# Game theory-surreal/dyadic/ordinal numbers

In summary, the post discusses a game where players take turns choosing surreal numbers and the goal is to keep the sum of the chosen numbers greater than or equal to 0. The post also mentions finding the birthday of a dyadic number using an algorithm where numbers are split into two sets on each day. To find the birthday of a dyadic number q, we continue this process until q is born on a certain day.
First of, a simple question. If we have 3 games with surreal equivalents, let's say *1/2 *-1/2 and *1/4, (where positive means in favor of player left) if left goes first he wants to keep that sum of those 3 games greater than or equal to 0 to win right?

## Homework Statement

For every dyadic number q, fi nd it's birthday

## Homework Equations

A dyadic rational is a number of the form n/2^k where n and k are integers.

Day 0: On day zero, the number 0 is born, and we identify 0 with fjg which we think of as
a split of the empty set into two empty sets.
Day n + 1: On day n + 1 we take all the numbers that have been born on days 0; 1; : : : n,
and we consider all possible ways to split these numbers into two sets fL j Rg where every
x 2 L and y 2 R satisfy x < y (note that we also allow L or R to be empty). We call L
the Left set and R the Right set. For each such split, a new number is born. If both L
and R are nonempty, then this new number is halfway in between the largest element of L
and the smallest element of R. If R is empty, then the new number is the smallest counting
number (positive integer) greater than everything in L, and similarly, if L is empty, we get
the largest negative counting number smaller than everything in R.

## The Attempt at a Solution

I really have no idea where to start..

I would first clarify some points in the forum post. It seems that the person is describing a game where players take turns choosing a number from a set of surreal numbers (numbers of the form n/2^k) and the goal is to keep the sum of the chosen numbers greater than or equal to 0. However, it is not clear how the players are choosing these numbers or what the rules are for the game. I would ask for more information on the game before attempting to answer the question.

As for finding the birthday of a dyadic number q, I would follow the given algorithm. On day 0, the number 0 is born. On day 1, we split 0 into two empty sets, which gives us the numbers -1/2 and 1/2. On day 2, we can split these numbers into two sets, giving us the numbers -3/4, -1/4, 1/4, and 3/4. We continue this process, splitting the numbers that have been born on previous days, until we reach the desired dyadic number q. The day on which q is born will be its birthday. For example, if q = 3/8, it will be born on day 4 since on day 4 we split the numbers 1/4 and 3/4 to get 3/8.

## 1. What is game theory-surreal/dyadic/ordinal numbers?

Game theory-surreal/dyadic/ordinal numbers is a branch of mathematics and economics that studies strategic decision-making in situations where the outcome of one's choices depends on the choices of other individuals. It uses surreal, dyadic, and ordinal numbers to represent and analyze these situations.

## 2. How do surreal, dyadic, and ordinal numbers differ from regular numbers?

Surreal, dyadic, and ordinal numbers differ from regular numbers in that they allow for the representation of infinitely large and infinitely small numbers. They also have specific properties and rules for arithmetic operations that differ from regular numbers.

## 3. What are some real-world applications of game theory-surreal/dyadic/ordinal numbers?

Game theory-surreal/dyadic/ordinal numbers have many real-world applications, including economics, political science, evolutionary biology, and psychology. They are used to analyze and predict strategic decision-making in various situations, such as negotiations, auctions, and conflicts.

## 4. How are surreal, dyadic, and ordinal numbers used in game theory?

Surreal, dyadic, and ordinal numbers are used in game theory to represent and analyze the strategic choices of players in a game. They allow for the calculation of optimal strategies and the prediction of game outcomes. They are also used to study the behavior of players in different types of games and to find solutions to game-related problems.

## 5. What are some challenges or limitations of using game theory-surreal/dyadic/ordinal numbers?

One of the challenges of using game theory-surreal/dyadic/ordinal numbers is that it requires a high level of mathematical understanding and expertise to apply. It may also be difficult to accurately model and predict the behavior of players in real-world situations. Additionally, the assumptions and simplifications made in game theory may not always reflect the complexity of human decision-making.

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