ME 403N

PRODUCTION, PLANNING and CONTROL

 

 

Lecture notes for:

First Price AUCTION Theory

 

Date : 1^st NOV 2002

 

Consider a bidding strategy b(X) that gives the bidding value “b” depending on our value X, and all bidders follow this strategy.

 

That is, for any bidder “i”, the bidding value is “b_i­­­ = b(X_i­)”

Then the payoff is given by:

 

Π_i         = X_i   - b_i ,        for b_i > max b_j

                = 0                   otherwise

 

Note that, everybody else follows the b(X)

 

Then the value of the second highest bid is :

 

Y­₁ = max(X₂, X₃,…,X_N) 

 

Where ,           b > β(Y₁)

Or,                   β^-1(b) > Y₁

 

Here β is an increasing function of Y.

 

 

Expected payoff        = Probability of winning * payoff

                                    = P( Y₁ ≤ β^-1(b))  * (X-b)

                                    = G(β^-1(b)) * (X-b)

 

{ b is your bid and X is the value }

 

now one has to choose b such that the expected payoff is maximized.

 

G(y)     = P(Y₁ ≤ y)

            = P(X₁ ≤ y)^N

            = F(y)^N-1

 

g(y)      = (N-1) * F(y)^N-2 * f(y)

 

differentiate the above equation to …………..

 

db

d (β^-1(b))  * g(β^-1(b)) * (X-b) - G (β^-1(b)) = 0                                                   ….. 1

 

 

Note that “ β(X) = b “ at the optimal point.

 

X = β^-1(b)

 

db

d (β^-1(b))  =  dX / db = 1 / (db / dX) = 1 / β’(X) = 1 / β’(β^-1(b))                       ….. 2

 

 

Form equations 1 and 2, we get

	g(X) * (X-b) / β’(X) – G(X) = 0

 

 

 

 

Note that, this equation should hold good at optimal point for all values of X.

 

G(X) β’(X) + g(X) B(X) = X g(X)

 

d[ G(X) β(X) ] = X g(X)

 

If value is zero then bid is also equal to zero.

Hence,             X = 0 , β(0) = 0

 

Integrating the above equation

	X

 

 

0

G(X) β(X) = ∫ y g(y) dy

 

 

 

β(X)     =    1      ∫ y g(y) dy                   here, G(X) = p(y₁ ≤ X)

0

               G(X)

 

           

            = E [ y₁ / y₁ < X ]

 

 

Nash Equilibrium

 

If everybody is following this above stated policy and one particular person is not following this policy, then he is bound to loose.

 

Nash equilibrium gives the proof for it.

 

If the value is “X”, and you bid “b”,

such that β(Z) = b

For value “X”, your optimal bid should have been β(X), but you have bid “b ≠ β(X)” and “b = β(Z)”, which should have been done for value Z (≠ X).

 

The expected payoff is given by :

 

Π(b,X) = P(Y₁ ≤ Z) (X – β(Z))

 

Z

             = G(Z) (X – β(Z))

 

0

             = G(Z) X – G(Z)  1         ∫ y g(y) dy

                                       G(Z)

 

Now we have to show that this policy is optimal, only and only if Z = X.

 

Now, adding and subtracting G(Z) to the above equation.

	Z

 

0

Π(b,X) = G(Z) (X –Z) +  ∫ (Z – y) g(y) dy                                                                    ….. 3

 

	Z

 

Now G(Z) =

0

∫ y g(y) dy                                                                                                ….. 4

 

 

Z

Solving Eqns 3 and 4

 

0

Π(β(Z),X) = G(Z) (X –Z) +  ∫ G(y) dy

 

 

X

   Π(β(X),X) - Π(β(Z),X) = G(Z) (X –Z) -  ∫ G(y) dy    ≥ 0

 

 

This equation would always be ≥ 0. that is the best solution is to have Z = X.

 

As shown in the adjoining figure,

 

G(y) = P (Y₁ ≤ y )

 

More the y, more is the probability.

X

                                                                                                     

 

0

β(X)     =    1      ∫ y g(y) dy                  

               G(X)

	X

 

0

            = X -   ∫ G(y)  dy         

X

                         G(X)

0

0

{as   ∫ y g(y) dy  = X G(X) -  ∫ G(y) dy }                                             

 

 

Now    G(Y) / G(X) = (F(Y) / F(X))^N-1

	X

 

 

0

β(X)     = X -   ∫ (F(Y) / F(X))^N-1 dy

 

 

as N à ∞ , F(Y) / F(X) à 0

	β(X) = X

 

 

 

That is as number of bidders’ increase, the margin of profit decreases and you bid close to your value “X”.

 

Revenue comparison ( Revenue equivalence theorem)

 

The revenue for the seller, in both 1^st price and 2^nd price cases, is same, for “n” number of bidders, in expected sense.

Variance may be different.

F[.] is uniform [0,1]

β(X)= X ((N-1)/N)

 

Suppose there are 2 bidders

X₁ & X₂

 

 

 

 

If X₁ > X₂  à 1 wins

            he pays X₁/2

 

So if X₁> X₂> X₁/2 à 2^nd price auction will pay more.

 

    if  X₁ > X₁/2 > X₂ à 1^st price auction will pay more.

 

For any specific realization, different forms of auction pay different.

On an averageà similar amount.

 

(to be continued…..)