Breath of Fire IV Rock-Paper-Scissors Guide v1.0 Written by Zylo2 (zylo@wideopenwest.com) CONTENTS I. Updates II. Introduction III. Conclusions IV. Optimum Strategy V. Exceptions to Optimum Strategy VI. Data VII. Calculations ------------------------------------------------------------------------------- I. UPDATES ------------------------------------------------------------------------------- v1.0 - 30 May, 2004 Initial version. ------------------------------------------------------------------------------- II. INTRODUCTION ------------------------------------------------------------------------------- In Breath of Fire IV (copyright Capcom 2000-2001), the Faerie Village minigame offers a Game Parlor as one of its jobs. The Game Parlor’s first faerie offers the game of Kecak, a test of your ability to press buttons according to a dynamic rhythm. The Parlor’s second faerie lets you rename your faeries. The Parlor’s third faerie offers the game of Rock-Paper-Scissors, a game of both chance and strategy. This guide will use basic principles of probability and counting (college-level mathematics) to determine the chances of success at Rock-Paper-Scissors. For those of you who don’t want to examine the gory details, I have placed the Conclusions sections before my methodology. If you trust my calculations, you can skip the methodology and form your own judgment about how much time you want to spend in Rock-Paper-Scissors. The prizes for Rock-Paper-Scissors (RPS) are: Points Item ------------------------------------------------------------------------------- 1 MultiVitamin 2 Dress Shoes (Defense 10. Weight 0. Can be used by Nina and Ursula) 4 1,000 zenny 8 Midas Stone (exclusive to this minigame only. Raises zenny gained at the end of battles. Weight 10) 16 10,000 zenny 30 Soul Ring (raises user’s CP by half) Each game of RPS costs 300 zenny to play, if you have no points yet. If you win a game, then continuing play doesn’t charge you. Faeries’ playing styles vary according to their personalities. For the purpose of this guide, assume that the assigned faerie is of Ordinary personality. You can play in Normal or Random mode. Normal assigns you two cards of Rock, two cards of Scissors, and two cards of Paper. It then assigns the same cards to your opponent. This leads to a somewhat predictable game. Random mode assigns cards randomly, which can lead to some bizarre combinations. I don’t recommend playing Random. We will assume Normal Mode here. In the game, your opponent and you will simultaneously flip cards. Rock beats Scissors, Scissors cuts Paper, and Paper wraps Rock. If you win outright, your opponent will lose his card while you keep yours. If it’s a draw, both of you will lose your cards. If you lose, your opponent keeps his card while you lose yours. The game ends when at least one of you runs out of cards. If you run out at the same time, the game is a draw, and your opponent wins by default. The number of cards that you have at a victory determines your points gained: 1 point for 1 card. After winning a game, you will be offered a chance to redeem your points for a prize. Whenever you leave a game session, all of your points are forfeited, so don’t leave unless you have your points cashed in for something. To win the big prizes, you must consecutively win games until you have the necessary points. If at any time you lose or draw a game, your opponent will reset your points back to 0, and you will need to pay another 300 zenny to play again. ------------------------------------------------------------------------------- III. CONCLUSIONS ------------------------------------------------------------------------------- According to the data and calculations written at the end of this guide, the chances of an Ordinary faerie drawing a certain card at the beginning of a game are as follows: 64.8% Rock 26.7% Scissors 8.5% Paper Also, your chances of winning such a game using optimum strategy (as described in the next section) are: 92.4% for a game that began with Rock 20.8% for a game that began with Scissors 40.0% for a game that began with Paper If you multiply the chances of a particular game occuring with the odds of your victory, and then add the results, you obtain your total chance of victory: (0.648) (0.924) + (0.267) (0.208) + (0.085) (0.4) = 68.8%. Therefore, if you use optimum strategy, you have a nearly 70% chance of obtaining a MultiVitamin in exchange for 300 zenny. To win the bigger prizes, you must determine how many successive games you must win in order to obtain that many points. For our first example, let’s shoot for the 8-point Midas Stone with the assumption that we will win 2 points at the end of each game. Therefore, 8 points / 2 points per game means that we must win four successive games. To obtain the probability for successive games, you must multiply a value of 68.8% for each game. For example, two successive games have a chance of 0.688 * 0.688 = 0.473 (47.3%). For four successive games, we must multiply 0.688 four times (quadrupling it), to obtain 0.224 (22.4%). That means that assuming you win 2 points per game, you will, on average, gain enough points for a Midas Stone every 1 out of 5 attempts. Since the average number of attempts for a Midas Stone is 5, and each attempt costs 300 zenny, the Midas Stone has an expected cost of 1,500 zenny -- cheap by this game’s standards. However, the big impact of raising 0.688 to powers doesn’t bode well for higher prizes. Using the same assumption of 2 points per game (which is rather liberal -- I would put it at 1.5), we will next shoot for the 16-point 10,000 zenny. 16 points / 2 points per game = 8 successive victories. 0.688 to the eighth power = 0.050. Your chances of winning eight successive games are 5%. That means only 1 in 20 games will proceed that far. Multiplying 20 by 300 zenny yields an expected value for the 10,000 zenny of 6,000 zenny. So, you gain 4,000 zenny (on average), but you must sit there and play through 20 games (on average) carefully reading your optimum strategy. At about a minute per game, that’s 20 minutes of gameplay for 4,000 zenny. Although if you adjust the points per victory to 1.5 rather than 2, the figures don’t look that rosy. There are much easier ways to gain that many zenny in this game. Those ways also may offer collateral benefits, such as experience. For example, slaying a Gold Gang near Fou-lu's Tomb with only magic will earn 10,000 zenny. Besides, any Rock-Paper-Scissors attempt could go much longer than 20 games, actually costing you zenny! So, in conclusion, this isn’t that great of a deal. For a Soul Ring, you must win (30 / 2) = 15 successive games. Your chance of doing that is 0.688 to the fifteenth power = (0.3%). You will achieve this, ON AVERAGE, in every 1 out of 300 games. At a game per minute, that’s 300 minutes (5 hours) of your life for a Soul Ring, ON AVERAGE. There is an easier way to obtain one: trading for it at the Checkpoint Manillo shop. The expected value here is 300 games * 300 zenny = 90,000 zenny (Ouch). The last word on the RPS game is: get the Midas Stone and lower prizes. You can play for the 10,000 zenny according to your judgment, but I advise you to find a better way to raise money. The Soul Ring attempt is a waste of your time, unless you receive a miracle of luck while trying for lesser prizes. ------------------------------------------------------------------------------- IV. OPTIMUM STRATEGY ------------------------------------------------------------------------------- My suggested optimum strategy is based on the principles of counting and my own version of Occam’s Razor: All other things being equal, your opponent has an equal chance of playing any of his cards -- he doesn’t favor any particular symbol. My statistical data indicate that the Razor doesn’t hold in all situations, but it does in most. As you can easily determine by playing a few games, your opponent does have SOME strategy. He’s not simply a random number generator. He won’t make blatantly stupid moves often. However, counting and the Razor make for a simpler strategy than playing 1,000 games and subjecting the data to hours of analysis. Let me give you some examples of situations and how you should determine your optimum move. I will describe exceptions afterward. Let’s say that in the first round, your opponent plays Rock, while you play Paper (which will happen often if you follow my broader strategy). Then the remaining cards are as follows: Opponent You R RR SS SS PP PP Now, we analyze this situation according to what could possibly happen if you play any of the three symbols. Let’s assume according to the Razor that your opponent has an equal chance of playing any of his cards (which fits with my data for this situation). Using counting, we see that if you play Rock, 2 out of his 5 cards will result in you winning (Scissors), 1 out of his 5 will result in a draw (Rock), and 2 out of his 5 will result in you losing (Paper). Similarly, if you play Scissors, 2 / 5 win (Paper), 2 / 5 draw (Scissors), and 1 / 5 loss (Rock). Finally, if you play Paper, 1 / 5 win (Rock), 2 / 5 draw (Paper), and 2 / 5 loss (Scissors). We write this out in a little chart for fast scanning: Win Draw Loss Total Rock 2 1 2 5 Scissors 2 2 1 5 Paper 1 2 2 5 By looking at the chart, we can see that by playing Scissors, you only risk a 1 in 5 chance of loss. According to probability, Scissors is the best move to play in this situation. Of course, it’s still possible that your opponent will hit that 1 in 5 chance and beat you. But that’s not the expected result. This strategy may result in an occasional loss, but over time, it is an optimum strategy. It’s better than guessing. We can draw the analysis to the third round of play by investigating the possibilities of you playing Scissors in the second round. If you play Scissors and you win (your opponent plays Paper), then we have the following situation and chart: Opponent You Win Draw Loss Total R RR Rock 2 1 1 4 SS SS Scissors 1 2 1 4 P PP Paper 1 1 2 4 Here, Paper is no good because it has 2 losses. Rock and Scissors both have only a single loss, but Rock looks better in the Win column than Scissors. Rock is the optimum move here. Now, we could continue analyzing this game to the fourth round, fifth round, etc. until we reached a situation in which you can’t lose due to chance anymore. Your victory is guaranteed by chance unless you make a mistake. My advice is that when you have the clear upper hand, don’t take any risks. A victory by 1 point is always preferable to risking an unnecessary loss. A loss resets your points to 0, which is a big Ouch. A single point usually isn’t worth that risk. Play conservatively. It pays in the long run. For the sake of brevity, let’s say that all of the results in Round 3 after playing Rock result in a guaranteed win (probability 1). Obviously that’s not true in the case of possible losses, but you can calculate those yourself, if you wish. We take the chance of winning for each result of Rock, and multiply it with its actual chance of occuring after Rock: (1) (2 / 4) + (1) (1 / 4) + (1) (1 / 4) = 1. So if Rock always leads to guaranteed wins, Rock itself is a guaranteed winner. Now let’s plug that back into the Round 2 chart, in which you played Scissors. Let’s assume that playing Scissors in Round 2 then Scissors in Round 3 had a 0.5 chance of resulting in an ultimate win, while Scissors -> Paper had a 0.25 chance of an ultimate win. We multiply those chances of victory with the chances of your opponent playing those symbols in Round 2, according to the Scissors row in the Round 2 chart: (1) (2 / 5) + (0.5) (2 / 5) + (0.25) (1 / 5) = 13 / 20 = 0.65. That number, 0.65, is your ultimate chance of victory if you choose to play Scissors in Round 2. Using these methods of analysis, you can determine the optimum strategy and chance of success for any situation, assuming that the Razor holds. Of course, some situations will have no optimum move. For example: Opponent You R R S S P P In this case, pure chance has taken over your strategy. Sometimes your strategy beats chance, and sometimes chance beats your strategy. Take your pick of symbols. ------------------------------------------------------------------------------- V. EXCEPTIONS TO OPTIMUM STRATEGY ------------------------------------------------------------------------------- Your opponent will not always play according to the Razor. Your opponent has a personality. During the game, you will notice that your opponent utters comments that may or may not reflect his decision-making. If you wish, you can play a bunch of games, writing down the particular comments and what card your opponent played after that comment. Some comments will result in 33% Rock, 33% Scissor, and 33% Paper over time. These comments indicate that your opponent has decided to use a purely random method. Other comments may show tendencies toward certain symbols. For example, you may find that, “Maybe I will play a Rock next” results in a Rock 60% of the time. Is 60% a significant number, or is it merely an anomaly of randomness? I could play 10,000 games and analyze the comments’ results over time. However, as described above, I’m not interested in items above the Midas Stone. If you want to do that, you can collect those types of stats. I find that the Razor holds up rather well, though. Note that on the first round of play, your opponent hasn’t uttered any comments yet. My data suggest that the first round of play isn’t purely random. There are percentages for Rock, Scissors, and Paper based on the faerie’s personality. As stated in the beginning of the Conclusions section, I currently have these percentages for an Ordinary faerie’s first round: 64.8% Rock 26.7% Scissors 8.5% Paper I also listed the ultimate odds of winning for each: 92.4% for a game that began with Rock 20.8% for a game that began with Scissors 40.0% for a game that began with Paper This is a good deal, since the most common case (your opponent plays Rock) has a 92.4% chance of success if you use optimum strategy. In this case, I calculated these numbers assuming that your optimum strategy on Round 1 versus an Ordinary faerie was to ALWAYS play Paper. Since the Ordinary doesn’t employ the Razor on his first round, neither should you rely on the Razor for your first symbol, since you have better data. Of course, always playing Paper will result in you losing the first round in those 26.7% of cases in which your opponent plays Scissors. After that, you only have a 20.8% chance of ultimate victory, which is rather bleak. Nevertheless, in the first round of any given game, the Ordinary is much more likely to play Rock than Scissors. It doesn’t follow any pattern such as Game 1- Rock, Game 2- Rock, Game 3- Scissors, as far as I can tell. Any such pattern that I tried eventually broke. I believe each game’s first round is determined by a set chance, regardless of the order of games or points obtained. Although I could be wrong on that. So, the best broad strategy is to always play Paper and hope that you will hit the common Rock response rather than the uncommon Scissors response. Over time,you will eventually encounter sequences of Game-Rock, Game-Rock, Game-Rock, Game-Rock, that will catapult you into the higher points. Using this broad strategy, you will often find yourself in the situation of having won Round 1 by Paper over Rock. You will be in a very familiar Round 2. Unfortunately, my data indicate that the Razor applies in Round 2, not the faerie’s personality. So I can’t give you any strategies for that common Round 2, other than analysis provided in the Optimum Strategy section. Good luck! Now for my data and calculations. The sample size isn’t that large, but it was enough for me to write this guide. You can develop your own results and add them to my sample, in order to create a larger and more representative sample. ------------------------------------------------------------------------------- VI. DATA ------------------------------------------------------------------------------- In order to obtain the Round 1 figures for an Ordinary faerie, 64.8% Rock 26.7% Scissors 8.5% Paper I used these data: 46 / 71 Rock 19 / 71 Scissors 6 / 71 Paper After playing my Paper wraps his Rock on Round 1, here are my Round 2 data: 13 / 37 Paper = 35% 12 / 37 Rock = 32% 12 / 37 Scissors = 32% As you can see, the distribution is even. I suggest the Razor-based strategy in Round 2. I often encountered his Scissors cutting my Paper on Round 1, leading to this Round 2: 7 / 15 Paper = 47% 6 / 15 Rock = 40% 2 / 15 Scissors = 13% This would indicate a non-random tendency on the part of the faerie, but a sample size of 15 is rather small. You can add that to your own sample. If you hit a Round 1 Paper versus Paper draw, then you continue in a drawish situation. Although I listed the ultimate chance of winning this at 40.0%, that’s a guesstimate on my part. The deep analysis of those scenarios creates a bunch of draw situations that are difficult. I figure that since either an ultimate Draw or Loss result in your effective loss, 35-40% is a good range for your victory. Paper vs. Paper is a rare situation, so a few points won’t affect the 0.688 figure all that much. I had the following data for what follows a “Hmm... what to do next?” comment by the Ordinary faerie. 11 / 28 Paper = 39% 9 / 28 Rock = 32% 8 / 28 Scissors = 29% That even distribution suggests that “Hmm ... what to do next?” activates a purely random choice on the faerie’s part. ------------------------------------------------------------------------------- VI. CALCULATIONS ------------------------------------------------------------------------------- To clarify where I obtained the ultimate odds of winning after you playing Paper in Round 1, 92.4% for a game that began with Rock (by opponent) 20.8% for a game that began with Scissors 40.0% for a game that began with Paper The 40.0% was a bit arbitrary but relatively insignificant in the 8.5% Paper vs. Paper cases. The 92.4% was based on a deep analysis assuming that you played Scissors on Round 2, as you should: Win Draw Loss Total Rock 2 1 2 5 Scissors 2 2 1 5 Paper 1 2 2 5 With the chance of victory after winning by Scissors being 95.2%, the chance of victory after drawing being (23 / 24), and the chance of victory after losing (4 / 5). (0.952) (2 / 5) + (23 / 24) (2 / 5) + (4 / 5) (1 / 5) = 92.4%. This is a very sunny situation. The bleak side is when in Round 1, your opponent’s Scissors cuts your Paper. I then assume that you play Scissors in Round 2, according to my data that indicate non-randomness: 7 / 15 Paper = 47% 6 / 15 Rock = 40% 2 / 15 Scissors = 13% Then take the deep analysis results of your opponent playing Paper ( 17 / 50 chance of victory), Scissors ( 3 / 25), and Rock ( 7 / 75). As a final thought on this whole guide, one of the programmers at Capcom put a great deal of effort into a minigame that reaps relatively few rewards. :) Nevertheless, it’s a fun game.