Chapter 3: Game Theory

When is an activity worth doing?

If benefit > cost

External benefit

A benefit received by people other than those directly involved in the activity

External cost

A cost imposed on people other than those directly involved in the activity

How does the existence of external benefits + costs influence the logic of “do an activity benefits > costs”

• Means private benefit > private costs doesn’t always mean an activity is socially desirable

Game

• A set of players, a list of possible strategies and the payoffs that correspond to each combination of strategies

• Strategic situations where your outcome depends partly on what others do

What can the theory of games model?

The interdependency between individuals, firms + government

3 elements common to all games

1. Players (ie who makes the decisions)

2. Strategies (ie possible actions each player can choose)

3. Payoffs (ie the outcome / cost associated w each combo of strats)

Dominant strategy

The strategy in a game that produces better results regardless of the strategy chosen by one's opponent

Do optimal individual decisions always lead to the best collective outcome?

No

Social dilemma

Type of game where individual incentives conflict with efficiency

• Eg prisoners dilemma

• ie individual rational decisions lead to a collectively worse outcome

• Individual + collective interests don’t coincide!!

Issue with the dominant strategy

1. Dominant strat is individually best

2. But if all players follow it → leaves everyone worse off

3. Individuals incentives conflict w Pareto efficiency!!!

What do individuals incentives conflict with?

Pareto efficiency

Cooperate + defect strategy in a social dilemma

1. Cooperate strat- do what is in the collective interest

2. Defect strat- do what is in a person’s individual interest

What strategy does pareto efficiency require all players to choose + what is the issue with this?

PE- all players should cooperate

• But each individual has an incentive to defect

Does each player have a dominant strategy to defect or cooperate?

Defect

• BUT this leads to a Pareto inefficient outcome (so welfare isn’t maximised)

What would increase everyone's payoff?

If every player were to cooperate

When all players use their dominant strategy, the outcome is not…?

Pareto efficient

Commitment problems

When people would be better off if they collectivelly commit themselves to behaviour that is against their future self-interest

• (inconsistent with their own material interests)

Example of commitment problem?

Social dilemmas eg prisoner’s dilemma

Players in a social dilemma would do better if they…?

Cooperate

• Despite the incentive to defect

Commitment device

• A device that commits a person to behave in a certain way in the future, even though they may wish to behave otherwise when the time comes.

• Ie something that changes incentives to make someone keep a promise

• Eg rules, fines, blackmail

Examples of social dilemmas

• The prisoner's dilemma

• Public good game

What is the key conflict in a social dilemma?

Individual incentives vs collective wellbeing

How can commitment solve a social dilemma?

It changes incentives so people have a reason to cooperate rather than defect

• Makes cooperation credible

• Gives people reason to follow thru on promises

What is the invisible hand?

Acting w self-interest can unintentionally produce socially beneficial outcomes

• In competitive markets

Example of the invisible hand in competitive markets

• Firms compete to make better / cheaper products → consumers benefit

• Higher profits enjoyed by firms temporarily → other firms copy product / new tech → puts downward pressure on price

• Ultimately consumers benefit by firms acting in self interest (max profit)

What can free markets result in?

Pareto efficiency

Key assumption of the invisible hand

Assumes buyers + sellers are price takers

• Only works when this is satisfied

Price taker

No individual buyer / seller can influence price

• They take the market price

• Eg individual supermarket customer → can’t negotiate the price

Does the invisible hand always work?

No

• It can fail

When can the invisible hand go awry?

• When external effects exist

• + people can influence others’ outcomes

Influence → external effects → strategic interdependence

(don’t need to know this right now- will go into more detail in a later chapter)

Influence vs external effect (dw abt remembering this)

• Influence: The ability of one person's actions to affect the price/cost faced by others.

• EE: your action affects someone else

Strategic interdependence

Your best action depends on what other people are likely to do

What happens when external effects create strategic interdependence?

Self-interest may not produce Pareto efficiency

What do social dilemmas show about the invisible hand?

Individual self-interest (rational choices) can produce a collectively worse outcome

• Aka pareto inefficient outcome

Effect of non-selfish motives (don’t focus on this, this is explained in a later chapter)

• Can improve efficiency

• Encourage cooperation

• Help solve commitment problems (bc people more willing to cooperate / keep promises)

• Align individual interests w collective interests

Is there a restriction on no. of players in a social dilemma?

No

What condition makes it possible to sustain cooperation in a social dilemma?

If an interaction is repeated an indefinite number of times

Why do repeated games encourage cooperation?

Defection can lead to future punishment

• So cooperation = worthwhile

• But in 1-time social dilemmas, it is hard to punish defection

Tit for tat strategy

Strategy to:

• Cooperate at first

• And then do whatever the other player did in the previous interaction

• Eg if they defect → you defect next time

Effective strategies to sustain cooperation

Tit for tat

• Bc it punishes defection

3 characteristics of tit for tat

• Nice → starts by cooperating.

• Tough → punishes defection.

• Forgiving → returns to cooperation when the other player cooperates.

What makes tit for tat a forgiving strategy?

A player is willing to cooperate w a former defector

• Once they show evidence of willingness to cooperate

Grim trigger

• A strategy to cooperate unless the other player defects

• BUT defect for ever more if the other player defects

Ie

• Cooperate initially

• Continue cooperating, unless the other player ever defects

• Once they defect → defect forever

Grim trigger vs tit for tat

Tit for tat is forgiving

• Grim trigger = single defection → no chance of mutual cooperation

What conditions help tit-for-tat succeed (ie repeated game cooperation)?

• Players are stable/recognisable.

• They remember past behaviour

• They expect future interactions

• Future outcomes matter enough that punishment is costly (ie they have a significant stake in what happens in the future)

What happens when future interaction is unlikely?

There is little reason to cooperate

• Bc threat of future punishment loses its force

How can repeated games encourage business cooperation?

Firms cooperate bc they need future relationships with suppliers/customers

• Eg firms pay suppliers on time bc they need future shipments

If interaction will be repeated a known number of times…

Cooperation is harder to sustain

Repeated interaction →

Cooperation

• Bc threat of future punishment

In a game, if every player has a dominant strategy, which strategy would you expect them to use?

Their dominant strat

• So easy to derive an equilibrium prediction

Nash equilibrium

• The combination of strategies in a game such that neither player has any incentive to change strategies given the strategy of his opponent.

• (in simple terms- each player is doing the best they can given others' choices)

How to identify if a situation is in Nash equilibrium?

Ask: “Does either player have an incentive to change their strategy, assuming the other player stays the same?”

• If neither wants to change → Nash equilibrium

• If at least one wants to change → not a Nash equilibrium.

When is a combination NOT a Nash equilibrium?

When at least one player would benefit from changing strategy.

Best response function

Says what strategy a player should choose (to maximise their payoff) conditional on the strategy of other players

• In simple terms- the strategy that gives a player the highest payoff given what the other player does

Example of payoff matrix

Best response function table

How do you find Nash equilibrium using best responses?

• Where both players are chosing their best response SIMULTANEOUSLY

• (in a table- find the cell w all players best responses)

• (where best function responses of ALL players coincide)

In a social dilemma, what happens if every player follows their dominant strategy (given each player has a dominant strat)?

The outcome is a Nash equilibrium

Is every Nash equilibrium Pareto efficient?

No

• Nash equilibrium does NOT = pareto efficiency

Does Nash equilibrium require dominant strategies?

No

• Can exist w/o dominant strategies

What does Nash equilibrium predict?

What players will do when they act in their own interests

• It does NOT necessarily give the best collective outcome → hence can be pareto inefficient

How many Nash equilibria can where be in a game?

1 or many

Coordination game

• A game where players need to coordinate their choices to achieve a good outcome

Is Nash equilibrium a recommendation for collective welfare?

No

• It predicts individual-interest behaviour, not collective-interest behaviour

Why can coordination be difficult?

• Players may want different coordinated outcomes

• Or may not trust others to coordinate

What is the Battle of the Sexes? (idts u need to know this- skip)

A coordination game where both want to coordinate but prefer different outcomes.

Distinguish betw these 6 concepts

• Dominant strategy

• Best response

• Nash equilibrium

• Pareto efficient

• Social dilemma

• Coordination game

Most common equilibrium concept used in game theory

Nash equilibrium

• But there are others (eg maximin)

Maximin strategy

Choosing the strategy that maximises the lowest possible value of their own payoff

• Ie makes the lowest payoff one can receive as large as possible

Easy way to remember what maximum means

Maximises ur minimum payoff

How do you find a maximin strategy?

• Look at the worst payoff from each strategy

• Then choose the highest of those

Choose strat w highest minimum payoff

When is maximin useful?

When there is strategic uncertainty (u dk what other player will do)

• If you don’t have a dominant strat → need to predict what other player will do

• If u r uncertain → maximin gives a safe strat

Strategic uncertainty

Uncertainty about what other players will choose

• (if u don’t have a dominant strat)

Cons of maximin

Might miss the opportunity to earn a much higher payoff

• Bc it is a v cautious strat (protects u from worst ourcome)

Maximin vs Nash equilibrium

2 diff ways of deciding what to do in a game

• Maximin: protects against the worst case

• Nash: is optimal given others' strategies.

Alternatives to Nash equilibrium

Playing the maximin strategy

• Eg if unsure abt if opponent will act rationally

(get more clarity on this)

• Firm 2 has a dominant strategy to stay out. I

• If firm 2 stays out then firm 1 does best to enter.

• So, firm 1 enter and firm 2 stay out is the Nash equilibrium.

• The maximin strategy of firm 1, however, is to stay out.

• But, note this only guarantees the relatively low payoff of 10.