Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Suppose, in $G_{\omega}(\mathbb{R})$, a player's two strategies are equivalent, if, for any strategy of his opponent, the outcome incurred are the same. It can be shown that in $G_{\omega}(\omega)$ and $G_{\omega}(2)$, the cardinalities of partition of strategy spaces derived from equivalence relation are $2^{\aleph_0}$, since cardinality of set of strategies that are constant functions(which are not equivalent)and that of strategy space are both $2^{\aleph_0}$. But in $G_{\omega}(\mathbb{R})$,the cardinalities of two corresponding sets are $2^{\aleph_0}$ and $2^{2^{\aleph_0}}$.


(Apologize for just copy and paste some content from the question I asked before)


Formally, a two-player game $$G(\omega,(X_n)_{n \in\omega},(Y_n)_{n \in\omega},)$$ where

  • $\omega$ is the number of the moves of each of the two players I and II.
  • $(X_n)_{n \in \omega}$ (or $(Y_n)_{n \in \omega} $) is a $\omega$-sequence of $X_n$(or $Y_n$), which is the action space of $n$th move of first (second) player.
  • The game is played in an alternating fashion.An outcome is a $\omega$-sequence:$$a_0 \in X_0, a_1 \in Y_0, a_2 \in X_1, a_3 \in Y_1, a_4 \in X_2……$$
  • A player moves at any stage contingent on history. So a strategy for player I take the form as s sequence of functions $\{\sigma_i\}_{i \in \omega}$: $$\sigma_i : \prod_{j<i}{X_j \times Y_j} \to X_i$$ Player II's strategy's form $\{\tau_i\}_{i \in \omega}$is defined similarly. Player I and Player II's strategy spaces are denoted as $S_{\text I}$ and $S_{\text {II}}$ respectively. Denote the binary operation $\star$ as the function that sends a pair of strategies of player I and player II to the outcome they give rise to.
  • Player I and Player II's strategy spaces are denoted as $S_{\text I}$ and $S_{\text {II}}$ respectively. Denote the binary operation $\star$ as the function that sends $\sigma$ and $\tau$, a pair of strategies of player I and player II to the outcome $a$ they give rise to$$\sigma \star \tau \mapsto a$$
  • Strategy $\sigma$ and $\sigma'$ for player I are equivalent, if for any player II's strategy $\tau$, $$\sigma \star \tau = \sigma' \star \tau$$ The equivalence class $[\sigma]$ is denoted as$\{\sigma' \in S_{\text{I}}:\forall \tau \in S_{\text{II}}(\sigma \star \tau = \sigma' \star \tau)\}$, and the corresponding partition of $S_{\text{I}}$ as $S_{\text{I}}'$. We define the equivalence relation for Player II in the same way.

What is cardinality of $S_{\text{I}}'$, when $X_n = Y_n = \mathbb{R}$?

share|improve this question

1 Answer 1

up vote 3 down vote accepted

Here are $2^{2^{\aleph_0}}$ pairwise inequivalent strategies for player II (and it's easy to modify them to work for player I instead). Fix two distinct reals $x$ and $y$. For each set of reals $A$, let $\sigma_A$ be the strategy for player II that outputs $x$ on the first turn if Player I's first move is in $A$, and $y$ otherwise. On each subsequent turn we let $\sigma_A$ output $x$.

share|improve this answer
    
Aleph and alpha are both he first letters of their respective alphabets, and they represent the same sounds. But they are not the same, more so mathematically. :-) –  Asaf Karagila Feb 15 '13 at 4:01
    
@Asaf Oops, I always do that. Thanks for pointing it out. –  Trevor Wilson Feb 15 '13 at 4:08
    
@Asaf I need a keyboard with $\aleph$ on it so I don't have to type something so similar to "alpha". –  Trevor Wilson Feb 15 '13 at 4:09
    
Oh, you still have to type \aleph. LaTeX and Hebrew aren't very good friends... :-) –  Asaf Karagila Feb 15 '13 at 4:32

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.