In a one-shot game, if your rival advertises and you advertise, he will earn $7 million and you will earn $2 million in profits. If neither of you advertise, you will make $4 million and he will make $2 million. If you advertise and your rival does not, you will make $3 million and your rival will make $8 million. If your rival advertises and you do not, he will make $1 million and you will make $3 million
. Write the above game in normal form.
b. Do you have a dominant strategy? If yes, what is it?
c. Does your rival have a dominant strategy? If yes, what is it?
d. What is the Nash equilibrium for the one-shot game
a. The game can be represented in normal form as follows:
| Your Rival
| Advertise | Don't Advertise
--------------------|----------------------
Advertise | (2, 7) | (3, 8)
---------------------|----------------------
Don't Advertise | (3, 1) | (4, 2)
b. A dominant strategy is a strategy that results in the highest payoff for a player, regardless of the other player's strategy. To determine if you have a dominant strategy, you need to compare your payoffs in each row and choose the strategy that maximizes your payoff regardless of your rival's choice.
Looking at the first row, if your rival advertises, your payoff is 2 if you advertise and 3 if you don't advertise. Since 3 is higher, not advertising is your dominant strategy in this case.
In the second row, if your rival doesn't advertise, your payoff is 3 if you advertise and 4 if you don't advertise. Since 4 is higher, not advertising is again your dominant strategy in this case.
Therefore, not advertising is your dominant strategy regardless of your rival's choice.
c. To determine if your rival has a dominant strategy, you need to compare their payoffs in each column and choose the strategy that maximizes their payoff regardless of your choice.
Looking at the first column, if you advertise, your rival's payoff is 7 if they advertise and 1 if they don't advertise. Since 7 is higher, advertising is your rival's dominant strategy in this case.
In the second column, if you don't advertise, your rival's payoff is 8 if they advertise and 2 if they don't advertise. Since 8 is higher, advertising is again your rival's dominant strategy in this case.
Therefore, advertising is your rival's dominant strategy regardless of your choice.
d. The Nash equilibrium is a combination of strategies where neither player has an incentive to unilaterally deviate from their chosen strategy. In this game, the Nash equilibrium occurs when both you and your rival choose your dominant strategies.
The Nash equilibrium for this one-shot game is when neither of you advertises (Don't Advertise, Don't Advertise) which results in a payoff of (4, 2) for both players. In this scenario, both players have chosen their dominant strategies and have no incentive to switch since any other combination would result in a lower payoff for at least one player.