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Crunching the numbers: Mixed strategies (view source)
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The Pokémon series is no stranger to Rock-Paper-Scissors. The {{bp|Team Rocket trio|Rocket Trio}} will sometimes solve their disputes with a quick match, and in {{bp|Accumula Town}}, a girl will challenge the player to a Pokémon-themed version of the game with {{t|Fire}}, {{t|Water}}, and {{t|Grass}}. | The Pokémon series is no stranger to Rock-Paper-Scissors. The {{bp|Team Rocket trio|Rocket Trio}} will sometimes solve their disputes with a quick match, and in {{bp|Accumula Town}}, a girl will challenge the player to a Pokémon-themed version of the game with {{t|Fire}}, {{t|Water}}, and {{t|Grass}}. | ||
A few {{bp|Pokémon Trading Card Game|TCG}} cards also feature Rock-Paper-Scissors: {{ | A few {{bp|Pokémon Trading Card Game|TCG}} cards also feature Rock-Paper-Scissors: {{TCG ID|Gym Heroes|Misty's Duel|123}}, {{TCG ID|Mysterious Treasures|Team Galactic's Wager|115}}, and {{TCG ID|Dragon Selection|Fast Ticket|19}}. With Team Galactic’s Wager, it was ruled that people couldn’t flip a coin (which really means roll a die at a tournament) in order to simulate the effects of the game. This forces both players to try and quickly outwit their opponent. So what’s the best way of going about this? | ||
Before I ever learned about game theory, the process in my head went something like, “I think someone told me that 50% of people start out with rock, so I should go with paper. But if my opponent is thinking one step ahead of me, than s/he will go with scissors, so should I go one step ahead of that by going with rock?” I’d go on and on and on much like the Sicilian from ''The Princess Bride''. The thought of ever finding a solution to the game was, to say the least, inconceivable. | Before I ever learned about game theory, the process in my head went something like, “I think someone told me that 50% of people start out with rock, so I should go with paper. But if my opponent is thinking one step ahead of me, than s/he will go with scissors, so should I go one step ahead of that by going with rock?” I’d go on and on and on much like the Sicilian from ''The Princess Bride''. The thought of ever finding a solution to the game was, to say the least, inconceivable. | ||
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Mixed strategies show up all the time in Pokémon games. Deciding on switching your active to something better on your bench? Bam! Use mixed strategies. Thinking about what the best Pokémon are to make a good team? Bam! More mixed strategies. Figuring out what deck to bring to a tournament? Bam! Even more mixed strategies. Trying to kick it up a notch? Bam! Ask Emeril. | Mixed strategies show up all the time in Pokémon games. Deciding on switching your active to something better on your bench? Bam! Use mixed strategies. Thinking about what the best Pokémon are to make a good team? Bam! More mixed strategies. Figuring out what deck to bring to a tournament? Bam! Even more mixed strategies. Trying to kick it up a notch? Bam! Ask Emeril. | ||
Let’s look at a real life Pokémon example. Your friend and you once again are arguing about what movie to watch. Your friend is fairly tired of getting beaten with the videogame all the time, so he or she has given up achieving victory digitally (at least for now), and has moved on to the TCG. Both of you have been up to date with the metagame and even have decks based on CMT ({{bp|Celebi (Triumphant 92)|Celebi Prime}}, {{ | Let’s look at a real life Pokémon example. Your friend and you once again are arguing about what movie to watch. Your friend is fairly tired of getting beaten with the videogame all the time, so he or she has given up achieving victory digitally (at least for now), and has moved on to the TCG. Both of you have been up to date with the metagame and even have decks based on CMT ({{bp|Celebi (Triumphant 92)|Celebi Prime}}, {{TCG ID|Next Destinies|Mewtwo-EX|54}}, and {{TCG ID|Emerging Powers|Tornadus|89}}), ZekEels ({{TCG ID|Black & White|Zekrom|47}} and {{TCG ID|Noble Victories|Eelektrik|40}}), and Terrakion (the only Pokémon in this deck are four {{TCG ID|Noble Victories|Terrakion|73}} cards) archetypes. | ||
These three decks follow a Rock-Paper-Scissors type set up. Terrakion demolishes | These three decks follow a Rock-Paper-Scissors type set up. Terrakion demolishes ZekEels. ZekEels has the advantage over CMT. CMT laughs in the face of Terrakion. Unlike Rock-Paper-Scissors, each deck only has a chance to win over the other. On top of that, there are no ties. Also, one Terrakion deck could be very different from another, so one deck could have an advantage in the mirror match. | ||
Between the two of you, the win percentages between your decks are like so: | Between the two of you, the win percentages between your decks are like so: | ||
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*Note: I pulled these numbers out of thin air. Please don’t take them seriously. | *Note: I pulled these numbers out of thin air. Please don’t take them seriously. | ||
Giving this a quick glance, there are no pure strategies. Ergo, there must be a mixed strategy. We want to minimize our opponent’s chances of winning, so we’ll assign a probability that we’ll play Terrakion, | Giving this a quick glance, there are no pure strategies. Ergo, there must be a mixed strategy. We want to minimize our opponent’s chances of winning, so we’ll assign a probability that we’ll play Terrakion, ZekEeels, and CMT (p<sub>1</sub>, p<sub>2</sub>, p<sub>3</sub>) and look at our opponent’s expected values. | ||
[[Image:Mixedstrategies10.png||center]] | [[Image:Mixedstrategies10.png||center]] | ||
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[[Image:Mixedstrategies23.png||center]] | [[Image:Mixedstrategies23.png||center]] | ||
Here’s the breakdown of our numbers: if we play Terrakion 22% of the time, | Here’s the breakdown of our numbers: if we play Terrakion 22% of the time, ZekEels 46% of the time, and CMT 32% of the time, we’ll force our opponent to have a 48.8% chance to win. That’s good for us because that means we have a 51.2% chance to win. | ||
While mixed strategies are powerful, it should be noted that both humans and computers are notoriously bad at being perfectly random (hence why the random number generators for computers are called psuedorandom number generators). That’s why the best Rock-Paper-Scissors algorithms are random at first, but then analyze the past choices of the opponent to spot patterns and exploit them. | While mixed strategies are powerful, it should be noted that both humans and computers are notoriously bad at being perfectly random (hence why the random number generators for computers are called psuedorandom number generators). That’s why the best Rock-Paper-Scissors algorithms are random at first, but then analyze the past choices of the opponent to spot patterns and exploit them. | ||