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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.

##### MSC:
 91A20 Multistage and repeated games 91A10 Noncooperative games 91B52 Special types of economic equilibria 91A06 $$n$$-person games, $$n>2$$
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##### 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
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