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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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