Institute of Applied Mathematical Research, KarRC of the RAS, Russia
Equilibrium in the strategic bargaining
The Rubinstein’s bargaining model, proposed in 1982, provided a convenient tool for solving game-theoretic problems about bidding by two persons with alternating offers on an infinite time axis. The main feature was the discounting factor d, which did not allow the game to last indefinitely, i.e. the closer d is to 0, the more impatient the players are and the faster they will agree to any offer. On the contrary, if the value d is close to 1, then the players are patient and will negotiate until they come to the most favorable offer for them. In [Baron, Ferejohn, 1989], a model of sequential multilateral negotiations with a majority rule was proposed. The game that was reviewed is a standard game “split the dollar”, in which n players, whose turn is chosen randomly, make suggestions on how to divide a pie of a fixed size, and agreement requires the consent of a simple majority. It is shown that a sub-game perfect equilibrium exists in the class of stationary strategies. Then, articles [Eraslan, 2002; Cho, Duggan, 2003; Banks, Duggan, 2006; Predtetchinski, 2011; Cardona, Ponsati, 2007, 2011] were devoted to the expansion of this direction for different applied problems.
We present a game-theoretic model of competitive decision on a meeting time. There are n players who are negotiating the meeting time. The players' utilities are represented by linear unimodal functions. The maximum values of the utility functions are located at the points i/(n-1), i=0,...,n-1. The final decision will be done if all participants accept the proposal. Players take turns 1, then 2, then 3, …, n. We assume that after each negotiation session, the utility functions of all players will decrease proportionally to d. We will look for a solution in the class of stationary strategies, when it is assumed that the decisions of the players will not change during the negotiation time, i.e. the player i will make the same offer at step i and at subsequent steps n+i, 2n+i, etc. We use the backward induction method. To do this, assume that player n is looking for his best responce, knowing player 1's proposal, then player (n-1) is looking for his best responce, knowing player n's solution, etc. The subgame perfect equilibrium in the class of stationary strategies is found in analytical form.