Extending the alternating-offers protocol in the presence of competition: models and theoretical analysis

The automation of bargaining is receiving a lot of attention in artificial intelligence research. Indeed, considering that bargaining is the most common form of economic transaction, its automation could lead software agents to reach more effective agreements. In the present paper we focus on the best-known bargaining protocol, i.e., the alternating-offers protocol. It provides an elegant mechanism whereby a buyer and a seller can bilaterally bargain. Although this protocol and its refinements have been studied extensively, no work up to the present provides an adequate model for bargaining in electronic markets. A result of these settings means that multiple buyers are in competition with each other for the purchase of a good from the same seller while, analogously, multiple sellers are in competition with each other for the sale of a good to the same buyer. The study of these settings is of paramount importance, as they will be commonplace in real-world applications. In the present paper we provide a model that extends the alternating-offers protocol to include competition among agents.1 Our game theoretical analysis shows that the proposed model is satisfactory: it effectively captures the competition among agents, equilibrium strategies are efficiently computable, and the equilibrium outcome is unique. The main results we achieve are the following. 1) With m buyers and n sellers and when the outside option (i.e., the possibility of leaving a negotiation to start a new one) is inhibited, we show that it can be reduced to a problem of matching and that can be addressed by using the Gale-Shapley’s stable marriage algorithm. The equilibrium outcome is unique and can be computed in $O(l \cdot m\cdot n \cdot \overline T + (m+n)^2)$, where l is the number of the issues and $\overline{T}$ is the maximum length of the bargaining. 2) With m buyers and one seller and when the seller can exploit the outside option, we show that agents’ equilibrium strategies can be computed in $O(l \cdot m \cdot \overline{T})$ and may be not unique. However, we show that a simple refinement of the agents’ utility functions leads to equilibrium uniqueness.