Chapter 19
Understanding the Game: Credible Threats and Signals
In This Chapter
Seeking an equilibrium concept that removes non-credible threats
Finding and using an equilibrium for positive signals
One wicked problem (as we use the term in Chapter 14) is that people rarely have perfect information, particularly when making strategic decisions in a game-like situation. Sometimes information is asymmetric — one party or player knowing more than another, as discussed in Chapter 15). Sometimes a player is unable to figure out the other’s best moves because the player is unsure about the type of opponent it faces in the game. In the Prisoner’s Dilemma from Chapter 16, for instance, neither party knows what the other one will do, but infers what is in the player’s interest from their payoffs. But what if a player is unsure about the payoffs of the other player because the other player could be a good guy with one set or a bad guy with another set of payoffs?
In reality, people typically do have imperfect or limited information. In a game-like situation they don’t know entirely what the other party will do, and so they look for information to go on in order to work out the right bargain or the right move.
Suppose, for example, you’re deciding whether or not to employ someone. Taking a look at her resume, you notice that she worked for a prestigious employer. You see that information as a sign that the person is a good bet to take on, and so you decide to call her in for an interview. A candidate with a less prestigious history may not fare so well.
Economists call the type of information in a resume a signal. It’s an imperfect piece of information about a person’s productivity in a new job, but it can be helpful to you in figuring out your next move.
In this chapter, we discuss signaling games where a player wants to signal or convey information to the other player about what type of player they are. In other words, these are games of asymmetric information, where one player knows their type but the other player does not. In each case, game theory looks at the signal and sees what payoffs you can infer from it.
Refining the Nash Equilibrium to Deal with Threats
In general, game theory looks at threats by dividing them into two categories: credible and non-credible. Distinguishing the two types of threat is difficult in real life, because it depends on you knowing quite a bit about the other party and what they might do.
In using a game theoretic model, we work from the assumption that the player issuing the threat of a future move is rational and plays the game in a way to secure the best payoff for that player. Therefore, you can evaluate a threat to see whether it leaves the player worse off — in terms of payoff — if carried out than not carrying out the threat. If so, we say the threat is non-credible.
In turn, this approach means using a stronger definition of the Nash equilibrium from Chapter 17 that eliminates non-credible threats: This equilibrium concept is called a subgame-perfect Nash equilibrium. If, after eliminating any non-credible threats, you still have a Nash equilibrium, it’s subgame perfect.
Finding a subgame-perfect Nash equilibrium by elimination
The best way to check whether a Nash equilibrium is subgame perfect is by backward induction (as explained in Chapter 17). To do so, it is best to write out the game in its extensive or tree form and work backwards from the end, eliminating any branch that a player has no incentive to move down.
If a rational person or entity wouldn’t do something, that action or behavior is non-credible from the standpoint of game theory.
A non-credible threat is a threat in a sequential game that a rational player would actually not carry out, because it would not be in their best interest to do so. For example, a frustrated parent driving the family to a summer vacation destination may threaten the kids quarrelling in the car by saying, “If you don’t start behaving, I am going to turn around the car and cancel the vacation.” The game-theoretic–savvy kids may reason that it is most unlikely that the parent would turn around and cancel the vacation — lose the deposit and so on. A better threat would be no dessert at lunch.
A non-credible threat is made in the hope that it will be believed, and therefore the threatening undesirable action will not have to be carried out. For a threat to be credible, it is one that should the occasion arise it will be fulfilled. In solving a game, we eliminate non-credible threats through backward induction, and the remaining equilibria are subgame-perfect Nash equilibria.
Exploring subgame perfection in an entry deterrence game
Here we explore a simple game and discuss how to eliminate non-credible actions. A common application of subgame perfection in economics is within games of entry deterrence — for instance, when a competitor in a market is trying to prevent an aggressive entrant from coming into that market. This game is appropriate in such a case because the working assumption about a business is that it is rational (otherwise it’s unlikely to stay a business very long), so weeding out irrational strategies is relatively easy.
In this game, NewCo is deciding whether to enter a market currently held by the monopolist OldStuff. NewCo gets to move first, deciding whether or not to enter the market — if it does, OldStuff decides whether or not to retaliate. Given the payoffs in the tree (see Figure 19-1), what’s the Nash equilibrium? In the figure, the numbers give the payoffs to the two companies from following the strategies in each branch of the tree.
© John Wiley & Sons, Inc.
Figure 19-1: Entry-deterrence game concluding no subgame-perfect equilibrium.
One Nash equilibrium is for NewCo not to enter, because if it does, OldStuff has threatened to cut its prices and punish NewCo. But is this threat credible? No, because it would leave OldStuff worse off, so the equilibrium isn’t subgame perfect; therefore, NewCo can see that it’s not credible — and therefore, NewCo can enter knowing that OldStuff can’t punish it without hurting itself.
In Chapter 13 (on monopoly), we say that a monopolist may decide to take losses in the short run to keep a competitor out. This could happen because the entrant may not know what kind of incumbent firm it will face if it enters — a high-cost one or a low-cost one. This is a situation in which one player does not know the payoff matrix of its opponent. Taking losses in the short run can be a way to convince the entrant that it is a low-cost rival and to keep out. We explore this strategy in the next section.
Employing game-theoretic reasoning in real-world situations can be challenging. You may not know all the necessary information. What if a company is deploying the madman strategy described in Chapter 16 to confuse a potential purchaser? Or what if you are dealing with an entity that does not behave rationally, such as a rogue nation or terrorist group?
Deterring entry: A guide to the dark arts
Given that we rule out strategies that involve taking a loss — because they’re non-credible threats — let’s dig deeper to see how a firm with a dominant position in a market could signal to a would-be entrant that it should stay out.
Firms can use a number of deterrent strategies in the real world, some of which may be illegal for larger companies under competition or antitrust laws. All these strategies have their basis in the entry-deterrence game, and so they all provide some reason to believe that they may signal a retaliatory action:
-
Limit pricing: A monopolist can price below its profit-maximizing monopoly price and make lower profits in the short run in order to deter entry in the longer run. This pricing behavior signals to an entrant that if it tries to grab market share, it’s going to be up against a low-cost competitor that can support lower prices, and it’s therefore unlikely to enter the market profitably.
Two problems exist with this tactic:
- Although economically sensible, it might attract the attention of regulators — who could be on the lookout for entry-deterring strategies. If you went as far as pricing below your average variable costs, you’d be acting illegally.
- The entrant may not interpret the pricing signal unless it’s accompanied by a statement of commitment to continue that pricing when it enters. Low prices means selling more output, so a commitment could be an investment in a new factory to support limit pricing.
- Creation of brand loyalty, and advertising: Investment in creating a brand name is a type of sunk cost — when the money is spent, it’s unrecoverable whether the investment is successful or not. The intention is to make the entrant see the cost of entry as including the cost of making a comparable effort in advertising, branding, and marketing. If the effort seems sufficiently high, the entrant stays away.
- Raising the cost of exit: Again, the way to achieve this is by creating a market environment that requires investments incurring large sunk costs. In that case, leaving the market will be costly — an entrant will have to think carefully about whether entering it is worthwhile because those investments will be lost if it leaves the market. The issue for policy makers is whether those investments benefit consumers or just raise rivals’ costs.
All these strategies require some degree of pre-commitment to be effective — that is, committing yourself to a course of action, usually by making an investment. The value in strategic terms is that it signals to another party that you’ve eliminated other possibilities, so that you can be sure you’re committed to whatever action you’re likely to take. If that action is costly to a rival, it acts as a deterrent signal.
The legendary Chinese strategist Sun Tzu recommended that if you want your army to win, you should burn your ships after landing. The idea is that if you cut off your own retreat, you’re signaling your determination to your opponent and therefore helping to break their morale — this is a very strong example of pre-commitment.
Signaling your good intentions
Of course, signals of pre-commitment aren’t always negative in the sense that they deter or limit competition — although the main examples used in the preceding sections are. Perhaps two economic agents want to facilitate market transactions and use signals positively, such as is often the case with product quality. A mark or certification of product quality can be a signal to the customer that your product is of superior value, for example.
The value of this type of mark to a consumer depends on how many sellers are able to fake a similar mark:
- If faking is hard: The quality guarantee helps to avoid the adverse selection problem detailed in Chapter 15.
- If faking is easy: The quality guarantee, if anything, supports the adverse selection problem and helps eliminate the market. This provides one other rationale for companies making strenuous efforts to defend the value of their brand. If the brands are fakeable, any quality signal is lost, and bad competitors drive out good ones.
If that isn’t sufficient, signaling quality by offering an expensive warranty program does the trick. The key thing that makes it doubly effective is for the warranty to be expensive. That way, the signal works in the right way and less scrupulous competitors are deterred from trying to fake it. (If in doubt, pull up the Groucho Marx quote from Chapter 16 for some sage advice.)
Responding to Positive Economic Signals
One of the important features of a market is that decision-makers respond to conditions without knowing for certain how things may turn out. As a result, participants in markets are often looking for information — to give them a sign of how things are going to be as much as for knowledge of how markets are right now.
Signals become most important when people and firms are looking for some guide to others’ intended actions. Often the best guide to deciding what to do for yourself is to look for something that gives you a sign that you may be making the right choice. In predicting markets, for instance, people look for signs of what direction prices may move in; in assessing unknowns, people look for any information that confirms their view of that unknown.
The decision whether to hire someone is a classic example of this problem. Only in retrospect do you know for sure what contribution a person is able to make to your firm’s value. When hiring, you have some information to go on, but there are a lot of unknowns about whether you are making the right hire.
Consider the level of an applicant’s education. You hope that it includes all kinds of useful skills to indicate that hiring the person will enhance your company’s profitability, but you don’t know for certain until you hire the person. But even if you assume the worst — that education gives little or no benefit — looking at the applicant’s education level and choosing the candidate with the most completed years may still be worthwhile.
Why? Completing a certain number of years of education is a signal that the person has a certain amount of stickability in the job — that she won’t bail when the work looks too hard — and that she’s reliable, as her continued attendance in education demonstrates.
This section describes a model and equilibrium to show you how signaling a person’s ability to stick at a job would affect wages for two different groups. This is, if you like, a use of a positive signal — a signal of employability. (We could, in fact, reverse some of the assumptions and use a similar model to show how, for instance, a criminal record signals poorer ability.) In this case, though, we use a classic model to show you how signals affect a market and how the use of these positive signals may not lead to an outcome that’s better for society.
Investigating signaling with a model
Economists use a framework developed in the 1970s by Michael Spence to analyze many signaling problems, the most famous of which is looking at the value of a university degree. In Spence’s model, asymmetric information exists (see Chapter 15 for more on that) because one party knows more about its value than another. In this case, the applicant knows more about her potential effort and ability or value than the hirer. The employer therefore looks for a signal — something that captures, albeit imperfectly, the ability of the applicant.
Spence’s framework has two types of applicants who are distinguished by their marginal product — that is, the amount of value that they add to the organization, denoted by a:
- Good workers have a marginal product of a2.
- Bad workers have a product of a1 with the constraint that a2 > a1.
This framework provides a simple way of making a clear distinction between the workers, and also leads to a first intuition: If you can clearly differentiate between the workers, working out what wages each type will receive in a competitive market is easy. Competition will drive wages w1 = a1 for the poor workers and w2 = a2 for the good workers (where w = wages of any given worker). A wage equal to the marginal product of labor in the two groups is also efficient.
Setting up the basic model
We continue to assume a competitive labor market, but now a firm in that market cannot distinguish between the two groups of workers beforehand. But suppose the firm knows that the proportion of good workers in the population is b and the proportion of poor workers is (1 – b). Here’s what happens if it offers a wage relative to the average quality of the workers:
This may look familiar to the problem from Chapter 15 — where a population or a product is of two types, most of which end in market failure. Here, though, the good workers may be able to find a way of telling the potential employer which group they’re in — in other words, their positive signal.
The signal in this case is how much education they’ve undergone, and it’s related to a cost of getting an education — and specifically to the opportunity cost to the two different types of workers. Suppose that type 1 workers incur a cost c1 per level to acquire an e1 level of education, and that the respective figures for type 2 workers are c2 and e2. That means the total cost incurred by a type 1 worker is e1c1. Firms can observe a worker’s education level before hiring.
Now each party needs to make a decision:
- The firm needs to decide what to pay workers with different levels of education.
- The worker needs to decide what level of education to get.
To look at these decisions in the context of signaling, assume that there is no particular gain in productivity from any level of education. This seems ridiculous, we know, but the important part of the model at this point is the signal, not the productivity.
Supposing that the marginal cost of getting educated is lower for better workers means that c2 < c1. Also, a2 > a1, so a unique level of education e* must exist that satisfies the following:
Finding a signaling equilibrium
Assume that the signals hold perfectly and education is a perfect signal of the worker’s marginal product. In that case, the wage must equal the marginal product, so a perfect signal is tantamount to the equilibrium where firms can observe workers directly. Now consider an outcome in which a poor worker chooses zero education, whereas a good worker chooses e*.
Is this an equilibrium? For the firm, yes. It has a clear way, based on the choice of signal, to choose between workers, and can set a wage based on their marginal product.
What about for the poorer quality workers, those with product of a1? Is it optimal for them to acquire e* of education? The benefit would be a2 – a1 but the cost would be c1e*. So a poor quality worker will have an incentive to get e* levels of education only if
But you know from the choice of e* that c1e*is bigger than a2 – a1. So, in this model, the benefits of getting e* of education are less than the cost incurred, and so the poorer quality workers will choose equilibrium, zero education.
For the better quality workers, it’s worth getting educated if a2 – a1 > c2e*. The choice of e* implies that this is true, so this logic works for better workers too.
Therefore, the equilibrium holds. Under these conditions, neither type of worker has any reason to change behavior.
This equilibrium is called a separating equilibrium, one based on subgroups of people within a population — whether producers or consumers — behaving differently from each other. The opposite, where they make the same choice, is called a pooling equilibrium. A pooling equilibrium would be if both types of workers acquire the same level of education e' and receive the pooling wage of ba2 + (1-b)a1, and no worker has an incentive to deviate and acquire a different level of education.
Evaluating the signaling equilibrium
The signaling equilibrium has some interesting implications for policy makers. The most important is that even if you believe that education has no effect on productivity, it’s still worth getting if you’re a high-productivity worker and want a higher wage.
In a Yale lecture, microeconomist Ben Polak describes the model as having a pessimistic conclusion. His points are as follows:
- For the signal to work, it has to be costly, and so c1 > c2. Within Spence’s framework, that means that the cost of education is its key feature. So what you studied doesn’t matter, only that you paid — in money or sweat — to study it.
- If that’s the point of education, spending money on teachers’ salaries is socially wasteful. (We prefer that you don’t believe that conclusion.)
- Given that the value of the signal is related to its cost, a separating equilibrium like this one tends to exclude any high-productivity worker who starts out too poor to afford the opportunity cost c2. So, signaling can hurt the poor and increases inequality.
Those conclusions are indeed pessimistic — for educators at least. Are they justified? Well, as long as you believe that the sole purpose of education is in signaling, you’d be entirely justified in holding those conclusions.
But education isn’t only a signal. Many reasons exist for valuing education and they are not related to signaling. For instance, education can provide important skills that improve productivity. In addition, education confers social benefits such as active citizenship and better health outcomes. The signaling framework leaves these features of education out of the model.
So here’s another lesson: Models can be garbage in, garbage out. If the thing you want to know about isn’t built into the model, you’re unlikely to be able to draw robust conclusions from it.