Game theory:- value of a game?

Question

I haven't found any suitable explanation or even definition for this concept. What is the value of game in game theory? Can anybody explain it to me with an example.

Answer 1

The value of a game is the expected value to a given player. For example, a game where you flip a coin and win $2$ for heads and lose $1$ for tails has a value to you of $\frac 12\cdot 2 + \frac 12 \cdot (-1)=\frac 12$. If you have to pay $\frac 12$ to play the game you will break even in the long run.

Answer 2

One definition of the "value of a game" is as the nim-value or "nimber" of a game. The Sprague-Grundy Theorem says that all games (satisfying a few standard assumptions true of most combinatorial games) are equivalent to a single nimheap, i.e. they behave the same way under game addition. The number of stones in the nimheap is the "nimber," which is perhaps what the value of a game means.

More reading:

https://en.wikipedia.org/wiki/Nimber

https://mathworld.wolfram.com/Nim-Value.html

https://en.wikipedia.org/wiki/Sprague–Grundy_theorem

Answer 3

As in general game theory, the Value of game is to be the minimax of the payoff. Symbolically, $$V(x)=\min_{\phi(x)}\max_{\psi(x)}(\text{ payoff }).$$

From Rufus Isaacs - Differential Games_ A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization