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Alternating-Offers Bargaining under One-Sided Uncertainty on Deadlines

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.

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Alternating-Offers Bargaining under One-Sided Uncertainty on Deadlines

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  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)

  4. Revision of Complete Information Solution [Napel, 2002]

  5. The Model of the Alternating-Offers with Deadlines • Players • Player function • Actions • Preferences

  6. Complete Information Solution • Equilibrium notion • Subgame Perfect Equilibrium[Selten, 1972], it defines the equilibrium strategies of anyagent 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

  7. x3[b] x2[b] xs xb x2[s] x3[s] Backward Propagation x x t-3 t-2 t-1 t t-3 t-2 t-1 t

  8. time Tb Ts RPb price RPs (buyer) (seller) (buyer) (seller) (buyer) (seller) (buyer) (seller) (buyer) (seller) (seller) Backward Induction Construction 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)bs RPs RPs

  9. 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)

  10. One-Sided Uncertainty Over Deadlines Solution (exclusively with pure strategies)

  11. 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 probabilitydistribution over the buyer’s possibledeadlines • Formally:

  12. Equilibrium of a Imperfect Information Extensive Form Game • Assessment(µ, ) • System of beliefs µthat 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

  13. 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

  14. 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 seller’s beliefs and the probabilities are normalized by Bayes rule

  15. 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

  16. Backward Induction with m (2) • Defining • Equivalent price e of an offer x: Us(e,t) = EUs(x,t) • Deadline functiond(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

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

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

  19. time 1 price 0 0 0 (buyer) (seller) (buyer) (seller) (buyer) (seller) (buyer) (seller) (buyer) (seller) (seller) Agent b Acting in a Possible Deadline of Her Tb,e Tb,l Ts be construction 1 1 e2[sb] esb e(offer 1) = 1·ω + (1 - ω) · (0b2[s]) e 0bs2[b] esbs es 0bsb e(offer 0bsb) = 0bsb bl construction 0b 0bs 0b2[s]

  20. time 1 e e es es price 0 0 0 (buyer) (seller) (buyer) (seller) (buyer) (seller) (buyer) (seller) (buyer) (seller) (seller) Agent b Acting in a Possible Deadline of Her Tb,e Tb,l Ts be construction 1 1 0bs2[b] 0bsb bl construction 0b 0bs 0b2[s]

  21. 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 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

  22. 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

  23. 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

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