Private-value auctions: theory and experimental evidence (Part I)

1. Private-value auctions: theory and experimental evidence (Part I) Nikos Nikiforakis The University of Melbourne

2. Outline of talk • What is an auction? • Why auctions? • Basic auction types • How “should” I bid? • Result from experiments

3. 1. What is an auction? • An auction is a mechanism used to sell (or buy) an object. • Potential buyers (or sellers) make bids. • The bids determine who obtains (or sells) the object. • The final price is determined by the auction type and the bids. • Implications: • Anonymity: only bids matter • Universality: any object can be sold

4. 2. Why auctions? • Method most likely to allocate object to the one who values it the most • Reveals information about value of object • Credibility and transparency • Maximize revenue • Understanding auctions is important given volume of goods traded through auctions

5. 3. Private- & Common-Value Auctions Two broad classes: • Uncertainty about the value of the object for each bidder (but each bidders know his valuation) Private Value Auctions • PVA: Buying a painting based only on how much you enjoy it

6. 3. Private- & Common-Value Auctions • Uncertainty about the value of the object for all bidders (but object has the same value for all bidders) •  Common Value Auctions • CVA: Buying an oil field; the price for oil is well known, but not the oil reserves of the particular field. Many cases are a hybrid: e.g. buying a house to live in for 10 years and then reselling it.

7. Basic auction types • First-price sealed-bid • Second-price sealed-bid • English (ascending) auction • Dutch (descending) auction Part 1 Part 2

8. More auction types • Many types of auctions • “Deadline” auctions • “Candle” auctions • “Anglo-Dutch” auctions • “Dutch-English” auctions • Auctions were winner pays average of bids • Multi-unit auctions etc.

9. First-price sealed bid auction (FPA) • Each bidder makes a single secret (“sealed”) bid by a given deadline • The object is awarded to the highest bidder • The winner pays his bid • Common for auctions run by mail

10. Second-price sealed bid auction (SPA) • Each bidder makes a single secret (“sealed”) bid by a given deadline • The person/firm submitting the highest bid wins the auction • The winner pays the second highest bid • Does this make sense?

11. Optimal bidding strategy (SPA) Let h denote the highest of the other bids (A) Assume you bid b < v • If h < b • you win and you pay h • But then by bidding v, you also win and also pay h. • If h > v • then you do not win with either b or v. • If b < h < v • then with byou do not win, but with vyou win and make a profit v - h> 0. • Thus b = vweakly dominates bidding b < v.

12. Optimal bidding strategy (SPA) (B) Assume you bid b > v • If h < v < b • you win and pay hboth if you bid bor v. • If h > b > v • you do not win in either case. • If v < h < b • you win and earn v – h < 0 ! It’s better you bid b=v (in which case you don’t win and have zero profit). • Thus bidding b = vweakly dominates bidding b > v • Proposition: In a SPA the best thing to do (the weakly dominant strategy is) to bid your value, i.e. b=v.

13. Optimal bidding strategy (SPA) • Apart from learning what’s the best thing to do we found out that… • … if bidders are rational • Then their bid tells us how much they truly value the object! • Moreover, as the winner is the one with the highest bid (and since b=v)… • … The SPA makes sure the object goes to the bidder who values it the most.

14. Optimal bidding strategy (FPA) • The bidder has to choose a bid that maximizes his expected profit. • The expected profit in FPA is Pr(win) * (v-b) • As b increases so does the probability that your bid is the highest and thus win • However, conditional on winning, as b increases your profit (v - b) decreases • What is the optimal b ?

15. Optimal bidding strategy (FPA) • Assume for simplicity that • There are only 2 bidders • Values are uniformly distributed between [0, 10] • It is reasonable to assume that the higher the value of the object for a bidder the higher the bid • In specific, let’s assume that bi=α* vi, for i=1,2 • Then, Pr(b1>b2) = Pr(b1> α* v2) = Pr(v2 <b1/α) = b1/α(by assumption 2)

16. Optimal bidding strategy (FPA) • Therefore, the expected profit of i =1 can be written as • Pr(b1>b2)*(v1-b1) = b1/α * (v1-b1) • Remember, choose b1such that expected profit is maximized. • If you know calculus (and still remember) finding the optimal bid is now a simple task. • If you don’t… let’s remember assumption 2  values are between 0 and 10)

17. Optimal bidding strategy (FPA) • If you bid 10 you win with probability (almost) 1 • If you bid 0 you lose with probability (almost) 0 • This implies, that the probability of winning is bi/10 • Therefore, the expected profit of i =1 can be written as • Pr(b1>b2)*(v1-b1) = b1/10 * (v1-b1) • Let’s look what the expected profit looks like for v1=8 and v1=10.

18. Optimal bidding strategy (FPA) Expected Profit when v1=8 b*= 4

19. Optimal bidding strategy (FPA) Expected Profit when v1=10 b*= 5

20. Optimal bidding strategy (FPA) • Proposition 1: The optimal strategy with 2 (risk-neutral)bidders is to bid half of the object’s value, i.e. b*=vi/2. • It can also be shown that … • Proposition 2: The optimal strategy with N (risk-neutral)bidders, with N>1, is to bid b*=(N-1)vi / N

21. Predictions for first 3 experiments • Summarizing… • Experiment 1&3: First Price Sealed Bid Auction • If you are risk-neutral then when n=3, you should be bidding according to the following rule b*=0.67vi • When n=6, you should be bidding according to the following rule b*=0.83vi • In other words, all else equal, you should increase your bids when there are more bidders

22. Predictions for first 3 experiments • Summarizing… • Experiment 2: Second Price Sealed Bid Auction • Regardless of your risk preferences you should all be bidding your values that is b*= vi • That is, bidding should be more aggressive under SPA than under FPA.

23. Predictions

24. Results • First scatter plot (y:bid, x:value) with fitted line for experiment 1, 2, and 3. • (Should include dotted line for “best-fit” in 1&3.) • Efficiency: (% of times highest-value bidder got the object) • “A natural measure of efficiency is the ratio of the valuation of the winning bidder to the highest valuation any bidder holds

25. Results b=0.81v

26. Results b=1.01v

27. Results b=0.93v

28. Efficiency • FPA n=6  14/15 in first 3 periods • Similar for other FPA with n=3 and SPA although some losses due to spite.

29. Optimal bidding strategy (FPA) revisited: Risk Aversion • Overbidding is consistent with risk aversion (RA) • In simple terms, if you exhibit RA you tend to undervalue the monetary rewards of winning in favour of the probability of winning • e.g. U(x)= x 1-r, where x is amount you earn (and 0≤r≤1).

30. Optimal bidding strategy (FPA) revisited: Risk Aversion • Remember: you “have” to maximize your expected payoff. • If you are risk neutral this implies: Pr(b>h) * (v-b) • Your optimal bidding strategy is:b*=(N-1)vi / N • If you are risk averse this implies: Pr(b>h) * (v-b)1-r • Your optimal bidding strategy is:b*=(N-1)vi / N - r

31. Optimal bidding strategy (FPA) revisited: Risk Aversion • Assume that r=0.5 • This implies, that for groups of 3 • b*= 0.8vi (instead of 0.67vi) • For groups of 6 • b*= 0.91vi (instead of 0.83vi) • r=0.5 is a reasonable assumption which organizes data well

32. Predictions with risk aversion

33. What did we learn? • Auctions can increase efficiency and revenue • Information revelation • Theory predicts behavior well • Experiments help us with adjustments to theory and implementation of auctions.