bayesian games

Bayesian Games

Microeconomics CAmine Ouazad

Who am I• Assistant prof. at INSEAD since 2008.• Teaching Prices and Markets in the MBA

program, Econometrics A, B, Microeconometrics, in the PhD program.

• Research:– Applied empirical work on Urban Economics.– Economics of Discrimination.– Banking/Competition.– Econometric Forecasts.

• I tend to cold call.

Goals of my Micro C classes

1. Economics and psychology have a large number of common interests, but use different toolboxes.– Subjective perceptions, gender, culture.– Economics and individual rationality.– Formation of perceptions using Bayes’ framework.

2. Economics and strategy use very similar tools and have a large number of common interests:– Strategic interactions.– Strategic interactions with imperfect information.

Two maths/econ tools for today

• Bayes’ formula(s):– P(A)= P(A|B) P(B) + P(A|not B)P(not B)– E(A)= E(A|B) P(B) + E(A|not B) P(not B)

• Risk neutrality, risk aversion:– Do you prefer : 0 with 50% chance, 10 euros with 50%

chance or 5 euros with certainty?– Risk neutral: indifferent between the two choices. What

matters for your choice is the expected payoff.• Assumption throughout: players are risk neutral.

Outline1. Recap on games, strategies,

and Nash equilibria.2. Guess a number3. Prisoners’ Dilemma

1. Perfect information2. Uncertainty

4. Entry Game.1. Basic Entry Game2. With Uncertainty3. Multiple Periods4. Multiple Periods with Uncertainty

5. Recommended Books and Papers.

Remember: “Economists do it with models”

1. Recap on games, strategies, and Nash Equilibria

• Key concepts: Players, Strategies, Payoffs.• Simultaneous-move and sequential games.• Sequential games: Nash Equilibrium by backward

induction.• Simultaneous move game: 1. Nash Equilibrium by finding

mutual best responses. 2. Nash equilibrium by finding strategies where no player has an incentive to deviate unilaterally.

• Typical games:– The prisoner’s dilemma.– The battle of the sexes.

2. Guess a number

• Each person gives me a number between 0 and 100.

• The person who is closest to 2/3 of the average gets a bottle of champagne.

• Number?

• What’s the reasoning?• Typical outcomes?

2. Guess a numberThe Bayesian Approach

• Assumption of perfect rationality is not consistent with the empirical observations…

• Assume that players are of one of two types: either rational or random.• The random players choose a number between 0 and 100 randomly.• What should be the choice of the rational players?• Note first that all rational players will choose the same number.

– Call this number x.

• Then we use Bayes’ formula.– E(numbers) = E(numbers|rational players). P(rational players) + E(numbers|random

players).P(random player).

• Solution?

2. Guess a number• Another approach to the problem.• “Iterated Elimination of Dominated Strategies”• Anyone playing a number between 67 and 100?• Anyone playing a number between 44 and 100?• Etc…• What is the number left?• But is everybody thinking so deeply?

(Nagel, 2002)• Can we explain our empirical results in the MBA

classroom? What is students’ depth of thinking?

3. Prisoners’ Dilemma

• Example #1: Prisoners.• Example #2: Price Competition.

Example #1: Prisoners.Roadmap• Players, Strategies, and Payoffs.• Write the payoff matrix.• Are there dominant strategies?• What is the Nash equilibrium?• Where is the uncertainty?• Write the payoff matrix(ces) with uncertainty.• What is one Bayesian Nash equilibrium?


Prisoners • Confess/Not Confess• Simultaneous or sequential move game?• Dominant strategy? Weakly dominant strategy?• Nash equilibrium?

Jim/John Not Confess Confess

Not Confess -2,-2 -8,0

Confess 0,-8 -5,-5

Prisoners• The psychology of the game is essential.• How does that affect the game?

Players’ types?Players’ beliefs?

The psychological cost of confessing. If both players have a cost of confessing:


Not Confess -2,-2 -8,0

Confess 0,-8 -5,-5


Not Confess -2,-2 -8,0-c

Confess 0-c,-8 -5-c,-5-c

Golden Balls

Bayesian game:Types, Beliefs, Strategies, Payoffs.

• Type is either {high cost c,low cost c}.• Beliefs about the other player’s type are

represented by the subjective probability of being of a high cost c of deviation/low cost.

• Simultaneous move game.• Strategy: one action for each type.• Payoffs: the payoff matrix for each pair of

types of players.

Bayesian Nash equilibrium

• is a strategy for each player, for each type, such that:each player’s strategy is a best response to the other player’s strategy given (a) his beliefs about the other player’s type and (b) given the other player’s strategy for each type.

• We check that the following is a Bayesian Nash equilibrium:– The high cost of deviation player does not confess.– The low cost of deviation player confesses.

Checking this is an equilibrium:• What is Jim’s best response?

– when he is of a high cost of confessing?– when he is of a low cost of confessing?

… and when he believes that John is of a high cost with probability p.… and when he assumes the above strategy (blue box) for John.• Same question for John.

What fraction of games see both players cooperating?

Bayesian Nash equilibrium

Key concepts for this session (1/2)

• Simultaneous move games with imperfect information.

• Players, Strategies, Payoffs.• Beliefs, Types.• Bayesian Nash Equilibrium.

Example #2: Price competition.• Airline pricing.• Capacity Constraints?• Players, Strategies, Payoffs.• Write the Payoff Matrix.• Are there dominant strategies?• What is the Nash equilibrium?• Where could be the uncertainty?


Price competition:Tiger vs. Singapore Airlines

Tiger/Singapore Airlines

High price Low price

High price $3600,$3600 0,$5200

Low price $5200,0 $2600,$2600

Flight at 10am on January 23rd

At 4pm the previous day… what should the Tiger and Singapore Airlines pricing people displayon the website? Two pricing points: $200 or $150.Demand for seats: 40. Marginal cost: $20 per seat.Airline with the lowest price sells 40 seats.If equal prices: customers indifferent between the two airlines.

What if… Tiger does not have 40 empty seats?

• If Tiger only has 10 seats unbooked…• When both set the same price, Singapore sells 30 seats, Tiger

sells 10 seats. (Total demand is 40).

Tiger/Singapore Airlines

High price Low price

High price $5400,$1800 $5400,$1300

Low price $5200,$0 $3900,$1300

Singapore Airlines does not know for sure Tiger’s remaining capacity

• Tiger can be of one of two types. Either Unconstrained, or Constrained

• Prior p=P(Constrained).• Singapore’s capacity is common knowledge.• Check whether the following is a Bayesian Nash

equilibrium:– The unconstrained Tiger Airways deviates, the constrained

Tiger Airways does not deviate; Singapore Airlines does not deviate.

– “deviate”=“sets a low price.”– Under what constraint on p is this a Bayesian Nash equilibrium?

4. Entry Game• Example #1: The flatmate.• Example #2: Apple vs Samsung.

Roadmap for this section• Write the sequential game.• What is the subgame perfect Nash equilibrium?• Where is the uncertainty?• Consider the game with no uncertainty, repeated multiple

times. What is the subgame perfect Nash equilibrium?• What about uncertainty with multiple periods?

Takeaways?

Apple vs Samsung

• Rivals: Handsets are (imperfect) substitutes in the eyes of consumers.

• Entrant and incumbent?• Fighting against the entrant?• Cost of fighting?• Benefit of fighting?

• “A little less Samsung in Apple sourcing.” Beyondbrics, Financial Times, Sep 10, 2012.

• “Trade Judge backs Apple in Samsung fight.”Oct 24, Financial Times.

• “Tension on Display: Samsung may end Dwindling LCD Panel Deal with Apple.” Wall Street Journal, Oct 22, 2012.

• “Samsung, Apple, amass 4G Patents for Battle,” Wall Street Journal, Sep 12, 2012.

"I'm willing to go thermonuclear war on this“-- Steve Jobs

Entry deterrence

• Predatory pricing.– Walmart.– But

• Increases in output (commodity markets, close substitutes).

• Lawsuits.– Apple vs Samsung.

Entry Game, “Soft” Incumbent

• Discuss the payoffs. Give at least 2 examples of market competition to which this sequential game may apply.• Notice the order of the payoffs. The first mover comes first.• What is the subgame perfect Nash equilibrium?

Entrant

Incumbent

(0,10)

(-5,4)

Stay out Enter

AccommodateFight

(5,5)

Entry Game, “Tough” Incumbent

• What is the subgame perfect Nash equilibrium? Such an equilibrium justifies talking about a “tough” incumbent.

Entrant

Incumbent

(0,10)

(-5,6)

Stay out Enter

AccommodateFight

(5,5)

What if we don’t know the incumbent’s type?

• Prior about the incumbent.• We represent this prior with a probability p: The entrant believes that the incumbent is tough with probability p.\• Fill in the payoffs below.• When does the entrant choose to enter? When does he choose to stay out?

Entrant

Incumbent

( , )

( , )

Stay out Enter

AccommodateFight

( , )

Playing the entry game twice…knowing that the incumbent is soft.

• Would the incumbent fight?

Entrant

Incumbent

Stay out Enter

AccommodateFight

Entrant

Incumbent

Stay out Enter

AccommodateFight

Round 1 Round 2

(0,10)

(-5,4) (5,5)

(0,10)

(-5,4) (5,5)

Playing the entry game twice…knowing that the incumbent is tough.

• Would the incumbent fight?

Entrant

Incumbent

Stay out Enter

AccommodateFight

Entrant

Incumbent

Stay out Enter

AccommodateFight

Round 1 Round 2

(0,10)

(-5,6) (5,5)

(0,10)

(-5,6) (5,5)

Playing the entry game twice…not knowing the incumbent’s type.

• Would the incumbent fight?• What information does the fight (or not fighting) give?

Entrant 1

Incumbent

Stay out Enter

AccommodateFight

Entrant 2

Incumbent

Stay out Enter

AccommodateFight

Round 1 Round 2

( , )

( , ) ( , )

( , )

( , ) ( , )

Reputation management

• Fighting tells potential entrants that you are either tough or a soft guy trying to build his reputation.

• Accommodating tells potential entrants that you are soft with certainty.

• ➭One discordant piece of information is enough to destroy one’s reputation.– “it takes a lifetime to build a reputation and one

second to destroy it.” Warren Buffett and many other “wise” guys.

• The tough incumbent fights in every period.• The soft incumbent fights if…– The cost of fighting is smaller than the benefits of

building a reputation.– What is this cost of fighting? – What is the benefit of having a reputation?

• With a discount factor?– What is the meaning of the discount factor?

Playing the entry game twice…not knowing the incumbent’s type.

Perfect Bayesian Nash Equilibrium

• Pooling equilibrium:– Tough and soft incumbents fight in the first period.– Soft incumbents find it rational to fight in the first

period.• Separating equilibrium:– Tough incumbents fight.– Soft incumbents accommodate.– Soft incumbents do not find it rational to fight in

the first period.

All types play the same strategy.Observing the actions does not bring information on the types.

Different types play different strategies.Observing the actions gives information about types.

Playing the Entry game n times… not knowing the incumbent’s type.

• When there are k periods (think years, quarters), the reputational benefits are multiplied by k (if discount factor is 1), so the earlier the entry, the larger the reputational benefits of fighting.

• Confident of being present in the market for a large number of years/quarters?The longer the time horizon, the more important reputation is.

• Solve this with 3 periods.


• Sequential games with imperfect information.• Players, Strategies, Payoffs.• Beliefs, Types.• Perfect Bayesian equilibrium.

In a Perfect Bayesian equilibrium, players “update” their beliefs according to Bayes rule.

5. Recommended Books and Chapters

Strategic Thinking• Dixit and Nalebuff’s

“The Art of Strategy”and “Thinking Strategically.”

• David Besanko’s “Economics of Strategy.”More than Strategic Thinking• “The Armchair Economist.”• “The Undercover Economist.”


• Simultaneous move games with imperfect information.

• Players, Strategies, Payoffs.• Beliefs, Types.• Bayesian Nash Equilibrium.

Make sure you know the meaning of these concepts.


• Sequential games with imperfect information.• Players, Strategies, Payoffs.• Beliefs, Types.• Perfect Bayesian equilibrium.

Make sure you know the meaning of these concepts.

bayesian games

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