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>Tetra quark
 
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{{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]]

Latest revision as of 06:34, 11 January 2015

Ok, that’s the great, so right here’s the unhealthy. A small downside to this knife is all of the holes in the deal with. I’ll admit, I like the way in which they look. They have a basic retro airplane rivet look to them that appeals to me. Nevertheless, these holes appeal to lint and filth, and I can see them filling up with mud in wet weather so maintaining this knife clear is going to be a problem. The autoLAWKS system, whereas maintaining your hand safe from the blade closing on it, is type of a backup to the liner lock on this knife.



First we must always discuss what you need a knife for. The answer to that's easy, isn’t it? Knives have a million and one uses. From reducing shavings off a stick to make tinder for a hearth to chopping paracord or other cordage to lash your survival shelter together, you just can’t really match the utility of a superb Best Fillet Knife For The Money knife. All kidding aside, knives are extremely helpful tools and no self-respecting Prepper could be caught with out one. A sharp instrument and the knowledge of the way to use it are one factor separating us from animals, proper?

This sharpener is barely six inches in size, two inches in width, and has a height of 0.75 inches. It's splendid for home use but it will also be used in shops to cater to the assorted edge-care necessities. You can simply remodel a uninteresting edged knife right into a razor sharp blade with the aid of coarse diamond. In comparison to different standard stones, the DMT monocrystalline diamond floor can sharpen knives rapidly. You do not have to use any water or oil. You possibly can sharpen knives even when the knife is dry.

Victorinox has made a name for itself by manufacturing a number of the most interesting Swiss knives to be found out there. The Midnite Minichamp is among the best utility knives in the market. It comes equipped with sixteen tools including a screwdriver, tweezers, toothpicks, and even a ballpoint pen! The blade is just over 2″ in length, crafted out of chrome steel. The highlight of the Midnite Minichamp is the LED light put in at its finish which capabilities as a brilliant flashlight. If you are going for max utility, you cannot get a greater option than the Victorinox Midnite Minichamp.

How a couple of nice Pocket Knife ? Bizarre? I do not assume so. Just give it some thought for a minute. It's a present that's carried around in a pocket (or a purse or pocket e-book) more often than not. So if you happen to have a look at it like that, then your considered more often than not. How cool is that? And there are many totally different kinds of pocket knives for a lot of completely different folks. Let's begin with Case Knives and Buck Knives They actually make some cool (and high quality) pocket knives that can allow you to match that onerous to search out particular person.

The 1 st one is the CKRT M1614DSFG made by the Columbia River Knife and Tool Firm. I’ve been using CKRT knives for fairly a while now and they actually do produce some top quality knives. The CKRT M1614DSFG isn't any exception to the rule and offers probably the most strong grips I’ve ever seen in a tactical folding knife. It has a real hilt upon full extension and the knife itself has a particularly sharp edge. One other thing about CKRT knives is that they all include a patented auto locking system that gives you with security in opposition to the blade closing upon your fingers.

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