Puzzle time - modified dots and boxes

[jon-nyc]

The game of Dots and Boxes begins with a rectangular grid of dots. Players alternate drawing edges that connect neighboring dots, vertically or horizontally. When a player completes a box, he or she labels it with his or her initial and moves again; when all the edges are drawn, the player with the most boxes wins.

Suppose we change the rules to allow a player who makes a box either to move again, or decline to do so. Show that the second player has a winning strategy if this game is played on a 4x6 grid (35 dots).

[Klaus]

Before I consider the changed rules, I want to understand the original game.

Is there a winning strategy here? If so, what would it be?

Also, can there be more than one edge between two dots?