Game theory articles (new article usually posted on Tuesday)

How to divide up chores fairly using game theory
This post is based on the following email I got from Cheryl: Hi Presh~ Could you write a blog post on game theory tips for fairly dividing household chores/tasks (for roommates, partners, and perhaps coworkers)? Bonus points for providing easy division methods that people will actually use. In any case, keep up the ...

The Colonel Blotto game
This is an interesting game I found in the book Introduction to game theory by Peter Morris that reminds me about the board game Risk. The Colonel Blotto problem is a zero-sum game about how to best position resources. While Colonel Blotto games are described in a military context, I will explain ...

Why your crazy girlfriend always gets what she wants: battle of the sexes game theory
You want to watch the football game. She wants to go out for a movie. Guess which one you end up doing. It's not exactly a scientific observation, but time and again, I have seen friends whipped by crazy girlfriends. (I would equally say there are a lot of crazy boyfriends, ...

A simple game that makes you think, “Wow, people can be really petty”
I came across a very interesting game theory example in the book The Survival Game. Let's you and I play this very simple game and analyze the best strategy.

A determinant game
This is a fun combinatorial game theory problem I came across. This problem appeared in the 2008 Putnam exam, an annual North American college math competition:

Video: strategy in a TV game show
Game theory concepts can often help when trying to win prizes on a TV game show. I have previously written about a couple of the scenarios from the U.S. show The Price is Right: Optimal strategy in spinning the wheel Strategy for Pay the Rent Strictly dominated strategies in Lucky Seven I came across ...

Video: Braess Paradox demonstrated with springs
A couple years ago, a few roads in my town were closed due to flooding. I was at first worried the road closures would make roads congested. But to my surprise, traffic seemed to be moving faster. Why was that?

Auction theory puzzle: finding the right number of bidders
If you liked yesterday's puzzle about optimizing with uncertain demand, you'll definitely enjoy today's post. One of my favorite topics related to game theory is the subject of auction theory. The results from auction theory are very interesting, but I have yet to cover much about it because the math can be ...

Cannibal game theory – a cool math puzzle
I came across a really interesting game theory problem at David Cowan's blog. The problem not only is about strategy, but its proof is interesting mathematically too. Here is the puzzle:

Alice and Bob play chess – a logic puzzle
Even if you don't like chess, you can appreciate today's puzzle. The problem does involve playing chess, but at its heart, the problem is a logic puzzle. Here is the setup:

A free throw game
In honor of the NBA lockout ending, today's post is related to basketball. It's a fun and relatively easy math problem about shooting free throws:

Puzzle: odds of a comeback victory
You're favorite sports team is down at halftime, but you are hoping they can pull it together and eek out a victory. What are the odds of that happening? The probability obviously depends on the exact game in question. But I came across an interesting probability problem that gives an interesting ...

Company size vs talent: which one is more important? (Math problem)
There are always tradeoffs when picking one company over another. Imagine for a second you are thinking about picking between a standard position at a straight-forward, large company versus a fun position at an exciting smaller business. Which one is more appealing to you? There are many reasons people pick one ...

The game of Morra: a fun alternative to rock-paper-scissors
People have long thought up ways to solve small disputes at random, from flipping a coin to playing a game of rock-paper-scissors. The other day I came across another game that dates all the way back to the ancient Greeks and Romans. The game is called Morra. While there are many variations ...

Ping pong probability
I came across a fun problem in this math book: Suppose A and B are equally strong ping pong players. Is it more likely that A will beat B in 3 out of 4 games, or in 5 out of 8 games? It's worth doing the math on this one. And once ...

Game Theory 101: an excellent introduction to game theory, and interview with William Spaniel
People often ask me to recommend a book that gives an introduction to game theory. Up until now I strangely did not have a proper answer. Most books either have too little math and miss out on the theory aspect, or they have way too much math and were just ...

How to split a cab fare fairly using game theory
I came across a fantastic game theory article that appeared in the Wall Street Journal Number's Guy blog all the way back in 2005. The article is about three friends who agree to share a cab, and the possible ways they can split the costs. I highly recommend you read the article. The ...

Dividing up Halloween candy: the strategy to get the best candy for yourself
One of the things I used to enjoy about trick or treating was the surprise factor. Each house offered different candy, resulting in a nice mix. But the variability lead to some issues too. Many times my brother and I would go to the exact same houses, but we would end ...

Monday Puzzle: Alice and Bob race to 1 million
This is a fun game in which Alice and Bob race to 1 million in a mathematical sense. Alice and Bob start with the number 1. Alice multiplies 1 by any whole number from 2 to 9. Bob then multiplies the result by any whole number from 2 to 9, and ...

A math game of dodgeball
Let's analyze a math game called dodgeball that's a sort of twist on tic-tac-toe. Here is how the game works. It's a two player game with the following set-up. Player 1 gets a 6x6 grid of squares as follows: Player 2 gets a 6x1 grid of squares: Here are the rules: 1. Player 1 begins ...

The ‘magic’ number 495
Today I want to talk about an interesting mathematical curiosity. First, I want you to think of any three digit number where all the digits are different. I will work through this example using my number of 729. What I want you to do is the following routine: 1. Rearrange the digits to make ...

Puzzle: who wins this dart game?
Alice and Bob play the following game with their friend Charlie. Charlie begins the game by secretly picking a spot on the dartboard. The spot can be anywhere on the board, but once picked it does not change. Then Alice and Bob each get to throw one dart at the board. At this ...

$10 if you guess who will be fired next – boss demonstrates how not to motivate employees
There are proper ways to use incentives to motivate employees. I have written about how incentives can help improve productivity and even reduce employee theft. Then there are the abysmal incentives that are dreamt up by sadistic bosses. This week there were news reports of a boss who made a contest ...

Game theory quiz: efficient scheduling
The other day I was running errands when I suddenly realized I needed to visit my bank. The issue was the time was 5:02pm, and I was not sure if my bank was open or if it had closed for the day. I was debating whether to take a few minutes to ...

A fun chess puzzle
Chess isn't something I normally write about, but this puzzle was too interesting to pass up. Why? This puzzle stumped some very smart people, including Ken Thompson (one of the people behind Unix and C), and even world class chess players like Garry Kasparov, Anatoly Karpov, and Mikhail Botvinnik, according to an ...

Warren Buffett’s other political proposal
Warren Buffett has recently been making headlines with the controversial "Buffett Rule" calling for a tax raise on millionaires. But this isn't the first time Buffett has said something interesting politically. Now is a good time to recount an amusing proposal he made in an op-ed in 2000 about political campaign finance ...

When game theory backfires: a case study of Robert Campeau’s takeover bid
In 1988, Canadian financier Robert Campeau was attempting a takeover of Federated, a retail congomerate that owned Bloomingdales. The negotiations were quite interesting, and in 1990 Fortune magazine went so far as to call it "The Biggest, Looniest Deal Ever." One of the notable strategic elements was a "two-tiered offer" that Campeau ...

Probability question: would you play this card game?
Today I want to share a fun card game that is a twist on drawing to the high card. The game is played with a deck of cards numbered 1 to 100, and its rules are as follows. First, you and I each draw one card from the deck and show them ...

Why are Coke and Pepsi never on sale at the same time? An answer from game theory
I regularly check ad papers, and it always struck me as odd that competing brands were never on sale at the same time. One week Coca Cola products might be on sale, but the next week it would be Pepsi. But rarely would the two ever be on sale at the ...

How stupid people make fools of us all
People are so familiar with survival of the fittest, we often forget there are situations that favor the weakest. Four years ago I wrote about survival of the weakest and discussed a game in which it can be better to be weak: your very weakness lets you hide as tougher players ...

A fun number elimination game: who will win?
Bored at the airport, Alice and Bob decide to play the following mathematical game. Alice writes the numbers 1, 2, . . . , N on a piece of paper. Bob goes first, and he picks two numbers x and y from the list. Bob crosses out these numbers from the ...

Motivating a gym class using game theory
One of my favorite parts about writing this blog is hearing stories of how people have used game theory in real life. The other day I got an email from Jimmy who noticed game theory in action while at a group exercise class. Here is his story: Hey Presh, Thanks for taking the ...

Coin flipping game: how to make a fair toss from an unfair coin
Alice and Bob play a game as follows. Alice spins a coin on a table and waits for it to land on one side. If the result is heads, Alice wins $1 from Bob; if tails, Alice pays $1 to Bob. While the game sounds fair, Bob suspects the coin may be biased ...

The Price is Right Wheel Game: what is the optimal strategy?
The Price is Right is a wonderful game show that lends itself to game theory analysis. Earlier I have posted about the games Pay the Rent and Lucky Seven. My friend Aaron emailed me about another interesting problem from the show: I've got a good problem for you to throw at your ...

Can you guess what I am thinking? Focal / Schelling point experiment
[Update 7-28-11: ] I've learned this site needs some upgrading on the backend, and also there was an issue with the game theory archive page--for the meantime: try the following link: game theory archives. I'm working to fix these things, and new posts to resume on August 1. Today I want ...

How to reduce employee theft without nagging: 4 tips from behavioral economics
image by Nisha A Employee theft is extremely common. I would hazard that even the best of us has stolen something from an employer. Most people do not view taking a few pens or printing personal material as a big deal. In fact, some people view it as an unpaid benefit ...

The game theory of choosing flight seats
I've been traveling a lot and thinking about the strategy of picking the best seat on a plane. Here is an interesting question that came across my mind. If you are flying with a friend, which two seats should you pick on a plane? Assume you want to sit next to your ...

Birthday laws probability puzzle
Wouldn't it be fun if you could get a holiday every year on your birthday? Wouldn't it be even better if your friends got the same holiday too? Such a labor law would make for some fun parties. But it might have some interesting macro effects on the economy as a whole. Here's ...

Using logic to solve a test question
Strategic thinking is so powerful you can often find the answer just by reasoning carefully. I have written before how I have used strategic thinking when hiring a plumber and also when negotiating a job raise. The following is another example showing the power of strategic thinking. It comes from an email ...

A game theory problem that puzzled Marilyn vos Savant
Marilyn vos Savant is most known for being listed as the "Highest IQ" in the Guinness Book of World Records. She is also famous for her column "Ask Marylin" in Parade magazine (that she has written since 1986) that tackles interesting questions and puzzles, the most controversial being the Monty Hall ...

Game theory in The Matrix – the mathematics of dueling
One of my favorite scenes in The Matrix is the subway fight scene (clip below). Rather than run away, Neo chooses to fight Agent Smith head on. The scene begins with a duel which I find particularly interesting due to the strategies they take.

Another way to explain the Prisoners’ Dilemma to high school students
The Prisoners' Dilemma is one of the most interesting problems in game theory. It demonstrates the idea of Nash equilibrium in a stark fashion, and the mathematical requirement is very minimal. Perhaps its only flaw, if there is one, is the subject matter of criminals and police, which is sexy but ...

Negotiating with the mob – Game theory in the Dark Knight part 2
In my original post about Game theory in The Dark Knight, I discussed the opening bank robbery scene and compared it to the pirate game. I mentioned the film was full of strategic thought and a great way to illustrate game theory concepts. In light of recent Batman news, like Facebook ...

Mutual vs Common Knowledge in course difficulty
A little bit of knowledge can be a powerful thing. There is a useful distinction that comes up a lot in game theory about knowledge. The two concepts are: Mutual knowledge: all people in a group know about an event Common knowledge: all people in a group know about an event, they ...

How do you split coupons when paying a restaurant bill?
It's Friday night and you're out to dinner with your friends. You are ready to split the bill, and you offer up a coupon to save money. How should the coupon affect how much you pay? Should the coupon reduce the entire group bill, or should the coupon reduce only how ...

A dilemma in dividing up chocolates
The day after Valentine's Day, a father ran into a math dilemma. He had bought a heart-shaped box of chocolates on clearance to be divided amongst his three daughters. The issue came when he tried to divide the chocolates. The box contained 17 pieces of chocolates. He had promised 1/2 of ...

The Wallet Paradox
Bill Gates meets Warren Buffett at a dinner party and the host tells them to play a game. Each person will place his wallet on the table. The person with less money in his wallet wins all the money. Is anyone favored to win this game? What happens when the game ...

Broken sticks puzzle, and a seemingly paradoxical ratio
image via flickr, CC by 2.0 This is a neat little problem that is an extension of a puzzle from the Richard Wiseman's blog. The puzzle A warehouse contains thousands of sticks, each 1 meter long. One day a bored worker breaks each of the sticks in two, with each of the breaks ...

Why repeating a digit may improve security on your iPhone’s 4-digit lockscreen PIN
image by flomiscuous, CC by 2.0 My good friend from college sent me an interesting email. Presh, real-life question for you: What is the safest way to lock my iphone? Let me explain. A friend unlocked his phone once and I grabbed it and said "so, 9,6,0, and 1, huh?" because the bulk of ...

Pay the Rent strategy on The Price is Right
I just learned about a new game on The Price is Right. In the premiere of the 39th season, the show introduced a game called "Pay the Rent." The game is easiest to learn if you watch the clip (after the break). Once you learn the rules, I challenge you to try ...