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

In summary, the problem is that there is a game where Player I can choose between two options, l or r, at the first move. If l is chosen, there is a chance move that selects L or R with probabilities p and 1-p respectively. If L is chosen, the game ends with a loss, while if R is chosen, a subgame identical to G is played. If r is chosen, there is also a chance move that selects L or R with probabilities q and 1-q respectively. If L is chosen, the game ends with a win, while if R is chosen, a subgame is played that is identical to G except that the outcomes win and loss are interchanged, along with the roles of
  • #1
mathlete
151
0
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?
 
Physics news on Phys.org
  • #2
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. :frown:


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?
 
Last edited:
  • #3
are L,R the nodes and l,r are the branches??
 
  • #4
The Possible answer is v=(1-1)(-1=1)
mathlete said:
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?
 
  • #5
The posible answer is v=(1-1)(-1+1)
 

What is game theory?

Game theory is a branch of mathematics that studies decision-making in strategic situations. It analyzes the actions of multiple individuals or agents in order to determine the best course of action for each participant.

What is the value of a game in game theory?

The value of a game in game theory is a measurement of the expected payoff or outcome for each player in a given game. It takes into account the strategies and decisions of all players involved, and is used to determine the optimal strategy for each player.

What factors influence the value of a game?

The value of a game can be influenced by a variety of factors, including the number of players, the available strategies, and the potential outcomes. Other factors may include the preferences and rationality of each player, as well as any external factors that may impact the game.

How is the value of a game calculated?

The value of a game is typically calculated using mathematical models and algorithms, such as the minimax theorem or Nash equilibrium. These methods take into account the potential payoffs for each player and determine the best course of action for each player in order to maximize their expected outcome.

What is the practical application of game theory and the value of a game?

Game theory and the value of a game have many practical applications, particularly in economics, politics, and business. They can be used to analyze and predict the behavior of individuals and groups in decision-making situations, and can inform strategies and policies in various fields.

Similar threads

  • Calculus and Beyond Homework Help
Replies
8
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
774
  • Precalculus Mathematics Homework Help
Replies
1
Views
1K
Replies
9
Views
2K
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
  • Precalculus Mathematics Homework Help
Replies
11
Views
2K
Replies
9
Views
885
  • Calculus and Beyond Homework Help
Replies
9
Views
1K
  • Special and General Relativity
Replies
8
Views
915
  • Calculus and Beyond Homework Help
Replies
1
Views
970
Back
Top