Bargaining and boldness.

*(English)*Zbl 1023.91003
Games Econ. Behav. 38, No. 1, 28-51 (2002); corrigendum ibid. 41, No. 1, 165-168 (2002).

The author studies a multiperson bargaining problem described in the form of the following Shaked’s game: There are \(n\geq 2\) bargainers (players). In each period \(t=1,2,\dots\) , player \(i=t\pmod n\) submits a nonnegative vector proposal \(x_t = (x_t^1,x_t^2,\dots , x_t^n)\) satisfying \(\sum_{j=1}^n x_t^j =1\). The other players \(j \neq i\), either accept or reject that proposal. If all of them accept \(x_t\), then the game concludes and each player \(k\) receives the share \(x_t^k\). In other case, two subcases can occur:

(1) with a constant probability \(1-\rho\) the game concludes with zero proposal \(x_t = (0,0,\dots , 0)\); (2) with probability \(\rho\) the play passes to the next period \(t+1\), and player \(j=t+1\pmod n\) becomes a new proposer, and so on (here \(\rho\) is fixed for all periods). The payoff of a player \(k=1,2,\cdots , n\) in the game is \(V^k(\rho^s,x_s^k)\), where \(V_k\) are continuous functions, and \(s\) is the period in which the proposal \(x_s\) has been accepted by all the players.

The game is studied in context of the existence and uniqueness of Stationary Subgame-Perfect Equilibria (SSPE), and their form is considered. Also the limit behavior of SSPE as \(\rho \rightarrow 1\), and their limit outcomes (called equally marginally bold) are analyzed. The paper ends with a comparison of such outcomes with Nash outcomes.

(1) with a constant probability \(1-\rho\) the game concludes with zero proposal \(x_t = (0,0,\dots , 0)\); (2) with probability \(\rho\) the play passes to the next period \(t+1\), and player \(j=t+1\pmod n\) becomes a new proposer, and so on (here \(\rho\) is fixed for all periods). The payoff of a player \(k=1,2,\cdots , n\) in the game is \(V^k(\rho^s,x_s^k)\), where \(V_k\) are continuous functions, and \(s\) is the period in which the proposal \(x_s\) has been accepted by all the players.

The game is studied in context of the existence and uniqueness of Stationary Subgame-Perfect Equilibria (SSPE), and their form is considered. Also the limit behavior of SSPE as \(\rho \rightarrow 1\), and their limit outcomes (called equally marginally bold) are analyzed. The paper ends with a comparison of such outcomes with Nash outcomes.

Reviewer: Tadeusz Radzik (Jelenia Gora)

##### MSC:

91A20 | Multistage and repeated games |

91A10 | Noncooperative games |

91B52 | Special types of economic equilibria |

91A06 | \(n\)-person games, \(n>2\) |

PDF
BibTeX
XML
Cite

\textit{A. Burgos} et al., Games Econ. Behav. 38, No. 1, 28--51 (2002; Zbl 1023.91003)

Full Text:
DOI

##### References:

[1] | Allias, M., Le comportement de I’homme rationnel devant le risque, critique des postulats et axiomes de l’école américaine, Econometrica, 21, 503-546, (1953) · Zbl 0050.36801 |

[2] | Aumann, R.; Kurz, M., Power and taxes, Econometrica, 45, 1137-1161, (1977) · Zbl 0367.90016 |

[3] | Camerer, C., Individual decision making, () · Zbl 0942.91021 |

[4] | Chew, S.H., A generalization of the quasilinear Mean with appliations to the measurement of income inequality and decision theory resolving the allias paradox, Econometrica, 51, 1061-1092, (1983) |

[5] | Chew, S.H.; Karni, E.; Safra, Z., Risk aversion in the theory of expected utility with rank dependent probabilities, J. econ. theory, 42, 370-381, (1987) · Zbl 0632.90007 |

[6] | Cubitt, R.P.; Starmer, C.; Sugden, R., Dynamic choice and the common ratio effect: an experimental investigation, Econ. J., 108, 1362-1380, (1998) |

[7] | Grant, S, and, Kajii, A. 1994, Bargaining, Boldness and Nash Outcomes, University of Pennsylvania, CARESS, Working Paper #94-03. |

[8] | Grant, S.; Kajii, A., A cardinal characterization of the rubinstein – safra – thomson axiomatic bargaining theory, Econometrica, 63, 1241-1249, (1995) · Zbl 0837.90130 |

[9] | Gul, F., A theory of disappointment aversion, Econometrica, 59, 667-686, (1991) · Zbl 0744.90005 |

[10] | Hanany, E.; Safra, Z., Reformulation of the bargaining problem with induced utilities, J. econ. theory, 90, 254-276, (2000) |

[11] | Houba, H.; Tieman, X.; Brinksma, R., The Nash bargaining solution for decision weight utility functions, Econ. letters, 60, 41-48, (1998) · Zbl 0922.90149 |

[12] | Kahneman, D.; Tversky, A., Prospect theory: an analysis of decision under risk, Econometrica, 47, 263-291, (1979) · Zbl 0411.90012 |

[13] | Karni, E.; Safra, Z., Ascending bid auctions with behaviorally consistent bidders, Ann. oper. res., 19, 435-446, (1989) · Zbl 0707.90029 |

[14] | Machina, M.J., Choice under uncertainty: problems solved and unsolved, J. econ. perspect., 1, 121-154, (1987) |

[15] | Machina, M.J., Dynamic consistency and non-expected utility models of choice under uncertainty, J. econ. lit., 27, 1622-1668, (1989) |

[16] | Merlo, A.; Wilson, C., A stochastic model of sequential bargaining with complete information, Econometrica, 63, 371-399, (1995) · Zbl 0834.90142 |

[17] | Milnor, J.W., Topology from the differentiable viewpoint, (1965), The University Press of Virginia Charlottesville · Zbl 0136.20402 |

[18] | Nash, J.F., The bargaining problem, Econometrica, 18, 155-162, (1950) · Zbl 1202.91122 |

[19] | Nash, J.F., Two-person cooperative games, Econometrica, 21, 128-140, (1953) · Zbl 0050.14102 |

[20] | Osborne, M.J.; Rubinstein, A., Bargaining and markets, (1990), Academic Press San Diego · Zbl 0790.90023 |

[21] | Osborne, M.J.; Rubinstein, A., A course in game theory, (1994), MIT Press Cambridge · Zbl 1194.91003 |

[22] | Puppe, C., Distorted probabilities and choice under risk, (1991), Springer-Verlag New York · Zbl 0759.90005 |

[23] | Quiggen, J., A theory of “anticipated” utility, J. econ. behavior organ., 3, 323-343, (1982) |

[24] | Rubinstein, A.; Safra, Z.; Thomson, W., On the interpretation of the Nash bargaining solution and its extension to non-expected utility preferences, Econometrica, 60, 1171-1186, (1992) · Zbl 0767.90094 |

[25] | Safra, Z.; Zilcha, I., Bargaining solutions without the expected utility hypothesis, Games econ. behavior, 5, 288-306, (1993) · Zbl 0776.90094 |

[26] | Segal, U., The ellsberg paradox and risk aversion: an anticipated utility approach, Int. econ. rev., 28, 175-201, (1987) · Zbl 0659.90010 |

[27] | Segal, U., Two-stage lotteries without the reduction axiom, Econometrica, 58, 349-377, (1990) · Zbl 0728.90013 |

[28] | Sutton, J., Non-cooperative bargaining theory: an introduction, Rev. econ. stud., 53, 709-724, (1986) · Zbl 0641.90093 |

[29] | Valenciano, F.; Zarzuelo, J.M., On the interpretation of nonsymmetric bargaining solutions and their extension to nonexpected utility preferences, Games econ. behavior, 7, 461-472, (1994) · Zbl 0827.90140 |

[30] | Yaari, M.E., The dual theory of choice under risk, Econometrica, 55, 95-115, (1987) · Zbl 0616.90005 |

[31] | Zeuthen, F., Problems of monopoly and economic welfare, (1930), Routledge and Kegan Paul London |

[32] | Zwick, R.; Rapoport, A.; Howard, J.G., Two-person bargaining behaviour with exogenous breakdown, Theory decision, 32, 241-268, (1992) · Zbl 0825.90817 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.