Okay. Now let me explain how those two gas stations can maintain high price in long-term relationship. Okay. So high price may be sustained in long-term relationship by the following kind of consideration. Okay. So, well, I can undercut the price today to steal customers from the, you know, next gas station, and I can certainly increase my payoff today. But if I do that, that may trigger a fierce price competition in the future. And after all that is not good for me, so let me just stick to high price. This could be a very plausible mechanism. And I'm going to show you how to formulate this basic idea in the formal model of repeated game played by two gas stations. Okay, so here is a strategy in repeated game that formalizes these basic, plausible considerations. Okay. So you start with high price, $3 per gallon, and if you keep on charging the same high price, $3 per gallon, if nobody has deviated before. Okay. But if anyone has deviated from $3, then you set price equal to cost. Cost is $2. So you apply this $2 forever. Okay. So this is one strategy which formalizes the basic orientation. Okay, so let's examine the nature of this strategy. What happens if two gas stations follow this strategy, okay? So, item number two- number one and number two says that, in the equilibrium, okay, nobody deviates and the high price of 3 is maintained every day, okay. In the equilibrium, cooperation, high price, $3 is maintained every day. But if anybody deviates from $3, if anybody undercuts supplies, then they set price equal to cost forever. Okay? Okay, deviation triggers cut-throat price competition. So this wonderfully captures our intention. And this is what is known as the Trigger strategy. Cheating triggers, cut-throat competition in this example. So let me try to explain generally what the Trigger strategy is. So Trigger strategy, in general, start with cooperation. In the first period players choose cooperative behavior and the keep on cooperating if nobody has deviated. And if en- everybody has deviated, they do something forever. Okay. In the gas station example after a deviation, they set price equal to cost. What is this? Well, this is the Nash equilibrium of the price competition played in every single day. In other words, this is the Nash equilibrium of stage game. Stage game is the price competition game which they play every day. Okay. So Trigger strategy states that, they start with cooperation, they should keep on cooperating if nobody deviated. But if anybody deviates, they go to the, back to Nash equilibrium in the stage game. Okay. So let me show you a diagram to understand the nature of trigger strategy in general. So oftentimes Nash equilibrium is not socially optimal. Nash equilibrium is inside in this grey area. And you have a room to improve both players' payoffs. So Nash equilibrium is quite often inefficient, and the trigger strategy says that, well, you should start with good outcome. In equilibrium, players should cooperate in every period to achieve this wonderful point here. And this point should be better than this black point, the Nash equilibrium for both player. For both player A and B, okay. So, cooperation point should be better than stage game Nash equilibrium for all the players. So, on equilibrium they play this wonderful you know, red point every day. But if anybody deviates, then after that they play the bad Nash equilibrium in the stage game forever. Okay, so if anyone deviates, play Nash equilibrium of the stage game. Like, cut-throat price competition in gas station example forever. So this is the nature of trigger strategy. So basically, this good point here, cooperation point is not Nash equilibrium. So by deviating you can gain something. In gas station example, if you undercut you can increase your payoff. So you have an incentive to deviate from this good point. But if you do that, your future payoff is decreased from this wonderful red point to this bad black point. In the future, players lose their future payoff. Okay. So that's how, trigger strategy enforces good outcome in the society. You can gain today, but if you do that, outcomes shift from a good one to a bad one in the future. Okay, so, let me closely examine the condition under which trigger strategy is in equilibrium. Okay. So let's suppose that cooperation that means charging a high price of three in gas station example. So let's, let's go back to the gas station example and let's suppose cooperation charging a high price of 3, gives each gas station $100 every day. Okay. So if they cooperated, they can get $100 every day. So let's examine the benefit and the cost of cheating. Well, so if you consider, say, gas station A, this is what gas station one, let's say, gas station one is going to earn. If gas stations stick to cooperation. So, any period starting from, say, March 1, March 2, March 3, this gas station is earning $100, okay. So, this is cooperative situation. Okay, so let's examine what happens if he deviates, gas station 1 deviates on March 1st. By slightly undercutting the price, okay, gas station can almost double its profit by stealing customers from gas station two. Okay. So, his profit almost doubles if, he slightly undercuts the price. Okay. What happens in the future? Well, this cheating triggers cut-throat competition and gas stations start charging very low price. Price of 2 which is equal to cost. Oh, I'm, I'm sorry. So I have to explain the gain from defection now. Okay. So before going to what, what's going to happen in the future let's calculate the game today. So cooperation profit is a 100. If you undercut the price by cheating your profit doubles so the gain from defection is the difference of those two numbers, and you can gain $100 today by cheating, okay. Now let's examine what's going to happen in the future, okay. This cheating triggers cut-throat price competition in the future, and all future benefit is gone. Okay. Your future benefit is, your future profit is zero in every period. So the future loss is $100 in everyday. So how can we evaluate those stream of losses coming on March 2, March 3 and March 4 and so forth? Okay. So, let's think about the value of $1 in March 1st and dollar of $1 in the distant future, say December 31st. Okay. Obviously, $1 on December 31st is not as good as $1 on March 1st, today. Okay. So, that means we actually discount future payoff. So, let me formalize this idea of discounting future payoff. Future payoff is not as good as today's payoff. So, let me measure magnitude of discounting by this number, d. So d is the value of $1 coming tomorrow. Okay. So, $1 tomorrow is less, you know, is not as good as $1 today. So, usually d is less than one. This means that we discount future payoff. So d may be 0.995 or something. Okay. So, $1 tomorrow is slightly worse than $1 today. And this discounting is, is measured by this number d. Okay, so you are losing $100 everyday in the future if you cheat. So, the future loss is calculated as follows. So now, we are on March 1st. And tomorrow $100 is lost. And since this is tomorrow's $100, it's discounted by the factor of d. The day after tomorrow, March 3rd, again, you lose $100, but since this is day after tomorrow, you discount twice. So d times d is applied here, and, the day after tomorrow's loss is 100 times d squared, and so on. Okay. So this is the total discounted value of future loss. Okay, so the future loss is here, d is the value of $1 in tomorrow. And today by cheating you can gain $100. And if future loss is larger than today's gain from defection, defection is not profitable under trigger strategy and therefore trigger strategy sustains cooperation. Okay, so therefore, gas stations can cooperate and maintain a high price by possibly by a trigger strategy in their long-term relationship.