801. taust1 - March 30, 1999 - 1:47 AM PT
Re Pellenilsson #800
Some interesting thoughts in that Economist article, only the sub-dominant cooperate and only a few of those.

802. FreetoChoose - March 30, 1999 - 4:07 PM PT
The article on horse friendship is here

803. Slackjaw - April 3, 1999 - 8:59 PM PT
well, I know everyone has been anxiously awaiting the next installment, so I won't delay any longer. I will return to a post about the repeated pd after some posts on the solution concept that made game theory famous (well, relatively, anyway).

So far, we have talked about zero sum games and dominant strategies. Both are very special types of games, and their solutions are very simple to characterize. In zero sum games, the sum of the payoffs to all the players from any given combination of strategies is 0. In games with dominant strategies, a player has a choice that is best from her point of view, regardless of what the other players do.

But, because they are so special, the criteria used to find solutions in these games do not apply to the vast majority of games one might conceivably dream up. What if there is no dominant strategy in a non-zero sum game? Nothing we have said so far offers much guidance for predicting the outcome of this game when it's played by rational players.

I will first talk about moving to 1-shot games, or static games, that are represented in normal form. Also, for now I will be talking about games where every player knows every other player's utility function, they all know they all know them, and so on ad infinitum.

But there is a tradeoff: introducing the machinery to obtain solutions for general non-zero sum games requires introducing more restrictions on decision making abilities, and more importantly knowledge of other players' decision making abilities. All that we needed to justify our prediction in a game where each player had a dominant strategy (like the pd) was a belief that each player wanted to maximize utility. Now that is not trivial, as experimental evidence on the prisoners' dilemma suggests, but it is about as weak as you can get in standard game theory.

804. Slackjaw - April 3, 1999 - 9:00 PM PT
The next step we must make is to assume not only that every player is rational, and not even that every player knows every other player is rational, but they are all rational, know they are all rational, know that they all know they are all rational, and so on ad infinitum. That is, we must assume common knowledge of rationality. The difference is not trivial. (Note then that we are also assuming, per above, that preferences are common knowledge--also something we didn't need to do with dominant strategies.)

After the publication of _Games and Economic Behavior_ in 1944 by von Neumann and Morgenstern, game theory languished. Zero sum games were well understood, as were games with dominant strategies, but general n-player non-zero games were hopelessly complicated in the vN-M framework.

Along came a young lunatic-to-be named John Nash, who in the early '50s proposed a devlishly simple solution concept, since dubbed Nash equilibrium. It is simply this: for a strategy combination--a prescription of a strategy choice for every player--to be the solution to a game played by rational players, it must be the case that each player, *given what all the others are doing*, maximizes his utility by choosing his element of the prescribed strategy combination.

For many people, this really gets at the essence of game theory more than dominant strategies. It's a simultaneous utility maximization problem, where one of the constraints is the choice of the other player. Nash equilibrium is also a very elegant way to cut what used to be known as 'the problem of conjectural variations,' which is a phrase that most economists today never even here!

cont.

805. Slackjaw - April 3, 1999 - 9:01 PM PT
In a game, where the best thing to do depends on what the other player does, you can get into an infinite regress of the form, "if he thinks that I think he'll do A, he should do B, but if I think he thinks that I think that he'll do A, I should plan for B, so if he thinks that I think that he thinks that I think he'll do A, he should then do C, but..." etc. You can actually read papers where people get all worried about this, and Nash equilibrium bypasses, by automatically in a sense taking you to the infinity of that regress.

As an example, consider the following very simple game, called 'right side/left side.' The players must simultaneously and without communicating decide whether to drive on the right side or left side of the road. If they both choose 'right' or 'left,' they each get a payoff of 1. If they make opposite choices, they each get a payoff of -1. (So note that this is *not* a zero sum game! But I should point out, Nash equilibrium applies perfectly well to zero sum games, and games solvable by dominant strategies, and leads to the same predictions in those games as the more specialized solution concepts developed just for them.)

What does Nash equilibrium say about it? Well, a strategy combination is a Nash equilibrium if everyone does the best she can, given what everyone else is doing.

What if player 1 chooses R? Then 2 does best by choosing R. But if 2 chooses R, 1 does best by choosing R too. So R is a Nash equilibrium. So, obviously, is L, by the same logic. Simple as that.

But wait, there's more! One thing I have alluded to only briefly is the possibility of randomization. We allow players not only to play R or L, but to play R with some probability and L with 1 minus that probability. This is called a 'mixed strategy,' and playing L for sure or R for sure is a 'pure strategy.'

cont.

806. Slackjaw - April 3, 1999 - 9:01 PM PT
I won't actually go through the arithmetic, but it turns out that this game also has one Nash equilibrium in mixed strategies. If 1 plays R and L each with probability .5, then 2 maximizes utility by playing L and R each with probability .5 and vice versa. The essential thing to see about mixed strategies is that one player's randomization has to make the other player just indifferent between at least two of her possible pure strategies (which is all she has in this game). Otherwise, that other player would be unwilling to randomize. If you have two choices, and playing one of them for sure makes you better off than placing any probability on the other one, why would you ever randomize?

I am not spending a lot of time on mixed strategies not because they are unimportant in game theory--they are very important for development of theoretical constructs like Nash equilibrium--but because it is a largely arithmetical exercise, and going through doesn't add such a profound insight.

To be precise, I should say that Nash equilibrium is not actually a solution concept, in that it does not necessarily pick out one strategy combination as *the* solution, as the above game illustrates. It only tells you what characteristics a solution ought to have--if someone holds up a strategy combination as the solution to the game, and it does not meet this condition, you'd be rightly skeptical. "But why should that be the solution?" you'd say. "It is self defeating--if the players knew that was the solution, at least one of them wouldn't want to play it, given the strategies of the others!" In that sense, Nash equilibrium is very compelling.

But there are some reasons people are unsatisfied with it, not least of which is that there is no reason why a game should have only 1 Nash equilibrium in general.

cont.

807. Slackjaw - April 3, 1999 - 9:02 PM PT
Indeed, some games have Nash equilibria that are intuitively unreasonable. This has spawned the vast literature on refinements of Nash equilibrium--strengthening the concept to pick out fewer candidate solutions.

Nonetheless, the concept caught on. The reason is two fold, I think. First, Nash equilibria are usually very straightforward to work with. In applications they are often easy to find, and theoretically the construct has some nice mathematical features that made further research possible.

Second, while Nash equilibrium doesn't always make as specific a prediction as we'd like, it can almost always be used to make some prediction--to reduce, sometimes by a lot, the potential candidates for a solution to a game. The reason is Nash's theorem, which states that all finite games in normal form have at least 1 Nash equilibrium, possibly in mixed strategies. (By finite game, I mean a game with a finite number of players and a finite number of pure strategies for each player.)

808. AzureNW - April 3, 1999 - 10:01 PM PT

slackjaw -

What kinds of problems is this theory applied to in practice?

809. Slackjaw - April 4, 1999 - 12:35 AM PT
A darn good question in light of the preceding posts, Azure. I was just getting to that...seems good to present some elementary theory and then talk about applications. These applications will actually make some use of dominant strategies, but only for one of the players, so I think the logic of Nash equilibrium still comes across. (This doesn't mean that we can't still talk about commons stuff, by the way.)

So, here are some very simple, well known examples using game theory to analyze legal rules. Specifically, this is about tort liability regimes.

Imagine two players, pedestrian (P) and motorist (M). They are each approaching an intersection where they each have two choices: exericise due care (dc or 'careful') or no care (nc or 'careless').

Note already how the modeling discipline has winnowed the situation down to a very bare structure. There is nobody else around, neither player has the possibility of being 'too cautious,' there is a binary choice of levels of care, and neither player thinks there is a chance the other is just a lunatic who never exercises care. Whether this is a good representation is where the art of modeling rears its head. Obviously the assumptions lead to a highly stylized scenario, yet it can be useful if it gives us some unexpected kernel of insight into incentives in torts.

So let's suppose that exercising care is costly, say it carries a utility of -1 for each player. Being careless is costless. Further suppose that if at least one player is careless, there will definitely be an accident. If both are careful, there is a 10% chance of an accident. Suppose an accident carries a utility of -10 for P but M is a reprobate with no intrinsic feelings one way or the other about an accident--it carries a utility of 0 for her.

(cont.)

810. Slackjaw - April 4, 1999 - 12:36 AM PT
Consider first "no liability." Then if both players are careless, payoffs are (-10,0), where (x,y) notation means a payoff of x to P and y to M. If P is careful and M careless, payoffs are (-11,0); if P careless and M careful, (-10,-1); and if both are careful, (-2,-1). (All payoffs except this last one are obvious--but if both are careful, there's a 10% chance of an accident, which costs P -10 in the event that it occurs, so it carries an expected cost of -1. Plus, there is the -1 cost of being careful, leading to -2.) Clearly, M has a strictly dominant strategy to be careless. If M were to be careful, P would also want to be careful. But P knows M's dominant strategy. Given that, P's best choice is to be careless. So here we have not only a Nash equilibrium, but a solution by iterated elimination of dominated strategies. So neither player is careful, and the aggregate payoff is -10--much worse than the -3 obtaining when both are careful. Not a good rule.

Suppose we change the liability rule to one of "pure strict liability." Now changing the liability regime is modeled simply as a change in the allocation of the 'pie' from any given combination of strategies. That is, for every strategy combination above, a new liability rule will not change the sum, but only how it's distributed among the players. Now in pure strict liability, any accident is the fault of M, who must pay the cost of an accident, -1, to P. So, if neither is careful, payoffs are (0,-10); if P is careful and M careless, (-1,-10); if P careless and M careful, (0,-11); and if both careful, (-1,-2) (again, this is because M has a 1 in 10 chance of having to transfer 10 to P, and faces the -1 cost of being careful). Now it's P who has a dominant strategy: to be careless. If P were careful, M would want to be too; but given that P is going to be careless, M does best by being careless too--facing a cost of -10 instead of -11. Again, no care, and an inefficient outcome.

811. Slackjaw - April 4, 1999 - 12:36 AM PT
How about a third regime, "contributory negligence"? In this case, P pays the cost of the accident unless she took due care, in which case, if there's an accident, M transfers 10 to P. Now, payoffs are like so: both careless, (-10,0); P careful, M careless, (-1,-10); P careless, M careful, (-10,-1); both careful, (-2,-1). Now P has a dominant strategy again, only this time to be careful. If M is careless, P gets -1 from DC, -10 from NC; if M is careful, P gets -10 from NC, -2 from DC. But given what P is going to do, namely play DC, what should M do? Clearly, play DC: it pays -1, compared to -10. So we have gotten to the efficient outcome: both exercise due care.

There are many, many different possible liability regimes to analyze, but I will stop here. These examples are good because: (1) they are very simple (and don't involve mixed strategies) yet they illustrate the logic of solving games; (2) they illustrate the application of game theory to the study of institutions, not just economics; (3) they give a basic flavor of what implementation theory is like too. Notice that we examined several different games, and looked for games with rules that lead to the outcome we like.

812. Slackjaw - April 4, 1999 - 12:45 AM PT
what is the intuition about legal rules?

(i) the good one--contributory negligence--is a compensatory rule. Parties bear their own cost of care, but are exactly compensated for the cost of injury.

(ii) under contributory negligence, no damage is paid by M when she execises due care. Likewise, P is fully compensated for an accident caused by a careless motorist.

What about when an accident happens when both are duely careful? This contriutory negligence regime assigned all cost to P in that case, but other regimes are possible. As long as (i) and (ii) are met, a liability regime can divide according to any fixed proportion between M and P the cost of an accident when both are duely careful--it can be shown that the resulting game still has the nice equilibrium.

813. Slackjaw - April 4, 1999 - 12:47 AM PT
The proof of Nash's theorem on the existence of equilibrium is elegantly simple, which leads me to a digression on mathematics in economics. Not long ago in Slate, a book called _A Beautiful Mind_ was reviewed. The book is about John Nash, who shared the 1994 Nobel Prize in economics with two other game theorists, and his decades long bout with dementia among other things. The reviewer, a woman named Sylvia Nassar whom I believe does "economics" writing for the NY Times, offhandedly remarked how simple was Nash's theorem, but that its simplicity didn't stop economists from thinking it profound, because "economists have been known to win Nobel Prizes for rediscovering theorems from elementary calculus."

Well, that kind of made me mad, not because of what it said about economics--Nassar's comment about Nobels awarded for elementary calculus is just wrong--but for what it tried to say about mathematical discovery. Credit is given not just for the difficulty of working with the formal objects, but for recognizing connections and applications never before seen. An example will illustrate my point.

Mathematical statistics is a difficult subject that could not be impugned even by an idiot on the grounds that its practitioners are too easily impressed with formalism (even though one its main branches, decision theory, is actually a special case of game theory).

(cont.)

814. Slackjaw - April 4, 1999 - 12:48 AM PT
One of the most important theorems in that subject is called the Cramer-Rao theorem, which is very useful in helping to identify estimators (for population quantities) with very nice properties (like smallest possible variance in a given class of estimators).

As anyone who has seen the theorem knows, the proof is sublimely and famously simple. It is a *direct* application of something called the Cauchy-Schwartz inequality, which in turn is essentially a restatement of an elementary *axiom* known to everyone who's every studied mathematics: the triangle inequality. It says simply that for 3 points A, B and C, the distance from A to B plus from B to C is not less than the distance from A to C.

The triangle inequality isn't even a THEOREM! It's just true because we say so. It's an assumption about a distance metric that appears in any basic treatment of metric space topology. Not only did Cauchy (one of the more famous mathemeticians in history) and Schwartz get a restatement named after them because they saw it in a different way, but Cramer and Rao got one of the most important results in mathematical statistics named after them because they saw it again in a different way!

But of course nobody disputes that the Cramer-Rao theorem is profound. The fact that it was obtained with very simple machinery is an *admirable* aspect of the theorem.

The same is true, I think, with Nash's theorem. The proof of the result is indeed very simple, but if anyone doubts that it took some real insight to come up with not only the proof but the right construct that allowed for such a simple proof, just remember that John von Neumann, arguably the most brilliant person of the century and certainly in the top 10, never even got close.

End rant.

815. Pseudoerasmus - April 4, 1999 - 3:13 AM PT
Hey, Slackjaw, EXCELLENT EXCELLENT analogy with Cramer-Rao Inequality! I will appropriate this analogy to illustrate the utter stupidity of such ignorant criticisms as Jim Holt's against economics.

By the way, the NYT "economics" reporter, Sylvia Nasar, is the author of _A Beautiful Mind_, not its reviewer. Jim Holt, an actual mathematician, reviewed the book for Slate.

816. taust1 - April 4, 1999 - 6:48 PM PT
Please you guys keep in mind, whilst the asides are good as well, that some out here are waiting for the next installement.
It is an excellent primer summary. It is great communication. (The asides are too good to miss so keep them coming as well).

817. Slackjaw - April 4, 1999 - 7:35 PM PT
Message #816 I find that singularly amazing. But, fear not--more to come later.

Thanks PE. Jim Holt is a mathematician?! Well, that is really sad. Even worse than some journalist saying things like that--he's supposed to know better.

818. Slackjaw - April 4, 1999 - 7:39 PM PT
Here is the offending review of the book about Nash.

819. AzureNW - April 5, 1999 - 10:16 PM PT

*skat*

820. Slackjaw - April 5, 1999 - 10:37 PM PT
okay, I'll bite. Azure, what is this skat business all about?

821. AzureNW - April 6, 1999 - 12:30 PM PT

*skat* is just a test word. It's a kind of dropping.


I copied some of the thread, then posted *skat* on exiting to see if the Fray was still working.

822. chloel - April 6, 1999 - 1:56 PM PT
A review of a book that might be interesting.

823. CalGal - April 6, 1999 - 2:05 PM PT
Hey, Slack. I've archived this disucssion as follows:

Game Theory: Prisoner's Dilemma.
Game Theory: Nash Equilibrium.

And then here is the bibliography.

This is all still in rough form, so ignore the links for now (except the ones in the bibliography to books, which should work.) And for some reason, Geocities is having problems with the Geoguide manager, so pardon the annoying windows. I'll get rid of them soon.

This is part of an Economics TOC I'm putting together--I'm going through the old Economics archive and breaking it down by discussion. So not all the books have to do with game theory.

824. CalGal - April 6, 1999 - 2:43 PM PT
Slack,

I hadn't gotten caught up here in a while; I'm sorry for not acknowledging your posts sooner.

In reading about Nash's equilibrium, I found myself automatically thinking, well, how do you design for the best outcome? So I was relieved to see you say that your example illustrated implementation theory as well.

It doesn't surprise me that there are fewer books on implementation theory--in fact, I imagine it is the poor stepchild in the field. This feels somehow related to the fact that management systems are destined to not progress. Or the other way round.

Did you ever read that article in the Atlantic about the Valuejet crash? That also seems related. And in all cases, I can't say why I feel the same tug of recognition. But it is something to do with the human tendency to ignore reality--to ignore the fact that the game they want to play will not provide the solution they want to have.

825. doogie - April 6, 1999 - 2:51 PM PT
Has anyone read Cass Sunstein's newest, "The Cost of Rights?" Is it worth acquiring?

826. uzmakk - April 6, 1999 - 4:14 PM PT
Slackjaw:

Did you ever read the Harper's piece on Monsanto re: genetically altered seeds, their control, 3world farmers etc.? I mentioned it a week or two ago. It sounded very gamey to me.

827. Slackjaw - April 7, 1999 - 10:21 AM PT
CalGal: many thanks for putting the archive together. Mighty neighborly of you.

Poor stepchild? Well, it is certainly is not as well known as the parent, inside the field and out. But, the Nobel in 1996 was awarded for work on this subject (more or less), and I'm betting another will be awarded soon for a different part of it.

I think the lack of coverage has a lot to do with the fact that the interesting stuff in implementation requires so much game theory behind it, and calculus of variations. That's why it's hard to have a book on basic stuff for the uninitiated. It's true, it's applicability is certainly more limited than its parent, so it gets integrated into graduate game theory texts...but there is definitely a sense in which you're right, because it's one of those chapters towards the back most people never read.

Uzmakk: ak, no, I haven't gotten it yet! I am a little busy for the rest of this week but I will be able to get it this weekend. What's on the cover?

doogie: haven't seen that one. I've heard good things about Sunstein, but mostly from lawyers.

828. Slackjaw - April 7, 1999 - 10:23 AM PT
Myerson's _Game Theory_ is an exception, as it does a fantastic job of integrating the logic of mechanism design.

829. thoughtful - April 8, 1999 - 1:36 PM PT
I have a question for any/all who have an interest in such things. I'm looking to buy a pc-based statistical package and was wondering if anyone can recommend something. The two names I've got so far are E Views and StatGraphics. If anyone has any sense about which is better or any other suggestions would be most appreciated.

Thanks,
thoughtful

830. uzmakk - April 8, 1999 - 1:55 PM PT
Slackjaw:

William Shakespeare"s frill collar is on the front of the (Monsanto) Harper's

831. jayackroyd - April 9, 1999 - 8:44 AM PT
Message #829

The DOS based NCSS was fully featured, but my copy is at least 8 years old. Presumably there's something more current. You could check out the site: NCSS


My favorite was always RATS, just for the name, although it's a little specialized. The scariest was GLIM.

832. jayackroyd - April 9, 1999 - 9:39 AM PT
Slackjaw--

A while ago there was a discussion in another thread about promiscuity and adultery. More recently in the GG&S thread someone mentioned Jared Diamond's _The Third Chimpanzee_. I'm reading it now. (thanks, whoever you were.)

There is an interesting chapter on adultery, with game-theoretic implications.

It turns out that blood typing studies show that cuckolding is common. Diamond cites in detail an unpublished study done in the 1940s that "was straighforward: go to the obstretics ward of a highly respectable US hospital, collect blood samples from one thousand newborn babies and their mothers and fathers, identify the blood groups in all the samples; and then use standard genetic reasoning to deduce the inheritance patterns....(T)he blood groups revealed nearly 10 percent of these babies to be the fruits of adultery!" [emphasis in the original pp85-86] This obviously underreports the rate of cuckoldery, because blood types are not conclusive; a type O dad with a type O baby might have been cuckolded by a type O adulterer.

He goes on to cite a range of (similarly understated) values from 5% to 30% of cuckolded dads in the US and Britain in other studies.

Here's the question. Suppose men have two strategies--broadcast or nurture. Women have two strategies, faithfulness with a nurturer or cuckold the nurturer with a broadcaster. The benefit of the latter, if you don't get caught, is you get the genes of the broadcaster, who may well pass on broadcasting genes, while getting the resource assistance of the cuckolded nurturer. This kind of scenario seems to me to lend insight into the benefits of concealed ovulation. It lets women, cuckoo-like, trick a nurturer into raising a child that is not his own.

How could we model this in a game theoretic frame from the female perspective?

833. jayackroyd - April 9, 1999 - 9:46 AM PT
I may not have been clear here. Women who have male children genetically disposed to broadcast will have a greater likelihood of more next generation offspring, as the male children broadcast among the cuckolded nurturers' families. This is clearly beneficial, as long as she and the child are not killed by an irate cuckold. Concealed ovulation greatly reduces the risk of detection by the cuckold.

834. thoughtful - April 9, 1999 - 10:45 AM PT
RE #831, Thanks for the suggestion, Jay. I'll check it out.

835. Slackjaw - April 9, 1999 - 11:34 AM PT
jay--clarification: what do you mean by broadcast?

what can the cuckolded male do to his mate if she commits adultery? That will affect the tradeoff she faces between her two strategies. As it stands, it looks like the female can only benefit from cuckoldry.

Concealed ovulation enters this specification indirectly--say, it will influence the probability of detection by the cuckolded. So you can use two paramters for that probability, one with concealed ovulation and one without. Then find the equilibria under each parameterization, and show that the female's expected payoff is higher under the case to be interpreted as 'concealed ovulation.'

One problem with that is that, loosely speaking, it's a 'partial equilibrium' model. In one sense, one of the agents in the model is not strategic: males cannot respond to concealed ovulation. This is because concealed ovulation is treated parametrically. In a model in which concealed ovulation is a product of the model rather than an input or parameter in it, one would have to reckon with the fact that (based on the intution of the story above) males WANT to be able to detect ovulation, even though females want to be able to hide it. Just like some germs would love to be able to hide their presence, but humans and other critters have developed the ability to perceive them.

Of course, in equilibrium, it can happen that some virulent germs are undetectable by human senses; similarly we can't rule out that this sort of model would produce concealed ovulation. But, there is both a 'supply side' and a 'demand side' to detection in these situations and in the long run both have to be dealt with together.

836. Slackjaw - April 9, 1999 - 11:58 AM PT
incidentally, there is a whole offshoot of game theory called evolutionary game theory. I don't know nearly as much about it as 'standard' game theory (for lack of a better word), but I will say a few things about it later.

I still have some applications of Nash equilibrium in 1-shot games to present, then it's on to dynamic games (played over time, modeled in extensive form), then static (i.e., 1 shot) games of incomplete information, then dynamic games of incomplete information. Then a few words about evolutionary game theory and anything else I said I'd cover but have forgotten about. In the meantime, don't anyone be bashful about lingering on any topics covered so far.

837. Slackjaw - April 9, 1999 - 11:58 AM PT
But, right now I want to mention a neat little story. Turns out that the engagement ring custom is not that old, maybe 70-75 years. Before that, it seems there was such a thing as an engagement lawsuit. Women were confined to a given pool of suitors, and had fairly short courtship lives--if one made it to 25 without getting married, she was damn near an old maid. So if a man had a 6 or 8 month courtship and a 6 or 8 month engagement, but decided to break it off, he had 'used up' a good part of the woman's courting life. Legal action was a recourse for the woman's family to get compensation. (It sounds outrageous, but an eligible woman was like a machine with a life of 6 years or so. If you used if for a year, you had to pay for the depreciation.)

Well, for one reason or another, the engagement lawsuit fell out of favor with courts and women started winning them less and less. Now, women in that era had no trouble committing to a relationship: it was costly for them to break off an engagement, because they had to return to the same pool of suitors with a much lower chance of finding a mate. Besides, breakup was not nearly as costly to a man, who was not an old maid at 25, so he had a lot less to guard against.

But suddenly a man's committment was a matter of cheap talk. There was essentially no legal recourse waiting at the back end to enforce committment. One way around this is for a man to post a bond when he enters an engagement, the value of which he loses if it fails. That way, he has an incentive to take into account his use of the woman's eligible life and reputation.

Indeed, such a custom is exactly what emerged. We call the bond an 'engagement ring.'

This is why (one reason, anyway) it was so important that a ring, once given to the woman, could not return to the man upon breakup. That would totally destroy the incentive effects created by the bond.

838. Slackjaw - April 9, 1999 - 11:59 AM PT
Of course, in this era of mobility and relative sexual freedom, the woman does not have to re-enter the same engagement pool or suffer any harm to her 'reputation' if the engagement fails. So to achieve parity in committment, it is a lot less important (economically) that the engagement ring stay in the hands of the woman upon breakup.

Naturally, we don't know for sure whether this is helped the engagement ring custom to take off, but it's not only consistent withthe custom; it fits very nicely the circumstances under which it arose. It fits much better, in fact, than simply saying, "women liked nice gifts so men gave them" or something like that.

839. thoughtful - April 9, 1999 - 12:49 PM PT
When a woman told ZsaZsa Gabor that she was getting a divorce and asked if she should give back the ring, ZsaZsa replied, "But of course, dahlink, give back the ring -- but keep the stone!"

840. thoughtful - April 9, 1999 - 12:54 PM PT
My limited understanding is that the importance of the female cycle is not concealed ovulation, but the ability to have sex throughout the cycle. The females able to have sex more frequently were able to attract/retain male suitors thus ensuring a steadier supply of food, healthier offspring, and a longer life.

841. jayackroyd - April 10, 1999 - 4:13 AM PT
Slackjaw--

Broadcast is to have sex with a lot of women.

Why can't the male treat the concealed ovulation probabilisitically? There's a one in thirty chance of impregnation in a given act, say. If he can take thirty cuckolding shots a year, he's got an even chance of placing a cuckoo in a nurturer's nest.

The equilibrium of broadcasters has to be a minority; they're free-riding on the nurturers. Still, there have to be values for the variables involved that makes it beneficial for women to raise broadcast-fathered children with nurturing cuckolds.

842. jayackroyd - April 10, 1999 - 4:29 AM PT
Message #840

Same thing. Sex throughout the cycle, rather than entering heat. That _is_ concealed ovulation. That story is one story--that concealed ovulation is designed to keep nurturers at home and assisting in childrearing. Variations include:

1. Concealed ovulation evolved to reduce conflict among men for access to females. People live in packs, work collectively to gather resources. Conflict over access to females in heat would undermine collective activity.

2. Human females might have evolved constant estrus to bribe hunters with sex in exchange for meat. (Here, your focus on continuous availability is treated similarly.)

3. Women evolved concealed ovulation to force men into a permanent marriage bond, by exploiting male paranoia about paternity. Not knowing when she is fertile, the male must copulate frequently, leaving less time for dalliance. Moreover, he can't leave the female alone without running the risk of being cuckolded. (If she went into heat, he could safe dally, knowing she's not fertile.)

4. Women evolved concealed ovulation to confuse men about paternity A woman who distributed her favors widely would enlist many men to help feed a child that might be theirs.

5. Assume that some proto humans figured out that copulation produced children before ovulation was concealed. Suppose there were occasional mutations where ovulation was concealed. Child birth is very risky for proto human females. Those who could detect ovulation may well have chosen to avoid copulation, thus creating selection pressure for concealed ovulation.

(This is a paraphrase of _The Third Chimpanzee_ pp79-82.)

843. MsIvoryTower - April 10, 1999 - 7:28 AM PT
Hmm, the discussion on ovulation and cuckolding is interesting in the context of game theory. It suggests that male violence in the face of hidden ovulation is one (maximizing) strategy for reducing cuckolding, and rather than being an artifact of territoriality wrt women, it is an ONGOING response to continued female reproduction strategies.

844. jayackroyd - April 10, 1999 - 8:45 AM PT
I don't follow you, MsIT. The general claim seems to be that because hidden ovulation produces uncertainty about paternity, there is less male violence. Of course, if women were entirely monogamous, there would be no violence at all. It turns out that gibbons are, but they live in pairs, not packs.

Certainly the threat of death to woman and child is the downside of female cuckoldry. That they still practice it is an indication that something interesting is going on. Diamond points out that all the rules we know about up to about the last century are rules (preemptive--like female circumcision and punitive--like allowing the offending woman to be killed or compensation taken from her family) that prevent women from engaging in cuckoldry, but do not punish men for being cuckolds.

BTW, the issue of sexual dimorphism among primates came up in our previous discussion. It turns out that among our closer relatives (gorillas, gibbons, orangutans, chimps etc.) that we're second least dimorphic, with gibbons the least. There is apparently a rule of thumb for degree of dimorphism and degree of polygyny. The bigger the gap, the bigger the harems.

(And if someone can explain when to say "polygyny" and when to say "polygamy", I'd be grateful.)

845. MsIvoryTower - April 10, 1999 - 8:53 AM PT
"There is apparently a rule of thumb for degree of dimorphism and degree of polygyny. The bigger the gap, the bigger the harems."

Yes, I've come across this as well. I found it comforting, actually.


"The general claim seems to be that because hidden ovulation produces uncertainty about paternity, there is less male violence. Of course, if women were entirely monogamous, there would be no violence at all. It turns out that gibbons are, but they live in pairs, not packs."

Yes, I know, but I was speaking about male violence toward WOMEN not toward other males. I was actually mixing theories and issues, evolutionary theories about male violence generally, and sociological/psychological theories about domestic violence (or more particularly, male to female). My understanding is that some of the latter theories focus on male violence toward women as being dysfunctional, and as being a personality flaw. But, if we look to evolutionary theory, male to female violence isn't necessarily that at all, but a strategy to control/limit female reproduction options, by males IN THE FACE OF UNCERTAINTY.

846. MsIvoryTower - April 10, 1999 - 8:55 AM PT
errata

If we look to evolutionary AND game theory......

847. Slackjaw - April 11, 1999 - 12:18 AM PT
Jay,

"Why can't the male treat the concealed ovulation probabilisitically?" Sure, that could work, but regardless it's still parametric and a male does not have a chance to adapt a response. So in one sense, any explanation based on it will be wanting: why have males not adapted some method for identifying their own offspring?

But what are you trying to explain--the benefits of concealed ovulation, right? Then there is a probabilistic element, but the important part about it is the effect on the rate of detection of cuckoldry. Concealed ovulation lowers that rate.

Also, as you mention, "Certainly the threat of death to woman and child is the downside of female cuckoldry," but this cannot play a role because "kill the female and child" is not in the male's strategy space. You could treat that as the parameter, and treat this as a 1 person decision problem under uncertainty, rather than a multiperson decision problem. A woman has two choices: conceive with the nurturer in her own nest, and conceive with a broadcaster. Conceiving with a broadcaster provides a fitness benefit, the one you mentioned, but carries a cost because say there is a chance p of detection, leading to the death of the child and some utility loss to the woman. Mating with your nurturing husband leads to a smaller fitness gain, but carries a smaller cost. Then the best choice depends on your attitude toward risk. (For example, suppose some demented rapscallion shows up at your house and tells you that you must either (a) play a gamble where you win $1 million if a fair coin comes up heads, and lose $1 million if it comes up tails, or (b) pay said rapscallion a fee of $1000 to avoid the gamble, in which case he will stop haranguing you. People with different attitudes toward risk choose differently; similarly they will choose differently in the mating problem.)

(cont.)

848. Slackjaw - April 11, 1999 - 12:19 AM PT
This then is the reason for a mix of cuckolding and fidelity. An observable implication is that women who try to pull this off are more tolerant of risk (in principle you could round them up, sit them in a lab and present them with various gambles designed to measure that tolerance). (And, I am assuming you are right about the fitness benefit of cuckoldry.)

And, the effect of concealed ovulation on the woman's utility can be examined by allowing the parameter p to vary. Lower values are to be interpreted as concealed ovulation.

Actually, a man has a decision problem too: he can mate with a receptive woman (not his mate) and possibly impregnate her, but also possibly suffer the repercussions of a pissed off husband. Or he can pass it up and play it safe. That is, it is not true that, for a man, mating with a receptive woman is always better than not mating with her. Again, the optimal decision depends on the man's attitude toward risk; less risk averse men should be more likely to take advantage of a willing partner.

Now, the choice in the woman's problem certainly influences the choice set in the man's problem--if she elects not to be receptive to a broadcaster, his problem is degenerate as he has only one choice. But, the utilities are not mutually dependent. That is, unless the very fact of a man's willingness to mate with another's wife conveys some information about the man's traits that one wants in a child--say you want risk averse children, because they are more likely to make it to maturity and mate. In that case, you don't want the men who'd have affairs with you. We have not yet discussed the theoretical tools to deal with games where infomation is revealed as the game progresses.

849. Slackjaw - April 11, 1999 - 1:16 AM PT
Before presenting the liability games, I said that game theory could be used outside a context that is usually regarded as 'economic.' Here is an application of Nash equilibrium in static (1-shot) games that are 'economic.' This is long, and I have to introduce some basic econ to set the context. Please, ask if something isn't clear, or you object to anything that follows. Since this is so long, I won't post any new material on technical aspects of game theory for a few days so anyone who might care to read these words can do so at a leisurely pace.

Economics has a large body of received wisdom about competitive markets, with a lot of firms, and monopoly. Results center around the quantity transacted in the market, the market price, profits of the firms, the welfare of consumers, etc. For example, if a market is served by a monopolist who does not know anything about the customers other than the market demand, and for whose goods there is no resale market, then the allocation of goods will be inefficient--quantity will be lower than under perfect competition--and the price will be higher than under perfect competition. What often pisses people off about monopoly is the price issue--but a monopolist cannot unilaterally determine its price and quantity; it does face a market demand constraint. For example, if soda was sold in a monopolized market, the seller could not charge $600 per can and expect to sell much soda.

What upsets economists about this standard sort of monopoly is the quantity restriction--the monopoly results in less output than is efficient. (The reason is precisely the demand constraint: a profit maximizing monopolist reduces quantity relative to the competitve output, because doing so allows a higher price to be had per unit. The monopolists profit maximization problem comes down to a tradeoff: selling fewer units means less quantity at your given profit margin, but a higher price means a higher profit margin per unit...

850. Slackjaw - April 11, 1999 - 1:17 AM PT
...The monopolist's optimal output resolves this tradeoff so that increasing price just a little bit would increase profit from the one blade of the sword exactly as much as it would decrease it from the other.) So, under monopoly, the benefit to consumers of another unit of output is higher than the cost to the firm of another unit; that's obviously inefficient. Under perfect competition, market outcomes lead to a situation in which another unit of output leads to benefit to consumers exactly equal to its cost to the firm. This is the efficiency benchmark.

So far, none of this involves any game theory. And before game theory came around, the theory of oligopoly--markets served by a few firms--was not well developed (though it was not totally absent). Standard economics is very good with a whole ton of firms or with 1; it's not so good with somewhere in between. The reason is that with a few firms, each must take account of the production decisions of others in making its own choices. In monopoly this isn't true; in perfect competition there are so many firms that you can basically ignore each one: no individual has any effect on market outcomes. Not so in oligopoly.

One intersting question is whether oligopoly matters. Chicago economists were once fond of saying that 1 is monopoly; 2 is perfect competition. It turns out that the effects of oligopoly on market outcomes depend on the nature of the competition. So we'll consider two stories.

First: Suppose two firms produce an identical product at identical cost, and must simultaneously decide how much to bring to market, where the unit price will be determined by the market demand, which is known to both firms. Then the first firm's profit maximizing choice of output will depend on its prediction of the second firm's choice, and its prediction--which partially determines its own choie--in turn affects the profits of that other firm.

851. Slackjaw - April 11, 1999 - 1:19 AM PT
What we want is a Nash equilibrium, a configuration of quantities such that each firm is choosing the profit maximizing quantity, given the quantity choice of the other firm. But this game is interesting technically, as well as economically, because we cannot use the same simple algorithm as before to identify a Nash equilibrium. The reason is that the strategy space is infinite: each firm has an infinity of possible quantities to choose, unlike the finite games considered so far. It is still a very straightforward problem mathematically, but does require some basic calculus.

Indeed, this exact scenario was identified some 100 years before Nash wrote, by a French mathematician named Antoine Augustine Cournot, and so duopoly with quantity competition is still called 'Cournot competition.' Cournot not only identified the game, but found the "Nash" equilibrium! He made two fatal mistakes, however. First, he did not really consider the implications of what he said beyond simple duopoly games, for interactive decision problems in general; second, he wrote in French. As a result, his work was neglected for a long time, and never had the impact it could have.

Back to the game. Suppose for expositional purposes that demand is linear: P(Q)=a-Q, where Q is the total market quantity supplied, or q1+q2 (ie, quantity supplied by firm 1 plus q. supplied by firm 2), a is a known parameter, and P(Q) tells you the market price as a function of quantity. Then each firm's profit is total revenue minus total cost--say they both produce at cost c per unit, so profit for firm 1 say is T=(q1)(a-q1-q2)-(c)(q1). Taking q2 as given, firm 1 wants to maximize profit, or solve: a-2q1-q2-c=0, or q1=(a-c-q2)/2. But of course firm 2 solves an identical problem, and gets q2=(a-c-q1)/2.

852. Slackjaw - April 11, 1999 - 1:21 AM PT
(The calculus is over; it was used to determine the equation each firm must solve to obtain its profit maximizing quantity choice.) Here are two equations in two unknowns which can be solved to determine profit maximizing quantity choice, and in turn market prices, profits earned, consumers surplus, etc. (Consumers surplus, the difference between total utility obtained by consumers from a good and their expenditure on it, is a measure of consumer welfare arising from some market.) You can in turn perform the relevant analysis for monopoly, competitive markets, etc., and see how they compare in terms of consumers surplus.

It turns out that the profit maximizing (equilibrium) quantity choice for each firm is between the monopoly and competitive levels (competitive output is higher; monopoly lower), as are market prices (higher under monopoly; lower under competition), firm profits (higher under monopoly), and consumer's surplus (higher under competition). It also turns out that if you consider the identical scenario as the number of firms grows very large, the equilibrium quantity choices tend to the competitive levels. But it is manifestly false that 2 is the same, in terms of market outcomes, as perfect competition.

853. Slackjaw - April 11, 1999 - 1:21 AM PT
Now consider a second scenario: again, there is a duopoly-two firms compete. But instead of deciding how much quantity to deliver to market, they must decide on the price p to charge for their output. They simultaneously declare their prices to the market organizer. Then the market organizer will inform each how much they must produce and bring to market, based on the commonly known market demand (which gives quantity demanded as a function of price) and each firm's stated price. Suppose once again that they each produce at a cost of c per unit. Suppose also that consumers will buy only from the supplier with the lowest price.

This scenario-duopoly with price competition-is known as Bertrand competition, in honor of another French mathematician who criticized Cournot's work.

Notice one thing about this game straight away: for each firm, charging a price of p = c, which leads to zero profit (NOTE: reserve judgment and see the end of this sequence of posts if the notion of 0 profit is unsettling), is a weakly dominated strategy. I.e., you can sometimes do strictly better, and never do worse, than by charging c + e, for some small number e. You can never do worse because the price c + e never leads profits less than 0. If the rival firm charges a lower price, you simply get 0 profit, same as when you charge c.

When will you do better? If the other firm charges some price q > c + e. In that case, you will supply the whole market, and will earn a profit margin of e on every unit sold, instead of a profit margin of 0. As long as market demand is not infinitely sensitive to changes in price, so that increasing price by any tiny amount causes demand to plummet to 0, you make more profit by choosing c + e than e. Also notice that any choice p < c is strictly dominated, or worse than some other available choice, regardless of what the rival does. p < c yields negative profit, but p = c yields 0 profit. So choices less than c can be ignored.

854. Slackjaw - April 11, 1999 - 1:23 AM PT
Before, when talking about STRICTLY dominated strategies, we said these could be deleted from the game without loss of generality. This is how the prisoners' dilemma was solved before we had Nash equilibrium to work with. Part of the reason the Bertrand game is interesting is not just the economics, but the technical feature that it shows emphatically that the same is NOT true of WEAKLY dominated strategies like p = c. For, that is exactly the equilibrium choice for each firm.

Why? Suppose prices were something else, say p and q for 1 and 2 respectively, both above c, and p > q. Then for some small number e, 1 can charge q - e and capture the whole market; this leads to lower profit than the whole market at price p, but that was never an option. It could have nothing at p or the whole market at q - e. The latter is obviously better. But of course now 2 can simply charge q - 2e, say, making firm 2 better off. Indeed, for any choice of prices not equal to c for each firm, exactly this sort of analysis shows that the higher price firm would like to deviate, and charge a different price. That is, for any configuration of strategies other than p = c and q = c, it is not the case that each player is maximizing utility, given the choice of the other player. This is to say nothing but that no other price configuration can be a Nash equilibrium.

Is p = c and q = c a Nash equilibrium? Yes. Given the choice of firm 2, firm 1 cannot make any choice that pays more than 0 profit. A higher price will shut it out of the market; lower price yields negative profit. By charging p = c when firm 2 charges c, firm 1 earns 0 profit--the best it can do given 2's choice. The same goes for firm 2 vis a vis firm 1's choice, and we have a Nash equilibrium.

But then the market price is p = c, exactly what would be the market price under perfect competition! That is, market outcomes under Bertrand competition are identical to competitive ones.

855. Slackjaw - April 11, 1999 - 1:25 AM PT
(last)

So, under Bertrand competition, unlike Cournot competition, 2 DOES equal perfect competition.

The moral is, how much we should wring our hands about oligopoly depends on the nature of the competition between the firms. Both of these stories can be complicated to allow for various uncertanties (say market demand is uncertain to each firm), more complicated (say nonlinear) demand functions, non-equal costs, etc. But the basic kernel of insight is present even in these very simple, stripped down games.

(Note: in economics, saying that a firm earns 0 profit does not mean it is wasting its time by producing. It simply means that, after accounting for all its costs-labor, including the owner's; rate of return on capital, traditionally called 'profit' in accounting-the firm has no excess left over. The firm does still earn a profit in the usual accounting sense; it just does not earn a 'supernormal' return on its capital.)

856. Slackjaw - April 11, 1999 - 1:32 AM PT
ack

In Message #853,

"I.e., you can sometimes do strictly better, and never do worse, THAN BY charging c + e, for some small number e."

should read

"I.e., you can sometimes do strictly better THAN P = C, and never do worse, BY charging c + e, for some small number e."

857. Slackjaw - April 11, 1999 - 1:36 AM PT
Uzmakk:

I just discovered that the library at my school which keeps current issues of Harpers is closed on weekends. I'll get it Monday.

858. uzmakk - April 11, 1999 - 9:48 AM PT
Thanks, Slackjaw.

859. AzureNW - April 11, 1999 - 9:53 AM PT

Yeah, thanks a lot guy. As if I need another reason to hang around in the Fray all weekend.

860. Slackjaw - April 13, 1999 - 1:08 AM PT
The Bertrand game, I think, is reasonably descriptive of competition among gasoline stations, except with multiple firms. Out here in California, we have particularly high gas prices of late. This has led to the usual spate of news stories about why this has happened, the lumpen answer being nothing more profound than this: gas stations are gouging us.

Of course, gas stations would love to gouge us. They would like to charge the monopoly price, and reap higher profits. Now, outright collusion is against the law, so no contract to collude can be binding. In short, we have a noncooperative game. The problem is that every firm has an incentive to undercut the others--the whole market at the monopoly price minus some small number e is better than 1/n of the market at the monopoly price. This is nothing but an embedded collective action problem, of course: everyone prefers to be the one defector to all other options; but prefers all cooperating to all defecting. But there are a lot of firms involved, typically, and monitoring is difficult, so usually we expect cooperation to fail. And, as an empirical matter, it usually does in such settings.

The moral is, if prices at the pump are rising, probably it's because of rising costs to the gas stations. Certainly stations would love to charge a high price, but they are constrained by the fact that each firm wants to undercut the others and capture the market.

Naturally, the AdamSelenes and FTCs of the world, who believe that cooperation is the regular state of affairs in repeated collective action settings, should object to free market organization: outcomes will be inefficient, and consumers will be gouged.

Now, collusion among the oil refineries, or oil drillers--that's a whole different story. Definitely some Cournot-style quantity competition there, because those guys make huge durable purchases of drilling and refining capacity; that is a long run strategic variable in their game.

861. jayackroyd - April 13, 1999 - 4:50 PM PT
Message #848

"And, I am assuming you are right about the fitness benefit of cuckoldry."

Please understand that is purely speculative--of the evil, evo-psycho, "it evolved, so it must be good" variety.

Yes, I see that game theory won't help us much here, but there still is an equilibrium question of the optimal mixed strategy, with risk as a parameter. I assume that men and women both adopt mixed strategies--the cuckolding wife still sleeps with the nurturer. The broadcaster may also raise kids by a woman he spends more time than others.

In fact, that would seem to be a dominant male strategy--pretend to pair bond and deceptively canoodle. And, likewise, this should hold for women. But that endangers the initial assumption. There needs to be a predominance of nurturing strategies for the broadcaster to free-ride on, and I've assumed broadcasters have more offspring and broadcasting is heritable.

Other issues might motivate a woman to cuckold--like fertility problems with her nurturing mate. In that story, you'd see delivery boy adultery, rather than alpha male adultery--safety with a young, fertile mate rather than a riskier attempt to acquire alpha genes.

862. chloel - April 13, 1999 - 6:13 PM PT
Seems to me the balance between these strategies, for both sexes, is considerably affected by how likely it is that a child will survive with only one nurturing parent.

Slack, could you rephrase that in game-theoretic terms?

There are societies where it takes several adults to reliably raise a child; they are in very harsh climates, and include both the openly polyandrous societies I know of.

863. jayackroyd - April 14, 1999 - 5:27 AM PT
The best mixed strategy is clearly related to resource availability. And, yes, there is an implicit assumption that offspring are more likely to survive to reproduce with two rearers. That assumption is made explicit in The Third Chimpanzee, where Diamond discusses, at some length, the uniquely long period of dependence that human infants have on their parents. We, for example, are the only mammal that doesn't do its own food gathering after weaning, according to Diamond.

And any of these stories have to work for hunter/gatherer societies, and only hunter/gatherer societies.

864. pellenilsson - April 14, 1999 - 9:34 AM PT
Slackjaw

A couple of weeks ago we discussed Elinor Ostrom (she must be of Swedish descent) and her Governing of the Commons. I have finally received the book. Have only browsed so far, but it looks very promising. Thanks for the tip! Thanks also for you kind words about my history of Sweden.

865. AzureNW - April 15, 1999 - 1:15 PM PT

This is by far the most elegant thread in the Fray, Slackjaw.

I hope it's still here later, when the world were not such a noisy place.

866. AzureNW - April 15, 1999 - 1:16 PM PT

were = is

where did that come from?

867. Slackjaw - April 15, 1999 - 2:26 PM PT
Thanks Azure, I'm glad you think you.

Chloel, I am waiting to address your question until I have some more time to post a smallish digression on expected utility theory.

Uzmakk, I have that Harper's article--now I just have to read it.

Baby steps.

868. MsIvoryTower - April 15, 1999 - 2:35 PM PT
Slack

Please, please read my last post in the Technical thread and weigh in if you can. I'd be humble and grateful. Oh, so grateful.

869. chloel - April 15, 1999 - 5:36 PM PT
Slack

You're apologizing for not making up for my laziness? This thread *is* something special in the Fray. You're a gent.

870. AzureNW - April 16, 1999 - 2:51 PM PT

Wow, Slackjaw, somehow I managed to page back far enough or not far enough to miss that whole chunk of conversation on the game theory of cuckoldry and Diamond's _The Third Chimpanzee_ last weekend and this week. Ha, I wondered what those bits of references to marriage and infidelity were about as I scrolled past them, thinking I don't have time for this. _The Third Chimpanzee_ sounds like an interesting book. I am curious about how strongly matrilineal societies, where women have extensive reprodutive freedom anyway, fit into Diamond's theoretical framework of mating strategies.

871. AzureNW - April 16, 1999 - 2:58 PM PT

From the comments, it sounds like he may have omitted a range of human behaviors from his analysis.

872. AzureNW - April 16, 1999 - 4:02 PM PT




back to Message #785

873. AzureNW - April 16, 1999 - 4:24 PM PT

Message #803 4/3

874. uzmakk - April 16, 1999 - 4:34 PM PT
Goodie on the Harper's article, Slackjaw.

875. Slackjaw - April 17, 1999 - 4:52 AM PT
Message #862yes, as I've been thinking of this you can model it in terms of a game with nature. Alternatively, this is known as decision theory, the 1-person analogue of game theory. Same sorts of issues, except the "strategy" of the other "player" no longer depends on your strategy choice. But, the same basic machinery can actually be used to examine both types of problems. Please allow a digression on that machinery before I get to the specific question of that post. But this is much longer than I thought it would be.

That machinery is expected utility theory (or, if you are a Bayesian and don't believe in objective probability, subjective expected utility theory). Expected utility theory allows a rational decision maker's (DM's) preferences to be represented in a very convenient linear form, with each term decomposed into two other terms: take payoff from the action in a given state of the world, times the probability of that state of the world. Sum over states of the world, and you get expected utility. Or, if a is a possible action, and 1,2,...,k are states of the world,

EU(a) = p(1)u(a,1) + p(2)u(a,2) + ... + p(k)u(a,k)

What are "states of the world"? Suppose, for example, that you are deciding to buy a lottery ticket. It's a very simple lottery: there are 10 possible numbers, and you have to pick one and write it on your ticket. Then a number is drawn from a hat. If you wrote down the one actually picked, you win a prize, $X. If you wrote down another number, you win $0. You must pay a fee $F to enter. Then a "state of the world" is the number drawn--generally, it's some possible random occurrence, which depending on the action you've chosen (the number you chose in this case), determines the prize you receive in that state.

876. Slackjaw - April 17, 1999 - 4:52 AM PT
For example, if you choose 2 and the state is 2, you win $X. Then in this decision, EU(2) = p(2)u($X-$F) + (1-p(2))u(-$F). The probability used in the second term, 1-p(2), is legitimate because the probability of *something* happening is 1--*some number* is drawn 100% of the time.

The expected utility theorem says that IF: {(1) preferences are complete and transitive (i.e., either A preferred to B or vice versa, for all A, B; and if A preferred to B and B to C, then A to C), (2) there are no infinitely bad or infinitely good outcomes, and (3) if you prefer outcome X to Y, and p, q and r are probabilities, with p > q, then you prefer (pX + rY) to (qX + rY)}, THEN a DM prefers choice A to choice B if and only if the expected utility of A is higher than that of B. This is useful because the axioms are plausible (though approximately descriptive of only about 50% of lab subjects--axiom 3 in particular causes trouble), and because they yield an exceptionally simple, linear construct. It's very convenient to work with such a thing.

But that's not all it says. Remember that little guy u(.) that appears in the terms of the sum that determines expected utility? It's called a utility function, a mathematical representation of the benefit the DM receives from a particular prize. The EU theorem also says that a DM with u(.) is formally equivalent to a DM with m*u(.) + c, where m is a positive number and c is any number. That is, expected utility obeys a certain scale invariance property--exactly as different temperature scales do. So, you can find the best possible prize B, and arbitrarily declare that u(B) = 1. And you can find the worst possible prize W, and declare that u(W) = 0. Then, using these as baselines, you can ascertain the payoffs of all other possible prizes, and figure out expected utility.

877. Slackjaw - April 17, 1999 - 4:53 AM PT
(The first expected utility theorem was actually proved in passing by von Neumann and Morgenstern, in an APPENDIX to their _Theory of Games and Economic Behavior_, the first extended treatment of game theory.)

So. In the cuckoldry problem, the prize is a fitness benefit. The factor I mentioned, detection, occurs with some probability and affects the fitness benefit achieved from any particular decision. But so does the factor chloe mentioned--if there is only 1 nurturing parent, say because cuckoldry was detected, and the offspring wasn't killed but the cuckolded male just takes off, will the offspring survive?

To understand this we must think about the possible states of the world. We must consider: whether cuckoldry is deteceted; if so, whether the cuckolded male kills the offspring, or whether he leaves the offspring to the mother, or whether he nurtures the offspring anyway. Call these D, K, L, and N respectively. If D = 1, the cuckoldry is detected; if K = 1, the offspring is killed; if N = 1 the cuckolded male nurtures the mother anyway. If N = 0 and K = 0, then L = 1 for sure. If L = 1, the offspring might perish (P = 1) because the mother cannot care for it.

Denote the set of states of the world as follows:
1 corresponds to D = 0
2 to D = 1, K = 1
3 to D = 1, K = 0, N = 1
4 to D = 1, L = 1, P = 0
5 to D = 1, L = 1, P = 1.

Suppose further that if D = 0, the offspring survives for sure and confers a fitness benefit (prize) B (for Best) on the mother; if D = 1 and K = 1, the mother's "prize" is T (Tragedu); if D = 1 and N = 1, the mother's prize is H (sHame); if D = 1, L = 1 and P = 0, the prize is S (Survival); if D = 1, L = 1 and P = 1, the prize is W (Worst).

Then, given that the probabilities of the 5 states are known by assumption**, we evaluate expected utility of cuckoldry as follows:

(cont.)

878. Slackjaw - April 17, 1999 - 4:54 AM PT
EU(cuckoldry) = p(1)u(B) + p(2)u(T) + p(3)u(H) + p(4)u(S) + p(5)u(W)

On the other hand, fidelity results in a different association of prizes with states. Namely, regardless of the state, the prize is F (Fidelity). EU(fidelity) = u(F).

If we assume that B > F > W, then, depending on the probabilities, and on a DM's attitude toward the risk inherent in the probabilities (an attitude that is captured in the curvature of the u(.) function (see next message)), a rational DM may prefer cuckoldry to fidelity.

** If this assumption bothers you, it can be done away with. That's what subjective expected utility is all about: it's based on probabilities that don't depend on any extrinsic randomness in states of the world.

879. Slackjaw - April 17, 1999 - 4:54 AM PT
What does it mean that attitudes toward risk are captured in the u(.) function? Consider the St. Petersburg paradox, an old game considered about 250 years ago by the famous mathematician Daniel Bernoulli--arguably the real father of expected utility theory.

Suppose you are offered the following gamble. A fair coin is flipped, over and over again until a Head occurs. If it occurs on flip N, you will be paid 2^N. So, if it occurs on the first head, you get paid $2. If it occurs on the second head, you get paid $4. If on the third, you get $8, etc.

How much would you pay to play this game?

Obviously only a fool would answer "less than $2," for that is strictly dominated. But in experiments, the right to play this game never sells for more than about $40.

Well what is so shocking about that? For a long time, mathematicians argued that the appropriate decision criterion is expected *value* maximization, not expected *utility* maximization--they recommeded simply choosing the decision available to you with the highest expected payoff, rather than the one with the highest expected utility from the payoffs.

Turns out that the expected value, the average of all payoffs you can get, in the St. Petersburg game is infinity! If you play that game, you *expect* a prize of infinity. Why? Well, consider your prize if the first flip is a head, 2, times the probability that the first flip is a head, (1/2). The product is 1.

Big deal. Well, consider the prize if the first head occurs on flip 2: it's 2^2, or 4. The probability of receiving that prize is, naturally, the probability that the first head occurs on flip 2, or (1/2)^2, of (1/4). The product is 1 again.

880. Slackjaw - April 17, 1999 - 4:55 AM PT
Yes, this is a trend. For *every* possible state of the world (i.e., number of the flip on which the first head occurs), the probability of that state times the prize in that state is 1. But there are infinitely many states possible! It is exceptionally unlikely, but possible, that the first head occurs on flip 100 (probability = (1/2)^100). But IF IT DOES, you get an exceptionally high payoff, 2^100. This is true for every possible N!

So the expected value of the gamble is a sum of infinitely many 1's, or infinity.

As I said, nobody pays quite that much to play it. Bernoulli first noticed this game, and proposed that it's because for very unlikely states of the world, the sure loss from having to pay to play swamps the possible gain.

Expected value maximization corresponds to u(prize) = prize. This is called "risk neutrality." Bernoulli proposed u(prize) = log(prize), which actually fits the data fairly well. This is called "risk aversion." While risk neutrality corresponds to u(.) that is a straight line as a function of the prize (so increases at a constant rate), log(.) increases at a decreasing rate--it is always increasing, but starts out parallel to the vertical axis curves to parallel the horizontal axis.

881. Slackjaw - April 17, 1999 - 4:55 AM PT
To make a long story short, illustrate the concepts thusly: a risk NEUTRAL gambler, who only cares about expected value, is indifferent between a prize of $10 for sure and a prize of $1010 half the time, and -$990 the other half. A risk AVERSE gambler strictly prefers the first one to the second. Both have the same expected value, however.

882. Slackjaw - April 17, 1999 - 5:01 AM PT
urf

in Message #879, in "So, if it occurs on the first head, you get paid $2. If it occurs on the second head, you get paid $4. If on the third, you get $8, etc.",

the "head" should be replaced with "flip." that's important.

883. Slackjaw - April 17, 1999 - 5:06 AM PT
This is why in game theory we talk about the utilities from various combinations of strategies. Then if you have expected utility maximizers, you are justified in working with the average value of those utilities, for a given player, arising from some randomization in the game (say, mixed strategies of other players).

884. jayackroyd - April 17, 1999 - 6:07 AM PT
Message #878

And the values of those probabilities are resource dependent, which is why you see different mating patterns in different parts of the world, even the rare polyandry case, which isn't in our model, but does arise, as AzureNW points out.

(On polyandry and the Third Chimpanzee, he does refer to Indian Nayars, where the women were promiscuous, and the men supported their sisters and their sisters' children. He doesn't refer to the South American group that has multiple men rearing kids of uncertain parentage. I don't remember the people now, nor, whether like lions, the men tended to be related to each other. If the sociobiologists are right, they should be. Paternal uncertainty is less of a problem if all the potential dads share genes.)

Oh, and I wouldn't recommend the book to anyone who has read Guns, Germs and Steel or many of Diamond's Natural History essays. I found myself skipping about half the essays because I'd read the content before. However, I would be sure to sit down in my local bookstore or library and read the essay on genocide.

885. AzureNW - April 17, 1999 - 11:46 AM PT

jayackroyd -

Re: Message #884

The social organization I'm referring to is not polyandry, but matrilineal. It's a social organization where women have the means to support a child independent of the child's biological father, have the right to divorce the father and remarry at will, and even have the right to kill his child if she chooses to. If the father kills the child, it is murder. If the mother kills the child, it is not murder. The house, everything in it and all the kids belong to the wife, period. The kids are raised by her brothers, who will kick your ass if you bother her.

This was not a rare phemonenon, but the prevailing social organization throughout Northeastern America prior to the arrival of Europeans, (or so I've been reading. I've seen some great quotes, but of course, right now I can't find anything but pages of kinship charts.)

This discussion of cuckholdry caused me to wonder what kind of men would be selected for in such a society, where women made the important reproductive decisions. Smart guys, I suspect. Guys with a great sense of humor and a big heart. Those are the ones women can't live without.

886. jayackroyd - April 17, 1999 - 11:52 AM PT
So what were these society's mating patterns?

And that's not what I mean by matrilineal. Matrilineal just means that the lines of inhertitance follow the mother's line, not the fathers. The idea of children as chattel of either parent is independent of which parent(s) get the right to dispose of the child.

On what guys women would select, you need to tell a little fancier story, and show how guys with big hearts are more likely to have offspring that reproduce in a hunter gatherer environment.

887. AzureNW - April 17, 1999 - 12:03 PM PT

jayackroyd -


This is a mixed economy combining farming and trade with hunting and gathering. The people live in scattered small towns organized around a larger principle town, organized into chiefdoms.

The mating patterns were elaborate and the bonds of matrimony were fragile. It sounds like a lot of mating was going on, and probably a lot of soap-opera gossip about who was breaking which taboo.

888. AzureNW - April 17, 1999 - 12:11 PM PT

I read a funny story about how one tribe in the Southeast ended up with so many clans. When a well-liked couple broke a rule forbidding sex with a particular clan member, the woman was declared to be of a new clan, usually with an unflattering association, like toad or mole. So, the couple would be accepted as married, but all her children would thereafter be toads or moles.

889. FreetoChoose - April 17, 1999 - 1:19 PM PT
In 240 of the Playpen thread, Slackjaw said (material in square brackets added):
“Don't know where you got the idea this is the "normal" state of affairs. Please, please forget anything you may have heard about what Robert Axelrod "proved." [1]First of all, mutual perpetual defection is *always* an equilibrium. [2]Second, cooperation *can be* (i.e., "might be," not "will be") sustained in an infinitely repeated PD. It cannot be sustained in a finite PD with rational players. [3]Third, re. your first possibility, 'irrationality' in every treatment I've ever read makes cooperation "easier" to sustain, not harder. [4]Fourth, what does your second possibility ("maybe we've played enough times") mean? [5]Fifth, as to the third possibility, punishment can most certainly be part of an equilibrium strategy profile sustaining cooperation in the infinitely repeated (multilateral or bilateral) PD.

But most importantly, mutual defection is always an equilibrium. Maybe not Pareto dominant, but an equilibrium just as sure.”

890. FreetoChoose - April 17, 1999 - 1:19 PM PT
My response:
I used the word “normal” where I intended “equilibrium”

[1] Granted
[2] Granted
[3] This is surprising to me. Can you elaborate?
[4] A typo, I mean to say “maybe we haven't”
[5] I didn't realize this. It maybe that I have a narrower definition of what I consider to be the classic definition of PD.

891. FreetoChoose - April 17, 1999 - 1:30 PM PT
Elaboration:

When I refer to the classic PD, I mean the original problem, as well as the isomorphic conversion to a two-by-two payoff matrix.

What is less clear to me is how to characterize the various extensions of this problem. For example:

• Finite repetition
• Infinite repetition
• Assumptions of rationality
• Actions of third parties

When you talk about “punishment” being part of the classic PD, are you arguing that punishment can be incorporated in the payoff matrix, or are you allowing a punishment, over and above the payoff matrix?

Sorry if this isn't clear.
Let me take a different tack.
I think of PD as involving two players. The fact that the decisions may have impact on third parties was, I thought, generally ignored. In my attempt to analogize to the Fray, decisions do have an impact on third parties. Is it acceptable to consider those in a PD game?
Frankly, I hope the answer is yes. I was simply trying to pre-empt the possibility that you would argue I was going out of bounds.

892. Slackjaw - April 17, 1999 - 1:33 PM PT
[3] Yes, I will when we get to Bayesian game theory
[5] I don't know what that could be. You have a prisoners' dilemma, you repeat it infinitely many times. By the folk theorem, any combination of per-period payoffs yielding more than the level a player gets from cooperating while the other defects can be supported as a Nash equilibrium. That says nothing about the strategies used to support that payoff profile. Some of them require a punishment phase to be invoked in the event of defection. Of course, the point is that the threat of that punishment is supposed to deter defection.

893. Slackjaw - April 17, 1999 - 1:39 PM PT
Oh, if that's what you mean by classic PD, i.e. the 1-shot game, then no, cooperation can never be supported as a Nash equilibrium with rational players.

By punishment, I don't mean a new strategy is added to the game. You still have only the two choices. However, you know that one of them inflicts harm on the other player, so the threat of using it to beat him about the head if he ever stiffs you can function as a punishment.

third parties can be brought in, in a multilateral PD. Otherwise, they have no place. If the effects on their utility matter to the players, the payoff structure must be modified to account for that and you may have a different game altogether.

894. Slackjaw - April 17, 1999 - 1:41 PM PT
Message #893 And in that sense you can see straight away why irrationality always makes it weakly easier to get cooperation, i.e., never makes it harder. After all, rational players never cooperate. Hard to get further from cooperation than that! In the finite PD, they still never cooperate.

I know of two rigorous treatments of irrationality, one relevant to the finite PD, one to the 1 shot, that I will talk about later.

895. FreetoChoose - April 17, 1999 - 1:50 PM PT
Slackjaw

In your version of PD, can the participants exchange information(other than that imparted by observing prior results)? Crudely, can they talk to each other?

896. FreetoChoose - April 17, 1999 - 1:52 PM PT
Slackjaw

“By punishment, I don't mean a new strategy is added to the game. You still have only the two choices. However, you know that one of them inflicts harm on the other player, so the threat of using it to beat him about the head if he ever stiffs you can function as a punishment.”

What about punishment of other persons?

To be specific, if the decision is whether to post in Playpen, it may be that the refusal doesn't harm the other player, but does impart punishment to other Fraygrants. Is that acceptable to consider in PD?

897. FreetoChoose - April 17, 1999 - 1:53 PM PT
I need to get some lunch. I'll return.

898. Slackjaw - April 17, 1999 - 2:10 PM PT
First, let me be perfectly clear. "My version" of the PD is the PD.

"Can they talk to each other" No, but in a finite PD (including the 1-shot) it doesn't matter with rational players. They have a strictly dominant strategy to defect in each period and talking doesn't change that. I will prove all this in our next installment on dynamic games. In the infinitely repeated PD, it might matter; also it might matter with a particular irrationality if one player is allowed to declare he is "irrational." I have to think about that, however. I don't think it will help.

"Punishment of other persons" = externality. Self interested (not the same as rational) people ignore it; "altruistic" people might consider it and roll it into their own payoffs.

Probably the actual game being played is different from PD, and different in an important way. For example, those who suffer the loss of the externality, even though they aren't party to the original embedded PD between the two people deciding whether to go to playpen, can mete out some just desserts of their own if people don't go. They can vilify people who don't take it to playpen, which alters the payoffs if the participants don't do it--presuming they are sensitive to vilification.

This is actually a game with 3 players, in two stages. First stage is a PD; second stage, player 3 gets to inflict some utility loss on the players in the event that they don't "cooperate" in stage 1 and go to playpen.

(I am assuming you are right about the first part being a PD with "cooperate" identified with "go to playpen," but it might not be, because some people just like going to the playpen.)

We have not yet covered the tools to deal with dynamic games like this, i.e., games with stages, or that take place in a specified sequence. That is the next topic to cover.

"lunch": Don't you live on the east coast?

899. FreetoChoose - April 17, 1999 - 2:45 PM PT
Slackjaw


Tell you what.

I haven't kept up with the discussion in this thread. I find it annoying when other people ask questions already answered before, so let me take my own advice, and read all the material you posted before asking more questions.

(Yes, I'm on the east coast. I promised myself I would just check a few threads, then head to lunch at a reasonable time. You can see what happened.)

900. Slackjaw - April 17, 1999 - 7:07 PM PT
Review to date (I will associate post numbers with these topics later; for some of the early stuff see the archive):

* zero sum games
* dominance solvability (e.g., collective action, 1-shot prisoners' dilemma)
* Nash equilibrium in static (1-shot) games of complete information (e.g., Cournot-Bertrand oligopoly theory, simple analysis of liability regimes)
* Expected utility theory (latest cuckoldry stuff)

Next: Nash equilibrium in dynamic games of complete information

I want to stress again that "complete information" means that each player's utility function (i.e., her preferences) are common knowledge. There may be uncertainty about other things, e.g., the state of the world as per the cuckoldry stuff, or the market demand in the Cournot game, but there is no uncertainty about preferences. (More precisely, whatever one player knows about his preferences, every other player knows about them too; AND it is common knowledge that they all have this information.) This is what is meant by "complete information."

It is NOT the same thing as "perfect information," which is rougly a situation in which there is no random element--i.e., in which "nature" is not a "player" in the game.

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