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For this reason, my article today will be about making optimal plays and how application of economical and mathematical concepts play into a variety of aspects of your game. Deck building, choosing which cards to play, making the correct sequence of card plays, deck choice, and more can all be broken down into more comprehensible analyses than you might think. For this reason I will bring a somewhat unique application of statistics that has to do with playtesting.

The rest of the statistics will be done in the game theory section to give the article more continuity rather than having to reference between sections as I go deeper into one concept compared to another. Most of the best players use these analyses without thinking or knowing the underlying concepts, so I hope that this article will be able to appeal to a wide range of audiences.

These concepts can be applied to any level of gameplay: These situations may seem unrealistic, but the underlying concepts are what you can apply to your game. I will try to give as many ways to eliminate these assumptions as possible. Game theory, also known as interactive decision theory, allows us to look at the possible outcomes of a situation and pick the most optimal line of play.

This is one of the more advanced ways to look at in-game decisions, as well as deck choices. One thing to note is that we are assuming that both players are rational decision-makers, which is to say that they will both be making the most optimal play given the information that they have. One way to display a game is to use a tree diagram, which is usually used to display an extensive form game. Extensive form games are games that have a sequence of timing between moves.

This will be the form that we use to analyze optimal plays within a game. The next option of display is to use a matrix, which is primarily used when representing normal form or strategic form game; extensive form games can be put into a matrix, but it is usually easier option trading game theory pokemon visualize on a tree diagram.

This matrix shows the different results based on the choices that both players make, and we can use this to analyze deck choices, due to there being no inherent sequence of these moves. We have two more pairs of scenarios here: Player 1 can either have a Lysandre in their hand, which guarantees a win, or not.

Player 2 then has the option to either N or Professor Juniper. The probabilities of each player winning are made up for the purpose of the example. The numbers represent the odds of each player winning Player 1 is red, Player 2 is blue. A quick explanation on my reasoning for these numbers is that Juniper will have the same chances either way if Player 2 has the win in hand, but if Player 2 does not then they will have to draw into the win or lose.

The probability of the N outcome changes, because if Player 2 does not have the win in hand they will have 2 additional draws. Player 2 has a dominant strategy in both of these cases, which means that they will make the same play regardless of what Player 1 does. Playing N will always give them a probability of winning that is greater than or equal to probability of winning when playing a Juniper.

Player 1 knows this and will thus make the action that gives the greatest probability of winning given N being played second. Player 1 will pass on both scenarios, because the probability of winning is greater when passing if the opponent is going to N. After determining the equilibrium, which is the outcome that is going to occur, we can now figure out the probabilities of each player winning the game. The probability of Player 1 winning is going to be 0. It is difficult to apply this while playing a game due to the importance of having percentages as accurate as possible.

We can do rough estimates, though, which often times can lead you to finding the equilibrium play. Despite this, you can still try to find a dominant strategy, like Player 2 had in the example above. Basing your plays on these models will result in a higher overall likelihood of success if done correctly, so I would highly recommend thinking about extensive form games in situations like these. What we can do with this knowledge is use the idea of an extensive form game to think about processions of plays while in a game.

Comparing the possible outcomes your opponents possible lines of play based on a decision you make on your turn is an important concept when determining optimal plays. Each player is also as the same skill level as you, so that you neither gain an advantage nor a disadvantage based on who you play. The normal form matrix below shows the matchups between the three decks. The red numbers show the percent chance of the deck in the row winning and blue numbers show the percent chance of the deck in the column winning.

This is found by multiplying its probability of winning a certain matchup option trading game theory pokemon the percentage of the field that the other deck occupies. Or, as an equation:. Using this equation we can compare the probabilities of each deck winning any individual game by plugging in the variables that we have from the matrix and computing. Here are the values:. In this scenario we can predict that Yveltal, having the highest probability of winning, will be the most optimal play.

Remember what I said before about rational decision-makers? To find out the optimal distribution of option trading game theory pokemon in this situation we have to set up a option trading game theory pokemon system of equations. If you would like to do this calculation yourself there is a website where you can plug in these values and it will solve the system of equations for you. I see many players talking about which deck is the best and trying to focus on beating it.

In theory, this is the wrong approach, option trading game theory pokemon actually the metagame will shift away from option trading game theory pokemon best deck on its own. Although this idea of perfect information and rational decision making is quite obviously not how real-life events play out, we can still look at these values to see where the distribution of decks should be gravitating towards.

An example of this would be to calculate your values based on the distribution of decks in the weekends leading up to your tournament and then seeing where the distribution should be gravitating towards, as well as the trends of the proportions of each deck. There option trading game theory pokemon a multitude of factors that play into the distribution of decks on any given weekend, so keeping everything else in mind while looking at these computations is important.

Numbers are far from an absolute truth. This is a very basic metagame, but we can apply these concepts to more complex situations by adding more decks and calculating the values of the matchups to a more precise number.

Figuring out the proportion of the decks is much more difficult than finding the matchups, but it can usually be roughly approximated due to information from previous tournaments or from online hubbub. The use of game theory for metagame analysis can be applied more directly to real life situations than can the in-game option trading game theory pokemon in the last section.

Running the numbers for a few possible distributions of decks can be a useful way to determine which decks thrive in various metagames.

This is a fantastic way to improve your metagaming skill, which is the ability to choose a correct deck for the metagame. Closed environments such as City Championship marathons are a great way to apply these figures for a few reasons.

By making a more reasonable estimate of the option trading game theory pokemon of decks you can estimate which deck will be the optimal play with much more certainty. An interesting way to look at the distributions can also be to estimate a good scenario and a bad scenario of the matchups and look at the range of the win percentages.

This is mostly useful when you have very little idea of which decks are going to be at a tournament. The decks that have the lowest variance option trading game theory pokemon the difference in deck distribution, and the ones that have a high average win percentage will be the decks that would be your best choice when going into an unknown event.

This concept is difficult to display with only three decks in the format, so I would recommend you try it out yourself with the current format. Determining matchups is important when choosing a deck to play, and when trying to make changes to a deck to give it a better balance against the format. In order to roughly determine matchups, you option trading game theory pokemon play at least 10 games between all possible matchups.

The amount of games a certain deck wins divided by the amount of games played can be used in place of my estimations of matchups for a more accurate result most of the time. In order to make our model more realistic, we have to eliminate as many of the assumptions that we made as possible. First and foremost we will have to add in every possible deck in the format and the proportion of the metagame that each will occupy. These will have to be rough estimates, and any unknown decks will have to be ignored.

To eliminate the assumption that every deck has the same cards we can look at the matchup of your deck versus another deck that has either teched for you deck, teched for another matchup, or is purely consistent.

Usually option trading game theory pokemon going into an unknown metagame sticking with the deck you are most comfortable with is the best option, but if you have a multitude of options that you are indifferent to these equations can be an interesting way to pick option trading game theory pokemon deck.

Statistics is a fantastic way to analyze the data that we have accumulated thus far. For this section, I will talk about one of my favorite ways of playtesting and how to apply statistics to eliminate the busywork. Playing 5 timed option trading game theory pokemon in a row of the same matchup can be a great way to look at how option trading game theory pokemon would do in a tournament against a not-so-great matchup.

It also helps you test techs for certain matchups and see how a tournament setting would play out. Before getting on to the calculations, we first have to come up with the different records that will result in more than 10 match points.

We now need to apply these different possibilities to our equation. The rest of the numbers are denoted by the letter next to them. By plugging the values of all of our possibilities of getting about 10 match points into the equation we get the probabilities of each of these outcomes. By adding these together we get our final probability of 0. This data can be useful if you expect option trading game theory pokemon large amount of Virizion and want to change your Yveltal deck to counter Verizion.

This is one of the concepts that is harder to apply directly to your game. If you have figured out the probabilities of each outcome the multinomial distribution equation is a way to visualize how changes in your matchups due to teching or adding more consistency plays out in multiple rounds. You can simplify this setup to have n be the number of games you expect to play against the other deck in an event.

My calculating the probabilities of the possible win, loss, and tie outcomes you can set a goal or a option trading game theory pokemon of wanting to have a certain record or better against another deck. An easy-to-use spreadsheet that will automatically solve the above calculations above for you is available here.

This type of article was entirely new option trading game theory pokemon me, but I had a fantastic time writing it and I hope that you all find the concepts I covered as interesting as Option trading game theory pokemon do. Reading that some financial traders look at Magic: If I used something incorrectly please let me know so I can improve and possibly bring you all a more in-depth article later on.

Any and all feedback is option trading game theory pokemon as always, I especially like to hear from my readers when I do something new option trading game theory pokemon order to gauge option trading game theory pokemon, and to pick my topics with more confidence in the future.

Good luck to everyone at their coming City Championships, especially with the marathons coming up very soon. After 90 days we open up past UG content for public viewing to help preserve the history of the game. New articles are reserved for Underground members. Click here to view the registration page if you are interested in joining Underground and gaining full access to our latest content.

In-Game Analysis Tree Diagram 1: Player 1 does have Lysandre in hand. Player 1 does NOT have Lysandre in hand. Metagame Analysis Figure 1:

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