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Chapter 27: Problem 3

Complete the analysis of the second-price auction by showing that bidding one's valuation \(v_{i}\) is weakly preferred to bidding any \(x

Short Answer

Expert verified
Bidding one's valuation \(v_i\) is weakly preferred as it avoids losing and ensures maximum payoff if winning.

Step by step solution

01

Understand the Second-Price Auction

In a second-price auction, each bidder submits a bid without seeing others' bids. The highest bidder wins the item but pays the second-highest bid as the price. The key characteristic is that the payment is determined by the second-highest bid, not the highest.
02

Define the Bidder's Valuation and Bidding Strategy

Let the bidder's true valuation of the item be denoted by \(v_i\), and we consider whether they should bid this value (truthful bidding) or any lower value \(x < v_i\). The bidder wants to maximize their payoff, which is the difference between their valuation and the actual price paid if they win the auction.
03

Calculate Payoff for Bidding Truthfully

If the bidder bids their true valuation \(v_i\) and wins, they pay the second-highest bid, say \(b\). Their payoff is \(v_i - b\) if \(v_i > b\), otherwise, they don't win, and the payoff is 0.
04

Calculate Payoff for Bidding Less Than True Valuation

If the bidder bids \(x < v_i\) and wins (only possible if \(x > \) all other bids except one below or equal to \(x\)), they pay the second-highest bid which might still be \(b\) or less, resulting in a payoff of \(v_i - b\). However, bidding \(x\) could cause them to lose when they would've won by bidding \(v_i\).
05

Compare Both Scenarios

Bidding \(v_i\) ensures winning if \(v_i\) is the highest, with a payment of the highest other bid \(b\). If bidding lower at \(x\), there's a risk of losing to a bid between \(x\) and \(v_i\), resulting in a payoff of 0 compared to a potential \(v_i - b\) with a truthful bid. Hence, \(v_i\) weakly dominates any lower bid.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Second-Price Auction
A second-price auction is an interesting and somewhat counterintuitive type of auction. In this format, each participant submits a bid without knowing what the others are bidding. What's unique here is that the winner of the auction doesn't pay their own highest bid. Instead, they pay the second-highest bid. This means:
  • The person with the highest bid wins the item.
  • The winner pays the value of the second-highest bid.
This system encourages bidders to bid their true value for the item. Why? Because even if they bid a lot more than the second-highest amount, they'll only end up paying the second-highest price, not their actual bid. Understanding this peculiar rule is crucial when evaluating different bidding strategies.
Bidding Strategy
In a second-price auction, the smartest bidding strategy is to bid exactly what the item is worth to you, known as your "true valuation." Imagine the item being auctioned is worth $100 to you. Should you bid less? What if someone else places a bid higher than your low bid but still less than your true value?
  • Bidding your true valuation, such as $100, ensures that if the second-highest bid is lower than $100, you win and pay the second bid's amount.
  • Bidding lower could lead to losing the auction to someone bidding between your bid and your true valuation.
Thus, in a second-price auction, the incentive is to bid exactly what you believe the item is worth to you. Any deviation from this can lead to missing out or potentially receiving a payoff which is less optimal.
Auction Theory
Auction theory helps us understand why different types of auctions have different outcomes based on bidding behavior and the rules of the auction. In essence, it looks at how auctions can be designed to achieve particular goals, such as maximizing revenue or ensuring fair prices.
In the case of a second-price auction, auction theory shows why bidding your true valuation is a dominant strategy. The theory suggests that by bidding truthfully, bidders don't have to worry about overpaying. This idea contrasts with other auction formats, such as first-price auctions, where the highest bid is what the winner pays, possibly incentivizing bidders to underbid.
Overall, auction theory helps explain the rationale behind the second-price auction's rules and why it aligns the bidder's incentives with truthful bidding, which may lead to an efficient allocation of resources.
Payoff Calculation
Payoff calculation is a critical component in understanding why a particular bidding strategy is favorable. For any auction, the payoff represents the benefit a bidder gets from winning an item.
In a second-price auction, the payoff is determined by the difference between the bidder's true valuation of the item and the price they actually pay (which is the second-highest bid). Let's say:
  • Your true valuation of an item is $100.
  • The second-highest bid is $80.
If you bid your true valuation, $100, and win, your payoff is calculated as:
  • Payoff = True Valuation - Price Paid
  • Payoff = $100 - $80 = $20
Comparatively, bidding anything less than your true valuation might lead you to lose the item if someone bids between your bid and your true valuation. Hence, calculating the expected payoff helps reinforce that truthful bidding is typically the best strategy, as it ensures a non-negative or potentially maximum payoff.

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Most popular questions from this chapter

Consider an "all-pay auction" with two players (the bidders). Player 1's valuation \(v_{1}\) for the object being auctioned is uniformly distributed between 0 and 1. That is, for any \(x \in[0,1]\), player 1's valuation is below \(x\) with probability \(x\). Player 2's valuation is also uniformly distributed between 0 and 1, so the game is symmetric. After nature chooses the players' valuations, each player observes his/her own valuation but not that of the other player. Simultaneously and independently, the players submit bids. The player who bids higher wins the object, but both players must pay their bids. That is, if player \(i\) bids \(b_{i}\), then his/her payoff is \(-b_{i}\) if he/she does not win the auction; his/her payoff is \(v_{i}-b_{i}\) if he/she wins the auction. Calculate the Bayesian Nash equilibrium strategies (bidding functions). (Hint: The equilibrium bidding function for player \(i\) is of the form \(b_{i}\left(v_{i}\right)=k v_{i}^{2}\) for some number \(k\).)

Suppose that you and two other people are competing in a third-price, sealed- bid auction. In this auction, players simultaneously and independently submit bids. The highest bidder wins the object but only has to pay the bid of the third-highest bidder. Sure, this is a silly type of auction, but it makes for a nice exercise. Suppose that your value of the object is 20 . You do not know the values of the other two bidders. Demonstrate that, in contrast with a second-price auction, it may be strictly optimal for you to bid 25 instead of 20 . Show this by finding a belief about the other players' bids under which 25 is a best-response action, yet 20 is not a best-response action.

Suppose you and one other bidder are competing in a private-value auction. The auction format is sealed bid, first price. Let \(v\) and \(b\) denote your valuation and bid, respectively, and let \(\hat{v}\) and \(\hat{b}\) denote the valuation and bid of your opponent. Your payoff is \(v-b\) if it is the case that \(b \geq \hat{b}\). Your payoff is 0 otherwise. Although you do not observe \(\hat{v}\), you know that \(\hat{v}\) is uniformly distributed over the interval between 0 and 1 . That is, \(v^{\prime}\) is the probability that \(\hat{v}

Suppose that a person (the "seller") wishes to sell a single desk. Ten people are interested in buying the desk: Ann, Bill, Colin, Dave, Ellen, Frank, Gale, Hal, Irwin, and Jim. Each of the potential buyers would derive some utility from owning the desk, and this utility is measured in dollar terms by the buyer's "value." The valuations of the 10 potential buyers are shown in the following table. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline \text { Ann } & \text { Bill } & \text { Colin } & \text { Dave } & \text { Ellen } & \text { Frank } & \text { Gale } & \text { Hal } & \text { Irwin } & \text { Jim } \\ 45 & 53 & 92 & 61 & 26 & 78 & 82 & 70 & 65 & 56 \\ \hline \end{array} $$ Each bidder knows his or her own valuation of owning the desk. Using the appropriate concepts of rationality, answer these questions: (a) If the seller holds a second-price, sealed-bid auction, who will win the auction and how much will this person pay? (b) Suppose that the bidders' valuations are common knowledge among them. That is, it is common knowledge that each bidder knows the valuations of all of the other bidders. Suppose that the seller does not observe the bidders' valuations directly and knows only that they are all between 0 and 100 . If the seller holds a first-price, sealed-bid auction, who will win the desk and how much will he or she have to pay? (Think about the Nash equilibrium in the bidding game. The analysis of this game is a bit different from the [more complicated] analysis of the first-price auction in this chapter because here the bidders know one another's valuations.) (c) Now suppose that the seller knows that the buyers' valuations are 45,53 , \(92,61,26,78,82,70,65\), and 56 , but the seller does not know exactly which buyer has which valuation. The buyers know their own valuations but not one another's valuations. Suppose that the seller runs the following auction: She first announces a reserve price \(p\). Then simultaneously and independently the players select their bids; if a player bids below \(\underline{p}\), then this player is disqualified from the auction and therefore cannot win. The highest bidder wins the desk and has to pay the amount of his or her bid. This is called a "sealed-bid, first price auction with a reserve price." What is the optimal reserve price \(p\) for the seller to announce? Who will win the auction? And what will the winning bid be?

Consider the common-value auction described in this chapter, where \(Y=y_{1}+y_{2}\) and \(y_{1}\) and \(y_{2}\) are both uniformly distributed on \([0,10]\). Player \(i\) privately observes \(y_{i}\). Players simultaneously submit bids and the one who bids higher wins. (a) Suppose player 2 uses the bidding strategy \(b_{2}\left(y_{2}\right)=3+y_{2}\). What is player 1's best-response bidding strategy? (b) Suppose each player will not select a strategy that would surely give him/her a negative payoff conditional on winning the auction. Suppose also that this fact is common knowledge between the players. What can you conclude about how high the players are willing to bid? (c) Show that for every fixed number \(z>0\), there is no Bayesian Nash equilibrium in which the players both use the bidding strategy \(b_{i}=y_{i}+z\).
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