Games of pure conflict
two-person constant sum games
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.
Maximin strategy
• One way to play a game is to take a very cautious
view.
• Your payoff from any action depends on other’s
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 if the
other player always does what is worst for you
given your strategy.
Maximin in Constant Sum games
In a two-person constant sum game, if the other
player is doing his best response to your action,
he is taking the action that is worst for you,
given your action.
A) Yes
B) No
C) Maybe
Why so?
• If the sum of your payoff and your opponent’s
payoff is constant, then the larger is your
opponent’s payoff, the smaller is yours.
• So if he is choosing the highest possible payoff
for himself, given your action, he is taking the
action that is worst for you, given your action.
In a two-player constant sum game, suppose
that you know that the other player has a spy
who will find out your strategy before he
chooses his. Your best option is to use a
maximin strategy.
A) Yes
B) No
C) Maybe
Simple hide and seek
Player 2 (Seeker)
Look Upstairs
p upstairs
Hide
Player 1
(Hider)
Hide
downstairs
Look Downstairs
0,1
1,0
Is this a constant sum game?
A) Yes B) No
1,0
0, 1
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.
Clicker question
• Suppose that you are hider and you choose to
randomize, hiding 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
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? (Remember: Your payoff is 0 if he
finds you, 1 if he doesn’t.)
A) .6
B) .4
C) .5
D) .35
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?
• What is your expected payoff if he does that?
Where p is the probability you hide
upstairs:
If p>0, he will look upstairs always. He will not
find you with probability 1-p. Your expected
payoff is
p x 0 + (1-p) x 1=1-p
What if you hide upstairs with p<1/2?
• What is worst thing that seeker can do to you?
• (He’ll look downstairs for sure.)
• What is your expected payoff?
If p<1/2,
Seeker stategy that is worst for hider is
Look downstairs always.
Payoff to hider is
p x 1+(1-p) x 0=p
Maximin for hide and seek
The pessimist’s view
Maximin strategy in hide and seek
Randomize with p=1/2.
There is a Nash equilibrium where both play
p=1/2.
If each player knew that the other had a spy
who knew your strategy, both would choose to
randomize with p=1/2.
Battle of Sexes
Bob
Movie A
Movie A
Alice
Movie B
3,2
1,1
0,0
2,3
Is this a constant sum game?
A) Yes
B) No
C) Maybe
Movie B
Clicker Question
In the Battle of the Sexes game, where Alice and
Bob play only pure strategies, what is the strategy
profile in which each plays a maximin strategy?
A) Both go to A
B) Both go to B
C) Alice goes to A, Bob goes to B
D) Alice goes to B, Bob goes to A
E) There are two maximin strategy profiles.
(Both go to A and both go to B)
Maximin strategies
Bob
Movie A
Movie A
Alice
Movie B
Movie B
3,2
1,1
0,0
2,3
What would Bob do if he thinks Alice will always do what is worst for Bob?
His worst payoff if he goes to A is 0. His worst payoff if he goes to B is 1.
His maximin strategy is therefore go to B.
What is Alice’s maximin strategy?
If Alice and Bob both use maximin
strategies in this game, is the outcome
a Nash equilbrium?
A) Yes
B) No
In the Movie Choice game between Alice and
Bob, suppose that Alice knows that Bob has a a
spy who will find out Alice’s strategy before he
chooses his. Alice’s best option is to use her
maximin strategy.
A) Yes
B) No
What should Alice do if she knows that
Bob will find out her strategy before
she moves?
A) Go to A
B) Go to B
C) Use a mixed strategy
Working Some Problems
Penalty Kick
Goalkeeper
Jump Left
Jump Right
Kick Left
.3 , .7
.8, .2
Kick right
.9, .1
.5, .5
Shooter
Is this a constant sum game?
A) Yes B) No
The story
• When Goalie guesses wrong, Kicker either
scores or misses the goal.
• When Goalie guesses right, Goalie sometimes
but not always stops the ball.
• Kicker kicks harder and more accurately to the
right than to the left.
• Both players know this. (Fairly realistic in
professional soccer.)
Finding N.E. goalie strategy
• Suppose Goalie goes Left with probability p
• Payoff for Shooter from shooting Left is
.3p+.8(1-p)=.8-.5p
Payoff for Shooter from shooting Right is
.9p +.5(1-p)=.5+.4p
Shooter will randomize if .8-.5p=.5+.4p
Solve this equation to find p=1/3
Finding N.E. Shooter strategy
If Shooter goes Left with probability q:
Payoff to Goalie from jumping Left is
.7q+.1(1-q)=.1+.6q
Payoff to Goalie from jumping Right is
.2q+.5(1-q)=.5-.3q
Goalie will randomize if
.1+.6q=.5-.3q
which implies that
q=4/9.
Does this game have a Nash equilibrium in which Kicker mixes left
and right but does not kick to center?
• 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 can’t be a Nash equilibrium where
Kicker never kicks Middle. (See why?)
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?
• In a Nash equilibrium where mugger mixes
only between no gun and gun hide, mugger
must be indifferent between these strategies.
• If Victim mixes with probability p of resisting,
Expected payoff to mugger of no gun is
2p+6(1-p). Expected payoff to mugger of
gun-hide is 3p+5(1-p). These are equal when
P=1/2.
So in a mixed strategy equilibrium it must be
That expected payoff to gunman is 4.
This will be a N.E. only if x<4.
Problem 7.7, Find mixed strategy Nash equilibria
A mixed strategy N.E. strategy does
not give positive probability
To any strictly dominated strategy
c dominates a and y dominates z
Look at reduced game without these strategies
In this reduced game, b dominates d.
We are left with a game where each player has 2
Possible strategies—b and c for Player 1, x and y
For Player 2. Find a mixed strategy N.E. for this
game.
Expected Utility Theory of Choice
Under Uncertainty
• Suppose that you face random outcomes. You
assign a “utility” to each possible outcome in
such a way that your choices among uncertain
prospects are those that maximize “expected
utility”.
Expected utility Example:
Utility of money
• Suppose you have a lottery that will with
probability 1/4 win 10 million dollars and
with probability ¾ will be worthless. You get
just one chance to sell your ticket.
Would you sell it for 2.5 million dollars?
A) Yes
B) No
Question
• Suppose you have a lottery ticket that will
with probability 1/4 win 10 million dollars
and with probability ¾ will be worthless.
What is the expected value of this lottery
ticket?
Expected utility Example:
Utility of money
• Suppose you have a lottery ticket that will
with probability 1/4 win 10 million dollars
and with probability ¾ will be worthless. You
get just one chance to sell your ticket.
Would you sell it for 1 million dollars?
A) Yes
B) No
Expected utility Example:
Utility of money
• Suppose you have a lottery ticket that will
with probability 1/4 win 10 million dollars
and with probability ¾ will be worthless. You
get just one chance to sell your ticket.
Would you sell it for 500 thousand dollars?
A) Yes
B) No
Expected utility Example:
Utility of money
• Suppose you have a lottery ticket that will
with probability 1/4 win 10 million dollars
and with probability ¾ will be worthless. You
get just one chance to sell your ticket.
Would you sell it for 100 thousand dollars?
A) Yes
B) No
Expected utility Example:
Utility of money
• Suppose you have a lottery ticket that will
with probability 1/4 win 10 million dollars
and with probability ¾ will be worthless. You
get just one chance to sell your ticket.
Would you sell it for 10 thousand dollars?
A) Yes
B) No
Construct a utility scale
• Let u(10 million)=1 Let u(0)=0.
• Then ask question. How much money X for sure
would be just as good as having a ¼ chance of
winning 10 million and ¾ chance of 0?
• Then assign u(X)=(3/4)u(0)+(1/4)u(10,000,000)=
(3/4)0+(1/4)1=1/4.
Assigning utility to any income
• Lets choose a scale where u(0)=0 and u(10
million)=1.
• Take any number X. Find a probability p(X) so
that you would just be willing to pay $X for a
lottery ticket that pays 10 million with
probability p(X) and 0 with probability 1-p(x).
• Assign utility p(X) to having $X.
Field Goal or Touchdown?
• Field goal is worth 3 points.
• Touchdown is worth 7 points.
Which is better? Sure field goal or probability ½
of touchdown?
Finding the coach’s von Neumann
Morgenstern utilities
• Set utility of touchdown u(T)=1
• Set utility no score u(0)=0
The utility of a gamble in which you get a touchdown
with probability p and no score with probability 1-p is
pu(T)+(1-p)u(0).
What utility u(F) to assign to a sure field goal?
Let p* be the probability such that the coach is indifferent
between scoring a touchdown with probability p* (with
no score with prob 1-p*) and having a sure field goal.
Then u(F)=p*u(T)+(1-p*)u(0)=p*x1+(1-p*)x0=p*.
So long, for now...
Problem 10.
Each of three players is deciding between the pure
strategies go and stop. The payoff to
go is 120, where m is the number of players that
choose go, and the payoff to stop is 55 m
(which is received regardless of what the other players
do). Find all Nash equilibria in mixed strategies.
Let’s 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
40p2-120p+65=0
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 ¼, 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.
Advanced Rock-Paper-Scissors
Rock
Paper
Scissors
Rock
0,0
-1,1
2,-2
Paper
1,-1
0,0
-1,1
Scissors
-2,2
1,-1
0,0
Are there pure strategy Nash equilibria?
Is there a symmetric mixed strategy Nash equilibrium?
What is it?
Finding Mixed Strategy Nash Equilibrium
Rock
Paper
Scissors
Rock
0,0
-1,1
2,-2
Paper
1,-1
0,0
-1,1
Scissors
-2,2
1,-1
0,0
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. Let’s 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.