Three paradigms of maximization

Danny Finkelstein in the Times says game theory is often too complicated to solve real world problems. He's right, if you try to use it to get precise solutions. But it has another use. It reminds us that there are (at least) three paradigms in economics and it matters enormously that we know which paradigm is relevant for which problem*.

One paradigm is parametric maximization: we maximize expected utility subject to given parameters.

This, though, isn't universally applicable, as the two envelopes problem teaches us.

Imagine I give you and a colleague an envelope each. I tell both of you that one envelope contains twice as much money as the other. I then ask you if you want to swap, paying me a small sum if you do.

Conventional maximization suggests you should swap. You figure: "I have a 50% chance of a 100% return and a 50% chance of a 50% loss. That's an expected value of plus 25%."

If both of you think like this, you'll both pay me to swap. But having swapped, the same reasoning applies again. So you'll both swap again. Each time, I get richer and you two get poorer.   

The error you're making here is that parametric maximization doesn't apply. Your colleague is not a given parameter but a strategic thinker just like you. Once you see this you see that the game theory paradigm applies and so cease to be an egocentric framer. You figure "if he wants to swap envelopes, why should I do so?" You thus reject the trade and the logic that impoverishes you.

This is no mere thought experiment. One of the best-attested anomalies in finance is IPO under-performance; newly-floated stocks tend to do badly (pdf) in subsequent months. I suspect that one reason investors fall into this trap is that they apply the wrong paradigm. They think they are in the domain of parametric maximization and try to maximize expected returns for given prices but in fact they should apply the game-theory paradigm, and ask: if well-informed owners of the firm want to sell, why should I buy?

This isn't the only way in which getting the paradigm wrong can be costly. Governments or managers who set targets which are subsequently gamed make the same error. They think they are in the domain of parametric maximization when in fact they are in the game-theory domain: they forget that subordinates are not mere pawns to be moved, but strategic agents. (The Lucas critique makes a similar point about macroeconomic policy.)

However, there's a third paradigm - entrepreneurship. Introductory textbooks tell us that bosses try to maximize profits subject to given costs and technology. The entrepreneur, however, doesn't take these as given but rather tries to discover new technologies or newer cheaper suppliers. As Schumpeter said, he does things that "lie outside of the routine tasks which everybody understands." There's a thin line between the entrepreneur and the criminal because both try to break established rules.

Again, the failure to see which paradigm applies can be expensive. And not just in business but in politics too. The 1929-31 collapsed because it tried to impose austerity to keep the UK on the Gold Standard. When the subsequent coalition government took us off that standard, Sidney Webb, a Cabinet minister in the Labour government said: "nobody told us we could do that." This suggests the Labour government had been stuck in the parametric paradigm, when it should have applied the entrepreneurial one. Sometimes, constraints are only illusory.

Some of you might think the Labour party now is making the same error - of regarding public opinion and mediamacro as given parameters rather than trying to change them.

Maybe. But in business and in politics, entrepreneurship often fails. The parametric paradigm is often the right one. And sometimes - often - we just can't know whether it is or not.  

* I'm talking here about fully rational maximisation: let's leave behavioural economics aside for now.

Comments

[JS]

In the case of the envelope problem, conventional maximisation indicates that you do not swap.

The expected utility of swapping is ((u(x)0.5+u(2x)0.5) minus the cost of the swap. In this case, that's an identical envelope and the fee.

Swapping yields a negative expected return, so you won't swap.

Game theory is for strategic interaction, where your pay-offs are jointly determined by the actions of other players.

[Brendan]

Yes, I was about to post the same thing.

More concretely, if the envelopes contain x and 2x worth of money respectively, then by keeping half the time I'll get x and half the time I'll get 2x. The average is therefore 3x/2.

But with no information this is the same as for swapping, except now I have to pay a fee. So I'd never swap.

Surprised this error made it through. It's a pretty obvious one.

[Metatone]

Of course, there are things you can do to discover whether or not you the parametric paradigm is the right one. Not perfectly, but there are lots of tools that can give a sense.

What is notable about situations like 1929, Greece & the Labour party is that those who are most sure it is a parametric paradigm are those who have put no analysis time in using such tools.

Of course, the final thing to consider is that there are many ways for entrepreneurship to fail in execution. So the failure rate is not necessarily a good guide to the paradigm state.

[Matt Moore]

This is absolute hogwash - sorry Chris.

You don't know which envelope you have:

SWAP: half a chance at big, half a chance at small
DON'T SWAP: half a chance at big, half a chance at small

The prior that the envelopes are randomly assigned means that each has the same expected value, and you would never pay any amount to swap.

This isn't even game theory. Or if it is, as part of some extended game, then it is solved by a dominant strategy - which is the one equilibrium concept with very strong predictive power.

[e]

Labour’s error is in believing a public will maintain allegiance to a parametric paradigm. You would have thought that Syriza’s entrepreneurialism, which has sensationally highlight neoliberalism’s willingness to forsake all before it would jolt them, but it seems not.

[chris]

@ everyone. I agree that the correct solution to the envelopes problem is not to swap, and there are lots of ways of getting to this.
My point is merely that one of these ways is to adopt the game theory mindset, in the simple sense of asking: what is the other fellow doing? Yes, this is vague. But that's my point - a vague theory can be more useful than a precise one. (Do acedemics under-rate this point?).
@ Matt - dismissing the problem as hogwash is a bit harsh, considering that a lot of people have devoted a lot of quite sophisticated reasoning to the problem. And the fact that people overpay for IPO stocks suggests that many do commit an error that's very anologous to swapping envelopes.

[Sam]

'Conventional maximization suggests you should swap. You figure: "I have a 50% chance of a 100% return and a 50% chance of a 50% loss. That's an expected value of plus 25%." '

Only if you can't do math. (Of course, as everyone agrees, you shouldn't swap envelopes.)

The math error is that you're averaging a 100% gain from one basis and a 50% loss from a different basis, and calling it a 25% gain. Which is nonsense math.

JS correctly points out that parametric maximization, when performed correctly, leads to the correct answer.

Of course, people commit similar mathematical errors involving percentages and different bases on a regular basis.

[Rick Crawford]

With the two-envelope problem, one can easily show that swapping is irrational without a paradigm shift. Therefore, it was a bad example to illustrate your point.
A variant of the two-envelope problem does illustrate your point beautifully: Imagine if both parties knew which envelope was which.

[aragon]

So much for Hotelling (in Politics)

"Jeremy Corbyn "on course to come top" in the Labour leadership election"

http://www.newstatesman.com/politics/2015/07/jeremy-corbyn-course-come-top-labour-leadership-election

Most of Labour have internalised neoliberalism including the of the leadership candidates (especially Liz Kendal). Ed Miliband was still advocated Neoliberal economics, with Ed Balls.

Mortons fork is Red Tories vs Blue Tories, so why not stick with the Big Lie.

Therefore: Red Tories become irrelevant.

For the population who don't like austerity and there is no alternative (TINA) driven by pseudo economics.

They want an alternative, not the status quo.

p.s
You have a 50/50 chance of each outcome, and no information to make a swap or not swap decision (other than it incurs a cost). Are we talking about the Monty Hall problem?

https://en.wikipedia.org/wiki/Monty_Hall_problem

Which goat is that behind the opened door?
Tony Blair, Michael Foot etc.

[aragon]

No it's the two envelope problem!

https://en.wikipedia.org/wiki/Two_envelopes_problem

"Thus the fact that we are not told anything about how the envelopes are filled can already be converted into probability statements about these amounts. No information means that probabilities are equal."

"This means that even before you open your selected envelope you know that you will want to take the other envelope instead, because on average you will gain by the switch. This conclusion is obviously ludicrous."

Have fun ...

[aragon]

Read the post properly first (I never do, getting quickly to the fun bit, what I think), it gives the wikipedia link to the two envelope problem.

What we are presented with is the Ali Baba variant, or the Nalebuff asymmetric variant.

By swapping you gain 100%, or loose 50%.
It says so in the post.

Half x is less than x. The potential loss is less than potential gain.

The extra information is that the other envelope contains half x or two x and you hold x.

Is the fee greater than half x and do both parties have a veto on swapping (clearly not)? Not game theory though.

People are arguing over the extra information in politics!

What are the risks of the status quo (hotelling) vs change (product differentiation)

To what degree and who has the answers, management or radical reform (reject the framing)?

I am not sure Jeremy ha the answers. (Of course I do!)

Marketing Myopia (Levitt)?

"To continue growing, companies must ascertain and act on their customers’ needs and desires, not bank on the presumptive longevity of their products. In every case the reason growth is threatened, slowed or stopped is not because the market is saturated. It is because there has been a failure of management."

"Its theme is that the vision of most organizations is too constricted by a narrow understanding of what business they are in."

"The “new marketing myopia” occurs when marketers fail to see the broader societal context of business [political] decision making, sometimes with disastrous results for their organization and society."

https://en.wikipedia.org/wiki/Marketing_myopia

[rogerh]

Forget envelopes. The real problem is what should Labour do. As things stand the Tories are set to run till 2025. History tells us that incompetence, sleaze and arrogance will make them unelectable by then. So will Labour be fit to take over? Well, fit for what, a new socialist utopia? hardly, because remarkable claims need remarkable proof and there is none. We live in a capitalist world and looking forward I doubt 2025 will be any different. But work conditions will likely be rather less pleasant than today and more welfare (however you fudge it) will be needed. In the meantime developing an alternative source of funding and dumping the unions will, however distasteful a task, probably be a necessary preparation for Labour. Otherwise history will definitely repeat itself.

[Matt Moore]

@Chris. My apologies. That was needless. I wouldn't have said it face-to-face. I'll try to be more measured in my language in the future.

[Matthew Maloney]

Excellent post. This is must read material. Almost sums up the type of labour we see in the modern labour market - nerdy maximisers, entrepreneurs/creatives and political creatures.

[Christiaan Hofman]

As for the envelope theory, I also think I would never swap, because the average gain is $0 (0.5 * $50 + 0.5 * -$50) while you will pay the fine, so you will lose on average. But to the commenters already making this point above, it does not mean Chris's point is not valid, because he calculates an average gain (although he forgot to include the fine). How can this be? Simply: he uses a different utility function. I think Chris should have mentioned this: the choice is not only about your strategy and applying game theory, but the game theory outcome is also highly dependent on the choice of utility function. I think his utility function is not reasonable for this example (I would not use it), but it may well be one many investors use on the stock market. And that's really the point.

[NoniMausa]

Of course, the envelope problem isn't two agent, but three agent -- two traders, and a banker.

Moreover, as I always say "proportions matter." If the envelope contains, say, a years salary, and if the difference between envelopes is 2:1, with a small fine like 1%, then rational trading activity will be far different from an envelope discrepancy of 1:1.1 with a 12% fine.

If the discrepancy is not a flat amount, but a preference based on little information, being touted to each trader by the salesmanship of the banker, then both the traders could be persuaded to trade and pay the fine, even though the value of the envelopes is equivalent.

We have a real world example of this: the housing market. Except that the initial envelopes are filled not by the banker, but by the traders themselves.

[JS]

To deal with the case where one envelope has been opened (or indeed both have been opened but neither knows what the other envelope holds), from distant (very distant) memory you need to use Bayesian inference or put bounds on the contents.

[Billikin]

As for the two envelopes problem, there are an infinity of strategies that do better than always swapping or always holding. Here is one.

If the amount in the opened envelope is less than, say, £1,000, switch; otherwise, hold. You will always make the right decision when the smaller amount is greater than or equal to £500 and less than £1,000. Otherwise, your expectation is the same as always swapping or always holding.

[Billikin]

Well, I was a little quick on the trigger, but this is a -- pardon me -- silly variant of the two envelopes problem. Adding a second person and not having them open either envelope helps to avoid the cognitive error of the original problem. Putting a price on swapping makes the choice even easier. If conventional maximization suggests swapping, that's a problem for conventional maximization.

However:

"The error you're making here is that parametric maximization doesn't apply. Your colleague is not a given parameter but a strategic thinker just like you. Once you see this you see that the game theory paradigm applies and so cease to be an egocentric framer. You figure "if he wants to swap envelopes, why should I do so?" You thus reject the trade and the logic that impoverishes you."

No, you should reject the swap even if there is no other player with the other envelope. The problem is not parameterization per se. One envelope contains twice the amount in the other. Assign the parameter, L, to the lower amount. Then there are two envelopes, one containing L and the other containing 2*L. You do not know which one yours contains. If it is L, then switching gains you L. If it is 2*L, then switching loses you L.

The error comes from assigning the parameter, M, to the amount in your envelope, and assuming that the other envelope contains either M/2 or 2*M. This has nothing to do with whether the guy with the other envelope is sentient or not.