Mechanism Design for Grandma

by on October 15, 2007 at 9:12 am in Economics | Permalink

Ok, Grandma may still have some difficulty but in honor of today’s Nobelists, Hurwicz, Maskin and Myerson let’s give it a go.  Suppose that you are selling a rare painting for which you want to raise the maximum revenue.  There are two potential buyers, Tyler, who values the painting at $100,000, and Alex who values it at $20,000.  The problem would be simple if you knew this information – you would then set the price at $99,999 and Tyler would buy maximizing your revenue.  But how much Tyler and Alex value the painting is their own private information.  How then should sell the painting?

One possibility that springs quickly to mind is an auction.  In a standard English open-cry auction Alex and Tyler will bid for the painting and the bids will keep rising until Alex is forced to drop out at $20,001.  Thus the auction earns you $20,001.  Not bad but is this the maximum revenue possible?  Remember that Tyler values the painting at $100,000 so you could be leaving a lot of money on the table.

What else can you do?  Well, how about an auction with a reserve price, say $50,000 – think of a reserve price as a secret bidder who calls in his bids on the phone.  A reserve price of $50,000 works well in this case as Tyler will pay $50,001.  But note that you just got lucky, if Tyler had valued the good at $30,000 you would have earned nothing at all.  Thus you would like to know whether a reserve is always optimal and how to set it.  (Riley and Samuelson, and much more generally Myerson both show that a reserve price is always optimal and how to set it).

But why stop at a reserve price?  How about a reserve price and an entry fee?  But why stop at reserve prices and entry fees?  You can add any kind of requirement to the auction that you want but will these requirements help you to raise revenue?  Lets boil the problem down to its essence.  Think about an auction as a mechanism – bidders put information into the mechanism, their bids, and the mechanism tells them the outcome.  (Hurwicz was the first to really start thinking about mechanisms in these very general terms.)

You want to design the mechanism to achieve a certain outcome.  The mechanism can be as complicated as you want but it must satisfy certain conditions.  First, the bidders must participate voluntarily – you can’t boil them in oil – so there is a participation constraint.  At the end of the day the bidders must expect to be at least as well off as if they did not play the mechanism game (at least on average).

Second, there is an incentive compatability constraint.  You don’t know how much Alex and Tyler truly value the painting so suppose that Tyler mimics whatever Alex does – Tyler can do this since he values the painting at least as much as Alex does.  It follows that whatever outcome the mechanism assigns to Alex, Tyler must get at least as much.  This is a significant constraint because it means that if you want Tyler to do something different than Alex, and you do, you want Tyler to bid more, then you must give Tyler something in return.  Thus, even in the optimal mechanism you, the seller, are not going to get everything.  Tyler is going to walk away with some surplus.

We still haven’t solved for optimal mechanism, however.  And here is where the magic comes.  Not magic as in something wonderful but magic as in hand-waving.  Maskin and Myerson proved something very useful about mechanisms with these types of constraints.  It turns out that if you follow the constraints then you can restrict attention to mechanisms in which Tyler and Alex always tell the truth about their values, this is called the revelation principle.  (In a sense, this is obvious for imagine that we find the optimal mechanism given that Tyler and Alex submit whatever bids/information they want.  Then you tell Tyler and Alex – next time why don’t you tell the truth about your values and we promise to give you exactly the outcome that we would have given you under the previous mechanism.)

In the case of auctions the direct mechanism is well known, a second price auction.  In a second price auction the high bidder wins but pays the second highest-bid.  In this auction it makes sense for every bidder to bid his true value – see if you can work out why – and it turns out that as the revelation principle says, revenues in this direct auction are the same as in say a regular English auction (under certain conditions, of course).

Ok, I have gone on for a while.  Here’s the bottom line.  The basic set-up of agents with private information submitting "bids" which are then fed into a mechanism resulting in outcomes is very general.  How to raise taxes, regulate a monopolist, fund a public good (here’s my own contribution to mechanism design), allocate organs, assign interns to hospitals, split common costs, allocate electricity across a grid – all can be thought of as mechanism design problems.   The tools that Hurwicz, Maskin and Myerson developed and their methods of paying attention to participation and incentive compatability constraints and using the revelation principle helps us to design, at least in principle, the best solutions to all of these problems.

M. Hodak October 15, 2007 at 9:44 am

It’s been a mystery to me why my paper on cost allocations, using a unique mechanism, was my most downloaded paper.

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=728524

Maybe it’s a strange Nobel Prize prediction mechanism…

M. Hodak October 15, 2007 at 10:22 am

Sol,

…and that it gets all the players to reveal their true reservation prices–a major benefit in multiple transaction games.

josh October 15, 2007 at 11:25 am

jb,

If its just a sealed bid auction, Tyler might try to bid just a bit more than he thinks Alex will bid. Bidding his full value would assure him that he gets no net benefit from acquiring the painting.

jake October 15, 2007 at 12:37 pm

I must be missing something about the second price auction. If each bidder bids his top bid, Alex bids $20k, Tyler bids $100k, and Tyler gets the painting for $20k. So this is even worse than the open cry auction outcome.

What if the winner pays the average of the top and second place bids? then grandma can get $60k.

vorkosigan1 October 15, 2007 at 1:08 pm

Ah, but there is a great deal of real-world information on how reserve prices work in iterated and non-iterated transactions, on eBay. And consistently, auctions that start with non-reserve prices bring in higher bids than auctions with non-reserve prices.

Now, perhaps the eBay sellers don’t know “how to set it.” But such a large body of real-world data seems to undermine the foundations of the thesis set out above.

Phill October 15, 2007 at 1:24 pm

Why wouldn’t I want to start the auction at let say $500k and then continue to reduce the price and have the fist bidder who bid for the price “win”.

It would seem then I would receive the highest value for the item. This solution seems easy.

josh October 15, 2007 at 1:37 pm

“In the second bid auction, suppose my true value is $20,000. Okay, I bid $20,000. The next bidder, whose true value is $25,000, immediately bids $1 million. Doesn’t he walk away with the the painting for $20,000, having priced out any third bidder?”

If somebody bids $25,000, the guy who bid $1,000,000 has to pay more than his value. If you think about it, you can’t do better than to bid your true value.

By the way, you don’t know what other people bid.

josh October 15, 2007 at 1:43 pm

“Why wouldn’t I want to start the auction at let say $500k and then continue to reduce the price and have the fist bidder who bid for the price “win”.

It would seem then I would receive the highest value for the item. This solution seems easy.”

This depends on how close the first and second bidder are in their valuations, as well as what they estimate for the others valuation (as well as what they estimate the other estimates for their valuation). People want SURPLUS value, they may let the price drop at the risk of losing the item. In a second price auction you can’t improve your situation by giving a false valuation. The seller is then theoretically guaranteed the second highest valuation which may or may not be more than they would receive from the Dutch auction.

Nikkei October 15, 2007 at 1:45 pm

For Phill at 1:24:33PM: How do you know that the 500k are greater than the maximum value between the two bidders?

zzzzzzzz October 15, 2007 at 1:46 pm

The revelation principle aside, isn’t Alex in this post just summarizing the contributions of *Gordon Tullock*?

Ralph October 15, 2007 at 2:23 pm

Are you sure we can’t boil them in oil?

That’s the classic Neocon solution, of course, and the great thing about it is that no study or learning is required. All you need is a sturdy kettle and some kind of cheap oil, to be burned for the necessary heat and also inside the kettle.

Of course we wouldn’t *really* boil the bidders. Ha ha, silly. Just put their feet in, while they reconsider their bids. Then off to Gitmo lest they reveal our “interrogation” techniques.

Considering the trillion dollars they’ve stolen without any fear of prosecution, I’d say the Neos know a lot more about economics than they get credit for.

AMW October 15, 2007 at 3:03 pm

“I don’t think Tyler would bid at $100K in a Dutch auction – it would be worth the risk to hold off for a while. How long? Ask someone who knows, which ain’t me.”

If Tyler was risk neutral, knew that he had only one competitor, that his value and Alex’s were both drawn from a uniform distribution, he would wait until the clock ticked down to $50,000. His optimal bid (in Dutch clock and sealed bid) is V[(n-1)/n], where V = his value ($100,000) and n = the number of participants in the auction.

Also, English Clock and 2nd price sealed bid auctions only incentivize truthful revelation of values when there is only one unit of a good being auctioned, or when each participant wants a maximum of one unit. In multi-unit demand auctions, truth-revealing prices are trickier.

Phill October 15, 2007 at 3:16 pm

“If Tyler was risk neutral, knew that he had only one competitor, that his value and Alex’s were both drawn from a uniform distribution, he would wait until the clock ticked down to $50,000.” -AMW

Which is still considerably more than 20k.

I understand this, and I know it’s the proper way of figuring it out but it always bothers me to say when we don’t know the odds of something to assume thats it’s equally distributed.

I mean it lets you do the math but it really is kind of just making a number up. To me I think of the distribution more as undefined but that doesn’t let you do any math. I guess no knowledge can be considered a special case of partial knowledge. But in a system like auctioning you know that it is not going to be an evenly distributed system even though you might not know what that distribution is like.

Sorry, I play too much poker.

Noah Yetter October 15, 2007 at 3:55 pm

Almost everyone is missing the fact that our personal reserve prices are not known to us. Alex and Tyler may value the painting at 20 and 100 respectively, but they don’t know that when the auction begins. A great measure of the value of an auction is that it helps us discover our reserve prices.

It’s easy to say the optimal strategy is to bid your reserve price. The propensity of ebay bidders to bid multiple times, despite the presence of a robust proxy-bidding system, has caused a great deal of head-scratching, but it is easy to understand once you acknowledge that reserve prices are unknown. Each time we are outbid we ask ourselves if we’re willing to go higher. You could theoretically do that up front by playing the bidding out mentally and figuring out where you’d stop, but it is difficult to draw the line at an exact figure above which you would not go even a dollar without actually being in the moment.

Barkley Rosser October 15, 2007 at 5:35 pm

Umm, the virtue of second-price auctions, which I believe are used by the US Treasury,
was already honored in a previous Nobel by its discoverer, William Vickrey, for whom such
auctions are named. I also find it mildly amusing that Alex did not cite any of these three
winners in the paper that he linked to by himself. Oh well.

Alex Tabarrok October 15, 2007 at 6:02 pm

Let me clarify a few things. The optimal revenue maximizing auction in this context is any of the standard auctions, English, Dutch, First-Price, or Second-Price each of them with a reserve fee. A number of assumptions are required most importantly that as Slocum points out values are independent and not contingent on how others value the good.

The point about the second-price auction is not that it has virtues compared to the others but that the equivalence between the auction forms illustrates Myerson’s revelation principle – for any auction no matter how complicated you can always write this as a direct (“truth-telling”) mechanism. Writing things in this way typically makes solving for the optimum much easier.

As to Barkley’s point, mechanism design theory shows that there is a deep relationship between auctions (Vickrey), optimal tax theory (Mirrles), principal-agent problems (Myerson) and social choice theory (Gibbard-Sattherwaite) so illustrations in one field often cross over into the other fields. Next time I will be sure to cite Rosser. :)

Daniel October 15, 2007 at 6:11 pm

Sorry to have posted twice – the server timed out and I refreshed without realizing it had caught the post.

AZ October 15, 2007 at 10:00 pm

@Phill

In a typical auction problem the bidder has a value, v, for the item drawn from some distribution. If we assume a RN-SIPV environment (risk neutral symmetric independent private values – meaning bidders are risk neutral and all values are independently drawn from the same known distribution and are only known to the respective individuals who draw them) and have as the goal of the bidder to maximize expected consumer surplus then a bidder in the Dutch auction is attempting to maximize the following:

Prob(I win given the bid I choose)*(surplus I receive from the bid I choose) or

Pr(win|bid)*(v-bid)

It should be clear that the optimal bid does not equal one’s value, which you stated above. Moreover, one should be able to “correctly” infer the probability of winning with a particular bid b* if one (1) knows the distribution of values and (2) believes that all other bidders will follow the same strategy as you do (bidders are all the same and their values are all drawn from the same distribution so why shouldn’t they).

You are also focusing too much on the particular example. If Tyler’s value for the painting is $100,000 and Alex’s is $90,000 then a 2nd price auction does better (for this particular example) than a 1st-price or Dutch auction because the 2nd price auction generates $90,000 of revenue. The key is that in a RN-SIPV environment none of the auctions (1st-price sealed bid, 2nd-price sealed bid, Dutch, English clock) does better than any of the other auctions in terms of expected revenue (Revenue Equivalence Theorem). With specific numbers then yes, we’ve just shown a case where the Dutch (and 1st-price sealed) does better and a case where the English clock (and 2nd-price sealed) does better. If buyers are risk averse, then the Dutch should generate more revenue than the 2nd-price because the bidder’s strategy does not change in the 2nd-price when he is risk averse but the bidder will increase his bid in the Dutch auction if he is risk averse (relative to when he is risk neutral).

Also, the distribution does not have to be uniform – it’s just a lot easier to work out the analytical solution for a uniform distribution. Order statistics are much easier for that than for the normal distribution.

AZ October 15, 2007 at 10:06 pm

@ Daniel

If you don’t know the number of bidders in the auction you are screwed. Seriously – most results rely on a known number of bidders. Sorry.

Also, a “bad” auction design which is easy to understand and is likely to draw in a lot of bidders is probably better than one that is complicated and draws few bidders but is “good”. I recall people berating Ebay for its poor auction design – however, it is fairly simple and I think I read that last year or a few years ago Ebay did $1 billion of business – in its sports-related items categories. So “bad” auction design be damned if it brings in a bunch of bidders …

Stuart Cooper October 15, 2007 at 10:21 pm

Sorry, but I don’t understand finance at all.

G October 16, 2007 at 3:27 am

I don’t see how regulation is relevant. If markets (such as ebay or LendingTree) want to adopt alternative forms of auctioneering, the market (of markets!) will select the one that pleases buyers and sellers the most. I can’t understand how regulation would at all be justified by this work, although I suppose logic and reason has rarely stood in the way of regulation in the past.

Ina Irro October 16, 2007 at 8:35 am

Okay, I did minor and limited work in this field – particularly to do with enginering resources managment.

I always wonder if we can alter pay-off matrix in a n-PD game based using mechanism design ….and how can we ?

I am not just talking about trial and error which is exactly what centralized governance does.

Can we eveolve the rules? why just one iteration? or rather do they evolve?

ALSO Doesn’t this look like it refutes publich choice theory. Cause it employs some altruism by being fair a for the collective good, (rather than norm-maximizing as in taking 100,000 from Tyler)?

what say you ?

I am very curuous, lol:)

Anonymous October 16, 2007 at 10:41 am

NICE!!!

goody October 16, 2007 at 1:14 pm

My grandma will not have any problem b’cose she is an expert mathematician of the older vintage.Therefore don’t generalise across grandmas.

goody October 16, 2007 at 1:15 pm

My grandma will not have any problem b’cose she is an expert mathematician of the older vintage.Therefore don’t generalise across grandmas.

tommy October 16, 2007 at 2:28 pm

Interesting. Second price auctions: bid twice, pay twice.

Ed October 16, 2007 at 5:34 pm

I don’t want to seem too Philistine about this, but mechanism theory, like so much of overly mathematicized economic theory, is in pursuit of an elusive deterministic explanation — within a certain set of parameters, of course.

Most of this sort of reasoning is not all that distinct from: “If my aunt had balls she’d be my uncle, and, uh, guess what?”

What any poker player — or Rock, Paper, Scissors player, for that matter — can tell you is that, while playing for the most part according to so-called “rational expectations,” you should “plan” to break the mold from time to time, and thereby come out ahead in the short run, cash in your chips, and leave the economists at the table assuring themselves that they will make it up “in the long run” — if they happen to be around then.

I mean, really, they give Nobel prizes for this sort of thing?

CLT October 16, 2007 at 10:08 pm

Real life version of the problem:

A lawyer is about to take on a new client for what he hopes will be a continuing engagement. He has no idea what hourly rate the prospect paid its last lawyer or whether the rate was considered too high or low. He also has no idea how much the client has in the budget for legal fees. He can make only one proposal, but the fee, if accepted, will probably set the standard for future fees for the life of the relationship. What can he do to avoid leaving money on the table?

Anonymous Coward October 16, 2007 at 10:32 pm

If he bids 101,000, he may have to buy the painting for more than he values it. He would have been better off bidding 100,000.

If he bid 101,000 and nobody bids between his bid and his true value, he is no better off than if he had bid 100,000 as he will pay the price he would have paid anyway.

If he bids 99,000, somebody may bid between his bid and his true value. He will have lost the opportunity to buy the painting for less than his value earning a nice surplus. He would have been better off bidding 100,000.

If he bids 99,000 and nobody bids between his bid and his true value, he is no better off than if he had bid 100,000. He will pay the price he would have paid anyway.

I think I’m missing the point here, isn’t the whole purpose of the second auction to reveal the ‘reserve’ price of the bidders and the winner of the first auction is still the winner regardless of the results of the second auction?

I’m also wondering what this ‘true value’ is that is separate from the winning bid. Isn’t the whole point of an auction to determine the ‘true value’ of an object?

I think that’s where all this goes wrong, there is no inherent ‘true value’ in an object except what a buyer and seller agree to exchange it for. If the buyer’s price is less than the sellers price then the seller either overvalues the object or the buyer doesn’t value it enough.

I do of course realize the value that the central planners place on the ability to determine the ‘true value’ of something…

Hafiz October 17, 2007 at 4:58 am

I know that this may be missing the point that silent bid with the bid kept secret between the bidder and the auctioneer may help solve the auction. Since each bidder doesn’t know each other bid, and given that each bidder really want it, each bidder needs to bid as high as possible, as far as each bidder’s preference allows.

In this setup, the $20,001 scenario may not happen, unless there is a leak of information.

barry payne - economist October 17, 2007 at 10:49 am

EXAMPLE OF CONFLICT AND GAME THEORY

The paragraph below is by Michael Kinsley, posing an example of Schelling’s “madman” theory. The full article is at http://hnn.us/articles/17183.html. To Ina Irro: I’m not sure what defect-defect means. In the context of that below, it seems to imply that neither person gives up so neither is released, i.e. the cold war just continued to raise the ante on both sides to avoid a hot war.

MICHAEL KINSLEY
“So you’re standing at the edge of a cliff, chained by the ankle to someone else. You’ll be released, and one of you will get a large prize, as soon as the other gives in. How do you persuade the other guy to give in, when the only method at your disposal—threatening to push him off the cliff—would doom you both? . . . Answer: You start dancing, closer and closer to the edge. That way, you don’t have to convince him that you would do something totally irrational: plunge him and yourself off the cliff. You just have to convince him that you are prepared to take a higher risk than he is of accidentally falling off the cliff. If you can do that, you win. You have done it by using probability to divide a seemingly indivisible threat. And a smaller threat can be more effective than a bigger one. A threat to drag both of you off the cliff is not credible. A threat to take a 60 percent chance of that same thing might be credible. . . . Madness can be wickedly rational. If one of those two folks on the cliff can convince the other that he is just a bit nuts, that makes his threat to drag them both off the cliff much more plausible. Some defenders of Richard Nixon used to claim that the evidence of insanity that bothered a few Americans was actually a purposeful strategy to enhance the deterrent power of our nuclear arsenal.3″

TRUE VALUE AUCTION BIDS
Separately, to understand how an anonymous second-price auction causes bidders to bid their true value, collapse it to one item auctioned in a single-round blind simultaneous submission of bids. Everyone knows they will pay the second highest price bidded if they win by bidding the highest price.

Some will ask, why not bid ridiculously high prices to win? But rational actors will realize if everyone does this, the winner will end up paying a ridiculously high second-highest bid. To protect themselves, each bidder bids the true value to insure that if it is the winning bid, no matter how close the second-highest bid is to that bid, it does not exceed winning bidder’s true value.

In reverse, if less than true value is bid, it could fall below the next highest bid and cause regret for not bidding true value in the first place. The result is like the parable of one person cutting the pie for a second one to choose his or her piece – the first person will “cut truely equal pieces” to insure a maximum size piece for his or her self.

To Berkley Rosser: It seems a descending Dutch auction with many bidders may represent an outcome close to the perfect competition model, or perhaps what Jeffrey Banks did. The general idea is that the price paid by all bidders for all units would decline until the last unit’s price would just equal the marginal unit cost of the last producer – at that point, the last consumer and producer get no surplus, but everyone before them does.

Jeff Nardine October 17, 2007 at 10:32 pm

Isn’t the second auction like playing the card game Papoose or Indian (wish I knew nicer names for the games) where one slaps a card blindly to their forehead and then bets on everyone else’s cards? If the game is Indian and somebody is holding an ace on their forehead you bet away from him using a bluff to get him to fold early.

Anonymous October 21, 2007 at 4:13 am

In a PD game with a pay off matrix that rewards R to both playersif they both coop and P in case both defect. ( T>R>P>S )

if both sides defect (defect-defect) you would have them both rewarded P,P ====> Equilibruim.

Mad-man theory is simply suggests that the other side in a stochastic game such as PD doesn’t truly know what is rational for myside. A siple hypothesis to act irrationally thereby tricking the otherside.

here
https://irro.blog.asu.edu/

POLICY IMPLICATIONS OF GAME THEORY

I don’t believe that the understanding of uncertain retaliation in the context of Mad Man Theory can result pre-emptive wars. Mad Man Theory in fact was the reason for the Nash-Equilibria that we call Cold-War. “irrational policy choices that lead to collective desired out-comes of stability†.

I am not sure if it (mad-man reasoning) could be used in today’s world however, I also don’t think the correct interpretation of Mad-Man Theory would lead to a preemptive wars. But it could solicit undue misinterpretation of the whole theory and hence show of strength through war. (see the links below )

There is also another suspect; †the principle of Vorsorgeprinzip† ! one can’t help but suspect that it is this principle (or perhaps misuse of Vorsorgeprinzip) that lead to today’s’ wars , at least in case of Iraq.

Here are some fine article on game theory and its affect on US policy. It is clear that even mad-man theory is intended to lead to stability and prevent wars.

(1) Is Bush Using Schelling’s Mad Man Theory

(2) WSJ on Schelling

(3) Jeffrey Kimball on Schelling’s Mad Man Theory (*)

鑽石 April 2, 2008 at 10:43 pm
online pharmacy May 11, 2008 at 10:18 am

Generic Effexor drug is a potent inhibitor of the reuptake of serotonin and norepinephrine-two neurotransmitters, thought to play important roles in the pathophysiology of depression. Correcting the imbalance of these two chemicals may help relieve symptoms of depression. Order discount generic Effexor pharmacy online.

Generic Celexa drug (Citalopram) is used to treat depression. This medication works by helping to restore the balance of certain natural chemicals in the brain. Order discount generic Celexa pharmacy online.

KicksOnFire.com October 5, 2008 at 7:48 pm

It strikes me that a big problem with this line of analysis is that Tyler’s valuation may very often be contingent and variable (depending on other valuations) rather than fixed and independent. That is, my willingness to pay $100,000 for a painting may depend on whether there is a group art-experts willing to bid almost as much. This strikes me as particularly true with paintings where the both monetary and personal values are socially constructed (part — perhaps most — of the satisfaction of owning a rare, famous painting derives from the fact that is rare and famous and others would dearly love to be in one’s shoes. But if it was actually the case that nobody else wanted the damn thing, then…

KicksOnFire.com – Air Force Ones and news and more!

搬家公司 October 9, 2008 at 11:59 am

搬家費用最平價,搬家服務最專業的搬家公司。

ielts October 9, 2008 at 12:02 pm

計劃到英國唸大學或研究所,就必須通過ielts國際英語能力測驗

徵信 October 9, 2008 at 12:04 pm

辦理婚姻訴訟、外遇徵信及工商蒐證等服務的徵信社

徵信社 October 9, 2008 at 12:05 pm

推薦優良徵信社,以為解決現今社會所衍生個人及企業的危機,由專業人員提供專業的徵信服務。

日文 October 9, 2008 at 12:06 pm

找對日文補習班,日文檢定(JLPT)測驗成績就能提升

汽車 October 9, 2008 at 12:07 pm

有關新車、買車、二手車及改裝車的資訊,都在車訊網。

室內設計 October 9, 2008 at 12:09 pm

有室內裝潢需求時,可上網多參考室內設計作品,以找出自己喜歡的風格

托福 October 9, 2008 at 12:10 pm

托福考試為全球施行最廣的英語能力檢定toefl測驗

搬家公司 October 9, 2008 at 12:11 pm

大台北為一全方位專業搬家公司,以高效率,高熱忱,專業搬家技術,整齊.細心.確實的服務精神。

遊學 October 9, 2008 at 12:12 pm

出國遊學選哈佛,哈佛是許多語言學校的駐台代表,語言學校有付哈佛辦事費,所以免代辦費。

搬家公司 October 9, 2008 at 12:15 pm

最具規模之專業搬家公司,透過GPS管理搬家車隊,保證不會中途加價。

多益 October 9, 2008 at 12:16 pm

公民營機構、外商公司及跨國企業大多採用多益測驗作為評估訓練結果。所以toeic成為就業的必備。

多益 October 9, 2008 at 12:22 pm

新制多益測驗的出題方向,乃以「More Authentic(更實際)」的理念為基礎,參加toeic考試要注意

虛擬主機 October 9, 2008 at 12:27 pm

最專業的電腦維修及虛擬主機公司,讓網站穩如泰山,不再連不上線。

Comments on this entry are closed.

Previous post:

Next post: