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?

Question

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?

BlackTom AI · Accepted Answer

The question describes an extensive-form game with two players, A moves first choosing Up or Down, and B moves second choosing Left or Right after observing A's move. To identify a subgame perfect Nash equilibrium (SPNE), we use backward induction: determine B's best response to each possible move by A, then have A pick the initial move that yields the best outcome given B’s subsequent best responses.

Option by option analysis:
- Option 1: (Up, Left)
  • Consider what happens if A chooses Up. B observes Up and chooses between Left and Right. B will select the action that yields the higher payoff for B in the (Up, Left) vs (Up, Right) subgames. If the resulting subgame outcome (Up, Left) is a Nash best response for B given Up, then (Up, Left) can be part of an SPNE for the Up branch. We must also check A’s choice given B’s BRs for the Down branch to ensure A wouldn’t deviate.
  • In backward induction, this option is evaluated by comparing B’s payoff from Left vs Right after Up. If Left is B’s best response to Up, and A anticipates that Left will be chosen, A would compare the payoff from Up against the payoff A would receive if A chose Down and B responded optimally there as well. If Up followed by Left yields at least as good a payoff for A as Down followed by B’s best response, then this branch can be consistent with an SPNE.
- Option 2: (Down, Right)
  • Analyze B’s reaction after A chooses Down. If Right is B’s best response to Down, this must hold for the Down branch to be part of an SPNE. Then, A would weigh Up (with B’s best response to Up) versus Down (with B’s best response to Down) to decide A’s equilibrium choice. If Down with Right is a best reply to Down and still optimal for A given the alternative Up branch, this option could be part of an SPNE.
- Option 3: (Down, Left)
  • Similar logic: After Down, B must prefer Left to Right for this to be an SPNE path. Then A compares Up (with B’s Up-branch BR) to Down (with B’s Down-branch BR). If Down with Left is the BR for B and A’s payoff from Down given B’s BR is at least as good as the payoff from Up given B’s Up BR, this path could be an SPNE.
- Option 4: (Up, Right)
  • After Up, B must prefer Right to Left for this to be the SPNE path. Then A must prefer Up over Down given B’s BRs. If Right is B’s best response to Up and A’s payoff from Up given that B’s BR is at least as high as the payoff from Down with B’s BR, this could be part of an SPNE.

Key conceptual takeaway:
- An SPNE requires that, in every subgame (including the one following A’s Up and the one following A’s Down), B’s strategy is a best response to A’s action, and A’s initial choice is a best response anticipating B’s subsequent best responses.
- Because SPNE relies on backward induction, the branch where B’s responses align with A’s anticipations and yield mutual best responses under each subgame is the viable SPNE path.

Note on structure: since the actual numerical payoffs for each terminal node are not supplied here, we cannot compute exact numerical comparisons. Instead, we explain the logic you would apply once payoffs are known: determine B’s BR to Up and to Down, then determine A’s optimal initial move given those BRs. The listed option (Up, Left) is the one identified in your provided answer set as the SPNE candidate, which would be confirmed or refuted by checking the BRs and A’s choice as described above against the full payoff tree.

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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