perfect bayesian equilibria in bargaining under higher ...pvarkey/docs/talks/paul_mwtd_2010.pdf ·...

IntroductionFirst Order Uncertainty

Second Order UncertaintyConclusion

Perfect Bayesian Equilibria in Bargaining UnderHigher Order Uncertainty

Paul Varkey

Multi Agent Systems Group, Department of Computer Science, UIC

60th Midwest Theory Day, Apr 17th , 2010Indiana University Bloomington

Paul Varkey Bargaining PBE Under Higher Order Uncertainty

Outline

1 IntroductionThe Bargaining ProblemThe foundational literatureThe Bargaining Problem – an extension to higher orders

2 First Order UncertaintyThe Model: The protocol & the epistemologyThe agents’ reasoningAn “existential construction” of the PBE

3 Second Order UncertaintyThe Model: The protocol & the epistemologyConstruction of “a” PBESensitivity results and a discussion on uniquenessHigher Order Uncertainty & Longer Time Horizon

4 ConclusionSummaryOngoing & Future Work

The Bargaining ProblemThe foundational literatureThe Bargaining Problem – an extension to higher orders

Outline

The Bargaining Problem

A seller and a buyer are negotiating over an item for which they have avaluation of c and v , respectively. How do (should) they split the availableprofit?

solution paradigms

Game-theoretic: Solution concepts such as the Nash Equilibrium, PerfectBayesian Equilibrium (PBE), etc. oftentimes provide excellent predici-tive/explanatory/descriptive accounts of strategic interactions betweenrational agents

Decision-theoretic: Solution concepts such as MDPs, POMDPs, etc.have been developed on rigorous and principled foundations, provideconsiderable prescriptive power and can be operationalized as a controlparadigm for an Artificial Intelligence

solution paradigms

The Bargaining Problem – an elaboration

Offers: seller-offers, alternating offers, etc.

Delay costs: discounting, fixed costs, etc.

Horizon: finite, indefinite, etc.

Information: complete and incomplete (1-sided, 2-sided, higher-order,etc.)

The Literature – an (incomplete) outline

Infinite-horizon, alternating offers under complete informationHere the players valuations and discount factors are commonly known.Rubinstein’s (1982) seminal paper solves this case by obtaining a uniqueSubgame Perfect Equilibrium (SPE) in which

there is no bargaining (i.e. agreement is immediate), andfor instance, if the common discount factor is δ, the solution apportionsa share 1/(1 + δ) to the offerer

Infinite-horizon, seller offers under incomplete information“solved” to obtain Perfect Bayesian (Sequential) Equilibria by

Sobel and Takahashi (1983): 1-sided (assymmetric) incomplete informa-tion; buyer’s valuation is private informationCramton (1984): 2-sided incomplete information

This line of research continued for the next 20 years: Chatterjee, Samuel-son, Grossman, Perry, Admati, Cho, Gul, Sonnenschein, etc.

What is missing in the theory?

Bargaining Under Higher Order Uncertainty

What happens if first-order beliefs are not assumed to be commonly knownand, instead, the players maintain (commonly known) second-order beliefs?Or, (commonly known) higher order beliefs (up to arbitrary finite levels)?

In this talk, I will present new results for Bargaining Under SecondOrder Uncertainty

PBE in pure strategies (non-unique in general; unique in strategies?)

under time horizon = 3:

Non-existence of PBE in pure strategies

Bargaining Under Higher Order UncertaintyWhat happens if first-order beliefs are not assumed to be commonly knownand, instead, the players maintain (commonly known) second-order beliefs?Or, (commonly known) higher order beliefs (up to arbitrary finite levels)?

under time horizon = 3: Non-existence of PBE in pure strategies

The Model: The protocol & the epistemologyThe agents’ reasoningAn “existential construction” of the PBE

Outline

Model: The Protocol

Seller-offers & 2-Horizon

The seller makes a first offer x2, which the buyer may reject, followingwhich the seller makes a final offer x1

Model: The Epistemology

1-Sided Incomplete (Assymmetric) Information

Seller’s valuation 0 is commonly known

Buyer’s valuation v is such that 0 ≤ v ≤ 1

Seller’s belief about v is ∼ F (v) = v 2 for 0 ≤ v ≤ 1 and is commonknowledge (“v comes from a known distribution”)

Sobel & Takahashi, (1983); Cramton, (1984)

The agents’ reasoning

The agents’ reasoning –

epistemological

(non-linear) optimization

The buyer’s reasoning process

After sellerx2−→ buyer, the buyer forms expectations about the seller’s

second (i.e. last) offer as E[x1|x2]

A buyer with valuation v accepts x2 iff

(v − x2) ≥ δ(v − E[x1|x2])

i.e. iff

v ≥ (x2 − δE[x1|x2])

(1− δ)=: d(x2,E[x1|x2]) (1)

The seller’s reasoning process – epistemological

v ≥ (x2 − δE[x1|x2])

(1− δ)=: d(x2,E[x1|x2]) (1)

Using its “knowledge” of (1), the seller infers that, if the game reachedthe second stage, the buyer’s valuation could not possibly be greaterthan d (since such a buyer would have accepted x2)

Bayes’ theorem can be used to encode this refinement of the seller’sknowledge as a proportional redistribution of the prior density on theupdated support [0, d ].

The seller’s reasoning process – optimization

The seller’s last (i.e. second) stage optimization program can then beexpressed as a maximization of the appropriate objective function:

π1(d) = maxx1≤d

x1 ·F(d)− F(x1)

from which we obtain the optimal last stage profit and offer functionsas

π∗1 (d) =2d

3and x∗1 (d) =

=x2 − δE[x1|x2]√

3(1− δ)(2)

The seller’s first stage optimization can now be written as:

π2(d , x2) = maxx2≤1

[x2 ·

F(1)− F(x2)

F(1)+ δ · π∗1 (d) · F(x2)

The seller’s reasoning process – optimization

The seller’s last (i.e. second) stage optimization program can then beexpressed as a maximization of the appropriate objective function:

π1(d) = maxx1≤d

x1 ·F(d)− F(x1)

from which we obtain the optimal last stage profit and offer functionsas

π∗1 (d) =2d

3and x∗1 (d) =

=x2 − δE[x1|x2]√

3(1− δ)(2)

The seller’s first stage optimization can now be written as:

π2(d , x2) = maxx2≤1

[x2 ·

F(1)− F(x2)

F(1)+ δ · π∗1 (d) · F(x2)

an “existential construction” of the PBE

The general program for obtaining PBE when it is unique

In sequential epistemic reasoning, the unique PBE is obtained by postu-lating its existence, solving the resulting optimization program, and thenproving that the solution is indeed an equilibrium

Here we carefully analyze expression (2), to see how one may proceedfrom common knowledge assumptions to a (unique) PBE

x∗1 (d) =x2 − δE[x1|x2]√

3(1− δ)(2)

Key observation: If the equilibrium is unique, then the buyer’s expec-tations must be correct, i.e.

E[x1|x2] = x∗1 (d)

from which we obtain that

E[x1|x2] =x2√

3−√

3δ + δ(4)

Recall:

π2(d , x2) = maxx2≤1

[x2 ·

F(d)− F(x2)

F(d)+ δ · π∗1 (d) · F (x2)

E[x1|x2] =x2√

3−√

3δ + δ(4)

The optimal decision boundary d∗ and the optimal offers x2∗ and x1∗ are obtainedby substituting E[x1|x2] from (4) into d in (3) and then solving the optimizationprogram

d∗ =

√ √3(1−δ)+δ

3(1−δ)+δ

x∗2 = (1− δ)d∗ + δd∗/√

x∗1 = d∗/√

An ExampleLet δ = 0.8.Then, d∗ = 0.7895, x∗2 = 0.5225 and x∗1 = 0.4558

The Model: The protocol & the epistemologyConstruction of “a” PBESensitivity results and a discussion on uniquenessHigher Order Uncertainty & Longer Time Horizon

Outline

Model: The Protocol & The Epistemology

Higher Order Uncertainty

The seller can be one of two types, characterized by its first order beliefabout the buyer’s valuation –

Type w: F (v) = v and Type s: F (v) = v 2

The buyer is uniformly uncertain about the seller’s type; a fact that isreflected in its second order beliefs (which abscribe an equal probabilityof 1/2 to each possibility)

The buyer’s second order beliefs are commonly known

Model: The Protocol & The Epistemology

Higher Order Uncertainty

The seller can be one of two types, characterized by its first order beliefabout the buyer’s valuation –

Type w: F (v) = v and Type s: F (v) = v 2

The buyer is uniformly uncertain about the seller’s type; a fact that isreflected in its second order beliefs (which abscribe an equal probabilityof 1/2 to each possibility)

The buyer’s second order beliefs are commonly known

Model: The Epistemology, illustrated

How do we construct a PBE?

There are two natural subgames for this game – the cases where theseller’s first order belief is commonly known and is:

F (v) = v 2 (type s; same game considered earlier), andF (v) = v (type w)

Let us denote the PBE solution profile for these games, respectively, ass∗ := d∗,s , x∗,s2 , x∗,s1 = 0.7895, 0.5225, 0.4558

w∗ := d∗,w , x∗,w2 , x∗,w1 = 0.75, 0.45, 0.375

x∗,s2 and x∗,w2 do not form plausible perfect separating equilibria. Thisis because –

w incentivized to deviate by offerring x∗,s2 , if such deviation convincesbuyer that seller is sIf buyer is so convinced, it stands to loseTherefore, such an unanticipated deception is unsustainable in any equi-librium

w∗ := d∗,w , x∗,w2 , x∗,w1 = 0.75, 0.45, 0.375

a pooling PBE – an outline of the main theorem

Say that the buyer anticipates pooling

Then, the buyer’s belief about the seller does not change after the firstoffer

Given this, the seller can reason about the buyer’s expectation of thelast (i.e. second) offer and

Compute the optimal first offer that should be offered against such abuyer – denote as x∗,p2

The only remaining component is a specification of the buyer’s be-liefs off-the-equilibrium path that supports the optimality of the players’strategies: for e.g., if the first offer is not x∗,p2 , the buyer believes (withprobability 1) that the seller is the weak type

a pooling PBE – an outline of the proof

On the equilibrium path, the buyer’s belief “update” is trivial and con-sists of preserving its priors

On the equilibrium path, the strong seller plays the (unique) optimalstrategy given the buyer’s belief

Check: The weak seller prefers to pool (even when this is anticipated)as opposed to separating

Check: The players’ strategies are optimal, given the buyer’s off-the-equilibrium-path beliefs

a pooling PBE – results for δ = 0.8

d∗ = 0.8069x∗,p2 = 0.5091x∗,s1 = 0.46586x∗,w1 = 0.40345

a discussion on uniquenessE

Discount Factor0.2 0.4 0.6 0.8

Strong Seller (Separating)Strong Seller (Pooling)

(a) p(Strong) = 34

Weak Seller (Separating)Weak Seller (Pooling)

(b) p(Strong) = 34

Strong Seller (Separating)Strong Seller (Pooling)

(c) p(Strong) = 14

Weak Seller (Separating)Weak Seller (Pooling)

(d) p(Strong) = 14

PBE not unique – many off-the-equilibrium

path beliefs can support it

Is it unique in strategies? Is itunique in beliefs on the path?

If (anticipated) pooling is better for weakseller than (anticipated) separating, apooling PBE can be constructed (as shownearlier)

If (anticipated) separating is better, we

conjecture that a PBE in pure strategies

does not exist

If separating anticipated, weak selleris incentivized to decieveIf pooling anticipated, weak seller isincentivized to separate

Existence under time horizon of 3

Strong Seller (Pool/Separate)Strong Seller (Separating)Strong Seller (Pooling)

(e) p(Strong) = 12

Weak Seller (Pool/Separate)Weak Seller (Separating)Weak Seller (Pooling)

(f) p(Strong) = 12

We consider pure strategies suchas pool-separate, always-separate,always-pool, etc.

For low values of δ, both sellers preferto pool first; although, neither sepa-rating nor pooling can be supportedas an equilibrium pure strategy forthe second offer

For high values of δ, neither poolingnor separating can be supported forthe first offer

Conjecture: No pure strategy PBEexists!

SummaryOngoing & Future Work

Outline

What ��is was missing in the theory?

Bargaining Under Higher Order Uncertainty

What happens if first-order beliefs are not assumed to be com-monly known and, instead, the players maintain (commonly known)second-order beliefs? What is the combined effect an extended timehorizon and higher orders of beliefs?

In this talk, I presented new results for Bargaining Under Second OrderUncertainty

Bargaining Under Higher Order UncertaintyWhat happens if first-order beliefs are not assumed to be com-monly known and, instead, the players maintain (commonly known)second-order beliefs? What is the combined effect an extended timehorizon and higher orders of beliefs?

under time horizon = 3: Non-existence of PBE in pure strategies

What is still missing in the theory?

A general theorem connecting epistemology & equilibria: for any finiteorder of belief and time horizon (finite or infinite)

Existence proofs (or counterexample) in mixed strategies (when pureequilibria do not exist)

Constructive/algorithmic results needed (and not just existential)

Thank You! Any Questions?

perfect bayesian equilibria in bargaining under higher ...pvarkey/docs/talks/paul_mwtd_2010.pdf ·...

Documents