Interdependence, Game Theory, and Markets

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The Prisoners’ Dilemma

Suppose that two students from your school have just been apprehended by the police for allegedly possessing illicit substances, possibly with the intention of reselling (or "trafficking in") these substances. For identification purposes, let’s refer to the first student as Psmith. The second student will be known as Joans.

The police place Psmith and Joans in separate cells, not allowing them to communicate with each other, and interrogate them in separate interview rooms. During the interrogations, each prisoner is offered the following choice: if you confess to being a dealer in partnership with the other prisoner, you will go free and the other prisoner will be imprisoned for ten years. Of course, though, you cannot both be witnesses for the Crown; if you both confess, you will each receive prison sentences of six years. But if neither of you confesses to being a dealer, we will still be able to convict you of possession, and you will be imprisoned for one year.

These options generate the dilemma for Psmith and Joans. While each one would like to go free and avoid even one year in prison, they both have some misgivings about finking on their compatriot. At the same time, they have little or no desire to be the patsy in the case and serve the full ten years while the other goes free. If they could coordinate their behaviour, they would each likely remain silent and serve just one year. In the absence of coordination, though, they are both just as likely to confess and end up serving six-year sentences. The source of their dilemma, then, is their inability to communicate and coordinate their actions.

The choices available to Psmith and Joans, along with the possible outcomes, are shown in Figure 12-1, which is called a payoff matrix. Psmith’s two choices are listed down the left side of the matrix: confess and don’t confess. Joans’ two choices are listed across the top of the matrix: again, confess and don’t confess.

There are four possible outcomes of the choices the two students might make. These outcomes are outlined with dark lines; each possible outcome is called a cell in the payoff matrix.

 

 

 

Joans’

Choices

Don’t Confess

Confess

  Don’t

1 year

no time

Psmith’s Confess
1 year
10 years
Choices Confess

10 years

6 years

    no time 6 years

Figure 12-1 The Prisoners’ Dilemma Payoff Matrix

Within each cell is a diagonal line (in the text, not here on the website). The number above and to the right of the diagonal line shows the sentence that Joans will receive if the two of them choose that combination of strategies; Psmith’s sentence is shown below and to the left of the diagonal line. For example, if Joans were to remain silent but Psmith confessed, the outcome would be shown in the lower left-hand cell of the payoff matrix — Joans would receive a ten-year sentence, and Psmith would go free (i.e. receive a zero sentence — do no time).

Whether Psmith or Joans (or both or neither) will confess cannot be answered by relying only on the payoff matrix and whatever underlying mathematics one may wish to invoke to study the problem. The optimal choice for each of them can emerge only after we know something else about the situation. The additional information necessary to solve the dilemma falls into three broad categories: (1) the expected detection lag, (2) the expected retaliation lag, and (3) the expected size of the retaliation. Let’s examine each of these in turn.

Expected Detection Lag

If Psmith (for some unfathomable reason) believes that Joans will never find out that Psmith confessed, then Psmith might give very serious consideration to confessing. "After all", the reasoning might go, "If I confess, then I can go free and Joans will never figure out what happened." If, at the same time, Joans goes through the same inexplicable thought process, Joans will also see that there are large expected benefits from confessing and zero expected costs from confessing. And if they both reason through the problem in this way, they will both confess; and they will both end up serving six years. You might well wonder how either Psmith or Joans could be so stupid as to imagine the other would not figure out they had confessed. You are probably correct. But this scenario has considerable applicability later in the chapter.

Suppose that Psmith, knowing that he has an incentive to confess, realizes that Joans also has an incentive to confess due to the long (infinite, with our assumptions) detection lag. In this case even if Psmith, for some reason such as honour or loyalty, had wanted to remain silent, he will grow quite concerned about what Joans will do. If Psmith anticipates that with high probability Joans will confess, then Psmith can make himself better off by confessing. A six-year sentence is clearly better than the ten-year sentence he would receive if Joans confesses and he doesn’t. Joans will likely go through the same type of thought process: "If Psmith is going to confess, I’ll be better off if I do, too, so I might as well." Even if Psmith and Joans are wrong in their expectations, so long as they believe that the detection lag is long, then they have little or nothing to fear in the way of retaliation from the other, and they will be likely to confess. But if the detection lag is short, or if the expected probability of detection is high, then whether they decided to confess will depend on what they expect the consequences will be — what will happen to them if they confess and the other finds out they confessed.

Expected Retaliation Lag

The consequences that Psmith and Joans will anticipate if they confess have two components: time and size. The time component is called the retaliation lag. If Psmith believes that Joans will not be able to retaliate for a long time, then Psmith might be more likely to confess. For example, if Psmith thinks there is a chance Joans will not confess, Psmith can confess and anticipate being free from retaliation by Joans for ten years. Even when the detection lag is short, if the retaliation lag is this long, Psmith might consider confessing.

Of course, if Psmith considers confessing under these conditions, he must expect that Joans will do so, too. And expecting that Joans is likely to confess, Psmith will weigh the options of six years versus ten years and be even more likely to confess.

But just because Joans is locked away for six or ten years doesn’t mean that the retaliation lag would necessarily be that long. If Psmith breaks some "code of honour" by confessing, Joans may well have some friends or family members who could inflict some form of retaliation on Psmith or on his friends and family within a very short period of time. If, for example, Psmith and Joans are members of an organization that shares this code of honour, they might very well fear that retaliation would be swift. And if they have good reason to fear swift retaliation, they will probably think twice about confessing. The effect is that both Psmith and Joans will be better off, on average, if they believe that the other person would have both a short detection lag and a short retaliation lag; under these circumstances, neither would be as likely to confess, and they would each serve only one year in prison.

Expected Size of the Retaliation

Even if Joans can detect that Psmith has confessed, and even if Joans is able to retaliate quickly, Psmith may still choose to confess if he believes that Joans will not be able to inflict much punishment on him for his having confessed. If the worst that Joans can do in the way of retaliation is to say, "Well, the next time I’m going to confess," or "I’m not going to be your friend anymore," then Psmith may decide that this is not going to be very much deterrence and confess anyway. And if the possibilities are symmetrical, i.e. if Psmith can similarly impose no harsh penalties on Joans for confessing, then they will both have an incentive to confess, and they will both end up serving six years. Furthermore, each student, knowing that the other expects little in the way of deterrence, will not want to be left holding the bag, and will decide to confess as well. Witness protection programs are designed to reduce the size of the expected retaliation by making it extremely unlikely that Joans could inflict suffering on Psmith should Psmith decide to confess.

If, however, Psmith fears that Joans (or some of Joans’ associates) will inflict serious punishment on Psmith or possibly some members of Psmith’s family, should he confess, then even if the retaliation lag is fairly long, Psmith may have serious qualms about whether to confess. The phrase, "I’ll hunt you down, even if it takes the rest of my life," conveys this threat quite effectively. And if both Psmith and Joans believe the other will carry out substantial retaliation, they will both choose not confessing rather than confessing.

Notice that if the students are similar in their abilities and their expectations, then the likely outcome of the game is that either both will confess or neither will confess. If one of them thinks it advantageous to confess, it is quite likely that the other one will, too. And if they think the other one will confess, they will realize that their own best strategy will be to confess — sort of making the best of a bad situation. By the same token, if one of them thinks the other is likely to be able to detect the confession and retaliate swiftly and strongly, they will be unlikely to confess — and by symmetry they are both likely to assess the situation this way. The conclusion, then, is that the most likely outcome is that both will choose to do the same thing, either confess or not confess; it is unlikely that one will choose to confess while the other chooses to remain silent. And whether they both choose to confess will depend on the expected length of the detection lag, the expected length of the retaliation lag, and the expected size of the retaliation.

LAGS AND PROBABILITIES... (see the text)

INTERDEPENDENCE AND BUSINESS DECISIONS

Now let’s see how the prisoners’ dilemma game can help us understand business decisions when rivals are aware of their interdependencies. Suppose that Smith and Jones are direct rivals in business, selling very similar products to the same potential pool of customers. Up until now, Smith and Jones have been cooperating, and charging a price that would maximize their joint returns, each earning 14% on their invested capital. But because there are laws against explicit price coordination, as we shall see in the next chapter, Smith and Jones are not allowed to continue their explicit cooperation. What will they do?

To answer this question, we begin by setting out the options, and to keep the analysis simple, we will limit their choices to just two options: cut prices or keep prices at the current level. Smith considers cutting her price. She anticipates that if she does so, and if Jones doesn’t follow suit, she can earn a 20% rate of return, while Jones will earn only 8%. However, if Jones matches her price cut, they will each earn a 10% rate of return. These options, along with the expected payoffs to both players, are shown in the payoff matrix in Figure 12-2.

 

Jones’

Choices

Don’t Cut Prices

Cut Prices

 

Don’t Cut

14%

20%

Smith’s

Prices


14%

8%

Choices

Cut

8%

10%

 

Prices

20% 10%

Figure 12-2 A Price-Cutting Payoff Matrix

The problems of the prisoners’ dilemma appear again: both players realize that they can be better off if they cut prices while the other one doesn’t, but they realize they will be worse off if they both cut prices. Should they cut prices or keep them at their current levels?

DETECTION LAG (AGAIN)

If Smith can cut prices without Jones finding out about the price cut, Smith can move to the lower left-hand cell of the payoff matrix and earn a 20% rate of return, leaving Jones with only an 8% rate of return. Jones would have to be pretty stupid not to see the price change in this situation, though. All it would take would be a visit to Smith’s store to check prices now and then. But Jones could probably figure it out even without the visits: if Jones’ sales begin falling precipitously for no apparent reason, Jones might well begin to suspect that Smith has cut her prices. In fact some customers, trying to get a better price from Jones might say, "I’ll buy from you if you can match Smith’s lower prices; otherwise I’ll go to Smith."

To counter this possibility, firms sometimes do not announce their price cuts. By not issuing new price lists, the firms can then offer different discounts to different customers — price discrimination, as we saw in Chapter 11. But they also increase the detection lag. By offering secret discounts to some customers, and pleading with them not to tell Jones, Smith can increase her profits and delay the time when Jones becomes aware of her price cuts.

The customers may at first wonder why they should go along with Smith. After all, they might be able to get even lower prices from Jones by telling him about Smith’s new, lower prices. The only reason they will go along with Smith would be that they are involved in a game with Smith, too. If they tell Jones about Smith’s lower prices, they might be able to get an even lower price from Jones now. But then Smith (if she finds out they told Jones) might not offer the secret price cuts in the future. If they are frequent customers of Smith, this will be important to them over the longer run, and they will be more likely to honour Smith’s request. If, however, they purchase something from Smith only rarely, and especially if the item is high-priced, they might well be better off by violating Smith’s confidence, telling Jones, and trying to get an even lower price from Jones. Even if Smith tries to retaliate by not offering a price cut in the future, if the retaliation lag is long and the expected gains to the customer from telling Jones are large, customers will be likely to tell Jones. And knowing this, Smith will be less likely to try to offer secret discounts to these customers.

Even if Smith is able to grant off-list discounts to her customers, though, the chances are good that Jones will eventually learn about her price cuts (by noticing that his own sales are declining, if nothing else), and Jones is not likely to sit on his thumbs while Smith rakes in the extra profits.

RETALIATION LAG (AGAIN)

How long will it take Jones to cut his prices to match Smith’s? With computerized pricing, even in large bureaucracies Jones can key in a few numbers and have prices changed in all his branches within hours, if not minutes. The retaliation lag in this case is short. But if it takes some time to prepare new advertisements with the lower prices and to place the ads strategically, the retaliation lag might be as much as several weeks or even a month or two. Meanwhile Smith has a price advantage and is cleaning up.

Different marketing tactics have different expected retaliatory lags. Price cuts are often quite easy to match fairly quickly; they have a short detection lag and a short retaliatory lag. But unique ad campaigns, while easy to detect quickly, are more difficult to counter quickly. So when there are comparatively few rivals squaring off in head-to-head competition, we often see them engaging in what we call non-price competition. Price competition is much rarer, though not completely absent.

EXPECTED SIZE OF THE RETALIATION (AGAIN)

What is the worst that Jones can do to retaliate against Smith’s price cuts? What type of deterrent action can Jones take that will serve as an effective threat against further price cuts by Smith? Ruling out illegal and violent action, about the only options available to Jones are for him to change his own marketing program. Changing the price to match a price cut will often be effective in sending a signal to Smith — "cut prices, and I’ll have to match you even though we’ll both be worse off. At least I’ll be less bad off." Sometimes, if Jones fears that Smith might then cut prices again, Jones might want to make a pre-emptive move by cutting prices significantly at the outset, to send a strong signal to Smith not to cut prices again.

Changing his product design or changing an advertising campaign might or might not be a large retaliation. The effects of these types of changes often are highly uncertain; some work out very successfully, while others are dismal failures. Even though the businesses would not intentionally choose an unsuccessful marketing strategy, many plans do not work out as hoped.

Applying the conclusions from game theory to the business world, we can see that for choices involving long detection lags, long retaliation lags, and small retaliatory measures, rivals are likely not to cooperate with each other, and they will end up not maximizing their joint returns. The cooperation need not involve face-to-face conversations and meetings, although increased communications are sure to reduce the length of detection lags and to give the game players a better idea of the likelihood of different reactions from their rivals. But cooperation, even if it is tacit, will be more likely when the detection and retaliation lags are short or if the expected size of the retaliation is large.

Suppose that instead of cutting prices, Smith is considering introducing a new product line. We can use the prisoners’ dilemma to see why this might be the option many rivals would choose instead of price-cutting to increase their profits. If Smith can do the research and development on the product in secrecy and then launch the product with a giant marketing blitz, Jones will be left in the dust. Smith can earn big profits, and Jones will not be able to counter Smith’s moves for quite some time. The detection lag is long, when you think of how long it takes Jones to find out that Smith is planning the move. And the retaliation lag is long, too, once Jones finds out what Smith is up to. Jones must develop his own new product line and marketing campaign, and these things cannot ordinarily be done quickly.

Knowing that Jones will not be able to counter her moves very quickly, Smith gives serious consideration to this tactic. But at the same time, she knows that Jones is thinking the same thing. She never knows when Jones will launch a new product, and she doesn’t want to be left biding her time while Jones’ new product cuts into her profits. Consequently, even if Jones were not planning to develop a new product, Smith will anticipate that Jones might be working on one, and so she will continue new product research and development as a defensive strategy, if not as an offensive strategy.

The game theoretic approach seems to imply that rivals in these types of situations will not cut prices very often but will always be working on new ad campaigns or the development of new product lines. These implications are not always borne out, however. For some products, prices are changed frequently, and there are comparatively few new advertising campaigns. Nevertheless, game theory has a general applicability that helps us analyze people’s choices in many different situations. We will use it repeatedly in the next few chapters as we examine the roles of different people and institutions in the economy.

... continued in the textbook...

(sample) QUESTIONS FOR DISCUSSION

  1. Why are there gasoline price wars? Before you answer this question, what would the prisoners’ dilemma imply about price wars? If one dealer or supplier or refiner cuts the price, do you think they don’t anticipate that others in the industry will retaliate and match the lower prices? If they think their price cuts will be matched, why do they bother starting the price war in the first place?
  2. Do most drivers buy more gas when there is a price war? Do they end up "filling the top half instead of the bottom half" of their gas tanks? What happens to the amount of gasoline stored in customers’ gas tanks? If it is extremely costly for oil companies build more storage facilities, and if it is illegal to dump petroleum products into the ground, might an oil company be willing to start a price war just to encourage customers to store more gasoline?
  3. Experiments involving students playing forms of the prisoners’ dilemma game have found that the players are able to discover and play a joint maximizing strategy very quickly. The experiments typically run as follows: the players are told that they are playing with one other player who is in a different room. If they both push a blue button, they will each earn one dollar, but if they both push the red button, they will receive only 25 each. If one pushes the blue button while the other pushes the red button, the one pushing the red button will earn two dollars, but the one pushing the blue button will earn nothing.
  1. Complete the payoff matrix for this game in Figure 12-4.

    Player

    A’s Choices
    Blue Button

    Red Button

    Player B’s

    Choices

    Blue
    Button
     

     

     
    Red
    Button
     

     

    Figure 12-4. Experimental Payoff Matrix

  2. What do you think happens when the players are told, "This is the last play of the game."?
  3. If you expect that type of behaviour on the last play of the game, what do you think happens on the 19th play of game when the players know there will be only 20 plays in the game? The 18th? Why do players cooperate at all when they play games like this?
  4. There is some evidence that students from arts and humanities cooperate more and hence have greater total winnings, playing these games, than do students in economics and business. Why might this be?

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