11. Matrix Games - Concepts of Pure and Mixed Strategies
In games of perfect information, it is clear that unless a player is
indifferent between two actions, then he/she should never
randomise.
However, in games of imperfect information (e.g. when moves are
made simultaneously like rock-scissors-paper) it is clear that one
player may not want the other to ”guess” which action he/she is
going to take.
In such cases individuals choose the action they take at random,
i.e. they use a mixed strategy.
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Concepts of Pure and Mixed Strategies
The matrix form of the rock-scissors-paper game is:
R
S
P
R
(0,0)
(-1,1)
(1,-1)
S
(1,-1)
(0,0)
(-1,1)
P
(-1,1)
(1,-1)
(0,0)
Intuitively, at equilibrium both players choose each action with a
probability of 1/3.
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Concepts of Pure and Mixed Strategies
If a player always chooses the same action in a matrix game, then
he/she is using a so called pure strategy.
It is normally assumed that players choose their actions
independently of each other (i.e. there is no communication).
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Concepts of Pure and Mixed Strategies
Suppose that in the rock-scissors-paper game, Player 1 plays rock,
scissors and paper with probability pR , pS and pP , respectively.
Player 2 plays rock, scissors and paper with probability qR , qS and
qP . The probability distribution over the set of strategy pairs is
R
S
P
R
pR qR
pS qR
pP qR
S
pR qS
pS qS
pP qS
P
pR qP
pS qP
pP qP
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Concepts of Pure and Mixed Strategies
When players use mixed strategies, the expected rewards of the
players can be calculated by taking expectations with respect to
the probability distribution over the set of strategy pairs. Hence,
R1 (M1 , M2 )=pR qR R1 (R, R) + pR qS R1 (R, S) + pR qP R1 (R, P) +
+pS qR R1 (S, R) + pS qS R1 (S, S) + pS qP R1 (S, P) +
+pP qR R1 (P, R) + pP qS R1 (P, S) + pP qP R1 (P, P)
=pR qS + pS qP + pP qR − pS qR − pP qS − pR qP .
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Concepts of Solutions to 2-Player Matrix Games - Pure
Nash equilibria
A pair of actions (A∗ , B ∗ ) is a pure Nash equilibrium if
R1 (A∗ , B ∗ ) ≥ R1 (A, B ∗ );
R2 (A∗ , B ∗ ) ≥ R2 (A∗ , B).
for any action A available to Player 1 and any action B available
to Player 2.
That is to say that a pair of actions is a Nash equilibrium if neither
player can gain by unilaterally changing their action (i.e. changing
their action whilst the other player does not change their action).
The value of the game corresponding to an equilibrium is the
vector of expected payoffs obtained by the players.
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Concepts of Solutions to 2-Player Matrix Games - Strong
Nash equilibria
A pair of actions (A∗ , B ∗ ) is a strong Nash equilibrium if
R1 (A∗ , B ∗ ) > R1 (A, B ∗ );
R2 (A∗ , B ∗ ) > R2 (A∗ , B).
for any action A 6= A∗ available to Player 1 and any action B 6= B ∗
available to Player 2.
i.e. a pair of actions is a strong Nash equilibrium if both players
would lose by unilaterally changing their action.
Any Nash equilibrium that is not strong is called weak.
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Concepts of Solutions to 2-Player Matrix Games - Mixed
Nash equilibria
A pair of mixed strategies, denoted (M1 , M2 ) is a mixed Nash
equilibrium if
R1 (M1 , M2 ) ≥ R1 (A, M2 ); R2 (M1 , M2 ) ≥ R2 (M1 , B)
for any action (pure strategy) A available to Player 1 and any
action B available to Player 2.
It should be noted that the expected reward Player 1 obtains when
she plays a mixed strategy M against M2 is a weighted average of
the expected rewards of playing his pure actions against M2 , where
the weights correspond to the probability of playing each action.
It follows that player 1 cannot do better against M2 than by using
the best pure action against M2 , i.e. if Player 1 cannot gain by
switching to a pure strategy, then she cannot gain by switching to
any mixed strategy.
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The Bishop-Cannings Theorem
The support of a mixed strategy M1 is the set of actions that are
played with a positive probability under M1 .
Suppose (M1 , M2 ) is a Nash equilibrium pair of mixed strategies
and the support of M1 is S. We have
R1 (A, M2 ) = R1 (M1 , M2 ), ∀A ∈ S; R1 (B, M2 ) < R1 (M1 , M2 ), ∀B ∈
/ S.
This is intuitively clear, since if a player uses actions A1 and A2
under a mixed strategy, then at equilibrium these actions must give
the same expected reward, otherwise one action would be preferred
over the other.
It follows that at such an equilibrium all actions in the support of
M1 must give the same expected reward, which thus has to be
equal to the expected reward of using M1 (which is calculated as a
weighted average). Thus all mixed Nash equilibria are weak.
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Nash Equilibria - Results
Every matrix game has at least one Nash equilibrium.
If there is a unique pure Nash equilibrium of a 2 × 2 game (i.e. a
game in which both players have just 2 possible actions), then that
is the only Nash equilibrium.
If there are no or two strong Nash equilibria in such a game, then
there is always a mixed Nash equilibrium.
Mixed Nash equilibria can be found using the Bishop-Cannings
theorem.
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Symmetric Games
A game is symmetric if
1. Players all choose from the same set of actions.
2. R1 (A, B) = R2 (B, A).
Note that the symmetric Hawk-Dove game satisfies these
conditions.
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Symmetric Games
If (A, B) is a pure Nash equilibrium in a symmetric game, then
(B, A) is also a pure Nash equilibrium.
At a mixed Nash equilibrium of a symmetric 2 × 2 game, the
players use the same strategy as each other.
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Nash Equilibria - Example 11.1
Derive all the Nash equilibria and values of the following game
H
D
H
(-2,-3)
(0,4)
D
(4,0)
(3,1)
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Example 11.1
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Example 11.1
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Example 11.1
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Advantages of the Concept of Nash Equilibria
1. The Nash equilibrium concept makes the reasonable
assumption that both players wish to maximise their
own reward.
2. When using the concept of Nash equilibrium it is
normally assumed that players know the payoff
functions of their opponents. However, in order to
find a pure Nash equilibrium, it is only necessary to
be able to order the preferences of opponents, not
their actual payoffs.
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Disadvantages of the Concept of Nash Equilibria
1. There may be multiple equilibria of a game, so the
concept of Nash equilibrium should be strengthened
in order to make predictions in such situations.
2. It is assumed that the payoff functions of opponents
are known. This information is necessary to derive a
mixed Nash equilibrium.
3. The concept of a Nash equilibrium requires that a
player maximises his/her reward given the behaviour
of opponents. However, this behaviour is not known
a priori.
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Actions Dominated by Pure Strategies
Suppose that by taking action Ai Player 1 always gets at least the
same reward as by playing Aj , regardless of the action taken by
Player 2, and for at least one action of Player 2 she obtains a
greater reward. Action Ai of Player 1 is said to dominate action Aj .
i.e. action Ai of Player 1 dominates Aj if
R1 (Ai , Bk ) ≥ R1 (Aj , Bk ), k = 1, 2, . . . , n
and for some k0 , R1 (Ai , Bk0 ) > R1 (Aj , Bk0 ).
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Actions Dominated by Pure Strategies
Similarly, action Bi of Player 2 dominates action Bj if
R2 (Ak , Bi ) ≥ R2 (Ak , Bj ), k = 1, 2, . . . , m
and for some k0
R2 (Ak0 , Bi ) > R2 (Ak0 , Bj ).
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Successive removal of dominated actions
It is clear that an individual should not use a dominated action.
Hence, we may remove such actions from the payoff matrix
without changing the set of Nash equilibria.
It should be noted that an action that was not previously
dominated may become dominated after the removal of dominated
strategies.
Hence, we continue removing dominated strategies until there are
no dominated strategies left in the reduced game (see tutorial and
following example).
It should be noted that if an action is strictly the best response to
some action of the other player, then it cannot be dominated (until
the game is reduced in some way).
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Payoff Dominant Nash Equilibria
A payoff vector (v1 , v2 ) is said to Pareto dominate payoff vector
(x1 , x2 ) if v1 ≥ x1 , v2 ≥ x2 and inequality is strict in at least one of
the cases.
That is to say, Nash Equilibrium 1 of a game Pareto dominates
Nash Equilibrium 2 if no player prefers Equilibrium 2 to Equilibrium
1 and at least one player prefers Equilibrium 1.
A Nash equilibrium is payoff dominant if the value of the game
corresponding to this equilibrium Pareto dominates all the values
of the game corresponding to other equilibria.
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Example
Consider the following game
A
B
A
(4,4)
(0,0)
B
(0,0)
(2,2)
Such a game is called a coordination game, as both players would
like to take the same action.
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Example
There are 3 Nash equilibria
1. (A, A) - Value (4,4).
2. (B, B) - Value (2,2).
3. (1/3A + 2/3B, 1/3A + 2/3B) - Value ( 34 , 43 ).
The first equilibrium Pareto dominates the other two equilibria,
whilst the second equilibrium Pareto dominates the third.
(A, A) is the payoff dominant equilibrium.
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Risk Dominance
Suppose that there are two pure Nash equilibria (A1 , A2 ) and
(B1 , B2 ) and a mixed Nash equilibrium
(pA1 + (1 − p)B1 , qA2 + (1 − q)B2 ).
Then the risk associated with A1 and B1 (Player 1’s actions) are
defined to be q and 1 − q (probabilities associated with mixed
strategy of Player 2), respectively.
The risk associated with A2 and B2 (Player 2’s actions) are defined
to be p and 1 − p (probabilities associated with mixed strategy of
Player 1), respectively.
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Risk Dominance
It should be noted that for Player 1 to prefer A1 to B1 , then Player
2 has to play A2 with probability at least q.
Hence, The higher q, the less sure Player 1 should be about
playing A1 .
Thus q can be used a measure of the risk of playing A1 .
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Risk Dominance
The Nash equilibrium (A1 , A2 ) risk dominates the Nash equilibrium
(B1 , B2 ) when q ≤ 0.5 and p ≤ 0.5, and a least one of the
inequalities q < 0.5 or p < 0.5 holds.
i.e. the risk associated with A1 is less than the risk associated with
B1 and the risk associated with A2 is less than the risk associated
with B2 .
Similarly, the Nash equilibrium (B1 , B2 ) risk dominates the Nash
equilibrium (A1 , A2 ) when q ≥ 0.5 and p ≥ 0.5, and a least one of
the inequalities q > 0.5 or p > 0.5 holds.
If a Nash equilibrium (A1 , A2 ) both payoff dominates and risk
dominates Nash equilibrium (B1 , B2 ), then it is expected that
rational players will play the Nash equilibrium (A1 , A2 ).
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Example 11.2
Consider the following game
A
B
C
A
(4,4)
(0,-10)
(-2,5)
B
(-10,0)
(2,2)
(1,4)
C
(5,-2)
(4,1)
(7,2)
i) Remove dominated strategies from this game and derive the
Nash equilibria of this game.
ii) What is the payoff dominant equilibrium? Is there a risk
dominant pure equilibrium?
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Example 11.2
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Example 11.2
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Example 11.2
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