Final answer:
The best outcome for Becky and Sarah regarding room cleaning is mutual cooperation leading to a clean room; the worst is non-cooperation resulting in a dirty room. They might not reach the best outcome due to individual rationality leading to a temptation to shirk responsibility.
Step-by-step explanation:
The best outcome for Becky and Sarah, who are sisters sharing a room, would be one where they both contribute to the cleaning effort—each spending two hours to achieve a clean room. The worst outcome would be if both decided not to clean, resulting in a dirty room for both of them. Constructing a Prisoner's Dilemma table can illustrate the potential outcomes and help understand their choices.
Costs and benefits analysis shows that cooperation yields the best net benefit. However, this optimal outcome may not occur due to the temptation for individual rationality leading each sister to shirk responsibility, assuming the other will clean, which ultimately results in the worst outcome of a dirty room. This dilemma can be attributed to a lack of communication and a defecting strategy where each sister chooses to minimize their immediate effort without regard to the overall best outcome.