# CS代考 Mixed strategy – cscodehelp代写

Mixed strategy

Chances are you would play a mixed strategy. You would:

‚ sometimesplayrock,

‚ sometimesplaypaper;and ‚ sometimesplayscissors.

A fixed/pure strategy is easy for an adaptive player to beat.

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Mixed strategy

A mixed strategy is just a probability distribution across a set of pure strategies.

So, for a game where agent i has two actions a1 and a2, a mixed strategy for i is a probability distirbution:

MSi “ tPpa1q,Ppa2qu

Given this mixed strategy, when i comes to play, they pick action a1 with

probability Ppa1q and a2 with probability Ppa2q.

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Mixed strategy

To determine the mixed strategy, i can compute the best values of Ppa1q and Ppa2q.

These will be the values which give i the highest expected payoff given the options that j can choose and the joint payoffs that result.

We could write down the expected payoffs of different mixed strategies and pick the one that optimises expected payoff.

There is also a simple graphical method which works for very simple cases. Will look at this method next.

c -Trenn, King’s College London 4

Mixed strategy

Let’s consider the payoff matrix:

j

a3

a4

-3 3

1 -1

0 0

-1 1

a1 i

a2

We want to compute mixed strategies to be used by the players.

That means decide Ppa1q and Ppa2q etc.

c -Trenn, King’s College London 5

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Mixed strategy

i’s analysis of this game would be something like this.

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Mixed strategy

33

22 1 j picks a3 1

00

−1 −2 −3

−1 −2 −3

0 P(a1) 1

Consider it from i’s perspective. Let’s say you know that j plays a3. i’s payoff will be 3 or 0 depending on whether i picks a1 or a2.

The expected payoff therefore varies along the line, as Ppa1q varies from 0 to 1.

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Mixed strategy

33 22 11 00

−1

−2 −3

j picks a4 −1

−2 −3

0 P(a1) 1

Consider it from i’s perspective. Let’s say you know that j plays a4. i’s payoff will be ́1 or 1 depending on whether i picks a1 or a2.

The expected payoff therefore varies along the line, as Ppa1q varies from 0 to 1.

c -Trenn, King’s College London 8

Mixed strategy

33

22 1 j picks a3 1

00

−1 −2 −3

0.2

j picks a4 −1

−2 −3

0 P(a1) 1

Where the lines intersect, i has the same expected payoff whatever j does. This is a rational choice of mixed strategy.

c -Trenn, King’s College London 9

Mixed strategy

j can do the same kind of analysis:

33 22 11

0 −1 −2 −3

i picks a2

i picks a1

0 −1

−2 −3

0.4

0 P(a3) 1

c -Trenn, King’s College London

10

Mixed strategy

This analysis will help i and j choose a mixed strategy in zero-sum games.

(Archives of the Institute of Advanced Study, Princeton)

This approach is due to von Neumann.

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General sum games

Battle of the Outmoded Gender Stereotypes ‚ akaBattleoftheSexes

this that this

that

1 2

0 0

0 0

2 1

Game contains elements of cooperation and competition.

The interplay between these is what makes general sum games interesting.

c -Trenn, King’s College London 12

(Time-Life/Getty)

Negotiation

Interplay between cooperation and competition leads to negotiation See, for example, the work of .

(law-train.eu)

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Nash equilibrium

Earlier we introduced the notion of Nash equilibrium as a solution concept for general sum games.

(We didn’t describe it in exactly those terms.)

Looked at pure strategy Nash equilibrium.

Issue was that not every game has a pure strategy Nash equilibrium.

c -Trenn, King’s College London 14

Nash equilibrium

For example:

j

DC D

i

C

2 1

1 2

0 2

1 1

Has no pure strategy NE.

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15

Nash equilibrium

The notion of Nash equilibrium extends to mixed strategies.

And every game has at least one mixed strategy Nash equilibrium.

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Nash equilibrium

For a game with payoff matrices A (to i) and B (to j), a mixed strategy px ̊, y ̊q is a Nash equilibrium solution if:

@x, x ̊Ay ̊T • xAy ̊T @y, x ̊By ̊T • x ̊ByT

In other words, x ̊ gives a higher expected value to i than any other strategy when j plays y ̊.

Similarly, y ̊ gives a higher expected value to j than any other strategy when i plays x ̊.

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Nash equilibrium

Unfortunately, this doesn’t solve the problem of which Nash equilibrium you should play.

c -Trenn, King’s College London 18

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