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 Alternating- Offers with Deadlines Players Player function Actions Preferences

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 {T b, T s } Solution construction 1.The deadline of the bargaining is determined 2.From the deadline backward induction construction is employed to determine agents’ equilibrium offers and acceptances

Backward Propagation x  3[b] x  2[b] xbxb x x  3[s] x  2[s] xsxs x tt-1t-2t-3tt-1t-2t-3

Backward Induction Construction (buyer) (seller) RP b RP s time price TbTb TsTs (RP s )  b (RP s )  bs (RP s )  bsb (RP s )  3[bs]b (RP s )  2[bs] (RP s )  2[bs]b (RP s )  3[bs] RP s 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)

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:

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  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 T b,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{T b,1, …, T b,m }, T s } 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 RP b 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: U s (e,t) = EU s (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 RP b 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

Agent s Acting in a Possible Deadline of Agent b (buyer) (seller) 1 0 time price T b,l TsTs 0 0b0b T b,e 0 e  sb e  sbs e(offer 0  b ) = 0·ω + (1 - ω) · (0  b ) e  2[sb] e  2[sb]s e  3[sb] e eses

Agent b Acting in a Possible Deadline of Her (buyer) (seller) 1 0 time price T b,l TsTs 0 0b0b T b,e  bs 0  bsb 1 0  b2[s] 0  2[bs] e e(offer 1) = 1·ω + (1 - ω) · (0  b2[s] ) b l construction b e construction 0  2[bs]b 0  3[bs] 0  3[bs]b e(offer 0  bsb ) = 0  bsb

Agent b Acting in a Possible Deadline of Her (buyer) (seller) 1 0 time price T b,l TsTs 0 0b0b T b,e  bs 0  bsb 1 0  b2[s] 0  bs2[b] eses e e(offer 1) = 1·ω + (1 - ω) · (0  b2[s] ) b l construction b e construction e  sb e  sbs e  2[sb] e(offer 0  bsb ) = 0  bsb

Agent b Acting in a Possible Deadline of Her (buyer) (seller) 1 0 time price T b,l TsTs 0 0b0b T b,e  bs 0  bsb 1 0  b2[s] b l construction b e construction eses e eses e 0  bs2[b]

The Equilibrium Assessment Theorem: If for all t such that  (t)=b holds U s (x*(t- 2),t-2) ≥ U s (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/n 2 ) of performing the equilibrium action, and (1/n 2 ) 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