Collective Action Problems and Game Theory: Understanding Societal Deadlocks - Prof. Don R, Study Guides, Projects, Research of Economics

Download Collective Action Problems and Game Theory: Understanding Societal Deadlocks - Prof. Don R and more Study Guides, Projects, Research Economics in PDF only on Docsity! ECO 2007 / EC 330 Collective Action Games (based on Dixit & Skeath Chapter 12) 1. However frustrating they can sometimes be, games among small numbers of players can usually be successfully manipulated by those players if they’re clever. This ceases to be the case as numbers of players rise. For games played by thousands or millions of people in a society, the result can be complete deadlock: everyone sees there’s a problem and no one can do anything effective about it. Most of the major, enduring problems that afflict societies have game-theoretic explanations, and collective-action games are the subject of more day-to-day policy work by economists than any other general kind of problem. 2. A social problem caused by the behaviour of people is automatically a collective action problem. This means that the problem could be solved or alleviated if more than one person – but typically many people – took a certain sort of action. But if it were in each person’s private interest to take the kind of action in question, then they’d do so, and it would be mysterious why we had a social problem in the first place. Thus it must be the case that it’s not in each person’s private interest to take the action that would solve the problem. As the Prisoner’s Dilemma shows, this can hold even if all would be better off if each (or some) took the required action. Optimal social states may not be Nash equilibria; and some NE may be very bad for society. Examples: pollution; depletion of fish stocks; overcrowding of roads; over-armed societies; use of steroids in sports – etc., etc. 3. We’ll illustrate the underlying logic of collective action games by first illustrating how the majority of collective action social problems derive from the structure of three kinds of two-agent strategic-form game you’ve already studied: Prisoner’s Dilemmas, Chicken, and Assurance Games. (Remember Emily, Nina and Talia, who wanted a common garden? In the case of the NE where none contribute, we have a collective action problem that’s a case of a 3-person PD.) So we’ll start from further analysis of the 2-agent cases, then expand the number of players. 4. Consider two farmers who could each benefit from a drainage ditch on the boundary between their properties. If either builds the ditch, both will benefit. In technical language, we say the benefits are nonexcludable: an agent who doesn’t pay for her share of the benefits can’t be excluded from that share by those who do pay. The benefits here are also nonrival: no agent’s benefits are reduced by the benefits others draw. A good that has both these properties is referred to as a pure public good. Consider, for contrast, a pure private good like a pizza: if you buy it you can prevent anyone else from enjoying any of it, and whatever part of it you eat can’t be eaten by anyone else. (Note: I chose the example of a pizza because people usually share pizzas. I want to illustrate that what matters to a good’s being private is that benefits can be excluded; it doesn’t matter whether they’re actually excluded in typical cases.) Many goods – e.g., an education – fall between the pure cases, with agents who pay getting greater benefits than agents who don’t, but being unable to capture all the benefits. 7. Now we change the parameters again. Suppose economies of scale in labour are reduced, so that the two-person job takes 3 weeks of work from each and the one-person job 4 weeks of work. Return the benefit levels to those we started with. Now the matrix is: Farmer 2 Work Shirk Work Farmer 1 Shirk 5, 5 2, 6 6, 2 0, 0 The game isn’t a PD anymore, but Chicken. Thus it has two NE, (work, shirk) and (shirk, work), with each farmer preferring a different NE. Furthermore, both NE are inefficient, i.e., overall welfare is maximized by (work, work). (Note that if we instead use the low-specialization parameters from the second version of the PD, we can make the efficient outcome equivalent to either NE instead of better than it.) In Chicken cases, each farmer might wait for the other to build. 8. The textbook says that if externalities are negative rather than positive, we can get another variant of Chicken in which each farmer has an incentive to build but both farmers are better off with no building. The idea here is that each farmer is made worse off, instead of better off, if the other builds. However, Dixit and Skeath fail to distinguish between two different kinds of case: (i) a case in which a farmer who builds still gets some benefits from building but also suffers some harm if the other builds; and (ii) a case in which, if a farmer builds, the harm he inflicts on the other farmer completely wipes out the benefits accruing to the latter from building. You’ll be asked to show in your tutorial exercise that only the second kind of case produces a variant of Chicken. It is not enough, as Dixit and Skeath suggest, if each player’s payoff is negative in the event that the other builds. To get the payoff structure of this variant of Chicken in the farmers’ setup, you’d need to imagine that one farmer’s building completely ruined the other’s farm, say by inundating it. 9. Finally, suppose costs and benefits are as in the first version of the story, but now the benefits of a ditch built by only one person are reduced to the equivalent of 3 weeks of farming time: Farmer 2 Work Shirk Work Farmer 1 Shirk 4, 4 -4, 3 3, -4 0, 0 This is now an Assurance game with two NE: (work, work) and (shirk, shirk). Here the farmers will agree on which of the two NE is preferred, and this is also the Pareto optimum. This illustrates the general point that collective action problems having the structure of Assurance games are generally easier to institutionally solve. 10. As noted earlier, collective action problems of real interest involve more than 2 players – usually many more. So we now imagine the scenario above with n farmers involved. 11. Both costs and benefits to each player are functions of n. Thus we write the payoff to each farmer who participates as p(n) = b(n) – c(n), and the payoff to each farmer who shirks as s(n) = b(n). Each farmer deciding whether to participate or shirk will reason as follows. If n farmers participate and I shirk then my payoff is s(n). If I decide to participate there will be n + 1 participants and my payoff will be p(n + 1). Therefore, I will participate if p(n + 1) > s(n) and shirk if p(n + 1) < s(n). 16. Now here’s the graph for n-person Chicken. Payoff s(n) p(n + 1) 0 N - 1 n Here, where n is small, p(n + 1) > s(n) and so shirkers will switch to participation. But to the right of the point where the curves intersect, participants will switch to shirking. The intersection thus corresponds to the NE of the game. (Technically, this won’t be quite true if the intersection doesn’t correspond to an integer value of n. In that case, at NE someone is almost but never quite indifferent between participating and shirking and will switch back and forth; in mathematical language, the NE is an attractor but not a stable point.) Question: remember that our example of 2-person Chicken from an earlier lecture had two NE. Why are we getting only one NE in this n-person case? 17. Finally, here’s the graph for the n-person Assurance Game: Payoff p(n + 1) s(n) 0 N - 1 n Here, where n is small, p(n + 1) < s(n) and so participants will switch to shirking. To the right of the point where the curves intersect, shirkers will switch to participating. Here the intersection does not correspond to the NE, since at the intersection the right-most shirker will switch to participation. This will iterate until all are participating, and that is one of the game’s NE. However, if the game starts at or is exogenously shifted into a point to the left of the intersection then everyone will shirk; and that is the other NE. 18. In general, the larger N is the more likely it is that a collective action problem will have the structure of a PD, because in larger groups it is more typically the case that p(n + 1) < s(n) for all n than in small groups. 19. As noted earlier, collective action logic applies to many kinds of social situations, not just contributions to common pools. Thus p might designate ‘refrains from buying a gun’ and s might designate ‘buys a gun’ in a gun control dilemma. 20. N-person PDs with respect to use of finite, shared resources have a catchy name of their own. They are called ‘tragedies of the commons’ and are especially important in environmental economics. There are two broad kinds of solution to these: government control, in which people are incentivized not to take the free rider’s bonus by threat of punishment, and privatizing the resources so non-owners can’t free ride and owners are incentivized not to take the free rider’s bonus by the fact that returns on the asset are more valuable than the bonus. 21. Many collective action problems are solved by culturally evolved conventions, but people often don’t recognize conventions as solutions to such problems. When this is the case, people will tend to believe and repeat bogus accounts of the origins and justification of the conventions – that is, they will concoct myths, including religious ones. However, sometimes people solve collective action problems by deliberate policies aimed at changing incentives (often by changing institutional structures). Economists refer to such economic engineering as ‘mechanism design’. 24. Norms are culturally evolved mutual expectations in a group of people (or other social animals) that have a further property over and above those of customs or conventions. This is that people who violate norms often punish themselves by feeling guilt or shame. Religious stories (or bogus philosophical ones) are especially likely to be told in explanation of norms. 25. Collective action problems that instantiate Chicken, if they have NE in which some shirk and others participate, can be the hardest of all to solve because it is difficult to find institutions that will prevent resentment of the shirkers from overturning the equilibrium. Most societies include elites, many of whom don’t do very much work. Such elites must typically be careful to appear to make some visible contribution, even if this is merely providing entertainment by being regular targets of gossip and putting up with it. But of course human history is also filled with violent revolutions in which elites lost their heads. 26. Elites sometimes deal with this risk by oppressing non- elites. Why can a smaller group often oppress a larger one? The answer to this question itself adverts to collective action problems. To revolt against oppression, the oppressed must coordinate (e.g., attack the barricades at the same time). But whoever raises the call to revolt, so as to coordinate others, can expect to be shot at once. Thus all are incentivized to hang back, waiting for someone else to lead. Tyrants who are skilled at manipulating this dynamic are often able to remain in power for decades, even though hated by most of their unfortunate people. 27. Empirical surveys of collective action problems throughout history suggest that the following conditions are conducive to changing games and achieving Pareto improvements: (1) participants are identifiable and relatively stable; (2) benefits of cooperation are large enough to readily cover the enforcement bill; (3) members of the group are able to communicate directly with one another. 28. It is mainly because small groups are better able to solve collective action problems than large groups that small lobbies can often bend government policy to their will against the interests of much larger but disorganized sets of people. The outstanding example of this is trade policy, In almost all large countries on earth, well coordinated producer groups are able to take huge sums from the pockets of consumers – that is, everyone – by means of barriers to imports because consumers are too many and too dispersed to coordinate. 29. We now turn to slightly more technical discussion of spillovers, or externalities. 30. Recall that the total payoff T(n) to society of a collective- action game, when n of N people choose action P and (N – n) choose action S, is given by T(n) = np(n) + (N – n)s(n). Imagine one person switches from S to P. Then T(n + 1) = (n + 1)p(n +1) + (N – n + 1)s(n + 1). Then the difference in total social payoffs between these two states is T(n + 1) – T(n) = (n + 1)p(n +1) + (N – (n + 1)s(n + 1)) – [np(n) + (N – n)s(n)] = [p(n +1) – s(n)] + n[p(n + 1) – p(n)] + [N – (n + 1)] [(s(n + 1) – s(n)]. 31. This equation shows how the marginal gain to society brought about by an extra player of the socially preferred strategy is distributed among different groups of players. The first term, [p(n +1) – s(n)], gives the change in the payoff of the switcher herself. This is called the marginal private gain. 32. The switcher’s move changes the payoff of everyone else. These changes, imposed unilaterally on the others, are referred to as spillovers or externalities. Where they make others’ payoffs worse they’re called negative externalities; where they make others’ payoffs better they’re called positive externalities. In the equation, the second term, n[p(n + 1) – p(n)] represents the spillover from the switch to the n other people choosing P. Finally, the third term, [N – (n + 1)] [(s(n + 1) – s(n)], shows the spillover from the switch on the payoffs of the N – (n + 1) people still choosing S. 36. To derive the MP(n) curve, take the equation for total social gain and re-arrange it so it has one term containing the p variable but no s variables and one term containing the reverse, as follows: T(n + 1) – T(n) = {p(n + 1) + n[p(n + 1) – p(n)]} – {s(n) + [N – (n + 1)][s(n + 1) – s(n)]} The first term is the effect on the payoffs of the commuters who choose P; in this case, the marginal private gain of the switcher plus the sum of negative externalities. This is the marginal social payoff for the P-choosing subgroup when they go from n to n + 1; call it MP(n + 1). The second term gives the marginal social payoff for the S- choosing subgroup, MS(n). T(n + 1) – T(n) decreases when someone switches from S to P if MP(n + 1) < MS(n). In the example MP(n + 1) = 45 – (n + 1)  0.005 + n  (- 0.005) = 44.995 - 0.01n MS(n) = 15 = (in this case) s(n) We already worked out that these two curves meet at n = 2,999; so we represent this on the graph. 37. There are many ways of trying to prevent the 3,000th commuter from using the highway. But how might we equitably choose this commuter? If we try to come up with fair criteria, we must remember that people will then be incentivized to compete to satisfy the criteria; and that is itself a social cost. Another approach – though not necessarily fair if some people are wealthier than others – is to estimate the distribution of time preferences in the population and then set a toll such that the 3,000th commuter will be indifferent between using the highway or the local roads, the 2,999 commuters who value their time less than her will prefer to use the local roads to paying the toll, and the 2,999 commuters who value their time more than her will prefer to pay the toll to using the local roads. This means that each user of the highway will in fact pay for the negative externality created by his preference; this is called internalizing the externality (i.e., making it go away). Now we in effect have a market in time, with the more impatient commuters buying time from the others. 38. Wherever there’s a potential negative externality (in this case, switching from S to P) there is, as a matter of logic, a potential positive externality (in this case, switching from P to S). This means that the marginal social gain will be greater than the marginal private gain – so the good will tend to be undersupplied by selfishly motivated agents. In the example, there’s a positive externality associated with driving on back roads, and this good is undersupplied at NE. 39. Sometimes externalities can create feedback by enhancing the value to each individual of the good that carries the externality. (Dixit & Skeath say this is peculiar to positive externalities, but that is a confusion, as we’ll see shortly.) This creates bandwagon effects. If everyone wants to use the same computer operating system as everyone else, for the sake of cross-user flexibility, then all will end up using the same system in NE – and this is compatible with scenarios in which all would have been better off using the other system. The logic is most easily shown graphically: Users’ benefits U W All Windows I All Unix Number of Unix users Benefits from Unix Benefits from Windows