This file concerns problem 4 of Barry's assignment 4

          Note that we use the variables p and q rather than x and y



                The pay-off of player 1  is, 4 p q - 2 p - q + 3

                        Let's look at the Best response

                Our function of , p, q, is , 4 p q - 2 p - q + 3

We have to find, for every, q, between  0 and 1 the value(s) of, p,

    that will make it largest

 The coefficient of p is the following expression linear function of q, 4 q - 2

               The expression, 4 q - 2, has a zero when , q = 1/2

when q is striclty less than, 1/2,

    this expression is negative, hence the best response is p=0

when q equals, 1/2,

    this expression is zero, hence every possible p is a best response

when q is strictly larger than , 1/2,

    this is positive, hence the best response is p=1

               The pay-off of player 2  is, -4 p q + 2 p + q - 3

                        Let's look at the Best response

               Our function of , q, p, is , -4 p q + 2 p + q - 3

We have to find, for every, p, between  0 and 1 the value(s) of, q,

    that will make it largest

The coefficient of p is the following expression linear function of q, -4 p + 1

              The expression, -4 p + 1, has a zero when , p = 1/4

when q is striclty less than, 1/4,

    this expression is positive, hence the best response is p=1

when q equals, 1/4,

    this expression is zero, hence every possible p is a best response

when q is strictly larger than , 1/4,

    this expression is negative, hence the best response is p=0

                   Hence the Nash equilibrium is, [1/2, 1/2]