Game Theory with Applications to Marketing Strategy
Lecture 4: Subgame Perfect Nash Equilibrium
‧ Robert Louis Stevenson’s magic bottle
‧ Noncredible threat: Chen is the owner-manager of a firm. The debt due one year
from now has a face value equal to $10. The total assets in place worth only $8.
But, just now, Chen found an investment opportunity with NPV = x > 1, which
does not need any extra investment other than the current assets in the firm. Chen
comes to his creditor and asks the latter to reduce the face value of debt by $1. He
claims (he is really bad) that he will not take the investment project unless the
creditor is willing to reduce the face value as he wants.
(i) Suppose x > 2. Show that there is an NE in which the creditor agrees to reduce
the face value of debt and Chen makes the investment.
(ii) Show that the NE in (i) is not an SPNE, because it involves incredible threat
from Chen.
(iii) How may your conclusion about (ii) change if x  (1, 2]?
(iv) Define bankruptcy as a state where the firm's net worth drops to zero. In case
of (iii), conclude that Chen's company has not gone bankrupt.
‧ The basic idea of perfect equilibrium is to select Nash equilibria that do not
involve noncredible threats, by (roughly) requiring that the players’ behavior be
optimal even in situations that are not reached on the equilibrium path.
‧ A (subgame) perfect equilibrium (Selten 1965) is a set of strategies for each
player such that in any subgame (truncated to this subgame) form a Nash
equilibrium. Thus, perfection requires that strategies be in equilibrium whatever
the location (understand subgame) in the game tree, and not only along the
equilibrium path.
‧ Find the subgame perfect Nash equilibria for the games depicted in figure 1~3.
Figure 1
Figure 2
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Figure 3
‧ Consider the following simple-minded inspection game: A firm may pick an
anticompetitive action (i.e., may prey) or not. Preying yields it an (extra) monetary
payoff of g > 0. Antitrust authorities can inspect or not. The cost of inspection is c
> 0. If the firm has preyed and inspection occurs, the firm pays a penalty p > g;
the antitrust authorities’ payoff is then s – c > 0. If the firm has not preyed and
inspection occurs, or if no inspection occurs, no penalty is paid and the
authorities’ payoff is –c or 0, respectively.
(i) Suppose that the antitrust authority observes whether predation occurred before
choosing whether to inspect. Draw the game tree. What is the perfect equilibrium?
(ii) Assume that whether predation has occurred is not observed before the
inspection decision. Draw the game tree. Argue that there exists no pure-strategy
equilibrium. Compute the mixed-strategy one. How does a change in the penalty
affect this equilibrium?
‧ Consider an elaboration of the Cournot duopoly game in which firm 1 engages in
advertising before the firms compete in the market. Firm 1 selects an advertising
level a, which is a number greater than or equal to zero. Advertising has a positive
effect on the demand for the good sold in the industry, enhancing the price that the
consumers are willing to pay for the output of both firms. In particular, the market
price is p = a – q1 – q1, where q1 is the output of firm 1 and q2 is the output of firm
2. After firm 1 selects a, it is observed by the other firm. Then the two
simultaneously and independently select their production levels. Assume for
simplicity that the firms produce at zero cost. However, firm 1 must pay an
advertising cost of 2a3/81. Find the SPNE.
‧ A manufacturer of automobile tires produces at a cost of $10 per tire. It sells units
to a retailer who in turn sells the tires to consumers. Imagine that the retailer faces
the inverse demand curve p = 200 – q/100. The retailer has no cost of production,
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other than whatever it must pay to the manufacturer for the tires.
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Suppose that the manufacturer and retailer interact as follows. First, the
manufacturer sets a price w that the retailer must pay for each tire. Then, the
retailer decides how many tires q to purchase from the manufacturer and
sell to consumers. The manufacturer’s payoff is q(w – 10), whereas the
retailer’s profit is (200 – q/100)q – wq. Calculate the SPNE of this game.
Suppose that the manufacturer sells its tires directly to consumers,
bypassing the retailer. Thus, the manufacturer can sell q tires at price p =
200 – q/100. Calculate the manufacturer’s profit-maximizing choice of q in
this case.
Compare the joint profit of the manufacturer and retailer in the first part
with the manufacturer’s profit in the second part. Explain why there is a
difference. This is called the double-marginalization problem.
‧ The set of players: two firms, i =1, 2. The strategies: two firms’ output quantities:
qi  [0, ) , i =1, 2. The payoff of each player i:
πi(qi, qj) = qi[p(qi + qj) – c] - Fi,
where P(*) is the inverse demand function, c and Fi are respectively the variable
and fixed costs. Suppose that c = 0 and p = 1 – q1 – q2.
 Firm 1 and Firm 2 choose their quantities simultaneously. Find the NE.
 Firm 1 and Firm 2 choose their quantities sequentially. Find the SPNE.
‧ The value of commitment and the corresponding notion of “burning one’s
bridges” have widespread applicability beyond economics. An oft-quoted example
is that two armies wishing to occupy an island located between their countries and
connected by a bridge to both. Each army prefers letting its opponent have the
island to fighting. Army 1, which is somewhat knowledgeable in game theory,
occupies the island and burns the bridge behind it. Army 2 then has no option
other than to let army 1 have the island, because it knows that army 1 has no
choice other than to fight back if army 2 attacks. This is the paradox of
commitment: Army 1 does better by reducing its set of choices.（類似項羽的破釜
沈舟）
‧ Suppose two firms are considering whether and how to enter a new industry in
which a specialized electronic component will be produced. Industry demand is
given by the inverse demand function p = 900 – q1 – q2, where p is the market
price, q1 is the quantity produced by firm 1, and q2 is the quantity produced by
firm 2. To enter the industry, a firm must build a production facility. Two types of
facility can be built: small and large. A small facility requires an investment of
$50,000, and it allows the firm to produce as many as 100 units of the good at
zero marginal cost. Alternatively, the firm can pay $175,000 to construct a large
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facility the will allow the firm to produce any number of units at zero marginal
cost. A firm with a small production facility is called capacity constrained; a firm
with a large facility is called unconstrained.
‧ Consider a two-firm industry. Firm 1 (the existing firm) chooses a level of capital
K1, which is then fixed. Firm 2 (the potential “entrant”) observes K1 and then
chooses its level of capital K2, which is also fixed. Assume that the profits of the
two firms are specified by π1(K1, K2) = K1(1 – K1 – K2) and π2(K1, K2) = K2(1 –
K1 – K2).
 Find the SPNE.
 Suppose that Firm 2 incurs a fixed cost of entry, f < 1/16, into the model, i.e.
π2(K1, K2) = K2(1 – K1 – K2) – f if K2 > 0 and π2(K1, K2) = 0 if K2 = 0. Find
the SPNE.
‧ Suppose a single firm sells LCD monitors. Call the manager of this firm Tony. To
keep things simple, suppose there are only four potential customers: Hal, Hilbert,
Laurie, and Lauren. These five players interact over two periods of time, which I
call period 1 and period 2. Each customer’s benefit from owning an LCD monitor
depends on which periods he or she owns the product. Hal could really use such a
monitor at work – he is a high type. Hal would get a benefit of $1,200 by using an
LCD monitor in period 1, and he would get a benefit of $500 from its use in
period 2. Laurie, on the other hand, doesn’t really need an LCD monitor – she is a
low type. In period 1, Laurie would obtain a benefit of $500 from an LCD monitor;
in period 2, her benefit would be $200. Hilbert is a high type just like Hal; Lauren
is a low type just like Laurie. The customers’ values of owning an LCD monitor
are summarized in the following figure.
Period 1
Period 2
Benefit to Hal and Hilbert
1200
500
Benefit to Laurie and Lauren
500
200
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Pricing scheme A: Induce all of the customers to purchase in period 1.
Pricing scheme B: Induce all of the customers NOT to purchase in period 1.
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Pricing scheme C: Induce the high types to purchase in period 1 and the low
types to purchase in period 2.
Price guarantees as a commitment to high prices.
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‧ Consider a seller costlessly producing a durable good to serve two segments (H
and L) of consumers with unit demand. The population of each segment is one.
There are two dates (t = 1, 2). A type-H (respectively, type-L) consumer gets
utility 3 (respectively, 1) when he gets the durable good. The seller and the
consumers have a common discount factor ρ = 1/2. The game proceeds as follows.
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At time 1, the seller posts p1 to maximize the sum of discounted expected profits
over the two dates, and upon seeing p1, all consumers simultaneously decide to or
not to buy the durable good. At time 2, the seller posts p2 to maximize his date-2
expected profit. Show that there is no pure strategy SPNE for this game.
‧ Rubinstein’s (1982) bargaining game: Two players, who must share a pie of size 1,
make sequential, alternating offers. At date 1, player 1 makes offer x1 in [0, 1];
player 2 accepts or refuses x1. If he accepts, he obtains 1 – x1, leaving x1 for player
1. If he refuses, he make offer x2 in [0, 1] at date 2. If this offer is accepted, he
gets 1 – x2; if player 1 refuses this offer, he gets to make an offer x3 in the third
period, and so on. The players alternate making offers until one of them accepts
his opponent’s offer. The payoffs are δtxt for player 1 and δt(1 – xt) for player 2 if
they agree at date t on share xt for player 1. The discount factor δ belongs to (0, 1).
Impatience will be the driving force that leads to agreement in his model. Solve
for the bargaining game for T = 2, 3, …. Show that x1 tends to 1/(1+δ) when T
goes to infinity.
‧ 有一天Adam閒來無事，便和30位哈佛大學企管碩士班的學生玩一場紙牌遊
戲。Adam自己持有30張黑牌，並且分給每位學生一張紅牌。慷慨的院長同意
贊助這場遊戲，只要黑牌與紅牌配成一對（黑牌與紅牌最多可配成30對），
院長就會給Adam一百美元。遊戲的規則如下：學生可以和亞當自由談判，約
定Adam應出價多少來收購學生的紅牌。唯一的規定是學生必須以個人身份進
行談判，不得先形成一個團體再與Adam談判。
 你預測這個談判最後會出現什麼結果？
 若Adam只有25張黑牌，你預測這個談判最後會出現什麼結果？
 當Adam手中的牌從30張變成25張時，對Adam是好是壞？
‧ 你的附加價值＝（你在賽局時餅的大小）－（你不在賽局時餅的大小）。
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The National Football League (NFL)
Nitendo
DeBeers
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