Worksheet for \"Striking Black Gold: Oil Field Auction in the Gulf of Mexico\"* (Based on Prof. R. Ravi's (Tepper School ofBusiness, CMU) lecture notes) The objective of this case is to illustrate the two major tools we have for decision making under uncertainty. namely, decision trees and simulation. We do this in an auction scenario. Auction theory (and more generally, game theory) is a rich area ofstudy for economists and decision analysts, but is beyond the scope of this class. Here. we will introduce vastly simplied concepts ofauction theory as and when necessary. 1 Background It is hard to understate the importance ofauctions for both businesses and individuals. In this era ofbusinesses focusing on core competencies, many tasks are outsourcedr with the contracts awarded on the basis of an auction. The government regularly resorts to auctions for awarding contracts for various purposes, such as building highways to granting telecom licenses and oil exploration contracts. eBay has brought auctions to the average individual. Auction theory is a growing eld of research with a vast literature. Here, we will only introduce some basic concepts which will enable you to complete the case. lfyou wish to learn more about this subject, you are encouraged to ask your Economics professors. Auction types For most of us, what comes to mind when the word \"auction\" is mentioned is a room full of bidders. with an auctioneer raising the price of the object being auctioned until only one bidder is willing to pay the price. While the price is usually raised in discrete units, it helps to think that the price is continuously ascending until only one bidder remains. This is called the ascending or English auction. eBay uses the English auction with two small modications: (i) Bids are incremented in amounts depending on the last bid, and (ii) eBay provides the use ofa proxy to help you with the bidding process and not require you to monitor the auction at all times. The other type of auction often seen in practice, typically to award government or corporate outsourcing contracts, is called the sealed bid rst price auction. Here, each bidder submits a sealed bid of how much they are willing to pay for the contract. Bidders submit bids independently and unobserved by the others. After all bids have been collected (usually by a pre-specied deadline), the contract is awarded to the highest bidder at a price equal to the amount bid by the highest bidder. *This note was written by Amitabh Sinha for 45730 (Probability and Decision Making) taught in MiniL Fall 2003. at the Graduate School of Industrial Administration (now the Tepper School of Business), Carnegie Mellon University. The case is entirely fictional. and bears no resemblance to any actual events. The use of actual names of corporate entities is solely to infuse a sense of reality in the case and in no way purports to imply anything whatsoever about the corporate entities mentioned. 95 A third type of auction is the sealed bid second price auction. This is identical to the first price auction. Except that the winner (highest bidder) only has to pay the second highest bid to the auctioneer. Let us compare the second price auction with the English auction. Consider a bidder who is willing to bid no more than b for an object. In the English auction, she would stay in the auction while the price was below b, and drop out as soon as it crosses b. If she won the auction before dropping out, then the auction would end at the price which was the bid amount of the second highest bidder. In other words, ifshe won the auction, she would pay a price equal to the bid ofthe second highest bidder. On the other hand, ifboth participants had submitted their respective bids in the second price auction, exactly the same thing would have occurred. Hence, strategically, the English auction and the sealed bid second price auction are identical (assuming each bidder's valuation is independent of the others). Since a decision in a sealed bid auction only consists of submitting one number (the bid amount), they are easier to analyze from the perspective of decision making. Hence we will focus our attention on the two types of sealed bid auctions. What you learn from here will continue to be approximately useful for your adventures on eBay, since we have argued that the sealed bid second price auction is equivalent to the English auction done by eBay. Setting for this case (refer to Striking Black Gold memo also) You (representing Exxon) are one of 4 bidders interested in purchasing a plot of land from the government, for drilling for oil. The other bidders are Amoco, British Petroleum and Chevron-Texaco. Your research department has estimated that the expected net cash flow from the well (if you win the contract) is $1.6 billion. Initially, you will do the analysis assuming that the expected net cash ow will be realized exactly; we will later incorporate the uncertainty of the value of the contract into the analysis. Therefore. for now, you are willing to pay up to $1.63 for the contract. since the difference will be your profit. You do not know how much the other three bidders value the plot. Past experience leads you to assume that each of the other three companies has a valuation which is independently and identically distributed as a uniform random variable between zero and $2 billion. For most of this case. we will assume that the other three bidders are bidding honestly, that is. they are bidding exactly what their valuations of the plot are. For one portion of Part II of this case, we will allow bidders to bid differently from their valuation. For the rest of this discussion and Part I, assume that all other bidders are bidding honestly. 2 Simplifying to discrete version 96 This section is somewhat technical in nature and describes how one comes up with the values of the highest competing bid. You can skip this section without loss of continuity. In order to proceed with the analysis, you need to know what the highest of the three competing bids was. Let Xmax denote the highest competing bid. In order to proceed with the rest of the case, we need to compute the distribution of Xmax. Since the others are bidding honestly, we need to figure out the distribution of a random variable which is the maximum of three independent, identically distributed random variables which are U [0, 2]. The following steps will help you do this. . Let X1, X2, X3 denote the random variables for the valuations of Amoco, British Petroleum and Chevron-Texaco respectively. What is the cumulative density function (cdf) of Xi? Let Xmax denote the random variable for the maximum of the three bidders. What is the cdf Xmax? After you choose your bid, we will consider one of 4 possible outcomes for the variable Xmax, specifying which quartile it falls in. That is, we divide the range [0,2] into four intervals[0, Yo.25], [Yo.25, Yo.s], [Yo.5, Yo.7s] and [Yo.7s,2], such that the variable Xmax is equally likely to fall into any of these four intervals. . How will you compute these intervals? We will represent each quartile by its median. Therefore, the first quartile is represented by Y1, the second by Y2, the third by Y3, and the fourth by Y4. Compute these medians. That is, given that the highest competing bid lies in the first quartile, there is a 50% chance that it lies above Y, and a 50% chance that it lies below Y1. The other medians are defined analogously. Observe that the medians are not equal to the mid-points of the intervals computed above. You should have found that the medians are exactly the four numbers which Julia mentioned as the possible values of the highest competing bid. Therefore, we are going to assume that after you place your bid, the highest competing bid is going to be revealed to you, and this is equally likely to be one of the four values you computed above. We will proceed with the analysis under this assumption of discretization. 3 Part I: Analysis using decision trees As a simplification, we will first consider a decision-tree based on analysis of your bidding strategy under the two different types of auctions. Assume that the value of the oil contract is exactly $1.6 billion and that the competitors bid honestly (their true valuations). Assume that the highest competing bid is equally likely to be one of 1, 1.44,[.7] and l.9l ($B). Additionally, we will assume that you are considering making only one of four possible bids: 0.8, l.2. [.6 and 2 (in Billions ofdollars). Use a decision tree to compute your optimal bid to maximize your payoff in the case of a rst-price auction. Also compute your expected payoff Do the same for a second-price sealed bid auction. How do you explain the bidding strategies for the two types of auctions? Is it just a round-off error since we are only considering 4 possible bids? Do the same insights hold if we consider a more rened bidding strategy with many more possible bids? 3 Part II: Analysis using simulation In the second part ofthe case. we will do a rened analysis using simulation. That is, we will use simulation to actually generate bids for the other three bidders in accordance with their distributions. We continue to assume that the true value of the oil contract is exactly $l.6 billion and that the competitors bid honestly. We now consider modeling the highest competing bid exactly (instead of the discretized approximation). So we will consider the case that the three bidders have valuations chosen independently and identically at random from the U [0. 2] distribution. in billions of dollars. 3 Part 11.] Second price sealed bid auction Begin with an analysis ofthe second price sealed bid auction. First test the four bidding choices we tested earlier-namely. bidding 0.3, l.2. 1.6 and 2 billion dollars. What is the optimal bidding strategy? Now refine your search to compute a more precise estimate ofthe optimal bidding strategy. You don't need to refine too precisely. but you should search over steps of at least $100 million. What is the expected payoff ifyou follow your optimal bidding strategy? Do the insights from the earlier exercise carry over? Does your explanation remain valid ifthe other bidder's valuation follows some other distribution? 3 Part 11.2. First price sealed bid auction I Repeat the above exercise for the first price auction. Again, start with searching for the optimal bid among the four choices 0.8, [.2, 1.6 and 2 billion dollars, and then refine your search. Continue to assume that the other players are bidding honestly. What is your optimal bidding strategy? What is the expected payoff at this strategy? 0 Do the insights from the earlier exercise carry over? 4. Auctioneer's perspective So far, we have been focusing on optimal strategies for the bidders in the two auctions. We seem to have computed optimal strategies fairly accurately for both the types of auctions. We now turn our attention to the auctioneer. in this case, the government. Naturally, the government wishes to choose an auction which maximizes its revenue. From the government's perspective. while choosing an auction, it does not know the valuations of any of the bidders. Therefore. we will assume that the valuations of all 4 bidders are drawn independently and identically at random from U [0. 2]. o What is the expected revenue to the auctioneer in the rst price auction? Assume that the 4 bidders are bidding optimally. I Now compute the expected revenue in the second price auction. I Which auction raises more revenue? Remember to use condence intervals to answer this meaningfully. o In which auction is the variance ofthe revenue raised more? If' you were a risk- averse auctioneer. which auction would you prefer? Under what circumstances might you prefer the other auction? Lll