Alternating offers protocol

An alternating-offers protocol (AOP, also known as alternate-offers protocol or alternate-moves protocol) is a procedure for negotiation and bargaining.

Consider two people who want to arrive at an agreement. There is a fixed set of possible agreements. In an AOP, each person in turn offers one of the possible agreements. The other person then either accepts the offer (in which case the negotiation ends), or makes a counter-offer.

Often, the protocol rules disallow to offer the same agreement twice. Hence, if the number of possible agreements is finite, at some point all them are exhausted. In this case, the negotiation ends without an agreement.

Game-theoretic analysis

An AOP induces a sequential game. A natural question is: what outcomes are a Subgame perfect equilibrium (SPE) of this game? This question has been studied in various settings.

Dividing a Dollar

Ariel Rubinstein studied a setting in which the negotiation is on how to divide $1 between the two players.[1] Each player in turn can offer any partition. The players bear a cost for each round of negotiation. The cost can be presented in two ways:

  1. Additive cost: the cost of each player i is ci per round. Then, if c1 < c2, the only SPE gives all the $1 to player 1; if c1 > c2, the only SPE gives $c2 to player 1 and $1-c2 to player 2.
  2. Multiplicative cost: each player has a discount factor di. Then, the only SPE gives $(1-d2)/(1-d1d2) to player 1.

Finite set of agreements

Nejat Anbarci studied a setting with a finite number of outcomes, where the protocol rules disallow to repeat the same offer twice.[2] In any such game, there is a unique SPE. It is always Pareto optimal; it is always one of the two Pareto-optimal options of which rankings by the players are the closest. It can be found by finding the smallest integer k for which the sets of k best options of the two players have non-empty intersection. For example, if the rankings are a>b>c>d and c>b>a>d, then the unique SPE is b (with k=2). If the rankings are a>b>c>d and d>c>b>a, then the SPE is either b or c (with k=3).

The unique SPE outcome converges to the Area-Monotonic-Solution if the options are uniformly distributed over the bargaining set and their number approaches infinity.

For more references, see [3] and.[4]

References
  1. Rubinstein, Ariel (1982). "Perfect Equilibrium in a Bargaining Model". Econometrica. 50 (1): 97–109. CiteSeerX 10.1.1.295.1434. doi:10.2307/1912531. JSTOR 1912531.
  2. Anbarci, N. (1993-02-01). "Noncooperative Foundations of the Area Monotonic Solution". The Quarterly Journal of Economics. 108 (1): 245–258. doi:10.2307/2118502. ISSN 0033-5533. JSTOR 2118502.
  3. Anbarci, Nejat (2006-08-01). "Finite Alternating-Move Arbitration Schemes and the Equal Area Solution". Theory and Decision. 61 (1): 21–50. doi:10.1007/s11238-005-4748-9. ISSN 0040-5833.
  4. Erlich, Sefi; Hazon, Noam; Kraus, Sarit (2018-05-02). "Negotiation Strategies for Agents with Ordinal Preferences". arXiv:1805.00913 [cs.GT].
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