# CS代考 – cscodehelp代写

Nash.

(princeton.edu, Universal Pictures/DreamWorks)

c -Trenn, King’s College London 2

In general, we will say that two strategies s1 and s2 are in Nash equilibrium (NE) if: 1. under the assumption that agent i plays s1, agent j can do no better than play s2; and 2. under the assumption that agent j plays s2, agent i can do no better than play s1.

Neither agent has any incentive to deviate from a NE.

Eh?

c -Trenn, King’s College London 3

Let’s consider the payoff matrix for the grade game:

j

YX Y

i

X

Here the Nash equilibrium is pY, Y q.

If i assumes that j is playing Y , then i’s best response is to play Y . Similarly for j.

2 2

1 4

4 1

3 3

c -Trenn, King’s College London 4

i

D

s

5

A

” ”

cD 24

z

2

G y

LAID) (0,4XNE (Ad)

ft,17)NE (Adsl (QD) XUE

CAN

XNE

If two strategies are best responses to each other, then they are in Nash equilibrium.

c -Trenn, King’s College London 5

In a game like this you can find the NE by cycling through the outcomes, asking if either agent can improve its payoff by switching its strategy.

j

YX Y

i

X

Thus, for example, pX, Y q is not an NE because i can switch its payoff from 1 to 2

by switching from X to Y .

2 2

1 4

4 1

3 3

c -Trenn, King’s College London 6

More formally:

A pair of strategies pi ̊, j ̊q is a Nash equilibrium solution to the game pA, Bq if:

@i,ai ̊,j ̊ • ai,j ̊ @j,bi ̊,j ̊ • bi ̊,j

That is, pi ̊, j ̊q is a Nash equilibrium if:

‚ If j plays j ̊, then i ̊ gives the best outcome for i. ‚ If i plays i ̊, then j ̊ gives the best outcome for i.

c -Trenn, King’s College London 7

Unfortunately:

1. NoteveryinteractionscenariohasapurestrategyNE. 2. SomeinteractionscenarioshavemorethanoneNE.

c -Trenn, King’s College London 8

This game has two pure strategy NEs, pC, Cq and pD, Dq: j

DC D

i

C

In both cases, a single agent can’t unilaterally improve its payoff.

5 3

1 2

0 2

3 3

c -Trenn, King’s College London 9

This game has no pure strategy NE:

DC D

i

C

For every outcome, one of the agents will improve its utility by switching its strategy.

We can find a form of NE in such games, but we need to go beyond pure strategies.

j

2 1

1 2

0 2

1 1

c -Trenn, King’s College London 10

Nash equilibria?

Consider this scenario (again):

j

CD A

1 2

4 3

2 3

3 2

i

B Are there any Nash equilibria?

c -Trenn, King’s College London

11

Nash equilibria?

Consider this scenario (again):

j

CD A

1 2

4 3

2 3

3 2

i

B Are there any Nash equilibria?

c -Trenn, King’s College London

11