Lateral Thinking Puzzle: Deadly Poisons and Strategic Reasoning

Core Concepts

A lateral-thinking puzzle involving poisons with strange properties admits several unintended solutions that are just as interesting as the intended solution. Analyzing these alternative solutions using game theory yields surprisingly subtle results and several unanswered questions.

Abstract

The content presents a lateral-thinking puzzle by Michael Rabin involving a world where poisons have strange properties. In this world, a healthy person who ingests a poison will die within an hour unless they ingest a stronger poison, which will restore their health. There are two types of poisons - magical and medical - that are strictly linearly ordered in strength.

The King wants to find the strongest poison in the land, so he summons the Royal Magician and the Royal Physician and instructs them to each bring their strongest poison to a showdown. The catch is that they must first drink the other's poison before drinking their own. The survivor will be the one with the stronger poison.

The author analyzes several possible strategies the Servants could employ, including:

• Conventional strategy (C): Arrive healthy and bring the strongest poison
• Advanced strategy (A): Drink a weak poison in advance and bring water
• Blank strategy (B): Bring water and drink nothing in advance
• Double Dose strategy (D): Drink a weak poison in advance and bring a stronger poison

The author shows that there are three ways, other than the intended "A-versus-A" scenario, in which both Servants could die from poisoning. The author then applies game theory to analyze the optimal strategies, finding that there are multiple Nash equilibria with varying survival probabilities.

The content concludes by noting that the puzzle admits further subtleties and unanswered questions, particularly when more than two poisons are available, and suggests the possibility of an empirical investigation through a tournament-style setup.

Stats

There are no key metrics or important figures used to support the author's key logics.

Quotes

"If my opponent employs Strategy C, then employing Strategy A is a surer way to survive than employing Strategy C myself and gambling that I have the strongest poison. But my opponent is no fool, and will probably realize the same thing, and employ Strategy A rather than Strategy C. I need to outsmart my opponent and find a way to defeat Strategy A."
"To apply the tools of game theory to Rabin's puzzle, we need to specify the (expected) payoffs to each Servant in each possible scenario (A-versus-A, A-versus-B, etc.). It seems reasonable to assume that each Servant wishes to maximize their own chances of survival, so let us assign a payoff of 1 to a Servant who survives and whose opponent dies, and a payoff of 0 otherwise (so in particular, if both Servants die of poisoning, or if both Servants survive the drinking ordeal and are therefore executed by the King, then both Servants receive a payoff of 0)."

Key Insights Distilled From

Cooking Poisons

by Timothy Y. C... at arxiv.org 04-09-2024

https://arxiv.org/pdf/2404.05053.pdf

Deeper Inquiries

What additional strategies or variations of the puzzle could be explored if the Servants had access to more than two poisons?

If the Servants had access to more than two poisons, the puzzle would introduce a higher level of complexity. One possible variation could involve each Servant having a range of poisons with varying strengths, creating a more intricate decision-making process. This would lead to a wider array of strategies, such as selecting a combination of poisons to counter the opponent's choices effectively. Additionally, the introduction of more poisons could result in the possibility of one Servant having a clear advantage in terms of poison strength, leading to potential dominant strategies that could be explored.

How might the optimal strategies change if the Servants' objective was not solely to maximize their own survival probability, but also to minimize the opponent's survival probability?

If the Servants' objective shifted to not only maximizing their own survival probability but also minimizing the opponent's survival probability, the strategies would need to be more aggressive and focused on outwitting the opponent. In this scenario, the Servants would need to consider not only their own choices but also anticipate and counter the opponent's moves effectively. Optimal strategies would likely involve a mix of defensive and offensive tactics, aiming to not only ensure their own survival but also actively work towards eliminating the opponent. This change in objective would lead to a more dynamic and strategic gameplay, requiring a deeper understanding of the opponent's potential moves and intentions.

What insights could be gained from an empirical investigation of the game through a tournament-style setup, where experienced players repeatedly play against each other with randomized poison strengths?

Conducting an empirical investigation of the game through a tournament-style setup would provide valuable insights into the decision-making processes of the players and the effectiveness of different strategies. By having experienced players repeatedly play against each other with randomized poison strengths, researchers could observe how players adapt their strategies over time, learn from their opponents' choices, and potentially develop new tactics. The tournament setup would allow for the analysis of long-term gameplay trends, the emergence of dominant strategies, and the impact of randomness on decision-making. Additionally, such an investigation could shed light on the effectiveness of different strategies in a competitive environment and provide a deeper understanding of the complexities involved in solving the puzzle.