Game Theory and Oligopoly Profit

Alright, so now let's take what we've learned about game theory and apply it to an oligopoly situation, right? So what we've learned so far, we've seen that in oligopoly, a firm's profit will be dependent on the decisions on the output, decisions of their competitors, right? They're gonna be dependent on each other. And this is called interdependence, right? The firms are said to be whoops interdependent. Okay, Because they depend on each other to make their best choices, right? They can't make their best choice without considering what the other people are going to do. Okay, so let's dive into an example here, we've got Jack and Jill on the only small only wells in a small town. Right? So Jack and Jill are the only producers, we've got two producers. So this is a duopoly right? A form of an oligopoly. Okay, so in our duopoly, Jack and Jill, they have no cost of pumping water, right? No cost. We're just keeping it simple. So their marginal cost is equal to zero. Average total cost is equal to zero. We're just not gonna deal with costs because it's just gonna complicate um and it wouldn't add anything to our discussion. Okay, so we've got a table here with our demand schedule and noticed we've got some quantities and prices and just like we would expect as the quantity increases the price decreases, right? Or as the price decreases quantity increases, this is what we expect with the demand curve, right? We've got the law of demand, a downward sloping demand curve. And then we have these quantities and prices and then a column for our revenue. And in this case the revenue is going to be the same as profit because there are no costs. We're just leaving that out. So let's go ahead and figure this out. So this total revenue, right? This is just our price times our quantity. Right? So all we did was multiply across. So zero times 1, 21. 0, 10 times 1, 10, 1120 times 100 was 2000. Right? So we just multiplied across our price times quantity to get our total revenues. Okay, So I want to point out some key points here. First, let's talk about the maximum revenue. Notice here when the quantity is 60, the price is 60, we have the maximum revenue of 3600. Right? And this is what a monopoly would produce right here. So whether or not you've studied monopolies, monopolies have significant market power and they can produce wherever they're gonna maximize their profit. They're the only producer in the market. So they're gonna maximize their profit just like this. Right? So they would produce here 60 quantity at a price of 60 to maximize their profit, compare that to perfect competition. Right? Remember we learned perfect competition, They reach efficiency. There's so many competitors in that market, right? There would be so many suppliers that they would keep driving the price down because there's still money to be made. So more competitors would join and drive it all the way to a situation where price equals marginal cost. Right? That's a situation that occurs in perfect competition. So, we would basically end up where there's no profit, Right? And we're supplying this maximum amount of 1 20 here. Perfect competition would reach some efficient quantity. Where are marginal benefit equals our marginal cost. Perfect competition right down here. Okay, And if you're confused, you see a price of zero. Don't get hung up on that. That's because we have no cost in this situation. We're just saying that the perfect competition is gonna be some greater quantity. That's basically the efficient quantity, right? Where that price is equal to that marginal cost anyways. Um, so what we're gonna see here in an oligopoly is, how are we gonna find what's the best situation for an oligopoly? Right. So, the first thing we have to consider is collusion, right? If the if the Jack and Jill were able to work together, well, they could essentially act as a monopoly, right? They could each produce half of the monopoly quantity, right? If Jack produces 30 Jill produces 30 well, they're just gonna split the monopoly between the two of them, Right? So that's one situation, and let's start there. Okay, So that could be probably the ideal situation for them would be something where they're colluding, Right? So, let's start here in this left column, or we have some collusion where they both say, let's produce 30 gallons to get that maximum profit in total, Right? The total maximum profit will be highest, or the total profit will be highest. Okay, so let's see what happens if they're both producing 30 gallons. Well, the total quantity is gonna be the 30 that Jack produces the 30 Jill produces, we're gonna have 60 total gallons, Right? So what's the price when there's 60 gallons on the market, 60 gallons on the market? A price of 60 Right. That's what we have in our demand schedule. So the price equals 60. Knowing this, we can calculate their profit, right? We've got no costs. So their revenue will be their profit. In this case. Let's go ahead and check that out. So Jack's profit is gonna be the 30 units he sells times the $60 price. Well, that's going to give him 30 times 60, 1800 in profit. Okay, so Jack's profit is 1800. Jill also sells 30 units at a price of 60, and she also gets 1800. Right? So, let's go underneath this table right here, let's do a column for a row for total profit. Right? What's the total profit in between the two of them? Well, that's the sum of the 1800 plus the 1800. That's gonna give us, uh, 3600 in total profit, right? 3600 total profit. And guess what, that's equal to our monopoly profit right here, right? That's what we expected. They took that monopoly profit and split it down the middle by each, producing, 30 gallons. Right? But remember when were colluding, there's that incentive to cheat, right? Jill can consider expanding her output knowing that Jack plans on cooperating, Jill can say, hey, if instead of producing 30 maybe I can make more money if I produce 40 gallons instead. Okay, so let's see what happens if Jill decides to cheat. Right? So this is gonna be a situation where Jill cheats on their collusive agreement to each produce 30. So now, let's see what happens. Jack produces 30, Joe produces 40. Well, that's gonna be a total quantity of 30 from Jack, 40 from Jill gets us to a total quantity of 70. Right? So now the price is gonna change, right? There's a higher quantity on the market. There's gotta be a lower price to clear that. And what do we see when there's a quantity of 70? Our price is 50 over here. Right, So the price now is 50 Jill produced more than the collusive agreement, and now it dropped the price down to 50. Okay, so let's see what happens in this situation. Jack, uh, still produced 30 gallons. Right? So Jack's profit is gonna be the 30 gallons times this price of 50 Right? He's no longer getting $60 per gallon, he's getting 50 And he's gonna have 1500 in profit. 30 times 50 is 1500. What about Jill? Well, Jill produced 40, right? Her quantity is 40 that she sold at this $50 price. And guess what? Jill comes out with $2,000 in profit. Jill got more profit by cheating here. Right? So this is that incentive to cheat that we were talking about by increasing her quantity. She took a little bit from jack and now she has a higher profit. But notice our total profit in the industry, right? Total profit is going to be the 1500 plus the 2000. Well, total profit is now 3500. Right? The total profit decreased. There was a better situation for the producers in total when they colluded and there was 3600 in total profit. Right? And notice at a quantity of 70 we've got a price of 50 and profit of 3500. Just like we saw there except now it's split between Jack and Jill 3500. Right? So now what about this final situation? Jill thought about cheating. What if at the same time Jack had thought about cheating too. They both considered cheating. Alright, so now we're in this final column, we're both cheat and they both, instead of producing the 30 collusive uh the collusive agreement of 30 they're gonna each produce 40. So what's our total quantity? Now we've got 40 plus 40 equals 80 in total quantity. Right. What what price do we have when 80 is the quantity on the market? Well here's 80 A price of 40 when there's a quantity of 80. Right? So we're gonna have a price of 40. So let's check this out. What's gonna happen to profit now? Okay, let me get out of the way here, so we can fill in these boxes. Well, we've got Jack's profit, right? Jack produced 40 units and he's selling them at a price of $40. And he's gonna get $1600 right? 40 times 40 is 1600 and the same thing for Jill, right? 40 units at $40. She's also gonna get 1600. So what's the total profit in this case? Well, the total profit is 1600 plus 1630. 200. So notice that total profit for the industry keeps decreasing. And in this case where they both cheat, we end up where both of them are worse off. Right? If they had colluded, they would have both had 1800. If they both cheat, they're both down to 1600. What does this sound like to you? This is something we've gone over before. Right? This sounds like a prisoner's dilemma, Right? So let's pause here and then in the next video, we'll fill out a payoff matrix using this information and see how the whole game theory uh relates to everything we just talked about. Alright, let's do that. Now

Alright guys, so now we're here at the bottom of the page in our payoff matrix. So let's go ahead and take all that information that we just gathered in our different situations, and let's put it into our payoff matrix. Okay, So let's start with the situation where they were colluding, right? When they both produce 30 gallons. Jack's decision was to produce 30 gallons. Jill's decision was to produce 30 gallons. Well, what was their Prophet? Jack had 1800 in profit, and Jill had 1800 in profit. Right? So that notice how we're filling out a payoff matrix, using this information that we just came up with, we're gonna have those decisions. How much are they going to produce? And the payoffs based on those decisions, Okay, how about that second situation where Jack produces 30 and Jill produces 40? Well, when Jack produces 30 Jill produces 40. That was a situation where Jill's taking some of Jack's revenue, right? And coming out on top. So Jack's producing 30 Jill's produce 40. That's his bottom box down here. And we see that Jack's income or profit will be 1500. In that case, Jill's profit will be 2000, Right? That's what we calculated up here, uh for Jack and Jill's profit, when Jack's pros produces 30 and joe produces 40? Well, it would be the opposite, right? What if Jack produced 40 and Jill produced 30? We would have a similar situation, it would just be flipped, right? In that case, when Jack produces 40 and Jill produces 30. Jack comes out with the 2000, Right? And Jill gets 1500 over here, right? Because we're just flipping who, who was the cheater And then last but not least. We've got the situation where they both cheat and both produced 40 gallons, right? So that's gonna be this box right behind me in the bottom right corner, and we had profits over here, right? We calculated Jack and Jill's profit to be 1600 each. So that would be 1600 uh for Jack and 1600 for Jill. Right? So we filled in our payoff matrix. I'm gonna go ahead and get out of the way and let's use our check and X method to find dominant strategies and a nash equilibrium. Okay, let's check it out. Now, oops, wrong button. Here we go. Alright, so let's do this. Let's start with Jack. Okay, so for Jack's decision, he has to think, what would he do based on Jill's decisions, Right? So Jill decides to produce 30 gallons Jack, what will Jack do when Jill produces 30 gallons What we saw, Right? Jack wants to cheat, he's gonna get more more profit when he cheats, right? He'll produce 40 gallons and get 2000 rather than 30 gallons and get 1800. So, we're gonna put a check over here for Jack's decision when, when joe produces 30 what about when joe produces 40? If joe produces 40 Jack either has to choose between 1500 or 1600 right? He can produce 30 and get 1500 or produce 40 And get 1600. So his better choice is to produce 40, right? So what do we see? We already see that Jack has a dominant strategy to produce 40, right? We have this column right here with two check marks. So we know that's a dominant strategy for jack. Okay, let's try Jill now. So, Jill, she's first gonna think, what if Jack produces 30 so if Jack produces 30 I can produce 30 and get 1800 right here, or Jack can produce 30 and I'll produce 40 and get 2000. So Jill is gonna want to cheat, right? Jill will cheat and produce 40 and that is why we're gonna put an X in that box. What about if Jack produces 40 again, the same thing? If Jack produces 40 Jill has to choose between 1500 and 1600. Well, guess what Jill is gonna pick, Jill's gonna pick the 1600 right? Rationally, she's gonna pick the one with more. And what do we see? We've got two exes in the same row, right? So now that we've got two exes in the same row, we know that that's also a dominant strategy, right? So Jax dominant strategy, I'm just gonna put dominant, Jax, dominant strategy is 40 gallons, right, Jill's dominant Is also 40 gallons. And what about our nash equilibrium? Our nash equilibrium is a box that has both a check and an X and we do have a box there, right? There is a box with a check in an X right here, that is our nash equilibrium are nash our nash equilibrium, that's a bad area. Let me try that again. Nash equilibrium is going to be this box right here, right? Where they're both earning 1600 right? So they're both earning 1600 there. Let me write equilibrium below. Oh man, I'm a mess right now, let's let's get that in right down here. Alright, nash equilibrium, because I'm not on the screen, I don't have that pressure to to do it. All right, the first time. Right, okay, nash equilibrium in that box. Right, So I'm coming back in now. So what do we see the nash equilibrium was something like a prisoner's dilemma, right? If they had been able to cooperate, they would have both made 1800 but they both end up cheating and end up in this situation where they earned 1600. Right? And that's because they're making their best choices based on what their competitors would do. So what does that do? It leads us to a situation where an oligopoly is producing somewhere around here, right? This is gonna be the oligopoly quantity. Somewhere in between the monopoly and perfect competition. And this makes sense, right, because the monopoly, there's no competition at all. Where the only producer, So we have the maximum amount of profit. Oligopoly. We see already just with a few firms, even just to that extra firm, that extra layer of competition already pushes us away from that maximum profit that a monopoly can get right. So this little bit of competition, uncertainty what your opponent's gonna do leads us to this higher quantity and gets us closer to the efficient quantity, Right? So that's that's how you have it. The oligopoly is gonna be producing somewhere between the monopoly and perfect competition because of this interdependence and uncertainty of what your opponents are gonna do. Cool, Alright, let's go ahead and move on to the next video.