Game Theory Flashcards
a body of knowledge that deals with making decisions when two or more intelligent and rational opponents are involved under conditions of conflict or competition.
Game Theory
the game theory started by
John Von Neumann & Mongenstern
a competitive situation where there are structured rules and regulations and end in a victory of one
Game
The GAME referred to in this chapter is the
Competition between two business organizations
Finite number of competitors are called
Players
It is played when each player chooses one of his courses or actions.
Play
two players (persons) namely A and B play the game.
Two-finger morra
Based on the name itself, it is a game with 2 numbers of players.
Two-person game
also known as ‘single strategy’ is one of the most preferred position for each player to achieve and it is the best strategy for each player to play one strategy throughout the game.
Pure Strategy Game
is when a player decides in advance to choose one of his courses of action
in accordance with some fixed probability distribution. This strategy associates probability to each course of action or also called ‘pure strategy’.
Mixed Strategy Game
Find the _____. If the game has a _____, the game is solved. Indicate the optimal strategies and the values of the game.
Step 1: Saddle Point
Find the _____. If the game has a _____, the game is solved. Indicate the optimal strategies and the values of the game.
Step 1: Saddle Point
If no _____, try to _____ the size of the matrix given (m x n) to:
1) 2 x 2 matrix, which has a formula for optimal strategies and the value of the game. Use the formula to get the answer.
2) 3 x2 or 2 x 3 matrix and use the sub game method to get the answer. (The sub games are once again 2 x 2 games).
3) To m x 2 or 2 x n matrix and use a graphical method to get a solution. Graphical solution will give us a way to 2 x 2 matrix.
Step 2: No Saddle Point, Reduce the size of the Given Matrix
If no _____, try to _____ the size of the matrix given (m x n) to:
1) 2 x 2 matrix, which has a formula for optimal strategies and the value of the game. Use the formula to get the answer.
2) 3 x2 or 2 x 3 matrix and use the sub game method to get the answer. (The sub games are once again 2 x 2 games).
3) To m x 2 or 2 x n matrix and use a graphical method to get a solution. Graphical solution will give us a way to 2 x 2 matrix.
Step 2: No Saddle Point, Reduce the size of the Given Matrix
Step 3. Use an _____ _____to get the solution.
Step 4. Use _____-_____ approach to get the solution. Use the simplex method to get solution. (Duality principle in _____-_____ is used).
Step 5. Use the _____ or _____ method to get the solution.
Step 3: Algebraic Method
Step 4: Linear Programming
Step 5: Iteration or Approximate
Is called a game with saddle point. This entails that the players of the game always use pure strategy. The element at the intersection of their
pure strategy is known as _____ _____ Another name given to the _____ _____ is _____ _____ of the game and the corresponding strategies form the equilibrium pair of strategies.
Maximin (a^ij)=minimax (a^aj)
Saddle Point, Saddle Point
Equilibrium Point
Steps in finding the saddle point
- Select the minimums of each row and encircle them.
- Select the maximums of each column and square them.
- A point where both circle and square appear in the matrix at the same point is the saddle point.
The general rules of dominance
- If all the elements of a column (say 1st column) are greater than or equal to the corresponding elements of any other column (say 2nd column), then 1st column is dominated by 2nd column.
- If all the elements of 1st row are less than or equal to the corresponding elements of any other
row, say 2nd row, then 1st row is dominated by 2nd row. - A pure strategy of a player may also be dominated if it is inferior to some convex combinations of
two or more pure strategies, as a particular case, inferior to the averages of two or more pure strategies.
SOLUTIONS TO 2 X 2 GAMES
WITHOUT SADDLE POINT
(MIXED STRATEGIES)
In such cases, the best strategies are the mixed strategies. So in dealing with mixed strategies, we have to determine the probobilities with which each action should be selected.
- If one of the player adheres to his optimal mixed strategy and the other player deviates from his optimal strategy, then the deviating player can only decrease his yield and cannot increase in any case (at most may be equal).
- If one of the players adheres to is optimal strategy, then the value of the game does not alter if the opponent uses his supporting strategies only either singly or in any combination.
- If we add (or subtract) a fixed number say 1, to (from) each elements of the payoff matrix, then the optimal strategies
remain unchanged while the value of the game increases (or decreases) by 1.
When the given pay off matrix of a game cannot be reduced to 2 × 2, or 2 × 3 or m × 2 or 2 × n, then, we can solve the game by using the _____ _____. This is a straightforward and lengthy and time-consuming method. Here we have to write a system of inequalities and consider them as equations and solve the simultaneous equations as usual
Algebraic Method
One of the methods of determining the approximate solution is the method of _____. The principle of the approximate method is:
- The two players are supposed to play the game iteratively and at each play the players choose a strategy which is best to himself or say worst to the opponent, in view of what the opponent has done up to the iteration.
Iterative Method (For Approximate Solution)
If there is no saddle point, find the payoff (P) of A using his first strategy with a probability (x) and his second probability (1-x), when B plays his multiple strategies.
- Make a graph
- Substitute the x = 0 and x = 1
to the payoffs
- Find the optimal strategy
Graphical Method
It is a type of business game. Where a company uses a similar strategy to motovate the potential market to prefer the product of their company to fulfill their objective or win over the opponent.
Competitive Strategies
It started in 20th century but ____ ___ _______ and ________ have mathematically dealt with the theory and published the paper “_____ __ _____ and ________ ______” in 1944. The approach that they used is the ________ ________ which involves the fundamental idea of minimization of the maximum losses.
John Von Neumann & Morgenstern
Theory of Games & Economic Behaviour
Minimax Principle
Every play or combination of course of action is associated with an outcome which is called
Payoff
It is the maximum guaranteed gain to the player. It is denoted by “v”.
Value of the Game
It is the smallest element in the row and greatest element in the column.
Saddle Point
