纳什均衡哈佛第六讲

Lecture VI:Existence of Nash equilibrium

Markus M.M¨o bius

February26,2008

•Osborne,chapter4

•Gibbons,sections1.3.B

1Nash’s Existence Theorem

When we introduced the notion of Nash equilibrium the idea was to come up with a solution concept which is stronger than IDSDS.Today we show that NE is not too strong in the sense that it guarantees the existence of at least one mixed Nash equilibrium in most games(for sure in allfinite games). This is reassuring because it tells that there is at least one way to play most games.1

Let’s start by stating the main theorem we will prove:

Theorem1(Nash Existence)Everyfinite strategic-form game has a mixed-strategy Nash equilibrium.

Many game theorists therefore regard the set of NE for this reason as the lower bound for the set of reasonably solution concept.A lot of research has gone into refining the notion of NE in order to retain the existence result but get more precise predictions in games with multiple equilibria(such as coordination games).

However,we have already discussed games which are solvable by IDSDS and hence have a unique Nash equilibrium as well(for example,the two thirds of the average game),but subjects in an experiment will not follow those equilibrium prescription.Therefore,if we want to describe and predict 1Note,that a pure Nash equilibrium is a(degenerate)mixed equilibrium,too.

1

the behavior of real-world people rather than come up with an explanation of how they should play a game,then the notion of NE and even even IDSDS can be too restricting.

Behavioral game theory has tried to weaken the joint assumptions of rationality and common knowledge in order to come up with better theories of how real people play real games.Anyone interested should take David Laibson’s course next year.

Despite these reservation about Nash equilibrium it is still a very useful benchmark and a starting point for any game analysis.

In the following we will go through three proofs of the Existence Theorem using various levels of mathematical sophistication:

•existence in2×2games using elementary techniques

•existence in2×2games using afixed point approach

•general existence theorem infinite games

You are only required to understand the simplest approach.The rest is for the intellectually curious.

2Nash Existence in2×2Games

Let us consider the simple2×2game which we discussed in the previous

equilibria:

lecture on mixed Nash

2

Let’s find the best-response of player 2to player 1playing strategy α:

u 2(L,αU +(1−α)D )=2−α

u 2(R,αU +(1−α)D )=1+3α(1)

Therefore,player 2will strictly prefer strategy L iff2−α>1+3αwhich implies α<14.The best-response correspondence of player 2is therefore:BR 2(α)=⎧⎨⎩1if α<14[0,1]if α=140if α>14

(2)We can similarly find the best-response correspondence of player 1:

BR 1(β)=⎧⎨⎩0if β<23[0,1]if β=231if β>23(3)

We draw both best-response correspondences in a single graph (the graph is in color -so looking at it on the computer screen might help you):

We immediately see,that both correspondences intersect in the single point α=14and β=23which is therefore the unique (mixed)Nash equilibrium of the game.

3