This man is a genius
What is the Nash equilibrium and why does it matter ?
We can usually explain the past and sometimes predict the future. but, not without help. One of the most important tools at their disposal is the Nash equilibrium, named after John Nash, who won a Nobel prize in 1994 for its discovery. This simple concept helps economists work out how competing companies set their prices, how governments should design auctions to squeeze the most from bidders and how to explain the sometimes self-defeating decisions that groups make. What is the Nash equilibrium, and why does it matter?
John
Forbes Nash, Jr was an American mathematician born in 1928. At age
twenty he entered the Mathematics Department at Princeton with a one
sentence recommendation letter which simply said: “This man is a genius.”
The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1994….John C. Harsanyi, John F. Nash Jr., Reinhard Selten
The work of John Nash explained by Steffen Hoernig and Susana Peralta.
One of the best-known illustrations is the prisoner’s dilemma: two criminals in separate prison cells face the same offer from the public prosecutor. If they both confess to a bloody murder, they each face ten years in jail. If one stays quiet while the other confesses, then the snitch will get to go free, while the other will face a lifetime in jail. And if both hold their tongue, then they each face a minor charge, and only a year in the clink. Collectively, it would be best for both to keep quiet. But given the set-up, an economist armed with the concept of the Nash equilibrium would predict the opposite: the only stable outcome is for both to confess.
Prison Breakthrough
JOHN NASH arrived at Princeton University in 1948 to start his PhD with a one-sentence recommendation: “He is a mathematical genius”. He did not disappoint. Aged 19 and with just one undergraduate economics course to his name, in his first 14 months as a graduate he produced the work that would end up, in 1994, winning him a Nobel prize in economics for his contribution to game theory.
On November 16th 1949, Nash sent a note barely longer than a page to the Proceedings of the National Academy of Sciences, in which he laid out the concept that has since become known as the “Nash equilibrium”. This concept describes a stable outcome that results from people or institutions making rational choices based on what they think others will do. In a Nash equilibrium, no one is able to improve their own situation by changing strategy: each person is doing as well as they possibly can, even if that does not mean the optimal outcome for society. With a flourish of elegant mathematics, Nash showed that every “game” with a finite number of players, each with a finite number of options to choose from, would have at least one such equilibrium.
His insights expanded the scope of economics. In perfectly competitive markets, where there are no barriers to entry and everyone’s products are identical, no individual buyer or seller can influence the market: none need pay close attention to what the others are up to. But most markets are not like this: the decisions of rivals and customers matter. From auctions to labour markets, the Nash equilibrium gave the dismal science a way to make real-world predictions based on information about each person’s incentives.
<div style="position:relative;height:0;padding-bottom:50%"><iframe src="https://www.youtube.com/embed/t9Lo2fgxWHw?ecver=2" style="position:absolute;width:100%;height:100%;left:0" width="720" height="360" frameborder="0" allowfullscreen></iframe></div>
One example in particular has come to symbolise the equilibrium: the prisoner’s dilemma. Nash used algebra and numbers to set out this situation in an expanded paper published in 1951, but the version familiar to economics students is altogether more gripping. (Nash’s thesis adviser, Albert Tucker, came up with it for a talk he gave to a group of psychologists.)
It involves two mobsters sweating in separate prison cells, each contemplating the same deal offered by the district attorney. If they both confess to a bloody murder, they each face ten years in jail. If one stays quiet while the other snitches, then the snitch will get a reward, while the other will face a lifetime in jail. And if both hold their tongue, then they each face a minor charge, and only a year in the clink
There is only one Nash-equilibrium solution to the prisoner’s dilemma: both confess. Each is a best response to the other’s strategy; since the other might have spilled the beans, snitching avoids a lifetime in jail. The tragedy is that if only they could work out some way of co-ordinating, they could both make themselves better off.
The example illustrates that crowds can be foolish as well as wise; what is best for the individual can be disastrous for the group. This tragic outcome is all too common in the real world. Left freely to plunder the sea, individuals will fish more than is best for the group, depleting fish stocks. Employees competing to impress their boss by staying longest in the office will encourage workforce exhaustion. Banks have an incentive to lend more rather than sit things out when house prices shoot up.
Crowd trouble
The Nash equilibrium helped economists to understand how self-improving individuals could lead to self-harming crowds. Better still, it helped them to tackle the problem: they just had to make sure that every individual faced the best incentives possible. If things still went wrong—parents failing to vaccinate their children against measles, say—then it must be because people were not acting in their own self-interest. In such cases, the public-policy challenge would be one of information.
Nash’s idea had antecedents. In 1838 August Cournot, a French economist, theorised that in a market with only two competing companies, each would see the disadvantages of pursuing market share by boosting output, in the form of lower prices and thinner profit margins. Unwittingly, Cournot had stumbled across an example of a Nash equilibrium. It made sense for each firm to set production levels based on the strategy of its competitor; consumers, however, would end up with less stuff and higher prices than if full-blooded competition had prevailed.
Another pioneer was John von Neumann, a Hungarian mathematician. In 1928, the year Nash was born, von Neumann outlined a first formal theory of games, showing that in two-person, zero-sum games, there would always be an equilibrium. When Nash shared his finding with von Neumann, by then an intellectual demigod, the latter dismissed the result as “trivial”, seeing it as little more than an extension of his own, earlier proof.
In fact, von Neumann’s focus on two-person, zero-sum games left only a very narrow set of applications for his theory. Most of these settings were military in nature. One such was the idea of mutually assured destruction, in which equilibrium is reached by arming adversaries with nuclear weapons (some have suggested that the film character of Dr Strangelove was based on von Neumann). None of this was particularly useful for thinking about situations—including most types of market—in which one party’s victory does not automatically imply the other’s defeat.
Even so, the economics profession initially shared von Neumann’s assessment, and largely overlooked Nash’s discovery. He threw himself into other mathematical pursuits, but his huge promise was undermined when in 1959 he started suffering from delusions and paranoia. His wife had him hospitalised; upon his release, he became a familiar figure around the Princeton campus, talking to himself and scribbling on blackboards. As he struggled with ill health, however, his equilibrium became more and more central to the discipline. The share of economics papers citing the Nash equilibrium has risen sevenfold since 1980, and the concept has been used to solve a host of real-world policy problems.
One famous example was the American hospital system, which in the 1940s was in a bad Nash equilibrium. Each individual hospital wanted to snag the brightest medical students. With such students particularly scarce because of the war, hospitals were forced into a race whereby they sent out offers to promising candidates earlier and earlier. What was best for the individual hospital was terrible for the collective: hospitals had to hire before students had passed all of their exams. Students hated it, too, as they had no chance to consider competing offers.
Despite letters and resolutions from all manner of medical associations, as well as the students themselves, the problem was only properly solved after decades of tweaks, and ultimately a 1990s design by Elliott Peranson and Alvin Roth (who later won a Nobel economics prize of his own). Today, students submit their preferences and are assigned to hospitals based on an algorithm that ensures no student can change their stated preferences and be sent to a more desirable hospital that would also be happy to take them, and no hospital can go outside the system and nab a better employee. The system harnesses the Nash equilibrium to be self-reinforcing: everyone is doing the best they can based on what everyone else is doing.
Other policy applications include the British government’s auction of 3G mobile-telecoms operating licences in 2000. It called in game theorists to help design the auction using some of the insights of the Nash equilibrium, and ended up raising a cool £22.5 billion ($35.4 billion)—though some of the bidders’ shareholders were less pleased with the outcome. Nash’s insights also help to explain why adding a road to a transport network can make journey times longer on average. Self-interested drivers opting for the quickest route do not take into account their effect of lengthening others’ journey times, and so can gum up a new shortcut. A study published in 2008 found seven road links in London and 12 in New York where closure could boost traffic flows.
Game on
The Nash equilibrium would not have attained its current status without some refinements on the original idea. First, in plenty of situations, there is more than one possible Nash equilibrium. Drivers choose which side of the road to drive on as a best response to the behaviour of other drivers—with very different outcomes, depending on where they live; they stick to the left-hand side of the road in Britain, but to the right in America. Much to the disappointment of algebra-toting economists, understanding strategy requires knowledge of social norms and habits. Nash’s theorem alone was not enough.
A second refinement involved accounting properly for non-credible threats. If a teenager threatens to run away from home if his mother separates him from his mobile phone, then there is a Nash equilibrium where she gives him the phone to retain peace of mind. But Reinhard Selten, a German economist who shared the 1994 Nobel prize with Nash and John Harsanyi, argued that this is not a plausible outcome. The mother should know that her child’s threat is empty—no matter how tragic the loss of a phone would be, a night out on the streets would be worse. She should just confiscate the phone, forcing her son to focus on his homework.
Mr Selten’s work let economists whittle down the number of possible Nash equilibria. Harsanyi addressed the fact that in many real-life games, people are unsure of what their opponent wants. Economists would struggle to analyse the best strategies for two lovebirds trying to pick a mutually acceptable location for a date with no idea of what the other prefers. By embedding each person’s beliefs into the game (for example that they correctly think the other likes pizza just as much as sushi), Harsanyi made the problem solvable. A different problem continued to lurk. The predictive power of the Nash equilibrium relies on rational behaviour. Yet humans often fall short of this ideal. In experiments replicating the set-up of the prisoner’s dilemma, only around half of people chose to confess. For the economists who had been busy embedding rationality (and Nash) into their models, this was problematic. What is the use of setting up good incentives, if people do not follow their own best interests?
All was not lost. The experiments also showed that experience made players wiser; by the tenth round only around 10% of players were refusing to confess. That taught economists to be more cautious about applying Nash’s equilibrium. With complicated games, or ones where they do not have a chance to learn from mistakes, his insights may not work as well.
The Nash equilibrium nonetheless boasts a central role in modern microeconomics. Nash died in a car crash in 2015; by then his mental health had recovered, he had resumed teaching at Princeton and he had received that joint Nobel—in recognition that the interactions of the group contributed more than any individual.
In a Nash equilibrium, every person in a group makes the best decision for herself, based on what she thinks the others will do. And no-one can do better by changing strategy: every member of the group is doing as well as they possibly can. In the case of the prisoners’ dilemma, keeping quiet is never a good idea, whatever the other mobster chooses. Since one suspect might have spilled the beans, snitching avoids a lifetime in jail for the other. And if the other does keep quiet, then confessing sets him free. Applied to the real world, economists use the Nash equilibrium to predict how companies will respond to their competitors’ prices. Two large companies setting pricing strategies to compete against each other will probably squeeze customers harder than they could if they each faced thousands of competitors.
The Nash equilibrium helps economists understand how decisions that are good for the individual can be terrible for the group. This tragedy of the commons explains why we overfish the seas, and why we emit too much carbon into the atmosphere. Everyone would be better off if only we could agree to show some restraint. But given what everyone else is doing, fishing or gas-guzzling makes individual sense. As well as explaining doom and gloom, it also helps policymakers come up with solutions to tricky problems.
Armed with the Nash equilibrium, economics geeks claim to have raised billions for the public purse. In 2000 the British government used their help to design a special auction that sold off its 3G mobile-telecoms operating licences for a cool £22.5 billion ($35.4 billion). Their trick was to treat the auction as a game, and tweak the rules so that the best strategy for bidders was to make bullish bids (the winning bidders were less than pleased with the outcome). Today the Nash equilibrium underpins modern microeconomics (though with some refinements). Given that it promises economists the power to pick winners and losers, it is easy to see why.
prisoners’ dilemma:
The
prisoners’ dilemma is the best-known game of strategy in social
science. It helps us understand what governs the balance between
cooperation and competition in business, in politics, and in social
settings.
In the traditional version of the game, the police have arrested two suspects and are interrogating them in separate rooms. Each can either confess, thereby implicating the other, or keep silent. No matter what the other suspect does, each can improve his own position by confessing. If the other confesses, then one had better do the same to avoid the especially harsh sentence that awaits a recalcitrant holdout. If the other keeps silent, then one can obtain the favorable treatment accorded a state’s witness by confessing. Thus, confession is the dominant strategy (see game theory) for each. But when both confess, the outcome is worse for both than when both keep silent. The concept of the prisoners’ dilemma was developed by RAND Corporation scientists Merrill Flood and Melvin Dresher and was formalized by Albert W. Tucker, a Princeton mathematician.
The prisoners’ dilemma has applications to economics and business. Consider two firms, say Coca-Cola and Pepsi, selling similar products. Each must decide on a pricing strategy. They best exploit their joint market power when both charge a high price; each makes a profit of ten million dollars per month. If one sets a competitive low price, it wins a lot of customers away from the rival. Suppose its profit rises to twelve million dollars, and that of the rival falls to seven million. If both set low prices, the profit of each is nine million dollars. Here, the low-price strategy is akin to the prisoner’s confession, and the high-price akin to keeping silent. Call the former cheating, and the latter cooperation. Then cheating is each firm’s dominant strategy, but the result when both “cheat” is worse for each than that of both cooperating.
Arms races between superpowers or local rival nations offer another important example of the dilemma. Both countries are better off when they cooperate and avoid an arms race. Yet the dominant strategy for each is to arm itself heavily.
On a superficial level the prisoners’ dilemma appears to run counter to Adam Smith’s idea of the invisible hand. When each person in the game pursues his private interest, he does not promote the collective interest of the group. But often a group’s cooperation is not in the interests of society as a whole. Collusion to keep prices high, for example, is not in society’s interest because the cost to consumers from collusion is generally more than the increased profit of the firms. Therefore companies that pursue their own self-interest by cheating on collusive agreements often help the rest of society. Similarly, cooperation among prisoners under interrogation makes convictions more difficult for the police to obtain. One must understand the mechanism of cooperation before one can either promote or defeat it in the pursuit of larger policy interests.
Can “prisoners” extricate themselves from the dilemma and sustain cooperation when each has a powerful incentive to cheat? If so, how? The most common path to cooperation arises from repetitions of the game. In the Coke-Pepsi example, one month’s cheating gets the cheater an extra two million dollars. But a switch from mutual cooperation to mutual cheating loses one million dollars. If one month’s cheating is followed by two months’ retaliation, therefore, the result is a wash for the cheater. Any stronger punishment of a cheater would be a clear deterrent.
The following five points elaborate on the idea:
1. The cheater’s reward comes at once, while the loss from punishment lies in the future. If players heavily discount future payoffs, then the loss may be insufficient to deter cheating. Thus, cooperation is harder to sustain among very impatient players (governments, for example).
2. Punishment will not work unless cheating can be detected and punished. Therefore, companies cooperate more when their actions are more easily detected (setting prices, for example) and less when actions are less easily detected (deciding on nonprice attributes of goods, such as repair warranties). Punishment is usually easier to arrange in smaller and closed groups. Thus, industries with few firms and less threat of new entry are more likely to be collusive.
3. Punishment can be made automatic by following strategies like “tit for tat.” This idea was popularized by University of Michigan political scientist Robert Axelrod. Here, you cheat if and only if your rival cheated in the previous round. But if rivals’ innocent actions can be misinterpreted as cheating, then tit for tat runs the risk of setting off successive rounds of unwarranted retaliation.
4. A fixed, finite number of repetitions is logically inadequate to yield cooperation. Both or all players know that cheating is the dominant strategy in the last play. Given this, the same goes for the second-last play, then the third-last, and so on. But in practice we see some cooperation in the early rounds of a fixed set of repetitions. The reason may be either that players do not know the number of rounds for sure, or that they can exploit the possibility of “irrational niceness” to their mutual advantage.
5. Cooperation can also arise if the group has a large
leader, who personally stands to lose a lot from
outright competition and therefore exercises restraint, even though he knows that other small players will cheat. Saudi Arabia’s role of “swing producer” in the opec cartel is an instance of this.
prisoners’ dilemma
We can usually explain the past and sometimes predict the future. but, not without help. One of the most important tools at their disposal is the Nash equilibrium, named after John Nash, who won a Nobel prize in 1994 for its discovery. This simple concept helps economists work out how competing companies set their prices, how governments should design auctions to squeeze the most from bidders and how to explain the sometimes self-defeating decisions that groups make. What is the Nash equilibrium, and why does it matter?
John
Forbes Nash, Jr was an American mathematician born in 1928. At age
twenty he entered the Mathematics Department at Princeton with a one
sentence recommendation letter which simply said: “This man is a genius.”The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1994….John C. Harsanyi, John F. Nash Jr., Reinhard Selten
The work of John Nash explained by Steffen Hoernig and Susana Peralta.
One of the best-known illustrations is the prisoner’s dilemma: two criminals in separate prison cells face the same offer from the public prosecutor. If they both confess to a bloody murder, they each face ten years in jail. If one stays quiet while the other confesses, then the snitch will get to go free, while the other will face a lifetime in jail. And if both hold their tongue, then they each face a minor charge, and only a year in the clink. Collectively, it would be best for both to keep quiet. But given the set-up, an economist armed with the concept of the Nash equilibrium would predict the opposite: the only stable outcome is for both to confess.
Prison Breakthrough
JOHN NASH arrived at Princeton University in 1948 to start his PhD with a one-sentence recommendation: “He is a mathematical genius”. He did not disappoint. Aged 19 and with just one undergraduate economics course to his name, in his first 14 months as a graduate he produced the work that would end up, in 1994, winning him a Nobel prize in economics for his contribution to game theory.
On November 16th 1949, Nash sent a note barely longer than a page to the Proceedings of the National Academy of Sciences, in which he laid out the concept that has since become known as the “Nash equilibrium”. This concept describes a stable outcome that results from people or institutions making rational choices based on what they think others will do. In a Nash equilibrium, no one is able to improve their own situation by changing strategy: each person is doing as well as they possibly can, even if that does not mean the optimal outcome for society. With a flourish of elegant mathematics, Nash showed that every “game” with a finite number of players, each with a finite number of options to choose from, would have at least one such equilibrium.
His insights expanded the scope of economics. In perfectly competitive markets, where there are no barriers to entry and everyone’s products are identical, no individual buyer or seller can influence the market: none need pay close attention to what the others are up to. But most markets are not like this: the decisions of rivals and customers matter. From auctions to labour markets, the Nash equilibrium gave the dismal science a way to make real-world predictions based on information about each person’s incentives.
<div style="position:relative;height:0;padding-bottom:50%"><iframe src="https://www.youtube.com/embed/t9Lo2fgxWHw?ecver=2" style="position:absolute;width:100%;height:100%;left:0" width="720" height="360" frameborder="0" allowfullscreen></iframe></div>
One example in particular has come to symbolise the equilibrium: the prisoner’s dilemma. Nash used algebra and numbers to set out this situation in an expanded paper published in 1951, but the version familiar to economics students is altogether more gripping. (Nash’s thesis adviser, Albert Tucker, came up with it for a talk he gave to a group of psychologists.)
It involves two mobsters sweating in separate prison cells, each contemplating the same deal offered by the district attorney. If they both confess to a bloody murder, they each face ten years in jail. If one stays quiet while the other snitches, then the snitch will get a reward, while the other will face a lifetime in jail. And if both hold their tongue, then they each face a minor charge, and only a year in the clink
There is only one Nash-equilibrium solution to the prisoner’s dilemma: both confess. Each is a best response to the other’s strategy; since the other might have spilled the beans, snitching avoids a lifetime in jail. The tragedy is that if only they could work out some way of co-ordinating, they could both make themselves better off.
The example illustrates that crowds can be foolish as well as wise; what is best for the individual can be disastrous for the group. This tragic outcome is all too common in the real world. Left freely to plunder the sea, individuals will fish more than is best for the group, depleting fish stocks. Employees competing to impress their boss by staying longest in the office will encourage workforce exhaustion. Banks have an incentive to lend more rather than sit things out when house prices shoot up.
Crowd trouble
The Nash equilibrium helped economists to understand how self-improving individuals could lead to self-harming crowds. Better still, it helped them to tackle the problem: they just had to make sure that every individual faced the best incentives possible. If things still went wrong—parents failing to vaccinate their children against measles, say—then it must be because people were not acting in their own self-interest. In such cases, the public-policy challenge would be one of information.
Nash’s idea had antecedents. In 1838 August Cournot, a French economist, theorised that in a market with only two competing companies, each would see the disadvantages of pursuing market share by boosting output, in the form of lower prices and thinner profit margins. Unwittingly, Cournot had stumbled across an example of a Nash equilibrium. It made sense for each firm to set production levels based on the strategy of its competitor; consumers, however, would end up with less stuff and higher prices than if full-blooded competition had prevailed.
Another pioneer was John von Neumann, a Hungarian mathematician. In 1928, the year Nash was born, von Neumann outlined a first formal theory of games, showing that in two-person, zero-sum games, there would always be an equilibrium. When Nash shared his finding with von Neumann, by then an intellectual demigod, the latter dismissed the result as “trivial”, seeing it as little more than an extension of his own, earlier proof.
In fact, von Neumann’s focus on two-person, zero-sum games left only a very narrow set of applications for his theory. Most of these settings were military in nature. One such was the idea of mutually assured destruction, in which equilibrium is reached by arming adversaries with nuclear weapons (some have suggested that the film character of Dr Strangelove was based on von Neumann). None of this was particularly useful for thinking about situations—including most types of market—in which one party’s victory does not automatically imply the other’s defeat.
Even so, the economics profession initially shared von Neumann’s assessment, and largely overlooked Nash’s discovery. He threw himself into other mathematical pursuits, but his huge promise was undermined when in 1959 he started suffering from delusions and paranoia. His wife had him hospitalised; upon his release, he became a familiar figure around the Princeton campus, talking to himself and scribbling on blackboards. As he struggled with ill health, however, his equilibrium became more and more central to the discipline. The share of economics papers citing the Nash equilibrium has risen sevenfold since 1980, and the concept has been used to solve a host of real-world policy problems.
One famous example was the American hospital system, which in the 1940s was in a bad Nash equilibrium. Each individual hospital wanted to snag the brightest medical students. With such students particularly scarce because of the war, hospitals were forced into a race whereby they sent out offers to promising candidates earlier and earlier. What was best for the individual hospital was terrible for the collective: hospitals had to hire before students had passed all of their exams. Students hated it, too, as they had no chance to consider competing offers.
Despite letters and resolutions from all manner of medical associations, as well as the students themselves, the problem was only properly solved after decades of tweaks, and ultimately a 1990s design by Elliott Peranson and Alvin Roth (who later won a Nobel economics prize of his own). Today, students submit their preferences and are assigned to hospitals based on an algorithm that ensures no student can change their stated preferences and be sent to a more desirable hospital that would also be happy to take them, and no hospital can go outside the system and nab a better employee. The system harnesses the Nash equilibrium to be self-reinforcing: everyone is doing the best they can based on what everyone else is doing.
Other policy applications include the British government’s auction of 3G mobile-telecoms operating licences in 2000. It called in game theorists to help design the auction using some of the insights of the Nash equilibrium, and ended up raising a cool £22.5 billion ($35.4 billion)—though some of the bidders’ shareholders were less pleased with the outcome. Nash’s insights also help to explain why adding a road to a transport network can make journey times longer on average. Self-interested drivers opting for the quickest route do not take into account their effect of lengthening others’ journey times, and so can gum up a new shortcut. A study published in 2008 found seven road links in London and 12 in New York where closure could boost traffic flows.
Game on
The Nash equilibrium would not have attained its current status without some refinements on the original idea. First, in plenty of situations, there is more than one possible Nash equilibrium. Drivers choose which side of the road to drive on as a best response to the behaviour of other drivers—with very different outcomes, depending on where they live; they stick to the left-hand side of the road in Britain, but to the right in America. Much to the disappointment of algebra-toting economists, understanding strategy requires knowledge of social norms and habits. Nash’s theorem alone was not enough.
A second refinement involved accounting properly for non-credible threats. If a teenager threatens to run away from home if his mother separates him from his mobile phone, then there is a Nash equilibrium where she gives him the phone to retain peace of mind. But Reinhard Selten, a German economist who shared the 1994 Nobel prize with Nash and John Harsanyi, argued that this is not a plausible outcome. The mother should know that her child’s threat is empty—no matter how tragic the loss of a phone would be, a night out on the streets would be worse. She should just confiscate the phone, forcing her son to focus on his homework.
Mr Selten’s work let economists whittle down the number of possible Nash equilibria. Harsanyi addressed the fact that in many real-life games, people are unsure of what their opponent wants. Economists would struggle to analyse the best strategies for two lovebirds trying to pick a mutually acceptable location for a date with no idea of what the other prefers. By embedding each person’s beliefs into the game (for example that they correctly think the other likes pizza just as much as sushi), Harsanyi made the problem solvable. A different problem continued to lurk. The predictive power of the Nash equilibrium relies on rational behaviour. Yet humans often fall short of this ideal. In experiments replicating the set-up of the prisoner’s dilemma, only around half of people chose to confess. For the economists who had been busy embedding rationality (and Nash) into their models, this was problematic. What is the use of setting up good incentives, if people do not follow their own best interests?
All was not lost. The experiments also showed that experience made players wiser; by the tenth round only around 10% of players were refusing to confess. That taught economists to be more cautious about applying Nash’s equilibrium. With complicated games, or ones where they do not have a chance to learn from mistakes, his insights may not work as well.
The Nash equilibrium nonetheless boasts a central role in modern microeconomics. Nash died in a car crash in 2015; by then his mental health had recovered, he had resumed teaching at Princeton and he had received that joint Nobel—in recognition that the interactions of the group contributed more than any individual.
In a Nash equilibrium, every person in a group makes the best decision for herself, based on what she thinks the others will do. And no-one can do better by changing strategy: every member of the group is doing as well as they possibly can. In the case of the prisoners’ dilemma, keeping quiet is never a good idea, whatever the other mobster chooses. Since one suspect might have spilled the beans, snitching avoids a lifetime in jail for the other. And if the other does keep quiet, then confessing sets him free. Applied to the real world, economists use the Nash equilibrium to predict how companies will respond to their competitors’ prices. Two large companies setting pricing strategies to compete against each other will probably squeeze customers harder than they could if they each faced thousands of competitors.
The Nash equilibrium helps economists understand how decisions that are good for the individual can be terrible for the group. This tragedy of the commons explains why we overfish the seas, and why we emit too much carbon into the atmosphere. Everyone would be better off if only we could agree to show some restraint. But given what everyone else is doing, fishing or gas-guzzling makes individual sense. As well as explaining doom and gloom, it also helps policymakers come up with solutions to tricky problems.
Armed with the Nash equilibrium, economics geeks claim to have raised billions for the public purse. In 2000 the British government used their help to design a special auction that sold off its 3G mobile-telecoms operating licences for a cool £22.5 billion ($35.4 billion). Their trick was to treat the auction as a game, and tweak the rules so that the best strategy for bidders was to make bullish bids (the winning bidders were less than pleased with the outcome). Today the Nash equilibrium underpins modern microeconomics (though with some refinements). Given that it promises economists the power to pick winners and losers, it is easy to see why.
prisoners’ dilemma:
The
prisoners’ dilemma is the best-known game of strategy in social
science. It helps us understand what governs the balance between
cooperation and competition in business, in politics, and in social
settings.In the traditional version of the game, the police have arrested two suspects and are interrogating them in separate rooms. Each can either confess, thereby implicating the other, or keep silent. No matter what the other suspect does, each can improve his own position by confessing. If the other confesses, then one had better do the same to avoid the especially harsh sentence that awaits a recalcitrant holdout. If the other keeps silent, then one can obtain the favorable treatment accorded a state’s witness by confessing. Thus, confession is the dominant strategy (see game theory) for each. But when both confess, the outcome is worse for both than when both keep silent. The concept of the prisoners’ dilemma was developed by RAND Corporation scientists Merrill Flood and Melvin Dresher and was formalized by Albert W. Tucker, a Princeton mathematician.
The prisoners’ dilemma has applications to economics and business. Consider two firms, say Coca-Cola and Pepsi, selling similar products. Each must decide on a pricing strategy. They best exploit their joint market power when both charge a high price; each makes a profit of ten million dollars per month. If one sets a competitive low price, it wins a lot of customers away from the rival. Suppose its profit rises to twelve million dollars, and that of the rival falls to seven million. If both set low prices, the profit of each is nine million dollars. Here, the low-price strategy is akin to the prisoner’s confession, and the high-price akin to keeping silent. Call the former cheating, and the latter cooperation. Then cheating is each firm’s dominant strategy, but the result when both “cheat” is worse for each than that of both cooperating.
Arms races between superpowers or local rival nations offer another important example of the dilemma. Both countries are better off when they cooperate and avoid an arms race. Yet the dominant strategy for each is to arm itself heavily.
On a superficial level the prisoners’ dilemma appears to run counter to Adam Smith’s idea of the invisible hand. When each person in the game pursues his private interest, he does not promote the collective interest of the group. But often a group’s cooperation is not in the interests of society as a whole. Collusion to keep prices high, for example, is not in society’s interest because the cost to consumers from collusion is generally more than the increased profit of the firms. Therefore companies that pursue their own self-interest by cheating on collusive agreements often help the rest of society. Similarly, cooperation among prisoners under interrogation makes convictions more difficult for the police to obtain. One must understand the mechanism of cooperation before one can either promote or defeat it in the pursuit of larger policy interests.
Can “prisoners” extricate themselves from the dilemma and sustain cooperation when each has a powerful incentive to cheat? If so, how? The most common path to cooperation arises from repetitions of the game. In the Coke-Pepsi example, one month’s cheating gets the cheater an extra two million dollars. But a switch from mutual cooperation to mutual cheating loses one million dollars. If one month’s cheating is followed by two months’ retaliation, therefore, the result is a wash for the cheater. Any stronger punishment of a cheater would be a clear deterrent.
The following five points elaborate on the idea:
1. The cheater’s reward comes at once, while the loss from punishment lies in the future. If players heavily discount future payoffs, then the loss may be insufficient to deter cheating. Thus, cooperation is harder to sustain among very impatient players (governments, for example).
2. Punishment will not work unless cheating can be detected and punished. Therefore, companies cooperate more when their actions are more easily detected (setting prices, for example) and less when actions are less easily detected (deciding on nonprice attributes of goods, such as repair warranties). Punishment is usually easier to arrange in smaller and closed groups. Thus, industries with few firms and less threat of new entry are more likely to be collusive.
3. Punishment can be made automatic by following strategies like “tit for tat.” This idea was popularized by University of Michigan political scientist Robert Axelrod. Here, you cheat if and only if your rival cheated in the previous round. But if rivals’ innocent actions can be misinterpreted as cheating, then tit for tat runs the risk of setting off successive rounds of unwarranted retaliation.
4. A fixed, finite number of repetitions is logically inadequate to yield cooperation. Both or all players know that cheating is the dominant strategy in the last play. Given this, the same goes for the second-last play, then the third-last, and so on. But in practice we see some cooperation in the early rounds of a fixed set of repetitions. The reason may be either that players do not know the number of rounds for sure, or that they can exploit the possibility of “irrational niceness” to their mutual advantage.
5. Cooperation can also arise if the group has a large
leader, who personally stands to lose a lot from
outright competition and therefore exercises restraint, even though he knows that other small players will cheat. Saudi Arabia’s role of “swing producer” in the opec cartel is an instance of this.
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