1001. Slackjaw - May 15, 1999 - 9:59 PM PT
elliot's paradox of the unexpected hanging appears in Message #86
1002. Slackjaw - May 15, 1999 - 10:00 PM PT
though I presume he didn't invent it.
1003. CalGal - May 15, 1999 - 10:14 PM PT
Slack,
If person 1 offers nothing and person 2 rejects it, shouldn't both of them get nothing?
1004. Slackjaw - May 15, 1999 - 10:38 PM PT
yeah--are you thinking that it wouldn't cost 2 anything to deviate from the SPNE in ultimatum?
1005. CalGal - May 15, 1999 - 11:10 PM PT
Um. Yeah. That's what I'm thinking. Sure.
Or maybe it's this--that offering #2 nothing is *not* the best for #1, since it doesn't give him the whole pie.
1006. Slackjaw - May 16, 1999 - 12:43 AM PT
but that depends on 2's strategy. If it is "accept anything," then by offering player 2 nothing, player 1 does get the whole pie. "Accept anything" is in turn part of a Nash equilibrium in the subgame where only player 2 moves, because he cannot do better by deviating.
Ask yourself the following: if you are playing "accept anything," and nothing is offered to you, could you do better by deviating? No, you couldn't--thus "accept anything" is a Nash equilibrium in the (1 person) subgame starting with 2's move after any particular offer.
1007. Slackjaw - May 16, 1999 - 12:44 AM PT
Actually, I wanted to mention something about ultimatum. The outcome where 1 gets the whole pie is the unique SPNE only in case currency is perfectly divisible--so 1 can offer, say, 1/pi cents if she chooses. If there is some minimum unit, like pennies, then there are actually 2 SPNE outcomes: one where 1 gets the whole pie, and one where 1 gets the whole pie minus the minimum unit of currency.
Suppose 2's strategy was "accept anything except an offer of 0." Clearly this is a Nash equilbrium in the subgame after any particular offer--in particular, if offered 0, in which case the strategy prescribes rejection, 2 doesn't mind carrying out the threat because he gets 0 by rejecting and does not do strictly better by deviating. Thus, it's a NE. The same reasoning holds if the strategy is "accept anything."
However, and this often trips up even people who know a bit of game theory for some reason, there is *only one* SPNE if currency is perfectly divisible. And it does *not* involve 1 offering "some arbitrarily small share greater than zero" to player 2. Nash equilibria require maximization of utility, and it is a mathematical impossibility to maximize an increasing function on an open set. That is, 1 would want to offer the smallest possible "small share greater than zero," but there is no such thing if a continuum of choices is available.
That may seem kind of arcane, but playing with toy examples in seemingly bizarre situations is how one learns the limits of the theory.
1008. Slackjaw - May 16, 1999 - 12:49 AM PT
again, SPNE stands for subgame perfect Nash equilibrium.
Notice another thing about the SPNE in ultimatum: while any allocation can arise from a (unrefined) Nash equilibrium, all of them except the SPNE involve weakly dominated strategies for player 2. That is, they require player 2 to play a strategy with the characteristic that he has another strategy that always makes him at least as well off, and sometimes better off (but never worse off) than the strategy he plays leading to the non-subgame perfect Nash equilibrium. (Recall that a strictly dominated strategy is one with the property that there is another strategy that always makes you strictly better off, never worse off, and never even just as well off).
This is not an accident, the relationship between SPNE and Nash equilibria in weakly undominated strategies. No SPNE can involve weakly dominated strategies.
1009. Slackjaw - May 16, 1999 - 2:00 AM PT
uh, I misspoke above. Selten's first article on subgame perfection appeared in 1965, not 1975. In 1975 he refined his refinement even further, because even it leaves too many "unreasonable" equilibria in some games.
His 1975 refinment, called "trembling hand perfection," has to do with whether a Nash equilibrium would still be a Nash equilibrium in a game where another player had some infinitessimally small chance of making a mistake in implementing his strategy--literally like placing a chess piece in the wrong square by accident.
Basically, you look at the limit of an infinite sequence of games wherein the probability of a mistake gets smaller and smaller, and finally converges to zero at the limiting game. A Nash equilibrium is trembling hand perfect if it's an equilibrium in every game in this sequence. (The case of non-infinitessimal error probabilities is still a very active area of research.)
These refinements really get to one of the practical problems with Nash equilibrium: there are too damn many in a lot of games. Despite some universally accepted refinements like these, no satisfactory refinement always winnows down the candidate equilibria to just one. And the refinements literature is now HUGE. Now that basic equilibrium concepts exist for all four types of canonical games (complete vs. incomplete information and static vs. dynamic), pure (noncooperative) game theory literature consists largely of refinements in various classes of games. Indeed, as I said earlier, even the basic equilibrium concepts for all games beyond static complete information, where we simply start with basic Nash equilibrium, are refinements.
1010. Slackjaw - May 16, 1999 - 3:30 PM PT
game theory posts to date:
* Introduction, general material, further reading: Message #666, Message #688, Message #697 to Message #699, Message #770, Message #791 and Message #792
* Expected utility theory: Message #875 to Message #883
* Zero sum games: Message #689, Message #692 and Message #693, Message #785
* Dominance solvability: Message #694 to Message #696
* 1-shot prisoners' dilemma, as a special case of collective action: Message #702, (the story CalGal refers to is described in the link in Message #666), Message #705 and Message #706, Message #711 to Message #722
* Static games of complete information, theory (Nash equilibrium): Message #803 to Message #807, and Message #813 and Message #814
* Static games of complete information, examples & applications: liability regimes (Message #808 to Message #812), oligopoly (Message #849 to Message #856 and Message #860), beauty contest (Message #988 to Message #994)
* Dynamic games of complete information (subgame perfection): Message #994 to present
Think that's most of them. Obviously I have reodered some posts, so that they correspond better with the logical development of game theory.
all through the 640's and 650's is some discussion of folk theorems in game theory, but they are out of context and it may be difficult to figure out exactly what it's all about. I'll cover them again later.
1011. uzmakk - May 17, 1999 - 6:50 AM PT
Slackjaw:
Just to let you know that I did give a big chunk of your thread to my sister the mathemetician. She seemed quite interested. Who knows when she will get back to me on it though. Busy busy girl.
1012. uzmakk - May 17, 1999 - 12:52 PM PT
Slackjaw:
How about a few sentences on that Harper's article re: Monsanto?
Liked your posting in dreams. I can relate.
1013. uzmakk - May 18, 1999 - 5:54 PM PT
Slackjaw:
Where are you?
1014. Slackjaw - May 18, 1999 - 11:50 PM PT
Uz,
I'm in and out--dealing with some pressing real life matters that have popped up recently. Going to be a real crunch, especially over the next week or so. Can you hold off just a little while longer on the Monsanto piece? It's an interesting article and I do want to talk about it...
1015. uzmakk - May 19, 1999 - 7:22 AM PT
Actually, I should do precisely the same thing. I have some concerns that I should set in order and from which I should not be distracted.
Later, then.
1016. Slackjaw - May 21, 1999 - 4:11 PM PT
so a while ago the Uz told me about a Harper's article on agribusiness giant Monsanto, and its practices in biotech. There are several interesting strategic aspects of this article, but one in particular relates to subgame perfection and credible committments.
One of the things the article (which is not exactly pro-Monsanto) mentions is that some farmers have been capturing seeds produced by plants grown with Monsanto seeds. They've got some amazing properties, these plants, like pest or pesticide resistance; the seeds they produce have these traits too. Thing is, Monsanto owns these seeds, and when you collect them, you're supposed to give them back. Some farmers evidently haven't been, but instead have been reselling them on the black market--a lucrative endeavor because these things are mighty expensive.
Monsanto has gone after these people like gangbusters, or at least wants to convey the image that it has. It has sent out a cordially threatening letter to stacks of farmers, telling them of specific instances where Monsanto has gone after some re-planters and levied or won large fines as a result.
Strategically, though, Monsanto's efforts aren't free. It has to pay a cost to monitor re-planting, and it does not appear to be a trivial cost. Once replanters are discovered, it has to pay to fight them. That's also not free.
Monsanto is essentially playing a game with a series of "small" players. It wants to establish a reputation for being rabid about this contractual breech, so that farmers in the future will be deterred from doing it.
1017. Slackjaw - May 21, 1999 - 4:11 PM PT
Basically Monsanto has sought to establish this reputation by making threats. We know from subgame perfection that if a threat is to be credible, it must be worth it for the threatener to actually carry out the threat should that contingency arise. If it's not, the threatenee will know it (assuming common knowledge of preferences) and disregard the threat. For example, suppose Monsanto threatened to nuke the US if it found anyone replanting. Nobody would take this seriously, because Monsanto would not want to do it if it ever came time.
Also important is the communication. This is fairly obvious, as a focal part of the article is the letter Monsanto sent, but it's hard to build a reputation unless you can convey to other players what has happened in your interactions with replanters. But the need for communication implies that some uncertainty is important in this game--Monsanto knows things other players don't know, and is trying to communicate it to them. This highlights an important aspect of the models considered so far--for all the material, we still have not gotten to models that allow one to understand communication very well. Part of the problem is that Monsanto has an incentive to lie about its effectiveness, and indeed this may be what the anecdotes in its letter may be all about.
One important aspect of this game is that there is more or less always a future (assuming that the date on which Monsanto ceases to exist is not known to anyone). We have actually not yet covered the techiques to deal with such situations. This is the subject of infinitely repeated games. Under certain conditions on the structure of the game, it is possible to say some interesting things, but mostly what we know about these games is that if you believe in equilibrium reasoning, it is sometimes difficult to say anything interesting a priori about the outcomes of the games.
1018. AuNaturel - May 22, 1999 - 1:24 AM PT
"that some farmers have been capturing seeds produced by plants grown with Monsanto seeds."
Are you familiar with "Terminator" seeds?
1019. stamper - May 22, 1999 - 10:12 PM PT
To any one
I have come over here to ask you to settle an agument between me and Andy Weigart. We got to talking today about playing dice up in Reno. The subject got around to odds and probability. Now Andy claims that there is no difference.
For example, if you flip a coin, the odds of it coming up heads is fifty-fifty or even. Now if you flipped that coin 100 times and every time it came up heads, what are the odds on the next toss. If you say fifty-fifty, go to the head of the class. But what is the probabilty of it coming up heads. I have no idea, but i suspect from common sense that it is not the same as the odds. Am i right of is Andy right?
1020. Slackjaw - May 22, 1999 - 11:10 PM PT
Odds and probabily are closely related but are not exactly the same thing. However, once you know one, you automatically know the other. For any occurrence or "event," the odds of the event are the probability of anything *but* the event, divided by the probabily of the event.
For example, if a horse's odds to win are quoted at 3:1, that means the horse is three times as likely *not* to win as to win. That means the probability of the horse winning is 1/4 or 25%.
Ol' Andy is right about the odds, they don't change after 100 consecutive heads--coins don't have memories. Since the odds are still 1:1 or fifty-fifty, the probability of a head is as likely as a probability of a tail--1/2 each. Regardless of the history of coin flips.
1021. AuNaturel - May 22, 1999 - 11:16 PM PT
"Since the odds are still 1:1 or fifty-fifty, the probability of a head is as likely as a probability of a tail--1/2 each. Regardless of the history of coin flips."
But if by then you don't suspect a two headed coin or other such dodge you'd be a fool.
1022. Slackjaw - May 22, 1999 - 11:43 PM PT
true enough--the assumption that the parameter of the distribution generating outcomes of a coin flip is known to you is very important.
With all this talk about horse races and probabilities, we get tantalizingly close to the wonderful world of Bayesian probability. What does it mean to talk about the probability of a horse winning a race? The standard (if you prefer, old fashioned) view of probability is as a long run frequency in repeated trials or something like that, but you start to strain the metaphor if you think of a horse race under some exact conditions. You could imagine it repeated over and over, but of course odds are quoted before it is even run once under the exact conditions under which it will be run. What do the probabilities mean that underly odds quoted at race tracks?
To a Bayesian, probability is not an objective construct. It is necessarily subjective, and is measured not as a long run frequency but as the willingness to bet. Odds at a track, then, aggregate the subjective probabilities of all the individuals participating in the betting market. Every individual has a different estimate, and that determines their willingness to bet at going odds.
1023. Slackjaw - May 22, 1999 - 11:43 PM PT
Consider the following wager: you get $1 if the population of Irian Jaya is larger than the population of Java, and $0 otherwise. To many people, this is an uncertain prospect, because they don't know the answer. But some people do know. So if you ask someone how much they'd be willing to pay to take this wager, you have a measure of their estimate of the probability that Irian Jaya has a larger population than Java. If someone *knew* Java to have the larger population, he wouldn't pay anything to take this wager. On the other hand, suppose someone was very unsure--and thought either possibility equally likely. Such a person would be willing to pay about $.50 to play this game, which is the expected value given her beliefs. (I am assuming no moral objections to gambling, of course.) Finally, suppose someone was roughly certain that Irian Jaya had the larger population. Then if he took this wager, he'd expect to gain about $1. So he should be willing to pay about that much to play this game.
(I have deliberately chosen small payoffs to avoid complications of risk aversion, in which case the value of a gamble is not simply its expected value--recall the St. Petersburg Paradox.)
But even though the prospect is uncertain, what does it mean to repeat the experiment over and over and get a long run frequency? One must go through some bizarre funambulism to fit this into the standard frequentist framework, while the Bayesian framework captures it quite naturally.
1024. Slackjaw - May 22, 1999 - 11:46 PM PT
(what I mean by risk aversion is this: are you indifferent between a "prize" of $0 for sure, and taking a gamble in which you lose all your wealth and possessions if a coin comes up heads, but double the value of your wealth and possessions if the coin comes up tails? If not, you are risk averse. The expected value of both "gambles" (one of which is degenerate) is $0, but one has substantially more risk associated with it. That's the one you avoided, conditional on a yes answer to the above question.)
1025. AuNaturel - May 22, 1999 - 11:46 PM PT
Does Bayesian statistics deal with paramutual betting?
1026. Slackjaw - May 22, 1999 - 11:47 PM PT
Yes, I have heard of terminator seeds, but for some reason Monsanto hasn't developed them for these products yet.
1027. Slackjaw - May 22, 1999 - 11:52 PM PT
well, Bayesian statistics deals with anything that classical statistics deals with. So if you are trying to make an inference, or test a hypothesis about some population quantity based on a sample quatity, then yes. Or, if you want to study general procedures for answering these questions (statistical decision theory), then yes.
The underlying worldview of Bayesian probability is certainly applicable if one wants to understand what the odds quoted at a track measure.
1028. stamper - May 23, 1999 - 12:05 AM PT
This is all very interesting to someone a whole lot smarter than me. Look, we know if someone is throwing dice that a certqain mumer can come up a number of times that simple defies the odds. for example, 12 should come sup once in X nymber of throws, i've had too much to drink to give that a number. So, i'm at the dice table and i happen to notice that 12 has not come up in the last throws in x+75. Now knowing that 12 pays a certain payoff, should i lay on 12 with the idea that since 12 has not come up since 12+ 75, it should come up before my odds and i will make money, at least as far as probability is concerned. No sure think for sure because the odds do not change, but probability theory gives me a better chance.
1029. stamper - May 23, 1999 - 12:07 AM PT
please excuse the typos i meant to preview but it screwed up.
1030. Slackjaw - May 23, 1999 - 1:08 AM PT
stamper: some terms are used inconsistently and/or incorrectly in your post, but I think your gist is this: since 12 hasn't come up for a long time, should you lay money on it because it's bound to come up rather soon?
The technical answer is: nononononononononononononono.
No
No
No
Dice DO NOT HAVE MEMORIES. They do not know 12 hasn't come up. The probability of a 12 on any given throw of 2 fair dice is 1/36. Period. That is all. It doesn't matter what the history is. Technically, fair dice, coins, roulette wheels, etc. all give rise to probabilities that are independent across trials (or throws if you like). The probability of a certain outcome in any throw of the dice has NOTHING at all to do with anything that's happened on any other throws. You are not more likely to win just because it's been a long time since 12 came up. My mother in law still thinks so, however, after many conversations and losing trips to Vegas.
But 12 does not become a better bet just because it hasn't come up for a long time.
The belief that any sample drawn from a random process must display the average behavior of that process, or that samples are "mean reverting," is called the "gambler's fallacy."
By the way, since the probability of a 12 is 1/36 on any given throw, the expected number of throws *before* a 12 occurs is 35 (so on average a 12 occurs on the 36th throw). If you let 12 denote "success" and anything else "failure," and p=1/36 is the probability of success on any given throw, we want to know the expected number of faiures before the 1st success. The number of failures before a given number r of successes follows a negative binomial distribution, and its expected value is r(1-p)/p or 35 in this case.
1031. stamper - May 23, 1999 - 7:14 PM PT
Slakjaw
I now have a better understanding than i had before. Thank you. I have not had much education in math, just my life experience. I'm good with % 'cause i have worked a great deal on commission selling various things such as vacuum cleaners door to door. I was good at it too, but it gets tiresome after all.
for a while there i thought i could make my living playing poker in the clubs. I could out play most guys seven ways to breakfast but i could not beat the rake. And another thing. It ain't like having a job with a steady job where you can figure where you are. For a few weeks you're way up and living like a big shot and you have a bad run for awhile and all that money is gone and you're scratching to pay the rean.
Are you a poker player? What of the dumbest things you will hear poker players say is never draw to an inside strait. Well that just depends. Say you're in a $5 to $25game with seven of jacks or better and you're the last to draw. Opener bets $5 and only one guy folds before you. That inside strait is somewhere around 12/1 odds. You already got half that and if you hit you might make alot more. Say nobody even you makes a hand and it gets checked to you. You can throw your hand in knowing you lost or you can make a bet. Bet $12, which looks like a come on if you are playing with average players. If your playing with either sharpies or palookas, throw it in, because you cannot psych a pooloka for sure and the sharpies will test you with a call.
I could give lessons in either jacks or better or low ball. I play with the guys now, but i don't play hard, 'cause i don't want their money anyway, just their company.
I enjoy talking to you Slackjaw and i sure did read you wrong when i thought you were a hells angel.
1032. Slackjaw - May 23, 1999 - 10:18 PM PT
Actually, you may be right, but the inside straight thing always made sense to me. If you have 3,4,5,6, then either a 2 or a 7 makes a straight. If you have 3,4,6,7, only a 5 will help you. The relative likelihood of making a straight in each case depends on what you know about what other cards are out there, I guess, but in general you're more likely to hit in the first case than the second.
I don't gamble much myself but I like to watch other people do it, to see how they play. Even at the racetrack or a casino I prefer to watch how other people estimate odds and react to them. Fascinating process.
I also just kind of like to watch random events unfold, whether as an interested participant or not.
1033. stamper - May 23, 1999 - 11:54 PM PT
Slackjaw
You are mathamatical wizard compared to me, but please do not get in a poker game for high stakes. In either case, with an open ended straight or inside it depends on the action. In the first case there are eight cards to make your hand. all you see are five cardsm which leaves 47 unknown. your chang\ce of gettting the right card is 8/47 or just under 6 to 1 Now if that pot cannot pay you better that six to one, fold. In the second case there is only 4 cards to make your hand. so 4/47, or just over 11 to 1. It doesn't matter what you are drawing to, the odds are easily calculated. the next is all psycology. You have to know the palookas you're playing with. Are they smart, tight as a cat's ass, macho, mouse, or brill. I try to be brill but there is always the possibility someone is smarter than you are.
I've won a lot of money playing chess because the opponent most always underestimates my ability. After all, i a unlearned guy, right? But chess ain't too complicated if you have the killer instinct. Please forgive any words misspelled for i am too tired to look them up.
1034. stamper - May 23, 1999 - 11:58 PM PT
Slackjaw
You are mathamatical wizard compared to me, but please do not get in a poker game for high stakes. In either case, with an open ended straight or inside it depends on the action. In the first case there are eight cards to make your hand. all you see are five cardsm which leaves 47 unknown. your chang\ce of gettting the right card is 8/47 or just under 6 to 1 Now if that pot cannot pay you better that six to one, fold. In the second case there is only 4 cards to make your hand. so 4/47, or just over 11 to 1. It doesn't matter what you are drawing to, the odds are easily calculated. the next is all psycology. You have to know the palookas you're playing with. Are they smart, tight as a cat's ass, macho, mouse, or brill. I try to be brill but there is always the possibility someone is smarter than you are.
I've won a lot of money playing chess because the opponent most always underestimates my ability. After all, i a unlearned guy, right? But chess ain't too complicated if you have the killer instinct. Please forgive any words misspelled for i am too tired to look them up.
1035. Slackjaw - May 24, 1999 - 12:36 AM PT
yes, I neglected pot considerations entirely (which as you obviously know determines your willingness to bet at any given odds) and thought only about probabilities.
Indeed stamper, very good for a self proclaimed unlearned feller. Did they teach combinatorics in your grammar school?
1036. rhartmann - May 29, 1999 - 4:24 PM PT
1
1037. Slackjaw - May 30, 1999 - 2:51 AM PT
here is another game that is primarily of theoretical interest, which should not be translated as "irrelevant." If we want to use games to model social interaction (and not just among humans), we should be confident in the predictions made by game theory. So it is important to push the limits of the theory and see where and why it is unsatisfactory. As far as real world application goes, think of it as simply working at a different stage of an r&d pipeline.
This game is called the centipede game. It has a unique subgame perfect Nash equilibrium; again that just means that we require strategies to be Nash equilibria in *every* subgame, even those that are never reached in equilibrium. If that sounds strange, remember that a strategy is a *complete* plan of action, a decision for a player about what to do in any circumstance in which she might get to move.
Who cares about the subgames that will never be reached? Remember, as with threats, it is what might happen outside of equilibrium that determines the incentives players have to play a particular way. This is one fundamental insight of game theory.
1038. Slackjaw - May 30, 1999 - 2:51 AM PT
Centipede proceeds as follows: there are 2 players. The game proceeds for a fixed number of rounds, for concretenes let's say 5. At the start of round 1, player 1 (she) gets a pile of money, say $2. She can either take, and end the game, at which point she gets $2 and player 2 gets $0. Or she can pass to player 2 (he). If she passes, round 2 starts; it is 2's turn and he has double what player 1 had at the start of round 1.
So, if she passes on round 1, he can either take or pass. If he takes, he gets $4 and she gets $1 and the game ends. If he passes, round 3 starts and player 1 gets to move again. If she takes on round 3, she gets $8 and he gets $2 and the game ends. If she passes, it's on to round 4, and you guessed it: he can either take or pass again.
If he takes, he gets $16 and she gets $4. If he passes, it's on to the last round, round 5, where she actually faces a degenerate choice: she can only take, and if this round is reached she gets $32 and he gets $8.
It's called Centipede because if you draw out a tree with the sequence of rounds on it, it resembles a centipede.
Obviously, from the collective point of view, the best outcome is when they get to round 5: everyone passes at every chance they get. Total payoff is $40, the maximum possible. Not only that, but this outcome is best for each individual. Neither can possibly make more money than they do if they jointly agree to pass every chance they get. (Contrast this to the prisoners' dilemma, for example, where the problem was that each player wanted to defect while the other cooperated, thus leading to mutual defection.)
Well, this is another central insight of game theory: just because it's collectively best doesn't make it individually rational. But it is quite natural to focus on this last, best outcome.
1039. Slackjaw - May 30, 1999 - 2:52 AM PT
So what is the SPNE? As always, start at the very last nontrivial decision. It is the one where 2 is deciding in round 4 whether to take the money or pass. He faces a quite simple decision: by taking, he gets $16; by passing he gets only $8. So player 2 will take if the game reaches round 4.
But 1 knows this. In particular, she knows that, if round 3 is reached, she can either take, and get $8, or pass to player 2 and precipitate round 4. But we already know what will happen in that contingency: 2 will take, and 1 will be left with only $4. Clearly the better thing for 1 to do is take if round 3 is ever reached.
But 2 knows this too. So if round 2 is ever reached, 2 can either take, and get $4, or pass, in which case round 3 starts, player 1 will take, and player 2 gets only $2. So clearly it's better for 2 to take if round 2 is ever reached.
Now we are at the beginning: player 1 can either take and get $2 or pass, at which point 2 will take, and player 1 will be left with only $1. Clearly $2 is better than this, so on the very first turn of the game, player 1 takes.
The unique SPNE is for every player to take at every chance they get; in particular, 1 takes in round 1 and payoffs are $2 for 1 and $0 for 2. This is in some basic sense inferior to the outcome where 1 gets $32, 2 gets $8, and the everyone passes all the way until round 5.
That's one reason to be suspicious of this prediction. But, to a game theorist, it's not a very good one. It could simply be a surprising set of incentives, and sometimes they work that way.
A better reason to be skeptical is this: both players know that "take on every turn" is the only SPNE strategy profile. It is predicated, as are all SPNE, on what players anticipate happening later on in the game.
1040. Slackjaw - May 30, 1999 - 2:53 AM PT
If player 2 ever gets to move, it means that 1 is not following her SPNE strategy. But here's the question: what should player 2 infer about the rest of the game conditional on observing this violation of equilibrium?
It's as if we expect "rational" players to play their equilibria strategies, and all the more so if there is only one equilibrium as is the case here. If someone doesn't play their prescribed strategy, then, the other player has strong evidence that the other player is not, in this sense, "rational."
But "take" on any given turn is predicated on the belief "if I don't take now, the other player will follow the rational strategy and take later and I'll be left with a lower payoff." The very fact that you get to say this to yourself means that the suppostion underlying the statement is suspect at best!
How are you supposed to reason about how the other player will play later in the game--the backbone of subgame perfection--if you know they did something they weren't "supposed to" earlier in the game?
1041. Slackjaw - May 30, 1999 - 2:54 AM PT
There is no universally accepted answer to this, but there are a couple different approaches. The one I favor assumes that each player has a commonly known but nontrivial probability of making mistakes, and the players each account for this in their strategy choices. This leads to a different equilibrium concept, called "quantal response equilibrium," and requires relatively more advanced techniques than we have so far. Another approach has it that players assume there is some small chance their opponent is "irrational," even though each player knows that he himself is rational, and so do things a little differently than standard theory predicts. This approach is a little easier and we might actually get to it.
Experimentally, people don't usually take right away, but nor do they pass the whole time. The evidence appears more consistent with the first explanation above than the second.
1042. uzmakk - May 30, 1999 - 6:18 AM PT
Slackjaw:
Thanks for the analysis of the Monsanto article. I suppose that the business of a corporation claiming ownership of all the seeds produced on a farmer's field has some kind of precident? For instance, I believe that it is illegal to propagate many roses. The game aspect that you put forth is quite a twist for me. Easy to understand, but quite a twist nontheless. The legal system used as a scare tactic in a game. I have used the word "control" before. Could you say a few words about the "control" that the Monsanto entity will have when seeds, previously thought to be a gift of god or something, are owned by Monsanto? I don't think that it is difficult to imagine that in the future, if the use of these seeds becomes widespread, that normal seeds and the nonresistant crops they produce will be fodder for insects. If I recall, not only are the seeds robust, but one of the properties that the plants have is that they can stand up to very powerful insecticides made my Monsanto that normal plants cannot tolerate. So, in the future, will all the seeds, produced in all the fields of the world be owned by biotechchemical companies? It is my impression that all of the fields are owned by agribusiness giants already and the trend continues. Very interesting. Can you say a few words about the idea of "control" in game theory? Or is the "theory of control" part of some other branch of mathematics?
1043. FreeToChoose - May 30, 1999 - 7:13 AM PT
In Message #1019 stamper says:
"Now if you flipped that coin 100 times and every time it came up heads, what are the odds on the next toss."
Slackjaw was correct to point out that a coin has no memory, so the odds on the next toss are still 50/50. However, as I am sure he will admit, he assumed the standard assumptions, namely that the coin is fair and it is being tossed in a random fashion. AuNaturel is equally corrct to point out that you should become suspicious about those assumptions after that many heads in succession.
Interestingly (at least to me) we can quantify some of this.
The odds of 10 consecutive heads from a fair coin is one in 1024. This is close enough to one in 1000 to use as a useful approximation. The probability of independent events is the product (multiply the probabilities together) of the individual events. We see that 20 consecutive heads is 10 heads followed by ten more, so the probability of each is one in a thousand and the probability of 20 is one in a million. Thusly, the probability of 30 consecutive heads is one in a billion; 40 is one in a trillion, etc.
1044. FreeToChoose - May 30, 1999 - 7:14 AM PT
continued
A 1 in 1000 event is fairly common. For example, suppose Stamper and Dolly each sit down and flip fair coins, recording the results of each of them (combined). If they can flip a coin one each second, they probably will have a run of ten (either heads of tails) within ten minutes. However, if they want a string of 20, they would have to flip continuously, day and night, with no breaks, for about a week. A string of 30 would take about 20 years. A string of 40 would take more than all of recorded history. If every person on the face of the earth flipped coins once each second, no one would record a string of 100 for millions of years.
Suppose the coin to be flipped was chosen at random from all coins in the world. The number of coins is probably less than one trillion, and there must be at least one double-headed coin on earth. If you observe 100 heads in a row, it is far more likely that someone happened to have the double-headed coin in his pocket than getting this string from a fair coin.
There are a few billion people on earth, several of whom are magicians. The odds are better that the person flipping the coin is a magician (presumably able to force a result) than that the results was chance.
Someone might argue that double-headed coins are usually owned by magicians, and not likely to be in general circulation. We could make calculations to determine how much less likely such a coin would be in a person's pocket, but I contend that for plausible selections of odds, it is still more likely that such a coin found its way into circulation.
1045. FreeToChoose - May 30, 1999 - 7:17 AM PT
uzmakk
"Thanks for the analysis of the Monsanto article. I suppose that the business of a corporation claiming ownership of all the seeds produced on a farmer's field has some kind of precident?"
I doubt that precedent is necessary. Precedent implies that common law would side with Monsanto's claim. It is far more likely that the claim is covered by contract law. When the original seeds are sold to the farmer, the farmer signs a contract acknowledging that any seeds produced are the property of Monsanto. I don't know that this is what happened, but I can't think why it wouldn't be possible.
1046. uzmakk - May 30, 1999 - 9:45 AM PT
FTC:
Thank you. I just find the situation that seems to be evolving rather curious. Brave New Worldish. I accept the legality of it all. But the legality pales in the face of the reality.
1047. freetochOOse - May 30, 1999 - 10:43 AM PT
uzmakk
I think I have read all the posts on Monsanto, but if I missed a discussion of this, I apologize.
The interesting part about Monsanto is not that they claim ownership of the seeds produced, it is that Monsanto has produced seeds that will bear fruit, but the resulting seeds will be sterile. So they have solved the problem of how to deal with farmers that may consider saving the resulting seeds.
Slackjaw's description of the approach is still an interesting concept in game theory, but Monsanto has essentially cut the Gordian Knot.
Yes, Monsanto's seeds are resistant to their insecticide. The insecticide is called Roundup, and the seeds are called Roundup Ready.
1048. uzmakk - May 30, 1999 - 11:07 AM PT
Free to Choose:
I don't know that they have inserted the "terminator" gene yet. The situation seems to be that they have not. Roundup Ready seeds currently produce crops which produce more fertile Roundup Ready seeds. This is the problem from Monsanto's POV. This is why the lawsuits. Their valuable commodity goes on the market for free, or very much less, if the seeds are fertile. Have not reread the article in question, but it mentioned the possibility of a terminator gene "migrating". All of this genetic manipulation to keep Monsanto in control.
1049. Slackjaw - May 30, 1999 - 12:25 PM PT
Message #1043 Not only would I acknowledge this, I did acknowledge it.
Uzmakk,
control theory is the name of a different branch of mathematics, one where (say) you try to maximize something, but the choice you make is of a function or infinite sequence of vectors rather than simply a number or a single vector as in basic calculus.
But, that's not really the sense in which you use "control." I will address this, and I'm not done with the Harper's article in any case.
1050. Slackjaw - May 30, 1999 - 12:29 PM PT
Message #1047 As I said, I don't think Monsanto had terminator genes in these particular seeds they were taking legal action over.
Does anyone know whether the pesticide used by one farmer in his own fields spills over into the fields of neighboring farmers, affecting their fields too? As a rule I mean.
1051. FreeToChoose - May 30, 1999 - 12:53 PM PT
here is a site with info on Monsanto, and on the terminator seeds.
While there is obviously reason for concern, note that there are also positive aspects. For example, some people are concerned about GMO's (Genetically modified organism) getting out of the farmland and into the wild. A terminator version of a seed would help reduce this concern.
1052. Slackjaw - May 30, 1999 - 1:13 PM PT
pop quiz: which of the following was genrerated by a (presumably fair) coin, and which did I concoct?
sequence 1: H T H H T H T T H H
sequence 2: H H H T H H T H H H
1053. FreeToChoose - May 30, 1999 - 1:20 PM PT
At the risk of being immodest, I'll leave my guess and reason until others have a chance to weigh in.
1054. Slackjaw - May 30, 1999 - 1:21 PM PT
thanks for the link FTC.
Here is something from the "analogy" page.
"For years, Nestle provided free samples of its milk formula to hospitals and clinics in developing countries. New mothers were given enough of a supply to ensure that they would no longer be able to lactate by the time their Nestle formula ran out, forcing them to buy more formula. Unsterile conditions and lack of money contributed to the deaths of untold numbers of babies due to diarrhoea and malnutrition. It took a worldwide boycott for Nestle to clean up its act.
"...I predict that North American seed companies will act in a similar way, providing the initial seed investment at a low price and building up a dependency. Farmers in developing countries have little cash to spare, and it's an ongoing tragedy that they are told they must buy expensive fertilizers and pesticides, rather than learning ecological agricultural practices. How will they be able to resist the marketing of terminator technology?"
Seems like a different situation to me, and a misleading analogy. Note the appeal in the very last line--it all comes down to "marketing" in the case of seeds. I don't understand how the dependency will work the same way. If seeds and fertlizer are too expensive, farmers will certainly find it in their interest to learn other methods. The trick is for Monsanto to price seeds just low enough so they still reap some surplus from the deal but don't induce farmers to switch practices. I don't know how low that price will be.
1055. FreeToChoose - May 31, 1999 - 6:17 AM PT
Slackjaw
I agree. At best, they are trying to analogize a physical dependency with a weak psychological one. If, however, Roundup affects the soil such that it is not feasible to return to ordinary seed, there may be a physical dependence, but I haven't even heard that alleged, much less proven.
(BTW, I saw Bite the Bullet last night. I don't know whether you were planning to join the FrayFilmFest, but there is a modicum of game theory in the movie. I know, not much, but how many movies have any?)
1056. cdm1110 - May 31, 1999 - 7:08 AM PT
FTC Re: 1043, 1044. What you say is quite reasonable. If we observed something that seems so nonrandom as a sequence of 100 heads, it makes sense to question the assumption of a fair coin, fairly tossed. Don't forget, though, that the probability of *any* given sequence of outcomes is a low probability event. For example, consider the following sequence:
THTHTHHTHTTTHTHTHHTTHTHHTTTHT
HTTTHHHHTTHTTHTTTHTHTTTHTHTHH
HHHTTTHTHTTTHTTTTTTTHTHTTHHHT
TTHHHTHTHTTHTHTTHHHTTHTHHTTHH
which I generated arbitrarily (not the same as randomly). Now, to paraphrase you, "if every person on the face of the earth flipped coins once each second, no one would record [this string] of 100 for millions of years."
As for game theory in movies, my favorite is Dr Strangelove.
1057. FreeToChoose - May 31, 1999 - 7:39 AM PT
cdm1110
"Don't forget, though, that the probability of *any* given sequence of outcomes is a low probability event."
Absolutely. It is often assumed (incorrectly) that a long sequence with a clear pattern (such as all heads, or alternating heads and tails) must be less likely than a pattern such as yours. While you are absolutely correct that the sequence you listed would take as long as a sequence of all heads to generate randomly, the notion that a sequence of all heads seems more unlikely than your sequence isn't all that misguided.
When we look at a sequence, we tend to categorize it into some group. This will tend to be the most restrictive group that can contain the sequence. For example, 100 heads would be mentally grouped into "all heads" or possibly "all the same", but not "at least 65 heads" and certainly not "more than 40 heads with no more than 3 tails in the first 20"
Similarly, HTHT...might be viewed as essentially the same as THTH..., and the pattern HHHTTTHHHTTT... might be grouped with HHHHTTTTHHHH... (Roughly speaking, we tend to group based on information content).
The sequence you gave doesn't seem so unusual, because we mentally group it in the category "no discernible pattern", and this group has many members, in fact, well over 99% of all sequences.
1058. FreeToChoose - May 31, 1999 - 8:00 AM PT
cdm1110
Not a bad sequence. It passes the test I used to select the non-random sequence from Slackjaw's pair. It is a bit high on "sign" changes (H followed by T, or T followed by H). With 59, this is a bit high. I'm too lazy to do the math to determine how unlikely this is.
Here is a decidedly non-random sequence:
HHHTHHHHTHHHHHTTTTTTTTTHHTT
What is the next entry?
1059. Slackjaw - June 1, 1999 - 10:07 PM PT
T
1060. CalGal - June 1, 1999 - 10:10 PM PT
FTC,
Game Theory in Bite the Bullet? You don't mean the ending, do you? That's the *opposite* of game theory (unless you think Hackman did it on purpose, and ain't no way.)
1061. Slackjaw - June 1, 1999 - 10:22 PM PT
oh, I just saw that post. Well, I don't think I'd make any new friends by blathering on about game theory in *two* threads.
And *all* movies are *packed* with strategic interaction. Haven't seen Bite the Bullett, though. Frankly, much of plot tension can be understood either as a game between the director and the audience, or as tension created when characters interact strategically with each other. Yes, it's the new craze sweeping art criticism: literary game theory.
There is a famous example of Bayesian game theory drawn from _The Maltese Falcon_.
My favorite game theory in art is _Othello_.
1062. FreeToChoose - June 2, 1999 - 5:10 AM PT
CalGal
No not the ending (although I did think it was on purpose. I was going to say it was too early to comment further, but I guess it is now Wednesday, so the discussion has started).
I meant earlier, when Coburn proposes that Hackman agree not to win. I was going to say that it was a very weak instanc eof game theory, that I mostly posted it to encourage Slack to see the movie. Even if he retorted that there was no game theory, we would have another participant. (Don't tell him, though)
But as I think about it more, there are interesting questions, probably well-settled by game theorists, but perhaps of interest to those who have only a bit of knowledge about game theory. For example, in a game with monetary prizes, I presume that game theorists want the participants to maximize their monetary income. But Hackman makes a choice that does not maximize his expected income. And I don't think it was irrational.
There's also the matter of the odds. The odds on Coburn were 7-1. This implies that he had an average chance of winning (if I am right that there were 7 participants). But surely there were some (Bergen, the old guy, the Mexican) with higher odds. Was the English guy enough of a favorite to make these odds reasonable, or did the filmmakers not give it much thought?
1063. FreeToChoose - June 2, 1999 - 5:12 AM PT
Slack
I haven't seen the Maltese Falcon, but I will.
1064. jayackroyd - June 7, 1999 - 6:51 AM PT
Message #1052
One way to analyze this is bayesian-style. Specify your prior probability belief of heads is one half.
In that case, the posterior probability for heads will be higher for sequence 2 than for sequence 1.
However, that prior will be really peaked (say ranging from 0.48 to 0.52), so this little variation will have a small effect on the posterior.
OTOH, 100 consecutive heads (in the post that started this), is gonna lead to a posterior probability of heads close to 1.
1065. chloel - June 7, 1999 - 2:37 PM PT
Actually, industrial farming (monocultures, high use of pesticides and inorganic fertilizers) does create a dependency on the chemicals, a phrase widely used - e.g.,
"The introduction of the broad-spectrum soil fumigants after World War II led to the replacement of traditional diversified farming systems by large-scale monocultures. Soil fumigants provided reliable and excellent disease and pest control, increased yields, high quality produce, extended crop seasons and reliable economic returns. Consequently, present-day California agriculture can be characterized by increased use and dependency on synthetic pesticides, a reduction in crop rotation frequency, and a limitation in the number of crops grown (163).
The increased use and dependency on pesticides in high-yielding crops have not only led to high and stable yields but also to increased risk to soil, water, and air pollution. A reduced crop rotation frequency can increase the epidemiological potential of soil-borne diseases and pests which can increase pesticide use. In addition, soil fumigation leaves a biological "vacuum" suitable for re-infestation by plant pathogens, requiring that the soil be treated each growing season."
(source here.)
However, high yields from synthetic inputs are sustainable only with increasing synthetic inputs - syn. fertilizers are *much* faster than organic ones, but also leach out a lot faster, and are so 'hot' that they can kill off a lot of the natural microorganisms in the soil which would normally be making nutrients available. You therefore need more and more inputs every year. Eventually you're doing hydroponics, using killed soil as a medium and paying for all the nutrients. It takes several years to go back to low-input farming because the topsoil has to be rebuilt.
1066. chloel - June 7, 1999 - 2:38 PM PT
Interestingly, Cuba seems to
be doing so, because their monocultures were explicitly supported by foreign
aid - without the aid, they're returning to low-input, but high-knowledge,
farming. High-input farming in the States is most profitable when oil is
cheap here.
The Phillippines are working on reducing pesticide use - not only do they
have a lot of small farmers, but it's particularly stupid to allow poisonous
runoffs when you need the fish & shellfish from your shores - and they have
several sets of data showing that educated farming can use less chemical
inputs and maintain or improve yields; see this & search for "Tangible gains of IPM".
1067. stamper - June 9, 1999 - 9:36 AM PT
Slackjaw
is there anyway that with mathematics that you can prove that ACEOFSPADES is not stamper? if that could be done, and you could put that on the Play Pen and clear up this mess, i sure would thank you.
1068. Slackjaw - June 10, 1999 - 1:32 AM PT
well stamper, if you weren't such an uneducated feller you would know that every school child can construct such a proof.
One simply needs to define a metric over stamperness, and invoke the Hahn-Banach theorem to show that the stamper function is measurable in a Banach space. It is then quite direct to show that in a Banach space the stamper function is discontinuous at the point AceofSpades, quod erat demonstrandum. It is also elementary, using a standard epsilon-delta argument, to show that the stamper function is continuous at all other points.
1069. stamper - June 10, 1999 - 7:15 PM PT
Slackjaw
now you are funning me now, aren't you. i guess you are on to my game anyway, of going all over the Fray making sure enough folks got over to the Play Pen to see my little trick on AceOfSpades. you are too smart to be fooled.
now on a serious not. as you may know, or maybe not, i am doing some research on fray posting. i may need some help on what to do with all my data, like frequency of certain posters, hours they post. and perhaps you could supply me with things i should be looking for. i don't expect you to do my work for me. but if you would be so good as to be a sort of mentor, i would forever be in your debt. i laid it out on the suggestions thread.
1070. stamper - June 10, 1999 - 7:17 PM PT
chloel
i have also noticed that most phillipinoes are small.
1071. Slackjaw - June 10, 1999 - 9:26 PM PT
Just got back from administering an experiment on the centipede game, the one analyzed theoretically in Message #1037. The subjects were from Pasadena City College, a local community college. It was a small pool actually, and the experiment was just a pilot to get them acclimated to the environment. They played a 6-turn version of centipede.
There were 8 subjects. They played a few practice rounds to make sure they got it, then played 10 real games of Centipede for real money. Each subject was randomly re-matched with a new subject after every game, so they do not learn how to play one particular person.
I have not analyzed the data yet, but here are a few quick observations of interest. First, in the early rounds, there is a marked tendency to "pass" or display cooperative behavior, which does not in this game happen to be a subgame perfect Nash equilibrium. In rounds 1-3, 3 of 12 games went all the way to the end--every player passed on every turn. 7 games out of the first 12 went for at least 3 rounds. Only 2 matches in the first 3 rounds ended before the 3rd turn, and only 1 of those 2 ended on the first turn.
However, the subjects definitely learned the equilibrium over time, which is a consistent pattern in game theory experiments, as I alluded to when describing the beauty contest. In rounds 6-10, toward the end of the experiment, 14 of 20 matches ended before round 3. Of the 6 that didn't, only 2 of occurred in rounds 8, 9 or 10. In the last 3 rounds--8 through 10--2/3 of matches ended on the first turn, the SPNE prediction. Clearly, there is a tendency to get to the equilibrium over time.
Payoffs to the subjects ranged from about $20 to $35, depending on their play. The experiment lasted about 1.5 hours. They were paid with your tax dollars by way of the National Science Foundation's Social and Behavioral Research program.
1072. Slackjaw - June 10, 1999 - 9:27 PM PT
The purpose isn't really to test straight ahead Nash equilibrium, but to discriminate against some more refined equilibrium concepts that do considerably better than Nash when confronted with data. One of these is the Quantal Response Equilibrium, which we haven't covered yet.
1073. Slackjaw - June 10, 1999 - 9:32 PM PT
stamper,
I'll not serve as your dissertation advisor in the U of the Fray's D. Conf. program unless you tell me your social security number or at least your true Fray identity.
I will, however, serve on your thesis committee. That means I won't read any of your research until it is officially time for you to present it, then I will subject you to relentless contumely for technical flaws and other things you should have done.
Unless you can find a chair for your thesis committee, you'll have to leave the program, by the way.
1074. Slackjaw - June 15, 1999 - 7:20 PM PT
well, looks like we've been RIPed. Thanks to all those who participated in this thread. Too bad I haven't had more time to push it along lately, but it looks as though I'll miss it the most.
I encourage anyone interested in pursuing any of this to bring it up in The Future of Capitalism. Hopefully we can also get into Bayesian vs. classical statistics over there, when I get around to it. I am much better equipped to have that discussion today than I was a year and a half ago.
1075. Slackjaw - June 15, 1999 - 7:21 PM PT
but really, it was only a matter of time before we got rid of the totally unjustifiable thread devoted to "game theory, Bayesian statistics, and related topics."
1076. pellenilsson - June 16, 1999 - 3:38 AM PT
slackjaw
Thank you for hosting this thread and for your contributions. I have been in lurking mode because I know so very little about game theory, although I know a little more now.