Question

consider battle of networks game with 3 networks instead of 2. the payoff matrices represent market share (%) for each network depending on whether they show sit come or sports events. solve pure strategy.

N3-Sitcom

N1/N2	Sitcom	Sports
Sitcom	50,40,10	42,38,20
Sports	35,35,30	40,50,10

N3-Sports

N1/N2	Sitcom	Sports
Sitcom	45,35,20	40,36,24
Sports	40,40,20	30,40,30

Answer 1

We can find the Nash equilibrium of this game by underlying the payoff approach.

for player 1

Let's start underlying the payoff for player 1, when player 3 has played strategy SITCOM that is to say left hand side of the payoff matrix. When player 2 plays SITCOM player 1 plays SITCOM hence underline the 50 and when player 2 plays SPORTS player 1 plays SITCOM hence underline 42.

when player 3 has played SPORTS that is to say right hand side of the payoff matrix, when player 2 plays SITCOM player plays SITCOM hence underline the 45 and when player 2 plays SPORTS player 1 plays SITCOM hence underline 40.

for player 2

Let's start underlying the payoff for player 2 ,when player 3 has played SITCOM that is to say left hand side of payoff matrix. When player 1 plays SITCOM player 2 plays SITCOM hence underline 40 and when player 1 plays SPORTS, player 2 plays SPORTS hence underline 50.

Now when player 3 has played SPORTS that is to say the right hand side of the payoff matrix, when player 1 plays SITCOM player 2 plays SPORTS hence we underline 36 and when payer 1 plays SPORTS player 2 is indifferent between playing either SPORTS or SITCOM hence we underline 40 in both columns.

for player 3

Now we need to check for the best strategy for player 3. when player 1 and 2 both have played SPORTS player 3 has two options either to play SPORTS or SITCOM that is to say right or left side of the payoff matrix. So, when player 1 and 2 both plays SPORTS player 3 chooses to play SPORTS hence we underline 30 on the right hand side of the payoff matrix where player 1 and 2 both plays SPORTS.

when player 1 has played SPORTS and player 2 has played SITCOM, player 3 will play SITCOM that is the left hand side of the payoff matrix hence we underline 30.

When player 1 has played SITCOM and player 2 has played SPORTS, player will play SPORTS that is the right side of the payoff matrix hence we underline 24.

When player 1 and player 2 both have played SITCOM, player 3 will play SPORTS that is to say the right side of the payoff matrix hence we underline 20.

And now we can clearly see the NASH equilibrium strategy profile is the one whose all payoff gets underlined in other words it's the best response for every player to other player's best strategy. And in this game the Nash equilibrium strategy profile is when player 1 plays SITCOM and player 2 and 3 plays SPORTS. (40,36,24).