Three firms (players I, II, and III) put three items on the market and adver- ti
ID: 3268922 • Letter: T
Question
Three firms (players I, II, and III) put three items on the market and adver- tise them either on morning or evening TV. A firm advertises exactly once per day. If more than one firm advertises at the same time, their profits are zero. If exactly one firm advertises in the morning, its profit is $200K. If exactly one firm advertises in the evening, its profit is $300K. Firms must make their advertising decisions simultaneously. Find a symmetric mixed Nash equilibrium Three firms (players I, II, and III) put three items on the market and adver- tise them either on morning or evening TV. A firm advertises exactly once per day. If more than one firm advertises at the same time, their profits are zero. If exactly one firm advertises in the morning, its profit is $200K. If exactly one firm advertises in the evening, its profit is $300K. Firms must make their advertising decisions simultaneously. Find a symmetric mixed Nash equilibrium Three firms (players I, II, and III) put three items on the market and adver- tise them either on morning or evening TV. A firm advertises exactly once per day. If more than one firm advertises at the same time, their profits are zero. If exactly one firm advertises in the morning, its profit is $200K. If exactly one firm advertises in the evening, its profit is $300K. Firms must make their advertising decisions simultaneously. Find a symmetric mixed Nash equilibriumExplanation / Answer
In one set of equilibria :
Player 1 always chooses morning,
Player 2 chooses evening,
Player 3 chooses morning with any probability.
Moreover pure strategy is at least used by one firm and these are the only Nash equilibria used.
For eg.
Let initially that firm 1 chooses the pure strategy of M (morning).
And if both firms 2 and 3 follow mixed strategies,
It can be seen that may be one of them could profit by changing to pure strategy E (evening).
To understand this , let the two mixed strategies be
lpha M + (1-lpha )E for firm 2
eta M+(1-eta )E for firm 3.
To find the mixed-strategy equilibria, let a, b, and c be the prob. of
advertisement in the morning for firms 1, 2, and 3.
The expected return to Firm 1 of advertising in the morning is:
(1 - b)(1 - c)
In the evening the expected return is 2bc.
If these are equal, any choice of a for firm 1 is Nash.
However equality is signified:
1 - b - c - bc = 0
or b = (1-c)/(1+c)
Now if it is repeated for firm 2 and firm 3
the equalities found are :
b = (1-c)/(1+c)
c = (1-a)/(1+a)
Solving simultaneously we get
a = b = c = sqrt{2}-1
thus a = (1-a)/(1+a) is a quadratic eqn with root betn 0 &1 is sqrt{2}-1
In order to prove that no other Nash equilibria exists, for eg
0 < a < 1 and
0 < b < 1. It must be shown 0 < c < 1, which reproduces equilibrium (b).
But 0 < a < 1 implies (1+b)(1+c) = 2 and 0 < b < 1 implies (1+a)(1+c) = 2
If c = 0, then a = b = 1, which is assumed not to be true. Similarly If c = 1 then
a= b = 0, which is also not true. Thus proves the equation regarding Nash Equilibria.