Auction Theory

Todays blog-post will be about Auction Theory. We will try to apply what we’ve learned and try to decide if we should rather sell our goods with a posted price or auction them off.

Warning: This blog-post will contain one thing we haven’t learned so far, which is order statistics. I promise I will write a blog-post about order statistics soon. Furthermore we will need some calculus background. For those who feel like their calculus background isn’t good enough, I recommend to have a look at the Open Course Ware of the MIT.

Auction Theory

The first question that probably comes up is; Why auction theory? Why does somebody care about this? Well the answer is quite simple. A good business man always cares about making the biggest possible profit.

The next question one could come up with is: Is there a difference between selling it for a posted price or auction it off? Yes of course there is and we can prove it mathematically.

Model

Our model will consist of:

  • n bidders or buyers
  • We know how their valuation is distributed
  • We consider that their valuations are uniform [0,1] distributed.

Should we choose a posted price or an auction?

Posted Price

We will first model the posted price as following:

The seller sets a price p and will just sell his product if at least one of the buyer valuations is greater or equal to that price, or in mathematical terms: V_{ i }\ge p

Our expected profit is then:

E(\pi(p))=\sum_{ i }{ p_{ i }\pi(p_{ i }) }=p\cdot P(V_{ i }\ge p for\;at\;least\;one\;i)

We have just two outcomes and therefore just two p. p is either the price we’ve set or 0 if we don’t sell our product. The first term p\cdot P(V_{ i }<p) falls away as p would be 0 in this case. Furthermore \pi stands for the profit function here.

We can simplify the above equation to E(\pi(p))=p(1-p^{ N }) , because the term V_{ i }\ge p in P(V_{ i }\ge p for\;at\;least\;one\;i) is nothing else then 1 – the cdf of the nth order statistic of the uniform[0,1] distribution. The nth order statistic tells us the probability that the given value is the maximum value of all our valuations.

Note: N is the amount of potential buyers.

We are interested in the maximum profit we can make and therefore in the maximum of the above equation. We take the derivative of the equation to find the maximum:

\frac{ d\pi }{ dx }=1-(N+1)p^{ N }

We then set the derivative equal to zero to get the optimal price. We won’t prove that this is the maximum, we will just consider it is this time.

0=1-(N+1)p^{ N }\Rightarrow p=\sqrt [ N ]{ \frac{ 1 }{ N+1 } }

Setting p=\sqrt [ N ]{ \frac{ 1 }{ N+1 } } in the equation of our expected profit gives:

E(\pi(p))=\sqrt [ N ]{ \frac{ 1 }{ N+1 } }\frac{ N }{ N+1 }

We can now visualise this in the following table:

Potential Buyers N Optimal Price p Expected Profit
1 0.5 0.25
2 0.577 0.385
5 0.699 0.582
10 0.787 0.715
15 0.831 0.779

Auction

We will now model the auction. We suppose we have an english auction so the price gradually increases until just one bidder is left. The last bidder then gets the product for the price of the second to last bidders valuation or in mathematical terms: p=V_{ (N-1) } . The parenthesis tell us that it is the N-1st oder statistic. That is because we don’t know the valuation of the last bidder and the auction is over when there is just one bidder left so the price won’t increase anymore. The english auction is also called “open outcry” auction and can be seen in many TV shows.

We assume again that the Valuations are uniform[0,1] distributed.

The N-1st oder statistic of a uniform[0,1] distribution is:

f_{ (N-1) }=N(N-1)(1-x)x^{ N-2 } for 0\le x \le 1

That leads us to our equation for our expected profit:

E(\pi(N))=\int_{ 0 }^{ 1 }{ N(N-1)(1-x)x^{ N-2 }x dx }=N(N-1)\int_{ 0 }^{ 1 }{ (1-x)x^{ N-2 }x }=N(N-1)(\frac{ 1 }{ N }-\frac{ 1 }{ N+1 })=\frac{ N-1 }{ N+1 }

We can now visualise our expected profit in a table:

Potential Buyers N Expected Profit
1 0
2 0.333
5 0.667
10 0.818
15 0.875

Comparing both shows that we can expect a higher profit at an auction if N is large. Furthermore the seller at an auction doesn’t need to know how the valuations are distributed as he doesn’t need to calculate the optimal price. Sure we needed to know the distribution but just because we wanted to compare the expected profits.

Potential Buyers N Expected Profit Auction Expected Profit Posted Price
1 0 0.25
2 0.333 0.385
5 0.667 0.582
10 0.818 0.715
15 0.875 0.779

Modeling error

Every model has some errors. Here are some potential errors for our model.

  1. We didn’t consider transaction costs for auctions. The expected profit could therefore be less in reality.
  2. We supposed that the seller for the Posted Price knew the distribution of the valuations. That might not be the case.
  3. We supposed that the valuations at an auction are independent. That might be the case in many scenarios but not in all. If we auction off a product that will be most likely resold, every individual valuation and bid tells us and the other bidders more about the real value of the product and the valuation of each bidder therefore depends on the already given bids.

When to set a posted price when auction off?

The above comparison might imply that auction products off is always the best way. That isn’t true. It is true under special circumstances. For instance if we don’t know the value of our goods or if it is a unique thing or very rare thing. On the other hand if we know the real value or if the supply is big, so we have a huge amount of the product, and it’s rather a common, every day product, then it’s probably better to sell things for a posted price.

Auction theory was the first blog-post of a new series of blog-posts in which we try to apply things we’ve learned. We will program some machine learning algorithms, analyse data sets, program our own valuation algorithms and do similar things like today.

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