Consider the following game: The Executive (e.g. President) is trying to push through the legislature (e.g. Congress) a series of bills that the latter is unsympathetic towards. The President proposes...

Consider the following game: The Executive (e.g. President) is trying to push through the legislature (e.g. Congress) a series of bills that the latter is unsympathetic towards.

The President proposes legislation and Congress must decide whether to make amendments. If it decides to amend, then the President must decide whether to fight the amendment or acquiesce. Looking at the President’s pay-offs it is obvious that, even though he or she prefers that the Congress does not amend the legislation, if it does, he or she would not want to fight on the floor of the House. The SPNE in the one-shot version of this game is simple enough: Congress amends and the President gives in. Now, suppose this game begins at time t = 0 (e.g. the President’s inauguration) and will end once and for all at time T (e.g. the compulsory end of his or her term). Moreover assume that, from the beginning, Congress entertains probabilistic doubt p0 that the President is dogmatically unbending and would thus ‘fight’ (irrationally) for his or her legislation regardless of pay-offs (i.e.
 is the probability that the President would fight even in the one-shot version of the game). Show that there exists a sequential equilibrium under which Congress will not amend the President’s legislation for a period of time k which is proportional to
.