Sequential version of all-pay auction

Question

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?

Answer 1

There are more NE equilibria in pure strategies. But they are all outcome equivalent to the ones you found. For example, Player 1 quitting in each period and Player 2 raising until $n$-th period with $n>5$ and quitting afterwards are also Nash equilibria. Any strategy profile, in which one of the players quits at her first move and the other player raises until the price is $5$ and quits at a later point, constitutes a NE.

However, only the two equilibria that you describe are subgame perfect. Backward induction does not work here as the game is infinite. Thus, use the definition of subgame perfection. That is, a pair of strategy profiles are subgame perfect if they constitute a NE equilibrium in any subgame. In the game at hand, subgames start in each period. The prices in each period are sunk costs. Thus, in each subgame the game going forward is the same. Therefore, one of Players always quitting and the other player always raising are the only subgame-perfect equilibria.

Side note: if you assume that players discount future payoffs at a discount factor $\delta<1$, the game is continuous at infinity and you can use the single-deviation principle to prove the result.