Download Decision-making Situations

CHAPTER 14
Markov Chains
Chapter 15
Theory of Games
• Deals with decision-making in situations where
 There are two or more rational players
 Who all have a set of strategies each
 Who are involved in conditions of competition and
conflicting interests, and
 They are all aware of the pay-offs resulting from the
play of various combinations of strategies by different
players
• The solution to a game calls for determining
optimal strategies for the players to play
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Two-person zero-sum Games
 There are two players
 Each player has a finite number of strategies to
play
 Conditional pay-offs resulting from play of
various combinations of strategies are known
 Each pay-off is a gain for one player and loss for
the other
 The solution calls for determining optimal
strategies for each of the players, whether pure
or mixed, and the resulting value of game
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Solution to Games
To check if a saddle point exists:
Find minimum pay-off in each row
Select the largest of the minimum pay-offs
This is maximin strategy of the maximising player
Find maximum pay-off in each column
Select the smallest of these pay-offs
This is minimax strategy of the minimising player
If the maximin and minimax strategies have
same pay-offs, the game has a saddle point
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Solution to Games
If a game has a saddle point, the maximin and
minimax strategies involved are optimal
strategies for the players and these are called
pure strategies
A game can have more than one saddle point,
resulting in multiple optimal strategies
If the game has no saddle point, the players have
to play mixed strategies
If value of the game, v= 0, it is called a fair game
If v > 0, the game favours maximising player and
if v < 0, it favours minimising player
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Solution to Games
with no Saddle Point
• If it is a 2 × 2 game
Player B
b1
b2
Player
a1
a11
a12
A
a2
a21
a22
If A plays a1 with probability x and a2 with probability 1-x, and B plays
b1 with probability y and b2 with probability 1-y, then
a22 – a21
a22 – a12
x = ----------------------------- y = --------------------------(a11 + a22) – (a21 + a12)
(a11 + a22) – (a21 + a12)
and
(a11 × a22) – (a21 × a12)
v = ----------------------------(a11 + a22) – (a21 + a12)
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Solution to Games
with no Saddle Point
• If it is a 2×n game or an m×2 game
 Plot expected pay-off of each strategy on a graph
 Locate the highest point in the lower envelop (in
case of a 2×n game) and the lowest point in the
upper envelop (in case of an m×2 game)
 Consider the pair of lines whose intersection
yields the highest/lowest point and use the
strategies represented by it
 This reduces the game to a 2×2 game and it is
solved accordingly
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Solution to Games
with no Saddle Point
• If a game is of the order m × n:
Attempt to reduce the order of the problem by
applying dominance rule
If it can be reduced to a 2×2 game, solve it
accordingly
If it can not be reduced to a 2×2 game, solve it
as an LPP
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Rule of Dominance
 If a strategy is inferior to another, it is said to be
dominated
 A dominated strategy can be deleted
 If each value in a row (say R1) is greater than, or
equal to, the corresponding value in another row
(say R2), then R1 dominates R2
 If each value in a column (say C1) is smaller than, or
equal to, the corresponding value in another column
(say C2), then C1 dominates C2
 A linear combination of two strategies [for example,
αR1 + (1 – α) R2 ] may also dominate a strategy
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LP Solution to Games
Games can be formulated and solved as
LPPs
The solution is obtained by simplex and
uses the concept of duality
Before formulation, make sure that all
pay-offs are non-negative
In case of negative values, add such a
constant to all values that leaves all values
non-negative
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LP Formulation of Games
• Sample formulation of a 3×3 game from
maximising player’s point of view
Minimise 1/U = X1 + X2 +X3
Subject to
a11X1 + a21X2 + a31X3 ≥ 1
a12X1 + a22X2 + a32X3 ≥ 1
a13X1 + a23X2 + a33X3 ≥ 1
X1, X2, X3 ≥ 0
where Xi = xi/U; xi is the probability that player
plays ith strategy and U is the value of game
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