In an isolated town two pizza shops open. Their owners Mario and Luigi are deciding what price to set on the menu. Before the opening night of both stores they discuss their pricing and work out that a price of $15 is going to bring the greatest benefit to them. Both know that there are 100 potential customers on any given night and that if they decrease their price to $14 then everyone would buy from their shop. Pizza's cost $10 to produce.

a) Draw the payoff matrix of the game for the first day of opening

b) What type of game does this represent? What are the key features (for example: Nash Equilibrium, Dominant Strategies)?

Changing the menu prices the night before open takes considerable time and so they would need to hire an additional casual staff member to assist with preparations in the morning costing $170.

c) Reconstruct the payoff matrix of the game using the above information

d) Do any of the key features change?

Reconsider the original case, now allow for a certain proportion of consumers to be loyal to one of the pizza shops (both have the same number of loyal customers). A loyal customer will always buy from their favourite pizza shop, even if the price is higher.

e) What proportion of people need to be loyal in order for the socially optimal outcome to be the only Nash Equilibrium in this game?

f) Is it possible to have a coordination game with any proportion of loyal customers? What proportion of customers need to be loyal in order for this to be a co-ordination game? Is it possible in the case where the cost of hiring the additional casual staff member to change prices is $100 instead of $170? If so for both, how do these two levels compare?