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
We game-theoretically study alternating-offers
protocol under one-sided uncertain deadlines
(exclusively in pure strategies)
• Original contributions
1. 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 AlternatingOffers with Deadlines
• Players
b (buyer )

 s ( seller )
• Player function
 (0)  i

 (t )   (t  1)
• Actions
• Preferences
offer( x )

accept
exit

U b ( NoAgreement )  U s ( NoAgreement )  0
( RPb  x)  ( b ) t
U b ( x, t )  
1

( x  RPs )  ( s ) t
U s ( x, t )  
1

t  Tb
t  Tb
t  Ts
t  Ts
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 construction
1.
2.
The deadline of the bargaining is determined
From the deadline backward induction construction is employed to
determine agents’ equilibrium offers and acceptances
Backward Propagation
xi  z : U i ( z, t  1)  U i ( x, t )
( xi , t  1) i ( x, t )
xb  RVb   b  ( x  RVb )
xs  RVs   s  ( x  RVs )
x3[b]
x
x2[b]
x s
x b
x2[s]
x
t-3
t-2
t-1
t
x3[s]
t-3
t-2
t-1
t
Backward Induction Construction
time
Tb
price
RPb
Ts
Infinite Horizon Construction
(RPs)3[bs]b
(RPs)2[bs]b
(RPs)bsb
(RPs)b
(RPs)3[bs]
(RPs)2[bs]
Finite Horizon Construction
RPs
(seller)
(buyer)
(seller)
(buyer)
(seller)
(RPs)bs
(buyer)
RPs
(seller)
(buyer)
RPs
(seller)
(buyer)
(seller)
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)
accept


 b* (t )  offerx * (t ) 

exit

accept


 s* (t )  offerx * (t ) 

exit

t  Tb
t  Tb
t  Ts
t  Ts
if  s (t  1)  offer( x) with x  x * (t  1)
otherwise
if  b (t  1)  offer( x) with x  x * (t  1)
otherwise
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 BTb0 concerning
buyer’s deadline: a finite probability distribution Pb0
over the buyer’s possible deadlines Tb
• Formally: Tb  {Tb ,1 ,, Tb ,m }
Pb0  { b0,1 ,  ,  b0,m }
BTb0  Tb , Pb0
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 method
1. We fix a (natural) system of beliefs m
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 m 1
• 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 m 2
• 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 m, 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
EU s ( RPb , t )  1  d (t )   d (t  1) U s ( RPb , t  1)  1  d (t  1)  U s (e * (t  1), t  1) 
EU s (e * (t  1) b , t )  1  d (t )  U s (e * (t  1) b , t  1)
Agent s Acting in a Possible
time Deadline of Agent b
Tb,e
price
1
Tb,l
Ts
e3[sb]
e2[sb]
esb
0 b
e(offer 0b) = 0·ω + (1 - ω) · (0b)
esbs
e
e2[sb]s
e s
0
(seller)
(buyer)
(seller)
(buyer)
(seller)
(buyer)
0
(seller)
(buyer)
0
(seller)
(buyer)
(seller)
Agent b Acting in a Possible
Deadline of Her
time
Tb,e
1
Tb,l
Ts
1
1
be construction
03[bs]b
price
02[bs]b
e(offer 0bsb) = 0bsb 0bsb
e(offer 1) = 1·ω + (1 - ω) · (0b2[s])
03[bs]
02[bs]
e
0 b
0bs
bl construction
0b2[s]
0
(seller)
0
(buyer)
(seller)
(buyer)
(seller)
(buyer)
(seller)
(buyer)
0
(seller)
(buyer)
(seller)
Agent b Acting in a Possible
Deadline of Her
time
Tb,e
1
esb
e(offer 1) = 1·ω + (1 - ω) · (0b2[s])
Ts
1
1
e2[sb]
Tb,l
be construction
e
0bs2[b]
esbs
e s
price
e(offer 0bsb) = 0bsb 0bsb
0 b
0bs
bl construction
0b2[s]
0
(seller)
0
(buyer)
(seller)
(buyer)
(seller)
(buyer)
(seller)
(buyer)
0
(seller)
(buyer)
(seller)
Agent b Acting in a Possible
Deadline of Her
time
Tb,e
1
Tb,l
Ts
1
1
e
be construction
e s
e
0bs2[b]
e s
price
0bsb
0 b
0bs
bl construction
0b2[s]
0
(seller)
0
(buyer)
(seller)
(buyer)
(seller)
(buyer)
(seller)
(buyer)
0
(seller)
(buyer)
(seller)
The Equilibrium Assessment
• Theorem: If for all t such that (t)=b holds Us(x*(t2),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 (11/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