Consider a sequential-move game with two players. The game tree with order of payoffs (Player 1, Player 2) is provided above. Choose the answer that is consistent with it.

Question

BlackTom AI · Accepted Answer

To evaluate the options, we first interpret the game from the tree. If Player 1 plays U, Player 2 responds by choosing between L (payoffs -1 for Player 1, 2 for Player 2) and R (payoffs 2 for Player 1, -1 for Player 2). Hence, after U, Player 2 prefers L since 2 > -1 for them. If Player 1 plays D, Player 2 faces L (1 for Player 1, 0 for Player 2) and R (0 for Player 1, 1 for Player 2). Here, Player 2 prefers R since 1 > 0.

Now consider Player 1’s decision anticipating Player 2’s responses. If Player 1 chooses U, the subsequent equilibrium payoff to Player 1 is -1 (since Player 2 will pick L after U). If Player 1 chooses D, the subsequent equilibrium payoff to Player 1 is 0 (since Player 2 will pick R after D). Since 0 > -1, Player 1’s best move is D, given Player 2’s best responses.

With Player 1 choosing D, the consistent continuation for Player 2 is R after D (as established), and with Player 1 choosing U, the consistent continuation for Player 2 is L after U. Therefore, the strategy profile where Player 1 plays D, and Player 2 plays L after U and R after D is the one that aligns with backward induction and subgame perfection.

Option a: Proposes D for Player 1 and Player 2 responds R after U and L after D. This contradicts the best responses described: after U, Player 2 would choose L, not R, and after D, Player 2 would choose R, not L.

Option b: States there is no Nash Equilibrium. The game has a well-defined sequential structure with best responses in each subgame, so a Nash Equilibrium (indeed a subgame-perfect one) exists. This makes the statement incorrect.

Option c: Proposes D for Player 1 and Player 2 responds L after U and R after D. This matches the backward-induction-based reasoning: P2 picks L after U and R after D, and P1 prefers D given those responses. This option is consistent with the game’s equilibrium analysis.

Option d: Proposes U for Player 1 with P2 choosing L after U and R after D. While P2 would choose L after U, Player 1 would not choose U because D yields a higher anticipated payoff (0 vs -1). Hence this profile is not a best response for Player 1 and is inconsistent with equilibrium behavior.

In summary, evaluating each choice against the players’ best-response logic and the subgames, only the option describing Player 1 choosing D with Player 2 responding L after U and R after D aligns with the equilibrium reasoning.

SESS0007_24-25 LSA Test

Consider a sequential-move game with two players. The game tree with order of payoffs (Player 1, Player 2) is provided above. Choose the answer that is consistent with it.

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