You have to win all 3 cat fight matches in a tournament 10 times. You can't cash out early after 1 or 2 wins. This is a tough trophy that will take some time.
You unlock the cat fight during chapter 6. You will run into a character who teaches you to throw money away to avoid a fight. He will then tell you about a special place to win money. This is the cat fights.
When you enter the cat fights, you have to pick a fighter to bet on and an amount to bet. Then you will play a game of Rock, Paper, Scissors and do some button mashing at times to try and help win the fight.
When picking your fighter, read the ticker at the bottom. It will give you a 1 line summary of each fighter. Read each line and look for the most positive sounding one. The ones with better fortune have much more chance to win. To be honest, I ignored odds of winning and strength of attacks and just picked judging on the ticker.
Your bet affects the outcome. I don't know how though. When I went for these wins, I started with 1,000,000 Yen bets and lost a lot. A lot of that was without my main plan though. I did win once with the million bet. The rest of the time I only bet 10,000 Yen which is the minimum allowed. Note that you have to win 100,000,000 Yen in Cat Fights if you are going for the 100%. I got an offer to predict the outcome of 8 fights after a few wins. I predicted each win and got a huge payout. This immediately covered the 100,000,000 I needed so I no longer had to bet big anyway.
Once you decide who to use, the next step is noting down your opponents strengths. The more stars in an attack option, the more chance of it being used. So you want to pick a counter move based on the most likely 2 attacks. If Rock is strongest, then Scissors, then Paper. You should mainly / only pick Rock. It will beat Scissors and have a button mash chance against Rock. I will list out each fighters stats and what I would use:
Francisca: Use Scissors
Rock 3
Scissors 3
Paper 4
Jennifer: Use Rock
Rock 5
Scissors 5
Paper 4
Maria: Use Scissors
Rock 1
Scissors 2
Paper 4
Momoko: Use Scissors
Rock 3
Scissors 5
Paper 4
Rei: Use Rock
Rock 3
Scissors 3
Paper 1
Sayuri: Use Paper
Rock 5
Scissors 1
Paper 3
Yuiko: Use Rock
Rock 3
Scissors 4
Paper 1
Yurie: Use Scissors
Rock 2
Scissors 2
Paper 4
The final step I did was note down every attack they used. I looked for a pattern and occasionally changed my attack. Not too much though, as more often than not, I found they countered my pick. There is no guaranteed win. Once I started using this system, I got 1 tournie win out of every 3 or so. After each win, I would leave and go collect my business payouts and also save. Just to mix things up so I didn't go crazy working on this. It took 3 or 4 nights of trying to get this one.