PS 172 Week 2 Notes; 04/09/2008
PS30 Material Review—Part 2
MSNE with more than 2 Strategies
Player 1
Player 2
X
Y
Z
A
0,
5
10^,
5
5,
10*
B
0,
10*
15^,
0
0,
5
C
5,
10
0,
15*
10^,
0
No PSNE found. Let’s try to find the MSNE. Remember the cardinal rule:
your payoff with
the other player’s probability
. This is because what you get depends on the fact that the
other player is going to play that strategy. Now, we attach probability values with each
strategy.
Player 1
Player 2
X
(q1)
Y
(q2)
Z
(1q1q2)
A
(p1)
0,
5
10^,
5
5,
10*
B
(p2)
0,
10*
15^,
0
0,
5
C
(1p1p2)
5,
10
0,
15*
10^,
0
Now we calculate the expected utility of each player’s strategy:
EU(1A) = 0(q1) + 10(q2) + 5(1q1q2)
= 10q2 + 5 – 5q1 – 5q2
= 5 – 5q1 + 5q2
EU(1B) = 0(q1) + 15(q2) + 0(1q1q2)
= 15q2
EU(1C) = 5(q1) + 0(q2) + 10(1q1q2)
= 10 – 5q1 – 10q2
Now we have 3 equations with 2 unknowns. We can solve for the probability values by first
equating 2 expected utilities together:
EU(1A) = EU(1C)
5 – 5q1 + 5q2 = 10 – 5q1 – 10q2
15q2 = 5
q2 = 1/3
The reason why I equate 1A and 1C is so that I can cancel as many terms as possible (in this
case, a variable). Note that variables don’t always cancel out this nicely though! Now we can
solve for q1:
EU(1A) = EU(1B)
5 – 5q1 + 5(1/3) = 15(1/3)
5q1 = 5/3
q1 = 1/3
We can now solve for 1 q1 – q2. Do the same for p1, p2, and 1p1p2 for exercise.
1
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Pareto Efficiency in MSNE
Multiple PSNE:
I
II
A (q)
B (1q)
A (p)
5*, 5*
0, 3
B (1p)
3, 0
1*, 1*
MSNE:
For player 1: EU (A) = 5q + 0(1q)
EU (B) = 3q + 1(1q)
5q = 3q + 1(1q)
= 3q + 1 – q
= 2q + 1
3q = 1
q = 1/3
For player 2: since the game has symmetric payoffs, p = 1/3
We want to see if using the MSNE strategy will yield a Pareto efficient outcome. So, we need
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 Spring '08
 Chwe
 Game Theory, Jessica dinner, Jessica concert tickets

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