Managerial economics
Lecture 1: game theory and competition
Game theory: fundamentals
Game theory = a mathematical tool used to represent strategic interactions in a
treatable way
Games are a stylized representation of agents’ goals, information and
capabilities
Our goal:
- Predict the likely outcomes when multiple players have conflicting paths
- Understand what factors can give an edge to different players or affect the
outcomes (comparative static)
- Identify pitfalls that undermine desirable outcomes and how to mitigate them
In real life: decision-makers have goals/preferences about the outcome of their
interactions (example: winning or losing, (not) getting a promotion)
In a game: we represent these preferences assigning to outcomes a numerical value
(payoff)
- We assign higher payoff to outcomes that are more preferred to them
- Players might disagree on what outcome is the most desirable (gives the
highest payoff)
- Strategies are actions or sequence of actions that agents can choose to achieve
their goals
- The outcome of a game is the result of all player’s strategies and chance if the
game involves stochastic/random elements
In game theory we assume that agents (players) are individually rational, that is:
1. They have rational preferences (over the outcomes)
2. They are payoff-maximizing (not necessarily selfish)
Strategic games
1. Static
o One-shot game: players interact only once
o Simultaneous choice: players choose their strategy without knowledge of
the other players’ choices
2. Complete information
o All players know the capabilities of every other player (strategies)
o All players know the consequences of every combination of strategies
(payoff)
Uncertainty (ea. stocks) ≠ asymmetric information (ea. Boss knows more)
Strategic games allow for uncertainty, but not asymmetric information
Complete information Asymmetric information
Static Strategic games Bayesian games
,Dynamic Extensive game, repeated Sequential (Bayesian)
game games
A strategic game is fully described by:
1. Set of players
2. Set of actions available for each player
3. Players’ payoff functions
An equilibrium in dominant strategy is a very strong notion, only relies on player’s
rationality
- Pros: if players are rational; it is the only possible outcome very reliable
- Cons: too restrictive, most games don’t have one
Nash equilibrium (NE)
= an action profile such that none of the players can increase their
payoff by choosing a different strategy, taking the other player’s
strategies for given
In other words: it is an action profile such that no player can
benefit from independently deviating from it
Notice: it is a resting point, there is nothing optimal about it
We say that each player is best responding to the other
player’s strategies
Demand, monopoly and competition
Market demand
Demand function tells us how many units of a product are purchased by consumers
depending on its price
- At price p1, customers buy quantity q1
- The slope of the demand dq/dp tells us the decrease/increase of units sold for a
small increment in the price. For normal goods, it is always negative
(opportunity cost)
∆ q/ p dq p
- Economists rather use price elasticity of demand ε ( p ) = = .
∆ p /q dp q
o Always negative for normal goods. Describes the ability/willingness of
consumers to substitute consumption with another good (or not spend)
o |ε| = 1: if price reduces by 5% then quantity increases by the same
percentage amount
o |ε| < 1: demand is inelastic => increasing price increases revenues
o |ε| > 1: demand is elastic => increasing price reduces revenues
Linear demand: q(p) = a – bp
- a is the quantity demanded when price is infinitesimal
- the slope of the demand is constant dq/dp = b
- elasticity is not constant
Inverse market demand: p(q) = A – Bq with a/b = A, 1/b = B
- A = maximum willingness to pay (WTP) among the consumers
,Perfect competition
does not exist! It’s an ideal archetype, but it is useful as a benchmark
- Inverse demand is a horizontal line at price p
- Consumers willing to buy any quantity but only at the price p
or lower
- Infinitely many firms willing to supply at that price
- Price elasticity = -∞
Single-agent problem. Each individual is too small to affect demand
(no strategic interactions)
- Can produce a quantity q at cost given by their cost function
C(q)
- Observe the exogenous price p
- Chooses to produce the quantity that maximizes its profits
Maxq { π ( q )= p∗q−C (q) }
Take derivative w.r.t. q for f.o.c.
−∆ C (q)
0=p+
∆q
∆ C (q)
p= = C’(q)
∆q
Perfect competition: p = C’
- Price-taking (alone) does not mean zero-profits
- If p is sufficiently high, profits are possible: triangle pAB minus fixed costs
- Wit free entry, in the longrun the price p drops to the minimum of the average
total cost curve (ATC), so profits are zero by construction
- Price taking + free entry = no profits. Not a good market to enter, but great for
consumers.
Monopolist’s problem
Still a single-agent (non-strategic) problem. The monopolist:
- Observes a downward sloping demand q(p) and can set the price
- Can produce its product at cost given by the cost function C(q)
- Choose the price (or quantity) that maximize its profits
Maxq { π ( p )= p∗q( p)−C (q ( p))}
Take derivative w.r.t. q for f.o.c.
∆ C ( q ( p ))
∗∆ q ( p )
0 = q(p) + p* ∆ q (p) ∆q
−
∆p ∆p
−∆ q ( p )
q(p) = ( p−C' ( q ) )
∆p
( )
−1 '
∆ q ( p ) q ( p ) ( p−C ( q )) ( p−C' ( q ) )
− = with the lerner index (%
∆p p p p
markup)
( ) 1
1+ p=C ' (q ( p ) )
ε
Imperfect competition: Bertrand duopoly
Now it’s a two-players game, assumptions:
- Two firms, firm 1 and firm 2, produce a homogeneous product
, - Both firms have the same costs function C(q)
- Each firm chooses its own price p1 and p2 simultaneously
- Consumers have a demand q(p) but only buy from the
cheapest firm
- If prices are equal, they split equally their purchases from
the two firms
- Firms only produce when/if units are sold
For simplicity, assume:
- Each firm has constant marginal cost C’(q) = c
- Consumer demand is linear: q = a - bp
Assume that firm 1 sets p1
- For prices above p1, the quantity sold is 0
a−bp 1
- At p1, the quantity sold jumps to
2
- For prices below p1 the quantity follows the consumer demand
- The discontinuity in demand carries over to profit
a+bc
If firm 1 sets p1 above the monopoly price pM =
2b
- Profits are negative if p2 < c (negative margins)
- Profits are increasing until pM
π ( p1 )
- After pM, profits are decreasing until p2 = p1, where it jumps to π 1=π 2=
2
- Zero profits for p1 > p2 (no sales)
If firm 1 sets p1 = pM
M
π
- Setting p2 = pM yields half monopoly profits:
2
- Setting p2 = pM – ε grants almost at most the full monopoly profits
M M M
π ( p −ε / p )=π
What if firm 1 sets c < p1 < pM?
- Setting p1 = p2 yields positive profits
- But again, undercutting can do better
- For p1 <= c, undercutting is bad. Set p1 = c instead
Lecture 1: game theory and competition
Game theory: fundamentals
Game theory = a mathematical tool used to represent strategic interactions in a
treatable way
Games are a stylized representation of agents’ goals, information and
capabilities
Our goal:
- Predict the likely outcomes when multiple players have conflicting paths
- Understand what factors can give an edge to different players or affect the
outcomes (comparative static)
- Identify pitfalls that undermine desirable outcomes and how to mitigate them
In real life: decision-makers have goals/preferences about the outcome of their
interactions (example: winning or losing, (not) getting a promotion)
In a game: we represent these preferences assigning to outcomes a numerical value
(payoff)
- We assign higher payoff to outcomes that are more preferred to them
- Players might disagree on what outcome is the most desirable (gives the
highest payoff)
- Strategies are actions or sequence of actions that agents can choose to achieve
their goals
- The outcome of a game is the result of all player’s strategies and chance if the
game involves stochastic/random elements
In game theory we assume that agents (players) are individually rational, that is:
1. They have rational preferences (over the outcomes)
2. They are payoff-maximizing (not necessarily selfish)
Strategic games
1. Static
o One-shot game: players interact only once
o Simultaneous choice: players choose their strategy without knowledge of
the other players’ choices
2. Complete information
o All players know the capabilities of every other player (strategies)
o All players know the consequences of every combination of strategies
(payoff)
Uncertainty (ea. stocks) ≠ asymmetric information (ea. Boss knows more)
Strategic games allow for uncertainty, but not asymmetric information
Complete information Asymmetric information
Static Strategic games Bayesian games
,Dynamic Extensive game, repeated Sequential (Bayesian)
game games
A strategic game is fully described by:
1. Set of players
2. Set of actions available for each player
3. Players’ payoff functions
An equilibrium in dominant strategy is a very strong notion, only relies on player’s
rationality
- Pros: if players are rational; it is the only possible outcome very reliable
- Cons: too restrictive, most games don’t have one
Nash equilibrium (NE)
= an action profile such that none of the players can increase their
payoff by choosing a different strategy, taking the other player’s
strategies for given
In other words: it is an action profile such that no player can
benefit from independently deviating from it
Notice: it is a resting point, there is nothing optimal about it
We say that each player is best responding to the other
player’s strategies
Demand, monopoly and competition
Market demand
Demand function tells us how many units of a product are purchased by consumers
depending on its price
- At price p1, customers buy quantity q1
- The slope of the demand dq/dp tells us the decrease/increase of units sold for a
small increment in the price. For normal goods, it is always negative
(opportunity cost)
∆ q/ p dq p
- Economists rather use price elasticity of demand ε ( p ) = = .
∆ p /q dp q
o Always negative for normal goods. Describes the ability/willingness of
consumers to substitute consumption with another good (or not spend)
o |ε| = 1: if price reduces by 5% then quantity increases by the same
percentage amount
o |ε| < 1: demand is inelastic => increasing price increases revenues
o |ε| > 1: demand is elastic => increasing price reduces revenues
Linear demand: q(p) = a – bp
- a is the quantity demanded when price is infinitesimal
- the slope of the demand is constant dq/dp = b
- elasticity is not constant
Inverse market demand: p(q) = A – Bq with a/b = A, 1/b = B
- A = maximum willingness to pay (WTP) among the consumers
,Perfect competition
does not exist! It’s an ideal archetype, but it is useful as a benchmark
- Inverse demand is a horizontal line at price p
- Consumers willing to buy any quantity but only at the price p
or lower
- Infinitely many firms willing to supply at that price
- Price elasticity = -∞
Single-agent problem. Each individual is too small to affect demand
(no strategic interactions)
- Can produce a quantity q at cost given by their cost function
C(q)
- Observe the exogenous price p
- Chooses to produce the quantity that maximizes its profits
Maxq { π ( q )= p∗q−C (q) }
Take derivative w.r.t. q for f.o.c.
−∆ C (q)
0=p+
∆q
∆ C (q)
p= = C’(q)
∆q
Perfect competition: p = C’
- Price-taking (alone) does not mean zero-profits
- If p is sufficiently high, profits are possible: triangle pAB minus fixed costs
- Wit free entry, in the longrun the price p drops to the minimum of the average
total cost curve (ATC), so profits are zero by construction
- Price taking + free entry = no profits. Not a good market to enter, but great for
consumers.
Monopolist’s problem
Still a single-agent (non-strategic) problem. The monopolist:
- Observes a downward sloping demand q(p) and can set the price
- Can produce its product at cost given by the cost function C(q)
- Choose the price (or quantity) that maximize its profits
Maxq { π ( p )= p∗q( p)−C (q ( p))}
Take derivative w.r.t. q for f.o.c.
∆ C ( q ( p ))
∗∆ q ( p )
0 = q(p) + p* ∆ q (p) ∆q
−
∆p ∆p
−∆ q ( p )
q(p) = ( p−C' ( q ) )
∆p
( )
−1 '
∆ q ( p ) q ( p ) ( p−C ( q )) ( p−C' ( q ) )
− = with the lerner index (%
∆p p p p
markup)
( ) 1
1+ p=C ' (q ( p ) )
ε
Imperfect competition: Bertrand duopoly
Now it’s a two-players game, assumptions:
- Two firms, firm 1 and firm 2, produce a homogeneous product
, - Both firms have the same costs function C(q)
- Each firm chooses its own price p1 and p2 simultaneously
- Consumers have a demand q(p) but only buy from the
cheapest firm
- If prices are equal, they split equally their purchases from
the two firms
- Firms only produce when/if units are sold
For simplicity, assume:
- Each firm has constant marginal cost C’(q) = c
- Consumer demand is linear: q = a - bp
Assume that firm 1 sets p1
- For prices above p1, the quantity sold is 0
a−bp 1
- At p1, the quantity sold jumps to
2
- For prices below p1 the quantity follows the consumer demand
- The discontinuity in demand carries over to profit
a+bc
If firm 1 sets p1 above the monopoly price pM =
2b
- Profits are negative if p2 < c (negative margins)
- Profits are increasing until pM
π ( p1 )
- After pM, profits are decreasing until p2 = p1, where it jumps to π 1=π 2=
2
- Zero profits for p1 > p2 (no sales)
If firm 1 sets p1 = pM
M
π
- Setting p2 = pM yields half monopoly profits:
2
- Setting p2 = pM – ε grants almost at most the full monopoly profits
M M M
π ( p −ε / p )=π
What if firm 1 sets c < p1 < pM?
- Setting p1 = p2 yields positive profits
- But again, undercutting can do better
- For p1 <= c, undercutting is bad. Set p1 = c instead
