Solution Concepts
How will a rational agent behave in any given scenario? Play. . .
‚ Dominantstrategy;
‚ Nashequilibriumstrategy;
‚ Paretooptimalstrategies;
‚ Strategiesthatmaximisesocialwelfare.
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Dominant Strategies
Given any particular strategy s (either C or D) agent i, there will be a number of possible outcomes.
We say s1 dominates s2 if every outcome possible by i playing s1 is preferred over every outcome possible by i playing s2.
Thus in this game:
j
DC D
i
1 2
4 2
1 5
4 5
C C dominates D for both players.
c -Trenn, King’s College London
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Dominant Strategies
Two senses of “preferred”
s1 strongly dominates s2 if the utility of every outcome possible by i playing s1 is
strictly greater than every outcome possible by i playing s2. In other words, ups1q ° ups2q, for all outcomes.
s1 weakly dominates s2 if the utility of every outcome possible by i playing s1 is no less than every outcome possible by i playing s2.
In other words, ups1q • ups2q, for all outcomes.
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Dominant Strategies
A rational agent will never play a dominated strategy.
So in deciding what to do, we can delete dominated strategies. Unfortunately, there isn’t always a unique undominated strategy (see later).
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Dominant Strategies
h t t pR
Rps
Game with dominated strategies
LCR U
M
D
%. s
l
O
1
O
3
O
O
1
0
O
O0
0
OO
110
O00
000
115
110
000
040
g
o
2
Can eliminate the dominated strategies and simplify the game Which strategy is dominated?
c -Trenn, King’s College London
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Dominant Strategies
Let’s look at the pay-off matrices A and B
̈ ̊1 1 0 ̨‹
For the column player j we get B “ ̊ ̋1 1 0‹‚ 110
̈ ̊1 ̨‹ ̈ ̊1 ̨‹ ̈ ̊0 ̨‹ We can think of this as three vectors ̊ ̋1‹‚, ̊ ̋1‹‚and ̊ ̋0‹‚
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Dominant Strategies
Let’s look at the pay-off matrices A and B
̈ ̊1 1 0 ̨‹
For the column player j we get B “ ̊ ̋1 1 0‹‚ 110
̈ ̊1 ̨‹ ̈ ̊1 ̨‹ ̈ ̊0 ̨‹ We can think of this as three vectors ̊ ̋1‹‚, ̊ ̋1‹‚and ̊ ̋0‹‚
We can see that every component of R is dominated by L (and actually also C) So we can remove R
c -Trenn, King’s College London 8
Dominant Strategies
Game with dominated strategies
LC U
M
D
Can eliminate the dominated strategies and simplify the game Remove R (dominated by L).
30 O1O 1
OO
11
11
OO
11
04
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Dominant Strategies
Let’s look at the pay-off matrices A
̈ ̊3 0 ̨‹
For the row player i we get A “ ̊ ̋1 1‹‚
04
– ́ ̄ ́ ̄ ́ ̄] Wecanthinkofthisasthree(row)vectors 3 0 , 1 1 and 0 4
c -Trenn, King’s College London
rewire remove
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c reinvent
10
Dominant Strategies
Let’s look at the pay-off matrices A
̈ ̊3 0 ̨‹
For the row player i we get A “ ̊ ̋1 1‹‚ 04
We can think of this as three (row) vectors ́3 0 ̄ , ́1 1 ̄ and ́0 4 ̄ No strategy here is dominated by any other …
So we cannot remove anything else
c -Trenn, King’s College London
11
Dominant Strategies
If we are lucky, we can eliminate enough strategies so that the choice of action is obvious.
In general we aren’t that lucky.
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Dominant Strategies
Consider this scenario:
j
CD A
1 2
4 3
2 3
3 2
i
B Are there any dominated strategies?
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13
Are there any dominated strategies?
D is dominating!
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