Game Theory Worked Example: AP Microeconomics Payoff Matrix

[00:00:00]what we have here is a free response question that you might see on an ap microeconomics type exam that deals with game theory and 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 baskets profits and the second entry shows quick lunches 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 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 the if bread basket chooses high prices prices and quick lunch also chooses high prices then what this tells us is the first one is bread baskets profit per day would be 105 dollars while quick lunches 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 profit 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 of 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 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 baskets 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 and so quick lunch goes high then bread basket could go high in charge 105 or it could go low and charge 120. so if quick lunch goes high bread basket should go low so let me write this quick lunch quick lunch if it chooses to go high then 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 a 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 so 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 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 so if bread basket goes high quick lunch should it go high or low so 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 actually 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 lower 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 dominant strategy if the two shops do not cooperate on pri 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 we're going to end up in this row right over here and if we are in that situation where bread basket goes low regardless 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 a day and quick lunch would be making 80 a day and this is a nash equilibrium because on its own no firm can change its decision to optimize its prices more because if you are in this if you are in this cell right over here quick lunch can't change bread baskets decision so 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 breadbasket they 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 well 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 and 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 the it's 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 that's not 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 forty dollars but in this situation right over here quick lunch is going to get a 20 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 so i would put that in and i'll 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 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 ninety five dollars ninety five dollars and quick lunch is going to make ninety 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 and 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 goes high what is quick lunch going to do so if bread basket goes high now in this situation quick lunch will either oh hi 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 quick lunch now has dominant dominant strategy for low price for low price and i could try to explain this out if 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 bread basket goes high quick lunch what is rational to make 150 instead of 110 would want to make 150.

[00:11:00]versus 110 by going low if if bread basket goes low goes low quick lunch makes ninety dollars versus versus eighty 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 use specific values so before subsidy actually pause this video and try to do this as well before subsidy and i'll do it together before subsidy quick lunch or i should say bread basket 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 is 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 dominant dominant strategy to go low resulting in 95 profit for bread basket and so 95 is less than 120 so bread baskets profit goes or i should say decreases decreases and we are done