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?

The solution

Scott also came up with the following well-written solution.

Reasoning Backwards

The last possible outcome is where Son 7 and Son 8 are voting on a plan. In this situation, Son 7 will be proposing the plan. Since only half of the votes are needed, whatever Son 7 proposes will be passed since Son 7 will naturally vote for his own plan. In this situation, Son 7 will end up with all the money and Son 8 will end up with none. Thus, it is in Son 8’s best interest to prevent it from coming to two people.

At three people, we have Son 6 proposing to Sons 7 and 8. Now, Son 6 knows that Son 8 does not want this vote to fail, since that will reduce it to two people, which is against his interests. Son 6 proposes that he gets all but $1, which goes to Son 8. Son 6 votes for this plan, being to his benefit. Son 7 votes against, since he gets nothing (and knows he stands to get everything if this continues). If Son 8 votes for the plan, he gets $1, if he votes against, he will ultimately get nothing. Thus, it is in Son 7’s best interest to prevent it from coming to three people.

At four people, we have Son 5 proposing to Sons 6, 7, and 8. Son 5 proposes that he gets all but $1, which goes to Son 7. Son 5 votes for his own plan. Son 6 votes against. Son 7, not wanting it to go any further, votes for, and Son 8 votes against. The 2-2 vote passes. Thus, Sons 6 and 8 will want to prevent it from coming to four people.

At five people we have Son 4 proposing to Sons 5, 6, 7, 8. He makes $1 offers to Sons 6 and 8, knowing they don’t want it to go any further, with the rest of the money staying with him. The vote ends up 3-2 (Sons 4, 6, and 8 vs. 5 and 7). Thus, Sons 5 and 7 will want to prevent it from coming to five people.

At six people, Son 3 is proposing. He makes $1 offers to Sons 5 and 7. The vote ends 3-3 with sons 4, 6, and 8 getting nothing. Thus, Sons 4, 6, and 8 will want to prevent it from coming to six people.

At seven people, Son 2 is proposing. He makes $1 offers to Sons 4, 6, and 8. The vote ends 4-3 with Sons 3, 5, and 7 getting nothing. Thus, Sons 3, 5, and 7 will want to prevent it from coming to seven people.

Lastly, with all eight sons, the eldest son, Son 1 will be proposing. Knowing all of the above he will propose $1 to Sons 3, 5, and 7. The vote will end 4-4, with Sons 4, 6, and 8 getting nothing.

Generalization

In a Pirate’s Game of N players–and gold pieces outnumber players–if the players are labeled 1, 2 … N, in order of of first to last proposer, player 1 should make a nominal offer to the odd-numbered players 3, 5, 7, etc., and keep the remaining wealth for himself.

It is interesting to note the Pirate’s Game can be extended to arbitrary number of pirates. The situation gets interesting when there are 100 gold pieces and 200 pirate or 500 pirates. The results are astonishing–the weaker pirates manage to survive–and the full write up is in an old issue of Scientific American.

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.

Could it be a fair price?

It was possible the plumber was quoting the fair price, not too high or not too low.

Some products are priced this way this because the brand exercises control over retailers, like Bose audio speakers or Apple products.

But plumbing is a different kind of product. There is a wide range in prices and a lot of discounts. I am constantly getting mailings with discounts for plumbers. Even my town calendar has a few coupons for plumbers.

The game is one of building trust and getting repeat customers. And most plumbers entice new customers with discounted service.

It seemed very unlikely to me that the plumber was simply quoting a fair price. I expected a price that was either discounted or marked up and I thought about each case.

Could it be a discount?

It was logically possible the plumber was giving a discount. I’ve been lucky like that before.

But I thought about how the plumber quoted me. He had said, “It’ll be $150 and I can do it today. What do you want to do?”

The important part is what he left out: he didn’t mention that he was giving a discount or a good price.

The omission is an important signal since businesses almost always emphasize discounts and promotions. It’s part of their strategy to get goodwill and repeat business. As I’ve written before, the game of discounts is very carefully calculated. A business that gives a discount without indicating it is doing something very, very odd.

That is why I could be very certain I was not getting a big discount or any discount at all.

How big was the markup?

I was then deciding how much I was being overcharged. If it was a slight markup, I was tempted to say yes and just get it done with.

So what is more likely: a slight markup or a big markup?

A bit of strategic thinking about repeated games can suggest an answer. The plumber knows the risk of quoting a high price. If I hired him now, and then later learned I was overcharged, I would obviously decide never to call that plumber again. (The game would end right there as I execute a “grim trigger” strategy.)

The plumber should anticipate my reaction, and reasoning one step further, I can figure out his motivation. If the plumber is going to overcharge, he has to make it worthwhile to compensate for losing future business. The one-shot deviation has to be sufficiently profitable or it’s not worth it.

Therefore, if the plumber is overcharging, he’s likely quoting a large markup. It can’t be outrageous like $1,000 or I would catch on right away. The plumber has to carefully craft a price that seems reasonable even though it is expensive.

Putting it all together, I highly suspected the $150 price was a big markup. I politely declined the service and tried to find someone else.

What happened next

I decided to call more plumbers. I was worried others might charge even more and that would signal a long search for me.

As it turned out, I got lucky. The very next plumber offered a price of $50–and he mentioned it was a very good price. I went through the logic and figured it was worth a shot. It was a win-win since I got a good price and the plumber won me over as a repeat customer.

In a way it all worked out and I was happy. It wasn’t an ideal situation: I wished I knew  plumbing prices in advance, and I wished I was just more handy. But in life it’s not possible to know everything. There are going to be situations where decisions have to be made with incomplete information.

It’s those times that strategic thinking is most valuable and game theory can really pay off.

Key lessons

–Think about the other person’s motivations
–In repeated business, don’t settle for full price on the first time
–An initial non-discounted quote may be a big rip-off

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

Dilbert Prisoner’s Dilemma

(If the above video is not working, there is also a version at gametheory.net: Dilbert clip)

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.

Fan-made trailers

The Dark Knight trailer Inception style

Batman Begins trailer Inception style

Matrix

Total Inception

Jurassic Park Inception Style

Toy Story 3

Titanic Inception Trialer

Up and Inception

Peter Pan + Inception

Willy Wonka and the Inception Factory

Stark Wayne Theatrical Trailer III

Lord of the Rings Inception Mashup

300 Inceptions

Harry Potter and the Deathly Hallows

Empire Strikes Back Inception

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?

The answer

From one perspective, the peasant is in a bind. The game is rigged so that he will pick a paper that says DEATH.

The trick is to exploit the knowledge of the onlookers and the common knowledge of the game.

The correct game should have one paper saying DEATH and the other MARRIAGE. This rule is known to the peasant, the king and all the audience.

So here’s the trick: the peasant doesn’t show which paper he picks!

The peasant picks a paper and hides it/destroys it. He says he’s too afraid to look at his choice, so let the king reveal to the audience the paper he didn’t pick.

When the paper is revealed to show DEATH, everyone will think the peasant must have picked MARRIAGE. The alternative is the king would have to reveal the game was rigged and that might ruin his reputation of fairness.

And so the peasant wins and lives happily ever after