Chapter
Fourteen
Game Theory
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Game Theory
• In this chapter, we examine three main
topics
– An Overview of Game Theory
– Static Games
– Dynamic Games
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An Overview of Game Theory
• Game theory
– A set of tools that economists, political
scientist, political scientists, military
analysts, and others use to analyze
decision-making by players that use
strategies
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An Overview of Game Theory
• Game
– Any competition between players (firms) in
which strategic behavior plays a major role
• Strategy
– A battle plan of the actions a player (firm)
plans to take to compete with other
players (firms)
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An Overview of Game Theory
• Strategic Behavior
— A set of actions a firm takes to increase
its profit, taking into account the possible
actions of other firms
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An Overview of Game Theory
• Game theory tries to answer two questions:
how to describe a game and how to predict
the game’s outcome.
• A game is described in terms of the players;
its rules; the outcome; the payoffs to players
corresponding to each possible outcome;
and the information that players have about
their rivals’ moves.
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An Overview of Game Theory
• The rules of the game determine the timing
of players’ moves and the actions that
players can make at each move.
• Games with complete information are games
where the payoff function is common
knowledge among all players.
• Games with perfect information are games
where the player who is about to move
knows the full history of the play of the game
to this point, and that information is updated
with each subsequent action.
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An Overview of Game Theory
• A static game is one in which each player
acts only once and the players act
simultaneously.
• A dynamic game is one where players move
either sequentially or repeatedly.
• Sequential move dynamic games: chess,
Stackelberg model.
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Static Games
• A static game is one in which the players
choose their actions simultaneously, have
complete information about the payoff
function, and play the game once.
• Examples: the Cournot and Bertrand models.
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Normal-Form Games
• A normal-form representation of a static
game of complete information specifies the
player in the game, their possible strategies,
and the payoffs for each combination of
strategies.
• A profit matrix (or payoff matrix), such as in
Table 14.1, shows the strategies the firms
may choose and the resulting profits.
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Table 14.1
Profit Matrix for a Quantity-Setting
Game: Dominant Strategy
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Normal-Form Games
• Because the firms choose their strategies
simultaneously, each firm selects a strategy
that maximizes its profit given what it
believes the other firm will do.
→a noncooperative game of imperfect
information
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Predicting A Game’s Outcome
• Dominant strategy
– A strategy that strictly dominates (gives
higher profits than) all other strategies,
regardless of the actions chosen by rival
firms
• Where a firm has a dominant strategy, its
belief about its rivals’ behavior is irrelevant.
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Predicting A Game’s Outcome
• If United choose the high-output strategy
( qU  64 ), American’s high-output strategy
maximizes its profit.
• If United chooses the low-output strategy
( qU  48 ), American’s high-output strategy
maximizes its profit.
• Thus the high-output strategy is American’s
dominant strategy.
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Predicting A Game’s Outcome
• They don’t cooperate due to a lack of trust:
Each firm uses the low-output strategy only
if the firms have a binding agreement.
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Predicting A Game’s Outcome
• Each firm has a substantial profit incentive to
cheat on the agreement.
• In this type of game—called a prisoner’s
dilemma game—all players have dominant
strategies that lead to a profit (or other
payoff) that is inferior to what they could
achieve if they cooperated and pursued
alternative strategies.
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Table 14.2
Profit Matrix for a Quantity-Setting
Game: Iterated Dominance
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Best Response and Nash
Equilibrium
• We can think of oligopolies as engaging in a
game, which is any competition between
players (such as firms) in which strategic
behavior plays a major role.
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Best Response and Nash
Equilibrium
• When is an oligopolistic market in
equilibrium?
• Nash equilibrium
– A set of strategies such that, holding the
strategies of all other firms constant, no
firm can obtain a higher profit by choosing
a different strategy
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Best Response and Nash
Equilibrium
• In a Nash equilibrium. No firm wants to
change its strategy because each firm is
using its best response —the strategy that
maximizes its profit, given its beliefs about
its rivals’ strategies.
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Table 14.3
Simultaneous Entry Game
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Multiple Nash Equilibria
• Pure Strategy: Each player chooses a single
action.
• Mixed Strategy: The player chooses among
possible actions according to probabilities it
assigns.
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Multiple Nash Equilibria
• This game has two Nash equilibria in pure
strategies: Firm 1 enters and Firm 2 does not,
or Firm 2 enters and Firm 1 does not.
• How do the players know which (if any) Nash
equilibrium will result? They don’t. It is
difficult to see how the firms choose
strategies unless they collude.
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Mixed Strategies
• These pure Nash equilibria are unappealing
because they call for identical firms to use
different strategies. The firms may use the
same strategies if their strategies are mixed.
When both firms enter with a probability of
one-half—say , if a flipped coin comes up
heads—there is a Nash equilibrium in mixed
strategies.
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Mixed Strategies
• One important reason for introducing the
concept of a mixed strategy is that some
games have no pure-strategy Nash equilibria.
• In the game with no dominant strategies,
neither firm has a strong reason to believe
that the other will choose a pure strategy. It
may think about its rival’s behavior as
random.
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Table 14.4(a)
Advertising Game
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Cooperation
• The game in panel (a) is a prisoners’ dilemma
game. Each firm has a dominant strategy: to
advertise.
• The sum of the firms’ profits is not
maximized in this simultaneous-choice oneperiod game.
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Cooperation
• Suppose the two firms meet in advance and
agree not to advertise. Each firm has a
substantial profit incentive to cheat on the
agreement.
• This is because that in this game, if one firm
advertises, its sales increase so that its profit
rises, but its rival loses customers and hence
the rival’s profit falls.
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Table 14.4(b)
Advertising Game
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Cooperation
• In panel (b), when either firm advertises, the
promotion attracts new customers to both
firms. Advertising is a dominant strategy for
both firms.
• Both firms advertises in the games in panel
(a) and panel (b). The distinction is that the
Nash equilibrium in which both advertise is
the same as the collusive equilibrium in
panel (b), but not in panel (a).
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Dynamic Games
• In dynamic games players move sequentially
or move simultaneously repeatedly over time,
so a player has perfect information about
other players’ previous moves.
• Extensive form of a game specifies the n
players, the sequence in which they make
their moves, the actions they can take at
each move, the information that each player
has about players’ previous moves, and the
payoff function over all possible strategies.
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Sequential Game - Stackelberg
Model
• Suppose, however, that one of the firms,
called the leader, can set its output before its
rival, the follower, sets its output.
• This type of game, in which the players make
decisions sequentially, arises naturally if one
firm enters a market before another.
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Figure 14.1 Stackelberg Game
Tree
• The game tree shows the order of the firms
move, each firm’s possible strategies at the
time of its move, and the resulting profits.
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Figure 14.1
Stackelberg Game Tree
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Stackelberg Game Tree
• If American choose 48, United will set 64, so
American’s profit will be $3.8 million.
• If American choose 64, United will set 64, so
American’s profit will be $4.1 million.
• If American choose 96, United will set 48, so
American’s profit will be $4.6 million.
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Stackelberg Game Tree
• Thus to maximize its profit, American choose
96. United responds by selling 48.
• This outcome is a Stackelberg (Nash)
equilibrium.
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Subgame Perfect Nash
Equiligrium
• A set of strategies forms a subgame perfect
Nsh equilibrium if the players’ strategies are
a Nash equilibrium in every subgame.
• We can solve for the subgame perfect Nash
equilibrium using backward induction, where
we first determine the best response by the
last player to move, next determine the best
response for the player who made the nextto-last move, and then repeat the process
until we reach the move at the beginning of
the game.
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Figure 14.2 Whether an Incumbent
Pays to Prevent Entry
• If the potential entrant stays out of the
market, it makes no profit,  e  0 , and the
incumbent firm makes the monopoly
profit,  i   m . If the potential entrant enters
the market, both firms make the duopoly
profit,  d . Entry occurs if the duopoly profit
is positive, d  0 .
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Figure 14.2 Whether an
Incumbent Pays to Prevent Entry
• Entry is blockaded (does not occur
regardless of actions by the incumbent) if the
duopoly profit is negative because of low
demand or high fixed costs of entering, both
of which lower profit, if entry is not
blockaded, the incumbent acts to deter entry
by paying for exclusive rights to be the only
firm at the rest stop only if  m  b   d.
Otherwise (if  m  b   d ), the incumbent
accommodates entry.
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Figure 14.2
Game Tree: Whether an Incumbent
Pays to Prevent Entry
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Credibility
• credible threat
– an announcement that a firm will use a
strategy harmful to its rival that the rivals
believe because the firm’s strategy is
rational in the sense that it is in the firm’s
best interest to use it
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Credibility
• By committing to produce a large quantity
whether or not entry occurs, the incumbent
discourages entry.
• Commitment as a Credible Threat. The
intuition for why commitment makes a threat
credible is that of “burning bridges.”
• Similarly, by limiting its future options, a firm
makes itself stronger.
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Repeated Game
• In a single-period game, one firm cannot
punish the other firm for cheating in a cartel
agreement.
• But if the firms meet period after period, a
wayward firm can be punished by the other.
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Repeated Game
• Supergame
– A game that is played repeatedly, allowing
players to devise strategies for one period
that depend on rivals’ actions in previous
periods
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Repeated Game
• In a repeated game, a firm can influence its
rival’s behavior by signaling and threatening
to punish.
• AA may signal UA by producing low output.
• In addition to or instead of signaling, a firm
can punish a rival for not restricting output.
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Repeated Game
• American will produce the smaller quantity
each period as long as United does the same.
• If United produces the larger quantity in
period t , American will produce the larger
quantity in period t  1 and all subsequent
periods.
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Repeated Game
• Thus United’s best policy is to produce the
lower quantity in each period unless it cares
greatly about current profit and little about
future profits.
• Thus if firms play the same game indefinitely,
they should find it easier to collude.
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Repeated Game
• Maintaining a cartel will be difficult if the
game has a known stopping point.
• If the players know that the game will end but
aren’t sure when, cheating is less likely to
occur.
• Collusion is therefore more likely in a game
that will continue forever or that will end at
an uncertain time.
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