Answers to Chapter 7 Exercises - Luis Cabralluiscabral.net/economics/books/iio2/answers/c07.games.sol.pdf · Nokia 100 90 100 800 500 1110 400 90 700 900 900 800 Now, based on price,

Answers to Chapter 7 Exercises

Review and practice exercises

7.1. Dominant and dominated strategies. What are the assumptions regarding playerrationality implicit in solving a game by elimination of dominated strategies? Contrast thiswith the case of dominant strategies.

Answer: See the discussion on pages 3 and following.

7.2. The movie release game. Consider the example at the beginning of the chapter.Suppose that there are only two blockbusters jockeying for position: Warner Bros.’s HarryPorter and Fox’s Narnia. Suppose that blockbusters released in November share a total of$500 million in ticket revenues, whereas blockbusters released in December share a total of$800 million.

(a) Formulate the game played by Warner Bros. and Fox.

Answer: The game in normal form is as follows, where payo↵s are in $ million:

Warner Bros.

Fox

November December

November250

250800

500

December500

800400

400

(b) Determine the game’s Nash equilibrium(a).

Answer: There are two Nash equilibrium in this game: (N,D) and (D,N). See also thediscussion on page 5.

7.3. Ericsson v Nokia. Suppose that Ericsson and Nokia are the two primary competitorsin the market for 4G handsets. Each firm must decide between two possible price levels:$100 and $90. Production cost is $40 per handset. Firm demand is as follows: if both firmsprice at 100, then Nokia sells 500 and Ericsson 800; if both firms price at 90, then sales

are 800 and 900, respectively; if Nokia prices at 100 and Ericsson at 90, then Nokia’s salesdrop to 400, whereas Ericsson’s increase to 1100; finally, if Nokia prices at 90 and Ericssonat 100 then Nokia sells 900 and Ericsson 700.

(a) Suppose firms choose prices simultaneously. Describe the game andsolve it.

Answer: First, it may help to write the demand curve as a matrix. (Notice this is not thegame firms are playing.)

Ericsson

Nokia

100 90

100800

5001110

400

90700

900900

800

Now, based on price, marginal cost and demand, we can write the payo↵ corresponding toeach strategy pair. This is now the normal form game played by firms

Ericsson

Nokia

100 90

10048

3055

24

9042

4545

40

Pricing at 90 is a dominant strategy for Nokia and Ericsson alike. The Nash equilibrium istherefore given by (90,90). Equilibrium profits are given by (40,45).

(b) Suppose that Ericsson has a limited capacity of 800k units per quarter.Moreover, all of the demand unfulfilled by Ericsson is transferred toNokia. How would the analysis change?

Answer: The new demand matrix is given by

Ericsson

Nokia

100 90

100800

500800

700

90700

900800

900

The new game is given by

2

Ericsson

Nokia

100 90

10048

3040

42

9042

4540

45

It is now a dominant strategy for Ericsson to price at $100. It is still a dominant strategyfor Nokia to price at $90.

(c) Suppose you work for Nokia. Your Chief Intelligence O�cer (CIO) isunsure whether Ericsson is capacity constrained or not. How muchwould you value this piece of info?

Answer: Nokia has a dominant strategy: price at 90. Therefore, it has no value for theinformation of whether Ericsson is or is not capacity constrained (as far as the present gameis concerned).

7.4. ET. In the movie E.T., a trail of Reese’s Pieces, one of Hershey’s chocolate brands,is used to lure the little alien out of the woods. As a result of the publicity created bythis scene, sales of Reese’s Pieces trebled, allowing Hershey to catch up with rival Mars.Universal Studio’s original plan was to use a trail of Mars’ M&Ms, but Mars turned downthe o↵er. The makers of E.T. then turned to Hershey, who accepted the deal.

Suppose that the publicity generated by having M&Ms included in the movie wouldincrease Mars’ profits by $800,000 and decrease Hershey’s by $100,000. Suppose moreoverthat Hershey’s increase in market share costs Mars a loss of $500,000. Finally, let b be thebenefit for Hershey’s from having its brand be the chosen one.

Describe the above events as a game in extensive form. Determine the equilibrium as afunction of b. If the equilibrium di↵ers from the actual events, how do you think they canbe reconciled?

Answer: The game’s extensive form is the following (payo↵s in millions of dollars):

...........................................................................................................................................................................................................................................................................................

[reject]...........................................................................................................................................................................................................................................................................................

[accept]

...........................................................................................................................................................................................................................................................................................[reject]

...........................................................................................................................................................................................................................................................................................

[accept]M

H

-.5, b - 1

0, 0

-.2, -.1

If b > 1, then Hershey is better o↵ by accepting Universal’s o↵er, were it ever asked to makethat choice; in which case Mars is better o↵ by accepting Universal’s o↵er. If b < 1, thenHershey is better o↵ by rejecting Universal’s o↵er, were it ever asked to make that choice;in which case Mars is better o↵ by rejecting Universal’s o↵er.

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7.5. ET (continuation). Return to Exercise 7.4. Suppose now that Mars does not knowthe value of b, believing that either b =$1,200,000 or b =$700,000, each with probability50%. Unlike Mars, Hershey knows the value of b. Draw the tree for this new game anddetermine its equilibrium.

Answer: The game’s extensive form is now given by the following (payo↵s in millions ofdollars):

..................................................................................................................................................................................................................................

[reject]..................................................................................................................................................................................................................................................................

[buy][bl][7pt][0pt]M

..................................................................................................................................................................................................................................

[b = 1200 (50%)]

..................................................................................................................................................................................................................................[b = 700 (50%)]

N

........................................................................................................................................................................................................

[accept]

........................................................................................................................................................................................................

[reject]

H

........................................................................................................................................................................................................

[accept] ........................................................................................................................................................................................................

[reject] H

0, 0

[l]

-500, -300

[l]

0, 0

[l]

-500, 200

[l]

-200, -100

[l]

-500

0

�500⇥ 50%+0⇥ 50% =

= �250

where M and H refers to Mars and Hershey, whereas N refers to the player Nature. Thevalues next to the H nodes correspond to M ’s expected payo↵ if we ever get to thatnode. Nature is not a strategic player: it simply chooses di↵erent branches accordingto predetermined probabilities. In the present case, Nature flips a fair coin and choosesb =$1,200,000 or b =$700,000 with equal probability. This implies that, from M ’s point ofview, the expected value given that we are in the N node is given by

�500⇥ 50% + 0⇥ 50% = �250

It follows that M ’s optimal choice is to accept Universal’s o↵er. To summarize, the equi-librium strategies are given by

• Mars: accept Universal’s o↵er

• Hershey: accept Universal’s o↵er if b is high, reject otherwise.

7.6. Hernan Cortez. In a message to the king of Spain upon arriving in Mexico,Spanish navigator and explorer Hernan Cortez reports that, “under the pretext that [our]ships were not navigable, I had them sunk; thus all hope of leaving was lost and I couldact more securely.” Discuss the strategic value of this action knowing the Spanish colonistswere faced with potential resistance from the Mexican natives.

Answer: By eliminating the option of turning back, Hernan Cortez established a crediblecommitment regarding his future actions, that is, to fight the Mexican natives should theyattack. Had Cortez not made this move, natives could have found it better to attack,knowing that instead of bearing losses the Spaniards would prefer to withdraw.

7.7. HDTV standards. Consider the following game depicting the process of standardsetting in high-definition television (HDTV).4 The US and Japan must simultaneously de-cide whether to invest a high or a low value into HDTV research. If both countries choose

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a low e↵ort than payo↵s are (4,3) for US and Japan, respectively; if the US chooses a lowlevel and Japan a high level, then payo↵ are (2,4); if, by contrast, the US chooses a highlevel and Japan a low one, then payo↵s are (3,2). Finally, if both countries choose a highlevel, then payo↵ are (1,1).

(a) Are there any dominant strategies in this game? What is the Nashequilibrium of the game? What are the rationality assumptions im-plicit in this equilibrium?

Answer: The game in matrix form looks like the following:

E↵ort by Japan

E↵ort by US

Low High

Low3

44

2

High2

31

1

It is a dominant strategy for the US to choose Low. Given that the US chooses Low, Japan’sbest response is to choose High. (Low, High) is thus the only Nash equilibrium of the game.

(b) Suppose now the US has the option of committing to a strategy aheadof Japan’s decision. How would you model this new situation? Whatare the Nash equilibria of this new game?

Answer: The most natural way to model this situation is by writing an extensive formgame as follows:

..............................................................................................................................................................................................................................................................................................................

[H]..............................................................................................................................................................................................................................................................................................................

[L]

...........................................................................................................................................................................................................................................................................

[H][br][0pt][+2pt]

...........................................................................................................................................................................................................................................................................

[L][tr][0pt][-2pt]

...........................................................................................................................................................................................................................................................................

[H][br][0pt][+2pt]

...........................................................................................................................................................................................................................................................................

[L][tr][0pt][-2pt]US

Japan

1, 1

3, 2

Japan

4, 3

2, 4

Japan’s optimal strategy is to choose H if the US choses L and to choose L if the US choosesH. Anticipating that strategy, the US optimal strategy is to choose H. The equilibrium istherefore (H,L).

(c) Comparing the answers to (a) and (b), what can you say about thevalue of commitment for the US?

Answer: In the simultaneous move game, US and Japan choose (L,H), respectively, whichgives the US a payo↵ of 2. In the sequential move game (with the US moving first), USand Japan choose (H,L), respectively, which gives the US a payo↵ of 3. It follows that the

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value of commitment for the US is 3� 2 = 1.

(d) “When pre-commitment has a strategic value, the player that makesthat commitment ends up ‘regretting’ its actions, in the sense that,given the rivals’ choices, it could achieve a higher payo↵ by choosinga di↵erent action.” In light of your answer to (b), how would youcomment this statement?

Answer: In the sequential choice game (with the US moving first), Japan ends up choosingL. Given that Japan chooses L, the payo↵ for the US would be higher if it chose L insteadof H. In this sense, there is ex-post regret. However, the sole reason for Japan choosingL is precisely the fact the US commits to H. Were such commitment not credible, that is,were the US able to change its choice easily, then Japan should anticipate that change andaccordingly chose H. In this sense, the US should not regret having committed to H in thefirst place.

7.8. Finitely repeated game. Consider a one-shot game with two equilibria and supposethis game is repeated twice. Explain in words why there may be equilibria in the two-periodgame which are di↵erent from the equilibria of the one-shot game.

Answer: When the game is repeated twice the strategy space for each player becomes morecomplex. Each player’s strategy specifies the action to be taken in period 1 as well as theaction to be taken in period 2 as a function of the outcome in period 1. The possibility oflinking period 2’s actions to past actions allows for equilibrium outcomes that would not beattainable in the corresponding one-shot game (for example, the use of a ’punishment’ actionin period 2 if one of the players deviates from the designated period 1 payo↵-maximizingaction).

7.9. American Express’s spino↵ of Shearson. In 1993, American Express sold Shearsonto Primerica (now part of Citigroup). At the time, the Wall Street Journal wrote that

Among the sticking points in acquiring Shearson’s brokerage operations wouldbe the firm’s litigation costs. More than most brokerage firms, Shearson hasbeen socked with big legal claims by investors who say they were mistreated,though the firm has made strides in cleaning up its backlog of investor cases. In1992’s fourth quarter alone, Shearson took reserves of $90 million before taxesfor “additional legal provisions.”5

When the deal was completed, Primerica bought most of Shearson’s assets but left the legalliabilities with American Express. Why do you think the deal was structured this way?Was it fair to American Express?

7.10. Sale of business. Suppose that a firm owns a business unit that it wants tosell. Potential buyers know that the seller values the unit at either $100m, $110m, $120,. . . $190m, each value equally likely. The seller knows the precise value, but the buyeronly knows the distribution. The buyer expects to gain from synergies with its existingbusinesses, so that its value is equal to seller’s value plus $10m. (In other words, there are

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Table 7.1Sale of business

Probability Exp. value ExpectedPrice of sale if accepted profit

100 10 110 1

110 20 115 1

120 30 120 0

130 40 125 -2

140 50 130 -5

150 60 135 -9

160 70 140 -14

170 80 145 -20

180 90 150 -27

190 100 155 -35

gains from trade.) Finally, the buyer must make take-it-or-leave-it o↵er at some price p.How much should the buyer o↵er?

Answer: We can write down Table 7.1, which summarizes, for each o↵er that the buyermakes, the probability that the o↵er gets accepted, the expected value (to the buyer)conditional on having the o↵er accepted, and the overall expected profit from any giveno↵er. From this we see that the seller should thus o↵er either $100m or $110m.

Suppose the buyer o↵ers p = 100 (in $m). Then, in most cases the o↵er is rejected.Specifically, 90% of the times the o↵er is rejected. O↵ering more would imply a higherprobability of sale, but the expected value of the unit would increase by less than the pricepaid. The intuition for this result is the force of adverse selection: the seller will only sellthe unit if its value is relatively low.

Challenging exercises

7.11. First-price auction. Consider the following auction game. There are two bidderswho simultaneously submit bids bi for a given object. Bidder i values the object at vi;it knows its own value but not the other bidder’s value. It is common knowledge thatvaluations vi are uniformly drawn from the unit interval, that is, vi ⇠ U [0, 1].

(a) Suppose that Bidder 1 expects Bidder 2’s bid to be uniformly dis-tributed between 0 and 1

2 . What is Bidder 1’s optimal bid function(that is, bid as a function of valuation v1)?

Answer: Suppose Bidder 1 believes that Bidder 2’s bid, b2, is some number between 0 and12 , with all numbers equally likely (that is, Bidder 2’s bid is uniformly distributed in the [0

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and 12 ] interval. By bidding b1, Bidder 1’s expected profit is given by

⇡1 = (v1 � b1)P(b1 > b2)

The higher b1, the lower then net gain from winning the auction, v1� b1; but the higher theprobability of winning the auction, P(b1 > b2). Specifically, if b1 = 0, then P(b1 > b2) = 0;whereas, if b1 =

12 , then P(b1 > b2) = 1. More generally, for b1 2 [0, 12 ],

P(b1 > b2) = 2 b1

It follows that⇡1 = (v1 � b1) 2 b1

Taking the derivative with respect to b1 and equating to zero, we get the first-order conditionfor profit maximization (see Section 3.2):

2 (� b1 + v1 � b1) = 0

or simply

b1 =v12

(7.1)

(b) If Bidder 2 expects Bidder 1 to follow the strategy derived in part (a),what is Bidder 2’s belief about Bidder 1’s bid levels?

Answer: Since Bidder 2 knows that v1 is uniformly distributed in [0,1], (7.1) implies thatb1 is uniformly distributed in the [0, 12 ] interval.

(c) Determine the bidding game Nash equilibrium (assuming there is onlyone).

Answer: From part (a), we know that the bidding function (7.1) is optimal given the beliefthat the other bidder’s bid is uniformly distributed in [0, 12 ]. From part (b), we know that, ifbidders bid according to (7.1), then, from the other bidder’s perspective, bids are uniformlydistributed in the [0, 12 ] interval. Together this implies that strategies and beliefs form aNash equilibrium.

7.12. Ad games. Two firms must simultaneously choose their advertising budget; theiroptions are H or L. Payo↵s are as follows: if both choose H, then each gets 5; if bothchoose L, then each gets 4; if firm 1 chooses H and firm 2 chooses L, then firm 1 gets 8 andfirm 2 gets 1; conversely, if firm 2 chooses H and firm 1 chooses L, then firm 2 gets 8 andfirm 1 gets 1.

(a) Determine the Nash equilibria of the one-shot game.

Answer: H is a dominant strategy, so the unique Nash equilibrium is (H,H).

(b) Suppose the game is indefinitely repeated and that the relevant dis-count factor is � = .8. Determine the optimal symmetric equilibrium.

Answer: The condition that (L,L) is an equilibrium is that

5

1� �� 8 + �

4

1� �

8

which is equivalent to � � 34 . Since � = .8, it follows that (L,L) is indeed an equilibrium.

(c) (challenge question) Now suppose that, for the first 10 periods, firmpayo↵s are twice the values represented in the above table. What isthe optimal symmetric equilibrium?

Answer: From the analysis in the previous answer, we conclude that, after t = 10, (L,L)is an equilibrium. Consider the situation at t = 9. Current payo↵s are doubled. It followsthat the no-deviation constraint is

10 + �5

1� �� 16 + �

4

1� �

which implies � � 67 ⇡ .86. If follows that (L,L) is not an equilibrium. By induction and

a fortiori, we also conclude that (L,L) is not an equilibrium for any earlier t. It followsthe best symmetric equilibrium is for firms to choose H during the first 10 periods and L

thereafter.

7.13. Finitely repeated game. Suppose that the game depicted in Figure 7.1 is repeatedT times, where T is known. Show that the only subgame perfect equilibrium is for playersto choose B in every period.

Answer: Suppose we are in period T , the last period of the finitely-repeated game. Subgameperfection implies that we look for a Nash equilibrium of this subgame. As we saw earlier,there exists a unique Nash equilibrium of this one-shot game: (B,R).

Now consider the subgame starting in period T � 1. This is e↵ectively a two-periodgame. Players correctly anticipate that, regardless of what happens in period T � 1, (B,R)will be played in period T . For this reason, they should treat choices in period T � 1 asif they were playing a one-shot game: nothing in the past or in the future depends on theoutcome of what takes place in period T � 1. Since there exists a unique equilibrium in theone-shot game, players choose (B,R) in period T � 1.

By induction, we conclude that, in a subgame perfect Nash equilibrium, players mustchoose (B,R) in every period.

7.14. Centipede. Consider the game in Figure 7.13.6 Show, by backward induction, thatrational players choose d at every node of the game, yielding a payo↵ of 2 for Player 1 andzero for Player 2. Is this equilibrium reasonable? What are the rationality assumptionsimplicit in it?

Answer: Starting from the right-most node, we observe that Player 2’s strategy, if thatnode is reached, is to play d, in which case its gets 101, whereas Player 1 gets 99. Thisimplies that, in the second to last node, Player 1 is better o↵ choosing d. In fact, by choosingr, Player 1 expects to get 99 (see sentence above) instead of 100 from d. And so forth. Weconclude that the unique sub-game perfect Nash equilibrium is for each player to play d

whenever it is called upon to make a move. The outcome of this equilibrium is Player 1getting 2 and Player 2 getting 0.

Obviously, one might question whether this result is reasonable or not. Here, the implicitassumption is that each player is rational, believes that the other player is rational, believesthat the other player believes that the first player is rational, and so forth.

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To see how important this assumption is, suppose that Player 1 chooses r in the firstperiod. Since this is not according to the equilibrium, Player 2 may not conjecture thatPlayer 1 is not rational. But then choosing d may no longer be in Player 2’s best interest.But then choosing r may be, after all, a rational strategy by Player 1 in the first place.

7.15. Advertising levels. Consider an industry where price competition is not veryimportant: all of the action is on advertising budgets. Specifically, total value S (in dollars)gets splits between two competitors according to their advertising shares. If a1 is firm 1’sadvertising investment (in dollars), then its profit is given by

a1a1 + a2

S � a1

(The same applies for firm 2). Both a1 and a2 must be non-negative. If both firms investzero in advertising, then they split the market.

(a) Determine the symmetric Nash equilibrium of the game whereby firmschoose ai independently and simultaneously.

Answer: Firm i’s profit is given by

⇡i =ai

ai + ajS � ai

where i 6= j. The first order condition for profit maximization with respect to ai is given by

(ai + aj)� ai(ai + aj)2

S � 1 = 0

In a symmetric equilibrium, we have a1 = a2 = ba. Thus

(ba+ ba)� ba(ba+ ba)2

S � 1 = 0

or simply

ba =1

4S

Each player’s payo↵ is then given by

b⇡ =1

2S � 1

4S =

1

4S

For aficionados: Note that in deriving the above solution I “cut some corners” by assumingthe solution is symmetric. I next follow a more complete line of reasoning. From thefirst-order condition, we can derive firm i’s best response mapping. From the first-ordercondition we get

aj S = (ai + aj)2

or simplyai =

paj S � aj

Solving the system, and imposing that ai � 0, we get ai = aj , as assumed in the earlierderivation.

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Figure 7.13The centipede game. In the payo↵ vectors, the top number is Player 1’s payo↵, the bottomone Player 2’s.

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...................................................... ...................................................... ...................................................... ...................................................... .................................... . . . .................................... ...................................................... ...................................................... ......................................................

......................................................

......................................................

......................................................

......................................................

......................................................

......................................................

......................................................

......................................................

1 2 1 2 1 2 1 2r r r r r r r r

d d d d d d d d

"2

0

# "1

3

# "4

2

# "3

5

# "6

4

# "97

99

# "100

98

# "99

101

#

"100

100

#

(b) Determine the jointly optimal level of advertising, that is, the level a⇤

that maximizes joint profits.

Answer:

⇡1 + ⇡2 = S � a1 � a2

It follows that a1 = a2 = a⇤ = 0 maximizes joint profits.

(c) Given that firm 2 sets a2 = a⇤, determine firm 1’s optimal advertisinglevel.

Answer: Given a2 = 0, any positive a1 gives firm 1 100% of the market. Since advertisingis costly, firm 1’s best response is to set an arbitrarily small but strictly positive value of a1(similarly to price undercutting under Bertrand competition).

(d) Suppose that firms compete indefinitely in each period t = 1, 2, ..., andthat the discount factor is given by � 2 [0, 1]. Determine the lowestvalue of � such that, by playing grim strategies, firms can sustain anagreement to set a⇤ in each period.

Answer: By choosing a⇤ each period, each firm gets S/2. The optimal deviation yields ap-proximately S. Finally, the static Nash equilibrium yields S/4 for each firm. The conditionfor a grim strategy equilibrium whereby firms set a = 0 in each period is then given by

1

1� �

1

2S � S +

�

1� �

1

4S

or simply

� � 2

3

Applied exercises

7.16. Laboratory experiment. Run a laboratory experiment to test a specific predictionfrom game theory. First, convene a group of willing subjects (you may need to clear theexperiment with the human subject review board at your institution). Second, write detailedinstructions to explain subjects what they are supposed to do. To the extent that it is

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possible, attach a financial reward to the subjects’ performance in the experiment. Third,run the experiment and carefully keep track of all of the subjects’ decisions. Finally, comparethe observed results with the theoretical predictions, and discuss any di↵erences there mightexist between the two. (If a dedicated laboratory does not exist in your institution, use theclassroom and your colleagues as a subject pool.)

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