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

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

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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

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2. Why auctions? 1. Method most likely to allocate object to the one who values it the most 2. Reveals information about value of object 3. Credibility and transparency 4. Maximize revenue Understanding auctions is important given volume of goods traded through auctions

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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

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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.

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Basic auction types 1. First-price sealed-bid 2. Second-price sealed-bid 3. English (ascending) auction 4. Dutch (descending) auction Part 1 Part 2

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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.

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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

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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?

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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 b you do not win, but with v you win and make a profit v - h > 0. Thus b = v weakly dominates bidding b < v.

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Optimal bidding strategy (SPA) (B) Assume you bid b > v If h < v < b you win and pay h both if you bid b or 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 = v weakly 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.

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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.

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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 ?

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Optimal bidding strategy (FPA) Assume for simplicity that 1. There are only 2 bidders 2. 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 b i =α * v i, for i=1,2 Then, Pr(b 1 >b 2 ) = Pr(b 1 > α * v 2 ) = Pr(v 2 **
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Optimal bidding strategy (FPA) Therefore, the expected profit of i =1 can be written as Pr(b 1 >b 2 )*(v 1 -b 1 ) = b 1 /α * (v 1 -b 1 ) Remember, choose b 1 such 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)

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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 b i /10 Therefore, the expected profit of i =1 can be written as Pr(b 1 >b 2 )*(v 1 -b 1 ) = b 1 /10 * (v 1 -b 1 ) Let’s look what the expected profit looks like for v 1 =8 and v 1 =10.

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Optimal bidding strategy (FPA) Expected Profit when v 1 =8 b * = 4

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Optimal bidding strategy (FPA) Expected Profit when v 1 =10 b * = 5

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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 * =v i /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)v i / N

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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 * =0.67v i When n =6, you should be bidding according to the following rule b * =0.83v i In other words, all else equal, you should increase your bids when there are more bidders

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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 * = v i That is, bidding should be more aggressive under SPA than under FPA.

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Predictions

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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

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Results b=0.81v

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Results b=1.01v

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Results b=0.93v

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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.

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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).

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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)v i / N If you are risk averse this implies: Pr(b>h) * (v-b) 1-r Your optimal bidding strategy is: b * =(N-1)v i / N - r

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

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Predictions with risk aversion

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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.

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(Single-item) auctions Vincent Conitzer v() = $5 v() = $3.

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