Consider the same "bank run" game as described in the previous question.

If we change this to a sequential game where Player B moves first, what is the subgame perfect Nash Equilibrium?

Similar to above, the answers below are written such that the first strategy listed is player A's and the second strategy listed is player B's.

There may be more than one answer, so choose all that apply.

Question

Consider the same "bank run" game as described in the previous question.

If we change this to a sequential game where Player B moves first, what is the subgame perfect Nash Equilibrium?

Similar to above, the answers below are written such that the first strategy listed is player A's and the second strategy listed is player B's.

There may be more than one answer, so choose all that apply.

BlackTom AI · Accepted Answer

Question restatement: The scenario is a sequential bank run game in which Player B moves first. We are asked to identify subgame perfect Nash equilibria (SPE), noting that there may be more than one correct answer. The answer options present ordered pairs (A's action, B's action) from the four possibilities: (withdraw, withdraw), (withdraw, leave until later), (leave until later, withdraw), (leave until later, leave until later).

Option analysis starts with understanding the structure: in a sequential game, an SPE requires that each player's strategy constitutes a best response at every possible subgame, given the subsequent actions of the other players. Since Player B moves first in this setup, B’s move determines the starting subgame for A, and A’s best response is evaluated given B’s chosen action. Conversely, in the full specification of an SPE for a sequential game, we also consider how B would respond to A’s contingent actions in any off-path subgames, ensuring sequential rationality.

Option 1: (withdraw, withdraw).
- If A withdraws first, and then B withdraws in response (as specified by the option), we would check whether B’s choice to withdraw after seeing A’s withdrawal is the best response given B’s beliefs and the resulting payoffs. However, since B moves first in the actual game, this notation is unconventional for a SPE in the stated order (A’s move first). The assessment hinges on the payoff structure: if withdrawing is costly or leads to worse payoffs when expecting mutual withdrawal, B might prefer to delay. Without the payoff details, we cannot confirm whether this is a best response for B after A’s action. Consequently, this option cannot be judged as SPE solely on the surface without payoffs.
- Conceptual why it could fail: If A withdraws and B is anticipating a worse outcome from withdrawal (e.g., bank failure or insurance), B might prefer to deviate to delay, depending on the model. Without explicit payoffs, we can’t adjudicate decisively.

Option 2: (withdraw, leave until later).
- Interpreting this as A withdraws and B delays: the crucial question is whether B’s delay after A withdraws is a best response given A’s withdrawal. If the structure gives B a strictly better payoff from delaying (perhaps due to anticipated future liquidity or capital buffers), then this could be plausible as part of an SPE path. Yet, since B moves first in the real game, the notation here may reflect a contingent plan, and its SPE status would depend on the continuation payoffs after B’s chosen delay. Again, without numeric payoffs, we cannot confirm.
- Potential pitfall: If B’s optimal move, given A’s withdrawal, would be to withdraw as well (to avoid being worse off), then this option would not be sequentially rational for B, and thus not part of an SPE.

Option 3: (leave until later, withdraw).
- This is the natural pairing if, given B moves first and chooses to withdraw, A’s best response is to delay. We must check whether A has a profitable deviation to withdraw immediately after B’s withdrawal. If delaying yields a higher payoff for A (perhaps due to better liquidity or insurance against bank failure), then delaying could be optimal for A. However, the SPE requirement also demands that B’s initial move was optimal anticipating A’s response. If B would have preferred to withdraw instead of delaying, this path would fail the SPE criterion.
- Key consideration: the sequential rationality hinges on whether the combination makes both players’ strategies best responses in the subgames that arise after each possible history.

Option 4: (leave until later, leave until later).
- If both players optimally choose to delay in response to what the other does, this could constitute a mutual-keep-money-situation path. For it to be an SPE, the initial move by B and the subsequent move by A must both be best responses given the continuation payoffs. If delaying yields higher survivability or payoff for both under the game’s dynamics, this could be sustained. However, this again depends on the precise payoff structure, and whether any unilateral deviation would yield a higher payoff for either player at any subgame.

Important caveat about missing information: The analysis above highlights that determining SPE requires the explicit payoff matrix and the timing of moves embedded in the bank run model. Without those numbers or a formal specification of the payoffs for each action and history, we cannot definitively certify which of the four action profiles are SPE. In many bank-run sequential models, sometimes the unique SPE involves specific delaying or withdrawing patterns, but this is not guaranteed here.

Observations about the provided answer hint: the given data lists the correct answer as (leave until later, leave until later). If we take that as the intended SPE path, it would imply that, given the sequential structure and the continuation payoffs, both players optimally choose to delay in equilibrium, and that neither has an incentive to deviate at their information sets. This would suggest that, after B’s first move, A’s best response is to delay, and anticipating this, B’s initial move also aligns with delaying. Yet, without payoff details, this remains a hypothesis rather than a guaranteed conclusion.

In summary, each option’s viability as part of an SPE hinges on the specific payoffs and the subgame structure after each history. The lack of explicit payoff information prevents a definitive, fully reasoned endorsement or refutation of any particular option, including the (leave until later, leave until later) profile noted in the provided answer list.

ECON_104B_001_25S Lecture Quiz #7

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Similar Questions

Consider an extensive form game between two players, A and B, where Player A plays first and Player B plays last. Player A can choose between Up and Down, and Player B can choose between Left and Right. If the payoffs are represented using this tree, what is the subgame perfect Nash equilibrium?

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