alternating-offers bargaining under one-sided uncertainty on deadlines
DESCRIPTION
Alternating-Offers Bargaining under One-Sided Uncertainty on Deadlines. Francesco Di Giunta and Nicola Gatti Dipartimento di Elettronica e Informazione Politecnico di Milano, Milano, Italy. Summary. - PowerPoint PPT PresentationTRANSCRIPT
Alternating-Offers Bargaining under One-Sided Uncertainty on
DeadlinesFrancesco Di Giunta and
Nicola GattiDipartimento di Elettronica e
InformazionePolitecnico di Milano, Milano, Italy
Summary
We game-theoretically study alternating-offers protocol under one-sided uncertain deadlines (exclusively in pure strategies)
• Original contributions1. A method to find (when there are) the pure
equilibrium strategies given a natural system of beliefs
2. Proof of non-existence of the equilibrium strategies (in pure strategies) for some values of the parameters
Principal Works in Incomplete Information
Bargaining• Classic (theoretical) literature
• [Rubinstein, 1985] A bargaining model with incomplete information about time preferences
• No deadlines (uncertainty over discount factors)• [Chatterjee and Samuelson, 1988] Bargaining under
two-sided incomplete information: the unrestricted offers case
• No deadlines (uncertainty over reservation prices)
• Computer science literature• [Sandholm and Vulkan, 1999] Bargaining with
deadlines• Non alternating-offers protocol (war-of-attrition refinement)• Continuous time
• [Fatima et al., 2002] Multi-issue negotiation under time constraints
• Non perfectly rational agents (negotiation decision function paradigm based agents)
Revision of Complete Information Solution [Napel,
2002]
The Model of the Alternating-Offers with
Deadlines• Players
• Player function
• Actions
• Preferences
)(
)(
sellers
buyerb
)1()(
)0(
tt
i
exit
accept
xoffer )(
s
st
sss
b
bt
bbb
sb
Tt
TtRPxtxU
Tt
TtxRPtxU
tNoAgreemenUtNoAgreemenU
1
)()(),(
1
)()(),(
0)()(
Complete Information Solution
• Equilibrium notion• Subgame Perfect Equilibrium [Selten, 1972], it defines
the equilibrium strategies of any agent in any possible reachable subgame
• Backward induction• The game is not rigorously a finite horizon game• However, no rational agent will play after his deadline• Therefore, there is a point from which we can build
backward induction construction• We call it the deadline of the bargaining T• It is: T = min {Tb, Ts}
• Solution construction1. The deadline of the bargaining is determined2. From the deadline backward induction construction is
employed to determine agents’ equilibrium offers and acceptances
Backward Propagation
)( bbbb RVxRVx )( ssss RVxRVx
x3[b]
x2[b]
xb
x x3[s]
x2[s]
xs
x
tt-1t-2t-3 tt-1t-2t-3
),()1,(
),()1,(:
txtx
txUtzUzx
ii
iii
Backward Induction Construction
(buyer)(buyer)(buyer)(buyer)(buyer) (seller)(seller)(seller)(seller)(seller)(seller)
RPb
RPs
time
pri
ce
Tb Ts
(RPs)b
(RPs)bs
(RPs)bsb
(RPs)3[bs]b
(RPs)2[bs]
(RPs)2[bs]b
(RPs)3[bs]
RPs RPs
Infinite Horizon Construction
Finite Horizon Construction
Equilibrium Strategies
• We call x*(t) the offers found by backward induction for each time point t
• Equilibrium strategies are expressed in function of x*(t)
otherwise)(*
)1(* with )()1( if
)(
otherwise)(*
)1(* with )()1( if
)(
*
*
s
s
b
s
b
b
s
b
Ttexit
Tttxoffertxxxoffertaccept
t
Ttexit
Tttxoffertxxxoffertaccept
t
One-Sided Uncertainty Over Deadlines Solution
(exclusively with pure strategies)
The Model Concerning Uncertain Deadlines
• We consider the situation in which buyer’s deadline is uncertain
• The seller has an initial belief concerning buyer’s deadline: a finite probability distribution over the buyer’s possible deadlines
• Formally:
00
0,
01,
0
,1,
,
},,{
},,{
bbb
mbbb
mbbb
PTBT
P
TTT
0bBT
0bP
bT
Equilibrium of a Imperfect Information Extensive Form
Game• Assessment (µ, )
• System of beliefs µ that defines the agents’ beliefs in each information set
• Equilibrium strategies that defines the agents’ action in each information set
• Equilibrium assessment• Equilibrium strategies are sequentially
rational given the system of beliefs µ• System of beliefs are somehow “consistent”
with equilibrium strategies µ
Notions of Equilibrium
• Weak Sequential Equilibrium (WSE) [Fudenberg and Tirole, 1991] • Consistency is given by Bayes consistency on
the equilibrium path, nothing can be said off equilibrium path, being Bayes rule not applicable
• Sequential Equilibrium (SE) [Kreps and Wilson, 1982]• Provide a criterion to analyse off-equilibrium-
path consistency• The consistency is given by the existence of a
sequence of completely behavioural assessment that converges to the equilibrium assessment
The Basis of Our Method
• The method1. We fix a (natural) system of beliefs 2. We use backward induction together with
the considered system of beliefs to determine (if there is any) the sequentially rational strategies
3. We prove a posteriori the consistency (of Kreps and Wilson)
• The considered system of beliefs• Once a possible deadline Tb,i is expired, it is
removed from the seller’s beliefs and the probabilities are normalized by Bayes rule
Backward Induction with
• The time point from which employing backward induction is T = min{ max{Tb,1, …, Tb,m}, Ts}
• Seller’s optimal offer• In complete information, it is the backward
propagation of the next buyer’s optimal offer• Under uncertainty, if the next time point is a possible
buyer’s deadline, the seller could offer RPb
• Seller’s acceptance• In complete information, it is the backward
propagation of the seller’s optimal offer• Under uncertainty, as the seller optimal offer could be
rejected, she will accept an offer lower than the backward propagation of her optimal offer
Backward Induction with
• Defining • Equivalent price e of an offer x: Us(e,t) = EUs(x,t)• Deadline function d(t): the probability, given at time t
according to , that time t is a deadline for the buyer • We summarize
• Seller’s optimal offer: the offer with the highest equivalent price between RPb and the backward propagation of the optimal offer of the buyer at the next time point
• Seller’s optimal acceptance: the backward propagation of the equivalent price of the seller’s optimal offer
• Expected utilities
)1,)1(*()(1),)1(*(
)1),1(*()1(1)1,()1()(1),(
tteUtdtteEU
tteUtdtRPUtdtdtRPEU
bsbs
sbsbs
Agent s Acting in a Possible Deadline of Agent b
(buyer)(buyer)(buyer)(buyer)(buyer) (seller)(seller)(seller)(seller)(seller)(seller)
1
0
time
pri
ce
Tb,l Ts
0
0b
Tb,e
0
esb
esbse(offer 0b) = 0·ω + (1 - ω) · (0b)
e2[sb]
e2[sb]s
e3[sb]
e
es
Agent b Acting in a Possible Deadline of Her
(buyer)(buyer)(buyer)(buyer)(buyer) (seller)(seller)(seller)(seller)(seller)(seller)
1
0
time
pri
ce
Tb,l Ts
0
0b
Tb,e
1
0
0bs
0bsb
1
0b2[s]
02[bs]
ee(offer 1) = 1·ω + (1 - ω) · (0b2[s])
bl construction
be construction
02[bs]b
03[bs]
03[bs]b
e(offer 0bsb) = 0bsb
Agent b Acting in a Possible Deadline of Her
(buyer)(buyer)(buyer)(buyer)(buyer) (seller)(seller)(seller)(seller)(seller)(seller)
1
0
time
pri
ce
Tb,l Ts
0
0b
Tb,e
1
0
0bs
0bsb
1
0b2[s]
0bs2[b]
es
ee(offer 1) = 1·ω + (1 - ω) · (0b2[s])
bl construction
be constructionesb
esbs
e2[sb]
e(offer 0bsb) = 0bsb
Agent b Acting in a Possible Deadline of Her
(buyer)(buyer)(buyer)(buyer)(buyer) (seller)(seller)(seller)(seller)(seller)(seller)
1
0
time
pri
ce
Tb,l Ts
0
0b
Tb,e
1
0
0bs
0bsb
1
0b2[s]bl construction
be construction
es
ees
e
0bs2[b]
The Equilibrium Assessment
• Theorem: If for all t such that (t)=b holds Us(x*(t-2),t-2) ≥ Us(x*(t),t), then the considered assessment is a sequential equilibrium
• The consistency proof can be derived from the following fully behavioural strategy:• Seller and any buyer’s types before their
deadlines: probability (1-1/n) of performing the equilibrium action, and (1/n) uniformly distributed among the other actions
• Buyer’s types after their deadlines: probability (1-1/n2) of performing the equilibrium action, and (1/n2) uniformly distributed among the other actions
Equilibrium Non-Existence Theorem
• Theorem: Alternating-offers bargaining with uncertain deadlines does not admit always a sequential equilibrium in pure strategies
• The proof reported in the paper• Is (partially) independent from the system of
beliefs• Assumes (only) that after a deadline, such a
deadline is removed from the seller’s beliefs
• It can be proved that the non-existence theorem holds for any system of beliefs, removing the above assumption
Conclusions and Future Works
• Conclusions• We have studied the alternating-offers
bargaining under one-sided uncertain deadlines• We provide method to find equilibrium pure
strategies when they exist• We prove that for some values of the parameters
it does not admit any sequential equilibrium in pure strategies
• Future works• Introduction of an equilibrium behavioural
strategy (which theory assures to exist) to address the equilibrium non-existence in pure strategies
• Study of two-sided uncertainty on deadlines and of other kind of uncertainty