What is the formula for the value of a game in game theory?

[mathlete]

The problem:
"Player I can choose l or r at the first move in a game G. If he chooses l, a chance move selects L with probability p, or R with probability 1-p. If L is chosen, the game ends with a loss. If R is chosen, a subgame identical in structure to G is played. If player I chooses r, then a chance move selects L with probability q or R with probability 1-q. If L is chosen, the game ends in a win. If R is chosen, a subgame is played that is identical to G except that the outcomes win and loss are interchanged together with the roles of players I and II"

*whew*

Now the question is... if the value of the game is v, show that v=q+(1-q)(1-v)

Now the game tree is so complicated... I really have no idea how to get the value of the game. Is there any easy way to do this that I'm missing?

 

[Hurkyl]

I don't understand the statement of the game.

What happens when player I picks `l', and `R' gets chosen? Is it now player II's turn? Does "win" always mean a win for player I? et cetera.

If I sat down and tried to teach this game to someone else so we could play, I'd have no idea what the rules are.


Anyways, the analysis should be straightforward.

What is the expected value of the game if player I picks `l'?
What is the expected value of the game if player I picks `r'?
What is the expected value of the game if player I picks optimally?

 

[neurocomp2003]

are L,R the nodes and l,r are the branches??

 

[Geer_J]

The Possible answer is v=(1-1)(-1=1)

The problem:
"Player I can choose l or r at the first move in a game G. If he chooses l, a chance move selects L with probability p, or R with probability 1-p. If L is chosen, the game ends with a loss. If R is chosen, a subgame identical in structure to G is played. If player I chooses r, then a chance move selects L with probability q or R with probability 1-q. If L is chosen, the game ends in a win. If R is chosen, a subgame is played that is identical to G except that the outcomes win and loss are interchanged together with the roles of players I and II"

*whew*

Now the question is... if the value of the game is v, show that v=q+(1-q)(1-v)

Now the game tree is so complicated... I really have no idea how to get the value of the game. Is there any easy way to do this that I'm missing?

 

[Geer_J]

The posible answer is v=(1-1)(-1+1)