# Game theory

1. Jan 13, 2008

### ehrenfest

http://en.wikipedia.org/wiki/Game_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?

2. Jan 13, 2008

### DaveC426913

Well, tic-tac-toe comes to mind.

3. Jan 13, 2008

### Hurkyl

Staff Emeritus
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.

4. Jan 13, 2008

### ehrenfest

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?

5. Jan 13, 2008

### EnumaElish

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.

6. Jan 13, 2008

### ehrenfest

I don't think rock paper scissers is a game in the game theory sense.

7. Jan 14, 2008

### EnumaElish

Why not?

8. Jan 14, 2008

### ehrenfest

Okay, I guess its an example of a simultaneous game.