lecture 1: commitment payoff theorem...backgroundcommitment payoff theoremperfect...

Background Commitment Payoff Theorem Perfect Monitoring Proof Discussion

Lecture 1: Commitment Payoff TheoremLong-Run Short-Run Models with Perfect Monitoring

Harry PEIDepartment of Economics, Northwestern University

Spring Quarter, 2020


Rules

Please ask questions.

Let me know if I am going too slow or too fast.

Interrupt me if:

• I made a mistake (highly likely)

• There is something unclear,

• There is something you don’t understand.

There is no stupid listener, there are only bad speakers.

• Also applies when you give talks :)

Email me if you have suggestions.

Email me if you want to chat about work/life/whatever.


What is a reputation?

Google: A widespread belief that someone or something has a particularhabit or characteristic.

Two approaches to study reputations:

1. The habit view: Players convince their opponents that they will behavein a particular manner (e.g., always cooperate, tit-for-tat).

2. The characteristic view: Players signal payoff-relevant characteristicsover time (e.g., low production cost, high ability, high quality).

Similarity: Dynamic games with incomplete information, one informedplayer facing one/multiple uninformed opponent(s).

Difference: Nature of the informed player’s private info.


Overview of this course

Mostly focus on the habit view (commitment-type models):

• Different versions (how patient the uninformed players are).

One lecture on a combination of the two:

• Pei (2018): Reputation Effects under Interdependent Values.

One lecture on the characteristic view (rational-type models):

• Pei (2020): Reputational Payoffs and Behaviors without Commitment.

One lecture on reputational bargaining:

• Cool application of reputation models to bargaining problems.


Intellectual History: Chainstore Paradox

• A monopolist has branches in T ∈ N towns.

He faces one potential competitor in each town.

• In period s ∈ 1, 2, ...,T, monopolist plays against the competitor inthe s-th town.

In Out

F A

C

M(2, 0)

(0, −1) (1, 1)

• Monopolist’s total payoff is the sum of payoffs in T markets.

• Every competitor perfectly observes all actions chosen before.


Intellectual History: Chainstore Paradox

Unique subgame perfect equilibrium:

• Every competitor chooses In and monopolist chooses Accommodate.

What is wrong with this prediction?

• No matter how long the time horizon is, the monopolist never fights.

• Even if a competitor observes the monopolist fighting the past 1000entrants, he still believes that he will be accommodated with prob 1.

Something is missing in complete information game models.


Intellectual History: Commitment Type Models

How to fix this?

• Kreps and Wilson (1982), Milgrom and Roberts (1982).

Known as the Gang of four.

Idea: Perturb the game with a small prob of commitment type.

• With probability ε > 0, the monopolist is irrational,

doesn’t care about payoffs, and mechanically fights in every period.

• With probability 1− ε, the monopolist is rational,

maximizes the sum of his payoffs across periods.


Result: Gang of Four

Theorem: Gang of Four

For every ε > 0, there exists T∗ ∈ N such that when T ≥ T∗,

on the equilibrium path of every sequential equilibrium,

• Strategic monopolist chooses F & each potential entrant chooses Out

in the initial T − T∗ periods

Another way to say this: Monopolist fights (i.e., builds his reputation forbeing aggressive) in all except for the last T∗ periods.

Takeaway from this result: The option to build reputation dramaticallyaffects forward-looking agents’ incentives.


Proof: Take Home ExerciseBonus Question: Figure out why sequential equilibrium

Proof Idea: Characterize the equilibrium via backward induction.

Equilibrium Behavior:

• In the first T − T∗ periods, rational incumbent plays F and entrantstays out. No learning takes place.

• In the last T∗ periods, entrant enters with positive prob, strategicincumbent mixes between F and A. Learning happens gradually.

Probability of entry makes the strategic incumbent indifferent, andstrategic incumbent’s mixing probability makes the entrant indifferent.

Establish Uniqueness of Sequential Equilibrium Outcome: Pin down theentrants’ on-path beliefs in the last few periods.


Robustness of the Gang of Four Insight?

Gang of four result requires:

• Finite horizon, backward induction.

• A particular stage-game.

• Entrant can perfectly observe monopolist’s action.

• Sequential equilibrium.

More important concern: Does it rely on the nature of incomplete info?

• Fix G = (N,A, u), and let w ∈ Rn be a stage-game NE payoff.

Folk Theorem under Incomplete Information: Fudenberg and Maskin (1986)

For any ε > 0 and any payoff vector v > w, there exists T∗ ∈ N such that for

any T > T∗, there exists a strategy si for each player i such that in the T-fold

repetition of G where each player i is rational with probability 1− ε and

committed to si with probability ε, there is a sequential equilibrium where

players’ average payoff vector is within ε of v.


Fudenberg and Levine (1989, 1992)

How robust is the gang of four insight?

• both finite and infinite horizon models.

• general stage game payoffs.

• imperfect monitoring.

• weaker solution concepts.

• not sensitive to the details of incomplete info.

Will present all results in infinite horizon. Robust to long finite horizon.

• Proof: Not constructive but captures the logic behind reputations.


Infinitely Repeated Game with One Long-Run Player

• Time: t = 0, 1, 2, ...

• Long-lived player 1 (P1),vs an infinite sequence of short-lived player 2s (P2).

• Players simultaneously choose their actions a1 ∈ A1 and a2 ∈ A2.

* Actions in period t: a1,t and a2,t.

• Stage-game payoffs: u1(a1,t, a2,t), u2(a1,t, a2,t).

* P1’s discounted average payoff :∑∞

t=0(1− δ)δtu1(a1,t, a2,t).

• Public signal: y ∈ Y , with ρ(y|a1, a2) the prob of y under (a1, a2).

* yt: public signal in period t.


Introducing Commitment Types

P1’s type space Ω ≡ ωs⋃

Ωm, type is perfectly persistent.

1. ωs is the strategic type.

Can flexibly choose actions in order to maximize payoffs.

2. Each α∗1 ∈ Ωm represents a commitment type, with Ωm ⊂ ∆(A1).

Does not care about payoffs and plays α∗1 in every period.

P2’s prior belief: π ∈ ∆(Ω).

What can players observe?

• Player 1’s history: ht1 ∈ Ht

1 ≡ Ω× A1 × Yt.

• Player 2’s history: ht2 ∈ Ht

2 ≡ A2 × Yt.

Important: P2 in period t observes (y0, ..., yt−1).

Assumptions: A1,A2,Y and Ωm are finite, π has full support.


Fudenberg and Levine (1989)Commitment Payoff Theorem (Pure Commitment & Perfect Monitoring)

Let’s make two simplifying assumptions:

1. Perfect monitoring: Y = A1 × A2 and ρ(a1, a2|a1, a2) = 1.

2. There exists a commitment type that plays a pure action.

For every commitment action a∗1 ∈ Ωm, P1’s commitment payoff from a∗1 :

v∗1(a∗1) ≡ mina2∈BR2(a∗1 )

u1(a∗1 , a2).

Let u1 be P1’s lowest stage-game payoff.

Commitment Payoff Theorem: Fudenberg and Levine (1989)

For every ε > 0, there exists T ∈ N,such that when π attaches prob more than ε to commitment type a∗1 ∈ Ωm,strategic P1’s payoff in any Bayes Nash Equilibrium is at least:

(1− δT)u1 + δTv∗1(a∗1).


What does the commitment payoff theorem tell us?

Patient P1’s payoff lower bound (applies to all BNEs):

(1− δT)u1 + δTv∗1(a∗1).

What happens when the informed player is patient, i.e., δ → 1?

• P1’s payoff lower bound→ v∗1(a∗1).

• Patient P1 receives at least his commitment payoff from a∗1 .

Does it depend on the details of the type space?

• No. We just need type a∗1 to occur with positive prob.


Refinement for Repeated Complete Info Games

Let

A2 ≡ α2 ∈ ∆(A2)|α2 best replies against some α1 ∈ ∆(A1)

Patient P1’s lowest equilibrium payoff:

minα2∈A2

maxa1∈A1

u1(a1, α2).

P1’s highest equilibrium payoff:

maxα2∈A2,α1 that α2 best replies against

mina1∈supp(α1)

u1(a1, α2).

In many games of interest, the option to build a reputation selects a subset ofhigh payoffs for P1. Sometimes, it selects P1’s highest equilibrium payoff.


Commitment Payoff Theorem in Product Choice Game

A firm (P1) and a sequence of consumers (P2s).

– T NH 2, 1 −1, 0L 3,−1 0, 0

Repeated complete information game:

• P1’s payoff can be anything within [0, 2].

Positive prob of commitment type that mechanically plays H.

• Patient strategic incumbent guarantees payoff ≈ 2 in every BNE.

The option to build a reputation for playing H selects the highest equilibriumpayoff for the firm.


Commitment Payoff Theorem in Entry Deterrence Game

An incumbent (P1) and a sequence of entrants (P2s).

– O IF 2, 0 −1,−1A 3, 0 0, 1

Repeated complete information game:

• P1’s payoff can be anything within [0, 2].

Positive prob of commitment type that mechanically plays F.

• Patient strategic incumbent guarantees payoff ≈ 2 in every BNE.

The option to build a reputation for playing F selects the highest equilibriumpayoff for the incumbent.


Proof: Overview

Commitment Payoff Theorem: Fudenberg and Levine (1989)

For every ε > 0, there exists T ∈ N,

such that when π attaches prob more than ε to commitment type a∗1 ∈ Ωm,

strategic P1’s payoff in any Bayes Nash Equilibrium is at least:

(1− δT)u1 + δTv∗1(a∗1).

Fix the parameters (π, δ). For every Bayes Nash Equilibrium (σ1, σ2),

• Consider strategic P1’s payoff

if he deviates from σ1 and mechanically plays a∗1 in every period.

• Let this payoff be U∗1 .

• Strategic P1’s equilibrium payoff ≥ U∗1 .


Proof: P1’s Payoff under His Deviation

Strategic P1’s payoff if he deviates and plays a∗1 in every period.

If strategic P1 does this, then only two things can happen in a given period:

1. P2’s action is supported in BR2(a∗1).

2. P2 has an incentive to play actions outside BR2(a∗1).

In Case 1, P1’s stage game payoff ≥ v∗1(a∗1).

In Case 2, there exists γ > 0 such that:

• P2 believes that a∗1 is played with prob less than 1− γ in that period.

Such γ depends only on players’ stage-game payoff functions.

• After P2 observes P1 plays a∗1 in that period, Bayes Rule suggests that:

Posterior Prob of Commitment Type a∗1Prior Prob of Commitment Type a∗1

≥ 11− γ

.

• This can happen in at most T ≡ dlog ε/ log(1− γ)e periods.


Proof: Wrapping up

What is strategic P1’s payoff if he deviates and plays a∗1 in every period?

In periods where P2’s action is supported in BR2(a∗1).

• P1’s stage game payoff ≥ v∗1(a∗1).

In periods where P2’s action is not supported in BR2(a∗1).

• P1 may receive low stage-game payoff,

• But there can be at most T ≡ dlog ε/ log(1− γ)e such periods.

Lower bound on strategic P1’s payoff from playing a∗1 in every period:

(1− δT)u1 + δTv∗1(a∗1).

This is also a lower bound for strategic P1’s equilibrium payoff.


How to Interpret Commitment Type?

Commitment type(s) capture the intuition that:

• once we observe a player behave in a particular way for a long time,

we tend to believe that he will behave in the same way in the future.

• This logic is missing in complete information game models.

• Commitment types: a modeling shorthand to capture this logic.

The proof captures this logic:

• Either P2 believes that P1 will play a∗1 and best replies.

• Or P2 does not believe that P1 will play a∗1 ,

but after observing P1 plays a∗1 , she will be surprised

and the probability she attaches to commitment type a∗1 increases.


Some Common Misunderstandings

1. Can strategic P1 convince P2s that he is a commitment type?

Not with high prob on the equilibrium path! Belief is a martingale.

Example: Think about a pooling equilibrium.

2. Will strategic P1 build a reputation?

Not necessarily in the infinite horizon game. He may find it strictlyoptimal to separate from the commitment type in period 0.

3. Does it lead to sharp predictions about behavior?

Not in the infinite horizon game.

Strategic long-run player’s behaviors differ across equilibria.

4. Does it say much about short-run players’ welfare?

No. Because strategic P1’s behavior cannot be pinned down.


A Good Take Home Exercise

Take the product choice game with Ωm = H.

What can we say about the following summary statistic about P1’s behavior:

X(σ1,σ2)(H) ≡ E(σ1,σ2)[ ∞∑

t=0

(1− δ)δt1a1,t = H]

You can easily construct equilibrium s.t. X(σ1,σ2)(H) = 1.

What about the lower bound for this object?

inf(σ1,σ2)∈NE(δ,π)

X(σ1,σ2)(H)

What about general stage games with a pure-strategy commitment type?

• Any tractable formula for the lower bound on behavior?


Next Lecture

Commitment payoff theorem with imperfect monitoring:

• the public signals are noisy,

• commitment payoff from mixed commitment actions.

Papers to read:

• Fudenberg and Levine (1992), Gossner (2011).

• Kalai and Lehrer (1993), Sorin (1999).

lecture 1: commitment payoff theorem...backgroundcommitment payoff theoremperfect...

Documents