Question

Consider a game in which, simultaneously, player 1 selects a number x ∈ [0, 8] and player 2 selects a number y ∈ [0, 8]. The payoffs are given by: u1(x, y) = 2xy − 8x − x 2 u2(x, y) = 4xy − y 2 (a) Calculate and graph the best responses for each player. (b) What strategies are never played? (c) Find all Nash equilibriums? (d) What is the preferred equilibrium in society? (e) How would we classify this type of problem?

Answer 1

Solution:

Given payoff functions as: for player 1: u1(x,y) = 2xy - 8x - x²

For player 2: u2(x,y) = 4xy - y²

With x,y belonging to [0,8]

a) Calculating the best response functions for each player, using the first order conditions (FOCs): = 0 and = 0

= 2y - 8 - 2x

So, FOC gives us 2y - 8 - 2x = 0, giving us best response function of player 1 as x*(y*) = y* - 4 ... (i)

Similarly, = 4x - 2y

So, FOC gives us 4x - 2y = 0, giving us best response function of player 2 as y*(x*) = 2x* ... (ii)

Plotting the two, we obtain the following graph:

b) From the graph, we can see that the strategies never played are the ones belonging to the grey area:

Thus, the strategies never played include: Player 1 will never choose a point beyond 4, so optimum x belongs to [0, 4]

c) Nash equilibrium is the point where the two best response curves intersect. From the graph, it is clear that this happens for x* = 4, and y* = 8. So, (x*, y*) = (4, 8) is the Nash equilibrium.

Mathematically, we can simultaneously solve the two equations: (i) and (ii). By substituting (i) in (ii), we get

y* = 2(y* - 4)

y* = 2y* - 8

y* = 8

So, x* = y* - 4 = 8 - 4 = 4

d) Preferred equilibrium in society occurs where both players' payoffs taken together are being maximized.

Total/societal payoff = u1(x,y) + u2(x,y)

TP = 2xy - 8x - x² + 4xy - y² = 6xy - 8x - x² - y²

Now solving the FOCs as follows: = 0 and = 0

= 6y - 8 - 2x

Using FOC, we have 6y - 8 - 2x = 0

So, 3y - x = 4 ... (1)

And, = 6x - 2y

Again using FOC, we have 6x - 2y = 0

So, y = 3x ... (2)

Solving (1) and (2) by substitution, we get 3(3x) - x = 4

8x = 4

Or, x = 1/2

So, y = 3(1/2) = 3/2

So, preferred equilibrium in society is (x', y') = (0.5, 1.5)