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Because it costs 50G to play, but you get 50G for each round you win, losing the fifth battle nets you 150G, and you can try again. Winning the 5th match caps you at 200G. Of course, at 50G per round, it's hardly worth the effort.
Not to nitpick, but it only costs you 50G when you loose, not to play. So if you win 4 in a row and loose the 5th, you end up gaining 150G (200 for the wins, -50 for the loss). Winning 5 in a row gets you 250G (since you didn't loose any).
Although the better option than loosing the 5th round would be to leave the house after getting 4 in a row since that will reset the number of wins. So you could theoretically win 50 games in a row and still not have the badge and have gotten 2500G for it. (I only say theoretically because the odds of winning 50 times in a row are statistically negligible.)
Maybe I'm just lucky but I beat it after only a couple tries. Didn't even have to reload. I just kept switching it up. I think I used paper the least, but I don't know that I went with any particular strategy.