Game Theory: Analyzing Advertising Strategies of Three Companies

Three companies, C1, C2, and C3, put three items on the market and advertise them either on morning or evening TV. A company advertises exactly once a day. If more than one company advertises at the same time, their profits that day are zero. If exactly one company advertises in the morning, their profits are $200K. If exactly one company advertises in the evening, their profits are $300K. The companies must make their advertising decisions simultaneously.

Write down two payoff tri-matrices, one for the case C3 chooses to advertise in the morning and another for the case C3 chooses to advertise in the evening. Find all the pure Nash equlibria and the corresponding payoffs for C1, C2, and C3.

Payoff matrix when C3 advertises in the morning:

| | C1 (morning) | C1 (evening) | C2 (morning) | C2 (evening) | |--------|-------------|-------------|-------------|-------------| | C2 (m) | 0 | 0 | 0 | 0 | | C2 (e) | 0 | 0 | 0 | 0 | | C1 (m) | 200 | 0 | 0 | 0 | | C1 (e) | 0 | 0 | 0 | 0 |

Payoff matrix when C3 advertises in the evening:

| | C1 (morning) | C1 (evening) | C2 (morning) | C2 (evening) | |--------|-------------|-------------|-------------|-------------| | C2 (m) | 0 | 0 | 0 | 0 | | C2 (e) | 0 | 0 | 0 | 0 | | C1 (m) | 0 | 0 | 0 | 0 | | C1 (e) | 0 | 0 | 300 | 0 |

To find the pure Nash equilibria, we need to look for cells where each player has chosen their best strategy given the other players' choices. In this case, there are no pure Nash equilibria in either matrix.

In the first matrix, if C1 advertises in the morning, C2 is better off advertising in the evening, and vice versa. If C2 advertises in the morning, C1 is better off advertising in the morning, and if C2 advertises in the evening, C1 is better off not advertising at all.

In the second matrix, if C1 advertises in the evening, C2 is better off not advertising at all, and if C2 advertises in the evening, C1 is better off not advertising at all. If C1 advertises in the morning, C2 is better off advertising in the morning, and vice versa.

Therefore, there are no pure Nash equilibria in this game.