There is also some similarity between "insert another coin to continue", which means progress on coin-op games, and "buy a powerful new weapon" on F2P games, just to take vengeance on a player who is beating you all time.
Both are cheap. With coin-op games, a credit costs around 1 dollar, and with F2P games you can find virtual goods between 50 cents to 6 dollars.
When I was working on coin-op games, I always had the idea that “if the credits were cheaper, the player would spend more money”. It was around the year 2000 when I designed a short coin-op game where each credit would only cost 20 euro cents. The game was an Old West duel game (with two screens and two guns, for two simultaneous players.)
For 20 euro cents, both players would shoot the gun at the same time, one against the other, until one of them died. The surviving player would play the next round for free, and the new player that joined the game would pay the next 20 cent credit (two players per credit).
It was a crazy idea, because movement sensors to detect the player's movement were needed and didn't exist yet. Also, the idea of a short, cheap game was new, too -- plus there was the high cost of the cabinet, so the project was abandoned. These days, it could be possible with the Kinect.
While the game was never made, it illustrates the idea that the right gameplay hook and a cheap entry price could lead people to spend more than the cost of a current generation console game -- and that's what is currently happening in all kinds of F2P games.
Both are impulsive. Imagine you are playing a racing coin-op game. You are about to cross the goal and get time extended, but you don't have much time to go through, 3... 2... 1... 0! When the game climax is at the max, a "Time Up" message appears on screen... You think, "Oh no, If I only had one more second!" The next message that appears is an invitation: “Continue Playing?” and a countdown, 9...8... 7... As fast as you can, you put your hand into your pocket to grab a coin and insert it before the countdown finishes. That's impulse.
The same happens with F2P shooters. Imagine you are fed up that you keep getting killed by the same player... so you want vengeance at any cost. We can be assured that, at the end of the battle, you will be enraged and will enter the shop to buy a more powerful weapon just to kill your rivals.
Both are satisfying to the player. After the coin insertion, which means progress, or after buying a powerful new weapon, which means revenge, the player gets a feeling of satisfaction, as he instantly gets what he wants and needs. And that's good, because playing our game makes the player feel better. That helps us fulfill our desire of keeping the player playing and, therefore, spending money.