AlternatingOffers Bargaining under OneSided Uncertainty on Deadlines

1
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

2
Summary

• We game-theoretically study alternating-offers
  protocol under one-sided uncertain deadlines
  (exclusively in pure strategies)
• Original contributions
• A method to find (when there are) the pure
  equilibrium strategies given a natural system of
  beliefs
• Proof of non-existence of the equilibrium
  strategies (in pure strategies) for some values
  of the parameters

3
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)

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Revision of Complete Information Solution Napel,
2002
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The Model of the Alternating-Offers with Deadlines

• Players
• Player function
• Actions
• Preferences

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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
• The deadline of the bargaining is determined
• From the deadline backward induction construction
  is employed to determine agents equilibrium
  offers and acceptances

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Backward Propagation
x
x
t
t-1
t-2
t-3
t
t-1
t-2
t-3
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Backward Induction Construction
Infinite Horizon Construction
(RPs)?3bsb
(RPs)?2bsb
(RPs)?bsb
(RPs)?b
(RPs)?3bs
(RPs)?2bs
Finite Horizon Construction
(RPs)?bs
RPs
RPs
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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)

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One-Sided Uncertainty Over Deadlines Solution
(exclusively with pure strategies)
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The Model Concerning Uncertain Deadlines

• We consider the situation in which buyers
  deadline is uncertain
• The seller has an initial belief
  concerning buyers deadline a finite probability
  distribution over the buyers possible deadlines
• Formally

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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

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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-pat
  h consistency
• The consistency is given by the existence of a
  sequence of completely behavioural assessment
  that converges to the equilibrium assessment

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The Basis of Our Method

• The method
• We fix a (natural) system of beliefs m
• We use backward induction together with the
  considered system of beliefs to determine (if
  there is any) the sequentially rational
  strategies
• 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 sellers beliefs and the
  probabilities are normalized by Bayes rule

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Backward Induction with m (1)

• The time point from which employing backward
  induction is T min maxTb,1, , Tb,m, Ts
• Sellers optimal offer
• In complete information, it is the backward
  propagation of the next buyers optimal offer
• Under uncertainty, if the next time point is a
  possible buyers deadline, the seller could offer
  RPb
• Sellers acceptance
• In complete information, it is the backward
  propagation of the sellers 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

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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
• Sellers 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
• Sellers optimal acceptance the backward
  propagation of the equivalent price of the
  sellers optimal offer
• Expected utilities

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Agent s Acting in a Possible Deadline of Agent b
Tb,l
Ts
Tb,e
e?3sb
e?2sb
0?b
e?sb
e?2sbs
e(offer 0?b) 0? (1 - ?) (0?b)
e?sbs
e
e?s
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Agent b Acting in a Possible Deadline of Her
Tb,l
Ts
Tb,e
be construction
1
1
0?3bsb
0?2bsb
0?bsb
e(offer 0?bsb) 0?bsb
e(offer 1) 1? (1 - ?) (0?b2s)
bl construction
e
0?3bs
0?b
0?2bs
0?bs
0?b2s
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Agent b Acting in a Possible Deadline of Her
Tb,l
Ts
Tb,e
be construction
1
1
e?2sb
e?sb
e(offer 1) 1? (1 - ?) (0?b2s)
e
0?bs2b
e?sbs
e?s
0?bsb
e(offer 0?bsb) 0?bsb
bl construction
0?b
0?bs
0?b2s
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Agent b Acting in a Possible Deadline of Her
Tb,l
Ts
Tb,e
be construction
1
1
0?bs2b
0?bsb
bl construction
0?b
0?bs
0?b2s
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The Equilibrium Assessment

• Theorem If for all t such that i(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 buyers types before their
  deadlines probability (1-1/n) of performing the
  equilibrium action, and (1/n) uniformly
  distributed among the other actions
• Buyers types after their deadlines probability
  (1-1/n2) of performing the equilibrium action,
  and (1/n2) uniformly distributed among the other
  actions

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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 sellers beliefs
• It can be proved that the non-existence theorem
  holds for any system of beliefs, removing the
  above assumption

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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