Derive the mixed strategy equilibrium if both Intel and AMD act simultaneously in the game in the Managerial Solution.

Part $2$

Intel and$$ AMD, the dominant central processing unit$$ manufacturers, decide whether to set their advertising levels low or high. For$$ now, suppose that they play this game$$ once, act$$ simultaneously, and their profits are symmetric. If both choose low levels of$$ advertising, Intel's$$ profit,

pi Subscript Upper II$,$

and$$ AMD's profit,

pi Subscript Upper AA$,$

are each

$22.$

If both choose$$ high, each earns

$33.$

If$$ Intel's advertising is high and$$ AMD's is$$ low,

pi Subscript Upper IIequals$=99$

and

pi Subscript Upper AAequals$=44.$

If$$ Intel's advertising is low and$$ AMD's is$$ high,

pi Subscript Upper IIequals$=44$

and

pi Subscript Upper AAequals$=99.$

Part $3$

Let the probability that a firm chooses low be

alpha

for Intel and

beta

for AMD. If the firms choose their advertising$$ independently, then

alphabeta

is the probability that both choose low$$ advertising,

$(1$minus$$alpha$)(1$minus$$beta$)$

is the probability that both choose high$$ advertising,

alpha$(1$minus$$beta$)$

is the probability that Intel chooses low and AMD chooses$$ high, and

$(1$minus$$alpha$)$beta

is the probability that Intel chooses high and AMD chooses low.$$

Part $4$

A$.$

This game has no$$ mixed$-$strategy Nash equilibria.

B$.$

The$$ mixed$-$strategy Nash equilibrium is for both firms to select the low price

$($alphaequals$=$betaequals$=1).$

C$.$

The$$ mixed$-$strategy Nash equilibrium is for Intel to choose low with probability

$0\mathrm{.}1250\mathrm{.}125$

$($alphaequals$=0\mathrm{.}1250\mathrm{.}125)$

and for AMD to choose low with probability

$0\mathrm{.}1250\mathrm{.}125$

$($betaequals$=0\mathrm{.}1250\mathrm{.}125).$

D$.$

The$$ mixed$-$strategy Nash equilibrium is for both firms to select the high price

$($alphaequals$=$betaequals$=0).$

E$.$

The$$ mixed$-$strategy Nash equilibrium is for Intel to choose low with probability

$0\mathrm{.}8750\mathrm{.}875$

$($alphaequals$=0\mathrm{.}8750\mathrm{.}875)$

and for AMD to choose low with probability

$0\mathrm{.}8750\mathrm{.}875$

$($betaequals$=0\mathrm{.}8750\mathrm{.}875).$