Rollback equilibrium

(b) In the rollback equilibrium, A and B vote For while C and D vote Against; this leads to payoffs of (3, 4, 3, 4). The complete equilibrium strategies are A votes For, B always votes For, C votes For unless both A and B voted For in which case C votes Against, and D always votes Against.

(c) The rollback equilibrium in any situation like this, where only one vote needs to change to affect the outcome, will entail the last voter’s (of the group willing to change) being the one to change her vote. Observers seeing a player like B changing her vote could use this rollback equilibrium to show that if  B had not done so, C would have. This argument is similar to claiming that B did not use her equilibrium strategy, since there was a voter to follow who would have changed her vote if B had not done so.

(d) This argument builds on the analysts’ views of the payoff structure for the legislators. Those who had opposed the death penalty originally but switched before the preliminary vote could not afford to switch their votes a second time; such an action presumably would give the impression of indecision or lack of conviction which the electorate might find unappeal- ing. Legislators hoping to maintain their seats would not want to switch positions so often that their constituents could not identify their true beliefs