About Game theory
On this blog, our correspondents analyse and report on sports minor and major, addressing the politics, economics, science and statistics of the games we play and watch.
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Game theory models something very relevant to psychologists (in particular social psychologists): conflict and cooperation between decision-makers. Unfortunately, classical game theory demands that these decision makers are rational (in a mathematically precise sense). This definition of rationality is challenged empirically (by work like Tversky & Shafir) and on a theoretical level (by complexity results on finding or approximating Nash equilibria: a PPAD-complete problem).
Economists and mathematicians (and others) have taken two approaches to overcome this problem. The first is top-down approach of limiting the agent's abilities from an all-powerful rational agent down; this is the bounded rationality approach. The other is the bottom-up approach of evolutionary game theory: start with the simplest agents (that can't even make decisions) and have natural selection, imitation, or another simple dynamic process evolve the population over time. This seems to dove-tail nicely with the biogenic approach to cognition.
Are there examples of the tools of evolutionary game theory being used in social psychology or other sub-disciplines of the cognitive sciences? Is there a survey (or book) on EGT's impact on the cognitive sciences?
“Keeping up with him was all but impossible. The feeling was you were on a tricycle chasing a racing car”
Israel Halperin on John von Neumann
John von Neumann made huge contributions to many areas of mathematics and was also one of the founders of game theory.
His mathematical ability shone through from a young age. At the age of six he was able to divide two eight-digit numbers in his head and by the age of eight he had mastered calculus.
One of his major contributions to game theory was the Minimax theorem, which he proved in 1928.
The basic idea applies to two player zero-sum games. As it is a zero-sum game one player’s payoff is the opposite of the other’s payoff. So if one gets a payoff of +10 then the other will get a payoff of -10, the two payoffs add up to zero.
In any game you are trying to maximise your own payoff and your opponent is trying to do the same thing. In a zero-sum game, if you maximise your payoff then that is the same as minimising your opponent’s payoff. There is only a fixed amount to be shared between the players in a zero-sum game so if one wins an amount then the other loses an equal amount.
Given that you know that your opponent is going to play a strategy that will maximise their payoff, you want to play the strategy against them that will minimise their payoff (and therefore maximise yours).
Putting these two things together means that you want to play the strategy that minimises your opponent’ s maximum payoff. This is where the name ‘minimax’ comes from.
In zero-sum games the minimax solution is the same as the Nash equilibrium.
For students in New York and Boston, who have a range of options beyond their neighborhood school, choosing a high school used to be a maddeningly complicated guessing game. In Boston, for instance, many students would list their three top school choices — but were not guaranteed acceptance at any of them.
That made school selection a stressful quandary for many students and their families: Should they put highly rated but popular schools on their lists, despite the low odds of acceptance? Or should they list less desirable schools, to increase their chances of getting in?
Picking a school wasn’t just a matter of figuring out which schools were good: Because students had to think strategically and anticipate which choices others would make, it was a real-world exercise in game theory. And a frustrating one: At least 20 percent of Boston students, by some estimates, were making strategic errors; in New York, a third of students were shut out of the system without receiving any school assignments.
Just a decade ago, it seemed like an intractable problem. But that soon changed, thanks in part to a graduate student — now an MIT professor — named Parag Pathak.
Building a ‘strategy-proof’ system
In 2003, New York City schools chancellor Joel Klein, who wanted to revamp the school-choice system, approached a Harvard University professor named Alvin Roth about the problem. Roth had studied the method for matching medical students to their residencies; New York officials hoped something similar would work for their school system.
In turn, Roth asked Pathak, then a first-year PhD student in economics, to look into New York’s school-choice system: Was it a substantive and interesting problem? Pathak decided it was. A decade later, he is still producing new research on the topic, and in 2011 received tenure at MIT, in part because of his work in the area.
Moreover, that work has produced real-world results. Based on the research of Roth and his collaborators, New York City soon adopted what is known as a “deferred-acceptance algorithm” to assign places. Then, Roth’s group, now including economist Tayfun Sonmez, helped Boston review its choice system, leading the city to adopt a new method in 2005.
Using this method, schools first weigh all the students listing those schools as first-choice venues; then, the students who are rejected are essentially allowed to revise their lists, and the process repeats until every student has been matched with a school selection. The crucial difference is that students and families can simply pick the schools they most want to attend, in order.
“Our whole agenda is to try to make these systems strategy-proof,” says Pathak, now an associate professor of economics at MIT. “All these methods move in the direction of simplifying the system for students.” Complicated tactical guesses about popularity are moot; the entire process is based on the substantive merits of schools.
This positive outcome, Pathak says, is the fruit of “trying to think of economics as an engineering discipline,” in order to construct practical solutions to real-world problems.
Within economics, his growing area of specialization is known as “market design.” Beyond schools, market-design problems can be found in health care, financial markets, even the process of keyword searching on the Internet. “These allocation problems are everywhere,” says Pathak, who now also studies school-performance questions and has produced papers examining the quirks of housing markets.
What makes schools good?
Pathak is the son of Nepalese parents who immigrated to the United States in the 1970s. He grew up in Corning, N.Y., where his father is a doctor and his mother a writer, before attending Harvard as an undergraduate. A direct line can be drawn between Pathak’s career and a class he took during his senior year at Harvard in the spring of 2002, team-taught by Roth and Paul Milgrom, two leaders in market design; Milgrom advised the Federal Communications Commission on the design of their broadcasting-spectrum auctions.
Pathak, an applied mathematics major who graduated summa cum laude from Harvard, says that class allowed him to recognize the possibility of linking game theory with practical problems. He soon enrolled in graduate school in economics at Harvard, received his PhD in 2007 and joined MIT in 2008.
Since then, Pathak’s research on school-choice issues has expanded in part because other places, including Chicago and much of England, have adopted systems similar to the ones he endorses — but due to their own initiative. “It’s as if they followed the discussion in Boston, although there is no evidence of it,” Pathak says. “It’s a great story of how markets evolve.”
Although strategy-proof systems are gaining in popularity, many cities do not employ them. And yet Pathak believes that in addition to making the selection process simpler, the new systems can lead to a virtuous circle in assessing school quality: If administrators know what students’ real preferences are — as opposed to their tactics-based selections — they can examine what makes certain schools popular and try to institute those elements of good schools in other places, too.
“If we have programs that are oversubscribed, we should figure out why and consider replicating them,” Pathak says.
To be sure, it can be very difficult for people to assess whether or not schools are good in the first place, and for what reasons. In part because of this, Pathak’s interests have developed to include measuring school performance. Along with MIT economists Joshua Angrist and David Autor, he is a founding director of the School Effectiveness & Inequality Institute at MIT, a new center that launched this year.
Angrist, Pathak and a variety of co-authors have published multiple studies about the performance of charter schools in Massachusetts, for instance, using random samples of students from schools’ admissions lotteries. While recognizing that this can be a “politically charged” issue, Pathak says their aim is simply to shine some empirical light on the matter. So far, the results they have found are nuanced: Some charter schools in urban areas such as Boston have dramatically improved student performance, but charter schools in other parts of Massachusetts have generally performed worse than their non-charter public counterparts.
The researchers are still trying to determine exactly why this is, and aim to expand their studies geographically. But the technical expertise of Pathak and Angrist — a pioneer in developing and refining “natural experiments” in economics — makes them confident they can rigorously equitably assess thorny questions about student performance.
“Through school assignment, we have an engine to measure a lot of things about education production,” Pathak says. And now, students have a vehicle for choosing schools on their merits.
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This morning’s headline on BBC Breakfast was the news that yesterday RBS raised the interest rate on three of its mortgage products by a quarter of a percent to 4%. Three days ago the Halifax wrote to its mortgage holders saying that it intends to raise the cap on its Standard Variable Rate (SVR) to 3.75% above base rate, rather than the current 3%. As the Telegraph reports, although this doesn’t guarantee that Halifax would raise the rate itself, brokers”… believe otherwise and suggested that this would soon happen for a million Halifax borrowers” – and the BBC are now reporting an expectation that the Halifax will announce a rise in the SVR with effect from 1st May. For A level economists this story has several implications.
The first is clearly in macroeconomics, with the likely effect on consumer spending. We know well how squeezed consumers are, with rises in fuel and food prices, and the Bank of England has regularly expressed the importance of keeping base rates low in order to allow households to keep their heads above water. For mortgage holders with an interest-only mortgage, the RBS rate increase represents a 14% increase in their monthly mortgage payment – highlighted in this BBC video report.
The second concerns kinked demand curves and game theory. We have already seen RBS actually raising their rate and a likelihood that Halifax are about to do the same. Stuart Gregory from Lentune Mortgage Consultancy, who used to work for Halifax, told the Telegraph he expected to see a “chain of events” whereby other lenders also increased rates. “Once one does it, the others see it as an opportunity,” he said. This sounds like oligopoly behaviour to me. Mortgage lenders have all held their rates low for almost three years since Base Rate sunk to 0.5% in March 2009. This could be because, in close competition with each other, they have been stuck on the apex of a kinked demand curve, not feeling that the market would allow them either to raise rates and increase their returns – perhaps because competitors would not follow suit, and in any case their clients couldn’t afford the repayments - or to lower rates and try to gain larger market share – perhaps because they can’t afford the lower margins.
But something must have changed to cause RBS and Halifax to risk moving away from the established position, and guess that others will follow suit. There is certainly no signal from the Monetary Policy Committee or the Bank of England, who are more inclined at the moment to give the impression that 0.5% remains set in stone. The justification given by RBS is a rise in their borrowing costs. Partly due to rises in the LIBOR rate – as this graph shows, there may be some justification for this as there was a significant rise between November 2011 and the end of January 2012, although it has fallen back a little during February.
Source: thisismoney.co.uk
A second reason may be that savers are becoming more savvy in moving their funds between banks and accounts in order to seek the best possible rate, squeezing the margin between the rate that banks pay to savers and charge to borrowers – a new paradox of thrift. And a third was suggested by Paul Lewis on the BBC this morning; could it be that base rate, stuck so low for so long, is losing its power as a signal to other banks, who see official monetary policy as increasingly disconnected from the real world?
در زندگی بازیهای بسیاری انجام میدهیم ‘ خواه برای سرگرمی و کودکانه ‘ خواه برای سوداگری و سود آوری ‘ یا حتی عاشقانه‘