Centre for Discrete and Applicable Mathematics
	CDAM Research Report, LSE-CDAM-98-18 October 1998

Abstract

We determine the multiple sub-game perfect Nash equilibria arising in the Rubinstein bargaining game when both players make prior claims on the so-called cake i.e. pre-commit to accept at least a certain percentage of the cake subject to forfeiting a fixed percentage  c  of the cake (to a third party) if, setting aside their claim, they accept below the prior claim level. This is a study in the treatment of discontinuity, which requires the expanded apparatus of pseudo-fixed points and quasi Nash equilibria (permitting infinitesimal tolerance). We show how to interpret the smallest and largest equilibrium as divisions in a long-term bargaining game with the termination condition that Player-2 (respectively Player-1) is free to choose the final division. It is shown that under the prior claims regime it is not optimal to claim too much; moreover to sustain the standard equilibrium requires a limited penalty, and otherwise the Nash equilibrium shifts upwards for Player-1 to

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