Delay in Strategic Information Aggregation
Ettore Damiano1
Li, Hao2
1 University
2 University
Wing Suen3
of Toronto
of British Columbia
3 University
of Hong Kong
December 3, 2011—Contract Theory Conference
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Introduction
We do not study a pure bargaining problem (in which the size of
the total pie is fixed). The collective decision to be made involves
more than a pure transfer of income across negotiating parties.
Information aggregation is a key aspect of the model.
Examples: legislative bargaining, trade negotiations, adoption of
industry standards, workplace practices
Disagreements can be either preference-driven or
information-driven, and we examine how delay can help players
sort out these two types of disagreements.
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Key Results
Costly delay occurs on the equilibrium path.
Repeated voting with delay helps improve the quality of
decisions—an outcome which is otherwise unattainable in a
one-shot game.
Ex ante welfare is higher with delay than without when the degree
of conflict is moderate.
Players make increasing concessions over the bargaining
process, but the model can also exhibit non-trivial dynamics in the
hazard rate.
Delay is useful even when per-period delay cost goes to zero.
Optimal mechanism under limited commitment:
stop-go negotiations
negotiation deadlines are optimal
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Model
Three states: L, M, and R, with corresponding probabilities πL ,
πM , and πR .
Two players LEFT and RIGHT have to decide on choosing
between two alternatives l or r .
Payoffs:
Decision
l
r
L
State
M
R
(1, 1)
(1, 1 − 2λ) (1 − 2λ, 1 − 2λ)
(1 − 2λ, 1 − 2λ) (1 − 2λ, 1)
(1, 1)
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Information Structure
RIGHT is able to distinguish between R and {L, M}. LEFT is able
to distinguish between L and {R, M}. Such information is private
and unverifiable.
Terminology: Player RIGHT who knows the state is R is called an
“informed RIGHT.” If he knows the state is L or M, he is called a
“uninformed RIGHT.” RIGHT is said to “vote for his favorite” if he
votes r ; or to “vote against his favorite” if he votes l.
Let γ = πM /(πL + πM ) denote the uninformed RIGHT’s belief that
the state is M. We are going to focus on a symmetric setting in
which πR = πL . So the uninformed LEFT’s belief that the state is
M is also given by the same γ.
γ can be interpreted as the degree of conflict in the bargaining
situation. Sometimes we call M the “conflict state.”
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Impossibility of Information Aggregation
Let qi be the probability of choosing r in state i ∈ {R, M, L}.
Proposition: When γ > 12 , any incentive compatible mechanism
without side transfers entails qR = qM = qL .
Proposition: If players can commit to paying a delay cost δ when
they disagree, then the first best outcome is achievable when
δ ≥ (2γ − 1)λ/(1 − γ)
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Repeated Voting
Each player either votes for r or l simultaneously. If the votes
agree, that decision is implemented immediately and the game
ends. If the votes disagree, each player incurs a delay cost of δ
and votes again in the next round. The game can in principal go
on indefinitely.
We consider equilibria of this game in which
the players’ strategies are symmetric
the informed types always vote according to their preferences
strategies depend only on the current “state variable” γ
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Repeated Voting
There are two kinds of disagreement. In a “regular disagreement,”
the two players vote according to their favorites. In a “reverse
disagreement,” the two players vote against their favorites.
Suppose the informed types always vote according to their
favorites. Then whenever there is a reverse disagreement, the
updated belief that the state is M jumps to 1.
Let x be the probability that the uninformed players vote according
to their preferences. If there is a regular disagreement, the
updated belief that the state is M goes down to
γ0 =
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γx
γx + 1 − γ
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No-deadline Game
First, solve the subgame when γ = 1. In a symmetric mixed
strategy equilibrium,
U(1) = x(1)(−δ + U(1)) + (1 − x(1))(1)
= x(1)(1 − 2λ) + (1 − x(1))(−δ + U(1))
The solution gives x(1) ∈ ( 12 , 1) and U(1) < 1 − 2λ.
Next, consider the subgame when γ is sufficiently close to 0 (in
particular, γ < G1 = δ/(δ + 1 + δ − U(1))). It is an equilibrium for
the uninformed types to play against their preferences (i.e.,
x(γ) = 0). The corresponding value function in this range of value
of γ is U(γ) = 1 − (1 + δ − U(1))γ.
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No-deadline Game
Proposition: There exists an infinite sequence,
G0 < G1 < G2 < . . ., with G0 = 0 and limk →∞ Gk = 1, such that
if γ ∈ [G0 , G1 ], then x(γ) = 0
if γ ∈ (Gk , Gk +1 ] for k = 1, 2, . . ., then x(γ) ∈ (0, x(1)) and satisfies
γx(γ)/(γx(γ) + 1 − γ) ∈ (Gk −1 , Gk ].
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Proof
The proof involves:
conjecturing that the value function is piecewise linear of the form
U(γ) = ak − bk γ for γ ∈ (Gk , Gk +1 ];
using the indifference condition to solve for x(γ) and U(γ) in the
next step;
applying Bayes’ rule and the formula for x(γ) to find the Gk +2 that
maps to Gk +1 ;
verifying that the informed types have no incentive to deviate.
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Characterizing the Equilibrium
x(γ) is continuous and increasing.
Every time there is a regular disagreement, the moderate players’
belief that the state is M decreases, so they soften their stance
(x(γ 0 ) < x(γ)). This can be interpreted as gradual concession
during the negotiation process.
Any time there is a reverse agreement, the moderate players’
belief jumps to one, and they toughen their position by playing
x(1) > x(γ) for the rest of the game. The resulting payoff is
U(1) < 1 − 2λ. This can be interpreted as a break-down of
negotiations.
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Welfare Properties
In states L and R, the correct decision is always taken. In state M,
each decision is taken with probability 21 .
Ex ante welfare is decreasing in γ.
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Limit Results
What happens when δ goes to 0?
lim G1 = 0
lim(Gk +1 − Gk ) = 0
lim x(γ) = 1 for γ ∈ (0, 1]
lim δx(γ)/(1 − x(γ)) = 2λγ for γ ∈ (0, 1]
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Limit Results
Can obtain the limit value functions U 0 (γ) and V 0 (γ).
Use the value function definition:
U(γ) =γ[x(γ)(−δ + U(γ 0 ))
+ 1 − x(γ)] + (1 − γ)(−δ + U(γ 0 ))
Subtract U(γ 0 ) from both sides. Divide lhs by γ − γ 0 and divide rhs
by γ(1 − γ 0 )(1 − x(γ)):
U(γ) − U(γ 0 )
γx(γ) + 1 − γ
1 − U(γ 0 )
=
(−δ) +
0
0
γ−γ
γ(1 − γ )(1 − x(γ))
1 − γ0
Take limit as δ goes to 0:
dU 0 (γ)
−2λ
1 − U 0 (γ)
=
+
dγ
1−γ
1−γ
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Limit Results
Can show that U 0 ( 21 ) = 1 − λ and V 0 ( 12 ) > 1 − λ.
So there exists a range of moderate values of λ such that the
delay game gives higher expected welfare than any single round
mechanism without budget breaking.
Expected total delay cost is not 0 even though per period delay
cost goes to 0.
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Limited Commitment
What is the optimal mechanism if the delay cost each period
cannot exceed an upper bound ∆?
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Slack
There is “no slack” in round t if δt = ∆ and
Ut+1 (γt+1 ) = 1 − 2λγt+1 .
Lemma: Given γt , the lowest feasible xt prior to round T is
max{χ(γt ), 0} and the lowest feasible posterior belief γt+1 is
max{g(γt ), 0}. These lower bounds are achieved when there is no
slack in round t.
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Optimal Mechanism
Proposition: In an optimal mechanism, xT = 0.
Assumption 2: ∆ ≤ ∆ and γ1 ≥ γ(∆).
Given Assumption 2, in an optimal mechanism, γT = γ∗ .
It is optimal to make the uninformed players concede with
probability 1. And the game should stop immediately once it is
feasible to make the uninformed concede with probability 1.
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Optimal Mechanism
Proposition: Given Assumption 2, in an optimal mechanism there
is no slack in each round of randomization except for at most one
round (and this round cannot be the first round).
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Optimal Mechanism
In the case with no slack in every round of randomization, the
following mechanism is optimal:
Set δ1 = δ3 = . . . = δt(r ∗ ) = ∆
Set δ2 = δ4 = . . . = δt(r ∗ )−1 = λ∆/(λ + ∆)
PT −1
Set t=t(r ∗ )+1 δt = λγ∗
Set δT = ∆
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Optimal Mechanism
xt = χ(γt ) in odd periods
xt = 0 in even periods
There is “stalling” in period t(r ∗ ) + 1 up to period T − 1.
There is a “last minute concession” with xT = 0 before the
deadline expires.
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Optimal Mechanism
For ∆ close to 0, payoff from the optimal mechanism is arbitrarily
close to payoff from a repeated voting game with deadline.
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