Ad: Game Theory by Maschler, Solan and Zamir

Yisrael Aumann wrote “‘This is the book for which the world has been waiting for decades.”
Eric Maskin added “There are quite a few good textbooks on game theory now, but for rigor and breadth this one stands out.”
Ehud Kalai thinks that “Without any sacrifice on the depth or the clarity of the exposition, this book is amazing in its breadth of coverage of the important ideas of game theory.”
Peyton Young goes further and writes “This textbook provides an exceptionally clear and comprehensive introduction to both cooperative and noncooperative game theory.”

Covering both noncooperative and cooperative games, this comprehensive introduction to game theory also includes some advanced chapters on auctions, games with incomplete information, games with vector payoffs, stable matchings and the bargaining set. Mathematically oriented, the book presents every theorem alongside a proof. The material is presented clearly and every concept is illustrated with concrete examples from a broad range of disciplines. With numerous exercises the book is a thorough and extensive guide to game theory from undergraduate through graduate courses in economics, mathematics, computer science, engineering and life sciences to being an authoritative reference for researchers.

This book is the outcome of 8 years of hard work (and its almost 1000 pages attest for that). It was born in Paris, in February 2004, around Sylvain Sorin’s dinner table, where Shmuel Zamir and your humble servant had a great dinner with Sylvain and his wife. Michael Maschler joined the team several months later, and each month the book grew thicker and thicker. I use the book to teach 4 different courses (two undergrad, two grad), and since it contains so many exercises, there is no reason to spend much time on writing exams.

The book is only one click away. Don’t miss it!

The Effectiveness of Iron Dome

There is a debate underway about how effective Israel’s iron dome system is in protecting populated areas from missile attacks. On the pro side it is argued that somewhere between 85% to 90% of incoming missiles are destroyed. The con side argues that the proportion is much smaller, 40% or less. A large part of the difference comes from how one defines `destroy’. Perhaps a better term would be intercept. It is possible that about 90% of incoming missiles are intercepted. However, a missile once intercepted may not have its warhead disabled, making at least one of the fragments that falls to ground (in a populated area) dangerous.

While nailing down the actual numbers may be interesting, it strikes me as irrelevant. Suppose that any incoming missile has a 90% chance of being intercepted and destroyed (which is the claim of the builder of the iron dome technology). If the attacker launches N missiles and iron dome is deployed, the probability (assuming independence) not a single one making it through is (0.9)^N. Thus, the probability of at least one missile making it through the `dome’ is 1 – (0.9)^N. If N is large, this is large. For example, for N = 10, the probability that at least one missile makes its way through is at least 60% (thanks to anonymous below for correction). Thus, as long as the attacker has access to large quantities of missiles, it can be sure to get missiles through the dome.

Coalitional Games for Real

Israel is a parliamentray democracy; our president has but ceremonial role, and the prime minister (and his government) is the one who actually makes all important decisions. After elections, each party recommends to the president a candidate for prime minister, and the person who got most recommendations is asked to form a government. To this end, he/she should form a coalition with at least 61 parliament members out of the total of 120.
In the last elections, taking place on 22-January-2013, results where as follows:
Likkud (secular right), the party of the last prime minister Benyamin Netanyahu, got 31 out of 120 parliament members.
Two ultra orthodox parties, which were part of the last government, got together 18 seats.
An orthodox right party got 12 seats.
Three secular centrist parties got 19 + 7 + 2 = 28 seats.
Five secular left parties got together 32 seats.
The last prime minister, Benyamin Netanyahu, was recommended by most of the parties to be the new prime minister as well, and so was asked to form a coalition. The five left parties cannot be part of a coalition because they share an opposite point of view than that of Netanyahu. Still, Netanyahu has several possible coalitions, and his most preferred coalition was with the two ultra orthodox parties and the secular centrist party. As coalitional game theory (and past experience) tells us, he should retain most of the power. Unfortunately for him, the largest secular centrist party and the orthodox right party realized this, and they formed an alliance: either both are part of the coalition, or both are out of it (and they want to ultra orthodox parties out of the government). Since together they have 31 seats, and the five left parties that will anyway be out of the coalition have 32 seats, this means that they became a veto player. Thus, even though Netanyahu is supposed to be the prime minister, these two parties will determine the shape of the coalition.

The coalition is yet to be formed (it took Netanyahu 28 days to realize that the alliance between the two parties is for real and unbreakable), and of course it is yet to be seen how the next government will function. Yet the power of coalitions in coalitional games, and the motivation of various amalgamation axioms, is demonstrated nicely by the current negotiations.