Lecture Notes 16: Collective Action
A collective action problem is one in which the aims of society are best served if members take some particular action or actions, but these actions are not in the private best interest of individual members. In game theoretic language, a collective action problem is one where the socially optimal outcome is not an equilibrium. Strategically, collective action problems often take on the structure of large prisoners’ dilemmas: there is an outcome that leaves everyone in society better off, but each individual member of society can deviate from this outcome to improve his own well-being.
Excessive Competition
• Schools in the US spend outrageous amounts of money on sports teams. If all the schools would get together and agree to only spend half as much on sports, the total number of wins and losses across schools obviously wouldn’t change, but all schools would be saving money. Of course, it’s unlikely that the agreement will work. Each individual school is better off deviating and increasing its spending to grab more of the wins for itself.
• Firms spend huge amounts of money on advertising, most of which does not actually generate any net increase in the number of customers for the market. For example, advertising cars doesn’t make people buy a car, but it might make people switch among different brands of cars. Of course, on an aggregate level, the effects cancel each other out. All firms expend huge sums on advertising, with no overall increase in the number of customers. Why don’t they just make a deal to cut back on advertising? While all firms would make more profit if they did so, each firm would have an incentive to cheat on the deal, advertise, and increase its own profit at the expense of other firms.1
• A professor grades a class on a fixed curve. The students get together and make a deal that there’s no point in working hard since the total number of high grades is fixed anyway. But each student is better off working hard to try to get one of those high grades for himself.
The common feature of these problems is that each player prefers to improve his relative position, but the absolute totals are fixed. Problems of this structure are collective action problems. Everyone is better off if they can all agree to invest less, but then each individual’s interest is served by deviating and trying to improve his own position.
1 There is an interesting case study along these lines. When the US banned cigarette advertising on television in the
Traffic Congestion
10,000 people travel to work each day and can choose to take the train or drive over the bridge. Suppose that 𝑛𝑛 of these people choose to drive.
• The travel time for the train is 40 minutes.
• The bridge is subject to congestion, and the travel time for the bridge is 20 + 0.005𝑛𝑛 minutes. In other words, each additional driver on the bridge raises travel time for all other drivers by 0.005 minutes.
The Nash Equilibrium must equate travel time on the bridge and on the train. Otherwise, drivers would be better off switching to the one with the lower travel time.
40 = 20 + 0.005𝑛𝑛 ⇒ 𝑛𝑛𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 4000
The payoff diagram is shown below (the train in San Francisco is called BART). Notice that the equilibrium is stable. If slightly more than 4000 people drive, then the bridge takes longer and people prefer to take the train. But if fewer than 4000 people drive, then it’s better to use the bridge, so the number of drivers will increase. Travel time is equated at 40 minutes for both groups.
But this equilibrium is not socially optimal. The social optimum would minimize total travel time across all drivers.
• Total travel time for the 𝑛𝑛 drivers is 𝑛𝑛 ⋅ (20 + 0.005𝑛𝑛)
• Total travel time for the 10,000 − 𝑛𝑛 people who use the train is (10000 − 𝑛𝑛) ⋅ 40
𝑛𝑛 ⋅ (20 + 0.005𝑛𝑛) + (10000 − 𝑛𝑛) ⋅ 40 = 0.005𝑛𝑛2− 20𝑛𝑛 + 400,000
Choosing the number of drivers 𝑛𝑛 to minimize total driving time gives:
0.01𝑛𝑛 − 20 = 0 ⇒ 𝑛𝑛𝑜𝑜𝑜𝑜𝑜𝑜𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒= 2000
It is easy to see that the optimal 𝑛𝑛 = 2000 drivers is more efficient than the equilibrium 𝑛𝑛 = 4000 drivers. The 2000 people who switch from road to rail continue to take 40 minutes for the commute, just like they did when they were driving. But the travel time is lower for the remaining 2000 people on the road. Thus, the sum of total travel time is clearly lower.
Again, this is a collective action problem. The efficient outcome for society is not the equilibrium outcome.
In economic terminology, the essence of the problem in this example is that each driver is paying only his own cost of using the bridge. He is not considering the external costs imposed on other drivers by increasing traffic congestion. One solution is a road tax. Economists say that taxes like this lead to a “double dividend” since they not only lead to efficient usage of the road, but they also generate revenue for the government. Another solution is private ownership of the bridge. The owner would maximize revenue from toll collections by maximizing the value of time saved.
Coordination (“QWERTY”) Problems
The continued use of QWERTY keyboards is pure historical accident. The first people who developed typewriters needed to configure the keys to minimize the chance that the hammers on the typewriter would jam with each other. The QWERTY keyboard was not designed for the purpose of maximizing typing efficiency in any way.
The Dvorak Simplified Keyboard (DSK) was developed in 1936. Evidence is unequivocal that DSK increases typing speed, reduces errors and reduces strain on the wrists. Furthermore, typists can be retrained to use DSK in as little as 10 days.
Consider an evolutionary-type model where new typists can decide to learn QWERTY or learn DSK. The value of learning each depends on the fraction of typists who use QWERTY. Thus, the chance that a new typist will learn QWERTY depends on the fraction of existing typists who use QWERTY.
Notice that the diagram shows the superiority of DSK. Even if 70% of typists currently use QWERTY, there is only a 50% chance that the next typist will want to do so.
The stable equilibria are for everyone to use QWERTY or for everyone to use DSK. QWERTY is inferior and will tend to take over only when the initial condition is that about 70% of typists are currently using QWERTY.
It’s a bad equilibrium for everyone to use QWERTY, but it’s still an equilibrium. Sociologists might call it a “positive feedback loop”. When everyone else uses QWERTY, all the machines are designed to QWERTY specifications, so you start to use QWERTY yourself. All the manufacturers design QWERTY keyboards, which only makes the problem harder to overcome.
The only way to overcome this is some kind of coordinated, collective action. Perhaps the government (which employs a large number of people) or a major computer manufacturer bites the bullet and forces a switch to DSK. They don’t need to get everyone to switch. Once about 30% or so switch, then typists are better off learning DSK on their own. The problem organically fixes itself once we get over the initial hump.
There are some other interesting historical examples.
everyone uses Microsoft, so most software is designed to be used with Microsoft operating systems. Thus it’s better for new users to use Microsoft, etc… We might all agree that life would be better if we all switched to Unix, but no one person individually can change the world. The diagram below shows the problem. Benefits are higher in the equilibrium where everyone uses Unix, but we’re currently at the equilibrium where everyone uses Windows. Overcoming it requires some critical mass of people to switch.
• Natural gas vs. gasoline for cars: Oil was cheaper when cars first came into wide use, and nobody cared about pollution. Now, natural gas is cheaper and it’s cleaner. The problem is that switching would require a huge infrastructure change. People don’t want to buy cars with natural gas engines unless there are plenty of gas stations selling natural gas. But it’s not worth it to build stations unless there are lots of people driving cars with natural gas engines. What a discouraging problem!
Again, these are collective action problems. Society has gotten stuck with inferior outcomes by historical accident and an initial condition. But getting out of QWERTY-type problems requires collective, coordinated action.
Speeding Enforcement
Consider the incentives when you decide whether to speed.
• If almost everyone follows speed limits, then you should follow the speed limit as well. It’s better to drive with the flow of traffic, and you are likely to get caught if you speed when everyone else is obeying the speed limit.
The diagram below shows the incentive structure.
The stable equilibria are for everyone to speed or for nobody to speed.
In terms of public policy, this model has some interesting implications. The key is to get a critical mass of drivers to obey the speed limit, and then from that point on it’s self-enforcing. A good solution is to have periods of short, intensive enforcement that scare nearly everyone into obeying the speed limits. Then, once most people obey the speed limit, others naturally will and we’re settled at the equilibrium where everyone obeys. This might be better than a long-term enforcement policy that is moderate and catches only a few speeders here and there.
Racial Segregation
American neighborhoods are extremely segregated by race. By some estimates, 85% of black people would have to move to different areas in order for neighborhoods in American cities to be fully integrated.
The answer is no. We can interpret segregation as a bad outcome of a collective action problem. In other words, even if everyone in his heart truly believes that it would be better to live in an integrated neighborhood, unilateral action by individuals can never overcome the problem.
Here is a simple model. Consider a dynamic process where the probability that the next entrant to a neighborhood is white depends on the current percentage of whites.
Most whites do not need to live in a neighborhood that is 95% white, and indeed might prefer a more integrated neighborhood. But many whites or blacks would be uncomfortable living in a neighborhood where they only constitute, say, 5% of the population.
That is what this diagram illustrates. If more than 70% of the residents of a neighborhood are white, then there is a high chance that the next resident will be white as well. Similarly, if very few residents are white, then there is a low chance that the next resident will be white. Thus, the model has equilibria where the neighborhood has no whites or is 100% white.
Unfortunately, the only stable equilibria are segregated neighborhoods. One way to interpret the problem is as an externality. If a white family leaves, it makes the neighborhood slightly less attractive for other whites; vice versa if a black family leaves. But the family is not assessed a fine for the harm it is causing to its neighbors.
Here are a few solutions that might help to preserve an integrated racial balance.
• Charge a fine for people who leave.
• Ban the use of “For Sale” signs on houses. Signs like that can contribute to the tipping problem. If a black family sees that other black families are selling their homes, this might make them more likely to want to leave themselves.
• Insure homeowners against changes in property value. If people are afraid that their property values might change because of changes in the racial mix, this fear can accelerate the tipping. If a family anticipates a change, it might try to unload its house now to preserve its value. Of course, our family’s house sale actually contributes to the problem. Insurance can ease these kinds of fears.
Sequential Voting
A group of 10 junior law partners are ranked from 1 (worst) and 10 (best), then put in a room and told to decide for themselves how the cutoff should be set for promotion to senior partner. The only rule is that all decisions have to be made by majority vote of all 10.
Suppose that they start off agreeing to give everyone the promotion. Then, some bright guy has the idea to raise the cutoff to 2. Obviously person 1 votes no, but everyone whose rank is 2 and above votes for the change since it improves the quality of the pool.
Now someone proposes to raise the cutoff to 3. Person 2 votes no. But everyone 3 and above votes yes in order to increase the quality of the pool. Person 1 votes yes also! It’s not so embarrassing not to get the promotion when other people don’t get it either.2
Now you see the problem. The cutoff keeps getting raised one by one. The marginal person strongly votes no, but all the people above like it, and all the people below like it too. Every increase passes by majority.
2 If you’re going to fail, might as well fail at something big. I’d rather admit to being unable to climb Mount Everest
Finally, once only person 10 is left, the other 9 vote to keep him from getting the promotion as well! Ultimately, none of them end up with the promotion, even though they would have all been better off not to have embarked on these votes at all, in which case all of them would have gotten the promotion.
The problem is one of intensity. For each one-step increase in the cutoff, the person who votes no is strongly opposed to raising the cutoff, because he doesn’t get the promotion. But the ones who vote yes have a weak preference for raising it. Ultimately, nine small victories are more than outweighed by one huge loss for each person.
It might be better to look ahead and set up the rules to prevent these small steps. Perhaps the partners should just force a vote right away on whether to promote everyone. The small, sequential steps might look attractive one by one, but they can be dangerous at the end.
An interesting example of this problem is congressional pay raises. If you are a congressman, you want the pay raise to pass, but you look better if you yourself vote against it. Even if everyone wants the raise, one or two defectors might figure that they prefer to vote no, look good, and get the raise anyway. But having defectors makes it even more costly for others to vote in favor of the raise, so there will be more and more defectors. The raise might actually fail, even though everyone wants it!
Public Goods
Suppose that there are a large number of people in a society. 𝑛𝑛 of them choose to pay to provide some public good (𝑃𝑃). The rest of them shirk, and don’t invest in the public good (𝑆𝑆). Investing in the public good creates benefits for everyone in society, but is costly for the people who undertake the investment.
In the traditional setup, the best outcome for society is for everyone to choose 𝑃𝑃, but it is always in the individual best interest of each member of society to free-ride and choose 𝑆𝑆. This is effectively a prisoners’ dilemma.
But a different cost / benefit structure might lead to a different outcome. Suppose that there are diminishing returns to the number of providers. In other words, suppose that the benefits of each additional provider are not very large on the margin. An example might be supervising the neighborhood. It’s a public good, but as long as there are a few people supervising, there’s not much extra benefit to having additional people join in.
Finally, consider a case where the benefit of being a provider grows rapidly as the number of providers rises. An example is a changing social norm. For example, if other people help the police or accost people who are smoking in public (which creates a public goods effect), then it becomes easier for you to do it.
In this case, there are two stable equilibria. If nobody is providing the public good (𝑛𝑛 = 0) then you’re also better off shirking. But if everyone is providing the public good, then you too are better off going along. In other words, the stable equilibria are for everyone to invest in the public good or for nobody to invest in the public good. Getting to the better equilibrium just requires crossing some initial threshold of people to go along. Then everyone else does so voluntarily.
The Economics of Collective Action
Collective action is not a new problem in economics, although early economists who treated the problem did not realize how neatly it fits into game theoretic terms.
Classical economics (a la Adam Smith) asserts that leaving people free to make their own choices leads to the best outcome for society. But this section seems to present a whole bunch of problems for which this “invisible hand” result is flat-out wrong. In these examples, each individual pursuing his own best interest leads to an outcome that is worse for everyone.
But a problem occurs when benefits and harms are not priced in. What happens when there are spillovers? If one person’s choices can impact the welfare of other people – as in public goods provision or racial segregation – then individual decision making will generally not lead to an optimal outcome. This problem includes the well-known cases of public goods and externalities. The tragedy of the commons (overuse of common resources) has also been well-understood for a long time and falls into this general class of problems.
The invisible hand result is very profound and important, but it often leads people to a maniacal support of free markets without any real understanding of the limitations of Smith’s idea. Economists do support freedom and choice as a general principle, but you have to understand when the optimality result breaks down. When actions have spillovers, Smith’s result is generally false.
What is the best way to solve collective action problems? In other words, how can we compel people to act against their own best interests for a greater common good? The traditional answer given by social scientists is social norms. If it is custom or convention for everyone to behave a certain way, then people might be embarrassed to deviate for their own selfish purposes. Here are some considerations in the usefulness of social norms as a way to overcome collective action problems. Think about being hospitable to visitors or using good manners. It creates a cost for you to observe the social norm, but all society is better if people can agree to do it.
• You want some confidence, if you’re going to bear a cost to contribute to the common good, that others will do the same thing. A long history of widespread compliance is helpful.
• Education and experience living in a culture are a good way to solidify social norms. Repetition helps.
• It helps if there is some way to punish people for uncooperative actions. If we all agree not to smoke in public, perhaps a person who violates this social norm feels social stigma from his neighbors.
• Some economists talk about a “warm-glow contribution” where people feel a psychological benefit to making their own donation to public projects, above and beyond the good actually being provided. The idea is that you feel some kind of ownership.
Problems
1. Consider a state where residents can choose whether to live in Alphaville or in Betaville. Let 𝑥𝑥 denote the fraction of residents living in Alphaville. The payoffs to living in either town are shown below as a function of 𝑥𝑥. The solid function is the payoff from living in Alphaville and the dashed function is the payoff from living in Betaville.
Notice that Alphaville is most pleasant when 40% of the state’s residents live there (i.e. when 𝑥𝑥 = 0.4) and then payoffs fall when there are more or fewer residents. Similarly, Betaville is most pleasant when 40% of the state’s residents live there (i.e. when 𝑥𝑥 = 0.6) and then payoffs fall when there are more or fewer residents.
Identify all equilibrium values of 𝑥𝑥. For each equilibrium, indicate whether it is stable.
2. In a class with 30 students, a professor asks an interesting exam question. Each student chooses YES or NO. Students who choose NO get zero points. Students who choose YES get a 10 point bonus if fewer than 15 students chose YES. But students who choose YES have 10 points deducted from their score if 15 or more students chose YES.
a. What are the Nash Equilibria?
b. What do you think would actually happen if students had to make their choices individually with no communication?
3. A group of countries 𝑖𝑖 = 1, … ,12 is considering whether to form a monetary union. Each stands to gain more from joining the union when many countries join. If 𝑛𝑛 is the total number of countries joining, the payoff to country 𝑖𝑖 from joining (IN) and the payoff from not joining (OUT) are given below:
Π𝐼𝐼𝐼𝐼 = 2.2 + 𝑛𝑛 − 𝑖𝑖 Π𝑂𝑂𝑂𝑂𝑂𝑂 = 𝑖𝑖 − 𝑛𝑛
Note that each country 𝑖𝑖 = 1, … ,12 differs in its own payoffs.
a. Show that IN is a dominant strategy for country 𝑖𝑖 = 1.
b. If we eliminate OUT for country 𝑖𝑖 = 1, show that IN is now a dominant strategy for country 𝑖𝑖 = 2.
c. Argue that all countries will choose IN.
d. Contrast the payoffs when all countries choose IN with the payoffs when all countries choose OUT. Which is more efficient? How many countries are made worse off by the formation of the monetary union?
4. (Braess’ Paradox) Consider a road network with 4000 drivers who drive every day from start to end. 𝑡𝑡 is the travel time for the segment and 𝑇𝑇 is the total number of drivers using the segment. The diagram is shown below.
a. Suppose initially that there is no link between A and B. Drivers must choose the top segments passing through A or the bottom segments passing through B. How long does the trip from start to end take in equilibrium?
b. Suppose now that a city planner wants to add capacity to the network by completing the dashed link between A and B. Assume that the link is very short and that the link itself adds nothing to travel time. How long does the trip from start to end take in equilibrium now? (Big hint: Look for a dominant strategy for the first segment and the second segment separately).
