P.P.C. Lecture Held on 30/10/2002

Auction Theory

By: Rohan Dayal

 

Our basic aim is to reach a Stable ‘Nash’ Equilibrium strategy, a concept borrowed from game theory, in which all players in the game follow the same strategy, and if and when a player deviates from it, he loses out.

 

We restrict our analysis to single object auctions (i.e. only one object is being sold at a time).

 

The assumptions:

1.) Private values: The bidders value the commodity being sold in the auction at a particular price. This value is decided independently by every individual bidder and does not depend on or change with the values of other bidders.

2.) Sealed bid: The values are not known to other bidders. Till the time that the object is sold, nobody knows the valuation of other bidders.

 

We look for a Symmetric Nash Equilibrium (Given by THE John Nash).

(Further reading: refer http://helios.unive.it/~gottardi/NASH.PDF)

 

Auction Model:

• Single object for sale
• N potential buyers
• Bidder i assigns a random value X_i Î [0,u] to the commodity with a Cumulative Probability Distribution F( · ) and a Probability Distribution f( · ), u being the upper limit to the value of the commodity
• F( · ), f( · ) and u are the same for all the bidders (hence in our analysis we search for a symmetric equilibrium strategy)

 

F and f are obtained using:

• Past experience
• Intelligent judgement

 

Problem:

Knowing the value X_i for the bidder i we have to decide the value of our bid b, such that it maximises our expected value (payoff). This is known as a ‘Risk Neutral’ strategy, meaning that all the bidders do not take risks, and try to keep their payoff either positive, otherwise zero, whatever they bid. They do not bid anything that may result in a negative payoff even if they win. This is a reasonable assumption in real life auctions.

 

We look at the problem for the 2 types of auctions that have been discussed:

• First price sealed bid
• Second price sealed bid

 

Second Price sealed bid:

We first look at this type of auction as the theory behind the strategy is easier.

 

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:

 

 

 

For Second price auctions, the optimal strategy is given by:

b(X_i) = X_i

This is a ‘dominant’ strategy, and means that no matter which strategy other bidders follow (whatever be the probability distribution of the bidding values of other bidders), we just need to bid as much as we value the commodity.

 

Proof of optimality:

Suppose we value the commodity to be X₁ and choose to bid a value of Z₁.

Let the second highest bid.

Now we look at the various possibilities:

 

Case 1: X₁ > Z₁ (i.e. we bid Z₁ lower than our valuation X₁).

 

Case 1a: If P₁ > X₁ (somebody bids higher than our value) then we lose whatever Z₁ we choose (Z₁ < X₁), so it does not make a difference, and the payoff is zero.

 

Case 1b: If P₁ < X₁, (the second highest bid is lower than our value) then the payoff is given by,

i.e. if we had bid X₁ (instead of Z₁) then the payoff would have been the same (i.e. X₁ - P₁) but in certain cases, due to a lower bid value, we would have ended up losing when we could actually have won.

Therefore, it is better to have bid X₁.

 

Case 2: Z₁> X₁ (i.e. we bid Z₁ higher than our valuation X₁).

 

Case 2a: If P₁ < X₁, (second highest bid is lower than our value and hence our bid Z₁) we win whatever we bid, with a payoff of X₁- P₁.

Case 2b: If P₁> Z₁, then we lose with a payoff of 0.

 

Case 2c: Z₁> P₁> X₁, (second highest bid lies between our value and our bid) we do win the auction if we bid Z₁, but the payoff = -(P₁ - X₁) i.e. a negative payoff. It would have been better if we had bid X₁ as then we would not have won but would have had a payoff of zero.

 

The cases are shown in the chart below,

 

It does not make sense bidding a large value of Z₁ with a hope that everyone will bid a nominal value and you will have to pay a value close to X₁, as:

If P₁>X₁ we end up paying more than what we value the commodity, and

if P₁<X₁ we would have won even if we had bid X₁.

 

\It is best to set b(X_i) = X_i.

This strategy is known as a ‘Dominate Strategy’ in game theory.

 

 

How much does a bidder pay in Equilibrium?

(or how to find the expected amount a bidder pays).

Consider an auction with N bidders.

Let Y₁=max(X₂, X₃,…,X_N) (the value of the second highest bid).

 

Let G(y) be the probability of a bid y being the highest.

\ G(y)= P(Y₁£y) (the probability that the second highest bid is below y)

= P(max(X₂, X₃,…,X_N) £ y)   the probability that all the
                                                other bids are lesser than y

= P(X₂ £ y & X₃ £ y & … & X_N £ y)

 

As we have assumed that the values are independent of each other,

G(y)     = P(X₂ £ y) . P(X₃ £ y) … . P(X_N £ y)

 

As we have assumed that the distribution is identical, the probabilities are equal,

\ G(y)= P(X_i£y)^N-1 (Equation 1)

 

The expected amount a bidder pays is given by the product of the probability that the bidder wins and the expected value of the second highest bid.

= P(bidder wins) . E(Y₁ | Y₁<x)

= G(x) . E(Y₁ | Y₁<x)

 

 

Example:

If X_i is uniformly distributed in [0,1].

 

 

If x is the highest bid, then all the other (N-1) bids are

at equal intervals between [0,x] as the distribution is uniform.

 

 

 

 

 

 

The probability distribution is given by,

Probability that any X_i is lesser than any value y is,

  (X_i can take any value lesser than y)

From Equation 1, G(y) = F(y)^N-1 = P(X_i£y)^N-1

\

As x lies between [0,1], u=1 and G(x) = x^N-1.

 

Continuing, the cumulative distribution g(y) is given by,

From conditional probability,

 

Therefore, expected value = E(Y₁ | Y₁<x) =

 

Expected amount bidder pays        = G(x) . E(Y₁ | Y₁<x)

=

 

 

Sealed bid first price auctions will be looked at in the next lecture.