Game theory worked example from A P Microeconomics

What we have here is a free response question that you might see on an AP Microeconomics type exam that deals with game theory. It tells us Bread Basket and Quick Lunch are the only two sandwich shops serving a small town, so we're in an oligopoly situation where we only have a few firms. Each shop can choose to set a high price or a low price for sandwiches. The payoff matrix below shows the daily profits for each combination of prices that the two shops could choose. The first entry shows Bread Basket's profits and the second entry shows Quick Lunch's profits.

Assuming that both shops know the information shown in the matrix, answer the following. So just to make sure I understand what's going on here, this is saying that for example, if Bread Basket can either choose to charge high prices or low prices, Quick Lunch can either choose to charge high prices or low prices. If Bread Basket chooses high prices and Quick Lunch also chooses high prices, then what this tells us is the first one is Bread Basket's profit per day would be 105 dollars while Quick Lunch's profit per day would be 110.

And then, this is a situation where Bread Basket is low and Quick Lunch is high price, and then Bread Basket would make 120 and Quick Lunch would make 80. And then, when Quick Lunch has low prices, we can see what are the profits of either when Bread Basket charges high prices or Bread Basket charges low.

So with that out of the way, let's try to answer these questions: does each shop have a dominant strategy to set a high price, a dominant strategy to set a low price, or does it have no dominant strategy? So pause the video and try to figure that out and just as a bit of a hint or reminder, a dominant strategy is a strategy that, regardless of what the other player does, you would still be better off to make that choice.

So a dominant strategy of setting a high price would be that regardless of whether the other player decides to set a high or low price, a high price would always make sense for you. So pause the video and try to see if you can answer that.

All right, now let's see what Bread Basket's situation is here. So if we think, Bread Basket of course can either choose to go high or low. Now if Quick Lunch goes high, what should Bread Basket do? Well then we are in this column over here and these two numbers are for Bread Basket. So if Quick Lunch goes high, then Bread Basket could go high and charge 105, or it could go low and charge 120. So if Quick Lunch goes high, Bread Basket should go low.

Now, what happens if Quick Lunch chooses to go low? Well if Quick Lunch chooses to go low, the two options for Bread Basket are either 40 dollars profit per day if they have a high price, because a lot of their business would go to Quick Lunch in that situation, or they could go low price and make 75 dollars per day. So even if Quick Lunch goes low, it still makes sense for Bread Basket to go low.

So Bread Basket goes low. We see that Bread Basket, no matter what Quick Lunch does, it makes sense for Bread Basket to go low. So Bread Basket has a dominant strategy to set a low price. So dominant strategy to set... let me write that a little bit clearer... set low price.

All right, now Quick Lunch. So we could do the same type of analysis based on what Bread Basket might choose to do. So if we see, okay, if Bread Basket goes high, what should Quick Lunch do? Well, let's see if then Quick Lunch should go high or low. It would have an option of 110, or it would have an option of 130. So if Bread Basket goes high, Quick Lunch is better off going low.

So let me write that down: Quick Lunch should go low. And if Bread Basket goes low, what should Quick Lunch do? So if Bread Basket goes low, Quick Lunch can either make 80 if it goes high, or 70 if it goes low. So in this situation, if Bread Basket goes low, it makes sense for Quick Lunch to go high and make the 80.

So Quick Lunch would go high. So Quick Lunch has no dominant strategy. It doesn't make sense for them to always go low or always go high, regardless of what Bread Basket is doing. Depending on Bread Basket, it might make sense for them to go low if Bread Basket goes high or high if Bread Basket goes low. So no dominant strategy.

If the two shops do not cooperate on setting prices, what will be the profit for each shop? Well actually, just pause the video, see if you can answer that before we work through it. Well, we already know that Bread Basket has a dominant strategy to set a low price. So Bread Basket is going to go low regardless.

So that means we’re going to end up in this row right over here, and if we are in that situation where Bread Basket goes low, we know that it makes sense for Quick Lunch to go high because they can either make 80 or 70. Well, they're going to go high, and so we're going to end up in this column right over here.

So if the two shops do not cooperate on setting prices, the profit of each shop would be: Bread Basket would be making 120 dollars a day and Quick Lunch would be making 80 dollars a day. This is a Nash equilibrium, because on its own no firm can change its decision to optimize its prices more.

If you are in this cell right over here, Quick Lunch can't change Bread Basket's decision. Quick Lunch could say, "I'm either going to make 80 or I'm going to make 70." So it wouldn't make any sense for them to switch away. And then, Bread Basket can't make the decision for Quick Lunch; they can say, "Hey, we're either going to make 120 or we're going to make 105."

And so since we can't change what Quick Lunch is doing, of course, we're going to choose to make 120. So people will stay in this bottom left cell; we're in Nash equilibrium.

All right, so the next part. They ask us, or they tell us, the town government is concerned that food prices are too high. It decides to give a daily subsidy of twenty dollars to any shop that chooses to set a low price for its food items. Redraw the payoff matrix under the government subsidy system.

So like always, pause this video and go through that exercise; it'll be interesting. All right, now let's do this together. So we have Bread Basket... I'll try to write Bread. Actually, let me draw my little matrix first.

So it's a two by two. So almost done with my matrix. One more... that wasn't near the middle. There you go, right around there. Still maybe there. And then we have that there, and then we have high, low, high, low. And then you have Quick Lunch.

If I were actually doing this on the test, maybe I would write this a little bit lower so it’s a little bit neater. And then this is Bread Basket. And let’s see, the government is giving a daily subsidy of 20. So we could just do that; that adds to the profit of a firm that is selling at low prices.

So if they're both at high prices, they’re not going to be able to get that subsidy from the government. So we’re still going to be at 105 and 110 for Bread Basket and Quick Lunch respectively. Now if Bread Basket stays high in this situation, well then they’re still only going to make 40 dollars, but in this situation right over here, Quick Lunch is going to get a 20 dollar subsidy because they are choosing to go low.

So they’re going to make 130 plus 20. So I’ll write that in a different color. I would put that in and I’ll put this in red. So 130 plus 20, they're going to make 150 because they're choosing to go low price here; that’s what qualifies you for the subsidy.

And in this situation, if Bread Basket goes low price, they're going to make 120, what they would have made in that situation, plus 20. So this is going to be 140, but this is a situation where Quick Lunch is charging a high price, so they're not going to get the subsidy. So they're still going to make 80.

And then, this sells both of them charging a low price, so they're both going to get 20 more than what you see right over here. So Bread Basket would make 95 dollars, and Quick Lunch is going to make 90 dollars in this situation.

And let’s see, they ask us more questions. They say using your redrawn payoff matrix, answer each of the following: Would Quick Lunch choose to set a high price or low price? Explain using specific values from your redrawn matrix.

So pause the video and see if you can answer that. All right, so we’re going to look at Quick Lunch now. We could do the same type of analysis to see if they have a dominant strategy. We could say, okay, if Bread Basket goes high, what is Quick Lunch going to do?

So if Bread Basket goes high, now in this situation Quick Lunch will either go high and make 110, or could go low and make 150. So Quick Lunch is going to go low in this situation. And if Bread Basket goes low, Quick Lunch can make 80 if it goes high, or can make 90 if it goes low. So once again, Quick Lunch is going to go low.

So Quick Lunch now has a dominant strategy for low price. I could try to explain this out: if Bread Basket goes high, I’ll write it out here. Normally you have a little book to write this in, but if Bread... I’ll abbreviate. If Bread Basket goes high, Quick Lunch would rationally make 150 instead of 110.

Would want to make 150 versus 110 by going low. If Bread Basket goes low, Quick Lunch makes 90 dollars versus 80 dollars by going low.

All right, would the profits for Bread Basket increase, decrease, or stay the same? Explain with a comparison to your answer in part B1 using specific values. So before the subsidy, actually pause this video and try to do this as well. Before the subsidy, Quick Lunch, or I should say Bread Basket, was making... and we have it up here.

So if the two shops do not cooperate on setting prices, what will be the profit of each shop? We saw Bread Basket would make 120 dollars because that was that Nash equilibrium state. After subsidy, after subsidy, so let's look at it. We still have Bread Basket still has a dominant strategy to go low.

Because if Quick Lunch goes high, Bread Basket would want to make the 140 instead of 105. And if Quick Lunch goes low, Bread Basket would want to make 95 instead of 40. But now they both have a dominant strategy for going low, so we're going to end up in this bottom right with this bottom right cell after subsidy.

Both have a dominant strategy to go low resulting in 95 profit for Bread Basket. So 95 is less than 120, so Bread Basket's profit goes, or I should say decreases. Decreases. And we are done.