Sequential version of the all-pay auction. Two bidders alternate in bidding. A prize of \$5 is auctioned. At each move of the game, the bidding player decides whether to raise the current bid by \$1 or to quit. The auction ends when one of the players quits. The winner gets the prize and both players pay the amounts of their most recent bids. Player 1 begins the auction with the bid \$1 (unless Player 1 decides to quit, in which case the prize goes to Player 2 and both players pay nothing). If both players raise their bids forever, they both receive $-\infty$ as their payoffs. There is no discount factor in this case.
(a) Find all Nash equilibria in pure strategies.
(b) Find all subgame-perfect equilibria in pure strategies.
To answer this question, I understand that for finding the Nash Equilibria in pure strategies, these can include actions on information sets that might not occur.
In this sense, the strategy profile (Bidder 1 (B1) always raise, Bidder 2 (B2) always quit) is a pure strategy NE: If B1 raise in the first period, B2 quit in the second period, then the payoffs are P1=4 P2=0. Since B1 action is raise in the third period and B2 is quit in the 4th period (with payoffs of P1=2,-2), then neither of them has incentive to deviate. Similarly, the strategy profile (B1 always quit, B2 always raise) is also a NE for similar reasons.
Is my reasoning correct? Are these the only NE in pure strategies?
On the other hand, I was trying to use some sort of backward induction for answering the second question, but I'm really struggling with the logic. Any hint?