Long-time reader Scott found the following puzzle from this site:

Here is a scenario which occurred many millennia ago: The patriarch of a wealthy family was on his deathbed and wanted to divide his gold among his eight sons who were all very, very greedy. Wishing to favor the oldest son (as tradition would have it) but also to reward the more cunning of his progeny, he made the following decree:

The oldest son is to propose a plan for dividing up the gold. The sons are all to vote on this plan, and if it receives at least half of the votes (four or more) then that will be the way the gold is divided. If this plan does not receive half of the votes, the oldest son gets nothing, the next oldest proposes a plan, and there is another vote, now among the remaining seven. Again at least half of the vote (still four or more) is required, and failure removes this son from the process. This is to continue until some son’s plan receives at least half of the votes of the remaining heirs.

Assuming that these sons will do anything to get the most gold possible for themselves, how much (if any) will the oldest son be able to inherit?

This is very much like an extended pirate’s game. Can you figure out the answer?
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The other day I called a new plumber for a non-urgent job. He came to my house, and after inspecting said, “It’ll be $150 and I can do it today. What do you want to do?”

I had no idea how much the job should cost, so I had to make an informed guess. Which of the following scenarios was most likely?

a. the price was a big discount
b. the price was a small discount
c. the price was fair
d. the price was a small markup
e. the price was a big markup

It might seem like there’s not enough information, but that’s the beauty of game theory – it can help you find answers even if you don’t know the question. By thinking strategically, I arrived at the right decision. Here is the logic I went through.

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An amusing clip from the short-lived Dilbert TV series.

Dilbert and his friends find themselves in a three-way Prisoner’s Dilemma. See how it turns out:

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I’m somewhat obsessed with the music from Christopher Nolan’s latest film Inception, and it turns out I’m not the only one!

The intoxicating music is making waves as re-cut and re-edited “Inception style trailers” for movies like The Dark Knight, The Matrix, Toy Story 3, and Total Recall, among others.

I’ve shared some of the vids below. Which one is your favorite?

*And if you’re wondering, the Inception song is Mind Heist by Zack Hemsey.

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Someone sent me a fun riddle that I wanted to share:

In a distant kingdom lived a king and his beautiful daughter. The daughter was in love with a peasant which the king strongly objects. However, he decided to show his fairness.

He gave the peasant a chance. He said he will let the peasant draw from 2 slips of paper, one of which has the word MARRIAGE written and the word DEATH on the other. The peasant agreed. On the way to the castle, the peasant over-heard the king and his knight talking.

Knight: “Sire, how can you let someone like him have the chance to marry the princess?”

King: “Don’t worry, I have written the word death on both paper”

Being a clever boy, the peasant found a way and later that day, married the princess. What did he do?

Can you figure out the answer?
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[update 7-1]: I’m on vacation…will be back with posts the week of the 12th!

I came across a fun math puzzle that’s relates to the game theory of guessing.

The puzzle was posted by James Grime, a mathematician who has posted some nice videos under the name singingbanana.

Let’s get right into it.

The puzzle

I’ve transcribed the puzzle from the original video:

It’s the Salem witch trials and two villagers are accused of being witches.

Now the witch-finder general says “There is a very simple test to tell whether you are witches. Each pick a card from a deck, and you can look at it if you want. But what I want you to do is predict the color of the other person’s card.

I’m going to put you in separate rooms so there is no communication between you. And if you’re both wrong, or one of you is wrong, then you are free to go. But if you both correctly predict the color of the other person’s card, then you are in league with the devil and will be burned at the stake.”

….

You have one minute to discuss strategy before the game begins. What is the strategy to survive?

What do you think? Give it a try before reading the solution below.

The solution

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There’s a nice game theory cartoon over at Saturday Morning Breakfast Cereal

The cartoon is very well done, though I must address a common misunderstanding.

The Golden Rule is not opposed but rather entirely compatible with game theory. In fact, the golden rule emerges naturally as a strong strategy in the repeated Prisoner’s Dilemma, called by economists as the “Tit-for-Tat” strategy.

It is no surprise that the golden rule has analogs in many world cultures.

I know very little about NASCAR, so I defer to Advanced NFL stats which has a great piece on NASCAR and game theory

I loved reading about the strategy of the race, particularly about the decision to make a risky pass against an opponent versus defending position by threatening a wreck. Both the ultimatum game and the game of chicken are relevant models.

My favorite part is the connection of game theory to a famous NASCAR fistfight that erupted from a game of chicken–talk about high stakes game theory!

Check out the first minute of this linked video to see the highlights and interviews with the drivers:

Youtube video: the infamous fistfight Daytona

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Close soccer matches are usually interesting. Teams are pressing on offense or locking down on defense. There’s a lot of chaos and intense energy.

This is less so in close basketball games. Usually teams are deliberate on offense and defense, stalling with fouls and timeouts. The pace is maddeningly sluggish. In one notable college basketball game, the final 3:19 took a whopping 23 minutes. How anti-climactic.

Why are the endings to these two sports so different? It’s partly due to the game-ending timing rule.

SportsHistory.us by Daniel Lauve has a fantastic explanation in the article Suspense vs. Purity: A Game Theory Perspective on Timed Sports.

Daniel starts out by discussing how players respond to fixed timed endings. The analogy used is the repeated Prisoner’s Dilemma in which the dominant strategy is to defect in any finite iteration. It was seen that some integrity could be restored if the exact ending round was unknown or probabilistically determined, thanks to experimental work by Professor Robert Axelrod.

The game theory experiment suggests a solution for the poor ending of basketball games:

Timed sporting events can have their consistency restored if we play them under uncertainty, keeping the participants in the dark about when the game will end. Some sports already do this, with varied results. In most soccer matches, stoppage time is added to the end of the match in an amount known only to the referees. In boxing, the scores are kept from the participants, which means that a boxer may be less likely to go for a knockout late in the match because he does not know whether the scoring dictates such a strategy.

I would be excited if basketball or NFL football might experiment with new game-ending rules. Daniel is less enthused and ultimately says this is a theoretical solution that fans, players, and coaches may not enjoy: “We would all miss the sports moments created by a ticking clock.”

There is also some talk about limiting timeouts to improve basketball. For now, the game will be riddled with slow endings, fixed timings, and the occasional exciting buzzer-beater.

Yesterday I wrote about a weird World Cup game where nobody tried to score.

There is amazingly an even weirder game on record. The infamous game is about a team advancing in a tournament by scoring an own goal.

In the 1994 Shell Caribbean Cup, Barbados needed two clear goals to advance against Grenada. Losing or winning by one goal meant not advancing. The wrinkle was a rule that goals in overtime would count double to reward a team winning a close game.

The stage was set for an unusual match. Professor Mike Shor of Vanderbilt University writes about the resulting craziness where scoring an own goal was rational:

Barbados needed to win by two goals. With less than ten minutes left in the match, Barbados led by exactly two goals and began to play very defensively. In the 83rd minute, Grenada finally scored, making the score 2-1. Barbados tried to answer but, with only three minutes remaining, was unable to score. Members of the Barbados team contemplated their options. To advance, they needed either to score one more goal in the last three minutes (winning by two), or force the game to extra time where a goal would count as if they won by two. Barbados scored on their own net, tying the game at 2-2.

This is not yet the odd part of the match. The Grenada players, initial shock abating, developed their own strategy. If they could score on Barbados in the waning minutes, they would win the match and advance. But, if they could score a goal on themselves, they would lose by one goal which was still enough to advance. For two minutes, Grenada tried to score on either goal, with Barbados players split between defending their own goal and that of their opponents!

Normal time ended in a tie and the game did go to overtime, in which Barbados scored a game winner and advanced

[emphasis mine]

Professor Shor’s cleverly dubbed the game “which goal is mine.”

Barbados managed to advance but it was eliminated in the next round. No penalties were given to either team since they were both trying to win (albeit in an odd way akin to the puzzle about fixing a broken bet).

The lesson is that simple and seemingly smart rules can have a devastating impact on competition and good play. And much of this can be anticipated if one uses game theory to consider the incentives by thinking ahead and reasoning backwards.

There is a video which shows highlights and the own goal (sadly it omits the chaos where Grenada tried to score on either goal)

Youtube video: Barbados Grenada 1994 Shell Caribbean Cup

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