The Mystery Beeping Riddle

 

 

We came back from the airport with our daughter who has just landed after a four-month trip to India. The car was making a strange beep every so often.

Maybe it is an indicator signal that should have turned off automatically? No, this possibility was quickly eliminated.

Can it be the radio? We made sure the radio was off but the beeps continued.

I looked in the car manual. The only slightly similar symptom described there was a beeping indicating that the air bags are out of order and the air bag light warning signal is also out of order. Was this the reason? In this case there would be a 5-second beep every minute. But our beeps were once every 5 minutes and each beep was for one second. Was there some mistake in the translation of the manual to Hebrew?

I called the garage. Yes, they told me, if I bring the car they can check out what is wrong and fix it. No, they have not encountered this problem before. No, it is not dangerous to drive the car back to Jerusalem. And no, they were not familiar with translation problems in the manual.

Another breakthrough idea! Maybe the beeping came from a mobile phone in the car. Some mobile phones tend to beep when the battery is low or when there is an unread message. We turned off the two mobile phones in the car. This looked promising, Continue reading

Which Coalition to Form (2)?

Yair Tauman

(This post is a continuation of this previous post.)

Aumann and Myerson proposed that if political and ideological matters are put aside, the party forming the coalition would (or should) prefer to form the coalition in which its own power (according to the Shapley-Shubik power index) is maximal. They expected that this idea would have some predictive value  —  even in reality, where political and ideological considerations are of importance. A few days ago Yair Tauman, another well-known Israeli game theorist, mentioned on TV this recipe as a normative game-theoretic recommendation in the context of the recent Israeli elections. (For Yair’s analysis see also this article. (I even sent a critical comment.))

Over the years, Aumann was quite fond of this suggestion and often claimed that in Israeli elections it gives good predictions in some (but not all) cases. The original paper mentions the Israeli 1977 elections and how delighted one of the authors was that four months after the elections a major “centrist” party joined the coalition, leading to a much better Shapley value for the party forming the coalition.

I was quite skeptical about the claim that the maximum-power-to-the-winning-party rule has any predictive value and in 1999 with the help of Sergiu Hart I decided to test this claim. I asked Aumann which Israeli coalition he regards as fitting his prediction the best. His answer was the 1988 election where Shamir’s party, the Likud, had a very large Shapley value in the coalition it formed. We checked how high the Shapley value was compared to a random coalition that the winning party could have formed. Continue reading

Which Coalition?

The problem.

OK, we had an election and have a new parliament with 120 members. The president has asked the leader of one party to form a coalition. (This has not happened yet in the Israeli election but it will happen soon.) Such a coalition should include parties that together have more than 60 seats in the parliament.

Can game theory make some prediction as to which coalition will be formed or give some normative suggestions on which coalition to form?

Robert Aumann and Roger Myerson made  (in 1977) the following concrete suggestion.  (Update: A link to the full 1988 paper is now in place.)  The party forming the coalition would (or should) prefer to form the coalition in which its own power (according to the Shapley-Shubik power index) is maximal. Of course Continue reading

Basic Open Research and Failed Institutions – Imagine

Imagine if in the last ten years before the collapse, the huge failed financial and insurance institutions had had independent research units devoted to doing basic, open, and critical research on matters of relevance to the business, ethics, and future of these institutions. Might it have made a small difference for the better regarding the fate of these institutions?

 

[Full disclosure:] I make my living by doing basic research.

Majority Rules! – The Story of Achnai’s Oven

It is election day in Israel, and an opportunity to tell the beautiful and moving story of Achnai’s oven. 

Towards the end of the first century, a few decades after the big Jewish rebellion against the Romans, the sages of the “Sanhedrin” (The highest court in Jewish law) had to determine whether a certain oven “the oven of Achnai” is appropriate to use according to the Jewish law.

With the exception of one sage, Rabbi Eliezer, all sages declared that the oven can become ‘impure’ and therefire it is not appropriate for use.

Rabbi Eliezer who was convinced that his position was right declared:   “If the rule is as I say [That Achnai’s oven is in fact pure], then let this carob tree prove it!” Then the carob tree flew out of the ground and landed thirty yards away.

The sages were not impressed: “One does not bring evidence from the carob tree!”

Rabbi Eliezer continued: “If the rule is as I say, then let the stream of water prove it.”  And the stream of water flowed backwards.

“One does not bring evidence from a stream of water,” replied the other sages.

“If the rule is as I say then let the walls of this house  prove it!” continued Rabbi Eliezer, and the walls began to fall inward.  Rabbi Yehoshua, Eliezer’s main opponent, censored the walls for their interference and they did not fall but neither did they return to their previous position.

“If the law is as I say then let it be proved by Heaven,” continued Rabbi Eliezer and indeed a voice from Heaven came and asserted that Rabbi Eliezer was right! Rabbi Yehoshua stood up and said (quoting Deuteronomy 30:12) “It is not in Heaven, ” (לא בשמיים היא) and a later sage explained further (quoting Exodus 23:2): “it is for the majority to decide!” (כי אחרי רבים להטות). And so it was.

The Retaliation Game

 

We have two players playing in turns. Each player can decide to stop in which case the game is stopped and the two players can go on with their lives, or to act.

The player that acts gains and the other player loose twice as much. 

So in the first round: player I can stop the game, if he acts he gets 1 and the other player II gets -2.
If the games continues in the second turn player II can stop the game, if he acts he gets 1 and player I gets -2.
If the games continues in the third turn player I can stop the game, if he acts he gets +1 and player II gets -2.

How to play this game?

Debates

 

Debates are fascinating human activities that are a mixture of logic, strategy, and show. Not everybody shares this fascination. The German author Emil Ludwig considered debates to be the death of conversation. Jonathan Swift regarded debates as the worst sort of conversation, and debates portrayed in books as the worst sort of reading. Public debates pose various interesting dilemmas. Continue reading

Thomas Bayes and Probability

Decision Theory under Uncertainty

How can we assign probabilities in cases of uncertainty?  And what is the nature of probabilities, to start with?  And what is the rational mechanism for making a choice under uncertainty?

Thomas Bayes lived in the eighteenth century.  Bayes’ famous formula shows how to update probabilities given some new evidence. Following is an example for an application of Bayes’ rule:

Suppose that ninety percent of pedestrians cross a certain crosswalk when the light is green, and ten percent cross it when the light is red. Suppose also that the probability of being hit by a car is 0.1% for a pedestrian who crosses on a green light, but the probability of being hit by a car is 2% for a pedestrian who crosses on a red light. A pedestrian is hit by a car at this particular crossing and brought to the hospital. How likely is it that he crossed on a red light?

Well, to start with (or a priori), only ten percent of the people who cross the crosswalk cross it on a red light, but now that we are told that this person was hit by a car it makes the probability that he crossed illegally higher. But by how much? Bayes’ rule allows us to compute this (a posteriori) probability. I will not describe the mathematical formula, but I will tell you the outcome: the probability that this person crossed on a red light is 2/3.   

The Bayesian approach can be described as follows. We start by assigning probabilities to certain events of interest and, as more evidence is gathered, we update these probabilities. This approach is applied to mundane decision-making and also to the evaluation of scientific claims and theories in philosophy of science.

Bayes’ rule tells us how to update probabilities but we are left with the question of how to assign probabilities in cases of uncertainty to begin with. What is the probability of success in a medical operation? What is the chance of your team winning the next baseball game? How likely is it that war will break out in the Middle East in the next decade? How risky are your stock-market investments? Continue reading

Would you decide the election if you could?

One mental experiment I am fond of asking people (usually before elections) is this:

Suppose that just a minute before the votes are counted you can change the outcome of the election (say, the identity of the winner, or even the entire distribution of ballots) according to your own preferences. Let’s assume that this act will be completely secret. Nobody else will ever know it.

Will you do it?

You can answer in this poll: 

The Prisoner’s Dilemma, Sympathy, and Yaari’s Challenge

Correlation and Cooperation

In our spring school devoted to Arrow’s economics, Menahem Yaari gave a talk  entitled “correlation and cooperation.” It was about games as a model of people’s behavior, and Yaari made the following points:

It is an empirical fact that people (players in a game)  act in a correlated way,

It is unscientific not to take this into account (although this is not taken into account in game theory and economics).

 

The prisoner’s dilemma

A basic example in game theory (which also played a central part in Yaari’s lecture) is the Prisoner’s dilemma. Let’s talk about this example a little, before getting to Yaari’s claims. Continue reading