Incentive Compatibility and
the Bargaining Problem
By Roger B. Myerson
Presented by Anshi Liang
lasnake@eecs
Outline of this presentation
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6.
Introduction
Bayesian Incentive-Compatibility
Response-Plan Equilibria
Incentive-Efficiency
The Bargaining Solution
Example
Outline of this presentation
1.
2.
3.
4.
5.
6.
Introduction
Bayesian Incentive-Compatibility
Response-Plan Equilibria
Incentive-Efficiency
The Bargaining Solution
Example
Introduction
 Consider the problem of an arbitrator trying
to select a collective choice for a group of
individuals when he does not have complete
information about their preferences and
endowments.
 The goal of this paper is to develop a unique
solution to this arbitrator’s problem, based
on the concept of incentive-compatibility and
bargaining solution.
Introduction
 Describe by a Bayesian collective choice problem:
(C, A1, A2, …, An, U1, U2, …, Un, P)
C is the set of choices or strategies available to the
group;
Ai is the set of possible types for player i;
Ui is the utility function for player i such that Ui(c, a1, a2,
…, an) is the payoff which player i would get if cЄC
were chosen and if (a1, a2, …, an) were the true vector
of player types;
P is the probability distribution such that P(a1, a2, …, an)
is the probability that (a1, a2, …, an) is the true vector
of types for all players.
Introduction
 Assumptions:
a. C and all the Ai sets are nonempty finite
sets;
b. The response of each player is
communicated to the arbitrator
confidentially and noncooperatively;
c. The arbitrator cannot compel a player to
give the truthful response;
d. The arbitration is binding.
Introduction
 Choice mechanism is a real-value function
π with a domain of the form
CX(S1XS2X…XSn)—for some collection of
response sets S1, S2,…, Sn—such that
∑c,ЄCπ(c’|s1,…,sn)=1, and π(c|s1,…,sn) for all
c,for every (s1,…,sn) in S1XS2X…XSn.
 Ai is the standard response set.
Outline of this presentation
1.
2.
3.
4.
5.
6.
Introduction
Bayesian Incentive-Compatibility
Response-Plan Equilibria
Incentive-Efficiency
The Bargaining Solution
Example
Bayesian Incentive-Compatibility
 With a choice mechanism π, we have Zi(π,
bi|ai) represents the conditionally-expected
utility payoff for player i, here ai is his true
type, bi is the type he claims.
 A choice mechanism is Bayesian incentivecompatible if
Zi(π, ai|ai)≥ Zi(π, bi|ai) for all i, aiЄAi, biЄAi
Bayesian Incentive-Compatibility
 Define Vi(π|ai)=Zi(π, ai|ai) if choice mechanism π is
used and if everyone is honest.
 Define V(π)=((Vi(π|ai))a1ЄA1,…,(Vn(π|an)) anЄAn).
 The feasible set of expected allocation vectors:
F={V(π): π is a choice mechanism}
 The incentive-feasible set of expected allocation
vectors:
F*={V(π): π is a Bayesian incentive-compatible}
Bayesian Incentive-Compatibility
 Theorem 1: F* is a nonempty convex and
compact subset of F (proof in the paper).
 If Vi(π|ai)<Vi(π’|ai), for all i and aiЄAi, we say
that π is strictly dominated by π’.
Outline of this presentation
1.
2.
3.
4.
5.
6.
Introduction
Bayesian Incentive-Compatibility
Response-Plan Equilibria
Incentive-Efficiency
The Bargaining Solution
Example
Response-Plan Equilibria
 A response plan for player i is a function σi
mapping each type aiЄAi onto a probability
distribution over his response set Si. σi(si|ai) is the
probability that player i will tell the arbitrator si if his
true type is ai
 So we have Wi(π, σ1, …, σn|ai) to represent the
player i’s expected utility payoff; similarly to before,
we have a vector of conditionally-expected
payoffs:
W(π, σ1, …, σn)=(((Wi(π, σ1, …, σn|ai))aiЄAi)ni=1)
Response-Plan Equilibria
 (σ1, …, σn) is a response-plan equilibrium for the
choice mechanism π if, for any player i and type
aiЄAi, for every possible alternative response plan
σ’i for player i:
Wi(π, σ1, …, σn|ai)≥ Wi(π, σ1, …, σi-1, σ’i, σi+1,…,σn|ai)
 The equilibrium-feasible set of expected allocation
vectors:
F**={W(π, σ1, …, σn): π is a choice mechanism, and
(σ1, …, σn) is a response-plan equilibrium for π}
 Theorem 2: F**=F* (proof in the paper)
Outline of this presentation
1.
2.
3.
4.
5.
6.
Introduction
Bayesian Incentive-Compatibility
Response-Plan Equilibria
Incentive-Efficiency
The Bargaining Solution
Example
Incentive-Efficiency
 π is incentive-efficient if and only if it is a
Bayesian incentive-compatible choice
mechanism and is not strictly dominated by
any other Bayesian incentive-compatible
mechanism (remind: If Vi(π|ai)<Vi(π’|ai), for
all i and aiЄAi, we say that π is strictly
dominated by π’).
Outline of this presentation
1.
2.
3.
4.
5.
6.
Introduction
Bayesian Incentive-Compatibility
Response-Plan Equilibria
Incentive-Efficiency
The Bargaining Solution
Example
The Bargaining Solution
 Conflict outcome: it represents what would happen
by default if the arbitrator failed to lead the players
to an agreement. Examples:
Market
Politics
Students
 Conflict payoff vector:
t=((ta1)a1ЄA1, (ta2)a2ЄA2,…,(tan)anЄAn), where each tai is
player i’s conditional expectation, given that ai is
his true type, of what his utility payoff would be if
the conflict outcome occurred.
The Bargaining Solution
 Given the conflict payoff vector t our collective
choice problem becomes a bargaining problem,
with a feasible set F*, t is a reference point in F*.
 Let F*+ be the set of all incentive-feasible payoff
vectors which are individually rational:
F*+=F*∩{y:yai≥tai for all i and all aiЄAi}
 Theorem 3: Suppose that c* is not incentiveefficient, then there exist a unique incentivefeasible bargaining solution.
Outline of this presentation
1.
2.
3.
4.
5.
6.
Introduction
Bayesian Incentive-Compatibility
Response-Plan Equilibria
Incentive-Efficiency
The Bargaining Solution
Example
Example
1. Two players share the cost of a project which
benefit them both.
2. The project cost $100, the two players call an
arbitrator to divide it.
3. Project value:
Player1: $90 if he is type1.0, $30 if he is type1.1
Player2: $90
4. To the arbitrator and player2, P1(1.0)=.9 and
P2(1.1)=.1
Example
 Some observation points:
a. No matter what player 1’s type is, the project
appears to be worth more than it costs;
b. The decisions cannot be made separately.
 Some intuitive solutions:
a. 50-50 or 20-80
b. 47-53
c. 50-50 or 0-0
Example
 Formal solution:
Let C={c0, c1, c2}, A1={1.0, 1.1}, A2={2}. We have
P(1.0, 2)=.9 and P(1.1, 2) =.1.
c0 means “do not undertake the project”; c1 means
“undertake the project and make player1 pay for
it”; c2 means “undertake the project and make
player2 pay for it”.
(u1, u2)
c0
c1
c2
a1=1.0
(0, 0)
(-10, 90)
(90, -10)
a2=1.1
(0, 0)
(-70, 90)
(30, -10)
Example
 Strategies can be randomized.
 Use the abbreviations π0j=π(cj|1.0, 2) and
π1j=π(cj|1.1, 2).
 The incentive-compatible choice mechanisms
satisfies the following:
-10π01+90π02≥-10π11+90π12,
-70π11+30π12≥-10π01+90π02,
π00+π01+π02=1, π10+π11+π12=1
,
Example
 Expected benefits for all players:
x1.0=0π00-10π01+π02,
x1.1=0π10-10π11+π12,
x2=.9(0π00+90π01-10π02)+.1(0π10+90π1110π12),
 Then the incentive-feasible bargaining
solution is the solution that maximize
((x1.0).9(x1.1).1x2), x and π satisfy the
restrictions above.
Example
 Result:
x1.0=39.5, x1.1=13.2, x2=36
π01=.505, π02=.495, π10=.561 and π12=.439
 Meanings in English
Conclusion
 A great paper overall
 The mathematical derivation is complicated
but very clear
 This concept can be possibly extended to
our networking study. For example, say that
the arbitrator is the network designer; the
two players are network users, etc.
Thank you
very much!
Anshi Liang