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## Optimal Bidding

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**Optimal Bidding**Chris Lamoureux, PhD Head of Finance Estes/Neill Professor of Finance University of Arizona Optimal Bidding**The Setting**There is an asset whose value, V, is drawn from a Normal Distribution with a mean of 100, (V-bar) and a standard deviation of 30, (σ). There are N people who each get an independent “signal,” X, of the value of this asset. This signal is a draw from a normal distribution with mean V and standard deviation s. Optimal Bidding**Expectations**Suppose that we were in a game where the object was to guess the true value of the asset, V. Here, we would use Bayes’ Rule to weight the 2 pieces of information about V by relative variances. Optimal Bidding**Expectations (2)**So, if our signal is 65, then our best guess about the asset’s value is: Or 72. Optimal Bidding**Bidding**But, bidding is a very different game from trying to guess the true value. If all the bidders are using the same strategy, then we will only win in those cases when we have the highest signal. If our bidding strategy fails to account for this, then we will suffer from the winner’s curse. Note that our best guess – above – did not depend on the number of bidders. But the winner’s curse, hence our optimal bidding strategy, must. Optimal Bidding**Bidding 2**Let’s start with the first-price, sealed-bid auction. If everyone bid their signals, then the expected winner’s curse is equal to the expected value of the maximum of N independent draws from a normal distribution with mean of 0, and standard deviation of s. While we could find this analytically, I prefer to use Monte Carlo. Optimal Bidding**Bidding 3**So, it makes sense that we would never bid more than our signal minus the expected winner’s curse. Our instincts should be drawn to our monopsony power as well. Since N is much less than infinity, we might be able to exploit our “market power.” Can we extract the expected difference between the expected maximum signal and the expected second-highest? Optimal Bidding**Expected Winner’s Curse & Blessing**Optimal Bidding**Expected Winner’s Curse & Blessing 2 Cont’d.**Optimal Bidding**Summary of Order Statistics**• The Winner’s Curse is generally surprisingly large. • It increases in N for a given s. • It increases in s for a given N. Both of these results are intuitive since the winner’s curse is the maximum of N independent draws from a normal distribution with standard deviation, s. Optimal Bidding**Summary of Order Statistics 2**• The Winner’s Blessing: • Decreases in N for a given s. • Increases in s for a given N. This is the market power of the bidders. At the limit, when there are infinite bidders, this is 0 – bidders are competitive. Note too that this market power is 0 if the bidders are perfectly informed. This is a restatement of the result that sellers are always better off revealing more information about the good. Optimal Bidding**Application of the Order Statistics**Optimal Bidding in the Second-Price Sealed-Bid Auction or English Auction is to bid your expected value – conditional on winning the auction. Thus, you bid your signal minus the expected winner’s curse. This is a Nash Equilibrium. If I assume that everyone else is doing this, I have no incentive to change. Optimal Bidding**Application of the Order Statistics 2**Verify whether this is so. Assume that all other bidders are subtracting the expected maximum error from their signals. Do you have any incentive to deviate from this strategy? A key reason that you don’t is that while whether you win the auction depends on your bid, what you pay in that instance does not. In the second-price sealed bid auction the expected selling price is the true value minus the “winner’s blessing.” Optimal Bidding**First-Price Auction**The first-price sealed bid auction does not have such an easy solution because here both whether you win or not and how much you pay depend on your bid. We start our analysis by starting at 2 candidate strategies: • Bid your signal minus expected winner’s curse, • Bid your signal minus expected winner’s curse minus expected winner’s blessing. Optimal Bidding**First-Price Auction Bidding**Let’s assume that N = 15 and s is 15. As we know in this case the expected winner’s curse is 26.1, and the monopsony power is 7.3. Let’s assume that everyone bids their signal minus the expected winner’s curse. If we also do this, then we will win 1/15 of the time, and break-even on average. (The expected selling price is the true value.) Optimal Bidding**First-Price Auction Bidding 2**By subtracting $.10 more, we don’t change the probability of winning much, but each time we win, we are better off, so we would want to subtract more. I did a simulation (with 1 million draws) of this case, and found that our optimal bid in this case is to subtract $32.96, in which case our expected average profit is $0.0538. (Were we to subtract the ewc and the ewb, we would subtract $33.56, and our expected average profit is $0.0536.) Optimal Bidding**First-Price Auction Bidding 4**So, unlike in the second-price auction, everyone bidding their expected value – conditional on winning – is not an equilibrium. We would still win the auction the same number of times - and buy at lower prices in those cases by bidding less. (Our optimal bid balances these competing effects.) Optimal Bidding**First-Price Auction Bidding 5**Let’s consider the case where all bidders subtract the ewc and the ewb ($33.56) from their signals. We again ask whether we have any incentive to switch. Again, I use Monte Carlo to evaluate this. If we follow this strategy, our expected average profit is $0.484. We find our optimal bid in this case is to subtract $26.06 (the ewc only), and our expected average profit is $0.707. Optimal Bidding**First-Price Auction Bidding 6**So this is not an equilibrium. By bidding more aggressively than everyone else, we win even in many cases when we don’t have the highest signal (and in these cases make big profits). Optimal Bidding**First-Price Auction Bidding 7**These 2 extremes suggest that there may be an interior strategy that is an equilibrium. The Monte Carlo analysis reveals that this is to subtract $28.66 from our signal. If everyone does this, we have no incentive to change our strategy. In this case, our expected average profit is $0.173. (And of course we expect to win 1/15 of the time.) Optimal Bidding**Notes**• Bidders do bid more aggressively in the second-price auction than in the first-price auction. • (However,) The seller expects a higher selling price in the first-price auction than in the second-price auction. • The average profit figures above average over all auctions. Under the equilibrium strategies, our average profit – conditional on winning – is 15 times higher (since we win 1/15 of the time). Optimal Bidding**Notes 2.**• All of our analysis assumes that bidders are risk-neutral. • The optimal bidding strategies for a Dutch auction are the same as for a first-price sealed bid auction. • The optimal bidding strategies for an English auction are the same as for a second-price sealed bid auction. Optimal Bidding**Notes 3.**• All of our analysis has assumed that the number of bidders is fixed and known. A seller would typically try to design an auction to attract more bidders. • Note that we can verify the maxim that the seller will expect a higher selling price the smaller is sigma. So, for example, by revealing all information about the item, the seller increases the expected sales price. Optimal Bidding