Lecture 1: Game Theory and Competition
Lecture 1: Game Theory and Competition
Game theory: fundamentals
Game theory = mathematical tool to represent strategic interactions between firms
- Games: a stylized representation of agents’ goals, information and capabilities
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)
- Identify pitfalls that undermine desirable outcomes, and how to mitigate them (e.g.
opportunistic behaviour, asymmetric information)
Strategies and payoff
In real life: Decision-makers have goals/preferences about the of their interactions.
Example: “winning/losing elections”, “(not) getting a promotion", “quarterly revenues”
In a game: We represent these preferences assigning to outcomes a numerical value (payoff)
outcome
- We assign higher (lower) payoff, to outcomes that are more (less) preferred to them.
- Players might disagree on what outcome is the most desirable (gives them 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
Individual rationality
In game theory we assume that agents (players) are individually rational that is:
1. They have rational preferences (over the outcomes):
o Each player can rank outcomes from their most favourite to the least favourite.
(More precisely: we say that the “preference” relation defines a semi-order over
the set of possible outcomes.)
o This assumption is necessary in order to have a payoff representation.
2. They are payoff-maximizing:
o Players choose their strategies in order to achieve the highest payoff for
themselves.
o Note: Being "rational" doesn't mean you are a greedy person. It just means you follow your own
preferences.
example: Even if you give money away, you are still being "rational" because helping others is what
gives you the highest payoff (satisfaction).
→ To be individually rational, you just need to know what you like and act in a way that gets you as much of it as possible.
,Three main categories of games
Strategic
1. Static
One-shot game: Players interact only once.
o
Simultaneous choice: Players choose their strategy without knowledge of the other
o
players’ choices
o Note: Even if a player has to make several small decisions, the result of the game is only seen
at the very end.
2. Complete information
o All players know the capabilities (strategies available) of every other player.
= “ I know all the strategies that an other player can do, but I don’t know which he will specifically
choose”
o All players know the consequences (payoffs) of every combination of strategies
= “I know what will happen in each combination of strategies, a probability distribution”
!! Does NOT mean certainty
Games (and business) often involve quantifiable risk. If all the possible outcomes are known, and occur
with a known probability distribution we call this risk “uncertainty”.
Example: “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.”
However, sometimes agents a player might know something that other don’t. Then we say that players
have asymmetric information
Example: “Mark Zuckerberg secretly sets on fire the Meta headquarters and now he is short
selling Meta stocks. I know that he knows more than me, but I don’t know what he knows.”
Strategic games allow for uncertainty, but not asymmetric information.
Broad taxonomy of games
,Abstract form
A strategic game is fully described by:
1. Set of players
For example: P = {Player1, Player2}
2. Set of actions available for each player
For example: AP1 = {Defect, Cooperate}, AP2 = {Defect, Cooperate}
Remark: in strategic games “actions” and “strategies” coincide, since a strategy consists in a single
action.
3. Players’ payoff functions: a description of each player’s payoff for each combination of
actions
For example:
Very inconvenient! However, 2-players games, are usually represented in a much simpler and
parsimonious way through a payoff-matrix also known as their “normal form”.
Normal (payoff-matrix)
- Rows correspond to the row-player’s actions
In this case: AP1 = {D, C, H}
- Columns correspond to the column-player’s actions
In this case: AP2 = {X, Y , T}
- Each cell is an action profile i.e. a tuple of 1 action
per player
For example < C, X > is the action profile where P1
plays C and P2 plays X
- Within each cell, we find the payoffs associated with
each action profile
For example in < C, X > P1 receives -5 and P2 receives
+5
!! Payoffs represent player’s preferences for a outcome,
while action profiles are its cause. Do not confuse them!
For example, < C, Y > and < H, T > have the same payoffs (3, 3),
so players are indifferent between the two outcomes.
(Oefening dia 20)
General equilibrium
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 prediction!
- Cons: Too restrictive. Most games don’t have
, Nash equilibrium
Definition:
- “A Nash Equilibrium (NE) is 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.”
- “A Nash Equilibrium (NE) is an action profile such that each player’s strategy maximizes that
player’s payoff conditional on the other player’s strategies.”
- In other words: it is a less stronger equilibrium then a general equilibrium
- Resting point: once you are at a Nash Equilibrium, nobody has an incentive to move.
- Best Response: This means you are giving your best possible answer to the moves the
other players made.
(Oefening dia 26)
Statements:
- An equilibrium in dominant strategies is also always a NE.
- Players can never choose a strictly dominated strategy in a NE.
- There doesn’t have to be a NE in each strategic game with 2 players (ex. Rock paper scissors)
- A nash equilibrium is not unique
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 consumers buy quantity q1
- The slope of the demand dq/dp tells us the decrease
(increase) of units sold for a small increment
(reduction) in the price. For normal goods, it is always
negative
!! Not very useful for decisions!
= If dq/dp = -1, it’s a very different story if q are candies or
$100 mil battleships
- Economists use price elasticity of demand ε(p)
∆q/q is the percentage change in q, and ∆p/p is the percentage change in p
- Always negative (for normal goods). Describes the ability/willingness
of consumers to substitute consumption with another good (or not
spend).
- |ε|= 1: if price reduces (increases) by 5%, then quantity increases
(reduces) by the same percentage amount
- |ε|< 1: Demand is inelastic =>Increasing price increases revenues
- |ε|> 1: Demand is elastic => Increasing price reduces revenues
Lecture 1: Game Theory and Competition
Game theory: fundamentals
Game theory = mathematical tool to represent strategic interactions between firms
- Games: a stylized representation of agents’ goals, information and capabilities
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)
- Identify pitfalls that undermine desirable outcomes, and how to mitigate them (e.g.
opportunistic behaviour, asymmetric information)
Strategies and payoff
In real life: Decision-makers have goals/preferences about the of their interactions.
Example: “winning/losing elections”, “(not) getting a promotion", “quarterly revenues”
In a game: We represent these preferences assigning to outcomes a numerical value (payoff)
outcome
- We assign higher (lower) payoff, to outcomes that are more (less) preferred to them.
- Players might disagree on what outcome is the most desirable (gives them 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
Individual rationality
In game theory we assume that agents (players) are individually rational that is:
1. They have rational preferences (over the outcomes):
o Each player can rank outcomes from their most favourite to the least favourite.
(More precisely: we say that the “preference” relation defines a semi-order over
the set of possible outcomes.)
o This assumption is necessary in order to have a payoff representation.
2. They are payoff-maximizing:
o Players choose their strategies in order to achieve the highest payoff for
themselves.
o Note: Being "rational" doesn't mean you are a greedy person. It just means you follow your own
preferences.
example: Even if you give money away, you are still being "rational" because helping others is what
gives you the highest payoff (satisfaction).
→ To be individually rational, you just need to know what you like and act in a way that gets you as much of it as possible.
,Three main categories of games
Strategic
1. Static
One-shot game: Players interact only once.
o
Simultaneous choice: Players choose their strategy without knowledge of the other
o
players’ choices
o Note: Even if a player has to make several small decisions, the result of the game is only seen
at the very end.
2. Complete information
o All players know the capabilities (strategies available) of every other player.
= “ I know all the strategies that an other player can do, but I don’t know which he will specifically
choose”
o All players know the consequences (payoffs) of every combination of strategies
= “I know what will happen in each combination of strategies, a probability distribution”
!! Does NOT mean certainty
Games (and business) often involve quantifiable risk. If all the possible outcomes are known, and occur
with a known probability distribution we call this risk “uncertainty”.
Example: “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.”
However, sometimes agents a player might know something that other don’t. Then we say that players
have asymmetric information
Example: “Mark Zuckerberg secretly sets on fire the Meta headquarters and now he is short
selling Meta stocks. I know that he knows more than me, but I don’t know what he knows.”
Strategic games allow for uncertainty, but not asymmetric information.
Broad taxonomy of games
,Abstract form
A strategic game is fully described by:
1. Set of players
For example: P = {Player1, Player2}
2. Set of actions available for each player
For example: AP1 = {Defect, Cooperate}, AP2 = {Defect, Cooperate}
Remark: in strategic games “actions” and “strategies” coincide, since a strategy consists in a single
action.
3. Players’ payoff functions: a description of each player’s payoff for each combination of
actions
For example:
Very inconvenient! However, 2-players games, are usually represented in a much simpler and
parsimonious way through a payoff-matrix also known as their “normal form”.
Normal (payoff-matrix)
- Rows correspond to the row-player’s actions
In this case: AP1 = {D, C, H}
- Columns correspond to the column-player’s actions
In this case: AP2 = {X, Y , T}
- Each cell is an action profile i.e. a tuple of 1 action
per player
For example < C, X > is the action profile where P1
plays C and P2 plays X
- Within each cell, we find the payoffs associated with
each action profile
For example in < C, X > P1 receives -5 and P2 receives
+5
!! Payoffs represent player’s preferences for a outcome,
while action profiles are its cause. Do not confuse them!
For example, < C, Y > and < H, T > have the same payoffs (3, 3),
so players are indifferent between the two outcomes.
(Oefening dia 20)
General equilibrium
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 prediction!
- Cons: Too restrictive. Most games don’t have
, Nash equilibrium
Definition:
- “A Nash Equilibrium (NE) is 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.”
- “A Nash Equilibrium (NE) is an action profile such that each player’s strategy maximizes that
player’s payoff conditional on the other player’s strategies.”
- In other words: it is a less stronger equilibrium then a general equilibrium
- Resting point: once you are at a Nash Equilibrium, nobody has an incentive to move.
- Best Response: This means you are giving your best possible answer to the moves the
other players made.
(Oefening dia 26)
Statements:
- An equilibrium in dominant strategies is also always a NE.
- Players can never choose a strictly dominated strategy in a NE.
- There doesn’t have to be a NE in each strategic game with 2 players (ex. Rock paper scissors)
- A nash equilibrium is not unique
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 consumers buy quantity q1
- The slope of the demand dq/dp tells us the decrease
(increase) of units sold for a small increment
(reduction) in the price. For normal goods, it is always
negative
!! Not very useful for decisions!
= If dq/dp = -1, it’s a very different story if q are candies or
$100 mil battleships
- Economists use price elasticity of demand ε(p)
∆q/q is the percentage change in q, and ∆p/p is the percentage change in p
- Always negative (for normal goods). Describes the ability/willingness
of consumers to substitute consumption with another good (or not
spend).
- |ε|= 1: if price reduces (increases) by 5%, then quantity increases
(reduces) by the same percentage amount
- |ε|< 1: Demand is inelastic =>Increasing price increases revenues
- |ε|> 1: Demand is elastic => Increasing price reduces revenues
