Daniel Ambach, Patrick Vetter, Matthias Wenzel
Matching Pennies
Set-Up
p
Player A
Pla
ayer B
heads
heads
tails
1
tails
-1
-1
-1
1
1
1
1
-1
Solution - in mixed strategies
g
Play
yer B
Player
y A
he d
heads
tails
heads
tails
q
1q
1-q
p
qp
-(1-q)p
(1 )
1-p
-(1-p)q
(1-q)(1-p)
Solution - in mixed strategies
g
• Calculate the solution in mixed strategies
ua ( p, q ) = −ub ( p, q )
ua = qp − q (1 − p ) − p (1 − q ) + (1 − q )(1 − p )
ua = −(2 + 4q ) p + 1 − 2q
Choice of q if player a
take a certain value of
p into account
1
1
1
p=
p<
p>
2
2
2
q = 1 q ∈[ 0,1] q = 0
Examples
for matching
p
g pennies
p
1. Rock, Paper, Scissors
Plaayer 1
1
Player 2
"rock breaks scissors,
scissors cut paper
paper, and
paper covers rock"
Rock
Paper Scissors
Rock
[0, 0]
[0 0]
[‐1, 1]
[ 1 1]
[1, ‐1]
[1 1]
Paper
[1, ‐1]
[0, 0]
[‐1, 1]
Scissors [‐1, 1]
[1, ‐1]
[0, 0]
•Three-dimensional equivalent of
Matching pennies
•Equilibrium in mixed strategies with
equal probabilities
Examples
for matching
p
g pennies
p
2. Penalty Kick
Scorer vs. Goalkeeper
Sccorer ((P1)
Keeper (P2)
jjumps p jumps j p
left
right
shoots (1 ‐1)
(1, ‐1)
left
shoots ((‐1, 1))
right
(‐1 1)
(‐1, 1)
((1, ‐1))
•No Nash-equilibrium
in pure
strategies
q
p
g
•Equilibrium in mixed strategies with p*=0,5 and q*0,5
Application
pp
in Reality
y
Why did that guy then need a „„Cheat
Cheat Sheet“
for a zerozero-sumsum-g
game ?!!!
• Lehmann knew the preferences
of some players
Lehmann at the WC 2006
with a „Cheat Sheet“
• People are not good in creating
randomness, especially in
p
g
repeated
games
Daniel Ambach, Patrick Vetter, Matthias Wenzel
Thank you for your attention!