I first saw the following clip from Mankiw's blog a few weeks back. Please watch it, and think of the following issues:
1. Is everyone playing the game rationally?
2. Does the talking before decision time matter?
3. What do you think will happen if the game is played repeatedly between these two people?
4. What would be your rational strategy, if you know you're playing this game with an irrational opponent?
I hope you will find a very close analogy of this clip to the time inconsistency issue.
Showing posts with label game theory. Show all posts
Showing posts with label game theory. Show all posts
EE432 2009: Revision Series - Time Inconsistency
We have not quite finished the course yet, and are seriously behind the schedule thanks to the protest. Unfortunately, the exam cares not about politics and the final draws closer as we speak, so it's a good idea that we start the revision early. To start with, here are some questions from a student related to topic 5: Time inconsistency.
Q1: Phillips curve...is it a tradeoff between (1) inflation and unemployment, or (2) inflation and output? Does it mater?
Unemployment and output are both measures of the economic activity, so they are tightly correlated. The empirical relationship between the two is called the Okun's law. No surprise the correlation is negative, as in an economic expansion you'd expect unemployment to be low and output high, while in a recession the opposite happens. You can think of it in terms of production function - output is a function of labour input, so the correlation must be there.
So for me it is almost semantic whether we use unemployment or output in the Phillips curve...the choice doesn't really matter conceptually. What we're essentially doing is tracing a relationship between inflation and economic activity.
Q2: I don't understand abt the third solution in the term of Reputation Equilibrium and Subgame Perfect.
We say in the lecture that the Nash equilibrium in output and inflation is inefficient, as the Phillips curve allow the inflation to be closer to desired level while the output gap needs not change. Of course the problem is, the central bank cannot credibly commit to that desired inflation. We calculate this Nash equilibrium assuming that the game is played only once, or in the game theory term, it is a static game.
If we allow the game to be played many many times (i.e. we have a 'repeated game'), so that each player maximises the discounted sum of utility instead of just one single utility, then we can get all sorts of interesting Nash equilibria and a good Nash equilibrium may be achievable. The central idea is, cheating may only earn you a welfare gain in the short run, but cheating destroys the trust (or your reputation), and hence it can be bad for you in the long run. If the long-run loss is greater than the short-term gain, you will decide not to cheat. And we get a good equilibrium.
Let me sketch 2 examples.
Suppose the game is played for infinite number of times. The public may decide to adopt the following strategy: Expect low inflation to begin with, and continue to expect low inflation as long as the central bank implements low inflation. But as soon as the central bank chooses a high inflation even for one period, expect a high inflation forever. This is called a 'trigger strategy', i.e. I choose to be well-behaved as long as my opponent cooperates, and retaliate forever if my opponent cheats even only once. Under this strategy, the central bank may find that cheating only earns him one period gain (when he surprises the public), and loss of welfare in every period after that. Cheating to get short-term gain may not be worth it for the central bank, in which case, we can observe a low inflation every period. Good equilibrium rules.
In the 2nd example, suppose the game is played twice, and there are 2 types of central bank, one that is like in our lecture (i.e. wants to cheat), and the other who cares only about low inflation. The public doesn't know which type the central bank is. In this case, the outcome of the game in the 1st period tells the public something about the type of the central bank. If the central bank misbehaves in the 1st period, he gives away his type, and the public may expect a high inflation in the 2nd period, which is bad for him. Anticipating this, the central bank may implement a low inflation in the 1st period, to build a 'reputation' and pretends to be the 2nd type. This pretense allows the central bank to earn trust and manipulate the inflation expectation, and get a better inflation-output tradeoff in the 2nd period. You may find this line of reasoning and the model in Romer's book.
Q3: In solution 1, Can we derive other L preference to solve Time inconsistency Problem??
You certainly can. There is an infinite number of loss functions you can write down that will solve the problem just as well. Can you think of some?
Q4: In Policymaker Reaction function,if a=0 then this mean policymaker don't care inflation that made L=0 ?? I don't understand the economic intuitions behind this??
Yes, if a=0, that means the policy maker only cares about the output deviation from y_star. The loss then takes a parabola shape, with minimun point at y=y_star, at which point L=0 as you said. The exact value of L is of no importance...we're operating in ordinal utility world (and not cardinal), so we care only about the relative loss. When a=0, the minimum loss is L=0, which is the best that the central bank can hope for.
Q5: Policy maker must be let independent or discretion or commit the rule?? In my thought, CB should be independent to solve explosive inflation,but when CB commit the rule yield better solution than discretion.
When CB has no independence, the monetary policy is dictated by the elected government. In that case, you can safely assume the loss function will include the targeted output y_star term, as short-term growth brings votes. When CB has independence and retains discretion, you must ask what is the loss function. With the right loss function, efficient outcome may be achievable. Otherwise, committing to a rule (i.e. specifying a credible pre-commitment, which can take the form of legal commitment like a binding inflation-targeting regime), can improve the welfare.
Q6: In Journal, I found content is not the same in your lecture? I don;t understand so much so Need I understand this outside your lecture? Is Romer 9.4 is the same as we learn from you??
The Kydland-Prescott paper is a little hard, but the Barro one is easier. Romer's is the easiest one. The central idea is identical, whichever reading you consult.
To be continued.
Q7: We assume y*= natural rate output>th e reason is because Incentive Distortion- I don't understand this word. and its process and why CB must target y* > naturate output not eq output. Pleae example in term of income tax?
When you impose income tax, this distorts the people's incentives to work. They work less hard, and produce less. The resulted neutral rate of output is y-bar, the natural rate. But in an ideal world, give the technology and preferences for consumption/leisure etc, it is socially desirable for this economy to produce more output (optimal, without tax distortions, to sleep less and work m0re) at y_star. This is what the CB is after...the first-best solution.
Q8: I know that because y*> natural rate output that drive inflation bias (constant term right? May be I don't clear this word) this made CB to cheat but how?
Q1: Phillips curve...is it a tradeoff between (1) inflation and unemployment, or (2) inflation and output? Does it mater?
Unemployment and output are both measures of the economic activity, so they are tightly correlated. The empirical relationship between the two is called the Okun's law. No surprise the correlation is negative, as in an economic expansion you'd expect unemployment to be low and output high, while in a recession the opposite happens. You can think of it in terms of production function - output is a function of labour input, so the correlation must be there.
So for me it is almost semantic whether we use unemployment or output in the Phillips curve...the choice doesn't really matter conceptually. What we're essentially doing is tracing a relationship between inflation and economic activity.
Q2: I don't understand abt the third solution in the term of Reputation Equilibrium and Subgame Perfect.
We say in the lecture that the Nash equilibrium in output and inflation is inefficient, as the Phillips curve allow the inflation to be closer to desired level while the output gap needs not change. Of course the problem is, the central bank cannot credibly commit to that desired inflation. We calculate this Nash equilibrium assuming that the game is played only once, or in the game theory term, it is a static game.
If we allow the game to be played many many times (i.e. we have a 'repeated game'), so that each player maximises the discounted sum of utility instead of just one single utility, then we can get all sorts of interesting Nash equilibria and a good Nash equilibrium may be achievable. The central idea is, cheating may only earn you a welfare gain in the short run, but cheating destroys the trust (or your reputation), and hence it can be bad for you in the long run. If the long-run loss is greater than the short-term gain, you will decide not to cheat. And we get a good equilibrium.
Let me sketch 2 examples.
Suppose the game is played for infinite number of times. The public may decide to adopt the following strategy: Expect low inflation to begin with, and continue to expect low inflation as long as the central bank implements low inflation. But as soon as the central bank chooses a high inflation even for one period, expect a high inflation forever. This is called a 'trigger strategy', i.e. I choose to be well-behaved as long as my opponent cooperates, and retaliate forever if my opponent cheats even only once. Under this strategy, the central bank may find that cheating only earns him one period gain (when he surprises the public), and loss of welfare in every period after that. Cheating to get short-term gain may not be worth it for the central bank, in which case, we can observe a low inflation every period. Good equilibrium rules.
In the 2nd example, suppose the game is played twice, and there are 2 types of central bank, one that is like in our lecture (i.e. wants to cheat), and the other who cares only about low inflation. The public doesn't know which type the central bank is. In this case, the outcome of the game in the 1st period tells the public something about the type of the central bank. If the central bank misbehaves in the 1st period, he gives away his type, and the public may expect a high inflation in the 2nd period, which is bad for him. Anticipating this, the central bank may implement a low inflation in the 1st period, to build a 'reputation' and pretends to be the 2nd type. This pretense allows the central bank to earn trust and manipulate the inflation expectation, and get a better inflation-output tradeoff in the 2nd period. You may find this line of reasoning and the model in Romer's book.
Q3: In solution 1, Can we derive other L preference to solve Time inconsistency Problem??
You certainly can. There is an infinite number of loss functions you can write down that will solve the problem just as well. Can you think of some?
Q4: In Policymaker Reaction function,if a=0 then this mean policymaker don't care inflation that made L=0 ?? I don't understand the economic intuitions behind this??
Yes, if a=0, that means the policy maker only cares about the output deviation from y_star. The loss then takes a parabola shape, with minimun point at y=y_star, at which point L=0 as you said. The exact value of L is of no importance...we're operating in ordinal utility world (and not cardinal), so we care only about the relative loss. When a=0, the minimum loss is L=0, which is the best that the central bank can hope for.
Q5: Policy maker must be let independent or discretion or commit the rule?? In my thought, CB should be independent to solve explosive inflation,but when CB commit the rule yield better solution than discretion.
When CB has no independence, the monetary policy is dictated by the elected government. In that case, you can safely assume the loss function will include the targeted output y_star term, as short-term growth brings votes. When CB has independence and retains discretion, you must ask what is the loss function. With the right loss function, efficient outcome may be achievable. Otherwise, committing to a rule (i.e. specifying a credible pre-commitment, which can take the form of legal commitment like a binding inflation-targeting regime), can improve the welfare.
Q6: In Journal, I found content is not the same in your lecture? I don;t understand so much so Need I understand this outside your lecture? Is Romer 9.4 is the same as we learn from you??
The Kydland-Prescott paper is a little hard, but the Barro one is easier. Romer's is the easiest one. The central idea is identical, whichever reading you consult.
To be continued.
Q7: We assume y*= natural rate output>th e reason is because Incentive Distortion- I don't understand this word. and its process and why CB must target y* > naturate output not eq output. Pleae example in term of income tax?
When you impose income tax, this distorts the people's incentives to work. They work less hard, and produce less. The resulted neutral rate of output is y-bar, the natural rate. But in an ideal world, give the technology and preferences for consumption/leisure etc, it is socially desirable for this economy to produce more output (optimal, without tax distortions, to sleep less and work m0re) at y_star. This is what the CB is after...the first-best solution.
Q8: I know that because y*> natural rate output that drive inflation bias (constant term right? May be I don't clear this word) this made CB to cheat but how?
Is there other factors that driving this inflaton bias?
The central bank wants to raise output up beyond the natural rate, but the Phillips curve forbids it from doing it for free...the CB must sacrifice some inflation. So to get a higher output, they must allow inflation to rise. But the public catches up with the CB, and hence the CB ends up getting a higher inflation with no gain in output. The Nash equilibrium is inefficient, and is a consequence of the CB's incentive to raise output up at bit at the margin. Please make sure you understand the mechanics of the model well...this is the key point of this topic.
The central bank wants to raise output up beyond the natural rate, but the Phillips curve forbids it from doing it for free...the CB must sacrifice some inflation. So to get a higher output, they must allow inflation to rise. But the public catches up with the CB, and hence the CB ends up getting a higher inflation with no gain in output. The Nash equilibrium is inefficient, and is a consequence of the CB's incentive to raise output up at bit at the margin. Please make sure you understand the mechanics of the model well...this is the key point of this topic.
How Useful is Game Theory in Monetary Economics?
I get an interesting question from an ee432 student today, who asked about the application of game theory to monetary economics, or to any real-life problems for that matter. Given that I have more than a few things to say, I hope you don't mind me posting the answer up instead of replying to you privately.
The first thought that comes to my mind is a remark by one of my past teachers, that "Economics is really game theory". Of course it's no surprise that he IS a game theorist, but I can certainly relate to that view. Economics for the most part is really a study of how clever people get on with their lives in a tough world. And when you pitch more than 2 clever people to interact with each other, you have a game. It's no surprise then that game theoretical way of thinking has penetrated almost all fields in economics, in various disguises. And applications are not restricted to economics... biologists, diplomats, sociologists have all used the tool to a good measure of success.
Monetary economics is no exception. If you can recall, right from our first topic on the use of fiat money as a medium of exchange, we did employ the notion of Nash equilibrium, the good old concept in game theory. In that world, people are playing the game of 'let's coordinate our choice of money so that we can trade faster', and the result is that even an intrinsically worthless piece of paper can be regarded as money in Nash equilibrium...fiat money is born!
Or when we do rational expectations (either in Lucas or New Keynesian models), we are basically looking at agents who are strategically very smart and are reacting to any changes in aggregate demand pattern in a rational way. If you just add in another smart guy who's controlling the aggregate demand and reacting strategically to the public (a central bank!), then again you have a game! We will precisely do this in topic 6, the science of monetary policy. You will see then that the monetary policy in modern days is essentially a game between a mister Ben Bernanke and the public.
Or in our class tomorrow, when I will continue discussing the Diamond-Dybvig model of bank runs, you'll see that we use the Nash equilibrium concept again to describe the outcome under banking allocations. Whether or not there is a bank runs, it depends on your confidence of the bank. But your confidence depends on others confidence, so it's a strategic decision, and hence calls for a game theory as a framework. Or when we move on to topic 5, on time inconsistency, again we will look at a game played by the policy maker and the public. Game theory appears almost everywhere once you look closely.
And I can say that modern economic research in monetary economics has not had enough with game theory...in fact, it wants more! Examples of recent works which I like are by Susan Athey on the optimal degree of monetary policy discretion, Hyun Song Shin on the benefit/cost of policy transparency (social value of public info), Markus Brunnermeier on asset price bubbles formation for example. These works are probably too advanced for an undergrad project, but just to show you that the application is really endless. The challenge is really to master the skills and techniques, which may take time and patience. But I do believe the reward will be worth your while.
Also check out http://www.gametheory.net/ to get more ideas.
The first thought that comes to my mind is a remark by one of my past teachers, that "Economics is really game theory". Of course it's no surprise that he IS a game theorist, but I can certainly relate to that view. Economics for the most part is really a study of how clever people get on with their lives in a tough world. And when you pitch more than 2 clever people to interact with each other, you have a game. It's no surprise then that game theoretical way of thinking has penetrated almost all fields in economics, in various disguises. And applications are not restricted to economics... biologists, diplomats, sociologists have all used the tool to a good measure of success.
Monetary economics is no exception. If you can recall, right from our first topic on the use of fiat money as a medium of exchange, we did employ the notion of Nash equilibrium, the good old concept in game theory. In that world, people are playing the game of 'let's coordinate our choice of money so that we can trade faster', and the result is that even an intrinsically worthless piece of paper can be regarded as money in Nash equilibrium...fiat money is born!
Or when we do rational expectations (either in Lucas or New Keynesian models), we are basically looking at agents who are strategically very smart and are reacting to any changes in aggregate demand pattern in a rational way. If you just add in another smart guy who's controlling the aggregate demand and reacting strategically to the public (a central bank!), then again you have a game! We will precisely do this in topic 6, the science of monetary policy. You will see then that the monetary policy in modern days is essentially a game between a mister Ben Bernanke and the public.
Or in our class tomorrow, when I will continue discussing the Diamond-Dybvig model of bank runs, you'll see that we use the Nash equilibrium concept again to describe the outcome under banking allocations. Whether or not there is a bank runs, it depends on your confidence of the bank. But your confidence depends on others confidence, so it's a strategic decision, and hence calls for a game theory as a framework. Or when we move on to topic 5, on time inconsistency, again we will look at a game played by the policy maker and the public. Game theory appears almost everywhere once you look closely.
And I can say that modern economic research in monetary economics has not had enough with game theory...in fact, it wants more! Examples of recent works which I like are by Susan Athey on the optimal degree of monetary policy discretion, Hyun Song Shin on the benefit/cost of policy transparency (social value of public info), Markus Brunnermeier on asset price bubbles formation for example. These works are probably too advanced for an undergrad project, but just to show you that the application is really endless. The challenge is really to master the skills and techniques, which may take time and patience. But I do believe the reward will be worth your while.
Also check out http://www.gametheory.net/ to get more ideas.
Quote of the Week
Questioned whether the EU leaders discussed foreign exchange intervention, Jean-Claude Juncker said
"Would we have done so, I wouldn't answer the question, but we have not done so. Would we have done so, I would have denied that we did"
Cutting-edge Stuff
Knowledge flows freely in internet age...
- William Sandholm is writing a book on evolutionary game theory (specifically on population game). Get a free copy before it's printed!!!
- What's new in econometrics? A series of lectures in google videos by Imbens and Wooldridge on modern econometrics.
Champions League Afterthoughts
Here's the full story. Congrats to all the Man U fans! 
The game was decided after Nicholas Anelka's penalty was saved by Van de Sar. On this, Alex Ferguson commented..
On second thought, was he double-bluffing?
The game was decided after Nicholas Anelka's penalty was saved by Van de Sar. On this, Alex Ferguson commented..
"That wasn't an accident, his penalty save. We knew exactly where certain players were putting the ball, so great credit to him."If only someone had taught a bit of game theory to Anelka, he would have 'mixed his strategy' properly, and turned the tide for Chelsea.
On second thought, was he double-bluffing?
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