LECTURE 1:
GAME THEORY, COMPETITION AND MONOPOLY
INTRODUCTION TO GAME THEORY
Game theory is the study of mathematical models of strategic interaction between rational players.
→ Games are a stylized representation of agents’ goals, information and capabilities.
Its goal:
- Predict the likely outcomes when multiple players have conflicting goals
- Understand what factors can give an edge to different players or affect the outcomes
(comparative static: what happens if one factor changes)
- Identify pitfalls that undermine desirable outcomes, like a price war, and how to mitigate them
e.g. opportunistic behaviour, asymmetric information
In real life: decision-makers gave preferences about the outcome of their interactions
- winning / losing elections
- (not) getting a promotion
- …
In a game: we represent these preferences by assigning outcomes a numerical value (the payoff)
Note: agents can have conflicting objectives
- strategies are (sequence of) actions that agents choose to achieve their goals
- the outcome of a game is the result of all player’s strategies
We make 2 key basis assumptions:
1. the agents have rational preferences: they can rank outcomes
2. the agents are payoff-maximizing: they choose their strategies in order to achieve the
highest payoff for themselves (note: this doesn’t equal that all players are selfish, preferences
can be altruistic)
Elements of a game:
a. each decision or plan of action is called a strategy
b. a list of every strategy per player is called a strategy combination
c. each strategy combination determines the outcome of the game (profit, payoff,...)
d. number of players
In this course, we define three main categories of games: strategic, extensive & sequential and
bayesian
,1. STRATEGIC GAMES
example: Rock-Paper-Scissor
We define a strategic game as a game that has the following elements:
→ Static
- one shot: players interact only once
- simultaneous action: players choose their strategy without knowledge of the other players’
choices: also when multiple choices are being made, the outcome is only observed once at
the end → there is no time
→ Complete information
- all players know the capabilities (strategies available) of every other player
- all players know the consequences (payoffs) given every combination of strategies
OR can formulate a probability distribution over them: uncertainty is allowed
⇒ games involve quantifiable risk.
“If I buy a stock of Meta today, I don’t know the price it will have tomorrow, but I can form expectations
and quantify the likelihood of different price swings.”
Strategic games are described by:
- set of players: P = {Player1, Player2}
- set of actions for each player: AP1 = {Defect, Cooperate}, AP2 = {Defect, Cooperate}
Note: in strategic games “actions” and “strategies” coincide: a strategy is just one direct choice
- player’s payoff function:
1. EXTENSIVE GAMES
We define an extensive game as a game that has the following elements:
→ Dynamic
- players make decisions in a sequence.
- firms can react to each other
- outcome depends not only on what firms choose, but also on when they choose
→ Complete information
- all players know the capabilities (strategies available) of every other player
- all players know the consequences (payoffs) given every combination of strategies
OR can formulate a probability distribution over them: uncertainty is allowed
,PAYOFF MATRIX
○ Rows correspond to the row-player’s actions
here: P1 choosing between D,C or H
○ Columns correspond to the column-player’s actions
here: P2 choosing between X,Y or H
○ Each cell is an action profile
→ within each cell, we find the payoffs associated with each action profile
note: action profile ≠ outcome!
Payoffs represent player’s preferences for a outcome, while action
profiles are its cause.
eg) < C , Y > and < H, T > have the same payoffs (3, 3),
so players are indifferent between the two outcomes
DOMINANT AND DOMINATED STRATEGIES
A dominated strategy is a strategy that will never be rationally chosen. → as they do not occur, they
can never be part of the Nash equilibrium
A dominant strategy is a strategy that beats all of the agent’s other strategies.
→ a dominant strategy outperforms an agent's other strategies, no matter what the other players
choose
but this does not necessarily mean that the agent will attain more payoff
→ it is clear that a dominant strategy combination, if there is one, is always a Nash Equilibrium, but
you can also have a Nash Equilibrium without necessarily having a dominant strategy.
The process is called iterative elimination of dominated strategies, which brings us:
An equilibrium in dominant strategy (here < D, X >)
→ very strong notion, only relies on the player's rationality.
- pro: it’s the only possible outcome so very reliable (if players are rational)
- con: most games don’t have one
⇒ We need more of a general solution: the Nash Equilibrium.
NASH EQUILIBRIUM AS A SOLUTION CONCEPT
Nash Equilibrium: a strategy profile such that no agent has an incentive to change the strategy it is
currently using, given that no other agent changes theirs as well.
= none of the players can increase their payoff choosing a different strategy, taken the other player’s
strategies for given.
- It is a resting point (equilibrium), there is nothing optimal about it.
- There isn’t necessarily one (in a strategic game), but there can also be more than one
- We say that each player is “best-responding” to the other player's strategies
Is there an action profile such that neither player
would change their strategy?
→ Yes: < X, A >
In < X, A > both players are best-responding to
each other strategies.
Hence, is the only Nash Equilibrium.
, DEMAND, MONOPOLY AND COMPETITION
MARKET DEMAND
In Managerial Economics, we will mainly focus on firm profit-maximizing behavior and a resultant
market outcome that such behavior implies.
Market Demand Curve describes the relationship between how much money (aggregated) consumers
are willing to pay per unit of the good and the quantity (aggregated) of the goods consumed.
→ depends on:
- price
- income
- expectations of price movement…
- ⇒ Quantity = f(price, income, marketing, ...)
𝑑𝑞
- 𝑑𝑝
shows decrease/increase of units
sold for small increment/reduction in the
price.
⇒ for normal goods: always negative
- not very useful for decisions
𝑑𝑞
( 𝑑𝑝 = -1 is a very different story if q are
candies or $100 mil battleships)
→ economists use price elasticity of demand
∆𝑞/𝑞 𝑑𝑞 𝑝
ε(𝑝) = ∆𝑝/𝑝
= 𝑑𝑝 𝑞(𝑝)
PRICE ELASTICITY ε(𝑝)
- describes the ability/willingness of consumers to substitute consumption with another good (or
not spend).
- always negative for normal goods
- |ε| = 1 : if price reduces (increases) by 5%, then quantity increases (reduces) by the same
percentage amount
- |ε| < 1 : demand is inelastic ⇒ increasing prices increases revenues (∆𝑞/𝑞 < ∆𝑝/𝑝 )
- |ε| > 1 : demand is elastic ⇒ increasing prices reduces revenues (∆𝑞/𝑞 > ∆𝑝/𝑝 )
LINEAIR DEMAND
𝑞(𝑝) = 𝑎 − 𝑏𝑝
- a signifies the quantity demanded when
the price is infinitesimally small
- the slope of the demand curve is
constant and equals to b
- However, ε is not constant
GAME THEORY, COMPETITION AND MONOPOLY
INTRODUCTION TO GAME THEORY
Game theory is the study of mathematical models of strategic interaction between rational players.
→ Games are a stylized representation of agents’ goals, information and capabilities.
Its goal:
- Predict the likely outcomes when multiple players have conflicting goals
- Understand what factors can give an edge to different players or affect the outcomes
(comparative static: what happens if one factor changes)
- Identify pitfalls that undermine desirable outcomes, like a price war, and how to mitigate them
e.g. opportunistic behaviour, asymmetric information
In real life: decision-makers gave preferences about the outcome of their interactions
- winning / losing elections
- (not) getting a promotion
- …
In a game: we represent these preferences by assigning outcomes a numerical value (the payoff)
Note: agents can have conflicting objectives
- strategies are (sequence of) actions that agents choose to achieve their goals
- the outcome of a game is the result of all player’s strategies
We make 2 key basis assumptions:
1. the agents have rational preferences: they can rank outcomes
2. the agents are payoff-maximizing: they choose their strategies in order to achieve the
highest payoff for themselves (note: this doesn’t equal that all players are selfish, preferences
can be altruistic)
Elements of a game:
a. each decision or plan of action is called a strategy
b. a list of every strategy per player is called a strategy combination
c. each strategy combination determines the outcome of the game (profit, payoff,...)
d. number of players
In this course, we define three main categories of games: strategic, extensive & sequential and
bayesian
,1. STRATEGIC GAMES
example: Rock-Paper-Scissor
We define a strategic game as a game that has the following elements:
→ Static
- one shot: players interact only once
- simultaneous action: players choose their strategy without knowledge of the other players’
choices: also when multiple choices are being made, the outcome is only observed once at
the end → there is no time
→ Complete information
- all players know the capabilities (strategies available) of every other player
- all players know the consequences (payoffs) given every combination of strategies
OR can formulate a probability distribution over them: uncertainty is allowed
⇒ games involve quantifiable risk.
“If I buy a stock of Meta today, I don’t know the price it will have tomorrow, but I can form expectations
and quantify the likelihood of different price swings.”
Strategic games are described by:
- set of players: P = {Player1, Player2}
- set of actions for each player: AP1 = {Defect, Cooperate}, AP2 = {Defect, Cooperate}
Note: in strategic games “actions” and “strategies” coincide: a strategy is just one direct choice
- player’s payoff function:
1. EXTENSIVE GAMES
We define an extensive game as a game that has the following elements:
→ Dynamic
- players make decisions in a sequence.
- firms can react to each other
- outcome depends not only on what firms choose, but also on when they choose
→ Complete information
- all players know the capabilities (strategies available) of every other player
- all players know the consequences (payoffs) given every combination of strategies
OR can formulate a probability distribution over them: uncertainty is allowed
,PAYOFF MATRIX
○ Rows correspond to the row-player’s actions
here: P1 choosing between D,C or H
○ Columns correspond to the column-player’s actions
here: P2 choosing between X,Y or H
○ Each cell is an action profile
→ within each cell, we find the payoffs associated with each action profile
note: action profile ≠ outcome!
Payoffs represent player’s preferences for a outcome, while action
profiles are its cause.
eg) < C , Y > and < H, T > have the same payoffs (3, 3),
so players are indifferent between the two outcomes
DOMINANT AND DOMINATED STRATEGIES
A dominated strategy is a strategy that will never be rationally chosen. → as they do not occur, they
can never be part of the Nash equilibrium
A dominant strategy is a strategy that beats all of the agent’s other strategies.
→ a dominant strategy outperforms an agent's other strategies, no matter what the other players
choose
but this does not necessarily mean that the agent will attain more payoff
→ it is clear that a dominant strategy combination, if there is one, is always a Nash Equilibrium, but
you can also have a Nash Equilibrium without necessarily having a dominant strategy.
The process is called iterative elimination of dominated strategies, which brings us:
An equilibrium in dominant strategy (here < D, X >)
→ very strong notion, only relies on the player's rationality.
- pro: it’s the only possible outcome so very reliable (if players are rational)
- con: most games don’t have one
⇒ We need more of a general solution: the Nash Equilibrium.
NASH EQUILIBRIUM AS A SOLUTION CONCEPT
Nash Equilibrium: a strategy profile such that no agent has an incentive to change the strategy it is
currently using, given that no other agent changes theirs as well.
= none of the players can increase their payoff choosing a different strategy, taken the other player’s
strategies for given.
- It is a resting point (equilibrium), there is nothing optimal about it.
- There isn’t necessarily one (in a strategic game), but there can also be more than one
- We say that each player is “best-responding” to the other player's strategies
Is there an action profile such that neither player
would change their strategy?
→ Yes: < X, A >
In < X, A > both players are best-responding to
each other strategies.
Hence, is the only Nash Equilibrium.
, DEMAND, MONOPOLY AND COMPETITION
MARKET DEMAND
In Managerial Economics, we will mainly focus on firm profit-maximizing behavior and a resultant
market outcome that such behavior implies.
Market Demand Curve describes the relationship between how much money (aggregated) consumers
are willing to pay per unit of the good and the quantity (aggregated) of the goods consumed.
→ depends on:
- price
- income
- expectations of price movement…
- ⇒ Quantity = f(price, income, marketing, ...)
𝑑𝑞
- 𝑑𝑝
shows decrease/increase of units
sold for small increment/reduction in the
price.
⇒ for normal goods: always negative
- not very useful for decisions
𝑑𝑞
( 𝑑𝑝 = -1 is a very different story if q are
candies or $100 mil battleships)
→ economists use price elasticity of demand
∆𝑞/𝑞 𝑑𝑞 𝑝
ε(𝑝) = ∆𝑝/𝑝
= 𝑑𝑝 𝑞(𝑝)
PRICE ELASTICITY ε(𝑝)
- describes the ability/willingness of consumers to substitute consumption with another good (or
not spend).
- always negative for normal goods
- |ε| = 1 : if price reduces (increases) by 5%, then quantity increases (reduces) by the same
percentage amount
- |ε| < 1 : demand is inelastic ⇒ increasing prices increases revenues (∆𝑞/𝑞 < ∆𝑝/𝑝 )
- |ε| > 1 : demand is elastic ⇒ increasing prices reduces revenues (∆𝑞/𝑞 > ∆𝑝/𝑝 )
LINEAIR DEMAND
𝑞(𝑝) = 𝑎 − 𝑏𝑝
- a signifies the quantity demanded when
the price is infinitesimally small
- the slope of the demand curve is
constant and equals to b
- However, ε is not constant
