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Game Theory Strategies Study Material
Typology: Study notes
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Consider the matching pennies game:
get exploited.should randomize (or mix) between strategies so that we do notwe play this game, we should be “unpredictable.” That is, weThere is no (pure strategy) Nash equilibrium in this game. If
chance) can get an expected payoff of 0.75with 25% chance). Then Player 2 by choosing Tails (with 100%Heads and .25 Tails (that is, Heads with 75% chance and TailsBut not any randomness will do: Suppose Player 1 plays.
original mixed stategy.wants to play Tails (with 100% chance) deviating from the0.5. But that cannot happen at equilibrium since Player 1 then
at Since this game is completely symmetric it is easy to see that (^) mixed strategy Nash equilibrium
(^) both players will
choose Heads with 50% chance and Tails with 50% chance.
In this case the expected payoff to both players is 0.
(-1) = 0 and neither can do better by deviating to another
strategy (regardless it is a mixed strategy or not).
equilibrium.In general there is no guarantee that mixing will be 50-50 at
increase his performance by mixing Forehands and Backhands.How can the Server do better than that? The Server can
50-50). Then the Receiver’s payoff ischance and Backhands with 50% chance (or simply mixesFor example suppose the Server aims Forehand with 50%
20 = 55 if she moves Forehands and
60 = 45 if she moves Backhands.
already an improvement for the Server’s performance.will be 45. (Note that the payoffs add up to 100). This ispayoff will be 55. Therefore if the Server mixes 50-50 his payoffSince it is better to move Forehands, she will do that and her
can he get the best performance?The next step is searching for the best mix for the Server. How
Backhands with 1-q probability. Then the Receiver’s payoff isSuppose the Server aims Forehands with q probability and
× 90 + (1-q)
20 = 20 + 70q if she moves Forehands and
× 30 + (1-q)
60 = 60 - 30q if she moves Backhands.
(^) 60 - 30q,
Backhands if 20 + 70q
(^) 60 - 30q, and
either one if 20 + 70q = 60 - 30q.
60-30q.That is the Receiver’s payoff is the larger of 20+70q and
Payoff to Receiver
q
Moving Backhands: 60-30q
Moving Forehands: 20+70q
Next let’s carry out a similar analysis for the Receiver.
side.If the Receiver does not mix, then the Server will aim the other
Then her payoff isSuppose the Receiver moves Forehands with p probability.
× 90 + (1-p)
30 = 30 + 60p if the Server aims Forehands
and
× 20 + (1-p)
60 = 60 - 40p if the Server aims Backhands.
(^) 60 - 40p,
Backhands if 30 + 60p
(^) 60 - 40p, and
either one if 30 + 60p = 60 - 40p.
Payoff to Receiver
p
Aiming Backhands: 60-40p
Aiming Forehands: 30+60p
Important Observation:
If a player is using a mixed strategy at
strategy Nash equilibrium in 2from the strategies he/she is mixing. We can easily find the mixedequilibrium, then he/she should have the same expected payoff
(^) 2 games using this observation.
from playing D is q+4(1-q).4 with probability (1-q). Therefore her expected payoff E(D)If she plays D she’ll receive a payoff of 1 with probability q and
E expected payoffs are the same:She’ll mix between the two strategies only if these two ( U (^) ) =
q (^) +(
q ) =
(^) q (^) +4(
q ) (^) ⇒
(^4) q (^) = 3
(^) q (^) = 3
q=3/4. Therefore Player 1 will mix between the two strategies only if
from playing R is 2p+(-1)(1-p).-1 with probability (1-p). Therefore her expected payoff E(R)If she plays R she’ll receive a payoff of 2 with probability p and
E expected payoffs are same:She’ll mix between the two strategies only if these two ( L ) =
(^) E (R ) (^) ⇒ −
p +(
p ) = 2
p − ( − p ) (^) ⇒
(^7) p (^) = 2
(^) p (^) = 2
p=2/7. Therefore Player 2 will mix between the two strategies only if
Example:
There can be mixed strategy Nash equilibrium even if
there are pure strategy Nash equilibria.
q Player 2
(1-q)
Player 1
p
(1-p)
between their two strategies: At the mixed Nash equilibrium Both players should be indifferent
Player 1:
q (^) = 1
(^) q (^) ⇒
(^4) q (^) = 1
(^) q (^) = 1
Player 2:
(^) p (^) = 3
(^) p ) (^) ⇒
(^4) p (^) = 3
(^) p (^) = 3
(1/4L+3/4R) at mixed strategy Nash equilibrium.Therefore Player 1 plays (3/4U+1/4D) and Player 2 plays
mix between only U and D.the mixed strategy (0.5U+0.5D) and therefore Player 1 canFor example in the following game strategy M is dominated byeven if they are dominated by another mixed strategy.Dominated strategies are never used in mixed Nash equilibria,
L Player 2
Player 1
game:equivalent to finding the mixed Nash equilibria of the following In other words finding its mixed strategy Nash equilibria is
finding the mixed Nash equilibria of the following 2 by 2 game:Therefore we can find its mixed Nash equilibria by simply
Player 2 l
r
Player 1
