As you know, oligopolists pursuing their individual gains can, and often do, end up worse off than if they were to cooperate. The situation is analogous to divid- ing a pie. People are struggling to get a larger share of a pie, but in the struggle some of the pie gets knocked off the table and onto the floor, and they end up sharing a smaller pie. A solution preferred by all participants exists but is dif- ficult to achieve. And, even if the preferred situation is somehow reached, such as the results in cell A in the prisoners’ dilemma (Table 13.1) or point C in the air- line example (Figure 13.2), rivals have a strong incentive to change their actions, which leads back to the noncooperative Nash equilibrium: cell D in the prisoners’
dilemma or point N in the airline example. Cooperation occurs, then, when oli- gopoly firms make individual decisions that make every firm better off than they would be in a noncooperative Nash equilibrium outcome.
As we have now established, when prisoners’ dilemma decisions are made just one time, managers have almost no chance of achieving cooperative outcomes.
In many instances, however, decisions concerning pricing, output, advertising, entry, and other such strategic decisions are made repeatedly. Decisions made
promises
Conditional strategic moves that take the form: “If you do A, I will do B, which is desirable to you.”
Now try T echnical Problem 14.
cooperation When firms make decisions that make every firm better off than in a noncooperative Nash equilibrium.
over and over again by the same firms are called repeated decisions. Repeating a strategic decision provides managers with something they do not get in one- time decisions: a chance to punish cheaters. The opportunity to punish cheating in repeated decisions can completely change the outcome of strategic decisions.
One-Time Prisoners’ Dilemma Decisions
Prisoners’ dilemma scenarios, as we explained previously, always possess a set of decisions for which every firm earns a higher payoff than they can when they each choose to follow their dominant strategies and end up in the noncooperative Nash equilibrium.9 Cooperation is possible, then, in all prisoners’ dilemma decisions.
Cooperation does not happen in one-time prisoners’ dilemma situations be- cause cooperative decisions are strategically unstable. If a firm believes its rivals are going to choose the cooperative decision in a simultaneous decision, then the firm can increase its profit by choosing the noncooperative decision. For conve- nience in discussions concerning cooperation, game theorists call making non- cooperative decisions cheating. We must stress, however, that “cheating” does not imply that the oligopoly firms have made any kind of explicit, or even tacit, agreement to cooperate. If a firm’s rivals expect that firm to make a cooperative decision, they will have an incentive to cheat by making a noncooperative deci- sion. All oligopoly managers know this and so choose not to cooperate. While the incentive to cheat makes cooperation a strategically unstable outcome in a one- time simultaneous decision, cooperation may arise when decisions are repeated.
We can best illustrate the possibility of cooperation, as well as the profit incen- tive for cheating, with an example of a pricing dilemma. Suppose two firms, Intel and Advanced Micro Devices (AMD), dominate the wholesale market for high- speed microprocessor chips for personal computers. Intel and AMD simultane- ously set their chip prices. For now, let us consider this a one-time simultaneous pricing decision, and the prices set by the firms will be valid for a period of one week. (In the next section, we will reconsider this decision when it is repeated every week.) Table 13.5 shows the profit payoffs from setting either high or low
repeated decisions Decisions made over and over again by the same firms.
cheating
When a manager makes a noncooperative decision.
AMD’s price
High Low
A: Cooperation B: AMD cheats
High $5,$2.5 $2,$3
Intel’s price C: Intel cheats D: Noncooperation
Low $6,$0.5 $3,$1
Payoffs in millions of dollars of profit per week.
T A B L E 13.5 A Pricing Dilemma for AMD and Intel
9To keep things simple, we continue to restrict our discussion to decisions having only one Nash equilibrium. Game theorists so far have been unable to provide much useful guidance for decisions possessing multiple Nash equilibria.
prices for high-speed computer chips. You can verify from the payoff table that a prisoners’ dilemma situation exists.
When the semiconductor chip prices are set noncooperatively, as they will be when the pricing decision is made only once, both firms will choose to set low prices and end up in cell D. Setting a low price for computer chips is the dominant strategy for both firms and this leads to the noncooperative Nash equilibrium out- come (cell D). As in every prisoners’ dilemma game, both firms can do better by cooperating rather than by choosing their dominant strategy–Nash equilibrium actions. Intel and AMD both can choose high prices for their chips and both can then earn greater profits in cell A than in the noncooperative cell D: Intel earns an additional $2 million ($5 million − $3 million) and AMD earns an additional
$1.5 million ($2.5 million − $1 million) through cooperation.
The problem with cooperation, as we have stressed previously, is that the decisions Intel and AMD must make to get to cell A are not strategically stable.
Both firms worry that the other will cheat if one of them decides to cooperate and price high. For example, if AMD expects Intel is going to cooperate and set a high price, AMD does better by cheating and setting a low price in cell B:
AMD earns $0.5 million more ($3 million − $2.5 million) by cheating than by making the cooperative decision to price high. Similarly, Intel can cheat on AMD in cell C and earn $1 million more ($6 million − $5 million) by cheating than by cooperating.
Suppose the managers at AMD and Intel tell each other that they will set high prices. This is, of course, farfetched since such a conversation is illegal. We are only trying to explain here why such an agreement would not work to achieve cooperation, so we can go ahead with our examination of this hypothetical sce- nario anyway. If the simultaneous pricing decision is to be made just once, nei- ther firm can believe the other will live up to the agreement. Let us get into the head of AMD’s manager to see what she thinks about setting AMD’s price high.
If AMD cooperates and prices high, it dawns on her that Intel has a good rea- son to cheat: Intel does better by cheating ($1 million better in cell C than in cell A) when Intel believes AMD will honor the agreement. And furthermore, she thinks, “What’s the cost to Intel for cheating?” She won’t know if Intel cheated until after they have both set their prices (the prices are set simultaneously). At that point, Intel’s manager could care less if he ruins his reputation with her; it’s a one-time decision and the game is over. Any hope of reaching a cooperative outcome vanishes when she realizes that Intel’s manager must be thinking pre- cisely the same things about her incentive to cheat. There is no way he is going to trust her not to cheat, so she expects him to cheat. The best she can do when he cheats is to cheat as well. And so an opportunity to achieve cooperation collapses over worries about cheating.
Even though cooperation is possible in all prisoners’ dilemma situations, oli- gopoly firms have compelling reasons to believe rivals will cheat when the deci- sion is to be made only one time. In these one-time decisions, there is no practical way for firms to make their rivals believe they will not cheat. When there is no tomorrow in decision making, rivals know they have only one chance to get the
most for themselves. A decision to cheat seems to be costless to the cheating firm because it expects its rival to cheat no matter what it decides to do. Furthermore, firms don’t have to worry about any future costs from their decisions to cheat be- cause they are making one-time decisions. Our discussion of decision making in one-time prisoners’ dilemmas establishes the following principle.
Principle Cooperation is possible in every prisoners’ dilemma decision, but cooperation is not strategi- cally stable when the decision is made only once. In one-time prisoners’ dilemmas, there can be no future consequences from cheating, so both firms expect the other to cheat, which then makes cheating the best response for each firm.
Punishment for Cheating in Repeated Decisions
Punishment for cheating, which cannot be done in a one-time decision, makes cheating costly in repeated decisions. Legal sanctions or monetary fines for cheat- ing are generally illegal in most countries; punishment for cheating usually takes the form of a retaliatory decision by the firm doing the punishment that returns the game to a noncooperative Nash decision—the decision everyone wanted to avoid through cooperation.
To illustrate how retaliatory decisions can punish rivals for noncooperative behavior, suppose that AMD and Intel make their pricing decisions repeatedly.
AMD and Intel list their wholesale computer chip prices on the Internet every Monday morning, and the managers expect this to go on forever.10 Table 13.5 shows the weekly payoffs for the repeated pricing decisions, and these payoffs are not expected to change from week to week in this hypothetical example. Further suppose that AMD and Intel have been making, up until now, cooperative weekly pricing decisions in cell A. Now, in the current week—call this week 1—AMD’s manager decides to cheat by setting a low price. Thus during week 1, Intel and AMD receive the profit payoffs in cell B of Table 13.5. Intel’s manager can punish AMD in the next weekly repetition, which would be week 2, by lowering its price.
Notice that AMD cannot avoid getting punished in week 2 should Intel decide to retaliate by pricing low. In week 2, AMD can either price low and end up in cell D or price high and end up in cell C; either decision punishes AMD. Of course we predict AMD would choose to price low in week 2, because cell D minimizes AMD’s cost of punishment from Intel’s retaliatory price cut.
It follows from the discussion above that Intel can make a credible threat in week 1 to punish cheating with a retaliatory price cut in week 2 because cutting price is Intel’s best response in week 2 to cheating in week 1. You can see from the payoff table (Table 13.5) that Intel’s profit increases by making the retaliatory cut
Now try T echnical Problem 15.
punishment for cheating
Making a retaliatory decision that forces rivals to return to a noncooperative Nash outcome.
10Game theorists have studied a variety of repeated games: games repeated forever and games repeated a fixed or finite number of times. They even distinguish the finite games according to whether players do or do not know when the games will end. These subtle differences are important because they can dramatically affect the possibility of cooperation, and hence, the outcome of repeat- ed games. To keep our discussion as simple and meaningful as possible, we will limit our analysis to situations in which managers believe the decisions will be repeated forever.
from $2 million per week to either $3 million per week or to $6 million per week, depending on AMD’s pricing decision in week 2. You can also verify for yourself that AMD can similarly make a credible threat to cut price in retaliation for an episode of cheating by Intel.
In repeated decisions, unlike one-time decisions, cheating can be punished in later rounds of decision making. By making credible threats of punishment, stra- tegically astute managers can sometimes, but not always, achieve cooperation in prisoners’ dilemmas. We are now ready to examine how punishment can be used to achieve cooperation in repeated decisions.
Deciding to Cooperate
Recall from Chapter 1 that managers should make decisions that maximize the (present) value of a firm, which is the sum of the discounted expected profits in current and future periods. The decision to cooperate, which is equivalent to deciding not to cheat, affects a firm’s future stream of profits. Consequently, managers must gauge the effect of cheating on the present value of their firm.
Cooperation will increase a firm’s value if the present value of the costs of cheating exceeds the present value of the benefits from cheating. Alternatively, cheating will increase a firm’s value if the present value of the benefits from cheat- ing outweighs the present value of the costs of cheating. When all firms in an oligopoly market choose not to cheat, then cooperation is achieved in the market.
Figure 13.5 shows the stream of future benefits and costs for a firm that cheats for N periods of time before getting caught, after which it is punished by a re- taliatory price cut for P periods of time.11 The benefits from cheating received in each of the N time periods (B1, B2, . . . , BN) are the gains in profit each period from cheating rather than cooperating: πCheat − πCooperate. For simplicity, we assume the payoffs, and hence the benefits and costs of cheating, are constant in all repetitions of the decision. Of course, the benefits and costs from cheating occur in the future and so must be discounted using the appropriate discount rate for the cheating firm. The present value of the benefits from cheating when the discount rate per period is r can be calculated as
PVBenefits of cheating = _______ B1
(1 + r)1 + _______ B2
(1 + r)2 + . . . + BN
_______
(1 + r)N
where Bi = πCheat − πCooperate for i = 1, . . . , N. The cost of cheating in each period, after cheating is discovered and continuing for P periods, is the loss in profit caused by the retaliatory price cut that results in a noncooperative Nash
Now try T echnical Problems 16–17.
11The pattern of benefits and costs shown in Figure 13.5 assumes that the firm making the deci- sion to cheat in period 1 can receive the cooperative profit payoff in period 1 if it makes the coopera- tive decision rather than the cheating decision. The pattern also assumes that there are no further costs for cheating after the punishment ends in period (N + P). The figure also depicts equal benefits and costs of cheating in each time period. Other patterns of benefits and costs are certainly possible.
Figure 13.5 provides a general approach to modeling the benefits and costs of cheating.
equilibrium: πCooperate − πNash. The costs of cheating, like the benefits from cheating, occur in the future and must be discounted. The present value of the costs of pun- ishment for P periods (when the discount rate per period is r) can be calculated as
PVCost of cheating = _________ C1
(1 + r ) N+1 + _________ C2
(1 + r ) N+2 + . . . + CP
_________ (1 + r ) N+P
where Cj = πCooperate − πNash for j = 1, . . . , P. We have now established the follow- ing principle.
Principle Cooperation (deciding not to cheat) maximizes the value of a firm when the present value of the costs of cheating is greater than the present value of the benefits from cheating. Cooperation is achieved in an oligopoly market when all firms decide not to cheat.
Trigger Strategies for Punishing Cheating
In repeated decisions, punishment itself becomes a strategy. We will now discuss a widely studied category of punishment strategies known in game theory as trigger strategies. Managers implement trigger strategies by initially choosing the
Now try T echnical Problem 18.
trigger strategies Punishment strategies that choose cooperative actions until an episode of cheating triggers a period of punishment.
N 5 number of periods of cheating P 5 number of periods of punishment
Time period
Firm’s profit payoff per period (dollars)
B1 B2 BN
2 N N 1 1 N 1 2 N 1 P
C1 C2 CP
pCheat
pNash pCooperate
0 1 F I G U R E 13.5
A Firm’s Benefits and Costs of Cheating
cooperative action and continuing to choose the cooperative action in successive repetitions of the decision until a rival cheats. The act of cheating then “triggers”
a punishment phase in the next repetition of the game that may last one or more repetitions, depending on the nature of the trigger scheme. The firm initiating the trigger strategy can announce openly to its rivals that it plans to follow a trig- ger strategy. Or, where attempts to facilitate cooperation might bring legal ac- tion, firms can secretly begin following a trigger strategy and hope that rivals will recognize what they are doing and choose to cooperate.
Two trigger strategies that have received much attention by game theorists are tit-for-tat and grim strategies. In a tit-for-tat strategy, cheating triggers punishment in the next decision period, and the punishment continues unless the cheating stops, which triggers a return to cooperation in the following decision period. In other words, if firm B cheated in the last decision period, firm A will cheat in this decision period. If firm B cooperated last time, then firm A will cooperate this time. Hence the name “tit-for-tat.” Tit-for-tat is both simple to implement and simple for rivals to understand. A tit-for-tat strategy imposes a less severe punish- ment for cheating than a grim strategy. In a grim strategy, cheating triggers pun- ishment in the next decision period, and the punishment continues forever, even if cheaters make cooperative decisions in subsequent periods. This is “grim” indeed!
Many experimental studies of repeated games have been undertaken to see how decision makers actually behave and also to determine which punishment strategies make cooperation most likely. In a famous “tournament” of strategies, Robert Axelrod at the University of Michigan invited game theorists to devise strategies for competition in repeated prisoners’ dilemma games.12 Numerous strategies were submitted, and computers were used to pit the strategies against each other in prisoners’ dilemma decisions repeated hundreds of times. The strat- egies winning most of the time tended to be simple strategies rather than compli- cated or clever strategies. In a surprise to many game theorists, tit-for-tat emerged as the most profitable strategy because of its ability to initiate and sustain coopera- tion among oligopoly rivals. Other experimental studies and subsequent strategy tournaments have confirmed tit-for-tat to be the most profitable strategy to follow in repeated games. Eventually, game theorists, or perhaps oligopoly managers, may discover a better strategy for making repeated decisions, but for now, tit-for- tat is the winner.
Pricing Practices That Facilitate Cooperation
Cooperation usually increases profits, as well as the value of oligopoly firms;
thus managers of firms in oligopoly markets frequently adopt tactics or methods of doing business that make cooperation among rivals more likely.
Such tactics, called facilitating practices by antitrust officials, encourage cooperation either by reducing the benefits of cheating or by increasing the costs of cheating. And, sometimes, both of these things can be accomplished at
tit-for-tat strategy A trigger strategy that punishes after an episode of cheating and returns to cooperation if cheating ends.
grim strategy A trigger strategy that punishes forever after an episode of cheating.
facilitating practices Generally lawful methods of encouraging cooperative pricing behavior.
12See Robert Axelrod, The Evolution of Competition (New York: Basic Books, 1984).