Lecture 5AMixed Strategies and Multiplicity

Not every game has a pure strategy Nash equilibrium, and some games have more than one. This lecture shows that mixed strategies are sometimes a best response, and explains that every game has at least one Nash equilibrium in pure or mixed strategies.

Matching pennies

Not every game has a pure strategy Nash equilibrium.

In this zero sum game, each player chooses and then simultaneously reveals the face of a two sided coin to the other player.

The row player wins if the faces on the coins are the same, while the column player wins if the faces are different.

The chain of best responses

If Player 1 plays H, Player 2 should play H, but if 2 plays H, 1 should play T.

Therefore (H,H) is not a Nash equilibrium.

Using a similar argument we can eliminate the other strategy profiles as being a Nash equilibrium.

What then is a solution of the game?

Avoiding losses in the matching pennies game

If 2 plays Heads with probability greater than 1/2, then the expected gain to 1 from playing Tails is positive.

Similarly 1 expects to gain from playing Heads if 2 plays Heads more than half the time.

But if 2 randomly picks Heads with probability 1/2 each round, then the expected profit to 1 is zero regardless of his strategy.

Therefore 2 expects to lose unless he independently mixes between heads and tails with probability one half.

The Ware case10 years ago Ware received a patent for Dentosite that has since captured 60 percent share in the market. National had been the largest supplier of material for dental prosthetics before Dentosite was introduced.

A new material FR 8420 was recently developed by NASA.

If Ware develops a new composite with FR 8420 it will be a perfect substitute for Dentosite.

If the technique is feasible then Ware would have just as good a chance as National of proving it first.

If Ware develops it first they could extend the patent protection to this technique and prevent any competitors.

Strategic considerations

Ware’s problem is bound to National’s.

Ware does not want to develop a technology that would not be used if the competitor does not develop it.

If National develops the technology Ware cannot afford to drop out of the race.

It all depends how people at National see this situation. Are Ware and National equally as well informed?

Some factsThe Ware Case

10% Discount/year$1.9091 Value of $1 in years 1+2$3.4462 Value of $1 in years 3+4+5+6+7

$0.500 Ware entry cost/yr 50% Probability process feasible$1.000 National entry cost/yr 50% Ware chance of winning race

50% Ware market share if Nat'l enters mkt$15 Range of possible 20% Ware profit margin$20 future annual sales (millions)

Ware case in the extensive form

Using the facts we can present the case in the following diagram:

Simplifying the extensive form

Folding back the moves of chance that are related to developing a new technology we obtain the following simplification.

Ware case in the strategic form

The arrows trace out the best replies.

As in the Matching Pennies example, there is no pure strategy Nash equilibrium in the Ware case.

Rather than defining the solution as a particular cell, we now define the solution as the probability of reaching any given cell.

The probability of Ware choosing ”in”

Suppose Ware chooses “in” with probability p. Then National is indifferent between the two choices if the expected profits are equal.

The value to National from choosing “out” is 0, and the expected profits to National from choosing “in” are:

-0.401*p + 1.106*(1 – p)

Solving for p we obtain:-0.401*p + 1.106*(1 – p) = 0=> p = .734

The probability of National choosing ”in”

Suppose National chooses “in” with probability q. Then Ware is indifferent between the two choices if the expected profits are equal.

The expected value to Ware from choosing “in” is:

-2.462*q - 0.955*(1 – q)

The expected value to Ware from choosing “out” is:

-3.015*q

Solving for q we obtain: 2.462*q + 0.955*(1 – q) = 3.015*q=> q = .633

Solution to the Ware caseIf Ware sets p = 0.734, then a best response of National is to set q = 0.633. Likewise if National sets q = 0.633, a best response of Ware is to set p = 0.734.

Therefore the strategy profile p = 0.734 and q = 0.633 is a mixed strategy Nash equilibrium. It is unique.

TaxationWe return to solve the tax auditing game that we played in the first lecture of this course.

For convenience the strategic form of this simultaneous move game is presented.

Best replies in taxation game

Monitoring by the collection agencyEquating the expected utility for the collection agency:

1011+ 4 12 + 2(1- 11- 12)= -11+ 512 + 3(1- 11- 12)

and:

1011 + 4 12 + 2(1- 11- 12)= 212 + 4(1- 11- 12)

Solving these equations in two unknowns we obtain the mixed strategy:

11 =1/12=0.083

12 = 1/4 =0.250

13 = 2/3 =0.667

Cheating by the taxpayerEquating the expected utility for the taxpayer across the different choices:

-1221 = -621 - 622 - 2(1- 21- 22) and

-1221 = -4

Defining the only strategy that leaves the taxpayer indifferent between all three choices is threfore:

21 =1/3 =0.333

22 = 1/6 =0.167

23 = ½ = 0.500

Mixed strategy Nash equilibrium in the taxation game

12=0.25

11=0.083

13=0.667

21=0.333 22=0.167 23=0.50

Existence of Nash equilibrium

This brings us to the central result of this lecture. Consider any finite non-cooperative game, that is a game in extensive form with a finite number of nodes.

If there is no pure strategy Nash equilibrium in the strategic form of the game, then there is a mixed strategy Nash equilibrium.

In other words, every finite game has at least one solution in pure or mixed strategies.

The threat of bankruptcy

We consider an industry with weak board of directors, an organized workforce and an entrenched management.

Workers and management simultaneously make demands on the firms resources.

If the sum of their demands is less than or equal to the total resources of the firm, shareholders receive the residual.

If the sum exceeds the firm’s total resources, then the firm is bankrupted by industrial action.

Strategic form of bargaining game

To achieve a bigger share of the gains from trade, both sides court disastrous consequences.

This is sometimes called a game of chicken, or attrition.

We investigate more complicated bargaining games in 45-976, “Bargaining, Contracts and Strategic Investment”, the second course in this sequence.

Best responses illustrated

Multiple Nash equilibrium

In this game, there are three pairs of mutual best responses.

The parties coordinate on an allocation of the pie without excess demands. Shareholders get nothing.

But any of the three allocations is an equilibrium.

If the labor and management do not coordinate on one of the equilibrium, the firm will bankrupt or shareholders will receive a dividend.

Corporate plans

In every corporation departmental heads jostle for influence, promotion and higher compensation.

This creates rivalries between different departments.

Consequently the goals of departments are seldom perfectly aligned with each other.

Best replies in corporate plans

There are two Nash equilibrium in this game, one pure strategy and the other mixed.

Is one equilibrium more plausible than the other?

Now suppose the payoff elements in all the corner cells were magnified by a factor of a hundred. Is there a Jack Welch way?

Lecture summary

Not every game of imperfect information has a pure strategy equilibrium.

However every strategic form game has at least one pure or mixed strategy solution, and we showed how to derive them.

Strategic uncertainty arises in situations where the solution to the game is a mixed strategy.

When there are multiple Nash equilibrium, other criteria might be used to pick amongst them, as coordinated by management.