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THEORY OF GAMES

Strategy in game is the most significant aspect of game, since the strategy determines the

players fate in the game. It is the complete set of action that one player decides to bank on while

he/she is plating the game. This strategy aids the player in handling the situation the player is in.

the action one player takes during any stage of the game depends on the strategy. A strategy

profile is a set of fixed strategies that each player decides to employ. A strategy profile consists

on only one strategy for each player respectively.

Difference between a move and strategy

A move and a strategy differ in form though people tend to confuse between these. A move is a

single step a player can take during the game. On the other hand a strategy is a complete set of

actions, which a player takes into account while playing the game throughout. In fact a strategy

constitutes several moves. It is game plan, which the player avails during the play session.

Kinds of strategies

There are two types of strategies in the game. They are:

Pure strategy, which is a comprehensive list of the methods that will be adopted by the players.

It defines every move, weighs the pros and cons of every option before the player picks up those

moves. A players strategy space represents the complete set of moves that a player has at his/her

disposal.Mixed strategy which is a strategy that provides a probability against every pure move. It

describes a possible strategy over every pure strategy. Each pure strategy can be assigned a

probability. This strategy defines that a player rather than selecting a particular pure strategy will

select a strategy randomly from the mix and match provided by the mixed strategy. The

probabilities of 1 and 0 are normally used within the strategy.

What do you mean by zero sum game and two person zero sum games in game theory?

Zero sum game: It is the game in which the sum of payments to all the players after the play of

the game is zero.

Two person zero sum game: It is a game involving two players, in which the gain the loss player

equals the loss of other.

What is saddle point and game value in game theory?

Saddle point: Saddle point is the number, which is the lowest in its row and highest in its

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column.

Game value: It is the average winning per play over a long number of players.

Differentiate between pure and mixed strategies

Pure strategy: If a player knows exactly what the other player is going to do, a deterministic

situation is obtained and objective function is to minimize the gain. Therefore the pure strategy is

a decision rule always to select a particular course of action.

Mixed strategy: If a player is guessing as to which activity is to be selected by the other on any

particular occasion, a probabilistic situation is obtained and objective function is to maximize the

expected gain. Thus, the mixed strategy is a selection among pure strategies with fixed

probabilities.

Indicate the difference between decision making under risk and uncertainty in statistical

decision theory

Decision making under risk: In the case more than one states of nature exist and there is enough

information available to assign probability to each of the possible state.

Decision making under uncertainty: Here more than one state of exists but there is no

information about the various states, not even sufficient to assign probabilities to them.

Enumerate the various quantitative methods which are used for decision making under

risk

Decision making under uncertainty:

i. Criterion of optimism

ii. Criterion of pessimism

iii. Minimax regret criterion

iv. Criterion of realism

v. Laplace criterion

Decision making under risk:

i. Expected value criterion

ii. Expected opportunity loss criterion

iii. Expected value of perfect information

iv. Use of incremental analysis

List the steps in decision making approach

i. List all the alternatives

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ii. Identify the expected future events

iii. Construct a payoff table

iv. Select optimum criterion

Distinguish between game of strategy and game of chance

Game of strategy: If in the game the actions are determined by skills, it is called a game of

strategy.

Game of chance: If in the game the actions are determined by chance, it is termed as game of

chance.

What do you mean by optimal strategy?

Optimal strategy: The strategy that puts the player in the most preferred position irrespective of

the strategy of his opponents is called an optimal strategy. Any deviation from this strategy

would reduce his payoff.

Problem 1

Mr. Apam had to decide whether or not to drill a well on his firm. In his village, only 40% of the

wells drilled were successful at 200 feet of depth. Some of the formers who did not get water at

200 feet drilled further up to 250 feet but only 20% struck water at 250 feet. Cost of drilling is

GH50 per foot. Mr. Apam estimated that he would pay GH18000 during a 5 years period in

the present value terms, if he continues to buy water from the neighbour rather than go for well

which would have a life of five years. Mr. Apam has three decisions to make. (a) Should he drill

up to 200 feet? (b) If no water is found at 200 feet, should he drill up to 250 feet? (c) Should he

continue to buy water from his neighbor?

Solution

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Thus the optimal course of action for Mr. Apam is not to drill the well and pay GH18000 for

water to his neighbor for five years.

Problem 2

Find the value of games shown below. Also indicate whether they are fair or strictly determinable

solution.

Solution

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Saddle point = (I, IV)

Game value 0

Strategy of A = A (I)

Strategy of B = B ((IV)

Since maximum = minimax = 0

So the game = fair

Problem 3

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In a game of matching coins, player A wins GH2 if there are two heads, win nothing if there are

two tails and loses GH1 when there are one head and one tail. Determine the payoff matrix, best

strategies for each player and the value of game to A.

Solution

The payoff matrix for A will be

H T

Player A H 2 -1

T -1 0

There is no saddle point. By arithmetic method

Player A best strategy (0.25, 0.75)

Player B best strategy (0.25, 0.75)

Game value

Let B plays H; value of the game

Problem 4 (by dominance)

Two players P and Q play a game. Each of them has to choose one of three colours, white(W)

black(B) and red(R) independently of the other. There after the colours are compared. If both P

and Q have choosen white (W, W) neither win anything. If player P selects white and player Q

black (W, B), player P loses GH2 or player Q wins the same amount and so on. The complete

payoff table is shown. Find the optimum strategies for P and Q and the value of the game.

(P) W B R

W 0 -2 7

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B 2 5 6

R 3 -3 8

Solution

There is no saddle point

By dominance rule for column, 3rd column may be removed

(P)

W B

W 0 -2

B 2 5

R 3 -3

By dominance rule for row, 2nd row may be removed. The resulting matrix (2 X 2) is

(P)

W B

B 2 5

R 3 -3

Applying arithmetic method,

Optimum strategies for P (0, 6/9, 3/9)

Optimum strategies for Q (8/9, 1/9, 0)

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Problem 5

Solve the following games by reducing them to (2 x 2) games by graphical method.

Reduce by dominance rule for column 2 and 5th column may be deleted as dominated by 4 th and

3rd column. the resulting matrix is shown below. No saddle point.

Resulting matrix

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Expected payoff the two lines which intersect at lowest point of upper bound show the two

course of action A should choose in his best strategy. The resulting matrix is

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Problem 6

Solve the following game.

(B)

(A)

B1 B2 B3 B4

A1 2 1 0 -2

A2 1 0 3 3

There is no saddle point in the game. By rule of dominance for column 1st and 3rd column may

be deleted as dominated by 2nd and 4th column respectively.

Thus the resulting matrix is

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Problem 7

Obtain the optimal strategies for both persons and the value of the game for zero sum two

person game whose payoff matrix is given as follows:

Solution

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There is no saddle point m the game by rule of dominance for column 2nd, 4th and 5th column

are dominated by 1st column and 3rd column dominated by 6th column hence 2nd, 4th, 5th and3rd column may be removed. The resulting matrix is (2 x 2).