Game Theory: No Winning Strategies

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Discussion Overview

The discussion revolves around the concept of winning strategies in game theory, particularly focusing on games with perfect information where neither player has a winning strategy. Participants explore examples and definitions related to this topic, including the implications of the axiom of choice in such games.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant mentions that it can be proven that there are games with perfect information where neither player has a winning strategy, referencing the axiom of choice.
  • Another participant suggests tic-tac-toe as an example of a game without a winning strategy.
  • A different participant clarifies that they believe an example should meet specific criteria, particularly having only "win" or "lose" outcomes, and expresses confusion over the definitions of "game" and "winning strategy" as presented in the referenced article.
  • Questions arise about the nature of games that are not explicitly constructible and whether one can assume a winning strategy exists for explicitly constructible games.
  • A participant defines a winning strategy and provides examples, including a matrix game scenario, but another participant challenges the classification of rock, paper, scissors as a game in the game theory sense.
  • There is a reiteration of the challenge regarding the classification of rock, paper, scissors, leading to a consensus that it is an example of a simultaneous game.

Areas of Agreement / Disagreement

Participants express differing views on the examples of games and the definitions involved. There is no consensus on what constitutes a valid example of a game without a winning strategy, and the discussion remains unresolved regarding the implications of the axiom of choice.

Contextual Notes

Participants note the ambiguity in definitions of "game" and "winning strategy," which may affect the discussion. The reliance on the axiom of choice introduces additional complexity that remains unexplored.

ehrenfest
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http://en.wikipedia.org/wiki/Game_theory

It can be proven, using the axiom of choice, that there are games—even with perfect information, and where the only outcomes are "win" or "lose"—for which neither player has a winning strategy.) The existence of such strategies, for cleverly designed games, has important consequences in descriptive set theory.

I am confused about this. Can someone give me an example of a game with perfect information in which neither player has a winning strategy?
 
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Well, tic-tac-toe comes to mind.
 
I assume ehrenfest was asking for an example that satisfied the hypotheses in the quoted passage -- specifically, the only outcomes are "win" and "lose".

Alas, the page doesn't give a precise definition of "game" and "winning strategy"; without that, I couldn't really speculate. But since the article suggests the axiom of choice is needed, such games probably aren't explicitly constructible.
 
What kind of a game is not explicitly constructible?

Does that mean that if I get asked a question about a specific game on a test, I can assume that one player has a winning strategy? Can one prove that for explicitly constructable games?
 
A strategy is winning if the player following it must necessarily win, no matter what his opponent plays. (http://en.wikipedia.org/wiki/Determinacy#Winning_strategies)

Example 1: Rock, paper, scissors.

Example 2:
___________Column player__
___________Left ____ Right__
Row player:
Up.....(1, 0)...(0, 1)
Down...(0, 1)...(1, 0)

If CP plays L, RP wins by playing U, but if CP plays R, RP wins by D.
If RP plays U, CP wins by playing R, but if RP plays D, CP wins by L.
 
I don't think rock paper scissers is a game in the game theory sense.
 
ehrenfest said:
I don't think rock paper scissers is a game in the game theory sense.
Why not?
 
Okay, I guess its an example of a simultaneous game.
 

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