Game Theory, Part 2 Consider again the game that Sol and Tina were playing, but with a different...
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Game Theory, Part 2 Consider again the game that Sol and Tina were playing, but with a different payoff matrix: 4 2 3 1 H T Tina H T Sol
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Transcript of Game Theory, Part 2 Consider again the game that Sol and Tina were playing, but with a different...
Game Theory, Part 2
Consider again the game that Sol and Tina were playing, but with a different payoff matrix:
4 2
3 1
H TTina
HT
Sol
Game Theory, Part 2
Consider again the game that Sol and Tina were playing, but with a different payoff matrix:
4 2
3 1
H TTina
HT
Sol
Begin by finding the maximin and minimax.
Game Theory, Part 2
Consider again the game that Sol and Tina were playing, but with a different payoff matrix:
4 2
3 1
H TTina
HT
Sol
Begin by finding the maximin and minimax.
-2
-3
4 1
Game Theory, Part 2
Consider again the game that Sol and Tina were playing, but with a different payoff matrix:
4 2
3 1
H TTina
HT
Sol
This time the maximin and minimax are not the same, so the game is not strictly determined.
-2
-3
4 1
Game Theory, Part 2
Consider again the game that Sol and Tina were playing, but with a different payoff matrix:
4 2
3 1
H TTina
HT
Sol
What strategy should Sol and Tina employ?
-2
-3
4 1
Game Theory, Part 2
4 2
3 1
H TTina
HT
Sol
What would happen if both Sol and Tina flipped their coins, resulting in a random selection of heads or tails?
Game Theory, Part 2
4 2
3 1
H TTina
HT
Sol
We would expect each possible outcome 25% of the time.
Game Theory, Part 2
4 2
3 1
H TTina
HT
Sol
Sol’s expectation:
.25(4) + .25(-2) + .25(-3) + .25(1) = 0
Game Theory, Part 2
4 2
3 1
H TTina
HT
Sol
Sol’s expectation:
.25(4) + .25(-2) + .25(-3) + .25(1) = 0
It works the same for Tina, and so this game is considered a fair game.
Game Theory, Part 2
4 2
3 1
H TTina
HT
Sol
Now say Sol wants to consider options other than just flipping his coin. He might decide to play heads 60% of the time, or 30% of the time, or some other value. Let’s call that value p.
Game Theory, Part 2
4 2
3 1
H TTina
HT
Sol
If Sol plays heads p part of the time, then that means he will play tails 1-p part of the time.
Game Theory, Part 2
4 2
3 1
H TTina
HT
Sol
We can arrange this in a matrix:
Next, multiply by the payoff matrix.
1p p
Game Theory, Part 2
4 2
3 1
1p p
4 ( 3)(1 ) 2 1(1 )p p p p
4 3 3 2 1p p p p
7 3 3 1p p
=
=
=
Game Theory, Part 2
4 2
3 1
Well that was sure fun. But what does it mean?
1p p
4 ( 3)(1 ) 2 1(1 )p p p p
4 3 3 2 1p p p p
7 3 3 1p p
=
=
=
Game Theory, Part 2
This matrix tells us that if Tina always played heads, Sol’s expectation would be 7p – 3.
If Tina always played tails, Sol’s expectation would be –3p + 1.
7 3 3 1p p
Game Theory, Part 2
Say for example that Sol decided to play heads 60% of the time, so p = .6
If Tina always played heads, Sol’s expectation would be 7(.6) – 3 = 1.2
If Tina always played tails, Sol’s expectation would be –3(.6) + 1 = -.8
7 3 3 1p p
Game Theory, Part 2
Of course Tina is not likely to always play heads or always play tails. One strategy for Sol, then, would be to balance out the expectations of those two extreme cases. In other words, let
7p – 3 = -3p + 1
7 3 3 1p p
Game Theory, Part 2
Of course Tina is not likely to always play heads or always play tails. One strategy for Sol, then, would be to balance out the expectations of those two extreme cases. In other words, let
7p – 3 = -3p + 1
10p = 4
P = .4
7 3 3 1p p
Game Theory, Part 2
Of course Tina is not likely to always play heads or always play tails. One strategy for Sol, then, would be to balance out the expectations of those two extreme cases. In other words, let
7p – 3 = -3p + 1
10p = 4
P = .4
So Sol should play heads 40% of the time.
7 3 3 1p p
Game Theory, Part 2
For Tina we can use a similar approach. We’ll use q to represent the frequency that she plays heads.
As she is the column player, we’ll represent her strategy in a column matrix. This means we have to multiply the payoff matrix times her column matrix—pay attention to that order.
Game Theory, Part 2
4 2
3 1
1
q
q
4 2 2
3 1
q q
q q
6 2
4 1
q
q
= =
Game Theory, Part 2
4 2
3 1
1
q
q
4 2 2
3 1
q q
q q
6 2
4 1
q
q
= =
Again, balance out the two expectations:
6q – 2 = -4q + 1
10q = 3
Q = .3
So Tina should play heads 30% of the time.
Game Theory, Part 2
Sec. 7.5 Assignment:
Pp. 391-393
#3-6, 9
