Game theory articles (new article usually posted on Tuesday)

An addition/subtraction game
The numbers 1, 2, 3, and 4 are written on a board. Alice writes a "+" or "-" sign in front of one of the numbers, and then Bob and Alice take turns. Once the four signs are written, the arithmetic expression is then evaluated. Bob gets points equal to the absolute ...

Video: safety in numbers
Consider the following game. Two generals have 5 units each to deploy. Each person decides how many units to send to battle. The general who sends more troops will win, but it's a draw if they both send the same number. Each also has the option of "passing" which averts war and ...

The wallet game: a neat auction
Imagine you are taking a game theory class, and the professor announces an experiment. You and another student are called up to the front of the classroom to participate in an auction. Here is how the auction works: --the prize money equals the sum of money in both of your wallets --the winner is ...

How to beat the Prisoner’s Dilemma in the TV game show Golden Balls
Golden Balls is an amusing British game show. Especially interesting is the final contest which is a version of the Prisoner's Dilemma. If you're never seen the show, here is how it works. Each of two contestants independently chooses to split or steal the final prize. If both choose split, then ...

A location game on a triangle
A father bequeaths a triangular piece of land to his children, Alice and Bob. The father was known to be quirky, and in his will, he specified a rather unusual way for the land to be divided. On a map of the land, Bob is to mark a location of his choosing. ...

A game of Guts
Imagine a casino offers a new game called "Guts." A dealer wants to test the game, so he recruits two strangers (Alice and Bob) for an experiment. The game works as follows. Alice will secretly write an even integer on a piece of paper, and Bob will secretly write an odd ...

The supply game
(Apologies my site has been down today with technical issues. I am looking to get it resolved) Recently India caused a stir in global markets when it banned its cotton exports. The ban was a result of the unstable cotton global market, which has lead China to stockpile its cotton supply ...

Number matching game
This is a game I created after reading about a similar card game in Game Theory: a Very Short Introduction. It's a simple example of a zero sum game that's slightly more complicated than rock-paper-scissors. Here is the game: Alice and Bob play a number matching game. Each simultaneously picks an integer ...

What’s the probability of a pure strategy Nash equilibrium?
This is a problem I adapted from the free e-book Algorithmic Game Theory. I had some fun solving it, and so I wanted to share it.

Video: understanding the Shapley-Shubik voting power index
With the election coming up this year, I thought it timely to revisit some voting theory. One of the useful concepts in voting theory is having an index that determines how powerful particular voters are. One of the voting power indices used is called the Shapley-Shubik index, which I have talked ...

The card game le her
While game theory as we know it began about 60 years ago, there were game theory ideas dating back to hundreds and thousands of years ago (see how game theory solved a religious mystery). Another early example happened in 1713, when James Waldegrave provided a mixed strategy solution to a two-person ...

How not knowing game theory can cost you millions
A few months ago, The Economist ran a great article about some of the practical applications of game theory. My favorite part of the article was the following paragraph which highlights how ignorance of game theory can be costly:

Video: how to buy a car using game theory
This video is about a car buying technique from Professor Bruce Bueno de Mesquita. I previously enjoyed one of his TED talks about predicting political events. The video is short and offers great practical advice for buying a car in America. Give it a watch:

Amazon’s $23.6 million book: a hilarious bidding war
Nearly 1 year ago, an out of print book about flies was listed at over $23.6 million on Amazon.com. As good as Peter Lawrence’s The Making of a Fly was, no one could see the sanity in that price. (thanks Robert for sending me the story) The listing was noticed by ...

Video: Tom Haverford uses game theory in relationship advice
Tom Haverford has a lot of funny ideas on the show Parks and Recreation. In one scene, he dispenses relationship advice to his friend about how to handle his anger. It turns out Tom has something of a hidden agenda, as the following clip illustrates.

Attacking a target optimally: an example of war game theory from RAND 1957
The Prisoner's Dilemma. The doctrine of mutually assured destruction (MAD). These are just two of the game theory ideas that came out of the RAND corporation during the 1950s, forever associating game theory with the non-profit think tank. But there were other interesting game theory applications, too. I came across a ...

Video: splitting the bill
Excedrin has a funny commercial about the annoyances that come with splitting the bill. This poor diner is with a group of friends who ordered the same thing, and yet use every excuse in the book to avoid paying their fair share. How many have you experienced? See the video after the ...

Video: monkey cooperation and fairness
While game theory was originally designed to explain human strategy, it turns out that the predictions are often applicable in animal behavior. Here is a fun video that shows an experiment about monkeys and cooperation.

Shoot your mate – game theory on TV shows
You are undercover and about to make a breakthrough with a mob boss. But your partner, not knowing your mission, tries to save you and gets captured. The boss suspects you might be working with the authorities. He asks you to prove your allegiance. He gives you a gun and requests you shoot ...

Video: don’t use penny auction sites
I have been meaning to write about penny auction sites like Quibids and Skoreit. These sites got big a couple years ago, and there are some great articles at codinghorror.com about how they are a scam and basically a lottery. This would be fine if the sites were regulated like a lottery. ...

Game theory of dividing a pizza
Figuring out how to divide a regular pizza is easy. A standard pizza is usually cut into equally-sized slices, so two people can just eat the same number of slices. But what happens when you cut the pizza in odd ways, like the following division? Let's play a game to test your ...

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 ...