CHAPTER 11: Strategic interaction
In 1995, in Sanya City, China, the Rural Credit Union raised interest rates from 9.2% to 10.8% on one-year savings
accounts, hoping to attract new deposits to the bank. Their principle competitor, the Hainan Development Bank,
matched the rate increase to avoid losing deposits. Profits declined at both banks because each bank faced much higher
capital costs, without a corresponding increase in deposits. In 1996, the Central Bank of China ended this “excessive”
competition for deposits by mandating a uniform (and low) interest rate throughout the country.
The anecdote illustrates a central concern facing competitors—that the actions of one firm will affect the profits of
another, and provoke a hostile reaction. To analyze this interdependency, we use game theory. In a game, we identify
the major players, the options or moves available to them, and the payoffs associated with combinations of moves. If
players act optimally, rationally, and selfishly, we can compute the likely outcome or equilibrium of the game. The
point of studying game theory is not to teach you how to “win,” but rather to get you to understand the likely outcomes
of a game---and how to manipulate the game to your own advantage.
We partition our study of game theory into three areas: sequential-move games, simultaneous-move games, and
repeated games.
Sequential move games
In these games, the players take turns, and each player observes the moves of its rivals before it has to move. We
represent games using the “extensive” or tree form of a game. To compute the equilibrium of these games, look ahead
and reason back. By analyzing the moves of the second mover, the first mover knows the consequences of his moves.
Thus, we work forward, and compute optimal moves for the first player.
In the examples that follow, pay particular attention to the issues of timing and commitment, i.e. who gets to move first
and how a player can commit to a course of action by “eliminating” one of their options.
Entry deterrence as a sequential game
Entrant
enters
incumbent prices high
incumbent gets 70
entrant gets 60
incumbent prices low
incumbent gets 0
entrant gets -40
stays out
incumbent prices high
incumbent gets 100
entrant gets 0
incumbent prices low
incumbent gets 0
entrant gets 0
Figure 1: Entry accommodation
In this game, an entrant is trying to decide whether or not to enter an industry in competition with an incumbent. The
entrant thinks that the incumbent will act rationally, optimally, and in her self-interest. Beginning on the bottom
branches of the tree, if the entrant enters, the incumbent does better by pricing high. Likewise if he stays out, the
incumbent does better by pricing high. Knowing this, we can analyze the upper branches of the tree. The profit to
entry is $60 while the profit to staying out is $0. The entrant does better by entering. The equilibrium path of the game
is for the entrant to enter and for the incumbent to price high, hi-lighted in the figure above.
Note however, that the incumbent does much better if no entry occurs. If the incumbent can credibly threaten to price
low if the entrant enters, the incumbent can change the outcome of the game and deter entry because pricing low
renders the entry unprofitable. However, the credibility of the threat is weak because it requires the incumbent to make
an otherwise unprofitable move. However, if the entrant believes that the incumbent will price aggressively, then he
will stay out. In this case, we say that by “committing” to price aggressively in the event of entry, the incumbent
successfully deters entry. We denote the commitment not to pursue the price high strategy by a dotted line. If the
incumbent can commit, she changes the equilibrium by eliminating a branch.
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Entrant
enters
incumbent prices high
incumbent gets 70
entrant gets 60
stays out
incumbent prices low
incumbent gets 0
entrant gets -40
incumbent prices high
incumbent gets 100
entrant gets 0
incumbent prices low
incumbent gets 0
entrant gets 0
Figure 10: Entry deterrence with commitment (-----)
Discussion Question: How could the incumbent commit to pricing low following entry?
Answer: Suppose that the incumbent has a choice of two technologies – one with a high fixed cost but a low marginal
cost and one with a high marginal cost but a low fixed cost. The incumbent could commit to pricing low by choosing
the technology with the low marginal cost (recall that fixed costs are irrelevant to the pricing decision). With the low
marginal cost technology, the incumbent will find it profitable to price low. Instead of choosing the high marginal cost
technology as in the John Deere example in Chapter 4, the incumbent chooses the capital intensive Henry Ford style
factory.
Bargaining as a sequential game
In the game below, a company is bargaining with its labor union over the wages that it will pay its workers. To
simplify matters, assume that management and labor are bargaining over a fixed sum of $200 and that there are only
two possible strategies, a generous offer of $150, or a low offer of $50. The union can either strike or accept the offer.
If the union strikes, each party earns nothing. We diagram the offers and payoffs below. The equilibrium is one where
management makes a low offer and union accepts it because it is in union’s interest to do so.
Table 1: management vs. labor union game
Mgmt
low offer
union strikes
mgmt gets 0
union gets 0
union accepts
mgmt gets 150
union gets 50
generous offer
union strikes
mgmt gets 0
union gets 0
union accepts
mgmt gets 50
union gets 150
Note that there is a first-mover advantage in this game—by moving first, management essentially “commits” to a
position. This advantage can be negated if the other side finds a way to commit to a bargaining position. For example,
suppose that in the game above, the union threatens to strike unless it receives a generous offer from management. If
management believes the threat, then management does best by making a generous offer. By committing to a position,
the union forces management to make a generous offer, as in the diagram below. As in the entry deterrence game, by
committing to a position, the Union eliminates an option (dotted line) and changes the equilibrium of the game.
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Mgmt
low offer
union strikes
mgmt gets 0
union gets 0
union accepts
mgmt gets 150
union gets 50
generous offer
union strikes
mgmt gets 0
union gets 0
union accepts
mgmt gets 50
union gets 150
Figure 2: Labor commits to a position
In general, it is difficult to credibly commit to pursue an otherwise unprofitable strategy. Strikes often happen because
management does not believe the threat. Often the only way that the union can convince management that they are
committed to striking is to actually go on strike.
Bargaining ultimatums (commitments)
In general, if you can commit to a bargaining position, you can capture all of the gains from trade from your bargaining
opponent. In your HR or organizational behavior classes, this is called the “ultimatum game.” For example, suppose
that we are bargaining over how to split one dollar, and I have the ability to credibly commit to a take-it-or-leave-it
offer of $0.99 for me and $0.01 for you. If you reject the offer, we do not transact and both of us leave with nothing. If
you are acting optimally, rationally and in your self-interest, you will accept the small split because it is better than
nothing. This highlights another interesting feature of bargaining games, also evident in the entry deterrence game and
in the labor-management game, the alternatives to bargaining determine the terms of any bargain.
Your ability to credibly commit to a position is determined by the opportunity cost of accepting the agreement. If your
next best alternative is good, then you can credibly commit to a very favorable position and this helps you capture the
gains from trade.
Discussion Question: When is the best time to ask for a raise?
Answer: The best time to ask for a raise is when you have an attractive offer from another company. The opportunity
cost of staying in your current job becomes very high. Thus, you can credibly commit to leaving unless you receive a
generous offer to stay.
Discussion Question: When is the best time to buy a car?
Answer: The best time to buy a car is when the sales person’s opportunity cost of bargaining with you is low. Go at
the end of the month, when the sales person earn an immediate commission for any sale; and go at unpopular times,
like Christmas Eve, when there are no other customers.
Discussion Question: When the predictions of the ultimatum game are tested in an experimental setting, offers deemed
to be “unfair” are rejected by the player receiving the ultimatum. What can explain this behavior?
Answer: Players have notions of “fairness” that can be interpreted as commitments to reject any but an even split of
the surplus. This is an equilibrium because the player making the offer knows that any but an even split will be
rejected. In essence, the player receiving the offer has made a credible commitment to reject any offer that is not fair.
Seven ways to commit1
1
1.
Establish and use a reputation. Use your reputation as a “bond” of credibility. It does not pay to forfeit the
bond, e.g., would you eat at a restaurant that was going out of business?
2.
Write contracts. Use the courts’ ability to punish you to bind yourself to a course of action, e.g. marriage
contracts penalize post-investment hold-up, which makes opportunistic behavior less likely.
This advice is taken from Avinash Dixit and Barry Nalebuff, Thinking Strategically, New York: W.W. Norton, 1991.
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3.
Cut off communication. For example if you refuse to answer mail, no one can threaten you or make
counteroffers.
4.
Burn bridges behind you. For example, Polaroid used its commitment to instant photography (no
diversification into other lines of business) as a credible threat to aggressively pursue anyone who tried to
infringe on its patents.
5.
Move in small steps. Note the analogy to using your reputation. If the gains are large enough, it may be
profitable to “forfeit” your reputation. By offering small contracts to do certain work, and always promising
more work, you make it unprofitable to forfeit a reputation.
6.
Develop credibility through teamwork. The “peer pressure” of Saturn’s team labor structure makes it very
costly to shirk on the job.
7.
Employ negotiating agents. Have someone negotiate for you, like an agent, who is not allowed to lower the
price. Note the similarity to point 3.
Simultaneous move games
In these game, players move simultaneously. To analyze them, we use the matrix or “reduced” form of a game. To
compute Nash equilibrium, we have to check that neither player can do better by unilaterally changing their actions.
In the following examples, pay particular attention to tension between conflict and cooperation. Self-interest leads to
equilibrium outcomes where both players are worse off. The one lesson of business tells us to figure out how to avoid
ending up in these bad outcomes. Moving to a better outcome can be thought of as a wealth-creating transaction
between the players.
Prisoners’ dilemma
The prisoners’ dilemma is perhaps the oldest and most-studied game, and is motivated by the story of two parolees,
Frank and Jesse, who are caught riding in a car together shortly after a bank was robbed. The police suspect that Frank
and Jesse – who are known felons – robbed the bank, but there is no direct evidence tying them to the crime. However,
association with other felons is a violation of parole, so the district attorney can send them both back to jail to serve out
their remaining terms. The district attorney puts them in a dilemma by offering both a lighter sentence in exchange for
testimony against his partner. If they both confess, each receives five years in jail. We present these payoffs in the
following matrix.
Table 2: Prisoners’ dilemma
Frank confesses
Frank says nothing
Jesse confesses
Frank receives 5 years in jail
Jesse receives 5 years in jail
Frank receives 10 years in jail
Jesse goes free
Jesse says nothing
Frank goes free
Jesse receives 10 years in jail
Frank receives 2 years in jail
Jesse receives 2 years in jail
The only Nash equilibrium is in the upper left corner. To verify that it is an equilibrium, check whether any of the
players can unilaterally change action and make himself better off. If Frank confesses, Jesse cannot make himself
better off by saying nothing. Likewise, if Jesse confesses, Frank cannot make himself better off by saying nothing. So
the upper left corner, (confess, confess) is a Nash equilibrium. In equilibrium, each player is doing the best they can,
given what his opponent is doing.
Note the tension between conflict and cooperation inherent in the game. If Frank and Jesse could cooperate (both say
nothing), they could reduce their total sentence. However, the lower right box (say nothing, say nothing) is not an
equilibrium because if your rival is saying nothing, you can reduce your own sentence by confessing. In other words
one party can make themselves better off by unilaterally changing strategy. In a sense, this lack of cooperation
represents an unconsummated wealth-creating transaction between the players. Our study of the prisoners’ dilemma
will focus on how to consummate this transaction.
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Horizontal pricing dilemma
The following pricing dilemma has the same logical structure as the prisoners’ dilemma. Both Coke and Pepsi could
make more money by pricing high, but that is not a Nash equilibrium. Once Coke prices high, Pepsi does better by
undercutting Coke’s price. And once Pepsi prices low, Coke does better by matching Pepsi’s low price. The only
Nash equilibrium is for both to price low.
Table 3: Horizontal pricing dilemma
Coke prices low
Coke prices high
Pepsi prices low
Both receive low profit
Pepsi prices high
Coke makes very high profit
Pepsi makes very low profit
Coke makes very low profit
Pepsi makes very high profit
Both make high profit
To get out of this dilemma, Coke and Pepsi must find a way to coordinate their pricing. However, explicit coordination
is illegal under the antitrust laws.
Vertical pricing dilemma
Suppose you develop a genetically-engineered strain of corn seed that is resistant to a patented herbicide manufactured
by a rival firm. If farmers plant with your herbicide-resistant seed, they can dramatically increase crop yields by using
the herbicide to get rid of unwanted weeds. So the farmers demand both herbicide and the herbicide-resistant seed
together, much as consumers demand both left and right shoes together. If the price of the herbicide increases, demand
for the herbicide-resistant seed decreases, i.e. they are complementary products.
Table 4: Vertical pricing dilemma
Herbicide price high
Herbicide price low
Seed price high
Both receive low profit
Seed price low
Herbicide makes very high profit
Seed makes very low profit
Herbicide makes very low profit
Seed makes very high profit
Both make high profit
Again, the only Nash equilibrium is upper left. Both firms try to capture a bigger piece of the profit pie by pricing high
but, by doing so, they price above the joint profit-maximizing price for the herbicide and seed bundle. The parties
could raise joint profit by coordinating to reduce price. Fortunately, such coordination is legal and common, at least in
the United States. 2 By coordinating, the parties reduce price and increase profit.
Price discrimination dilemma
If you are a monopolist, then you can always raise profit by price discriminating. When you are an oligopolist,
however, this is not the case. If your competitors price discriminate in reaction to your decision to discriminate, then
everyone’s profit falls below what they would have been if no one had price discriminated. In essence, price
discrimination in an oligopoly has the same logical structure as the prisoners’ dilemma.
Consider a Kroger grocery store that sends discount coupons to customer offering a percentage reduction on their next
grocery bill. Customers located close to the store have less-elastic demands than those customers located farther away,
so the grocery store can raise profits by offering bigger discounts to those who live farther away – or nearer to a
competitor’s store.
If only Kroger does this, then its profits rise. However, if Kroger’s competitor follows suit then, In the end, all firms
wind up with about the same quantity, but at much lower prices. In the new equilibrium, everyone is worse off.
2
In Europe, the antitrust authorities are only beginning to treat such vertical constraints with
tolerance.http://europa.eu.int/rapid/start/cgi/guesten.ksh?p_action.gettxt=gt&doc=IP/99/286|0|RAPID&lg=EN
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Table 7: Oligopoly price discrimination dilemma
Kroger price discriminates
Kroger sets a single price
Safeway price
discriminates
Both receive low profit
Kroger makes very low profit
Safeway makes very high profit
Safeway sets a
single price
Kroger makes very high profit
Safeway makes very low profit
Both make high profit
Again, the only Nash equilibrium is upper left. Both firms try to increase profit by price discriminating. But by doing
so, they compete vigorously for every single customer in the area regardless of whether they are close to the store or
not. By contrast, in a single price equilibrium, the stores compete vigorously only for customers on the boundaries of
their market areas.
Advertising dilemma
The following advertising dilemma has the same logical structure as the prisoners’ dilemma.
Table 8: Advertising dilemma
RJR advertises heavily
RJR does not advertise
PM advertises
heavily
Both receive low profit
RJR makes very low profit
PM makes very high profit
PM does not
advertise
RJR makes very high profit
PM makes very low profit
Both make high profit
Both RJR and Phillip-Morris could make more money by not advertising, since advertising is predatory – that is, it
serves mainly to steal market share from rivals without increasing the size of the market. But the lower right hand
corner is not a Nash equilibrium. Once RJR stops advertising, Phillip-Morris does better by advertising heavily to steal
RJR’s customers. And once Phillip-Morris advertises, RJR does better by advertising to protect its market share. The
only Nash equilibrium is for both to advertise heavily and earn low profits.
When over-the-air cigarette advertising was banned by the government in the early 1970s, the profitability of the
cigarette industry increased because the government essentially moved the industry from the upper left corner to the
lower right corner of the payoff matrix. Ordinarily one cannot count on the government to be this helpful.
Free riding dilemma
The game below denotes the strategic interdependence typical of an MBA study group. This is also typical of the kinds
of payoffs you would expect in any group or team-based activity.
Table 9: Free riding
Sally shirks
Sally works hard
Joe shirks
Joe works hard
Both receive a C and leisure
time
Both receive a B, and Joe gets
some leisure time.
Both receive a B, and Sally
gets some leisure time.
Both receive an A, but no
leisure time
To determine the Nash Equilibrium of the game, you need to know how Sally and Joe value leisure and grades.
Assume that both students rank the outcomes as follows:
1.
2.
3.
4.
a grade of B and leisure is better than
a grade of A and no leisure which is better than
a grade of C and leisure which is better than
a grade of B and hard work.
With this set of preferences, they reach a Nash equilibrium where each shirks and each receives a C even though they
would jointly prefer to each get the A that comes with hard work. Like the other prisoners’ dilemma games, this
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illustrates the fundamental conflict between competition and cooperation. Successful study groups will figure out how
to reach the preferred cooperative outcome.
Dating game
The following dating game is designed to illustrate the conflict between group-interest and self-interest. Even though
they have different interests – Sally likes the ballet and Joe likes wrestling – they would prefer to attend events with
each other. The group would be best served if Sally and Joe could decide to attend an event together, but neither of the
coordination possibilities – both attend the ballet or both attend wrestling match – is a Nash equilibrium. The only Nash
equilibrium is lower left, where Joe goes to the wrestling match and Sally goes to the ballet. The Nash equilibrium
represents an unconsummated wealth creating transaction.
Table 10: Dating game
Joe goes to wrestling match
Joe goes to ballet
Sally goes to
wrestling match
Joe is extremely happy;
Sally is OK.
Joe is unhappy;
Sally is unhappy.
Sally goes to ballet
Joe is happy;
Sally is happy.
Joe is OK;
Sally extremely happy
The dating game is analogous to the behavior of divisions within a corporation. Suppose that two separate divisions,
like Saturn and Cadillac, within a single corporation, like General Motors will receive a volume discount if they
purchase common tires from a single supplier. However, if Saturn and Cadillac cannot agree on a single supplier
because they have different preferences, then the volume discounts – shown above in the upper left or lower right
corners – are not realized. This problem could occur if each division were run as a separate profit center and each
division had a strong preference for one of the suppliers.
Table 11: Corporate division dating game (externality)
Saturn chooses Goodyear
tires
Saturn chooses Michelin
tires
Cadillac chooses
Goodyear
Saturn earns 27;
Cadillac earns 17.
Saturn earns 10;
Cadillac earns 10.
Cadillac chooses
Michelin
Saturn earns 20;
Cadillac earns 20.
Saturn earns 17;
Cadillac earns 27
Game of “chicken”
In the classic game of chicken, two teenage boys drive large automobiles straight towards one another. If both go
straight, they crash, and die. If both swerve, they are better off because they live, although with the shame of
“chickening out.” The best outcome for a boy is to go straight, while the other swerves. And of course the boy that
swerves endures “loss of face.”
The game of Chicken has two equilibria, illustrated below. The lesson of the game is that coordination is very
important. Even though there are two equilibria, one equilibrium is preferred by one player, and the other equilibrium
is preferred by his rival. Coordinating on the choice of equilibrium is very important here to avoid the really bad
outcome where both players die.
Table 5: Game of chicken
Dean goes straight
Dean swerves
James goes straight
James swerves
James wins glory, but dies.
Dean wins glory, but dies.
James wins glory, and lives.
Dean faces shame, but lives.
James faces shame, but lives.
Dean wins glory, and lives.
James faces shame, but lives.
Dean faces shame, but lives.
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The game of chicken has business applications as well. In 2000, a biotechnology company had a choice of developing
hybrid grapes to grow in South Africa or in Italy. The company could afford to invest in development of only one
grape variety. Since the Italian market is much bigger than the South African market, it would prefer to serve the
Italian market. However, its only rival is also developing grapes for the two markets, and faces the same choices. Both
would prefer to be the sole entrant in a market, and both prefer Italy to South Africa. This game has the same logical
structure as the Game of Chicken above.
Table 6: Game of chicken
B goes into Italy
B goes into S.
Africa
A goes into Italy
A goes into S. Africa
A is unprofitable,
B is unprofitable.
A is very profitable.
B is moderately profitable.
A is moderately profitable.
B is very profitable.
A is unprofitable,
B is unprofitable.
Again, the lesson of the game is that coordination is very important. This is the also the kind of game that gives rise to
first-mover advantages. If A can move first, it can capture the Italian market for itself. By moving first, A turns the
simultaneous move game into a sequential move game. And it gets to “choose” the favorable equilibrium. We graph
this outcome below.
Firm A
S. Africa
B to Italy
A gets 50
B gets 100
B to S. Africa
A gets -50
B gets -50
Italy
B to Italy
A gets 0
B gets 0
B to S. Africa
A gets 100
B gets 50
We can model the sequential bargaining game between the labor union as a simultaneous game that has the same
logical structure as a game of chicken. The two strategies available to each player are “bargain hard” or
“accommodate”. If both bargain hard, no deal is reached; and if both accommodate, then they split the gains from
trade.
Table 7: Bargaining as a Game of Chicken
Mgmt. bargains hard
Mgmt. accommodates
Labor bargains
hard
Mgmt. gets 0.
Labor gets 0.
Mgmt. gets 50.
Labor gets 150.
Labor
accommodates
Mgmt. gets 150.
Labor gets 50.
Mgmt. gets 100.
Labor gets 100.
Shirking/monitoring game
The following game has no pure strategy equilibrium. Verify this for yourself. If the boss monitors, then the employee
does better by working because he is paid a flat salary. If the employee works, the boss does better by not spending
resources to monitor the employee. But if the Boss does not monitor, then the employee does better by shirking. And
so on.
Table 10: Shirking/Monitoring game
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Boss monitors
Boss does not
monitor
Page 9
Employee shirks
Employee works
Employee gets 0
Boss gets 1
Employee gets 2
Boss gets 0
Employee gets 1
Boss gets 1
Employee gets 1
Boss gets 2
In these kinds of games, the players wind up playing mixed strategies, i.e. randomizing over the strategies that they do
play. The idea is to use the element of surprise to keep your opponent from taking advantage of your strategy. By
randomly choosing actions, neither player can be taken advantage of by the other.
In this game, of course, one might decide that even random monitoring is too expensive, and change to an incentivecompensation scheme. That way, the employee does not have to be monitored as frequently.
Repeated games
In repeated games, one would expect that players find ways to cooperate among themselves, to get out of bad equilibria
because there are richer strategies available in a repeated game. For example, you can punish a rival for behaving
competitively.
Repeated prisoners’ dilemma
The Nash Equilibrium of a prisoners’ dilemma represents an unconsummated wealth-creating transaction between the
players. In the pricing dilemma, both would like to price high. In the advertising dilemma, both would like to advertise
less. In the free riding game, both would like to work harder. However, none of these outcomes is a Nash Equilibrium.
The point of studying the prisoners’ dilemma is to learn to avoid these bad outcomes, or alternatively, to learn how to
consummate these unconsummated wealth-creating transactions. If the game is played only once, it is difficult to find
your way out of such a dilemma. But if the game is repeated, only an idiot would stay stuck in a bad Nash Equilibrium.
To determine the best way to play a repeated prisoners’ dilemma, economist Robert Axelrod 3ran a tournament.
Economists submitted strategies as programmable functions and Axelrod ran simulated tournaments between these
strategies. For example, one strategy might be to price high unless your opponent prices low. If he prices low, then
punish him by pricing low for the next 10 periods. Axelrod was able to characterize the features of the strategies that
earned the highest profits.
1.
2.
3.
4.
5.
Be nice: no first strikes.
Be forgiving: do not try to punish competitors too much if they defect from a good outcome.
Be easily provoked: respond immediately to rivals.
Don’t be envious: focus only on your own slice of the profit pie, not on your competitor’s.
Be clear: make sure your competitors can easily interpret your actions.
Tit-for-tat – doing what your opponent did last period – was the winning strategy. It exhibits all of the characteristics
of a successful strategy.
Discussion Question: Company A and Company B produce carburetors. The industry is declining due to the adoption
of fuel injection technology, but carburetors are still demanded by those who need to repair older automobiles. There is
relatively little direct competition between the two companies because A produces a bronze-finish carburetor and B
produces a chrome-finish carburetor. Consumers do not consider the products to be close substitutes because chrome
carburetors are used externally in “muscle” cars, while bronze carburetors are used internally. Company A hired a
recent MBA from Kellogg who decided to go after B’s customers by producing a new chrome-finish carburetor.
Company B retaliated by offering a bronze-finish carburetor. Profits suffered. Show the implied game, compute Nash
equilibrium, and offer strategy advice to Company A.
3
Robert Axelrod, The Complexity of Cooperation: Agent-based Models of Collaboration and Competition, Princeton
University Press: Princeton, NJ, 1997). http://pup.princeton.edu/titles/6144.html
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Table 12: Carburetor companies
A offers both bronze and
chrome carburetors
A offers only bronze
carburetors
B offers both
bronze and
chrome
carburetors
Both earn low profit
A earns very low profit;
B earns very high profit.
B offers only
chrome
carburetors
A earns very high profit;
B earns very low profit.
Both earn high profit
Answer: The above table depicts what happened. Both companies were earning high profits, but player A was tempted
by the high profits it could earn from entering the chrome-finish carburetor market. To get out of the dilemma, A
should announce its withdrawal from the chrome market and hope that B follows suit. If B does not follow suit, A
should consider further actions against B that would clearly indicate to B that it must leave the bronze-finish market to
make high profits.
Big lesson of the prisoners’ dilemma
Do not get caught in one. It is a bad idea to compete using an easily copied strategy like lowering price.
To avoid getting caught in a dilemma, try to change the structure of your payoffs so that they are not so dependent on
what your rivals do. To do this, try one of the so-called “generic strategies”, differentiate your product, lower your
costs or do something that cannot be easily mimicked. If you have no other option, try the third option, and find ways
to control the competition without running afoul of the antitrust laws.
Discussion Question: Firm X is a large securities firm with two separate divisions located in Denver. Each division
operates as a separate profit center. Division A raised salaries by 20 percent in an attempt to attract more workers,
especially those with industry experience. To avoid losing most of its workforce to Division A, Division B was forced
to match the 20 percent salary increase. Neither division was able to attract workers with industry experience and
profits declined due to the high salaries. What would you recommend to the parent company?
Answer: The parent company adopted two rules to prevent competition among the divisions. Employees were not
allowed to transfer between divisions unless they had worked at least six months in their current job. And pay increases
were limited to 15 percent for transfers within the company. These two rules reduced competition among the divisions
for workers and led to a reduction in wages at the company. Note that these kinds of rules are very similar to the kinds
of restrictions placed on sports teams within a league to prevent them from bidding up player salaries.
Summary of main points
When players act rationally, optimally, and in their self-interest, then it is possible to compute the likely outcomes
of games. By studying games, we learn where the pitfalls are, and how to avoid them.
In sequential games, there is a first-mover advantage, or disadvantage, and players can change the outcome by
committing to a future course of action. Meaningful commitments are difficult to make because they require
players to act in an unprofitable way—against their self-interest.
In prisoners’ dilemma, there is typically a tension between conflict and cooperation—self-interest leads the players
to outcomes that no one likes. Studying the games can help you figure a way to avoid these bad outcomes.
In repeated games, it is much easier to get out of bad situations. Here are some general rules of thumb.
1.
2.
3.
4.
Be nice: no first strikes.
Be forgiving: do not try to punish competitors too much.
Be easily provoked: respond immediately to rivals.
Don’t be envious: focus only on your own slice of the profit pie, not on your competitor’s.
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5.
Be clear: make sure your competitors can easily interpret your actions.
Homework Problem & Solutions
Individual problems
Study Group Free Riding: In the game below, change the ranking of values that Sally and Joe place on leisure and
grades to change the game from a prisoners’ dilemma into a game of chicken (PNG calls this co-opetition) with
two equilibria. Give advice to Joe about how to change the game to his advantage.
Joe shirks
Joe works hard
Sally
shirks
Both receive a C and
leisure time.
Sally
works
hard
Both receive a B, and
Joe gets some leisure
time.
Both receive a B, and
Sally gets some
leisure time
Both receive an A,
but no leisure time.
Upper left is an equilibrium if both students rank the outcomes as follows:
1.
2.
3.
4.
a grade of B and leisure is better than
a grade of A and no leisure which is better than
a grade of C and leisure which is better than
a grade of B and hard work.
ANSWER
Assume that both students rank the outcomes as follows:
1.
2.
3.
4.
a grade of B and leisure is better than
a grade of A and no leisure which is better than
a grade of B and hard work.
a grade of C and leisure which is better than
Joe should try to commit to shirking, or change the game into a sequential game and move first, in order to force
Sally into working hard. In essence, this allows him to “choose” the best equilibrium for himself.
Coke v. Pepsi: In 1931, Pepsi was almost broke. The depression hit it hard, and Coke had most of the Duopoly market
for soft drinks in the U.S. It tried many things: marketing campaigns, label changes etc. Then it came up with the idea
of selling 12 oz bottles for 5¢, which had been the 6 oz price. Coke could have followed the price per unit down, but it
didn’t. Total soft drink demand increased, and Pepsi took a larger share of the demand. Why is the equilibrium of this
game different from that of a prisoners’ dilemma? (HINT: change the payoffs of the prisoners’ dilemma to reflect the
implied equilibrium.)
Answer:
Coke prices low
Coke prices high
Pepsi prices low
Both make low profit
Pepsi prices high
Both make low profit
Coke makes OK profit
Pepsi makes high profit
Coke makes high profit
Pepsi makes low profit
Labor vs. management: Consider the following game between labor and management:
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Mgt: Bargain Hard
Mgt: Be Nice
Labor: Bargain Hard
Labor: Be Nice
Mgt: 0
Mgt: 20
Labor: 0
Labor: 10
Mgt: 10
Mgt: 15
Labor: 20
Labor: 15
Circle the letter(s) that correspond to the Nash Equilibrium(ia):
A.
B.
C.
D.
Upper left
Upper right
Lower left
Lower right.
Nashville tax on hotels: Nashville area hotels have experienced a decline in demand since Opryland closed three years
ago. The hotels are lobbying the city council to increase the city’s hotel room tax to raise funds to advertise the
benefits of Nashville to attract more tourists. Suppose that there are only two hotels, A and B, and model the
advertising game between the hotels as a prisoners’ dilemma. Fill in the following table with strategies and payoffs.
Compute equilibrium, and comment on the tax as a solution to the free riding problem.
Table: Free riding dilemma
Strategy for A
Strategy for A
Strategy for B
Strategy for B
ANSWER:
Table: Free riding dilemma
Hotel B does not advertise
Hotel A does not advertise
Hotel A advertises
Both earn low profit
Hotel A earns very low profit
Hotel B earns high profit
Hotel B advertises
Hotel A earns high profit
Both earn moderate profits
Hotel B earns very low profit
Advertising by one hotel would benefit both hotels. Hotel A and B would like to advertise jointly, but this is not a
Nash equilibrium. Once Hotel A advertises, hotel B does better by not advertising, i.e. by free riding off A’s
advertising expenditures; and once hotel B does not advertise, Hotel A does better by not advertising. The Nash
Equilibrium is in the upper left where neither hotel advertises. The room tax “forces” both hotels to pay for advertising
and is thus a way out of the dilemma.
Repeated prisoners’ dilemma: Which of the following is not a good strategy in a repeated prisoner’s dilemma? Circle
all correct answers.
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a.
Retaliate immediately against your opponent.
b.
Do not be the first one to price low if you are selling a substitute product.
c.
Never let your opponent know what you are thinking.
d.
Do not punish your competitor too much.
Airline competition: Suppose that Eastern Airways and Aero Sol charge $100 for a round-trip economy class fare, and
that, presently, each airline has an equal market share. Each has a demand elasticity of -2.0. The cross-price elasticities
of demand are both 1.0.
a.
If both marginal costs are constant at $50, are the airlines currently in Nash equilibrium?
ANSWER: YES, MR=MC or (P-MC)/P=1/|elas| for each player. Neither can do better by unilaterally changing
price.
b.
If revenue sharing is introduced, will the airlines be able to raise price above $100? Why? [HINT: What
happens to MR under a revenue sharing deal?]
ANSWER: YES. At a price of $100, MR=MC. Under revenue sharing, MR is cut in half, so each firm finds it
beneficial to sell less, or equivalently, to raise price.
Ultimatum Game: You are given an offer to split a $20 bill. The other player offers you $1. If you accept the offer,
you keep the $1 and the other player keeps $19. If you reject the offer, neither of you will get anything. Do you take
the offer?
Answer: Yes, take the offer, since your payoff is $1 or nothing. In this game, the other player has the advantage.
Ultimatum Game continued: How could you take the advantage away from the other player in the ultimatum game?
Answer: In the last game, you had no influence on the game or its rules. If you could effectively signal before the
game that you would accept only a fair deal and would reject any other offer, you would take away his/her advantage.
Prisoners’ Dilemma: Nokia and Ericsson plan on introducing new handheld communication devices. However, they
each must decide whether to use their own software standard or a common third party developed standard. Here are the
respective payoffs:
Ericsson
Own Standard
Common Standard
Own Standard
$15M, $18M
$8M, $29M
Nokia
Common Standard
$25M, $10M
$20M, $23M
Answer:
The Nash equilibrium is the top left box (both developing their own standards).
Maze Problem: According to the following maze, draw a tree form diagram mapping out each decision point and the
possible outcomes with no backtracking.
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START
Answer:
For each of the following points, a decision must be made.
1
2
START
3
Therefore the tree form looks like the following:
Start
East
South
(DEAD END)
South
East
(DEAD END)
West
(DEAD END)
East
(GOLD)
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Chicken game: Two airlines are deciding to choose whether Atlanta or Chicago should be their major hub. What is/are
the equilibrium(a) of the payoffs?
Airline A
Airline B
Atlanta
Chicago
Atlanta
$40M, $40M
$60M, $85M
Chicago
$85M, $60M
$35M, $35M
Answer: since this is a game of chicken, there are two equilibria:
Airline A
Airline B
Atlanta
Chicago
Atlanta
$40M, $40M
$60M, $85M
Chicago
$85M, $60M
$35M, $35M
Group problems
Simultaneous Game: Describe a simultaneous game within your firm, or between your firm and a competitor, or
between your firm and a customer or supplier. Formally draw a 2 X 2 payoff matrix with the strategy choices clearly
labeled, and the payoffs to each of the parties (use number if you can estimate them, otherwise describe qualitative
rankings among outcomes.). Clearly identify the equilibrium by shading the cell of the table. What advice can you
derive from on your analysis? Compute the profit consequences of the advice.
Sequential Game: Describe a sequential game facing your firm. Plot the sequential game using Figures 10.2 and 10.3
in PNG as models. HINT: Insert the Microsoft Organizational Chart “object” into Word to represent the extensive
form game. Compute and analyze the equilibrium of the game by shading the equilibrium. What advice can you
derive from your analysis? Compute the profit consequences of the advice.
Repeated Game: Describe a repeated game facing your firm. Compute and analyze the equilibrium of the game and
explicitly show how it differs from the one-shot (non-repeated) equilibrium. What advice can you derive from your
analysis? Compute the profit consequences of the advice.
Appendix: Finding Nash Equilibria by Mikhail Shor
When a game does not have any dominant or dominated strategies, or when the iterated deletion of dominated
strategies does not yield a unique outcome, cell-by-cell inspection is used. Simply put, each cell of the game box is
checked, each time asking “does either player have incentive to change their strategy unilaterally” or “are both players
playing a best response.” The problem with this approach is that it can take quite a while. If there are two players, one
with N strategies and the other with M, we need to check NM cells which, for many games, can be quite large.
In this note, you will learn a simpler technique for conducting cell-by-cell inspection. This method, called “strategy-bystrategy inspection” requires only N+M inspections, and is less likely to result in mistakes.
Consider a two-player game with five strategies for each player:
Player 2
A
B
C
D
E
9
7
5
3
1
V
,
,
,
,
,
9
8
6
9
2
7
5
3
1
9
W
,
,
,
,
,
1
2
3
9
8
Player 1
X
5 , 6
3 , 6
1 , 8
9 , 4
7 , 7
3
1
9
7
5
Y
,
,
,
,
,
4
4
7
9
6
1
3
1
5
3
Z
,
,
,
,
,
1
3
5
9
7
Since each player has five strategies, in order to perform cell-by-cell inspection, you could have to check 55=25
inspections. Instead, using the best-response method, only 5+5=10 checks have to be performed.
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Start with Player 1. For each of Player 1’s strategies, select the strategy/strategies that would maximize Player 2’s
payoff. For example, if Player 1 plays V, Player 2’s best response is to play A, earning a payoff of 9. Underline this
payoff in the game box:
Player 1
Player 2
A
V
9 , 9
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Proceed with Player 1’s other strategies.
Player 2
A
B
C
D
E
9
7
5
3
1
V
,
,
,
,
,
9
8
6
9
2
7
5
3
1
9
W
,
,
,
,
,
1
2
3
9
8
Player 1
X
5 , 6
3 , 6
1 , 8
9 , 4
7 , 7
3
1
9
7
5
Y
,
,
,
,
,
4
4
7
9
6
1
3
1
5
3
Z
,
,
,
,
,
1
3
5
9
7
Now, we can do the same thing for Player 1’s best response to Player 2.
If Player 2 plays D, Player 1 is indifferent between V,W,Y, and Z. Note that a best response is not necessarily unique. A
player can have a number of strategies that are all best responses if they all earn the same payoffs, and earn greater
payoffs than other strategies. In cases such as these, underline all of the best responses. Continue for the rest of Player
2’s strategies:
Player 2
A
B
C
D
E
9
7
5
3
1
V
,
,
,
,
,
9
8
6
9
2
7
5
3
1
9
W
,
,
,
,
,
1
2
3
9
8
Player 1
X
5 , 6
3 , 6
1 , 8
9 , 4
7 , 7
3
1
9
7
5
Y
,
,
,
,
,
4
4
7
9
6
1
3
1
5
3
Z
,
,
,
,
,
1
3
5
9
7
Recall that an equilibrium is defined by both players simultaneously playing their best replies, so that neither player has
an incentive to deviate. This is equivalent to saying that a pair of strategies is in equilibrium if both payoffs are
underlined.
In the above, we find three equilibria:
{A,V} {E,W} {D,Z}
You should convince yourself that in all three cases, neither player has an incentive to deviate, or change their strategy
unilaterally.
17