Lecture 1: Game theory and competition
GAME THEORY: FUNDAMENTALS
Game theory = mathematical tool used to represent strategic interactions (= decisions while
considering the other players)
à GOAL = predict outcomes when players have conflicting goals
In a game:
- Preferences represented by numerical value = payo,
- Actions / sequence of actions = strategies
- Result of all player’s strategies = outcome
! Assumption: agents / players are individually rational:
1. Rational preferences
= players can rank outcomes: most preferred - least preferred -indiOerent
2. Payo,-maximizing
= players choose strategies to achieve highest payoO
≠ necessarily selfish!
In this course: assume that firms want more profit (not always true in reality)
3 main categories of games:
Complete / perfect information Asymmetric information
Static (1) Strategic games (3) Bayesian games
Dynamic (2) Extensive games & repeated games (3) Sequential games
Static
- One-shot
- Simultaneous choice = players have no knowledge of other players’ choice
ó Dynamic
- Not simultaneous = players know other players’ choice
Complete information
- All players know possible strategies of all other players
- All players know payoOs of every combination
! Complete information ≠ certainty
ó Asymmetric information
,1 Strategic games
Abstract form
( PayoO P1 , PayoO P2 ) = ( x , y ) if (action P1 , action P2 )
à Inconvenient!
Normal form / payoO-matrix
à Used for 2-player games
Player 2
Action A Action B Action C
Player 1 Action X (…;…) (…;…) (…;…)
Action Y (…;…) (…;…) (…;…)
Action Z (…;…) (…;…) (…;…)
Finding an equilibrium:
Option 1: elimination of dominated strategies
è Doesn’t always exist
Option 2: Nash-equilibrium
Nash-equilibrium NE = action profile such that none of the players can increase
their payoO by choosing a diOerent strategy, given the other player’s strategies
è No one can benefit by individually deviating
è Once there, no one has incentive to change
è Not necessarily optimal
! Equilibrium in dominant strategies = always a NE
! Players never choose strictly dominated strategy in a NE
2 Extensive games
Agents can observe and react to other’s decisions, or can anticipate their moves
Abstract form
- Players
{ P1 , P2 }
- Histories = state of the game
{ ø = start of game , (enter) , (notEnter) , (enter, enter) , (enter, notEnter) }
à Terminal if game ends when reached
à Non-terminal otherwise
, - Player function = what player acts in each non-terminal history
R(ø) = Burger King
R(enter) = McDonald’s
R(notEnter) = McDonald’s
- Player payoO = for each terminal history
Extensive form
Finding optimal strategy
Backward induction
! Always equilibrium using backward induction
à Result = subgame-perfect Nash equilibrium SPNE
à Finite games: always unique SPNE
à First-mover advantage / disadvantage possible (compare payo7 static NE to
payo7 SPNE)
DEMAND, MONOPOLY AND COMPETITION
Demand = q(p)
Inverse demand = p(q)
Price elasticity
𝒅𝒒(𝒑) 𝒑
𝜺= .
𝒅𝒑 𝒒
- | 𝜀 | < 1 à inelastic demand
- | 𝜀 | > 1 à elastic demand
- 𝜀 = 0 à perfectly inelastic
- 𝜀 = ∞ à perfectly elastic
! Linear demand ≠ constant elasticity
! Perfect competition = horizontal p(q)
, Perfect competition
- Profit π = p . q – c . q
= (p – c) . q ( c = marginal cost)
- Optimal condition: p = c
Monopoly
- Monopoly price = always best option if you are the only one in the market
!"#$
- pM = %#
when demand and costs are linear: q = a-bp & marginal cost = c
&'$
- Lerner’s index = %markup = &
Oligopoly
à In between extremes of perfect competition & monopoly
à Here, there is strategic interaction
- Best option = monopoly price IF YOU ARE THE ONLY ONE IN THE MARKET
- Otherwise: Bertrand game
o Firms undercut each other
o Until p = c
à Bertrand paradox: two firms are enough to achieve competitive outcome
! BUT, in many markets, firms DO make profit through:
- Increased marginal costs
- Capacity constraints
- Search / information frictions
- Product diOerentiation
o DiOerence in product = horizontal
o DiOerence in quality = vertical
ARTICLE
Markup µ = measure of market power
price
µ=
MC
à Average markup has increased over the years
! BUT only looking at markup says nothing about it being good or bad
o Maybe because of higher value?
o Maybe technology changed?
>> MC ↓
>> fixed & overhead costs ↑
>> higher markup then necessary to break even è no problem
>> BUT is profits also ↑ è more than necessary è problem
o Maybe policy allowed mergers?
GAME THEORY: FUNDAMENTALS
Game theory = mathematical tool used to represent strategic interactions (= decisions while
considering the other players)
à GOAL = predict outcomes when players have conflicting goals
In a game:
- Preferences represented by numerical value = payo,
- Actions / sequence of actions = strategies
- Result of all player’s strategies = outcome
! Assumption: agents / players are individually rational:
1. Rational preferences
= players can rank outcomes: most preferred - least preferred -indiOerent
2. Payo,-maximizing
= players choose strategies to achieve highest payoO
≠ necessarily selfish!
In this course: assume that firms want more profit (not always true in reality)
3 main categories of games:
Complete / perfect information Asymmetric information
Static (1) Strategic games (3) Bayesian games
Dynamic (2) Extensive games & repeated games (3) Sequential games
Static
- One-shot
- Simultaneous choice = players have no knowledge of other players’ choice
ó Dynamic
- Not simultaneous = players know other players’ choice
Complete information
- All players know possible strategies of all other players
- All players know payoOs of every combination
! Complete information ≠ certainty
ó Asymmetric information
,1 Strategic games
Abstract form
( PayoO P1 , PayoO P2 ) = ( x , y ) if (action P1 , action P2 )
à Inconvenient!
Normal form / payoO-matrix
à Used for 2-player games
Player 2
Action A Action B Action C
Player 1 Action X (…;…) (…;…) (…;…)
Action Y (…;…) (…;…) (…;…)
Action Z (…;…) (…;…) (…;…)
Finding an equilibrium:
Option 1: elimination of dominated strategies
è Doesn’t always exist
Option 2: Nash-equilibrium
Nash-equilibrium NE = action profile such that none of the players can increase
their payoO by choosing a diOerent strategy, given the other player’s strategies
è No one can benefit by individually deviating
è Once there, no one has incentive to change
è Not necessarily optimal
! Equilibrium in dominant strategies = always a NE
! Players never choose strictly dominated strategy in a NE
2 Extensive games
Agents can observe and react to other’s decisions, or can anticipate their moves
Abstract form
- Players
{ P1 , P2 }
- Histories = state of the game
{ ø = start of game , (enter) , (notEnter) , (enter, enter) , (enter, notEnter) }
à Terminal if game ends when reached
à Non-terminal otherwise
, - Player function = what player acts in each non-terminal history
R(ø) = Burger King
R(enter) = McDonald’s
R(notEnter) = McDonald’s
- Player payoO = for each terminal history
Extensive form
Finding optimal strategy
Backward induction
! Always equilibrium using backward induction
à Result = subgame-perfect Nash equilibrium SPNE
à Finite games: always unique SPNE
à First-mover advantage / disadvantage possible (compare payo7 static NE to
payo7 SPNE)
DEMAND, MONOPOLY AND COMPETITION
Demand = q(p)
Inverse demand = p(q)
Price elasticity
𝒅𝒒(𝒑) 𝒑
𝜺= .
𝒅𝒑 𝒒
- | 𝜀 | < 1 à inelastic demand
- | 𝜀 | > 1 à elastic demand
- 𝜀 = 0 à perfectly inelastic
- 𝜀 = ∞ à perfectly elastic
! Linear demand ≠ constant elasticity
! Perfect competition = horizontal p(q)
, Perfect competition
- Profit π = p . q – c . q
= (p – c) . q ( c = marginal cost)
- Optimal condition: p = c
Monopoly
- Monopoly price = always best option if you are the only one in the market
!"#$
- pM = %#
when demand and costs are linear: q = a-bp & marginal cost = c
&'$
- Lerner’s index = %markup = &
Oligopoly
à In between extremes of perfect competition & monopoly
à Here, there is strategic interaction
- Best option = monopoly price IF YOU ARE THE ONLY ONE IN THE MARKET
- Otherwise: Bertrand game
o Firms undercut each other
o Until p = c
à Bertrand paradox: two firms are enough to achieve competitive outcome
! BUT, in many markets, firms DO make profit through:
- Increased marginal costs
- Capacity constraints
- Search / information frictions
- Product diOerentiation
o DiOerence in product = horizontal
o DiOerence in quality = vertical
ARTICLE
Markup µ = measure of market power
price
µ=
MC
à Average markup has increased over the years
! BUT only looking at markup says nothing about it being good or bad
o Maybe because of higher value?
o Maybe technology changed?
>> MC ↓
>> fixed & overhead costs ↑
>> higher markup then necessary to break even è no problem
>> BUT is profits also ↑ è more than necessary è problem
o Maybe policy allowed mergers?
