Matching Grants

A philanthropist wants to raise $3,000,000 for his favorite charity. Though he is quite

wealthy, this sum is too much even for him. In order to spur others to contribute, he

establishes a matching grant whereby he will donate $1,000,000 if $2,000,000 is raised

from other donors. Anything less than $2,000,000 and he will contribute nothing.

Suppose that there are 10 prospective donors who are simultaneously deciding how much to

contribute. Assume that a donor's possible strategies are all real numbers between 0 and

$500,000, measured in dollars. The payoff of a donor is:

1/5 x (Total of Contributions) - Own Contribution

The first term in the payoff is the benefit derived from the money going to a worthy cause

(note that the coefficient 1/5 measures how much the donors care about the cause), and

depends on the contributions of all donors. The second term is the personal cost of making

a contribution.

Suppose that there is no matching grant. Is there a dominant strategy? If so,

find it, if not, explain why not. What will happen in this situation?