 So, today we are going to look at dynamics of love, each one of us is going to be affected by it at some point or already are affected, but how we are going to react to it depends on each of our personalities right. Some people are going to you know love your partners more if they love you more and you may get discouraged if they do not love you back or some people may get annoyed if their partners love them too much and they may be attracted to people who are not attracted to them right. So, what you are going to do is try and model the different personalities and understand how the relationship unfolds by building system dynamic models. So, that is what we are going to do in today's class and we are going to capture the personalities by using different parameters within the model and see how it goes ready let us go for it. Let us start with the classical love story, boy meets girl, sparks fly and they both like each other and his love grows which makes her love grows for him, she likes him even more which makes him likes her even more which keeps growing probably much of annoyance of others, but still it grows. Let us call this Romeo and Juliet model, so Romeo and Juliet are madly in love with each other with each meeting, Romeo's love for Juliet grows because he loves her, he does everything to woo her, Juliet is flattered by his attention and in return her love for Romeo also grows because Romeo senses that Juliet loves him, he loves his allows his passion to sow like. So, now what will happen to Romeo and Juliet's love, so now we need to see how we can model this in our as a system dynamic model, but before that, so what do you think is happening here, each other's love seems to reinforce the other's love on the partner, so what we are expecting here is it is a reinforcing loop which is going to result in a exponential growth in their love for each other, so the question is if we want to model it what is going to be the stock, love is the stock, so let us simply define it as let one lover's love for the other be the stocks, let us have two stocks, one called Romeo's love for Juliet and Juliet's love for Romeo, let us define two different stocks, now what will be the units for the stock, units has to match, we are engineers, we will just come up with some love units, let us say arbitrary let us call it love units, I do not know how it will be measured, maybe it is number of what can I say, messages they exchange per day or number of times they think about each other a day or yeah anything like that, so number of likes they put in Facebook on each other or something, so let us say that I will measure it as some love units which is somewhat measurable which we just saw and let us make some simple assumptions like positive love units represent how much they like each other and negative love units represent how much they dislike each other and if it is 0 they are indifferent to each other, so now we are getting somewhere, we have defined two stocks and we have defined the units, stocks change to changes through what, flows, so there has to be two stocks then there has to be two flows, it is not a material system it is not like one love stands for to other and there is no conservation of flows etc happening, each can grow as much as it wants and we just saw that Romeo's love for Juliet grows based on how much Juliet loves him and Juliet's love for Romeo grows based on how much Romeo loves her, so they are feeding of each other, we can represent it as a stock flow diagram as shown here, so let us define two stocks Romeo's love for Juliet and Juliet's love for Romeo and we have of course any flow that changes it we just simply call it as changes in Romeo's love and the changes in Juliet's love affects the stock of Juliet's love, now you can observe, so this is the basic I think you can get it, the key thing that you have brought in is something called as two parameters, one is Romeo's reaction and second is a Juliet's reaction, so here we are trying to capture the personalities where what is Romeo's reaction to Juliet's love for him and what is Juliet's reaction to Romeo's love for her, in this particular case we just saw that if Juliet's love is Romeo then Romeo's reaction is to get reinforced it increases, same similar with Juliet's reaction, so Romeo's reaction Juliet's love is positive, Juliet's reaction Romeo's love is also positive, so let us simply capture it by this parameter Juliet's reaction, let us say each unit of Romeo's love increases Juliet's love by one unit, so then we can just set Juliet's reaction is just one plus one, so now we have a nice looking SD model, the initial love of Romeo for Juliet and initial love of Juliet for Romeo they are just to denote the initial value of the stocks it does not change the stock it just sets the initial value I just wrote it outside for convenience, so what you are going to do is see what happens when their initial reactions are different when they meet, so the Romeo's reaction similar to Juliet's reaction, so both is plus one, so let us go to the pencil, so we just I just want to show one of the stocks Romeo's love for Juliet, initial values initial love of Romeo for Juliet and changes in Romeo's love is nothing but Juliet's love for Romeo multiplied by Romeo's reaction, so for each unit of Juliet's love his Romeo's reaction is going to multiply that by a factor of one in this case and Romeo's reaction is one and units is one per day, units of stock is love and initial love of Romeo is one, initial love of Juliet is one, so what you are assuming is when they meet each other first time sparks fly and they love each other, love at first sight both sides, let us go for it, let us simulate the model and what do you expect the behavior to be I am just going to plot the stocks, you can observe both the stocks increase, it is identical both the lines overlap so you do not see the other one but both are overlapping and in a matter of 12 days they have accumulated 150,000 love units and it has grown exponentially, so in this so why does it grow exponentially then there has to be a positive feedback otherwise it cannot grow exponentially, so let us just trace it, exponential growth, so here as Romeo's love for Juliet increases the change is in the positive direction, so Juliet's love for Romeo also increases and because of that Romeo likes Juliet even more, so it is a reinforcing positive feedback system that we have here which is causing an exponential growth in both their love. Let us take this case as we just saw initially both Romeo and Juliet attracted to each other, so we just do that by setting the initial value of Romeo's love to plus 1 and Juliet's love is also plus 1 units initially and we have seen this exact same behavior exponential growth and we trace the feedback so both has to be positive feedback loop because a reaction is positive, so whatever amount of love units you are just multiplying it by plus 1, so direction of change is still positive. Now let us do some interesting scenarios and plot changes in the basic storyline, what will happen if initially Romeo loved Juliet twice as much as she loved him? Will Juliet's love ever grow to equal Romeo's love? What do you think? If you simulate it, we can set the initial value of Romeo's love to plus 2 love units and Juliet's love to plus 1 love units and if you simulate it, actually dynamics is as seen here, both their love for each other is going to grow exponentially and it is going to still overlap that is because when we started Romeo's love was 2 units that caused Juliet's love to increase twice as much as before which caused Romeo's love to increase same one times more which again caused Juliet's love to increase, so both so whatever setting they are going to have the same amount of love for each other but since Romeo loved her twice as much initially, their love grows to 240,000 units, 240,000 love units much more than 150,000 love units we started with. Let us suppose more changes, let us suppose Juliet is initially indifferent to Romeo but Romeo finds her attractive at first sight that is Juliet's initial love units is 0 while Romeo feels one love unit for Juliet, what do we expect the dynamics to be? It has to increase exponentially, just think about it there is initial Romeo loves her so with one unit based on that the change in Juliet's love will also increase to 1. Initially Romeo's love may not change because Juliet is indifferent to her but once Juliet sees Romeo's love for him her love start or she will start reciprocating it maybe little later but once both have start having positive values it become a positive feedback system which starts reinforcing each other. Let us see how many love units accumulate in 12 days, not as much as original but still 70,000 love units is what they have because Juliet was initially indifferent to Romeo but Romeo feels one love unit for Juliet. Still we can experience exponential growth as long as a positive feedback system and we are going to have non-zero values system is going to grow exponentially it cannot grow otherwise. Let us flip it those who remember this story will know that Romeo's and Juliet's family fight with each other they do not like each other suppose influenced by their family they also initially dislike each other that is both initially both feel minus one love units towards each other they see each other and they start hating them immediately then what kind of dynamics can we expect people are all drawing in the air so it is also going to have exponential behavior but it is going to be accelerated decay or exponential collapse is what we can expect and just like we anticipated some of us their hate for each other will continue to feed off each other. So with every unit of time Juliet I mean Romeo hates Juliet so Juliet's hate for Romeo is going to increase and because that hate increases Romeo's hate for Juliet will further increase and that going to get reinforced each other so direction of change is continues to be the same which results in an exponential collapse among them both going to hate each other forever. How about if it is more complicated things although Romeo is initially enchanted by Juliet she feels a initial dislike for him I think we simply model it by saying initial value of Romeo's love is plus one love unit and Juliet is minus one love unit she dislikes him but he likes her then what do you think will happen will you cancel each other after some time it will you have seen too many movies if you try and simulate it we actually get a very surprising dynamics so the red line is Juliet's love for Romeo green line is Romeo's love for Juliet it is actually very surprising dynamics because here we have positive feedback system which is displaying a goal seeking behavior the entire semester we saw that positive feedback system is exponential growth negative feedback system results in goal seeking behavior now we see that love changes everything so let us see let us try to understand why that happens right. So as we just told or just saw Romeo's reaction is in the positive direction to whatever Juliet's love for him is right. So let us start there so Romeo likes Juliet but he figures out that Juliet does not like him so his stock of love falls he gets discouraged but for Juliet if you see she finds out that he loves him and her dislike only reduces and hence we are getting a and this happens over time and you find that they eventually reach point of indifference towards each other at least as per this model. So that is a very surprising behavior we just have to positive feedback system just by setting the initial parameters we never change the model we are only playing the initial parameters right that is quite fun just by changing parameter setting we are getting different dynamics some very hyper exponential growth or I mean exponential growth final values keep changing depending on where we are at the initial values and now with one parameter combination we are actually getting a goal seeking behavior which is counterintuitive and very interesting. So expanding that if both Romeo and Juliet are indifferent that is both feel 0 love units that means initial stock is 0 so net change is also going to continue to be 0 so they are going to be continued to be indifferent to each other they are going to continue to be indifferent to each other. So for this but this is a very unstable equilibrium even if one feels some like some amount of like to the other person they are going to start off a reinforcing clue or classical love story is going to start right whatever even if you assume whatever a small number assume Juliet is indifferent but Romeo feels a very very slight 1 by 100th feeling of whatever of love unit for Juliet very very tiny amount but still if you simulate it you will still get an exponential growth because it saturates at a much smaller love units but still nevertheless over time they are going to have exponential growth they love. So that is very unstable equilibrium and even a small change in the positive direction is going to result in a exponential growth in their love and works the other way also if you are going to have very slight dislike for the person that might also get reinforced over time.