Pea galaxy: Difference between revisions

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In 1st paragraph of "Description" star formation rate of peas said to be 13 million solar masses a year. Corrected it to 13 solar masses (explained in talk section).
en>Richard Nowell
Wrote about the HST observations at the end of 2013.
Line 1: Line 1:
{{no footnotes|date=January 2014}}
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{{too technical|date=January 2014}}
 
In [[game theory]], the '''price of stability (PoS)''' of a game is the ratio between the best objective function value of one of its equilibria and that of an optimal outcome. The PoS is relevant for games in which there is some objective authority that can influence the players a bit, and maybe help them converge to a good [[Nash equilibrium]]. When measuring how efficient a Nash equilibrium is in a specific game we often time also talk about the [[price of anarchy]] (PoA).
 
==Examples==
Another way of expressing PoS is:
 
: <math> \text{PoS} = \frac {\text{value of best Nash equilibrium}} {\text{value of optimal solution}},\  \text{PoS} \geq 0.</math>
 
In the following [[prisoner’s dilemma]] game, since there is a single equilibrium (B,&nbsp;R) we have PoS&nbsp;=&nbsp;PoA&nbsp;=&nbsp;1/2.
{| class="wikitable" style="text-align:center; width:100px; height:100px" border="1"
|+Prisoner's Dilemma
|-
!  !! Left !! Right
|-
! Top
| (2,2) || (0,3)
|-
! Bottom
| (3,0) || (1,1)
|}
On this example which is a version of the battle of sexes game, there are two equilibrium points, (T,&nbsp;L) and (B,&nbsp;R), with values 3 and 15, respectively. The optimal value is 15. Thus, PoS&nbsp;=&nbsp;1 while PoA&nbsp;=&nbsp;1/5.
{| class="wikitable" style="text-align:center; width:100px; height:100px" border="1"
|+
|-
! !! Left !! Right
|-
! Top
| (2,1) || (0,0)
|-
! Bottom
| (0,0) || (5,10)
|}
 
==Background and milestones==
The price of stability was first studied by A. Schulzan and N. Moses and was so-called in the studies of E. Anshelevich.  They showed that a pure strategy [[Nash equilibrium]] always exists and the price of stability of this game is at most the nth [[harmonic number]] in directed graphs. For undirected graphs Anshelevich and others presented a tight bound on the price of stability of 4/3 for a single source and two players case. Jian Li has proved that for undirected graphs with a distinguished destination to which all players must connect the price of stability of the Shapely network design game is <math>O(\log n/\log\log n)</math> where <math>n</math> is the number of players. On the other hand, the [[price of anarchy]] is about <math>n</math> in this game.
 
==Network design games==
===Setup===
Network design games have a very natural motivation for the Price of Stability.
In these games, the Price of Anarchy can be much worse than the Price of Stability.
 
Consider the following game.
* <math>n</math> players;
* Each player <math>i</math> aims to connect <math>s_i</math> to <math>t_i</math> on a directed graph <math>G = (V, E)</math>;
* The strategies <math>P_i</math> for a player are all paths from <math>s_i</math> to <math>t_i</math> in <math>G</math>;
* Each edge has a cost <math>c_i</math>;
* 'Fair cost allocation': When <math>n_e</math> players choose edge <math>e</math>, the cost <math>\textstyle d_e(n_e) = \frac{c_e}{n_e}</math> is split equally among them;
* The player cost is <math>\textstyle C_i(S) = \sum_{e \in P_i} \frac{c_e}{n_e}</math>
* The social cost is the sum of the player costs: <math>\textstyle SC(S) = \sum_i C_i(S) = \sum_{e \in S} n_e \frac{c_e}{n_e} = \sum_{e \in S} c_e
</math>.
 
[[File:Network-design-poa.svg|thumb|right|A network design game with <math>\Omega(n)</math> Price of Anarchy]]
 
===Price of anarchy===
The price of anarchy can be <math>\Omega(n)</math>. Consider the following network design game.
 
[[File:Network-design-pos.svg|thumb|right|Pathological Price of Stability game]]
 
Consider two different equilibria in this game. If everyone shares the <math>1 + \varepsilon</math> edge, the social cost is <math>1 + \varepsilon</math>. This equilibrium is indeed optimal. Note, however, that everyone sharing the <math>n</math> edge is a Nash equilibrium as well. Each agent has cost <math>1</math> at equilibrium, and
switching to the other edge raises his cost to <math>1+\varepsilon</math>.
 
===Lower bound on price of stability===
Here is a pathological game in the same spirit for the Price of Stability, instead.
Consider <math>n</math> players, each originating from <math>s_i</math> and trying to connect
to <math>t</math>. The cost of unlabeled edges is taken to be 0.
 
The optimal strategy is for everyone to share the <math>1+\varepsilon</math> edge, yielding
total social cost <math>1+ \varepsilon</math>. However, there is a unique Nash for this game.
Note that when at the optimum, each player is paying <math>\textstyle \frac{1 + \varepsilon}{n}</math>, and player 1 can decrease his cost by switching to the <math>\textstyle \frac{1}{n}</math>. edge. Once this has happened, it will be in player 2's interest to switch to the <math>\textstyle \frac{1}{n-1}</math> edge, and so on. Eventually, the agents will reach the Nash equilibrium of paying for their own edge. This allocation has social cost <math>\textstyle 1 + \frac{1}{2} + \cdots + \frac{1}{n} = H_n</math>, where <math>H_n</math> is the <math>n</math><sup>th</sup> [[harmonic number]], which is <math>\Theta(\log n)</math>. Even though it is unbounded, the price of anarchy is exponentially better than the price of anarchy in this game.
 
===Upper bound on price of stability===
Note that by design, network design games are congestion games.
Therefore, they admit a potential function <math>\textstyle \Phi = \sum_e \sum_{i=1}^{n_e} \frac{c_e}{i}</math>.
 
'''Theorem.''' Suppose there exist constants <math>A</math> and <math>B</math>
such that for every strategy <math>S</math>,
: <math> A \cdot SC(S) \leq \Phi(S) \leq B \cdot SC(S).</math>
Then the price of stability is less than <math>B/A</math>
 
''Proof.'' The global minimum <math>NE</math> of <math>\Phi</math> is a Nash
equilibrium, so
: <math> SC(NE) \leq 1/A \cdot \Phi(NE) \leq 1/A \cdot \Phi(OPT) \leq B/A \cdot SC(OPT).</math>
 
Now recall that the social cost was defined as the sum of costs over edges, so
: <math> \Phi(S) = \sum_{e \in S} \sum_{i=1}^{n_e} \frac{c_e}{i} =
\sum_{e \in S} c_e H_{n_e} \leq \sum_{e \in S} c_e H_n = H_n \cdot SC(S). </math>
 
We trivially have <math>A = 1</math>, and the computation above gives <math>B = H_n</math>, so we may invoke the theorem for an upper bound on the price of stability.
 
==References==
 
#''Algorithmic Game Theory'' by N. Nisan, T. Roughgarden, E. Tardos, and V. Vazirani (eds), Cambridge University Press, 2007. ISBN#0521872820
#L. Agussurja and H. C. Lau. ''The Price of Stability in Selfish Scheduling Games''. Web Intelligence and Agent Systems: An International Journal, 9:4, 2009. 
#Jian Li. ''An <math>O(\log n/\log\log n)</math> upper bound on the price of stability for undirected Shapely network design games with a common target vertex.'' Manuscript (arXiv:0812.2567v1), 2008.
 
{{DEFAULTSORT:Price of stability}}
[[Category:Game theory]]
[[Category:Fixed points (mathematics)]]
[[Category:Decision theory]]

Revision as of 15:44, 1 March 2014

Knife + Fork is a restaurant and wine bar on East 4th among 1st and 2nd, and has one of the ideal tasting menus in New York City. Higher finish chef knives will list all of these attributes for you to appear over and critique, so make positive you spend close consideration to them when you search for the ideal chef knife for you. The most prevalent components that high quality chef knives are created of are stainless and carbon steel, titanium and laminated or ceramic components. You do not want to invest income on a skilled set that is not worth it.



A leading-notch chef's knife set are going to be adequately weighted so it feels great in your hands plus the knives ought to preserve their edge as time passes. Which implies the knife had been "forged" employing a single section of steel and hammered beneath significant weight and higher temperature for the shape. This method of generating a chef's knife is somewhat more pricey, and only superior chef's knife makers will use forging all through building. Rust can ruin your knife more than night.

Some common excellent practices and safety measures for preserving your knives: A sharp knife is safer than a dull one particular due to the fact it requires less pressure in cutting, (also the knife will not slip as easily and your hand will not tire as rapidly). Come across terrific deals on eBay for steak knife case and steak knife case only. This knife functions a stainless steel blade that is riveted onto a plastic deal with. It's been several years, but that old set is worn out.

Overall, customers have been glad they purchased this Ginsu set simply because it is well balanced, sharp and with 14 various knives it presents a unique solution to any sort of slicing or dicing challenge. If you have any questions regarding exactly where and how to use Best Kitchen Knives Brand, you can get hold of us at the web-site. As is prevalent with all of Wusthof's collection, every single knife from this set is created from a unique blend of high carbon content material German made stainless steel that runs by means of the entire length of the knife delivering exceptional balance and strength.

Lengthy story quick - she placed the knife down on the counter and mentioned, "I can not reduce with this." Mind you, this is the knife I have utilised for years. But, the next day I went out and invested in a sharp knife... and a knife sharpener. I am missing a very good chef's knife and have been producing do with a dull one. I had to consult with a chef to make positive I got it correct. Right after speaking with the chef I recognize I have all the hand utensils and then some that I actually don't require. Let them know a knife is in the air.

Buying the correct knife the first time, and caring effectively so that it does not have to be replaced regularly can make up for the probable expense of purchasing a top quality knife. With a bit of information and suitable care, a excellent paring knife can be purchased and employed for years. Of course, if you are like me, and cannot follow a recipe, you are open to the whole globe of exciting possibilities. If you do not however have it, I strongly suggest a set of stainless steel cookware. Angle Knife Item # JAS010055B.

Sur La Table has 3 locations - in Northwest, Lake Oswego and Tigard (particulars here ) and they sharpen every little thing except serrated, ceramic and Cutco brand knives It really is a next-day service Monday to Thursday, and if you take in a knife more than the weekend it will normally be prepared on Monday. This Best Selling - Wusthof Grand Prix Ii two-piece Starter Knife Set Tends to SELL OUT Quite Rapid! Wusthof Grand Prix II two-Piece Starter Knife Set.

When asked for which knives they would recommend to clients, professional chefs about the world would be speedy to say the Global Professional knife set. Worth $449.95 on , $450 on , and $479.95 on , this set provides you the best value for your revenue. The set consists of an eight" slicer, an eight" chef's knife, and a specialty paring knife. A great all round knife, You will really like this knife.

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