THEORY OF GAMES
1. Explain the properties found in competitive games
Properties of game theory are classified as follows:
Chance of strategy: In a game, activities are determined by skills, it is said to be a game of strategy,
if they are determined by chance, it is a game of chance.
Number of Persons: A game is called an n -person game if the number of persons playing is n .
The person means an individual or a group aiming at a particular objective.
Number of activities: These may be finite or infinite.
Number of alternatives: These may be finite or infinite.
Information to the players about the past activities of other players: Information to the players
about the past activities of other players is completely available, partly available or not available at
all.
Pay – off: A quantitative measure of satisfaction a person gets at the end of each play is called a
pay-off.
2. Write a short notes on Zero-sum games.
Any game should involve some gains and losses to the players. Accordingly we have two types,
a) Zero sum game: In such a game, the sum of gains and losses is equal to zero, i.e. the gain of
one player means the loss of the other player. In this game player make payments to each
other mathematically if n -player game is played by n -players p1 , p2 ,......., pn whose payn
offer at the end of the game are v1 , v2 ,......., vn then the game will be zero. i.e.
v
i 1
i
0
at each play of the game.
b) If the sum of gains and losses is not equal to zero, it is called non-zero sum games.
Games having the zero-sum character that the algebraic sum of gains and losses of all the
players is zero are called zero-sum games. Zero-sum games with 2 players are called rectangular
games. In this case,, the loss/gain of one player is exactly equal to the gain/loss of the other.
The gain resulting from a two-person zero-sum game can be represented in the matrix form,
usually called ‘pay-off matrix’.
3. Explain the following terms:
(i) Pure Strategy
(ii) Mixed strategy
(v) value of the game
(iii) Optimal strategy
(iv) saddle point
Strategy: A strategy refers to the action to be taken by a player in various contingencies in
playing a game.
Pure Strategy: A pure strategy is a decision, in advance of all plays, to choose a particular
course of action always, which means each player knows exactly what the other is going to do.
Mixed strategy: As mixed strategy is a decision, in advance, of all plays, to choose a course of
action for each play in accordance with some particular probability distribution. This is a
situation where a player decides in advance, to use all or some of his available courses of action
in some fixed proportions. This means the other player does not know exactly what his
opponent is going to do and he keeps guessing.
Optimal Strategy: An optimal strategy is such as it provides the best situation in the game in the
sense that it involves maximal pay off to player. Any deviation from this strategy results in a
decreased pay off for the player.
Saddle point: it is the solution or the value of the game when the players adopt pure strategy.
It is obtained by computing maximum value for row player and minimax value for column
player. If these two values coincide, then the value so obtained is called the saddle point.
Value of the game: It is the expected pay-off of the game when all the players of the game
adopt their optimal strategies.
Value of the game:
Zero  Fair game
Value of the game: non-zero  Unfair game
4. Write the limitations of game theory:
Game theory which was initially received in literature with great enthusiasm as holding
promise, has been found to have a lot of limitations. The major limitations are summarized
below,
a) The assumptions that the players have the knowledge about their own payoffs and
payoffs of others is rather unrealistic. He can only make a guess of his own and his
rival’s strategies.
b) As the number of players increase in the game, the analysis of the strategies become
increasingly complex and difficult. In practice, there are many firms in an oligopoly
situation and game theory cannot be very helpful in such situations.
c) The assumptions of maximin and minimax show that the players are risk-averse and
have complete knowledge of the strategies. These do not seem practical.
d) Rather than each player in an oligopoly situation working under uncertain conditions,
the players will allow each other to share the secrets of business in order to work out a
collusion. Then the mixed strategies are not very useful.
5. Briefly explain the general rules for dominance:
In a game situation, sometimes a strategy available to a player might be found to be preferable
than some other strategy or strategies. Such a strategy is said to dominate the other ones. This
is called domination concept. Using the principle of dominance, the size of the pay-off matrix
can be reduced. The need for reducing the pay-off matrix size is to adopt the simpler methods
of solving the rectangular game.
Once the dominant strategy is identified, the unattractive strategy are deleted and the
deleted strategies will never be used by both the players for determining their optimum
strategy. The steps involved are
a) Check for Row-dominance: A row player tries to maximize his gains. If every element
(pay-off) in a particular row is greater than or equal to the corresponding elements of
another row, then the player will prefer to play the former row. He will never choose
the latter row irrespective of the course of action of the other player. Thus, the row
with lesser values can be deleted.
b) Check for Column-dominance: A column player tries to minimize his losses. If every
element (pay-off) in a particular column is less than or equal to corresponding elements
of another column, then the player will prefer to play the former column. He will never
choose the latter column irrespective of the course of action of the other player. Thus,
the column with greater values can be deleted.
c) Check for advanced row dominance: In the first step no row could be deleted, then
compare a row with average of a group of row. If every element is less than or equal to
average of corresponding elements of a group then the former row can be deleted.
d) Check for advanced column dominance: In the second step no column could be deleted,
then compare a column with average of a group of a columns. If every element in a
column is greater than or equal to average of corresponding elements of a group then
the former row can be deleted.