7.1. Dominant and dominated strategies.
What are the assumptions regarding player rationality implicit in solving a game by elimination of dominated strategies? Contrast this with 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 Harry
Porter 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:
November
Warner Bros.
November December
250 800
250 500
Fox
December
500 400
800 400
(b) Determine the game’s Nash equilibrium(a).
Answer: There are two Nash equilibrium in this game: (N,D) and (D,N). See also the discussion on page 5.
7.3. Ericsson v Nokia.
Suppose that Ericsson and Nokia are the two primary competitors in 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 firms price 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 sales drop to 400, whereas Ericsson’s increase to 1100; finally, if Nokia prices at 90 and Ericsson at 100 then Nokia sells 900 and Ericsson 700.
(a) Suppose firms choose prices simultaneously. Describe the game and solve it.
Answer: First, it may help to write the demand curve as a matrix. (Notice this is not the game firms are playing.)
Nokia
100
90
100
Ericsson
90
800 1110
500 400
700 900
900 800
Now, based on price, marginal cost and demand, we can write the payo ↵ corresponding to each strategy pair. This is now the normal form game played by firms
Nokia
100
90
30
100
Ericsson
90
48
24
55
42 45
45 40
Pricing at 90 is a dominant strategy for Nokia and Ericsson alike. The Nash equilibrium is therefore 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 to
Nokia. How would the analysis change?
Answer: The new demand matrix is given by
Nokia
100
90
100
Ericsson
90
800 800
500 700
700 800
900 900
The new game is given by
2

Nokia
100
90
30
100
Ericsson
90
48
42
40
42 40
45 45
It is now a dominant strategy for Ericsson to price at $100. It is still a dominant strategy for Nokia to price at $90.
(c) Suppose you work for Nokia. Your Chief Intelligence O ffi cer (CIO) is unsure whether Ericsson is capacity constrained or not. How much would you value this piece of info?
Answer: Nokia has a dominant strategy: price at 90. Therefore, it has no value for the information of whether Ericsson is or is not capacity constrained (as far as the present game is 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 by this 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 down the 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 would increase Mars’ profits by $800,000 and decrease Hershey’s by $100,000. Suppose moreover that Hershey’s increase in market share costs Mars a loss of $500,000. Finally, let b be the benefit 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 a function of b . If the equilibrium di ↵ ers from the actual events, how do you think they can be 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 make that choice; in which case Mars is better o ↵ by accepting Universal’s o ↵ er. If b < 1, then
Hershey 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.
3

7.5. ET (continuation).
Return to Exercise 7.4. Suppose now that Mars does not know the value of b , believing that either b =$1,200,000 or b =$700,000, each with probability
50%. Unlike Mars, Hershey knows the value of b . Draw the tree for this new game and determine its equilibrium.
Answer: The game’s extensive form is now given by the following (payo ↵ s in millions of dollars):
-500
........
........
........
........
[ reject]
.............
[buy][bl][7pt][0pt]M
.............
.............
.............
.............
.............
.............
b = 1200 (50%)
.............
.............................................................................................................................................
]
...............
...............
...............
...............
[ b = 700 (50%) ] [accept] [reject]
...............
[accept]
N
[reject] [l] [l] [l] [l] [l]
500 ⇥
......................
......................
H
= 250
⇥ 50% =
......................
......................
H
-200, -100 where M and H refers to Mars and Hershey, whereas N refers to the player Nature. The values next to the H nodes correspond to M ’s expected payo ↵ if we ever get to that node. Nature is not a strategic player: it simply chooses di ↵ erent branches according to predetermined probabilities. In the present case, Nature flips a fair coin and chooses b =$1,200,000 or b =$700,000 with equal probability. This implies that, from M ’s point of view, 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 equilibrium 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 Cort´ In a message to the king of Spain upon arriving in Mexico,
Spanish navigator and explorer Hernan Cort´ez reports that, “under the pretext that [our] ships were not navigable, I had them sunk; thus all hope of leaving was lost and I could act more securely.” Discuss the strategic value of this action knowing the Spanish colonists were faced with potential resistance from the Mexican natives.
Answer: By eliminating the option of turning back, Hernan Cort´ez established a credible commitment regarding his future actions, that is, to fight the Mexican natives should they attack. Had Cort´ez 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 standard setting in high-definition television (HDTV).
4
The US and Japan must simultaneously decide whether to invest a high or a low value into HDTV research. If both countries choose
4
0
-500, 200
0, 0
-500, -300
0, 0

a low e ↵ ort than payo ↵ s are (4,3) for US and Japan, respectively; if the US chooses a low level and Japan a high level, then payo ↵ are (2,4); if, by contrast, the US chooses a high level and Japan a low one, then payo ↵ s are (3,2). Finally, if both countries choose a high level, then payo ↵ are (1,1).
(a) Are there any dominant strategies in this game? What is the Nash equilibrium of the game? What are the rationality assumptions implicit in this equilibrium?
Answer: The game in matrix form looks like the following:
E ↵ ort by US
Low
High
E ↵ ort by Japan
Low High
3 4
4 2
2 1
3 1
It is a dominant strategy for the US to choose Low. Given that the US chooses Low, Japan’s best 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 ahead of Japan’s decision. How would you model this new situation? What are the Nash equilibria of this new game?
Answer: The most natural way to model this situation is by writing an extensive form game as follows:
........
........
............
............
............
............
............
............
............
.......................................................................................................................................................................................
......................
......................
......................
......................
...................................................................................................................................................................................
Japan
........
........
[H] [L]
...............
...............
...............
[H][br][0pt][+2pt] [L][tr][0pt][-2pt]
...............
...............
...............
...............
......................................................................................................................................................................................
[H][br][0pt][+2pt]
US
............
............
............
............
............
............
............
.......................................................................................................................................................................................
......................
......................
......................
......................
Japan
...................................................................................................................................................................................
Japan’s optimal strategy is to choose H if the US choses L and to choose L if the US chooses
H. Anticipating that strategy, the US optimal strategy is to choose H. The equilibrium is therefore (H,L).
(c) Comparing the answers to (a) and (b), what can you say about the value of commitment for the US?
Answer: In the simultaneous move game, US and Japan choose (L,H), respectively, which gives the US a payo ↵ of 2. In the sequential move game (with the US moving first), US and Japan choose (H,L), respectively, which gives the US a payo ↵ of 3. It follows that the
5
1, 1
3, 2
2, 4
4, 3

value of commitment for the US is 3 2 = 1.
(d) “When pre-commitment has a strategic value, the player that makes that commitment ends up ‘regretting’ its actions, in the sense that, given the rivals’ choices, it could achieve a higher payo ↵ by choosing a di ↵ erent action.” In light of your answer to (b), how would you comment this statement?
Answer: In the sequential choice game (with the US moving first), Japan ends up choosing
L. Given that Japan chooses L, the payo ↵ for the US would be higher if it chose L instead of H. In this sense, there is ex-post regret. However, the sole reason for Japan choosing
L 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 and accordingly chose H. In this sense, the US should not regret having committed to H in the first place.
7.8. Finitely repeated game.
Consider a one-shot game with two equilibria and suppose this game is repeated twice. Explain in words why there may be equilibria in the two-period game 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 more complex. Each player’s strategy specifies the action to be taken in period 1 as well as the action to be taken in period 2 as a function of the outcome in period 1 . The possibility of linking period 2’s actions to past actions allows for equilibrium outcomes that would not be attainable in the corresponding one-shot game (for example, the use of a ’punishment’ action in period 2 if one of the players deviates from the designated period 1 payo ↵ -maximizing action).
7.9. American Express’s spino ↵ of Shearson.
In 1993, American Express sold Shearson to Primerica (now part of Citigroup). At the time, the Wall Street Journal wrote that
Among the sticking points in acquiring Shearson’s brokerage operations would be the firm’s litigation costs. More than most brokerage firms, Shearson has been 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. In
1992’s fourth quarter alone, Shearson took reserves of $90 million before taxes for “additional legal provisions.”
5
When the deal was completed, Primerica bought most of Shearson’s assets but left the legal liabilities 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 to sell. 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 buyer only knows the distribution. The buyer expects to gain from synergies with its existing businesses, so that its value is equal to seller’s value plus $10m. (In other words, there are
6

Table 7.1
Sale of business
Price
100
110
120
130
140
150
160
170
180
190
Probability of sale
10
20
30
40
50
60
70
80
90
100
Exp. value if accepted
110
115
120
125
130
135
140
145
150
155
Expected profit
1
1
0
-2
-5
-9
-14
-20
-27
-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 buyer makes, 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 given o ↵ 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 higher probability of sale, but the expected value of the unit would increase by less than the price paid. The intuition for this result is the force of adverse selection: the seller will only sell the unit if its value is relatively low.
7.11. First-price auction.
Consider the following auction game. There are two bidders who simultaneously submit bids b i for a given object. Bidder i values the object at v i it knows its own value but not the other bidder’s value. It is common knowledge that
; valuations v i are uniformly drawn from the unit interval, that is, v i
⇠ U [0 , 1].
(a) Suppose that Bidder 1 expects Bidder 2’s bid to be uniformly distributed between 0 and
1
2
. What is Bidder 1’s optimal bid function
(that is, bid as a function of valuation v
1
)?
Answer: Suppose Bidder 1 believes that Bidder 2’s bid, b
2
1
2
, is some number between 0 and
, with all numbers equally likely (that is, Bidder 2’s bid is uniformly distributed in the [0
7

and
1
2
] interval. By bidding b
1
, Bidder 1’s expected profit is given by
⇡
1
= ( v
1 b
1
) P ( b
1
> b
2
)
The higher b
1 probability of winning the auction, P ( b
1 whereas, if b
1
, the lower then net gain from winning the auction, v
1
=
1
2
, then P ( b
1
> b
2
> b
2
). Specifically, if
) = 1. More generally, for b
1 b
2
1
= 0, then P ( b
1
[0 ,
1
2 b
1
],
; but the higher the
> b
2
) = 0;
P ( b
1
> b
2
) = 2 b
1
It follows that
⇡
1
= ( v
1 b
1
) 2 b
1
Taking the derivative with respect to b
1 and equating to zero, we get the first-order condition for profit maximization (see Section 3.2):
2 ( b
1
+ v
1 b
1
) = 0 or simply b
1
= v
1
2
(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 v
1 b
1 is uniformly distributed in the [0 ,
1
2 is uniformly distributed in [0,1], (7.1) implies that
] interval.
(c) Determine the bidding game Nash equilibrium (assuming there is only one).
Answer: From part (a), we know that the bidding function (7.1) is optimal given the belief that the other bidder’s bid is uniformly distributed in [0 ,
1
2
]. From part (b), we know that, if bidders bid according to (7.1), then, from the other bidder’s perspective, bids are uniformly distributed in the [0 ,
1
2
] interval. Together this implies that strategies and beliefs form a
Nash equilibrium.
7.12. Ad games.
Two firms must simultaneously choose their advertising budget; their options are H or L . Payo ↵ s are as follows: if both choose H , then each gets 5; if both choose L , then each gets 4; if firm 1 chooses H and firm 2 chooses L , then firm 1 gets 8 and firm 2 gets 1; conversely, if firm 2 chooses H and firm 1 chooses L , then firm 2 gets 8 and firm 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 discount factor is = .
8. Determine the optimal symmetric equilibrium.
Answer: The condition that ( L, L ) is an equilibrium is that
1
5
8 +
1
4
8

which is equivalent to
3
4
. Since = .
8, it follows that ( L, L ) is indeed an equilibrium.
(c) (challenge question) Now suppose that, for the first 10 periods, firm payo ↵ s are twice the values represented in the above table. What is the 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 follows that the no-deviation constraint is
5 4
10 + 16 +
1 1 which implies
6
7
⇡ .
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 follows the 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 repeated
T times, where T is known. Show that the only subgame perfect equilibrium is for players to choose B in every period.
Answer: Suppose we are in period T , the last period of the finitely-repeated game. Subgame perfection 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-period game. 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 as if they were playing a one-shot game: nothing in the past or in the future depends on the outcome of what takes place in period T 1. Since there exists a unique equilibrium in the one-shot game, players choose ( B, R ) in period T 1.
By induction, we conclude that, in a subgame perfect Nash equilibrium, players must choose ( B, R ) in every period.
7.14. Centipede.
Consider the game in Figure 7.13.
6
Show, by backward induction, that rational players choose d at every node of the game, yielding a payo ↵ of 2 for Player 1 and zero for Player 2. Is this equilibrium reasonable? What are the rationality assumptions implicit in it?
Answer: Starting from the right-most node, we observe that Player 2’s strategy, if that node is reached, is to play d , in which case its gets 101, whereas Player 1 gets 99. This implies that, in the second to last node, Player 1 is better o ↵ choosing d . In fact, by choosing r , Player 1 expects to get 99 (see sentence above) instead of 100 from d . And so forth. We conclude 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 1 getting 2 and Player 2 getting 0.
Obviously, one might question whether this result is reasonable or not. Here, the implicit assumption is that each player is rational, believes that the other player is rational, believes that the other player believes that the first player is rational, and so forth.
9

To see how important this assumption is, suppose that Player 1 chooses r in the first period. Since this is not according to the equilibrium, Player 2 may not conjecture that
Player 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 very important: 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 advertising investment (in dollars), then its profit is given by a
1 is firm 1’s a
1 a
1
+ a
2
S a
1
(The same applies for firm 2). Both a
1 and a
2 zero in advertising, then they split the market.
must be non-negative. If both firms invest
(a) Determine the symmetric Nash equilibrium of the game whereby firms choose a i independently and simultaneously.
Answer: Firm i ’s profit is given by
⇡ i
= a i a i
+ a j
S a i where i = j . The first order condition for profit maximization with respect to a i is given by
( a i
+ a j
) a
( a i
+ a j
) 2 i
S 1 = 0
In a symmetric equilibrium, we have a
1
= a
2
= b . Thus
( b + b ) b
( b + a ) 2
S 1 = 0 or simply b =
1
4
S
Each player’s payo ↵ is then given by b =
1
2
S
1
4
S =
1
4
S
For aficionados: Note that in deriving the above solution I “cut some corners” by assuming the solution is symmetric. I next follow a more complete line of reasoning. From the first-order condition, we can derive firm i ’s best response mapping. From the first-order condition we get a j
S = ( a i
+ a j
)
2 or simply a i
= p a j
S a j
Solving the system, and imposing that a i derivation.
0, we get a i
= a j
, as assumed in the earlier
10

Figure 7.13
The centipede game. In the payo ↵ vectors, the top number is Player 1’s payo ↵ , the bottom one Player 2’s.
" #
"
.............
...
..............
..
d
..
..
..
..
..
..
..
..
..
..
..
..
1
..
..
................
......
............
............
............
2
...
......................................................
........
r
# "
.............
...
d
..
..
..
..
..
..
2
............
............
............
............
......
..
..
1
..
..
..
..
..
..
...
r
......................................................
........
# "
.............
...
d
..
..
..
..
..
..
..
1
......
............
............
............
............
4
..
..
..
..
..
..
..
...
r
......................................................
# "
.............
d
..
..
..
..
..
..
..
2
............
............
......
............
............
3
..
..
..
..
..
..
..
...
r
......................................................
# "
.............
d
...
..
..
..
..
..
..
..
1
......
............
............
............
............
6
..
..
..
..
..
..
..
...
.....
# r
....................................
. . .
....................................
"
..............
.............
d
...
..
..
..
..
..
..
..
..
..
..
..
2
......
............
............
............
............
..
..
..
..
...
.....
........
.......
97 r
......................................................
# " d
...
..............
.............
..
..
..
..
..
..
1
......
............
............
............
............
..
..
..
..
..
..
..
..
..
...
.....
................
........
100 r
......................................................
# "
.............
...
..............
d
..
..
..
..
..
..
.............
............
......
............
............
............
..
2
..
..
..
..
..
..
..
...
.....
................
........
99
# r
......................................................
100
100
0 3 2 5 4 99 98 101
(b) Determine the jointly optimal level of advertising, that is, the level a ⇤ that maximizes joint profits.
Answer:
⇡
1
+ ⇡
2
= S a
1 a
2
It follows that a
1
= a
2
= a ⇤ = 0 maximizes joint profits.
(c) Given that firm 2 sets a
2 level.
= a ⇤ , determine firm 1’s optimal advertising
Answer: Given a
2
= 0, any positive a
1 gives firm 1 100% of the market. Since advertising is costly, firm 1’s best response is to set an arbitrarily small but strictly positive value of a
1
(similarly to price undercutting under Bertrand competition).
(d) Suppose that firms compete indefinitely in each period t = 1 , 2 , ...
, and that the discount factor is given by 2 [0 , 1]. Determine the lowest value of such that, by playing grim strategies, firms can sustain an agreement to set a ⇤ in each period.
Answer: By choosing a ⇤ each period, each firm gets S/ 2. The optimal deviation yields approximately S . Finally, the static Nash equilibrium yields S/ 4 for each firm. The condition for a grim strategy equilibrium whereby firms set a = 0 in each period is then given by
1
1 1
2
S S +
1
1
4
S or simply
2
3
7.16. Laboratory experiment.
Run a laboratory experiment to test a specific prediction from game theory. First, convene a group of willing subjects (you may need to clear the experiment with the human subject review board at your institution). Second, write detailed instructions to explain subjects what they are supposed to do. To the extent that it is
11

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, compare the observed results with the theoretical predictions, and discuss any di ↵ erences there might exist between the two. (If a dedicated laboratory does not exist in your institution, use the classroom and your colleagues as a subject pool.)
12