In a game with eight players, each player has x > 10 coins. Each player also cho

Question

In a game with eight players, each player has x > 10 coins. Each player also chooses the number of coins to contribute to a 'common fund'. The remaining coins are deposited into a 'bank'. Each coin contributed to the common fund reciveves 23 - 0.25a in return (where 'a' is the total number of coins contributed to the fund). Each coins contributed to the 'bank' gets a return of 5. The utility function of each player is the total return from all x tokens.

Formulate this as a simulteaneous-move game, and find the total amount contributed to the common fund in the unique symmetric (same contribution for each player) pure strategy Nash equilibrium of the game. Why is this outcome not socially optimal?

Explanation / Answer

Each player has x coins.

Let x1, x2, x3, x4, …, x8 refer to coins contributed to common fund by player 1, 2, 3, … , and 8, respectively.

Player 1’s return is-

            1         =          x1[23 – 0.25(x1 + x2 + x3 + x4 + … + x8)] + 5(x – x1)

            1         =          23x1 – 0.25x1 x1 – x2 x1 – x3 x1 – x4 x1 – … – x8 x1 + 5x – 5x1

            1         =          18x1 – 0.25x1 x1 – x2 x1 – x3 x1 – x4 x1 – … – x8 x1 + 5x

Equate to zero the derivative of the above function with respect to x1.

            18 – 0.5x1 – x2 – x3 – x4 – … – x8       =          0

            x1 =      36 – (x2 + x3 + x4 + … + x8)/2

So player 1’s best response function is

            x1 =      36 – (total of what all players except player 1 contribute)/2

Similarly, the response functions of other players are:

            x2 =      36 – (total of what all players except player 2 contribute)/2

            x3 =      36 – (total of what all players except player 3 contribute)/2

            x4 =      36 – (total of what all players except player 4 contribute)/2

            x5 =      36 – (total of what all players except player 5 contribute)/2

            x6 =      36 – (total of what all players except player 6 contribute)/2

            x7 =      36 – (total of what all players except player 7 contribute)/2

            x8 =      36 – (total of what all players except player 8 contribute)/2

Add all the eight response functions.

   x1 + x2 + x3 + x4 + … + x8    =          36(8)    – [(7x1 + 7x2 + 7x3 + 7x4 + … + 7x8)/2]

                                    a          =          288      – (7a/2)

                                    a          =          64

Therefore, total contribution is 64 coins, and each player contributes 8 coins.

This outcome is not socially optimal because here individual returns were maximized on the assumption that everyone was maximizing one’s own return. In a socially optimal outcome, the combined returns of all will be maximized.

Solve for the socially optimal outcome.

Given 8x coins, suppose a is the number of coins that are invested in the common fund and the rest of the coins are invested in the bank.

            Social returns =          a[23 – 0.25a] + 5(8x – a)

            s         =          23a – 0.25a2 + 45x –5a

            s         =          18a – 0.25a2 + 45x

Equate to zero the derivative of the above function with respect to a.

            18 – 0.5a + 45             =          0

            a = 126

Therefore, under socially optimal outcome a total of 126 coins will be contributed to the public fund.

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