How is silver a better conductor than platinum? Based on my understanding, platinum should be a better conductor than silver.
Here's my reasoning and assumptions. An atom's ability to conduct is based on the amount of energy required to displace electrons from it. The energy required is based (largely) on the number of electrons in the valence shell as well as the atomic radii (distance from valance shell to nucleus). There is a direct correlation between the magnetic attraction of the valence electrons to the protons and the radius of the atom. The further away the valence electrons, the lesser the energy required to displace the electron. 
Let's take a look at silver:

You can see that silver has 5 electron shells, with 46 (18+18+8+2) electrons shielding the valence electron. It has one electron in the valence, so it is a good conductor.
Now, let's look at platinum:

Platinum, also, has a single valence electron. However, there are 6 electron shells -- one more than silver. There are 77 (17+32+18+8+2) electrons shielding the valence electron. Platinum has a larger radius than silver and, similarly, has a single valence electron.
I cannot find any explanations for the ranking of conductivity, but, in general, the interwebs provides an order of conductivity such as: silver, gold, copper, platinum. If platinum were, indeed, less conductive than gold and copper, why in the world would there be a market for platinum audio cabling -- assuming that platinum is more expensive? 
Edit:
Thank you for all of the replies, but none offer a concrete reason as to why silver is a better conductor. I am intrigued by the 'crystal structure' argument, but no one has provided the actual structures for platinum and/or silver and why the structure makes a difference. My search results show that platinum, silver, and gold all have the same structure -- 'face centered cubic'. 
 A: The situation is a lot more complicated than you described. When calculating the conductance of a metal, you first need to evaluate the allowed energy states of electrons in the bulk material, which is done by placing nuclei in a regular pattern (depending on the crystal structure). Usually, various approximations are used, which may or may not include the interaction between the electrons. Then you count the number of states at a given energy and get the so-called density of states. Then you start putting the electrons into these allowed energy states, until all electrons belonging to that element are added. And only then will you know if something is a conductor at all. For insulators, there is an energy gap between the highest occupied and the lowest unoccupied state, i.e. there are no states just above the energy of the highest filled state (the so-called Fermi energy). For metals, there is not.
I'm aware that this might be hard to follow if you are not familiar with these terms, but it's just to show that from a single atom sketch you can not conclude the electronic properties of the bulk material.
As to why there is a market for platinum cables - there is a market for anything people will buy, whether it makes sense or not.
A: It appears that none of the responses so far have addressed the real reason silver is a better conductor than platinum.
The reason is electron scattering. The simplest model for conductivity in solid state physics is called the Drude model. This is a microscopic description of what happens to the electrons when you apply an electric field to get them moving. The Drude model explains Ohm's law of resistance, among other things, and it works surprisingly well for all simple conductors (and some not-so-simple ones).
The basic assumption of the Drude model is that electrons scatter off of things as they move around in the material. They basically scatter off the crystal lattice ions and can generate lattice vibrations (heat). Each material has a characteristic scattering time determined by the details of the crystalline structure, the temperature, and other things. In metals, electrons scatter around once every $10^{-14}$ seconds (!).
Ultimately, electrons in platinum scatter more rapidly than those in silver (by a factor of two or so). According to the Drude model, this means platinum has a lower conductivity.
As mentioned above, platinum does have a big practical advantage, though: it is a noble metal and does not oxidize (unlike silver). But gold is also noble and is a better conductor than platinum. The only reason I can think of why you might not prefer gold for audio cables is that it is soft. But the platinum cables are probably a gimmick.
A: I'll add my two cents to the other answers from the audio perspective that hopefully would clarify the "platinum cable" issue.
First of all, there is a belief in the audio field that an ideal conductor, aside from being highly conductive, stable, and reasonably chemically inert, would also have one s-electron on the top orbit with all lower orbits full. There are only 3 metals to my knowledge fulfilling this criteria: copper, silver, and gold with all 3 positioned right one under the other in the periodic table. 
Secondly, platinum plated connectors are extremely rare. I only know of a couple top brands in the world that use platinum for plating. Platinum is usually applied only on top of silver, whether silver is the base metal or the base metal is first plated with a silver underlayer.
There is a widespread misconception that silver oxidizes. This is incorrect. Solver does not oxidize at room temperatures. There also is an equally big misconception that silver oxide is a good conductor, thus tarnished silver connectors do not make a big difference. It is true that silver oxide is conductive, but the tarnish that easily develops on silver plated connectors is not silver oxide, because silver does not oxidize. Instead it is silver sulfide, brown to very dark brown (as opposed to pure black for silver oxide). Silver sulfide is not a conductor, but a semiconductor. Thus the idea that the silver tarnish does not affect the connection is a bad myth.
Silver aggressively reacts with hydrogen sulfide, a gas that naturally occurs in the natural gas and also comes from cooking eggs (even fresh) and from the car exhaust. If you have none of these three sources in your area (unlikely), then your silver plated connectors would stay shiny for much much longer. Platinum does not tarnish or oxidize at any temperature (e.g. platinum plated spark plugs), so it can protect silver from sulfur in the air.
It is a huge and wide spread misconception that the choice of metal for plating has anything to do with its conductivity. The typical plating thickness is between $0.3\mu m$ and $11\mu m$, give or take, plus has a substantial area of connection. Thus the contribution to the resistance of a thin cable several feet long is absolutely negligible.
The choice of metal for plating is based on the anti-tarnish properties, wear resistance, reasonable conductivity (usually at least 15% of copper), adhesion to the base metal, and marketing of noble metals. 
Typical gold plating is done over a layer of nickel and thus essentially is just a nickel plating with a bit of extra protection from oxidation. Gold over copper is rarely used, because gold dissolves in copper. This makes a thin plating (less than $1\mu m$) pointless while a thick gold plating is expensive. Test results show that a gold plating of less than $3\mu m$ does not protect well from harsh environments. Gold also is not very durable for frequent connections, especially as a thin plating.
One of the most popular metals for high end plating today is rhodium. It can be applied directly over copper or silver, has a reasonable conductivity (like platinum), and does not wear (also like platinum).
