Tragedy of the Commons | Game Theory | Nash Equilibrium Analysis

[00:00:09]hello welcome to another module in this online course strategy and introduction to game theory what we are going to look at today is we are going to look at a different game so we have seen a large number of games or we have seen several examples of games so far which are slightly simplistic what we are going to do right now is we are going to look at start looking at slightly more refined and slightly more sophisticated games in particular the kind of games the kind of game that we are going to look at today is also known as the tragedy of commons this game is right titled as the tragedy of the commons it is a very catchy title what it means is it relates to the usage or rather it relates to a game sort of a game interaction between competing agents or between competing people related to the utilization of a common resource such as for instance a mine minerals or a fishery or let us say a forest or so on so the environment for that is matter right so what this game is was this tragedy this game the tragedy of commons is about the usage of common resources or rather exploitation resources such as forests such as forests mines fisheries or environment or pasture environment with respect to the number of the amount of greenhouse gases that are released into the environment causing environmental pollution or pasture lands relating to the amount of grazing and also often over grazing that leads to depletion of these pastures so what we are looking at we want to model this sort of a competition or a strategic interaction between different agents who are using a common resource that is common to all the people for instance of a certain state or a certain country such as a forest or a mine fisheries which are often subject to over exploitation and leading to their eventual depletion and we are trying to model this as a game and try to understand the behavior of these agents in such a game or in such an environment well how do we start with it lets start by looking at a simple example in which we are taking a look at the usage of a forest lets say there are two timber agencies let us consider a scenario where there are two timber agencies each is involved in cutting logs or cutting trees or lumbering in a particular forest that is or each has a license for cutting trees in in a forest towards providing lumber right and each one can use an effort e i so each can use an effort ei can also be of trees that are cut for instance the number of possible trees that are logged by these two different timber agencies right and of course the number of trees that are cut is proportional to the effort e i right so we can look at two timber agencies the effort of each is denoted so the effort of timber agency one agency one has an effort of e one and agency two has an effort of e two so we are looking at this strategic interaction between these two timber agencies which are cutting or logging trees towards providing timber and their actions are their efforts effort e one of agent c one e 2 of agency e 2 of agency 2 and we are trying to understand the strategic interaction between these 2 timber agencies of course now we have to specify a payoff function corresponding to these 2 efforts we already said that the number of trees that are logged or cut is proportional to the effort put in by each timber agency so the payoff can be modeled as follows the payoff u1 of agency 1 as a function of its effort e 1 and the effort e 2 of the other timber agencies can other timber agency can be modeled as e 1 times 1 minus e 1 plus e 2.

[00:05:33]now of course this requires some explaining the first factor e one we are saying the utility or the payoff is proportional to its own effort because the number of trees cut or the number of trees logged is proportional to its own effect but also the payoff decreases with the total effort put in by both the timber agencies that is e one and plus e two because the payoff is also proportional as you can t c to one minus e one plus e two because the more number of trees that are logged by these temperatures together the less is left to be used in the future so the less is left for replenishment which means the future payoffs are going to be lower as compared to when they are both logging less so this is an interesting payoff function which takes into account not only the current payoffs it also takes into account the possible future payoffs related arising from over exploitation of of a certain certain resource such as a forest so it is proportional to e one that is it increases with e one but the payoff also decreases with the total effort that is one minus even plus e two reason being that the more effort but both of them put in together the more trees are cut the more trees are locked therefore less is left for replenishment and less left for future use so this let me just highlight the important aspects this shows the payoff of agency one as a function of its effort e one and effort e two of the other agency proportional to its effort but decreases with joint effort of agency one plus agency two so we are saying its proportional to its own effort but it also decreases with the sum or the net effort put by agency one and agency two the reason being the more the joint effort put by them the more is the exploitation of the resource leaving less for possible future use similarly now the payoff of the second agency u2 e2 remember we have to put the action of the second agency first while talking about h payoff u equals e 2 into 1 minus e 1 plus e 2 also again proportional to e2 and inversely and decreases with increasing e one plus e two so there are two components one which is proportional to e two another component which is decreasing with e one plus e two which is the reason being similar to what we had said in the case of the payoff function of agency 1. so as you can see this is a strategic interaction because the payoff for each agency or for the payoff of the timbering agency or the payoff of each logging operation depends not only on its own effort but also depends on the effort put in by its competitor agency right so this is naturally an example of a strategic interaction between these two timber agencies what is the best what is the best what is what is the how does this game evolve or what do you expect to see with respect to the efforts put in by both these agencies in practice and of course this game is now slightly more refined or of slightly more slightly more advanced compared to the other games that we have seen before because now these efforts e1 and e2 are not confined to a finite set but each e one e two can be any real number between zero and one so we are saying zero less than equal to e one less than equal to one zero less than equal to e two less than equal to 1 which means e1 and e2 can take any real value between 0 and 1.

[00:10:08]so we have a continuous set of efforts that can be put by these timber agencies so the action sets are in finite and continuous action sets compared to the discrete actions that remember in all the games that we have seen previously we had two players and each player had a finite set of actions in fact each player in fact most of the games each player had possibility of two actions to choose from so now we are moving to a slightly different scenario slightly more advanced scenario where the set of possible actions is in finite that is each timber agency can choose an effort and this effort can be any real number between 0 and 1. so this is a more advanced game right so where we have an in finite set of possible actions that also means because we have an infinite set of possible actions i can no longer draw the game table so i can since i can no longer draw the game table right because the game table i can have only a finite number of rows for a certain finite number of actions because the number of actions is in finite i can no longer draw a game table but therefore to analyze this game or to come up with a reasoning to interpret the behavior of the different agents in this game i have to come up with a different framework and that is what we are going to talk about in the next couple of minutes that is how to characterize the behavior or how to characterize the outcome in this particular game which involves an impossibly infinite which involves an infinite set of actions for both the players right so naturally as we said before that is to analyze the behavior of any game we have to first start by looking at the best response of each agent or the best response of each player well let us start by looking at the utility function of player 1 that is u1 to characterize its best response we have u 1 of e 1 comma e 2 equals e 1 into 1 minus e 1 minus e 2 right which is equal to i can expand this to write it as e one minus e one square minus e one e two now i have a payoff of u1 which is a function of both p1 and e2 i have to find the best response e1 for a given effort e2 or for a given strategy e2 of agency 2 which means i have to maximize this utility u1 i have to maximize to find the best response the best response is where the payoff is maximum for a given effort e2 by player 2 and therefore to maximize this continuous function most of you as most of you must be familiar from an introductory knowledge of differential calculus i can differentiate this utility function and set it equal to 0 to find the maximum to find the point at which this payoff is maximum therefore i am going to differentiate this with respect to u one and set it equal to zero so d u one by d e one when i differentiate this with respect to e one i am going to have let me write it down here that is d by d e one of e one minus e one square minus e one e two which is equal to derivative of e one is one derivative of e one square is two e one derivative of e one e two is t two and therefore to find the maximum that is the effort even at which it is maximum i have to equate it to 0 and now solving this equation i have even star which is the best response equals 1 minus e 2 divided by two so the optimal so the best response e one star equals one minus d two by two i hope everyone was able to follow that argument that is basically we have a payoff function which is a function of the f at e1 to find the best response i have to maximize this payoff function this is now a continuous function with respect to e one this is in fact a differentiable function so i am going to differentiate this with respect to e one and set it equal to zero to find the best response even star and the best response e one star is one minus e two by two in fact this can be written as e one star equals best response 1 with respect to e 2 which is equal to 1 minus e 2 divided by 2. so we have found the best response even star as a function of e 2 this is the best effort of agency 1 as a function for a given effort e 2 by the timber agency 2. so this is the best response this is the best response effort of agency but what do i do next next i have to find the best response effort of agency 2 right which i can obtain from the utility function u 2 u 2 of e 2 comma e 1 equals remember that is equal to e 2 into 1 minus e 1 minus e 2 that is its payoff your utility function is given as e 2 into 1 minus e 1 e 2 and remember now i have to find the best response e 2 that is that e 2 for which this payoff is maximized for a given effort e one by its competitor who is agency one or who is player one and therefore now i have to differentiate this with respect to e two and set it equal to zero to find that e two where this is maximized or to find the best response e two therefore differentiating this with respect to e two i have d by d e two of e two minus e one e two minus e two square which is equal to 1 minus e 1 minus 2 e 2 which i am now going to equate to 0 this is equal to 0 and solving this i am going to obtain e 2 star equals 1 minus e 1 by 2 that is the best effort e 2 star equals 1 minus e 1 by 2. so this is in fact e 2 star which is the best response of 2 given action 1 or effect e 1 of agent c1 so we have the best response of e2 star this is the best response best response e two star of agency two or player two right so what have we found so far we have found e one star which is the best response of agency one we have also found e two star which is the best response of agency two so we have now found the best responses of both the players now to find the nash equilibrium we of course have to find the point where the best responses intersect remember nash equilibrium were nash equilibrium is where is where best responses intersect which means each is playing best response to others action or strategy which means what we are saying is the nash equilibrium is where the best responses intersect therefore each is playing his best response action to the action of the other player right so naturally we have found remember e 1 star equals b r 1 e 2 e two star equals b r two e one now at the nash equilibrium everyone is playing his best response so what we have is in fact we have e one star equals v r one e two star right because at nash equilibrium player two is also playing his best response and similarly e two star equals b r two e one star this equations hold only at nash equilibrium hold only at these equations hold only at the nash equilibrium because both the players are playing their best response as a result what we have is we have e 1 star equals best response 1 e two star which is one minus e two star divided by two similarly we have e two star equals best response is two best response 2 of e 1 star equals 1 minus e 1 star by 2. so i have a system of linear equations in terms of e 1 star and t 2 star let me write them down again clearly i have e one star equals one minus e two star divided by two i have e two star equals one minus e one star divided by two this is the set of equations i have to find the nash equilibrium actions e 1 star and d 2 star i can substitute e 2 star from the second equation in the first and then i will have e 1 star equals half minus half e two star but e two star is one minus e one star divided by two so have substituted so let us call this the second equation let us call this the first equation i have substituted for e to star from the second equation in the first equation and therefore i have e one star equals half minus half one minus e one star divided by two which after simplification gives me e one star equals as you can clearly see one fourth plus one fourth e one star which means three fourths e one star equals one fourth and therefore e one star equals e 1 star equals solving this set of equations i get e 1 star equals 1 by 3 so that is the nash equilibrium effort e 1 star of agency 1 what about e two star well e two star equals one minus e one star divided by two which is one minus one by three divided by two you can also see this is one by three anyway we could also have guess that from the symmetry of the game therefore we get that the effort e two star equals one by three the nash equilibrium effort and therefore what we have is we have an expression we have interestingly found what the nash equilibrium effort for both the timber agencies is the nash equilibrium effort or the nash equilibrium outcome e one star comma e two star equals one by three comma one by three therefore this is the nash equilibrium this is the nash equilibrium outcome where each of them is using an effort e one star equals one by three e two star equals one by three that is to say that e one star is the best even star equal to one by three is the best effort of agency one to the effort e two equals e two star equals one by three by agent c two similarly e two star equal to one by three is the best response of agency two to the effort e one star equal to one by three by agency one and since both of them are playing their best responses therefore this is a nash equilibrium that is each one is playing the best response to the action of the other player which in turn is the best response to the action of all the other players right so this is the nash equilibrium outcome of this game and finally let us calculate what the nash payoff is what is the payoff to each at nash equilibrium at nash equilibrium that is what is v u 1 of e 1 star e 2 star equals u 1 of 1 by 3 comma 1 by 3 this we can say is equal to 1 by three one minus e one plus e two one by three plus one by three which is equal to one by three into one by three equals one by 9 similarly you can show that because of course both of the payoffs are symmetric u2 of e2 star e1 star equals 1 by 9 therefore what we can say is that the nash payoff the nash payoff or the payoff at nash equilibrium to both the timber agencies is u 1 or u 1 of e 1 star comma e 2 star equals u 2 of e 2 star comma e 1 star equals 1 by 9 which is the nash payoff of both these competing agencies or the payoff to both these competing agencies at the nash equilibrium so let me summarize what we have learnt in this game so far we have looked at an interesting game which is based on the tragedy of commons paradigm which is basically we are looking at interaction between two companies or two agents or individuals who are using a common resource for instance in this example we consider two timber agencies logging a forest for timber and their payoff is proportional to not only their effort but its inversely proportional to the joint effort put by everyone together because that leads to a depletion of the resource this is a novel game compared to what we have looked at previously because the efforts the possible set of efforts by each agent is infinite so their infinite set of possible actions of each player and we used differential calculus to compute the best response at the nash equilibrium each of them is using is each of the players is playing his or her or their best response and therefore we have used this principle to solve a simultaneous set of linear equations to compute the nash equilibrium shown the nash equilibrium the nash equilibrium efforts to be e one star equal to 1 by 3 e 2 star equal to 1 by 3 and also we found out the payoff to each of these firms at the nash equilibrium as a last note even though we have considered only two firms or two agents in this example this can easily be extended to a set of more such agencies that is more such agents or more such individuals involved in the depletion of this resource and that will lead to interesting conclusions i lead this i leave this as an effort or as an exercise to the people who are viewing this lecture so let us stop at this point and in the next module we are going to analyze what is the outcome of this game that is with what is the equilibrium outcome what is the effect on the resource and other aspects of this game the tragedy of commons thank you