Consider the following game:
Player B
IÂ Â Â Â Â Â Â Â Â Â II
| a11 | a12
|
| a21 | a22
|
I
Player A
II
Let us suppose this game does not have any saddle points, we assume that both the players use mixed strategies.
If player A choose strategy I with the probability p and strategy II with the probability 1 – p. For player B, if he chooses strategy I, then expected gain to the player A will be given by,
a11p + a21 (1 – p)
When player B chooses strategy II, expected gain to the player A will be given by,
a12p + a22 (1 – p)
Considering the optimal plan for player A; he requires to have equal expected gain for each strategies adopted by player B. In this case, we get-
a11p + a21 (1 – p) = a12p + a22 (1 – p)
p = a22 – a21/ (a11 + a22) – (a12 + a21)
In the similar way, if player B chooses strategy I with the probability q along with strategy with the probability (1 – q), the expected losses will be given by,
For player B:
a11q + a21 (1 – q) and a12q + a22 (1 – q)
Equating the expected losses, we get-
q = a22 – a21/ (a11 + a22) – (a12 + a21)
Now, value of the game, v can be calculated by substituting the value of p in any one of the above equations. On simplification, we get-
v = a11a22 – a12a21/ (a11 + a22) – (a12 + a21)
Links of Previous Main Topic:-
- Introduction to statistics
- Knowledge of central tendency or location
- Definition of dispersion
- Moments
- Bivariate distribution
- Theorem of total probability addition theorem
- Random variable
- Binomial distribution
- What is sampling
- Estimation
- Statistical hypothesis and related terms
- Analysis of variance introduction
- Definition of stochastic process
- Introduction operations research
- Introduction and mathematical formulation in transportation problems
- Introduction and mathematical formulation
- Queuing theory introduction
- Inventory control introduction
- Simulation introduction
- Time calculations in network
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