we are going to begin discussing Game Theory uh this is one of my favorite topics in the course and uh a lot of the students uh would agree I hope you really like it so we separated imperfect competition into two different models we talked about monopolistic competition where we have lots of firms so we thought for example about um companies making jeans uh restaurants things like that now we also mentioned an oligopoly which is we have a small number of f things like video game console manufacturers or cell phone service providers in oligopoly there a small number of competitors so your firm can think about their comp how your competitors will react to a change in your strategy which means that each firm's decision has an effect on the other firms so for example if Verizon Cuts their price how will AT&T respond how will AT&T act if it thinks that Verizon could cut its price how will Verizon react if it thinks at at& thinks that Verizon is going to cut its price things like that we want to model these sorts of interactions so game theory is as simple as saying my decision affects your payoff and your decision affects my payoff right and so that is when we get into Game Theory with monopolistic competition this wasn't a concern right there were in our model there were so many different genes companies it wasn't worth it for me to think about the actions of any other one other company all I had to think about was my were my decisions whereas if we think of something like chess chess great example of Game Theory good reason of why there's the word game in Game Theory right chess if you try to make chess moves without looking at what the other person's going doing that's a terrible idea so when are times when we see game theory in action um rock paper scissors is a classic example right rock paper scissors is a very simple game and it's one that we like is in Game Theory because it's it's a real clean example of how my decision my payoff is affected by both my decision and your decision right there's no one of these that always wins or always loses each one of them only the whether each one wins depends entirely on what the other player chooses uh soccer penalty kicks are another example so as I understand it in the very high level soccer um what it's basically the kickers kick so fast that the goalies cannot really they cannot really react very efficiently to the what the kicker is which direction the kicker is aiming so the goalie actually has to generally guess and try to jump to the left or the right and so there is Game Theory there because the goalie is anticipating which direction the kicker's going the kicker is anticipating which direction the goalie is going to jump there was actually an economics paper written about soccer penalty kicks and what they found was that kicker should actually kick the ball right down the middle more because goalies were typically jumping to the left or jumping to the right and leaving the middle open and the authors suggested that the reason we don't see more kicks down the middle is because if you miss kicking down the middle it's sort of embarrassing right that you got up there as the uh penalty kicker and didn't even try to do anything interesting or creative you just kick the ball right down the middle so you don't want to be embarrass for doing that if you kick it down the middle and gets stopped so you kick it to the left or the right and this way at least if you miss it sort of you know oh the goalie had to had to dive and make a good save so I I actually took this clip this is from the um penalty kick shootout at the End of the World Cup final and this kicker right here um he made the kick by kicking it right down the middle see the goalie dived out of the way right there's the ball going right down the middle and he made the kick another time when game theory is applicable is a nuclear war this is actually a lot of game theory was developed during the Cold War and you think about what was going on between the United States and the Soviet Union is both countries were sort of saying I'm going to make my Arsenal bigger so that I can respond to your attack with an even larger attack um advertising we're actually going to look at an advertising example today um is one where we think about how well do I want to have an ad well it's going to depend if you have an ad because I don't want to you maybe I don't want to give you all the spotlight but if we both have an ad then we've spent a lot of money and not really and Bas and just sort of poached each other's customers we both have a lot of outlay advertise without actually getting much benefit from it um pricing is another one right we kind of talk about that how all right do if you have do I want to undercut your price a little bit well if you think that I'm going to do that how will that affect your pricing and so on so here are the assumptions assumptions we're going to have in this class we're going to assume that we have two players each is trying to maximize something usually profits and this is a simultaneous game where both players move at the same time what we're going to do first is we're going to learn about the notation we use to show and explain and solve games and then after that we'll look at some more interesting games and how they can be applicable and how they uh why they're relevant for economics so here's our first example it's going to be Coca-Cola and Pepsi Coca-Cola and Pepsi are both considering buying a Super Bowl ad if Coca-Cola buys an ad and Pepsi doesn't buy an ad Coca-Cola will get $200 in profit and Pepsi will get $50 in profit so what's going on here is basically Coca-Cola has the ad gets all the buzz the day after you're at work it's like oh man Coca-Cola had the coolest ad Coca-Cola Gets 200 profit Pepsi gets 50 if Pepsi buys an ad and Coke doesn't day after work you get to work you're like oh man Pepsi had the coolest ad of the Super Bowl pepsy gets 150 in profit Coca-Cola Gets 50 so this suggests that maybe Pepsi's at isn't quite as good as Cox is why their profits are 150 versus 200 if neither company buys an ad both firms will save on Advertising costs and get $75 in profit if both companies launch an ad Coca-Cola Gets $100 in profit and Pepsi gets $10 in profit that basically Cocola has a better ad and so Pepsi will have spent all this money on Advertising but not actually gotten a not gotten a big bump from it because everyone noticed and talked more about the Coca-Cola ad it's it's useful to condense those words into a table right wouldn't it be nice if when we're trying to predict how Coke and Pepsi will behave it' be nice if we could find some way to turn all this text into something a little more Compact and that is called a payoff Matrix a payoff Matrix is a table showing the potential outcomes arising from the choices made by decision makers what do you mean by that well table we want to show the potential outcomes that's the profits for Coke and Pepsi and how they depend on the choices by Coke and Pepsi so here's what our Matrix is going to look like Coca-Cola is the row player or player one Pepsi is the column player or player two this is a simultaneous game so no significance to player order so it's not like Pepsi can see what Coke does and respond nope they both have to play at the same time like rock paper scissors each option for Coca-Cola is a row so we have Coca-Cola here they can choose add or no add each option for Pepsi is a column Pepsi could choose the left column for ad or the right column for no add and each of these four boxes shows the profit for each player with player one shown first so what we're going to do is we're going to go back through through all that text we just looked at and see how we can condense it into this one table so first we said if Coca-Cola buys an ADD and Pepsi doesn't cocacola will get $200 in profit and Pepsi will get $50 in profit so what we want to do then is find the which of these four boxes corresponds to this Coca-Cola buys an ad Pepsi buys no ad right Coca-Cola buys an ad Pepsi doesn't Coca-Cola Gets 200 in profit Pepsi gets 50 Coca Cola is player one we're listing them here so we show their payoffs first Pepsis are shown second if Pepsi buys an ad and Coca-Cola doesn't Pepsi will get $150 in profit and Coca-Cola will get $50 in profit all right so let's uh give you a chance to try to think about how we're going to put this in The Matrix so first question which box should we be filling in for this this is the example we just we were just talking went through Pepsi gets p Pepsi buys an add Coke doesn't which box should we fill in the answer here is B this is Coca-Cola doesn't buy an ad Pepsi does buy an ad so that's right here next question what should go in the Box answer here is a 50 comma 150 we show Coca-Cola's payoff first which is 50 Pepsi pay off second which is 150 so here's what it looks like next if neither company buys an ad both firms will save at advertising cost and get $75 in profit so for this we want to look there Coca-Cola doesn't buy an ad Pepsi doesn't buy an ad that's going to be right here right here what we find is that both companies get $75 in profit finally if both companies buy an ad Coca-Cola Gets $100 in profit Pepsi gets $10 in profit that's Coca-Cola buys an ad Pepsi buys an ad Coke gets 100 Pepsi gets 10 notice what we've done here is we have turned all of that text into one compact table isn't that nice and by doing so we can now instead of trying to read through all those paragraphs and sentences as to figure out what happened we can get all of that information from right here okay our next step well we want to solve it how do we expect Coca-Cola and Pepsi to behave and that brings us to this term which is best response best response says conditional on another player's action what is your best action so we want to think about given what another player does so if the other person you know if the other player plays Rock what is your best response in rock paper scissors like that uh one note is that we do call it a response even in simultaneous games in rock paper scissors the best response to scissors is Rock right the other player plays scissors you want to play Rock you're not actually going to have the opportunity to respond because it is a it is not it is a simultaneous game you're not actually looking at and looking and responding but the best but we we call it a best response to scissors we say that is Rock typically we underlined a best response payoff in a payoff Matrix a strategy is just what a player chooses to do so for example add is a possible strategy for Pepsi right Pepsi has two possible strategies they can buy an ad or they can not buy an ad so let's first think about Coca-Cola's best response so usually when we're doing this we want to find we find player One's best responses first then player two's it it doesn't matter what order you do them in it's just sort of we nice to get in a pattern a habit of doing it that of some of some sort of habits that we make sure we don't miss a step when we're solving these so let's look at Coca-Cola's best response first so the way we look at Coca-Cola's best response and we say suppose the other player chooses an action so suppose Pepsi plays ad what is Coca-Cola's best response well Coca-Cola has to decide if they want to buy an ad or not if Coca-Cola buys an ad they get $100 in profit if they don't buy an ad they get 50 in profit one way you could think about this is you could say all right once Pepsi chooses to play an ad we could just we could just cover up this half right ignore this half because we if Pepsi chooses to buy an ad we know we're going to be somewhere on the left hand side Coca-Cola could say all right I know I'm going to be on the left Pepsi chose that do I want to be the top and get a 100 by buying an ad or on the bottom row don't buy an ad and get 50 cocaa per Coca-Cola would rather get 100 than 50 so Coca-Cola's best response if Pepsi buys an ad is for Coca-Cola to buy an ad okay now I'll let you try one suppose Pepsi plays no ad what is Coca-Cola's best response the answer here is a the best response for Coca-Cola is to buy an ad let's look at how we do that notice we've already underlined Coca-Cola's best response if Pepsi buys an ad now we're going to cover this part up and say all right if Pepsi chooses to buy an ad that means we or chooses not to buy an ad Coca-Cola knows already said all right we're going to be on the right side of this of this Matrix because we're going to have to be responding to Pepsi not buying an ad so if Pepsi plays no ad Coca-Cola can buy an ad and get 200 in profit or not buy an ad and get $75 in profit Coca-Cola would rather have 200 75 so Coca-Cola's best response is to buy an ad all right so now we have um figured out Coca-Cola's best response if Pepsi buys an ad Coca-Cola's best response is also buy an ad Pepsi doesn't buy an ad Coca-Cola's best response is to buy an ad all right we looked at Coca-Cola's best responses now let's look at Pepsi's best response suppose Coca-Cola plays an ad so what's going on here is Coca-Cola said all right we are going to buy an ad so we are going to force you into the we are definitely going to be in the top row Pepsi gets to choose if they want to be on the left or the right Pepsi can buy an ad and get $10 in profit or not buy an ad and get $50 in profit so the best respon they would rather get 50 than 10 so Pepsi's best response to Coca-Cola buying an ad is for it to not buy an ad so Pepsi's best response to add is no ad so we underline this suppose Coca-Cola plays no ad and doesn't buy an ad Pepsi can either buy an ad and get 150 in profit or not buy an ad and get 75 in profit rather get 150 so Pepsi's best response to no add is to buy an ad so we'll underline the payoff there all right so based on this let's see if we could let's see if we could figure out how we expect this game to resolve itself how we expect Pepsi and Coke to behave so what do you think Coca-Cola is going to do well we expect Coca-Cola to buy an ad right no matter what Pepsi does cocacola is better off buying an ad here's a term we have buying an ad is a dominant strategy for Coca-Cola dominant strategy is a strategy that is always best regardless of what the other player does so if Pepsi buys an ad Coca-Cola's best response is to buy an ad if Pepsi doesn't buy an ad Coca-Cola's best response is to buy an ad right if Pepsi buys the ad Coca-Cola buys the ad and gets 100 instead of 50 if they don't buy the ad Coca-Cola buys the ad you get 200 instead of 75 so no matter what Pepsi does Coca-Cola is always better off buying ad Pepsi does not have a dominant strategy their best response is basically do whatever do the opposite of Coca-Cola if Coca-Cola buys an ad Pepsi Pepsi should stay out of the market so you know what we're not going to spend all our money on this ad that's not going to get the buzz anyway what because Coca-Cola is better but and their best response to Coca-Cola not buying an ad is say all right Coca-Cola is not buying a Super Bowl ad if we do it we'll get lots of Buzz so we should buy an ad all right so C Pepsi doesn't have a dominant strategy however we already know that Coca-Cola will buy an ad and one of our assumptions in all these models is that each player knows every other player's payoff that this is this is known to this whole Matrix is known to everybody which means that Pepsi knows Coca-Cola has a dominant strategy Pepsi knows that Coca-Cola is going to buy an ad so Pepsi knows that it's that we're going to be somewhere in here that Coca-Cola decided to buy an ad well Pepsi realizes that if they also buy an ad they'd get 10 if they don't they get 50 so Pepsi will not buy an ad we can abbreviate this no ad or we can abbreviate this as ad comma noad Coca-Cola strategy listed first just because we always do Coca-Cola first we we do whoever is the roow player first okay uh notice that we're trying to figure out what happens what we want we want to say that the strategy here the outome is that is ADD no add um we don't want to say just the numbers 250 is it's not as clear so again if I ask you for a strategy what we want is what each player does next we are going to learn this term Nash equilibrium most games do not have a player with a dominant strategy so for example which side of the road should you drive on right or left well there is no dominant strategy but we do know we want everyone on the same side right there it's not the case that it's always best for you to drive on the r or always but best for you to drive on the left all you really want is for everyone to pick the same well how do we predict an outcome in these sort of games so remember this this this information is going to be from a slide that we did way back at the beginning of class when we first started talking about equilibrium said an incentive is an opportunity to make yourself better off it's something that induces a person to act people respond to incentives by acting to make themselves better off so we had the example of that um if there's a long line and a short line at the grocery store people in the long line have an incentive to move to the Short Line an equilibrium is a situation where nobody has an incentive to do something different so all checkout lines are the same length for example a Nash equilibrium is a way of predicting what will happen in a game if a set of strategies in is such that no player has an incentive to unilaterally deviate it is a Nash equilibrium and by unilaterally we just mean that each player is acting on the run this idea that I can't control what you do given what you're doing if you know you made your choice am I doing the best I can so in a Nash equilibrium the following is true for both players given your strategy I'm doing the best I can or I have no regrets about what I play so for example um if we're both driving on the right side of the road that seems like it's probably going to be an ash equilibrium right I saw you driving on the right side of the road I also drove on the right side of the road I have no regrets about that we did not crash into each other the other person would say the same thing all right so now we we have our definition of an ash equilibrium if a set of strategies is such that no player has an incentive to unilaterally deviate it is an ash equilibrium and what we I suggested was that Pro we think that this is what's going to happen that Coca-Cola is definitely going to buy an ad Pepsi knows that so Pepsi will choose to not buy an ad all right is this a Nash equilibrium well let's the way we're going to solve this is say does either player have a profitable deviation is there a way that does either the play either player wish they had done something different if I had done something different I could have been better off well let's look at Coca-Cola Coca-Cola chose to buy an ad if Coca-Cola had unilaterally deviated so if they had changed their strategy they would have not bought an ad and only gotten 75 if Coca-Cola deviates from its strategy and doesn't buy an ad Coca-Cola will get a profit of $75 which is less than its current profit of $200 Coca-Cola does not have an incentive to unilaterally deviate Coca-Cola unilaterally deviated they would be worse off okay how about Pepsi if Pepsi deviates from its strategy and buys an ad Pepsi will get a profit of $10 which is lower than its current profit of $50 so Pepsi does not have an centive to unilaterally deviate right if Pepsi unilaterally deviated so if it changed its strategy which would mean that it bought an ad instead of not buying an ad they would be worse off so what we see is that Coca-Cola does not have an incentive to unilaterally deviate Pepsi does not have an incentive to unilaterally deviate that means that Coca-Cola buying an ad and Pepsi not buying an ad is a Nash equilibrium important to remember the Nash equilibrium is a strategy not an outcome the Nash equilibrium is ADD no add not 250 next thing we want to think about is finding a Nash equilibrium right what I just did was I said this is probably a Nash equilibrium let's confirm let's see how we're actually going to find it as we did before let's underline each player's best response to each possible strategy play by the other player right so we wanted to underline Coca-Cola we underlined Coca-Cola's best response for Pepsi buying an ad and for Pepsi not buying an ad we underline Pepsi's best response for Coca-Cola buying an ad and Coca-Cola not buying an ad all cells with both payoffs underlined represent the results of a Nash equilibrium strategy so let's look at this simple game of two cars where player one and player to are deciding whether to drive on the left side of the road or the right side of the road and we will write the game so that you get a payoff of one if you don't crash into each other and a pay off of negative one if you do crash and so first we design the The Matrix let's look at the payoffs whereas if both players drive on the left we each get a payoff of one because we didn't crash if both drive on the right pay off of one because we didn't crash if player one drives on the right and player two D drives on the left they crash into each other both get a payoff of negative one player one on the left player two on the right crash again both get negative one so that explains how we um how we designed this Matrix now let's go through and figure out everyone's best response so first we do player One's best response if player two plays Left player one can either play left as well and get a payoff of one or pay play right and get a payoff of negative one obviously rather get the payoff of one so best response to left is left what's player One's best response to a right well player two drives on the right player one can either drive on the left and get negative one drive on the right and get positive one they'd rather get positive one so we underline that now let's look at player two's best response if player one plays Left player two can either play left as well and get one play right and get negative one they would rather get one so the best response to player one playing Left is for player two to play left player one plays right player two can either play left and get negative one play right and get one so the best response to player one playing right is for player two to play right so what we find then is that there are two nashy equilibria to this game everybody driving on the left and everybody driving on the right and that makes sense right these are the two things that we would hope to happen is that people are not crashing into each other people have figured out that we should all drive on the same Hot Side whether it's both on the left or both on the right all right that's all we have for this time next time we're going to look at some more specific types of games