Game Theory Q: Proving Player Win w/Axiom of Choice

In summary, the conversation discusses an infinite game where players choose between 0's and 1's, and the resulting sequence can be seen as a binary expansion of a real number between 0 and 1. The game has two players, where player 1 wins if the sequence corresponds to a number in a given set E, while player 2 wins if it corresponds to a number not in E. The Axiom of Choice implies the existence of a set E where the game has no value. There is a discussion about ambiguities similar to 0.99999...=1 and the certainty that either player 1 or player 2 will win in the game.
  • #1
dhong
2
0
Hey PF!

Can you help me with something:

Players alternately choose 0's or 1's. A play of this infinite game is thus a sequence of 0's and 1's. Such a sequence can be considered as the binary expansion of a real number between 0 and 1. Given a set ##E## of real numbers satisfying ##0 < x < 1 \forall x \in E##, say that player 1 wins if the play corresponds to a number in ##E## and player two wins if the way corresponds to a number in ##[0,1] \backslash E##.

Evidently the Axiom of Choice implies there exists a set ##E## for which the game has no value. Can you help me out with showing this?
 
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  • #2
1. What about ambiguities similar to 0.99999...=1, for example the two sequences 011111... and 100000, which both correspond to the real number 0.1?

2. If this ambiguity is resolved, it is certain that either player 1 or player 2 wins, since every real number in [0,1] lies in either E or its complement. But you meant perhaps something else with the "value" of the game?
 

1. What is game theory and how does it relate to the Axiom of Choice?

Game theory is a branch of mathematics and economics that studies strategic decision-making. It involves analyzing the behavior of players in games in order to determine the most rational course of action. The Axiom of Choice is a fundamental mathematical principle that states that for any collection of non-empty sets, a selection function exists that allows one element to be chosen from each set. In game theory, the Axiom of Choice can be used to prove the existence of a winning strategy for a player.

2. What is the role of the Axiom of Choice in proving a player's win?

The Axiom of Choice is an essential tool in proving a player's win in game theory. It allows us to construct a strategy for the player that guarantees a win, even in games with infinite or uncountable options. Without the Axiom of Choice, it would be impossible to prove a player's win in some games.

3. What are some examples of games that can be analyzed using the Axiom of Choice?

The Axiom of Choice can be applied to a wide range of games, including simple games like Tic-Tac-Toe and more complex games like Chess. It can also be used in economic games, such as the Prisoner's Dilemma, and in social scenarios, such as bargaining and negotiation.

4. Are there any limitations to using the Axiom of Choice in game theory?

While the Axiom of Choice is a powerful tool, it does have some limitations. It can only be used to prove the existence of a winning strategy, but it does not provide a way to find or construct that strategy. Additionally, the Axiom of Choice is a controversial principle in mathematics, with some mathematicians arguing that it leads to counterintuitive and paradoxical results.

5. How does the Axiom of Choice impact real-world applications of game theory?

The Axiom of Choice is a theoretical concept that is primarily used in mathematical and economic models. Its impact on real-world applications of game theory is limited, as most real-life situations do not involve infinite or uncountable options. However, understanding the Axiom of Choice can provide valuable insights into the complexities of strategic decision-making and can be applied to a range of scenarios, such as business negotiations and political conflicts.

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