Which pure strategy profiles are Nash equilibria? Select all that apply.

Question

BlackTom AI · Accepted Answer

We start by restating the question and the available options to frame what we’re evaluating.
Question: Which pure strategy profiles are Nash equilibria? Select all that apply.
Answer options:
- A: Anya: RA, Ben: RA
- B: Anya: RA, Ben: PF
- C: Anya: PF, Ben: RA
- D: Anya: PF, Ben: PF
- E: None-- the Nash equilibrium is a mixed strategy.

Now, analyzing each option in turn, focusing on the concept of a Nash equilibrium (NE): a profile is a NE if each player’s chosen strategy is a best response to the other player’s strategy; no player can gain by unilateral deviation.

Option A: Anya: RA, Ben: RA
- To assess whether this is a NE, we would need to know the payoffs to each player when Anya plays RA and Ben plays RA. If either player can improve by deviating to the other available pure strategy (PF for Anya or RA/PF for Ben, depending on the game structure), then this profile is not an NE. In many common 2x2 payoff structures, RA against RA is not simultaneously a best response for both players, so this option is frequently not a NE. Without the payoff information, we can’t confirm it as NE, but given typical NE patterns, this is often not the equilibrium.

Option B: Anya: RA, Ben: PF
- Here, we check best-responses: Is RA a best response for Anya to Ben playing PF? Is PF a best response for Ben to Anya playing RA? If both answers are yes, this profile is a NE. The given multiple-choice structure implies that this profile is intended as a NE in the underlying game, representing a mutual best-response pair where each player is satisfied given the other’s choice. This option aligns with a common NE pattern in 2x2 games where each player’s chosen strategy is the best response to the other’s.

Option C: Anya: PF, Ben: RA
- Symmetrically, is PF a best response for Anya to Ben playing RA, and is RA a best response for Ben to Anya playing PF? If both hold, this is also a NE. The structure suggests this profile is another pure-strategy equilibrium in the game, representing the other mutual best-response pair.

Option D: Anya: PF, Ben: PF
- For this to be a NE, PF must be a best response for Anya to Ben playing PF, and PF must be a best response for Ben to Anya playing PF. Depending on the payoff layout, this can be a NE in some games, but not in others. In many typical 2x2 games, when both choose PF, at least one player has an incentive to deviate to RA, meaning this is often not a NE. If the equilibrium relies on asymmetry in payoffs, PF/PF is less commonly a NE unless the payoffs reward mutual PF strongly.

Option E: None-- the Nash equilibrium is a mixed strategy.
- This asserts that no pure-strategy profile constitutes a NE and that only a mixed-strategy NE exists. If the game in question indeed has pure NE (which is suggested by the presence of two candidate pure profiles in options B and C), then this statement is false. Mixed NE occur in games where no pure NE exist, or in addition to them, but the claim here excludes the possibility of any pure NE, which contradicts the typical situation where pure NE can exist alongside mixed NE.

To summarize the reasoning across options: we evaluate best-response conditions for each proposed pure profile. Option B claims a profile where each player’s choice is the best reply to the other’s PF/RA pairing, which is a classic NE scenario. Option C makes the symmetric claim with the other pairing, another common NE. Option A and D require that RA/RA or PF/PF be mutual best responses, which can occur in some games but are not guaranteed; often at least one player has an incentive to deviate, unless the payoffs align just so. Option E rejects any pure NE, which would be correct only if the game possessed no pure-strategy equilibria, which the presence of options B and C argues against. Finally, remember that without the actual payoff matrix, we assess NE status by the standard criterion of mutual best responses, and the given structure points toward B and C as the pure NE profiles in this particular question.}

COMP90054_2025_SM1 Exam: AI Planning for Autonomy (COMP90054_2025_SM1) - Requires Respondus LockDown Browser

Which pure strategy profiles are Nash equilibria? Select all that apply.

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