Question

Nash Equilibria with Multiple Players.

Hi,

I'm painfully struggling with practicing multiple nash equilibria problems.

The attached below are conditions and questions.

Please help me understand the concepts of it

- There are "n" investors - Each one has the option of investing $1million into a project. - project is funded if and only if it receives strictly more than $(k-1) millions ->in this case, the project revenue is $"m" millions, independent of how many investors decided to invest on it - The revenue is equally splitting among all investor who did invest. - If project is not funded, project fails and those who invested lose all their money - Investor who decided to not invest always keep their money. Questions 1. Assuming m>n > k, find all Nash Equilibria 2. Assuming n > m> k, find all Nash Equilibria

Answer 1

Given,

Number of investors = n

Project revenue = m

Project funding = Funding per investor X n = 1 million X n = n million

Revenue obtained per investor = r = m / n

For project to be funded, funding must be atleast k-1, going by assumptions given in both parts of the question, we see that n > k, hence, project funding is greater than k million, hence project will not fail and investors wont lose their money

For your knowledge, Nash equilibrium, in theory, is an equilibrium solution in which all the players ( read investors) are better off with a certain strategy, not necessarily it is the most optimum strategy for them as an individual.

Now answering the questions using simple logic,

Part 1

On the assumption m > n > k, say m= 10, n = 9, k=8. Now let us define 2 strategies

Strategy A: If all 9 investors invest, project funding would be 9 X 1million, i.e., 9 million which is greater than (k-1) = 8 millions. Project revenue split among all investors will be m / n millions = 1.11 million. Hence each investor would make a profit of 1.11 - 1 million = 0.11 million.

Strategy B: Now lets reduce k to 5. As discussed, project funding must be greater than (k-1) = 4 million and as each investor can invest 1 million, n >=5. Assuming n = 5, project revenue split among all investors will be m / n millions = 10 / 5 million = 2 million. Hence each investor would make a profit of 2 - 1 million = 1 million.

Note that strategy B is more profitable to each investor (infact it is the maximised profit solution), but as there is no restricted entry, investors would keep on joining in until m >n which occurs at n =9, beyond which the assumption would stop holding true. Hence Strategy A would be the only Nash Equilibrium solution.

Part II

On the assumption n > m > k, say m= 9, n = 10, k=8. Now let us consider the 2 strategies

Strategy A: If all 10 investors invest, project funding would be 10 X 1million, i.e., 10 million which is greater than (k-1) = 8 millions. Project revenue split among all investors will be m / n millions = 0.9 million. Hence each investor would make a profit of 0.9 - 1 million = - 0.1 million ( read loss of .1 million).

Strategy B2: Now lets increase n to 12. As discussed, project funding must be greater than (k-1) = 7 million and as each investor can invest 1 million. Project revenue split among all investors will be m / n millions = 9 / 12 million = 0.75 million. Hence each investor would make a profit of 0.75 - 1 million = -0.25 million ( read loss of 0.25 million).

Note that each of the above strategy A and B2 would lead to a loss which decreases as gap between n and m reduces. But the point is, every strategy would lead to loss. Hence, none of the investors should be interested to invest, and hence the Nash Equilibrium would be at n =0 which means project would not be executed.