Sunday, 31 July 2011

On the accessibility of the core (Review)

This paper by Yang (2010) belongs to a recent wave of literature that study the core of a cooperative game as a dynamic concept. Sengupta and Sengupta (1996) have shown that from any imputation the core can be accessed by a finite number of blocks. Kóczy (2006) provided an alternative blocking sequence and showed that the number of blocks required is bounded. The present paper relies on the proof of Sengupta and Sengupta by using z-dominance and provides an explicit bound on the length of z-dominance paths: the number of active coalitions, that is, coalitions with a payoff higher than the sum of their members' individual payoffs.

Yang observes that earlier methods considered dominance paths that often go in circles with the same coalitions blocking repeatedly. z-dominance targets a core element z, but also respects the history of the negotiations in the sense that earlier blocking coalitions are not made worse off. Yang does not prove that z-dominating sequences exist from each imputation, but refers to Sengupta and Sengupta (1996) for a proof. All z-dominating sequences end in the core and since a coalition may block at most once in such a sequence, their length is smaller than the number of active coalitions.

(Review written for Mathematical Reviews.)

Kóczy, László Á. The core can be accessed with a bounded number of blocks.  J. Math. Econom.  43  (2006),  no. 1, 56-64.
Sengupta, Abhijit;  Sengupta, Kunal. A property of the core. Games Econom. Behav.  12  (1996),  no. 2, 266--273.
Yang, Yi-You. On the accessibility of the core. Games Econom. Behav. 69 (2010), 194-199.

No comments:

Post a Comment