Sequential bargaining

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Sequential bargaining (also known as alternate-moves bargaining, alternating-offers protocol, etc.) is a structured form of bargaining between two participants, in which the participants take turns in making offers. Initially, person #1 has the right to make an offer to person #2. If person #2 accepts the offer, then an agreement is reached and the process ends. If person #2 rejects the offer, then the participants switch turns, and now it is the turn of person #2 to make an offer (which is often called a counter-offer). The people keep switching turns until either an agreement is reached, or the process ends with a disagreement due to a certain end condition. Several end conditions are common, for example:

Contents

Several settings of sequential bargaining have been studied.

Game-theoretic analysis

An alternating-offers protocol induces a sequential game. A natural question is what outcomes can be attained in an equilibrium of this game. At first glance, the first player has the power to make a very favorable offer. For example, in the Dividing the Dollar game, player #1 can offer to give only 1% of the money to player #2, and threaten that "if you do not accept, I will refuse all offers from now on, and both of us will get 0". But this is a non-credible threat, since if player #2 refuses and makes a counter-offer (e.g. give 2% of the money to player #1), then it is better for player #1 to accept. Therefore, a natural question is: what outcomes are a subgame perfect equilibrium (SPE) of this game? This question has been studied in various settings.

Dividing the 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.

Rubinstein and Wolinsky [2] studied a market in which there are many players, partitioned into two types (e.g. "buyers" and "sellers"). Pairs of players of different types are brought together randomly, and initiate a sequential-bargaining process over the division of a surplus (as in the Divide the Dollar game). If they reach an agreement, they leave the market; otherwise, they remain in the market and wait for the next match. The steady-state equilibrium in this market it is quite different than competitive equilibrium in standard markets (e.g. Fisher market or Arrow–Debreu market).

Buyer and seller

Fudenberg and Tirole [3] study sequential bargaining between a buyer and a seller who have incomplete information, i.e., they do not know the valuation of their partner. They focus on a two-turn game (i.e., the seller has exactly two opportunities to sell the item to the buyer). Both players prefer a trade today than the same trade tomorrow. They analyze the Perfect Bayesian equilibrium (PBE) in this game, if the seller's valuation is known, then the PBE is generically unique; but if both valuations are private, then there are multiple PBE. Some surprising findings, that follow from the information transfer and the lack of commitment, are:

Grossman and Perry [4] study sequential bargaining between a buyer and a seller over an item price, where the buyer knows the gains-from-trade but the seller does not. They consider an infinite-turn game with time discounting. They show that, under some weak assumptions, there exists a unique perfect sequential equilibrium, in which:

General outcome set

Nejat Anbarci [5] studied a setting with a finite number of outcomes, where each of the two agents may have a different preference order over the outcomes. The protocol rules disallow repeating the same offer twice. 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 a 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).

In a later study, Anbarci [6] studies several schemes for two agents who have to select an arbitrator from a given set of candidates:

In all schemes, if the options are uniformly distributed over the bargaining set and their number approaches infinity, then the unique SPE outcome converges to the Equal-Area solution of the cooperative bargaining problem.

Erlich, Hazon and Kraus [7] study the Alternating Offers protocol in several informational settings:

Experimental analysis

Laboratory studies

The Dividing-the-Dollar game has been studied in several laboratory experiments. In general, subjects behave quite differently from the unique SPE. Subjects' behavior depends on the number of turns, their experience with the game, and their beliefs about fairness. For details of specific experiments, see:

See also the survey by Güth and Tietz [13] at 1990.

Field study

A field study was done by Backus, Blake, Larsen and Tadelis [14] at 2020. They studied back-and-forth sequential bargaining in over 25 million listings from the Best Offer platform of eBay. Their main findings are:

They also report some findings that cannot be rationalized by the existing theories:

They suggest that these findings can be explained by behavioral norms.

Further reading

See also

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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. S2CID   14827857.
  2. ↑ Rubinstein, Ariel; Wolinsky, Asher (1985). "Equilibrium in a Market with Sequential Bargaining". Econometrica. 53 (5): 1133–1150. doi:10.2307/1911015. ISSN   0012-9682. JSTOR   1911015.
  3. ↑ Fudenberg, Drew; Tirole, Jean (1983). "Sequential Bargaining with Incomplete Information". The Review of Economic Studies. 50 (2): 221–247. doi:10.2307/2297414. ISSN   0034-6527. JSTOR   2297414.
  4. ↑ Grossman, Sanford J; Perry, Motty (1986-06-01). "Sequential bargaining under asymmetric information". Journal of Economic Theory. 39 (1): 120–154. doi:10.1016/0022-0531(86)90023-2. ISSN   0022-0531. S2CID   154201801.
  5. ↑ 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.
  6. ↑ 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. S2CID   122355062.
  7. ↑ Erlich, Sefi; Hazon, Noam; Kraus, Sarit (2018-05-02). "Negotiation Strategies for Agents with Ordinal Preferences". arXiv: 1805.00913 [cs.GT].
  8. ↑ Güth, Werner; Schmittberger, Rolf; Schwarze, Bernd (1982-12-01). "An experimental analysis of ultimatum bargaining". Journal of Economic Behavior & Organization. 3 (4): 367–388. doi:10.1016/0167-2681(82)90011-7. ISSN   0167-2681.
  9. ↑ Binmore, K.; Shaked, A.; Sutton, J. (1985). "Testing Noncooperative Bargaining Theory: A Preliminary Study". The American Economic Review. 75 (5): 1178–1180. ISSN   0002-8282. JSTOR   1818658.
  10. ↑ Güth, Werner; Tietz, Reinhard (1988). Tietz, Reinhard; Albers, Wulf; Selten, Reinhard (eds.). "Ultimatum Bargaining for a Shrinking Cake — An Experimental Analysis —". Bounded Rational Behavior in Experimental Games and Markets. Lecture Notes in Economics and Mathematical Systems. Berlin, Heidelberg: Springer. 314: 111–128. doi:10.1007/978-3-642-48356-1_9. ISBN   978-3-642-48356-1.
  11. ↑ Neelin, Janet; Sonnenschein, Hugo; Spiegel, Matthew (1988). "A Further Test of Noncooperative Bargaining Theory: Comment". The American Economic Review. 78 (4): 824–836. ISSN   0002-8282. JSTOR   1811179.
  12. ↑ Ochs, Jack; Roth, Alvin E. (1989). "An Experimental Study of Sequential Bargaining". The American Economic Review. 79 (3): 355–384. ISSN   0002-8282. JSTOR   1806850.
  13. ↑ Güth, Werner; Tietz, Reinhard (1990-09-01). "Ultimatum bargaining behavior: A survey and comparison of experimental results". Journal of Economic Psychology. 11 (3): 417–449. doi:10.1016/0167-4870(90)90021-Z. ISSN   0167-4870.
  14. ↑ Backus, Matthew; Blake, Thomas; Larsen, Brad; Tadelis, Steven (2020-08-01). "Sequential Bargaining in the Field: Evidence from Millions of Online Bargaining Interactions". The Quarterly Journal of Economics. 135 (3): 1319–1361. doi:10.1093/qje/qjaa003. ISSN   0033-5533.
  15. ↑ "Game-Theoretic Models of Bargaining | Microeconomics". Cambridge University Press. Retrieved 2021-02-05.
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