Games of pure conflict two person constant sum. Two-person constant sum game Sometimes called...
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Games of pure conflict two person constant sum
Games of pure conflict two person constant sum. Two-person constant sum game Sometimes called zero-sum game. The sum of the players’ payoffs is the same,
Two-person constant sum game Sometimes called zero-sum game.
The sum of the players payoffs is the same, no matter what pair of
actions they take. In a two-person constant sum game, one players
gain is the others loss.
Slide 3
Maximin strategy One way to play a game is to take a very
cautious view. Your payoff from any action depends on others
actions. In a two-player game, you might assume other player always
does what is worst for you. Given that assumption, you would choose
the strategy such that gives you the best payoff available if the
other player always does what is worst for you given your
strategy.
Slide 4
Simple hide and seek 1,0 0, 1 0,1 1,0 Look Upstairs Look
Downstairs Player 2 (Seeker) Player 1 (Hider) Hide upstairs Hide
downstairs p Is this a constant sum game? A) Yes B) No
Slide 5
Penalty Kick.9,.1.5,.5.3,.7.8,.2 Jump Left Jump Right
Goalkeeper Shooter Kick Left Kick right Is this a constant sum
game? A) Yes B) No
Slide 6
Going to the Movies 3,2 1,1 0,0 2,3 Bob Alice Movie A Movie B
Movie A Movie B Is this a constant sum game? A)Yes B)No
C)Maybe
Slide 7
Mixed strategies and maximin Suppose you are Hider, choosing a
mixed strategy, and you believe that Seeker will do what is worst
for you, given your mixed strategy. This is not a silly assumption
in a two-player zero sum game, because what is worst for you is
best for your opponent. The maximin player will choose her best
mixed strategy given that she believes opponent will respond with
the strategy that is worst for her.
Slide 8
Clicker question Suppose that you are Hider and you choose to
hide upstairs with probability.6. What strategy by SEEKER is worst
for you? A)Look upstairs with probability.6 B)Look upstairs and
downstairs with equal probability C)Look upstairs for sure D)Look
upstairs with probability.4
Slide 9
Clicker question If you are Hider and hide upstairs with
probability.6 and Seeker uses the strategy that is worst for you,
what is your expected payoff? A).6 B).4 C).5 D).35
Slide 10
More generally If you are Hider and you hide upstairs with
probability p>1/2, what is the strategy for Seeker that is worst
for you? Look upstairs What is your expected payoff if he does
that? You win only if you hide downstairs. Probability of this is
1-p. Expected payoff is (1-p)x1+px0=1-p
Slide 11
What if you hide upstairs with p
Slide 12
Maximin for hide and seek
Slide 13
The pessimists view
Slide 14
Penalty Kick.9,.1.5,.5.3,.7.8,.2 Jump Left Jump Right
Goalkeeper Shooter Kick Left Kick right Lets look from pessimistic
shooters view
Slide 15
Shooters View
Slide 16
Clicker question If shooter randomly chooses left with
probability p>4/9, what Goalie strategy is worst for shooter
A)Jump left B)Jump right C)Jump left with same probability that
shooter shoots left D)Jump left with probability , right with
probability .
Slide 17
Clicker question If Shooter shoots left with probability p,
what is the best response for Goalie. A)Jump left with probability
p B)Jump left with probability C)Jump left for sure if p>5/9,
right if p
Slide 18
Constant sum games and Maximin Note that when shooter uses
maximin strategy, his own payoff is the same for either response by
Goalkeeper. If shooters payoff is the same from both strategies, so
is goalkeepers. (Why?) If goalkeepers strategy is same from both
strategies, goalkeeper is willing to randomize.
Slide 19
Slide 20
Clicker Question If Goalie jumps left with probability , what
strategy by Shooter is worst for Goalie? A)Shoot left B)Shoot right
C)Shoot left or right with equal probability
Slide 21
Clicker question What strategy by Goalie makes Shooter equally
well off from shooting left or right? A)Jump left with probability
B)Jump left with probability 2/3 C)Jump left with probability
1/3
Slide 22
Summing up In Maximin equilibrium: Shooter shoots to left with
probability 4/9 Goalkeeper jumps left with probability 1/3 Shooter
scores with probability.663 Goalkeeper makes save with
probability.366 Maximin is also a Nash equilibrium in zero sum
games
Slide 23
Maximin and the movies 3,2 1,1 0,0 2,3 Bob Alice Movie A Movie
B Movie A Movie B This is not a constant sum game. Maximin
equilibrium is not a Nash equilibrium.
Slide 24
Alices View
Slide 25
Maximin equilibrium Symmetric story for Bob. In maximin
equlibrium each is equally likely to go to either movie.
Slide 26
If Alice is equally likely to go to Movie A or Movie B, what is
Bobs best response? A)Randomize with probability B)Go to Movie B C)
Go to Movie A
Slide 27
Is the maximin equilibrium for Alice and Bob a Nash
equilibrium? A)Yes B)No
Slide 28
Some more Problems
Slide 29
RockPaperScissors Rock0,0-1,12,-2 Paper1,-10,0-1,1
Scissors-2,21,-10,0 Advanced Rock-Paper-Scissors Are there pure
strategy Nash equilibria? Is there a symmetric mixed strategy Nash
equilibrium? What is it?
Slide 30
RockPaperScissors Rock0,0-1,12,-2 Paper1,-10,0-1,1
Scissors-2,21,-10,0 Finding Mixed Strategy Nash Equilibrium Let
probabilities that column chooser chooses rock, paper, and scissors
be r, p, and s=1-p-r Row chooser must be indifferent between rock
and paper This tells us that -p+2(1-p-r)=r-(1-p-r) Row chooser must
also be indifferent between rock and scissors. This tells us that
p+2(1-p-r)=-2r+p We have 2 linear equations in 2 unknowns. Lets
solve. They simplify to 4r+4p=3 and 4p=2. So we have p=1/2 and
r=1/4. Then s=1-p-r=1/4.
Slide 31
Problem 7.7 Find mixed strategy Nash equilibia For player 1,
Bottom strictly dominates Top. Throw out Top Then for Player 2,
Middle weakly dominates Right. Therefore if Player 1 plays bottom
with positive probability, player 2 gives zero Probability to
Right. There is no N.E. in which Player 1 plays Bottom with zero
probability, (Why?) (If he did, what would Player 2 play? Then what
would 1 play?)
Slide 32
More mechanically Suppose player 1 goes middle with probability
m and bottom with probability 1-m. Then expected payoffs for player
2 are: 1m+3(1-m) for playing left 3m+2(1-m) for playing middle
1m+2(1-m) for playing right We see that playing right is worse than
playing middle if m>0. So lets see if there is a mixed strategy
Nash equilibrium where Player 2 plays only left and middle and
Player 1 is willing to play a mixed strategy.
Slide 33
Does this game have a Nash equilibrium in which Kicker mixes
left and right but does not kick to center?
Slide 34
If there is a Nash equilibrium where kicker never kicks middle
but mixes between left and right, Goalie will never play middle but
will mix left and right (Why?) If Goalie never plays middle but
mixes left and right, Kicker will kick middle. (Why?) So there cant
be a Nash equilibrium where Kicker never kicks Middle. (See
why?)
Slide 35
Problem 4: For what values of x is there a mixed strategy Nash
equilibrium in which the victim might resist or not resist and the
Mugger assigns zero probability to showing a gun?
Slide 36
Muggers Game If there is a Nash equilibrium in which mugger
does not show gun and both mugger and victim have mixed strategies,
it must be that the muggers payoff in this equilibrium is at least
as high as that of showing a gun.
Slide 37
Mixed strategy equilibrium with no visible gun ResistDont
resist No Gun2,66,3 Hidden Gun3,25,5 Note that there is no pure
strategy N.E. If Victim resists with probability p then Muggers
expected payoff from having no gun is 2p+6(1-p)=6-4p Muggers
expected payoff from having a hidden gun is 3p+5(1-p)=5-2p Mugger
will use a mixed strategy only if 6-4p=5-2p, which implies p=1/2.
If p=1/2, the expected payoff from not showing a gun is 4.
Slide 38
Muggers Game If mugger shows gun, he is sure to get a payoff of
x. If victims strategy is to resist with probability 1/2 if he
doesnt see a gun, then expected payoff to mugger from not showing a
gun is 3x1/2 +5x1/2=4. So there is a mixed strategy N.E. where
mugger doesnt show gun if x 4.
Slide 39
Entry N players consider entering a market. If a firm is the
only entrant its net profit is 170. If more than one enter each has
net profit 30. If a firm stays out it has net profit 60. Find a
symmetric Nash equilibrium. In symmetric N.E. each enters with same
probability p.
Slide 40
Equilibrium Let q=1-p. If a firm enters, the probability that
nobody else enters is q N-1 If nobody else enters, your profit is
170. If at least one other firm your expected profit is 10. So if
you enter, your expected profit is 170q N-1 +10(1-q N-1 ) If you
dont enter your expected profit is 60. So there is a mixed strategy
equilibrium if 170q N-1 +10(1-q N-1 )=60, which implies that 160q
N-1 =50 and q=(5/16) 1/N-1 Then p=1-q=1-(5/16) 1/N-1
Slide 41
Saddam and UN (Lets Pretend Saddam had WMDs) Part a) Saddam is
hiding WMDs in location X, Y, or Z. UN can look either in X AND Y
or in Z. All Saddam cares about is hiding. All UN cares about if
finding. This reduces to a simple hide and seek game. Only trick:
Saddam has more than 1 N.E. mixed strategy
Slide 42
Saddam and UN Part b) Saddam is hiding WMDs in location X, Y,
or Z. UN can look in any two of these places. Think of UNs strategy
as where not to look. In N.E. probability of each strategy will be
equal. (Why?) Also in N.E. Saddams strategy of hiding missiles in
each place is the same. (Why?)
Slide 43
See you on Thursday
Slide 44
Hints on some more problems from Chapter 7
Slide 45
Problem 9. Each of 3 players is deciding between the pure
strategies go and stop. The payoff to go is 120/m, where m is the
number of players that choose go, and the payoff to stop is 55
(which is received regardless of what the other players do). Find
all Nash equilibria in mixed strategies. Lets find the easy ones.
Are there any symmetric pure strategy equilibria? How about
asymmetric pure strategy equilibria? How about symmetric mixed
strategy equilibrium? Solve 40p^2+60*2p(1-p)+120(1-p) 2 =55 40p 2
-120p+65=0
Slide 46
What about equilibria where one guy is in for sure and other
two enter with identical mixed strategies? For mixed strategy guys
who both Enter with probability p, expected payoff from entering is
(120/3)p+(120/2)(1-p). They are indifferent about entering or not
if 40p+60(1-p)=55. This happens when p=1/4. This will be an
equilibrium if when the other two guys enter with Probability , the
remaining guy is better off entering than not. Payoff to guy who
enters for sure is: 40*(1/16)+60*(3/8)+120*(9/16)=92.5>55.
Slide 47
Problem 7.7, Find mixed strategy Nash equilibria
Slide 48
c dominates a and y dominates z A mixed strategy N.E. strategy
does not give positive probability To any strictly dominated
strategy Look at reduced game without these strategies
Slide 49
A Nash equilibrium is any strategy pair in which the defense
defends against the outside run with probability.5 and the offense
runs up the middle with probability.75. No matter what the defense
does, The offense gets the same payoff from wide left or wide
right, So any probabilities pwl and pwr such that pwl+pwr=.25 will
be N.E. probabilities for the offense. Problem 8, Chapter 7