Bargaining Strategy

In looking at the Prisoners' Dilemma and related problems, we have assumed that collusion was limited by an inability to make an enforceable agreement. Clearly, alternative outcomes are possible (and likely) if firms or individuals can make promises that can be enforced. The Prisoners' Dilemma illustrated by the pricing problem shown in Table 13.8 is a good example of this. If there were no antitrust laws and both firms could make an enforceable agreement about pricing, they would both charge a high price and make profits of 50. Here, the bargaining problem is simple.

Other bargaining situations are more complicated, however, and the outcome can depend on the ability of either side to make a strategic move that alters its relative bargaining position. For example, consider two firms that are each planning to introduce one of two products, which happen to be complementary goods. As the payoff matrix of Table 13.17 shows. Firm 1 has an

ProduceA

Firm 1

ProduceB

Firm 2

Produce A Produce B

40,5

50,50

60,40

5,45

Example 153 in Chapter 15 examines in more detail the profitability of capital investment by a new entrant in the diaper market.

advantage in producing A, so that if both firms produce A, Firm 1 will be able to maintain a lower price and will make much higher profits. Similarly, Firm 2 has an advantage in producing product B. As should be clear from the payoff matrix, if the two firms could agree about who will produce what, the only rational outcome would be that in the upper right-hand corner. Firm 1 produces A, Firm 2 produces B, and both firms make profits of 50. Indeed, even without cooperation this outcome will result, whether Firm 1 or Firm 2 moves first or both firms move simultaneously The reason is that producing B is a dominant strategy for Firm 2, so (A, B) is the only Nash equilibrium.

Firm 1 would, of course, prefer the outcome in the lower left-hand corner of the payoff matrix. But in the context of this limited set of decisions, it cannot achieve that outcome. Suppose, however, that Firms 1 and 2 are also bargaining over a second issue-whether to join a research consortium that a third firm is trying to form. Table 13.18 shows the payoff matrix for this decision problem. Clearly, the dominant strategy is for both firms to enter the consortium, thereby obtaining increased profits of 40.

Now suppose that Firm 1 links the two bargaining problems by announcing that it will join the consortium only if Firm 2 agrees to produce product A. (How can Firm 1 make this threat credible?) In this case it is indeed in Firm 2's interest to agree to produce A (with Firm 1 producing B), in return for Firm l's participation in the consortium. This example illustrates how a strategic move can be used in bargaining, and why combining issues in a bargaining agenda can sometimes benefit one side at the other's expense.

Two people bargaining over the price of a house is another example of this. Suppose I, as a potential buyer, do not want to pay more than $200,000 for a house that is actually worth $250,000 to me. The seller is willing to part with the house at any price above $180,000 but would like to receive the highest price she can. If I am the only bidder for the house, how can I make the seller think I will walk away rather than pay more than $200,000?

I might declare that I will never, ever pay more than $200,000 for that house. But is such a promise credible? It is if the seller knows that I have a strong reputation for toughness and steadfastness and that I have never broken my word on a promise of this sort. But suppose I have no such reputation. Then the seller knows that I have every incentive to make the promise (making it costs nothing), but little incentive to keep it (since this will probably be our tabi f 13.18 Decision to Join Consortium

Firm 2

Work Alone Enter Consortium

Work Alone

Firm 1

Enter Consortium

10,10

10,20

20,10

40,40

only business transaction together). As a result, this promise by itself is not likely to improve my bargaining position.

The promise can work, however, if it is combined with a strategic move that gives it credibility Such a strategic move must reduce my flexibility-limit my options-so that I have no choice but to keep the promise. A possible move would be to make an enforceable bet with a third party-for example, "If I pay more than $200,000 for that house, Til pay you $60,000. "Alternatively, if I am buying the house on behalf of my company, the company might insist on authorization by the Board of Directors for a price above $200,000, and an' nounce that the board will not meet again for several months. In both cases, my promise becomes credible because I have destroyed my ability to break it. The result is less flexibility-and more bargaining power.

Summary

1. A game is cooperative if the players can communicate and arrange binding contracts; otherwise it is noncooperative. In either kind of game, the most important aspect of strategy design is understanding your opponent's position, and (if your opponent is rational) correctly deducing the likely response to your actions. Misjudging an opponents position is a common mistake, as Example 13.1, "Acquiring a Company," illustrates.26

2. A Nash equilibrium is a set of strategies such that each player is doing the best it can, given the strategies of the other players. An equilibrium in dominant strategies is a special case of a Nash equilibrium; a dominant strategy is optimal no matter what the other players do. A Nash equilibrium relies on the rationality of each player. A maximin strategy is more conservative because it maximizes the minimum possible outcome.

3. Some games have no Nash equilibria in pure strategies, but have one or more equilibria in mixed strategies. A mixed strategy is one in which the player makes a random choice among two or more possible actions, based on a set of chosen probabilities.

4. Strategies that are not optimal for a oner-shot game may be optimal for a repeated game. Depending on the number of repetitions, a "tit-for-tat" strategy, in which one plays cooperatively as long as one's competitor does the same, may be optimal for the repeated Prisoners' Dilemma.

5. In a sequential game, the players move in turn. In some cases, the player who moves first has an advantage. Players may then have an incentive to try to precommit themselves to particular actions before their competitors can do the same.

26 Here is the solution to company As problem: It should offer nothing for Company T's stock. Remember that Company T will accept an offer only if it is greater than the per share value under current management. Suppose you offer $50. Thus, Company T will accept this offer only if the outcome of the exploration project results in a per share value under current management of $50 or less. Any values between $0 and $100 are equally likely. Therefore the expected value of Company T's stock, given that it accepts the offer, i.e., given that the outcome of the exploration project leads to a value less than $50, is $25, so that under the management of Company A the value would be (1.5)($25) = $37.5, which is less than $50. In fact, for any price P, if the offer is accepted. Company A can expect a value of only (%)P.

6. An empty threat is a threat that one would have no incentive to carry out. If one's competitors are rational, empty threats are of no value. To make a threat credible, it is sometimes necessary to make a strategic move by constraining one's later behavior, so that there would be an incentive to carry out the threat.

7. To deter entry, an incumbent firm must convince any potential competitor that entry will be unprofitable. This may be done by investing, and thereby giving credibility to the threat that entry will be met by price warfare. Strategic trade policies by governments sometimes have this objective.

8. Bargaining situations are examples of cooperative games. As with noncooperative games, in bargaining one can sometimes gain a strategic advantage by limiting one's flexibility.

Questions for Review

1. What is the difference between a cooperative and a noncooperative game? Give an example of each.

2. What is a dominant strategy? Why is an equilibrium stable in dominant strategies?

3. Explain the meaning of a Nash equilibrium. How does it differ from an equilibrium in dominant strategies?

4. How does a Nash equilibrium differ from a gamers maximin solution? In what situations is a maximin solution a more likely outcome than a Nash equilibrium?

5. What is a "tit-for-tat" strategy? Why is it a rational strategy for the infinitely repeated Prisoners' Dilemma?

6. Consider a game in which the Prisoners' Dilemma is repeated 10 times, and both players are rational and fully informed. Is a tit-for-tat strategy optimal in this case? Under what conditions would such a strategy be optimal?

7. Suppose you and your competitor are playing the pricing game shown in Table 13.8. Both of you must announce your prices at the same time. Might you improve your outcome by promising your competitor that you will announce a high price?

8. What is meant by "first-mover advantage"? Give an example of a gaming situation with a first-mover advantage.

9. What is a "strategic move"? How can the development of a certain kind of reputation be a strategic move?

10. Can the threat of a price war deter entry by potential competitors? What actions might a firm take to make this threat credible?

11. A strategic move limits one's flexibility and yet gives one an advantage. Why? How might a strategic move give one an advantage in bargaining?

Exercises

1. In many oligopolistic industries, the same firms compete over a long period of time, setting prices and observing each other's behavior repeatedly. Given that the number of repetitions is large, why don't collusive outcomes typically result?

2. Many industries are often plagued by overca-pacity-firms simultaneously make major investments in capacity expansion, so that total capacity far exceeds demand. This happens in industries in which demand is highly volatile and unpredictable, but also in industries in which demand is fairly stable. What factors lead to overcapacity? Explain eachbriefly.

3. Two computer firms, A and B, are planning to market network systems for office information management. Each firm can develop either a fast, high-quality system (H), or a slower, low-quality system (L). Market research indicates that the resulting profits to each firm for the alternative strategies are given by the following payoff matrix:

FirmB H L

30,30

50, 35

40,60

20,20

a. If both firms make their decisions at the same time and follow maximin (low-risk) strategies, what will the outcome be?

b. Suppose both firms try to maximize profits, but Firm A has a head start in planning, and can commit first. Now what will the outcome be? What will the outcome be if Firm B has the head start in planning and can commit first?

c. Getting a head start costs money (you have to gear up a large engineering team). Now consider the two-stage game in which first, each firm decides how much money to spend to speed up its planning, and second, it announces which product (H or L) it will produce. Which firm will spend more to speed up its planning? How much will it spend? Should the other firm spend anything to speed up its planning? Explain.

4. Two firms are in the chocolate market. Each can choose to go for the high end of the market (high quality) or the low end (low quality). Resulting profits are given by the following payoff matrix:

Firm 2 Low High

Low

-20, -30

900, 600

High

100, 800

50, 50

a. What outcomes, if any, are Nash equilibria?

b. If the manager of each firm is conservative and each follows a maximin strategy, what will be the outcome?

c. What is the cooperative outcome?

d. Which firmbenefits most from the cooperative outcome? How much would that firm need to offer the other to persuade it to collude?

5. Two major networks are competing for viewer ratings in the 8:00-9:00 P.M. and 9:00-10:00 P.M. slots on a given weeknight. Each has two shows to fill this time period and is juggling its lineup. Each can choose to put its "bigger" show first or to place it second in the 9:00-10:00 P.M. slot. The combination of decisions leads to the following "ratings points" results:

Network 2 First Second

First

18, 18

23,

20

Network 1

Second

4, 23

16,

16

a. Find the Nash equilibria for this game, assuming that both networks make their decisions at the same time.

b. If each network is risk averse and uses a maximin strategy, what will be the resulting equilibrium?

c. What will be the equilibrium if Network 1 makes its selection first? If Network 2 goes first?

d. Suppose the network managers meet to coordinate schedules, and Network 1 promises tc schedule its big show first. Is this promise credible, and "what would be the likely outcome?

6. We can think of U.S. and Japanese trade policies as a Prisoners' Dilemma. The two countries are considering policies to open or close their import markets. Suppose the payoff matrix is:

Japan Open Close

Open

10, 10

5,5

Close

-100, 5

1,1

a. Assume that each country knows the payoff matrix and believes that the other country will act in its own interest. Does either country have a dominant strategy? What will be the equilibrium policies if each country acts rationally to maximize its welfare?

b. Now assume that Japan is not certain that the U.S. will behave rationally. In particular, Japan is concerned that U.S. politicians may want to penalize Japan even if that does not maximize U.S. welfare. How might this affect Japan's choice of strategy? How might this change the equilibrium?

7. You are a duopolist producer of a homogeneous good. Both you and your competitor have zero marginal costs. The market demand curve is

where Q - Qi + Qi. Q\ is your output, and Qi is your competitor's output. Your competitor has also read this book.

a. Suppose you are to play this game only once. If you and your competitor must announce your outputs at the same time, how much will you choose to produce? What do you expect your profit to be? Explain.

b. Suppose you are told that you must announce, your output before your competitor does. How much will you produce in this case, and how much do you think your competitor will produce? What do you expect your profit to be? Is announcing first an advantage or a disadvantage? Explain briefly. How much would you pay to be given the option of announcing either first or second?

c. Suppose instead that you are to play the first round of a series often rounds (with the same competitor). In each round you and your competitor announce your outputs at the same time. You want to maximize the sum of your profits over the ten rounds. How much will you produce in the first round? How much would you expect to produce in the tenth round? The ninth round? Explain briefly.

d. Once again you will play a series of ten rounds. This time, however, in each round your competitor will announce its output before you announce yours. How will your answers to (c) change in this case?

*8. Defendo has decided to introduce a revolutionary video game, and as the first firm in the market,' it will have a monopoly position for at least some time. In deciding what type of manufacturing plant to build, it has the choice of two technologies. Technology A is publicly available,and will result in annual costs of:

Ci(q) = 10 + 8q Technology B is aproprietary technology developed in Defenders research labs. It involves higher fixed cost of production, but lower marginal costs:

Cfi(^)=60 + 2q Defendo's CEO must decide which technology to adopt. Market demand for the new product is P -20 - Q, where Q is total industry output.

a. Suppose Defendo were certain that it would maintain its monopoly position in the market for the entire product lifespan (about five years) without threat of entry. Which technology would you advise the CEO to adopt? What would be Defendo's profit given this choice?

b. Suppose Defendo expects its archrival, Of-fendo, to consider entering the market shortly after Defendo introduces its new product. Offendo will have access only to Technology A. If Offendo does enter the market, the two firms will play a Cournot game (in quantities) and arrive at the Cournot-Nash equilibrium.

(i) .If Defendo adopts Technology A and Offendo enters the market, what will be the profits of both firms? Would Offendo choose to enter the market given these profits?

(ii) If Defendo adopts Technology B and Offendo enters the market, what will be the profit of each firm? Would Offendo choose to enter the market given these profits?

(iii) Which technology would you advise the CEO ofDefendo to adopt given the threat of possible entry? What will be Defended profit given this choice? What will be consumer surplus given this choice?

c. What happens to social welfare (the sum of consumer surplus and producer profit) as a result of the threat of entry in this market? What happens to equilibrium price? What might this imply about the role of potential competition in limiting market power?

9. Three contestants, A, B, and C, each have a balloon and a pistol. From fixed positions, they fire at each other's balloon. When a balloon is hit/ its owner is out When only one balloon remains, its owner is the winner, and receives a $1000 prize. At the outset, the players decide by lot the order in which they will fire, and each player can choose any remaining balloon as his target. Everyone knows that A is the best shot and always hits the target, that B hits the target with probability .9, and that C hits the target with probability .8. Which contestant has the highest probability of winning the $1000? Explain why.

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