Econ 300

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Econ 300 Intermediate Microeconomics University
of Illinois at Urbana Champaign Giovanni
Facchini Lectures 24-27 Game Theory and
applications
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Outline

• 1. Motivation Honda and Toyota
• Dominant and dominated strategies
• The Nash equilibrium
• 4. Limitations of the Nash equilibrium concept
• Multiplicity
• Non existence of pure strategy equilibria

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Outline
Outline

• 5. Duopoly
• Cournot (quantity) competition
• Bertrand (price) competition
• 6. Extensive form games and the value of
  commitment

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What is the likely outcome of this game?
Toyota vs. Honda Game
Matrix 1 A Capacity Expansion Game
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Elements of this game Players agents
participating in the game (Toyota,
Honda) Strategies Actions that each player may
take under any possible circumstance (Build,
Don't build) Outcomes The various possible
results of the game (four, each represented by
one cell of matrix) Payoffs The benefit that
each player gets from each possible outcome of
the game (the profits entered in each cell of the
matrix)
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Information A full specification of who knows
what when (in our example full
information) Timing Who can take what decision
when and how often the game is repeated
(simultaneous, one-shot) Behavioural prediction
"What is the likely outcome"? (Dominant
Strategy Equilibrium, Nash Equilibrium)
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Dominant and Dominated Strategies
In Toyota vs Honda, the payoff from playing
Build a new plant is always higher than the
payoff from Do not build, whatever the action
of the other player. Build a new plant is a
dominant strategy Definition A dominant
strategy is a strategy that is better than any
other strategy that a player might choose, no
matter what strategy the other player follows.
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• In Toyota vs Honda, if the players are rational,
  they will always choose Build
• Build, Build is the equilibrium in dominant
  strategies
• If a player has a dominant strategy, he does not
  need to form expectations on what the opponent
  will do ? this makes the analysis particularly
  simple

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The Prisoner's Dilemma
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• Equilibrium in dominant strategies is Confess,
  Confess, but it is Pareto dominated by the
  outcome Dont confess, Dont confess
• Prisoner Dilemma is a parable that can explain
  why cooperation often fails
• Same structure in many other games
• Toyota vs Honda
• Cartels
• Disarmament
• Provision of public goods

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Dominated Strategies
2
1
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• No player has a dominant strategy, but agent 2
  would never play Right. Playing center would
  always guarantee a higher payoff.
• Definition A strategy B is strictly dominated by
  another strategy A if A yields a higher payoff
  than strategy B, no matter what strategies are
  chosen by the other players
• A rational player would never choose a strictly
  dominated strategy
• Outcome up and center is the equilibrium

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2
1
No dominant/ dominated strategies!!!!
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• Look for best replies to the opponents
  strategy
• Do so for every strategy and every player
• The common joint reply (is) is (are) the
  equilibrium (a) of the game
• The concept has been introduced by John F. Nash,
  Nobel Laureate in Economics in 1994, in his Ph.D.
  thesis at Princeton

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Nash Equilibrium
Definition A Nash Equilibrium occurs when each
player chooses a strategy that gives him/her the
highest payoff, given the strategy chosen by the
other player(s) in the game. ("rational
self-interest") Toyota vs. Honda A Nash
equilibrium Each Firm Builds a
New Plant
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• Why?
• Given Toyota builds a new plant, Honda's
• best response is to build a new plant.
• Given Honda builds a new plant, Toyota's
• best response is to build a plant.

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Why is Nash Equilibrium a plausible outcome?

• A Nash equilibrium is a pair of mutual best
  replies
• But When a player decides which strategy to
  choose, he does not know what his opponent is
  going to do he will form expectations about his
  behavior. In equilibrium these expectations have
  to be consistent
• If the strategy profile actually played is not an
  equilibrium, at least one player must have made a
  mistake
• Either he doesnt play a best response to the
  expected strategy of his opponent
• His strategy is optimal, given his expectations,
  but his expectations are wrong.

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• A Nash equilibrium can be interpreted as a pair
  of mutually consistent expectations If each
  player chooses the strategy his opponents expect
  him to choose, no player has an incentive to
  deviate and the expectations will be fulfilled

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Why should we observe real players play Nash
equilibria?

• Rational analysis of the game ? but some
  philosophical issues here (Bernheim, Pierce
  rationalizability)
• External proposal (external consultant). The
  proposal could be a social norm
• Communication between players before the game is
  actually played if they agree to play Nash, the
  outcome is self-enforcing, otherwise not
• Learning by trial and error

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Note When a player has a dominant strategy,
that strategy will be the player's Nash
Equilibrium strategy.
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Limitations of Nash Equilibrium 1. The Nash
Equilibrium need not be unique
He
She
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• Multiple equilibria (Boxing, Ballet) and
  (Ballet, Boxing)
• If there are several Nash Equilibria, it is not
  obvious which equilibrium should be played by
  rational agents
• Several theories that try to predict which Nash
  equilibrium should be played
• Focal Points (Schelling The strategy of
  conflict)
• Refinements of the Nash equilibrium concept
  (Selten subgame perfect)

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2) Nash equilibrium in pure strategies may not
exist
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Matching Pennies
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• There is no equilibrium in pure strategies for
  this game
• Nash has shown that if we allow players to
  randomize between strategies, an equilibrium in
  mixed strategies will always exist

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Applications Oligopoly

• In many interesting economic applications, the
  strategy spaces available to the players are
  continuous
• We cannot just write down the payoff matrix, and
  mark the best responses there
• Instead we need to explicitly compute the best
  responses

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• Assume Many Buyers
• Few Sellers (two in most examples)
• Each firm faces downward-sloping demand because
  each is a large producer compared to the total
  market size
• There is no one dominant model of oligopoly we
  will review several.

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Cournot Duopoly
Assume Firms compete setting outputs
(quantities) Homogeneous Products
Agents move simultaneously
Noncooperative Definition In a Cournot game,
each firm sets its output (quantity) taking as
given the output level of its competitor(s), so
as to maximize profits. Price adjusts according
to demand. Residual Demand Firm i's guess
about its rival's output determines its residual
demand.
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Example Residual Demand
Price
Residual Marginal Revenue when q2 10
10 units
Residual Demand when q2 10
MC
Demand
Quantity
0
q1
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Profit Maximization Each firm acts as a
monopolist on its residual demand curve, equating
MRR to MC. MRR p q1(dp/dq) MC

• Best Response Function
• The point where (residual) marginal revenue
  equals marginal cost gives the best response of
  firm i to its rival's actions.
• For every possible output of the rival, we can
  determine firm i's best response. The sum of all
  these points makes up the best response
  (reaction) function of firm i.

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q2
Example Reaction Functions, Quantity Setting
Reaction function of firm 1
0
q1
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q2
Example Reaction Functions, Quantity Setting
Reaction function of firm 1

q2
Reaction function of firm 2
0
q1
q1
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Equilibrium No firm has an incentive to deviate
in equilibrium in the sense that each firm is
maximizing profits given its rival's
output. Example P 100 - Q1 - Q2 MC AC
10 What is firm 1's profit-maximizing output
when firm 2 produces 50? Firm 1's residual
demand P (100 - 50) - Q1 MR50 50 - 2Q1 MR50
MC ? 50 - 2Q1 10 Q150
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b. What is the equation of firm 1's reaction
function? Firm 1's residual demand P (100 -
Q2) - Q1 MRr 100 - Q2 - 2Q1 MRr MC ? 100 -
Q2 - 2Q1 10 Q1r 45 - Q2/2 firm 1's reaction
function
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c. Similarly, one can compute that Q2r 45 -
Q1/2. Now, calculate the Cournot
equilibrium. Q1 45 - (45 - Q1/2)/2 Q1
30 Q2 30 P 40 ?1 ?2 30(30) 900
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Bertrand Duopoly
Assume Firms set price
Homogeneous product
Definition In a Bertrand oligopoly, each firm
sets its price, taking as given the price set by
other firm, so as to maximize profits.
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• How will each firm set price?
• Homogeneity implies that consumers will buy from
  the low-price seller.
• Further, each firm realizes that the demand that
  it faces depends both on its own price and on the
  price set by other firms
• Specifically, any firm charging a higher price
  than its rivals will sell no output.
• Any firm charging a lower price than its rivals
  will obtain the entire market demand.

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Definition The relationship between the price
charged by firm i and the demand firm i faces is
firm i's residual demand In other words, the
residual demand of firm i is the market demand
minus the amount of demand fulfilled by other
firms in the market Q1 Q - Q2
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Price
Example Residual Demand Curve, Price Setting
Market Demand
Residual Demand Curve (thickened line segments)

Quantity
0
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• Assume firm always meets its residual demand (no
  capacity constraints)
• Assume that marginal cost is constant at c per
  unit.
• Hence, any price at least equal to c ensures
  non-negative profits.

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Price charged by firm 2
Example Reaction Functions, Price Setting
Reaction function of firm 1
p2
Price charged by firm 1
0
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Example Reaction Functions, Price Setting and
Homogeneous Products
Price charged by firm 2
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Reaction function of firm 1
Reaction function of firm 2

p2
Price charged by firm 1
p1
0
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Thus, each firm's profit maximizing response to
the other firm's price is to undercut (as long as
P gt MC) Definition The firm's profit
maximizing action as a function of the action by
the rival firm is the firm's best response (or
reaction) function Example 2 firms Bertrand
competitors Firm 1's best response function is
P1P2- e Firm 2's best response function is
P2P1- e
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Equilibrium
If we assume no capacity constraints and that
all firms have the same constant average and
marginal cost of c then For each firm's
response to be a best response to the other's
each firm must undercut the other as long as Pgt
MC Where does this stop? P MC (!)
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So 1. Firms price at marginal cost 2. Firms
make zero profits 3. The number of firms is
irrelevant to the price level as long as more
than one firm is present two firms is enough to
replicate the perfectly competitive
outcome! essentially, the assumption of no
capacity constraints combined with a constant
average and marginal cost takes the place of free
entry
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Sequential Move Games
So far, we have considered games in which both
parties decide simultaneously which strategy to
choose. Now we will consider situations in which
one player moves first, and the other player
observes what the first player did, and then
decides which strategy to choose. Definition A
game tree shows the different strategies that
each player can follow in the game and the order
in which those strategies get chosen.
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A modified version of the entry game Toyota
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• Game trees often are solved by starting at the
  end of the tree and, for each decision point,
  finding the optimal decision for the player at
  that point.
• Keeps analysis manageable.
• Ensures optimality at each point.
• The solution to the revisited game differs from
  that of the simultaneous game. Why?
• First mover can force second mover's hand
• Illustrates the value of commitment (i.e.
  limiting one's own actions) rather than
  flexibility
• Example Irreversibility of Business
  Decisions in the Airline Industry.

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Summary
1. Game Theory is the branch of economics
concerned with the analysis of optimal decision
making when all decision makers are presumed to
be rational, and each is attempting to anticipate
the actions and reactions of the competitors 2.
A Nash Equilibrium in a game occurs when each
player chooses a strategy that gives him/her the
highest payoff, given the strategies chosen by
the other players in the game.
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• When there are multiple Nash Equilibria, we must
  refine the equilibrium concept in order to
  choose the "likely" outcome of the game.
• Game theory is very useful in analysing markets
  where there are only a small number of producers
  (intermediate situations b/w competition and
  monopoly)
• 5. An analysis of sequential move games reveals
  that moving first in a game can have strategic
  value if the first mover can gain from making a
  commitment.