For the operating systems game, let us now assume the intrinisc superiority of M
ID: 1189075 • Letter: F
Question
For the operating systems game, let us now assume the intrinisc superiority of Mac is not as great and tat network effects are stronger for Windows. These modifications are reflected in different payoffs. Now, the payoffs from adopting Windows is 50*w and from adopting Mac is 15+5*m; n consumers are simultaneously deciding between Windows and Mac.
A. Find all Nash Equilibria.
B. With these new payoffs, let us now suppose that a third option exsists, which is to not buy from either operating system; it had a payooff of 1000. Consumers simultaneously decide among Window, Mac, and no operating system. Find all Nash equilibria
Explanation / Answer
Payoff from adopting windows = 50*w
Payoff from adopting mac = 15 + 5*m
Payoff from adopting windows = 50*(n-m)
since w+m = n
if the same number of buyers purchase a mac and windows => m = w
every individuals payoff is larger with windows 50w > 15+5m
network effect = the more people are using the same operating system that you use, the more valuable it becomes
A)
if all people buy mac , the payoff for any individual is 15+5*n ( m = n , equilibrium payoff)
if instead 1 deviates towards windows , 1 obtains only 50(n- (n-1)) = 50
since it is symmetric , we can extend the same argument to all consumers
Therefore there is a nash equilibrium when everybody buys a mac
Now, for the other equilibrium
if all buy windows , an individuals payoff is 50n ( w = n)
if, instead , 1 deviates towards mac , 1 obtains 15+5*1 = 20
In order for this extreme equilibrium to exist we thus need,
50n>=20 => n>= 20/50
Hence, if the population exists above 0.4 ( which is always true)
herefore there is a nash equilibrium when everybody buys windows
b)
Let us consider that all people decide to buy no operating system
Payoff = 1000*X = 1000*n
w+m+x = n
Minimum condition ,
if 1 deviates towards mac = payoff = 20
if 1 deviates towards windows = payoff = 50
Total payoff = 70
In order for this extreme equilibrium to exist we thus need,
1000n>=70
n>=70/1000
which is always possible
Therefore there is a nash equilibrium when everybody buys no operating system