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Bargaining in Markets with One-Sided Competition: Model and Analysis Nicola Gatti DEI, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milano, 20133, Italy

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Nicola Gatti Results Overview The paper aims at studying models for bilateral negotiations with rational agents when they are in competition Modeling result: extension of the alternating-offers protocol to the situation wherein: There are more buyers and one seller, and The seller can exploit the outside-option Algorithmic result: development of an efficient solving algorithm for computing agents’ equilibrium strategies (this result is not discussed in the presentation) 2

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Nicola Gatti 3 Bilateral Negotiation (Bargaining) BuyerSeller 10$ 12$ 11$...

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Nicola Gatti 4 Alternating-Offers Protocol Game theoretical mechanism for bilateral negotiation [Rubinstein, Econometrica, 1982] A buyer and a seller wants to negotiate in an alternate-fashion with discrete time Someone decides the agent that starts to make offers Time discounts agents’ utilities (so agents prefer to reach agreement as soon as possible) Agents are supposed to be rational, i.e., they act aiming at maximizing their expected utility The solution is a pair of strategies, a buyer’s one and a seller’s one, that are in equilibrium (typically, refinement of Nash)

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Nicola Gatti 5 The Formal Model [Di Giunta and Gatti, 2006] Players Player function Actions Preferences

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Nicola Gatti Bargaining in Markets More buyers and more sellers are present in a market More buyers are in competition over the purchase of an item from the same seller More sellers are in competition over the sale of an item to the same buyer The alternating-offers protocol does not capture this competition We extend the alternating-offers protocol to capture: The matching between a buyer and a seller The possibility of leaving a negotiation to start a new one with a different commercial partner We consider a situation with a seller and more buyers 6

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Nicola Gatti Bilateral Negotiation with Outside-Option 7 Buyer 3 Seller 11$ Buyer 1 Buyer 4 Buyer 2 14$ 15$ accept

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Nicola Gatti 8 The Formal Model situationstageagentstime pointsavailable actions agent i is not negotiating 1buyeranymatchable, nonmatchable 2selleranymatch(b j ) if b j has made matchable agent i is negotiating 2buyeralternatelyoffer(x), accept, exit 2selleralternatelyoffer(x), accept, exit, match(b j ) is b j has made matchable Devoted to matching Devoted to negotiation

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Nicola Gatti Closing Remarks The proposed model allows one to effectively capture competition among agents: With one seller and more buyers, the seller gains drastically more than against the weakest buyer The computation of agents’ equilibrium strategies is easy (linear in the size of the problem) In future, we will investigate: The comparison between the agents’ expected utilities in our protocol with respect to the ones in multilateral negotiations The analysis of negotiations with uncertainty 9

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Nicola Gatti Questions to: 10

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