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Strictly as formulated, this is not an ultimatum game, for ultimatum game specifies a particular protocol: one player proposes a price, the other has one chance of accepting or rejecting. The post assumes no such restriction, the players could for example go through 100 bargaining iterations of any nature, such as usually proposed in various bargaining protocols.

shokwave14y20

But as long as the payment itself is not iterated (ie there is still only one easement) then at any point during the bargaining both players can make more money for themselves by pushing for more money.

See in context

19 Econ/Game theory question

by Psychohistorian

11th May 2011

1 min read

19

This puzzled me. I'm pretty sure it's one of those unsolvable questions, but I'd want to know if it's not.

Two members of the species Homo Economus, A and B, live next to each other. A wants to buy an easement (a right to cross B's property, without which he cannot bring anything onto his lot) from B so that he can develop his property. B, under the law, has an absolute right to exclude A, meaning that nothing happens unless B agrees to it. The cost to B of granting this easement is $10 - it's over a fairly remote part of his land and he's not using it for anything else. A values the easement at $500,000, because he's got a sweet spot to build his dream house, if only he could construction equipment and whatnot to it. A and B know each others costs and values. They are "rational" and purely self-interested and bargaining costs zero. What's the outcome? I'm guessing it's "Between $5 and $500k," or "There is no deal unless one can credibly commit to being irrational." But I'm really not sure.

This could be asked as "In a bilateral monopoly situation where the seller's reservation price is $5 and the buyer's is $500,000, what is the predicted outcome?" But I figured the concrete example might make it more concrete.

Now that I've written this, I'm tempted to develop a "True price fallacy" and its implications for utilitarian measurement. But that's a separate matter entirely.

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[-]endoself14y120

$250,005 - the average.

Based on the proof at http://www.stern.nyu.edu/networks/phdcourse/Nash_Two_person_cooperative_games.pdf . You can check the assumptions used, but I think they match up to this scenario. It's an open problem to generalize this to an unlimited number of agents, with incomplete knowledge of each other, etc.

[-]Perplexed14y60

This is basically correct. It is a two-person cooperative game and the 'classical' solution is the Nash Bargaining Solution - introduced in the 1950 paper that you cite. Stuart_Armstrong has written several top-level postings on this standard topic in game theory recently. So it is shocking to me that so many people failed to identify the problem and even more shocking that so many of them incorrectly think that it is the ultimatum game.

I have one minor quibble with your solution, and one improvement. The quibble is that it is not necessarily the case that the marginal utility of money is the same for the two players. The Nash bargaining solution is based on maximizing the product of the utility gains - not the money gains.

The improvement follows Rubinstein. If the two parties do not reach agreement today, then they can still reach agreement tomorrow. (This is why it is different from an ultimatum game.) So the threat that each party holds over the other is to delay the (ultimately inevitable) agreement. But although the delay applies to each party (one has a delay in building the house, the other has a delay in receiving and then investing the cash), it may be that the two have different discount factors. The opportunity to build now is worth $500,000 to the one guy, but perhaps he is in such a hurry that the ability to build next year is worth only $400,000. But the other guy may figure that $500,000 next year is worth about $475,000 now. His discount factor is is only about a quarter of that of the eager home-builder. This puts him at a significant advantage in the bargaining - so much so that the solution of the Rubinstein bargaining model is close to $400,000 to be paid for the easement.

[-]endoself14y00

Stuart_Armstrong has written several top-level postings on this standard topic in game theory recently.

Actually, I think that's how I heard of this solution. Links for the interested.

[-]wedrifid14y30

Thanks for the reference.

You can check the assumptions used, but I think they match up to this scenario.

I read through to find the assumptions. It introduces the ability for either party to make precommitments (page 130) and a four stage negotiation process.

[-]endoself14y00

It introduces the ability for either party to make precommitments (page 130)

So it applies to perfectly rational agents.

and a four stage negotiation process.

A bit later he gives an alternate derivation.

[-]wedrifid14y-10

So it applies to perfectly rational agents.

The assumption in the actual document is that there is an umpire of some sort which can be relied on to enforce the pre-commitments.

[-]Skatche14y90

It seems to me that the easement will cost, at most, the amount of money that B could get from A in court for illegally crossing B's land. Given the additional expenditure of time and legal fees, not to mention the uncertainty of the legal outcome, it will probably be somewhat less than that.

[-]Vladimir_Nesov14y120

This clever point shouldn't distract from the intended sense of the post.

[-]wedrifid14y40

It seems to me that the easement will cost, at most, the amount of money that B could get from A in court for illegally crossing B's land. Given the additional expenditure of time and legal fees, not to mention the uncertainty of the legal outcome, it will probably be somewhat less than that.

I'm not sure how the parent managed to get to +11 votes. It introduces an additional external complication to the problem and then handles it incorrectly. The limit from this new mechanism is actually the amount that B could get from A in court for A trespassing every time A wishes to travel across B's land for the duration of the life of the easement - which appears to be indefinite. The value of a once off trespassing suit is not all that relevant.

[-]Skatche14y00

Well, of course. But assuming B is a rational agent, and assuming the expected damages awarded in court per trespass are additive, she's going to wait until A has finished building his house, then take him to court for all counts of trespassing, rather than fight each one individually, since that'll save her a great deal on time and legal fees.

[-]Psychohistorian14y00

B could get an injunction prohibiting the crossing of his land. Easements traditionally give rise to injunctive relief. That would make A criminally liable if he or his agents crossed his land - it wouldn't be too hard for B to prevent any construction company from working there. That the outcome of litigation is certain is stipulated to. That's actually why this problem is interesting - there is some dispute as to whether injunctions or damages are better solutions to these problems.

If you want to make it cleaner, I suppose you could say that B has put up a fence AND obtained a declaratory injunction already, and they're trying to bargain to have B invalidate the injunction. But I thought the original was clean enough.

[-]Richard_Kennaway14y00

B could do all of these things to keep the problem in the box you're trying to define, but if he does, it's clear that friendly relations have already broken down between A and B, and by acting in this way, B is reducing the value of the land to A. Does A still want to live next door to a neighbour who is going to be so obnoxious about trifling property disputes?

I understand that you're asking the question: how can prices be rationally decided in a bilateral monopoly? But the response that bilateral monopolies don't happen can't be brushed aside. Rational agents in this hypothetical situation will always be looking for alternatives, and the more is at stake the more creative they will get about it.

[-]Psychohistorian14y10

The actual solution to this in the real world, 99 times out of 100, is that B just says OK, or A insists on giving him $100 to cover the damages, or something generally amiable. The reason I asked this question is because I'm thinking about the efficiencies of injunctions (which result in bargaining) versus damage awards (which generally don't). So the only characters I care about are the ones who aren't neighborly.

Indeed, having confirmed my suspicions that this problem is insoluble, it favors a damage award in this context. B's actions are almost pure holdup. If all he were entitled to were damages and not injunctive relief, he wouldn't have nearly the same capacity for holdup, and the outcome looks more like the neighborly one (except with more bad will, perhaps).

In other words, I'm assuming that the agents are selfish and somewhat inhuman - irrational in a big picture sense - because occasionally these disputes do happen. There's a MAJOR case where a landlord sued over having to install a 1 cubic foot cable-box that increased their property value, and there's a case of a guy suing to stop someone from using an easement to get to a contiguous property (i.e. he had a right to cross A to get to B, but he was crossing A to get to B and then continuing on to C, and that was impermissible and went to court).

[-]Vladimir_Nesov14y90

This is essentially an ultimatum game (if we focus on the interesting bits). There is no theory that reliably helps with the process of picking a fair price, and the fair price depends on many factors which you didn't specify, including the details of how players' minds work, and what each player believes about the other. This makes intuition the only method that can take into account all the varieties of potentially relevant information, although there might be some explicit algorithms that show better performance in practice, especially if the other player doesn't know what algorithm you use.

There are some ideas from game theory that suggest certain algorithms for picking fair price, but their outcomes are mostly the product of privileging those algorithms as Schelling points for reaching agreement and not of clear a priori considerations for which price should be chosen. If players' brains are completely rotten by CDT thinking, they will additionally insist that A should accept whatever B is demanding, and conversely, depending on who gets the last say.

"There is no deal unless one can credibly commit to being irrational."

This is using "rationality" is a wrong sense. The word should refer to whatever it is they should do.

[-]Richard_Kennaway14y40

How is this an ultimatum game? There is no limitation in the problem of how long A and B can take to negotiate over the matter or what form that negotiation may take. Adding such limitations is not focussing on the interesting bits, it is focussing on a different problem.

[-]shokwave14y00

Nesov is right, this is just the ultimatum game.

Consider B granting the easement and charging A $10. B gets $0, A gets $499,990.
Consider B granting the easement and charging A $500,000 dollars. B gets $499,990, A gets $0.
Consider B not granting the easement. A and B get $0 each.

So they are playing a reversible ultimatum game for $499,990. Either A or B may make or reject offers. Any negotiation would immediately reduce to the ultimatum game - if A and B have a common Schelling point (say at 50/50) then they aren't homo economus, they are homo sapiens.

[-]Vladimir_Nesov14y00

[-]shokwave14y20

[-]Vladimir_Nesov14y00

I think it doesn't matter how long they have to decide, if we are resolved to ignore intuitive hunches for Schelling points and causal updating (which are important in practice, but not in principle). You can see any problem as one-step by deciding a whole (possibly infinite) strategy instead of just the next action. The questions governing the way this strategy should be chosen are similar (for the purposes of my comment) to what happens with simple ultimatum game.

[-]Richard_Kennaway14y00

The original problem is symmetrical: there is a potential trade which will benefit both A and B, and they need to strike a price. The Ultimatum game is asymmetrical: one player goes first. This seems to me a conclusive proof that this problem cannot be modelled as an Ultimatum game.

You can see any problem as one-step by deciding a whole (possibly infinite) strategy instead of just the next action.

I don't think this works in the large unless P=NP (or something of the sort). In the small, e.g. analysing chess, it reduces the problem to no steps at all: both players exhaustively analyse the game and know the outcome without playing a single move. (I'm using "small" and "large" in the sense of the dispute between small-world and large-world Bayesians.) If that worked for the bargaining problem, A and B would independently come up with the same price and no bargaining process would be necessary. No-one has posted a method of doing so.

[-]wedrifid14y-20

How is this an ultimatum game? There is no limitation in the problem of how long A and B can take to negotiate over the matter or what form that negotiation may take. Adding such limitations is not focussing on the interesting bits, it is focussing on a different problem.

You are correct. Vladimir is discussing an entirely different problem.

[-]Eugine_Nier14y20

I agree with this post.

Also an important idea implicit in the above post that I think deserves to get spelled out is that the mere act of thinking about a Schelling point can move its location.

[-]wedrifid14y-20

Also an important idea implicit in the above post that I think deserves to get spelled out is that the mere act of thinking about a Schelling point can move its location.

An additional point worth spelling out is that Homo Economicus has by definition already thought everything through so no such movement is possible here.

[-]drethelin14y30

why the fuck did a buy land that is apparently inside b's land?

[-]wedrifid14y40

why the fuck did a buy land that is apparently inside b's land?

It happened on "Round The Twist". In that case the precise boundaries of the lighthouse's plot technically didn't include their driveway and they only discovered this when a nefarious company bought out the neighboring property.

[-]gwern14y30

Exogenous shocks to his wealth. By which I mean his rich irrational uncle left him land.

It doesn't matter! Asking why is dangerously close to sunk cost bias - what matters is the situation he's in, not why he's in it.

[-]James_Miller14y30

It's theoretically indeterminate. To figure this out you would need to develop a specific bargaining structure such as A makes a take it or leave it offer to B.

[-]Vladimir_Nesov14y10

It's theoretically indeterminate.

Since the question is (partially) of descriptive nature, theoretically you can just rerun the universe up until 5 seconds after they struck the deal and check the outcome. If you mean that existing theories don't give any useful answers, I agree, although again for the descriptive question we can have some useful stats.

[-]Psychohistorian14y10

The problem is that if A is perfectly rational, in a sense, he can't make a credible take it or leave it offer. If he offers $10,000, B knows he would be willing to pay $11,000, so he rejects. On the other hand, A knows B would take less than $10,000, so why offer that much in the first place? That's why I suspect it's just intractable.

Do any of the elaborate decision theories popular around these parts solve this problem?

[-]cousin_it14y40

No, our decision theories don't have any new insights for solving bargaining games. But such games are widely studied elsewhere, so maybe you can find a model that solves your problem if you feed it some more detail. Sorry for the disappointing response :-(

[-]Vladimir_Nesov14y20

No, our decision theories don't have any new insights for solving bargaining games.

I think they do give the insight that there can be no simple solution. Each player does have to think about the other's thinking to get further advantage. Even if one of the players is (safely boxed) superintelligent and the other isn't, as ASP problem demonstrates.

[-]jsalvatier14y00

Don't they at least a little bit? Isn't pre-commitment a natural feature of TDT? That doesn't necessarily solve such problems, but it does seem like its relevant. I suppose that's not a new insight; guess I answered my own question.

[-][anonymous]14y00

Allowing players to precommit isn't enough to pin down a single outcome in a bargaining game. Different combinations of precommitments can lead to different deals, or even failure to make a deal at all if they are incompatible.

[-]wedrifid14y-10

Isn't pre-commitment a natural feature of TDT?

In this context what the pseudo precommitment does is allow B to metaphorically break out of an ultimatum game even if A was somehow able to limit the situation to an ultimatum. Note: this would apply even if only B was using TDT and A was using a CDT.

[-]bentarm14y10

The problem is that if A is perfectly rational, in a sense, he can't make a credible take it or leave it offer

Of course he can - he signs an enforceable contract to pay C $500,001 in the case that he is ever seen to be offering B more than $8191 for the land, and has done with it.

[-]Psychohistorian14y10

...but when he signs this contract, he may find out that B signed a contract refusing to accept anything less than $450,000 for the land, or else pay some large some to D. If there's any lag in communication between the two of them, this is an extremely risky strategy.

[-]wedrifid14y00

The problem is that if A is perfectly rational, in a sense, he can't make a credible take it or leave it offer.

I think James meant adding a new limitation to the situation such that there is only one chance to make the deal and one person goes first. ie. Turn it into an ultimatum game.

Do any of the elaborate decision theories popular around these parts solve this problem?

No. At least not if there isn't also a solution specified in Causal Decision Theory. The same problem exists in both. In fact using Timeless Decision Theory makes the problem apply even in the ultimatum game variant. Because even if I was perfectly rational if A offered me $11,000 I would tell him or her to go @#$@#$ @#$@#$.

[-]wedrifid14y-20

It's theoretically indeterminate. To figure this out you would need to develop a specific bargaining structure such as A makes a take it or leave it offer to B.

That would not help unless you also stipulate that a specific (broken) decision theory is to be used.

[-]saturn14y20

If both A and B are TDT-like and have no restrictions on their communication, I think the outcome would be $250002.50

[-]fubarobfusco14y10

How much money does A have? What are her other options for how to use it? How much would it cost to move? It's extremely uncommon for people to be in a situation where they have no choice but to deal with a monopolist; indeed, that pretty much only occurs as the result of outside-the-market constraints such as regulation.

[-]wedrifid14y-10

Replace dollars with utilons. That automagically incorporates all of those considerations into the utilon calculation we are provided.

[-]nerzhin14y10

This looks very much like an ultimatum game, with Player A playing the proposer (how much of her $500,000 will she share?) and Player B as the responder (his role is to either accept or decline).

[-]Psychohistorian14y00

As has been pointed out, it is not, because there is continuous interaction, bargaining, and perhaps the potential for pre-commitments, though as I mentioned, those could be risky.

[-][anonymous]12y00

I think it resolves as follows: If A can precommit more strongly than B can (and show than he can), the agreed upon price will $10. If B can precommit more strongly than A can (and show that he can), the agreed upon price will be $500,000. If they're sufficiently equal in mental ability to precommit (or neither can credibly show they've precommitted), the agreed upon price will be either $250,005 (if they're superrational) or nonexistent. Has homo economus developed the ability to precommit?

[-]Mass_Driver13y00

The way I was trained to think about this type of problem was to consider each party's best available alternative (BATNA). Even without trying to deal with psychological considerations, A could probably sell his property, buy a new property elsewhere, and build his dream house there. That might cost him $300,000 in transaction costs and in the decreased value of the second-best dream house site, but it would still provide an upper limit on how much he was willing to pay to escape a holdup.

Similarly, even if ultimate legal victory is certain, B's alternative to negotiation involves going to court to enforce a holdup -- the inconvenience is probably worth at least on the order of $400.

So you can narrow the likely range of settlement from [$5 $500,000] to something like [$400; $300,000]. Within that space I agree with the other commenters that the answer is more a matter of psychology than economics -- although economics might have something to say based on relative bargaining power if we add more details about, e.g., each party's timeline for construction or suit, each party's access to credit, and so on.

[-]TraderJoe13y20

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[-]Mass_Driver13y00

I think assuming away the secondary variables in a negotiation problem is less interesting/useful than assuming away friction and air resistance in a physics problem, for tworeasons.

First, air resistance usually explains only a small fraction of the variance in outcomes -- my numbers won't be quite right, but the distribution of physics parameters across all back-of-the-envelope physics calculations will probably be something like: log(air pressure in atms) = 0 +- 0.2, log(density in g/ml) = 0 +- 0.3, and log(initial velocity in m/s) = 1 +- 1. If you vary the air pressure by one standard deviation, you get maybe a 3% change in the distance traveled. If you vary the initial velocity by one standard deviation, you get maybe a 200% change in the distance traveled. So it's relatively safe to leave out air resistance. By contrast, bargaining power, precommitment, BATNA, transactional costs, etc. often explain as much or more of the variance in the agreed-upon price as does the deal's utility to each person. It's not at all safe to leave those factors out -- if you do leave those factors out, it's unlikely that you'll get even 4 full bits of Bayesian evidence about the final price as compared to a prior that allows any price between $0 and $10 million.

Second, air resistance is much harder to measure than initial velocity, and, even if you figure out what the level of air resistance is, introducing that variable makes the math considerably harder -- you might have to upgrade from algebra to calculus. Each additional variable superlinearly increases the risk of calculation error, because physics reasons from explicit mathematical equations whose counterintuitive results should be trusted, assuming the math and the inputs are correct. By contrast, it's often just as hard if not harder to measure the utility a person gets from a deal as it is to measure, e.g., their BATNA. Adding new variables to the calculation doesn't necessarily add much to the difficulty of the calculation, because there is no unified, complex mathematical equation -- just Bayesian updating on what each piece of data has to say about the overall likelihood of any particular deal. In other words, if the deal is worth a lot of money to you, that suggests that you're relatively more likely to agree to a higher price; if your BATNA is quite good; that suggests that you're relatively less likely to agree to a higher price. Total up all the evidence pro and con, and you'll get a sense of where the price is most likely to be. There's no need to factor complex mathematical expressions; at worst, the complexity of the calculation increases at a linear rate as new variables are added.

[-]wedrifid14y-10

(Pardon a couple of typo corrections.)

Homo Economus

Homo Economicus

I'm guessing it's "Between $5 and $500k,"

I'm pretty sure it is not in the range ($5, $10].

[-]Oscar_Cunningham14y-10

You mean [$5,$10), no?

[-]wedrifid14y40

You mean [$5,$10), no?

I do actually mean ($5, $10] but I can understand if you would prefer the other variation. My reasoning is that 'between' sounds to me to be non-inclusive ("between Fred and Joe" for example) while making the trade at $10 would just be pointless. The $10 in particular could be seen as ambiguous and if there was a symbol that meant "')' or ']', whatever" then I would use that instead.

[-]RolfAndreassen14y-20

Are we assuming that the two players have perfect knowledge of each others' prices? Because if so, it seems to me that the price is a simple 500k (minus epsilon). If A has something that B values at that price, and that can't be gotten anywhere else, he will charge what the market will bear; and the market will bear 500k, because that's what the phrase "B values the access at 500k" means. If B is not in fact willing to pay that sum, on the grounds that A's reservation price was much lower, then he did not genuinely value the access at 500k.

If the two parties don't have knowledge of each others' prices, then presumably A makes some offer greater than $5 and B accepts it, or vice-versa. In this case the price is basically random. It gets higher as you increase A's knowledge of B.

[-]Psychohistorian14y00

If A has something that B values at that price, and that can't be gotten anywhere else, he will charge what the market will bear; and the market will bear 500k.

Try:

If B wants to buy something that A obtained at a certain cost, and that can't be sold anywhere else, he will pay what the market will bear; and the market will bear $6.

If A refuses to pay $500k, B gets nothing. If there were multiple buyers and A had the highest reservation cost, your answer would work and the problem would be boring.

But as the reversal shows, if B offers $6, A would take it, under similar reasoning. That's what it means to say it costs A $5. No one is going to make a higher competing offer, because no one else can even legally buy the product (and the product is a legal construct, so that means no one else can buy the product, period). It would make as much sense for B to pay $499,999, as it would for A to accept $6.

A has other sources of money

This is immaterial. A has no other use for the easement - he either sells it to B (losing $5), or it doesn't exist (0$). Conversely, B could simply not build a house on her property ($0). The fact that each has other things they can do with their life is immaterial to the transaction at issue, because that transaction has no alternatives - either A & B come to an agreement, or they both get nothing.

[-]RolfAndreassen14y00

Ok, as a point of game theory you've convinced me. As a matter of human psychology, I think A has B over a barrel, although possibly not a half-million-dollar barrel. Although A gets nothing if B refuses to buy, A is not the one who wants a specific, very valuable change in the starting situation. B is the one who wants the status quo changed in a specific way; he has, so to speak, the burden of proof.

Although both parties have an opportunity cost from not making a deal, it seems to me that the opportunity cost "I don't get to do these specific things I had planned on" will weigh more heavily in a human mind than "I don't get some amount of free money, which may be small".

[-]Psychohistorian14y10

As a matter of psychology, the two are neighbors. They probably work it out amiably, and A probably doesn't end up charging much because it doesn't cost him anything, and because B will get really, really angry if A insists on some high price. Also, practically, if B is so inclined, he can punish A by litigating the issue - it'll cost A money and is just an unpleasant experience. It'll cost B the same, but we know that real people are willing to pay money to punish those they find uncooperative.

If these were two competing businesses, or if involved business more generally, I wouldn't be surprised if A did try to take advantage of his position. But the actual fact is that humans are not homo economicus, and will generally not bend other people over a barrel in such situations. If the costs to A were higher, it'd be a very different story.

Or perhaps I have an overly optimistic view of average human behaviour.

[-]wedrifid14y-20

Are we assuming that the two players have perfect knowledge of each others' prices?

A and B know each others costs and values.

That is a yes.

If A has something that B values at that price, and that can't be gotten anywhere else, he will charge what the market will bear; and the market will bear 500k, because that's what the phrase "B values the access at 500k" means.

This is not the case. In this scenario there is no special privilege for the resource that happens to be the service over the resource that happens to be money - the 'seller' doesn't arbitrarily get to dominate.

[-]RolfAndreassen14y20

no special privilege for the resource that happens to be the service

I don't understand why this should be the case. Presumably A has other sources of money, but B has no other sources of access; unless you are specifying otherwise, there is an obvious asymmetry. If the situation is intended to be symmetric, the example is a bad one; it is cross-grained to well-established intuition about how money works.

[-]wedrifid14y-20

Let me second Psycho's question. While I personally with my current decision making algorithm would just offer $225,005 to another wedrifid like entity and reject anything worse I am not sure of the conclusions of the formal literature on the subject. I'm sure a lot of intelligent theorists have tackled the question right down to the ground.

EDIT: Assuming linear utility for $, obviously.

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