game theory summary
TRANSCRIPT
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What Is Game Theory? -making in situations where two or
more intelligent and rational opponents are involved under conditions of
conflict and competition.
The approach of game theory is to seek to
counter-
Basic Terminology The models in the theory of games can be classified depending upon the
following factors:
Number of Players: If a game involves only two players (competitors),
then it is called a two-person game. However, if the number of players is
more, the game is referred to as n-person game.
Payoff: Outcome of a game when different alternatives are adopted by
the competing players.
Strategy: A set of rules or alternative courses of action available to a
player in advance, by which he decides the course of action that he
should adopt.
Strategy may be of two types:
Pure strategy: If the players select the same strategy each time. In
this case each player knows exactly what the other is going to do.
Mixed strategy: When the players use a combination of strategies
and each player always kept guessing as to which course of action
is to be selected by the other player at a particular occasion.
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Optimum strategy: A course of action or play which puts the
player in the most preferred position, irrespective of the strategy
of his competitors.
Value of the Game: The expected payoff of play when all the players of
the game follow their optimum strategies.
The game is called fair if the value of the game is zero and unfair if
it is non-zero.
Payoff Matrix: The payoffs in terms of gains or losses, when players
select their particular strategies, presented in form of matrix.
Two-person Zero-sum Game: A game of two persons, in which the
gains of one player are the losses of the other player.
The algebraic sum of the gains to both the players after a play is
bound to be zero
Maximin Minimax Principle: The maximum of minimum gains is the
maximin value of the game and the corresponding strategy is the
maximin strategy. In a similar way, the minimum of maximum losses will
be called the minimax value of the game and the corresponding strategy
is the minimax strategy.
Saddle point - The position in the matrix where the maximin is
equal to the minimax.
Assumptions/ Characteristics of a Game The number of competitors called players is finite.
The players act rationally and intelligently
Each player has available to him a finite number of choices or possible
courses of action called strategies.
The number of choices need not to be the same for each player
All relevant information is known to each player in advance.
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Application of Game Theory in Real Life Situations Oligopolistic Strategy
Two Players A and B, play the coin
In Football Match
Politics
Methods to Find Value of a Game under Decision
Making Environment & Calculation Process
Methods:
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Calculation process:
Pure Strategy Games (With Saddle Point )
A pure strategy game is a game whereby the players select the same
strategy each time.
There is a deterministic situation and the objective of the player is to
maximize gains or to minimize losses.
The maximizing player arrives at his optimal strategy on the basis of the
maximin
minimax criterion.
The Saddle Point which is the solution of the game is derived at the
point where the maximin value equals the minimax value
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Example: For the following payoff matrix for Firm A, determine the optimal
strategies for both Firm A and Firm B and the value of the game (using
maximum-minimax principle)
the value of the game
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Analysis of Pure Strategy Games
Develop the payoff matrix.
maximin strategy (for the maximizing player).
Identify column maximums and select the smallest of these as the
for the minimizing player).
If the maximin value equals the minimax value, the game is a pure
strategy game and that value is the saddle point
Principle of Dominance
payoff matrix by eliminating a course
of action which is so inferior to another course of action that can be left out
of the set of choices.
The concept of dominance is especially useful for the evaluation of two-
person zero-sum games where a saddle point does not exist.
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Example (no saddle point):
After the matrix is reduced, it can now be solved further, using the mixed
strategy methods.
The rules of dominance to reduce the size of a payoff matrix:
When all elements in a row of a payoff matrix are less than or equal to
the corresponding elements in another row, then the former row is
dominated by the latter and can, therefore, be deleted from the matrix.
When all elements in a column of a payoff matrix are greater than or
equal to the corresponding elements in another column, then the
former column is dominated by the latter and can, therefore, be deleted
from the matrix.
A pure strategy may be dominated if it is inferior to average of two or
more other pure strategies.
Nash Equilibrium
It is a set of strategies such that each player believes (correctly) that it is
doing the best it can, given the strategy of the opponents.
Since the player is satisfied that he has made the best decision possible,
he has no incentive to deviate from the chosen strategy. Thus, Nash
strategies are stable.
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It is possible only if we assume that all players understand the game and
are rational.
Illustration: PAYOFF MATRIX: ADVERTISING GAME
The Nash equilibrium for this game is Advertising by both Firm A and Firm B.
Distinction between Nash Equilibrium & Dominant-Strategy Equilibrium
In dominant strategy case, each player chooses his best strategy,
irrespective of the strategies of other players.
While in the case of Nash equilibrium, each player chooses a strategy
that is his best choice, subject to what strategies the opponent chooses
Mixed Strategy Games (Games without saddle point )
In a game without saddle point, the optimal policy is to use mixed strategies.
This is the combination of strategies which keep each player guessing on
what course of action is to be selected by the other player at a particular
occasion.
It is a selection among pure strategies with fixed probabilities.
To order to solve such a game, each player adopts the concept of chance move
and starts to play in a random manner and in such a way that his average
payoff over a large number of plays of the game should be optimal, even
though he may lose more in any individual play of the game.
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A mixed strategy can be solved by the following methods:
1. Algebraic/Arithmetic Method;
2. Graphical Method;
3. Matrix Method;
4. Linear Programming Method.
Algebraic Method (2 2 Strategies Game)
If this game is to have no saddle point, the two largest elements of the
matrix must constitute one of the diagonals.
We have assumed this and therefore both players use mixed strategies.
Our task is to determine the probability with which both players choose
their course of action.
Solve the following game:
, ) and for the player B, the
) respectively. The value of the game is 13
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Arithmetic Method (2 2 Strategies Game)
It is also known as odds method or shortcut method.
Example: Reduce the following game by dominance and find the game value.
, ) and for the player B, the
) respectively. The value of the game is
Steps in calculation:
Subtract the smaller payoff in each row from the larger one and the
smaller payoff in each column from the larger one.
Interchange each of these pairs of subtracted numbers found in Step 1.
Put each of the interchanged numbers over the sum of the pair of
number
Simplify the fraction to obtain the required strategies.
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Graphical Method (2 m or n 2 Strategies Game)
2, so that it can be possible to solve using Algebraic/Arithmetic Method.
Example: Obtain the optimal strategy to BOTH player and the value of the
game for two-person zero-sum game by using graphical method whose payoff
matrix is given as follows:
Optimal strategy for player A is: (0, 0, 0, ) and optimal strategy for player B
is: ( , ) and value of game, v =
Steps in Calculation:
State the expected payoff of the player with many strategies in line with
the other, if pure strategy is selected.
These expected payoffs will be plotted against the two strategy, they are
the parallel line in which the expected payoff line is drawn in.
For the maximum value, the innermost intersect on the graph and the
two expected payoff that results to it, will be extracted and use to form
a 22 matrix, similarly in terms of minimum value, the outermost point
is extracted and the payoffs are selected.
Finally, the two payoffs can be equated to get immediate probability (p
or q as the case maybe) and also the value of the game. Hence, the 22
matrix gotten will be used to solve for the remaining payoff.