Posted on Categories:Game theory , 博弈论, 经济代写

# 经济代写|博弈论代考Game theory代写|ECON3301 Top-down Induction

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## 经济代写|博弈论代考Game theory代写|Top-down Induction

When talking about combinatorial games, we will often use for brevity the word game for “game position”. Every game $G$ has finitely many options $G_{1}, \ldots, G_{m}$ that are reached from $G$ by one of the allowed moves in $G$, as in this picture:

If $m=0$ then $G$ has no options. We denote the game with no options by 0 , which by the normal play convention is a losing game. Otherwise the options of $G$ are themselves games, defined by their respective options according to the rules of the game. In that way, any game is completely defined by its options. In short, the starting position defines the game completely.

We introduce a certain type of mathematical induction for games, which is applied to a partial order (see the background material text box on the next page).
Consider a set $S$ of games, defined, for example, by a starting game and all the games that can reached from it via any sequence of moves of the players. For two games $G$ and $H$ in $S$, call $H$ simpler than $G$ if there is a sequence of moves that leads from $G$ to $H$. We allow $G=H$ where this sequence is empty. The relation of being “simpler than” defines a partial order which for the moment we denote by $\leq$. Note that $\leq$ is antisymmetric because it is not possible to reach $G$ from $G$ by a nonempty sequence of moves because this would violate the ending condition. The ending condition for games implies the following property:
Every nonempty subset of $S$ has a minimal element.
If there was a nonempty subset $T$ of $S$ without a minimal element, then we could produce an infinite play as follows: Start with some $G$ in $T$. Because $G$ is not minimal, there is some $H$ in $T$ with $H<G$, so there is some sequence of moves from $G$ to $H$. Similarly, $H$ is not minimal, so another game in $T$ is reached from $H$. Continuing in this manner creates an infinite sequence of moves, which contradicts the ending condition.

## 经济代写|博弈论代考Game theory代写|Background material: Partial orders

Definition $1.2$ (Partial order, total order). A binary relation $\leq$ on a set $S$ is called a partial order if the following hold for all $x, y, z$ in $S$ :
$x \leq y \quad$ and $y \leq z \quad \Rightarrow \quad x \leq z \quad(\leq$ is transitive $)$,
$x \leq x \quad(\leq$ is reflexive $)$, and
$x \leq y \quad$ and $y \leq x \quad \Rightarrow \quad x=y \quad$ ( $y$ is antisymmetric).
If, in addition, for all $x, y$ in $S$
$$x \leq y \text { or } y \leq x \quad \text { ( } \quad \text { is total) }$$
then $\leq$ is called a total order.
For a given partial order $\leq$, we often say ” $x$ is less than or equal to $y^{\prime \prime}$ if $x \leq y$. We then define $x<y\left(” x\right.$ is less than $\left.y^{\prime \prime}\right)$ as follows:
$$x<y \quad \Leftrightarrow \quad x \leq y \quad \text { and } \quad x \neq y .$$
This relation $<$ is also called the strict order that corresponds to $\leq$. Exercise $1.1$ asks you to show how a partial order $\leq$ can be defined from a relation $<$ on $S$ with suitable properties (transitivity and “irreflexivity”) that define a strict order. Then $\leq$ is obtained from $<$ according to
$$x \leq y \quad \Leftrightarrow \quad x<y \quad \text { or } \quad x=y .$$
An element $x$ of $S$ is called minimal if there is no $y$ in $S$ with $y<x$.

# 博弈论代写

## 经济代写|博恋论代考Game theory代写| Background material: Partial orders

$x \leq y \quad$ 和 $y \leq z \quad \Rightarrow \quad x \leq z \quad(\leq$ 是可传递的， $)$
$x \leq x \quad(\leq$ 是自反的，并且 $)$
$x \leq y \quad$ 和 $y \leq x \quad \Rightarrow x=y \quad(y$ 是反对称的）。

$$x \leq y \text { or } y \leq x \quad(\quad \text { is total })$$

$$x<y \quad \Leftrightarrow \quad x \leq y \quad \text { and } \quad x \neq y .$$

$$x \leq y \quad \Leftrightarrow \quad x<y \quad \text { or } \quad x=y$$

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