Game Theory 101 MOOC (#39): Duels

Summary

In this Game Theory lecture, William Spaniel discusses a duel scenario between two Gunslingers, each with one gun and bullet. Starting 100 yards apart, they must decide when to shoot, balancing the risks of firing too soon or too late. Despite the lack of assumptions about their relative accuracy, the conclusion reveals that both players will shoot at the same time from the same distance. Through a proof by contradiction, Spaniel demonstrates how the strategic optimization of accuracy leads to this unexpected but logical outcome, highlighting the fascinating dynamics of Game Theory in action.

Takeaways

• 😀 The game theory scenario involves two gunslingers, each with a gun and a single bullet, starting 100 yards apart and gradually moving closer.
• 😀 The key strategic decision in this duel is when to shoot: shoot too early and miss, or wait too long and become vulnerable.
• 😀 The accuracy of the shot increases as the gunslinger moves closer to the opponent, making the timing of the shot crucial.
• 😀 The game assumes that both gunslingers miss for certain from long distances, but can hit for certain from point-blank range.
• 😀 The relative accuracy of the two gunslingers is not assumed to be known or fixed, which makes the analysis more universally applicable.
• 😀 The surprising conclusion is that both players will shoot at the exact same time from the same distance, regardless of their relative accuracy.
• 😀 If one player shoots first from a greater distance, the other player can always take a small step forward to improve their accuracy.
• 😀 This leads to a dynamic where both players continuously move closer, seeking to optimize their shot without giving up any advantage.
• 😀 The solution to this game involves both players shooting from the same distance, resulting in a simultaneous shot for both.
• 😀 Mathematical modeling and calculations are needed to determine the exact optimal shooting distance, though these details are complex and require advanced game theory analysis.
• 😀 The conclusion highlights an interesting and counterintuitive result in game theory: both players will reach equilibrium by choosing the same strategy despite the absence of assumptions about their skill levels.

Q & A

What is the central game theory problem discussed in this video?

The central problem involves a duel between two gunslingers, each with a single bullet. They start a long distance apart and gradually move closer. The dilemma is about when to shoot, balancing the desire to shoot early with the risk of missing and leaving oneself vulnerable.

Why is there a trade-off between shooting early and shooting later in the duel?

Shooting too early risks missing the target and leaving the shooter defenseless, while waiting too long increases the risk of being shot first. The trade-off is about finding the optimal moment to shoot, where the player balances accuracy with safety.

What assumptions are made in the game theory analysis of the duel?

The assumptions include that gunslingers are more accurate at closer ranges, they miss for certain at 100 yards but will hit for certain at zero yards, and no assumptions are made about the relative accuracy between the two players.

How does the distance between the gunslingers affect their accuracy in the duel?

The closer a gunslinger is to the target, the more accurate their shot will be. This means the gunslingers aim to get closer to each other to maximize their chances of hitting their target.

What happens if one gunslinger decides to shoot from a farther distance than the other?

If one gunslinger shoots from a farther distance, the other can always move closer, thereby increasing their accuracy and reducing the effectiveness of the first player's shot. This leads to both players continuously adjusting their positions to optimize accuracy.

What is the surprising conclusion about the timing and distance of the shots?

The surprising conclusion is that, regardless of their individual accuracy, both players will end up shooting at the exact same time and from the same distance. This is proven through the process of contradiction.

How is the proof by contradiction used in this game theory scenario?

Proof by contradiction is used to show that if one player shoots from a farther distance, it would not be optimal, as the other player could always move closer and shoot more accurately. This leads to the conclusion that both players must shoot at the same time and distance.

What does the concept of 'optimal strategy' refer to in the context of this duel?

The optimal strategy refers to both players continuously adjusting their positions so they can shoot at the most accurate distance, which ultimately results in both players shooting simultaneously from the same distance.

What is the significance of the assumption that players miss for certain at a long distance but hit for certain at zero distance?

This assumption simplifies the analysis by setting clear boundaries for when a player can hit the target (at zero yards) and when a player is guaranteed to miss (at 100 yards). It helps define the strategy for how the players adjust their positions.

How does this duel scenario illustrate the principles of Game Theory?

This duel is an example of Game Theory because it involves strategic decision-making under uncertainty, where both players must anticipate and react to the other's actions. The players' optimal strategies, derived from their choices regarding when and where to shoot, are key aspects of game theory.