is collected?
c. Assume bi = 1 for all i. Find all Nash equilibria. How much is collected?
Now suppose the manager of these 20 employees has announced that she
will contribute d > 0 dollars for each dollar that an employee contributes.
The warm glow effect to employee i from contributing a dollar is now bi X
(1 + d) because each dollar contributed actually results in a total contribution of 1 + d. Assume bi = 0.1 for i = 1, . . . , 5; bi = 0.2 for i = 6, . . . , 10; bi = 0.25 for i = 11, . . . , 15; and bi = 0.5 for i = 16, . . . , 20.
d. What value must the manager choose for d in order to get her employees to contribute $100?
e. What value must the manager choose for d in order to raise $750 in
total from both her employees and her own matching contribution?
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