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David Blackwell & Game Theory

STEM Topic 29: VOCABULARY

Ascendancy: A position of dominance or superiority.

Circumstantial evidence: Evidence that suggests a fact indirectly or through inference, rather than directly proving it.

Conventions: Large meetings or gatherings, often involving people with shared interests or professions.

Distort: To alter or misrepresent the true nature of something.

Duelists: Individuals who engage in a duel, a formal combat or contest with deadly weapons, often to resolve a dispute.

Hypothesize: To propose a theory or explanation based on limited evidence or observations.

Intuitively: Based on instinct or common sense, without the need for explicit reasoning.

Loftiest: The highest or most elevated.

Outliers: Values that are significantly different from the majority of other values in a data set.

Prescribed: In this context, it means a specific or set number of paces or steps that the duelists must take away from each other.

Ramification: The possible outcomes or consequences of a decision or action.

Representative sampling: A subset of data that accurately reflects the characteristics of the entire population.

Simulate: To imitate or reproduce the conditions of something in order to study or understand it better.

Smooth bore pistols: These are pistols with a smooth inner barrel, as opposed to rifled barrels, used in older firearms.

Statistical noise: Irregularities or fluctuations in data that can obscure patterns or trends.

Statisticians: People who specialize in the collection, analysis, and interpretation of numerical data.

Unparalleled: Something that is unmatched or without equal; beyond comparison.

David Blackwell & Game Theory

“One day some of us were talking and this question arose: If two people were advancing on each other and each one has a gun with one bullet, when should you shoot? If you miss, you’re required to continue advancing… If you fire too early your accuracy is less and there’s a greater chance of missing… Then I got the idea of making each gun silent. With the guns silent, if you fire, the other fellow doesn’t know, unless he’s been hit. He doesn’t know whether you fired and missed or whether you still have the bullet.” - David Blackwell

In the classic pistol duel, the two duelists start back to back. On command they march a prescribed number of paces away from each other, and then turn to face each other, pistols at their sides. Then they raise their smooth bore pistols and each fire a single bullet at the other. If one duelist hits the other, he is considered the winner, while if both are hit or both are missed the duel is considered a draw. It seems to make sense that firing your pistol quickly might be advantageous as you could hit your combatant before he fires and hence throw off his shot. On the other hand, if you wait longer (and presumably aim more accurately) you stand a better chance of hitting your opponent. If you wait long enough to see that your opponent fired his weapon, and you are still standing, you could take as long as you wanted to line up your shot.

 

We will simulate two simplified versions of this dual. The first will be a “noisy duel” where each duelist can tell when his opponent fires his weapon and the second a “silent duel” where this information is not available. For both of these games we will make the following assumptions:

 

  • The duelists must fire their weapons after 1, 2, 3, 4, 5, or 6 seconds.

  • If a duelist fires after second i, the probability of hitting their opponent is i/6.

  • In the noisy duel, if the duelists do not fire their guns at the same time, the duelist who fires second will wait until after 6 seconds thus guaranteeing that he hits his opponent.

 

Game 1: The Noisy Duel

To simulate this game we need two duelists with a piece of paper and a pair of dice. Before the game starts, each duelist writes down a number from 1 to 6 representing when he plans to fire his weapon. The two numbers are then compared. If the numbers are not equal, the player with the lower number will roll his die to simulate firing his weapon. If his number is i and he rolls 1, 2, …,i then he hits his target and wins the duel. If he misses, the other player will win as he will wait until after 6 seconds to fire and will be guaranteed a hit. If both players picked the same number, they both roll their die to simulate firing their weapon and determine the outcome in the same manner.

 

 

Activity 1: Below are some strategies represented as ordered pairs. The pair (i ,j) represents Player 1 planning to fire after i seconds and Player 2 planning to fire after j seconds. Below each ordered pair is one or two die rolls. In each case, determine the outcome of the duel.

Screen Shot 2023-08-08 at 5.15.08 PM.png

Activity 2: Now get some dice and find a partner to challenge to a duel. Simulate 10 or 15 duels, and try to find a winning strategy. Hypothesize what you think the optimal strategy might be.

Game 2: The Silent Duel
 

This game is simulated in the same manner as the noisy duel unless the player who fires first misses. In this case, the other player does not know he has fired and will continue to fire his weapon as planned. Hence two rolls are always needed, and it becomes more likely that both players will miss.

 

Activity 3: Analyze the strategies and rolls given below and see if you can determine the outcome of each duel. The roll on the left is always Player 1’s roll and the roll on the right is Player 2’s roll, regardless of the order in which they fired.

Screen Shot 2023-08-08 at 5.17.07 PM.png

Activity 4: Get your dice and partner and prepare to duel. Try to hypothesize an optimal strategy as you simulate 10 or 15 duels.


Prisoner’s Dilemma*

Prisoner’s Dilemma Scenario: Imagine that you and your accomplice have robbed a bank. Outside of the bank you are apprehended by police, separated, and then taken to different interrogation rooms in the police station. The police offer you a deal. You have to choose whether or not to implicate your accomplice. If both of you implicate each other then you and your accomplice will each go to prison for 2 years. However, if one of you implicates the other but the other keeps silent, the one who has ratted out his accomplice will go free, while the other will rot in jail for 5 years on the maximum charge. If you both keep silent, only circumstantial evidence exists, and so you will both serve one year.

 

Question 1:  Fill in the following table to help organize the ramification of each option:

Screen Shot 2023-08-08 at 5.22.20 PM.png

Question 2: If you can talk to your accomplice and you trust him or her, what should you do to minimize the time that you both spend in jail? Explain why. This is called the co-operative strategy.

 

Question 3: Let’s say that you know that your accomplice is going to implicate you. What should you do to minimize your jail time? Compare your options and explain.

 

Question 4: Let’s say that you know that your accomplice is going to keep silent. What should you do to minimize your jail time?  Compare your options and explain.

 

Question 5: You should have gotten the same answer for Questions 3 and 4. Following this strategy is best for you if you can’t trust your accomplice (who, after all, is a criminal) because you come out ahead no matter what the other person does. This is called the selfish strategy. If both you and your accomplice follow the selfish strategy, how much time will you each spend in jail?

 

Question 6: What are some other real-life situations that have similar elements?

 

*Note: Think about police use of this dynamic in the Netflix film “How They See Us.”

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